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Contributing Authors
Gaetano Antinolfi
Department of Economics
Washington University in St. Louis
One Brookings Drive
St. Louis, MO 63130-4899
gaetano@artsci.wustl.edu
Costas Azariadis
Department of Economics
Washington University in St. Louis
One Brookings Drive
St. Louis, MO 63130-4899
cre@artsci.wustl.edu
Ravi Bansal
Fuqua School of Business
Duke University
Durham, NC 27708
ravi.bansal@duke.edu
James B. Bullard
Federal Reserve Bank of St. Louis
P.O. Box 442
St. Louis, MO 63166-0442
bullard@stls.frb.org
John H. Cochrane
Graduate School of Business
University of Chicago
5807 S. Woodlawn Avenue
Chicago, IL 60637
john.cochrane@chicagogsb.edu
Riccardo DiCecio
Federal Reserve Bank of St. Louis
P.O. Box 442
St. Louis, MO 63166-0442
Riccardo.DiCecio@stls.frb.org

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Martin Fukac̆
Reserve Bank of New Zealand
P.O. Box 2498
Wellington
New Zealand
martin.fukac@rbnz.govt.nz
Michael F. Gallmeyer
Mays Business School
Department of Finance
Texas A&M University
College Station, TX 77843-4218
mgallmeyer@mays.tamu.edu
Burton Hollifield
David A. Tepper School of Business
Carnegie Mellon University
5000 Forbes Avenue
Pittsburgh, PA 15213-3890
burtonh@andrew.cmu.edu
Peter N. Ireland
Department of Economics
Boston College
140 Commonwealth Avenue
Chestnut Hill, MA 02467-3895
irelandp@bc.edu
Narayana R. Kocherlakota
Department of Economics
University of Minnesota
271 19th Avenue South
Minneapolis, MN 55455
nkocher@econ.umn.edu.
Pamela A. Labadie
Department of Economics
The George Washington University
1922 F Street, NW
Old Main, Suite 208
Washington, DC 20052
labadie@gwu.edu

J U LY / A U G U S T

2007

iii

Edward Nelson
Federal Reserve Bank of St. Louis
P.O. Box 442
St. Louis, MO 63166-0442
edward.nelson@stls.frb.org

Brian Sack
Macroeconomic Advisers
231 South Bemiston Avenue, Suite 900
St. Louis, MO 63105
bsack@macroadvisers.com

Lee E. Ohanian
Department of Economics
University of California, Los Angeles
Box 951477
8283 Bunch Hall
Los Angeles, CA 90095-1477
ohanian@econ.ucla.edu

Thomas J. Sargent
Department of Economics
New York University
269 Mercer Street
New York, NY 10003
thomas.sargent@nyu.edu

Adrian Pagan
Queensland University of Technology
School of Economics and Finance
Queensland University of Technology
GPO Box 2434
Brisbane QLD 4001
Australia
a.pagan@qut.edu.au
Francisco J. Palomino
David A. Tepper School of Business
Carnegie Mellon University
5000 Forbes Avenue
Pittsburgh, PA 15213-3890
fjp@andrew.cmu.edu

Eric T. Swanson
Federal Reserve Bank of San Francisco
101 Market Street
San Francisco, CA 94120-7702
eric.swanson@sf.frb.org
Stanley E. Zin
David A. Tepper School of Business
Carnegie Mellon University
5000 Forbes Avenue
Pittsburgh, PA 15213-3890
zin@cmu.edu

Glenn D. Rudebusch
Federal Reserve Bank of San Francisco
101 Market Street
San Francisco, CA 94120-7702
Glenn.Rudebusch@sf.frb.org

iv

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F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

An Estimated DSGE Model for the United Kingdom
Riccardo DiCecio and Edward Nelson
The authors estimate the dynamic stochastic general equilibrium model of Christiano, Eichenbaum,
and Evans (2005) on U.K. data. Their estimates suggest that price stickiness is a more important
source of nominal rigidity in the United Kingdom than wage stickiness. Their estimates of
parameters governing investment behavior are only well behaved when post-1979 observations
are included, which reflects government policies until the late 1970s that obstructed the influence
of market forces on investment. (JEL E31, E32, E52)
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 215-31.

I

n this paper we estimate a dynamic stochastic general equilibrium (DSGE) model
with nominal rigidities for the U.K. economy. The model we estimate is due to
Christiano, Eichenbaum, and Evans (2005; CEE)
and has become a benchmark, matching important aspects of the U.S. data while also being
derived from optimizing behavior.
Interest in DSGE modeling of the United
Kingdom has been heightened in recent years
with the introduction of the Bank of England
quarterly model (BEQM) into the U.K. monetary
policy process. This model is based to a considerable degree on explicit optimizing foundations;
see Harrison et al. (2005) for the model and Pagan
(2005) for a discussion. BEQM is, however, dissimilar in important respects from the CEE model
of the United States and the variant of the CEE
model that Smets and Wouters (2003) estimate
for the euro area. These dissimilarities make it
difficult to use BEQM to compare the structure
of the U.K. economy with that of other economies.
For example, the estimation procedure for BEQM
is different from that used by CEE and by Smets

and Wouters; portions of the BEQM model are
estimated over a considerably shorter sample
than CEE consider for the United States, and
there are deviations from explicit optimization
in the dynamics of the BEQM model.
All in all, it is probably fair to say that there
has been considerably less work done for the
United Kingdom in terms of DSGE modeling
with systems estimation than there has been for
other economies. But U.K. data may contain a
type of information that is ideal for estimation
of a DSGE model—specifically, information on
private sector responses to policy actions. As the
present governor of the Bank of England, Mervyn
King, observed some 30 years ago,
Maintenance of the existing order and existing
rates produces no information, whereas more
information can be obtained by making
changes. In this respect the U.S....is at a disadvantage by comparison with the U.K. A good
illustration of this is afforded by the excitement generated amongst American economists
in the 1960s by the investment tax credit and
the attempts to assess its effects. A British
economist would have shrugged this off as a

Riccardo DiCecio is an economist at the Federal Reserve Bank of St. Louis, and Edward Nelson is an assistant vice president at the Federal
Reserve Bank of St. Louis and a research affiliate at the Centre for Economic Policy Research. The authors thank Adrian Pagan and participants at
the policy conference and in seminars at the Federal Reserve Bank of Atlanta for comments on an earlier draft and Nick Davey, Jennifer
Greenslade, and Kenny Turnbull for helpful information. Justin Hauke provided research assistance.

© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
only with prior written permission of the Federal Reserve Bank of St. Louis.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

J U LY / A U G U S T

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DiCecio and Nelson

Table 1
Parameters in the Model
Description

Parameter

Description

β

Discount factor

ξp

Degree of price stickiness

α

Production elasticity with respect
to capital

ξw

Degree of wage stickiness

δ

Depreciation rate

1/κ

µ

Steady-state money-growth rate

1/σa

Elasticity of investment to p̂K ′
Elasticity of utilization to rˆ k

ψl

Relative weight of leisure in utility

ρ

Interest rate smoothing parameter

ψq

Relative weight of real balances
in utility

rπ

Interest rate response to inflation

λw

Wage markup

ry

Interest rate response to output

1/σq

Interest semi-elasticity of money
demand

r∆π

Interest rate response to change
in inflation

b

Habits parameter

r∆y

Interest rate response to change in
output

λf

Price markup

σε

Standard deviation of monetary policy
shock

mere trifle compared to the changes he had
witnessed over the years. (King, 1977, p. 6)

This observation, though made with reference
to the changes wrought in U.K. fiscal policy up
to the 1960s, applies tenfold to monetary policy
experience in the period since the 1960s. Over
that period, the United Kingdom has undergone
great variation in inflation, interest rates, and
monetary regimes.1 It is true that for estimation
this is a mixed blessing because large regime
changes make it problematic to estimate a structural model over a long sample. But Christiano,
Eichenbaum, and Evans (1999) and Sims and Zha
(2006) argue for the United States that constantparameter policy reaction functions may be reasonable approximations even over long samples,
a view also implicit in CEE’s (2005) choice of a
1965-95 estimation period. In modeling the United
Kingdom using a DSGE model, we make a compromise between these positions by treating the
1

Parameter

Various advantages of the U.K. data for testing macroeconomic
hypotheses have been stressed by Ravn (1997) (evaluating a real
business cycle model against the behavior of U.K. real aggregates),
Nelson and Nikolov (2004) (using a small New Keynesian model
to evaluate different U.K. policy regimes), and Benati (2004)
(assessing the behavior of U.K. data moments over the postwar
period).

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J U LY / A U G U S T

2007

period since 1979 as a single regime,2 but also by
presenting results for pre-1979 and a long sample covering 1962-2005.3
We present in the following sections our
model, estimates for our main sample, and results
for the longer sample, with a discussion of other
regime-change issues.

MODEL
The model is the same as that in CEE (2005),
by now standard in the DSGE literature. The
model incorporates both nominal frictions (sticky
prices and wages) and dynamics in preferences
and production (habit formation in consumption,
investment adjustment costs, and variable capital
utilization). The pattern of timing in agents’ decisions is consistent with the VAR identification
2

Some work for the United Kingdom (e.g., Castelnuovo and Surico,
2006) focuses on 1992 as the start of the present policy regime.
But our use of a baseline sample period that starts in the late 1970s
matches the choices implied by some of the BEQM-equation estimation periods (see, e.g., Harrison et al., 2005, pp. 115-20).

3

The long-sample estimates are the U.K. analogue to the Del Negro
et al. (2005) treatment of the U.S. sample 1954-2004 as a single
regime (including an unchanged inflation target).

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

DiCecio and Nelson

Table 2
Variable Definitions
Description

Variable

Variable

m̂

Real money supply

π̂

Inflation

q̂

Households’ demand for money

ŵ

Real wage
Rental rate on capital

k̂
–
kˆ

Capital services

r̂ k

Physical capital

µ̂

Growth rate of nominal money stock

l̂

Labor

p̂K ′

Price of capital

ĉ

Consumption

ÛC

Marginal utility of consumption

î

Investment

R̂

Nominal interest rate

ŷ

Output

restriction that we use in the next section. In our
outline here of the linearized version of the model,
all variables are expressed in log-deviations from
their steady-state values. For convenience, model
parameters and variables are summarized in
Tables 1 and 2, respectively.
Prices are governed by Calvo (1983) contracts,
augmented by indexation to the previous period’s
inflation for those firms not allowed to reoptimize
their pricing decision. The implied inflation
dynamics are given by the following Phillips
curve:

the preceding period’s price inflation. This produces the nominal wage equation:

 ϖ 1wˆ t +1 + ϖ 2wˆ t + ϖ 3wˆ t −1 + ϖ 4πˆ t +1 
(2) E t −1 
 = 0,
π t −1 + Uˆ C ,t − lˆt
 + ϖ 5πˆ t + ϖ 6π̂

where

βξp
ξp (1 + β )


πˆ t +1 −
πˆ t 

1 − ξp 1 − βξp

 1 − ξp 1 − βξp


ξp
 = 0.
(1) Et −1  +
πˆ t −1

 1 − ξp 1 − βξp



 + α rˆ k + (1 − α ) Rˆ + wˆ 
t
t 
  t




)(

(

(

)

)(

)(

(

ϖ1 =


λw 
− βξw
1+

,
βξ
ξ
λ
1
−
1
−
(
w )(
w )
w − 1

ϖ2 =


λw 
λw
1 + βξw2
−
,
1+


βξ
ξ
λ
−
λ
1
−
1
1
−

(
w )(
w )
w
w −1

ϖ3 =


λw 
−ξw
,
1+

(1 − βξw )(1 − ξw )  λw − 1

)

)

(

ϖ 4 = ϖ 1, ϖ 6 = ϖ 3 ,

)

ϖ5 =
Here, hats on variables indicate the log-deviations from steady-state values. For the nominal
interest rate and inflation terms that appear in
the model, the hatted variables are effectively
the demeaned net inflation and interest rates,
because the log-deviations are computed using
gross rates.
Nominal wages are staggered along similar
lines to prices,4 with a clause for indexation to
4

Description

See Erceg, Henderson, and Levin (2000) for the development of
this form of staggered wage contracts.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

ξw (1 + β )

λw 
1+

.
βξ
ξ
λ
1
1
−
−
(
w )(
w )
w − 1

Firms’ optimality conditions imply that their
total payments for capital services equal their
total cost of hiring labor each period:
(3)

rˆtk + kˆ t = wˆ t + Rˆ t + lˆt .

Underlying this condition is the assumption
that firms finance their wage bill with funds borrowed one period earlier. Real unit labor costs are
therefore (in log terms) equal to the sum of the real
wage and the short-term nominal interest rate.
J U LY / A U G U S T

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DiCecio and Nelson

The typical household’s intertemporal Euler
equation for consumption and first-order condition for capital purchases are, respectively,

(

Uˆ C ,t = Et Rˆ t +1 − πˆ t +1+ Uˆ C ,t +1

(4)

(

E t −1 pˆ K ′, t + Uˆ C, t

(5)

)

)



1−δ
rk
= Et − 1 Uˆ C, t + 1 + k
pˆ K ′, t +1  .
rˆtk+1 + k
r +1−δ
r +1−δ



(9)

where δ denotes the depreciation rate. Though
physical investment is subject to adjustment costs,
equation (9) indicates that a unit of investment
adds to the physical capital stock in a standard
manner.
Households’ money demand function is
given by
(10)

Because of habit formation in preferences, the
household marginal utility of consumption that
appears in the above expressions is not a static
function of consumption. Instead, it depends on
the current, prior, and expected future levels of
consumption:

(1 − bβ ) (1 − b ) Et −1Uˆ C,t − bβ Et −1cˆ t +1
(6)

)

(

+ 1 + b 2β E t −1cˆ t − bcˆ t −1 = 0.

The economy’s technology allows additional
productive services to be generated, at a cost, from
an unchanged stock of physical capital. The
degree of capital utilization—that is, the difference between the physical capital stock (denoted
by an overbar) and capital services—is chosen by
households to equate marginal cost with marginal
benefit:
(7)

1
ˆ
E t −1 kˆ t − kt =
Et −1 rˆtk ,
σa

(

)

where 1/σa is the elasticity of the utilization
function.
The equilibrium condition for household
investment choices can be written as

ˆ K ′, t = κ Et −1 (ιˆ t − ιˆ t −1 ) − βκ Et −1 (ιˆ t +1 − ιˆ t ).
(8) Et −1 p
This condition indicates that the price firms pay
for capital services is a function of two parameters
that emerge from the behavior of households
(who are the producers and suppliers of capital
services): the households’ discount factor, β, and
the elasticity of their investment adjustment cost
function, 1/κ.
The stock of physical capital obeys the law
of motion:
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J U LY / A U G U S T

2007

ˆ
ˆ
kt +1 = (1 − δ ) kt + διˆ t ,

qˆ t = −

1  R ˆ

R + Uˆ C,t  ,
σ q  R − 1 t


a condition that indicates the standard choice
between holding money for the transaction services it provides5 or, instead, holding one-period
securities for interest income.
The following identity relates growth of
nominal money supply to inflation and changes
in real money supply:
(11)

ˆ t −m
ˆ t −1 + πˆ t .
µˆ t −1 = m

The aggregate demand for money in the
economy comes from two sources: demand by
firms (to finance their wage bill) and by households as given by condition (10). In equilibrium,
total money demand is equal to the aggregate
money stock:
(12)

)

(

ˆ t ).
qqˆ t + wl wˆ t + lˆt = µ m ( µˆ t + m

The resource constraint and the aggregate
production function can be written as

)

1
K ˆ
C
K
ˆ
(13)  + δ − 1
kt − kt + cˆ t + δ ιˆt = yˆ t
Y
Y
Y
β



(

(14)

yˆ = αλ f kˆ t + (1 − α ) λ f lˆt .

Equation (13) indicates that resources this period
can be consumed, invested, or used to generate
additional capital utilization. Equation (14) indicates that the two inputs in production are labor
and capital services.
Monetary policy follows a dynamic version
of the Taylor rule:
5

Because of habit formation, prior and expected future transactions
create a demand for real balances—i.e., money over and above the
demand generated by current transactions.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

DiCecio and Nelson

(15)

Rˆ t = ρRˆ t −1 + (1 − ρ )  rπ πˆ t + ry yˆ t 
+ r∆π (πˆ t − πˆ t −1 ) + r∆y ( yˆ t − yˆ t −1 ) − εt .

The short-term nominal interest rate is therefore
a smoothed function of inflation, output, and
changes in inflation and output. There is also a
monetary policy shock, εt . This rule is similar to
that in Smets and Wouters (2003) and Levin et al.
(2005).
Other than our use of an interest rate rule,
the model we use corresponds to the CEE benchmark. A limitation in our application to the U.K.
data is that the CEE model describes a closed
economy. But there are several reasons for using
a closed-economy model when analyzing the
United Kingdom; see Neiss and Nelson (2003)
for a discussion. For the present paper, the main
reasons why a closed economy of the DSGE model
might be suitable for the United Kingdom are as
follows: (i) Openness makes it difficult to model
capital formation endogenously, whereas the
presence of endogenous capital is a key feature
of the CEE model. And (ii) the simplest openeconomy models give counterfactual weight to
the exchange rate in consumer price inflation
dynamics; once the exchange-rate channel is
“tamed” by such approaches as assuming incomplete pass-through, imported intermediates, etc.,
the model’s properties become more like those of
a closed economy (see, e.g., Obstfeld, 2002).

VAR Estimates
We estimate our VAR on a U.K. dataset consisting of a subset of the variables studied in the
U.S. case by CEE. Our VAR contains the logs of
real gross domestic product (GDP), real consumption,7 real investment, and labor productivity, as
well as the nominal Treasury bill rate and the
quarterly (retail) inflation rate.8 As these choices
imply, our focus is on the response of the policy
rate, inflation, and aggregate demand to a monetary policy shock, as well as the split of aggregate
demand among its components and the division
of the output response between labor and other
inputs.
The sample period is 1979:Q2–2005:Q4. The
start date is the quarter corresponding to the
period (May 1979) in which the Thatcher government first took office—and so an important monetary policy regime change.9 It also corresponds
approximately to the date of some other significant
changes in government policy that are important
for the VAR responses, as we discuss in the next
section.
Figure 1 plots the estimated VAR responses
to a monetary policy shock and their bootstrapped
confidence intervals, along with the match to
each response made by our estimates of the CEE
model; the model-based responses are the blue
lines. Parameters fixed in estimation are given in
Table 3.

ESTIMATION

Parameter Estimates

To estimate the model, we first obtain data
responses to a monetary policy shock from a
vector autoregression (VAR) for the United
Kingdom. Then, as in CEE, we match these
impulse responses as closely as possible with
the CEE model, using a minimum-distance estimation procedure.6 Our analysis here is limited
to monetary policy shocks, but there is evidence
for the United States that estimates of the CEE
model are robust to incorporating technology
shocks into the analysis (see DiCecio, 2005, and
Altig et al., 2005).

The parameter estimates resulting from this
matching of impulse responses are given in
Table 4. Standard errors for the parameter estimates appear in parentheses and are calculated
by the asymptotic delta method.10

6

This procedure was also used with a smaller VAR by Rotemberg
and Woodford (1997).

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

7

We have not split consumption between durables and nondurables. VAR impulse responses in Aoki, Proudman, and Vlieghe
(2002, Chart 2), using a different VAR specification and sample
period from ours, found similar response functions for the two
types of consumption.

8

Our VAR does not include a time trend. Impulse responses look
similar regardless of whether a linear trend is included in the VAR.

9

See, e.g., Goodhart (1989) for a perspective on this regime change.

10

See Newey and McFadden (1994, p. 2145).

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DiCecio and Nelson

Figure 1
Estimated and Model Responses to a Monetary Policy Shock
Inflation

Output

0.2

0.2

0.1
0

0
−0.1

−0.2
0

5

10

15

20

0

5

Consumption

10

15

20

Investment
0.8
0.6
0.4
0.2
0
−0.2
−0.4

0.2
0
−0.2
0

5

10

15

20

0

5

Productivity

10

15

20

Interest Rate

0.2
0
0.1
−0.2

0

−0.4

−0.1

−0.6

−0.2
0

5

10

15

20

The parameter indexing habit formation in
consumption is larger than that estimated by CEE
and Altig et al. (2005) but is basically in line with
Fuhrer (2000). So the degree of habit formation
in the United Kingdom appears similar to U.S.
estimates.
The markup estimate is, at somewhat above 2,
high by the standards of calibrated and estimated
DSGE models. It is, however, roughly in line with
the estimate of the average U.K. gross markup (in
manufacturing) by Haskel, Martin, and Small
(1995, p. 30) of 2.0. Our high markup estimate
appears more standard if it is regarded as the
wedge between consumer prices and (principally)
nominal wages,11 including the impact of cost
11

Interest on the nominal wage bill also enters the cost expression,
with implications we discuss shortly.

220

J U LY / A U G U S T

2007

0

5

10

15

20

elements we have not modeled explicitly.12 It
should be remembered that the model is an
abstraction of a model with imported intermediate goods and indirect taxes. With these unmodeled elements built into the empirical price-level
series, the estimated markup of retail prices on
nominal wages is increased.13
The estimated interest rate policy rule has
responses to both the level and growth rate of
inflation as well as to the deviation of output
from the steady state. Because the interpretation
12

By contrast, Haskel, Martin, and Small’s (1995) markup estimate
allows for costs of materials, so our markup estimate should be
higher than theirs, other things equal.

13

Therefore, our high estimate may be consistent with Britton,
Larsen, and Small (2000) setting the U.K. steady-state markup
value closer to 1.0 when calibrating a model with explicit imported
intermediates.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

DiCecio and Nelson

Table 3

Table 4

Parameters Fixed in Estimation

Baseline Model Estimates, 1979:Q2–2005:Q4

Parameter

Value

β

(1.04)–1/4

α

0.36

δ

0.025

µ

1.017

ψl

Such that l = 1

ψq

Such that q/m = 0.25

λw

1.05

of the inflation responses is affected by the output
response, we deal with the latter response first.
As technology shocks are held constant, any output movements reflect the opening of the output
gap and so also inflationary pressure. It is precisely this type of output variation that a monetary
authority will have greatest interest in stabilizing.
This may account for the output response being
larger than is usual in estimated interest rate
rules, which typically do not remove from the
output measure the variation that is due to technology shocks.
The monetary policy reaction to inflation
consists of a standard level response and a negative response to the change in inflation. We find
that the inflation-change response, although not
very precisely estimated, is negative and economically sizable. Under some parameter values, an
estimated negative response to the change in
inflation implies that policymakers have lagged
inflation rather than current inflation in their rule.
Our estimated r∆π response is, however, too large
(in absolute value) for this to be the case. Instead,
policymakers actually make different-signed shortrun responses to inflation, initially allowing a
temporary reduction in the interest rate when
inflation rises. To understand this response, one
has to keep in mind the supply side of the CEE
model. In the standard sticky-price model, the
impulse responses of output and the output gap
to a monetary policy shock are identical, because
potential output depends on real shocks only. In
the CEE model, however, this is not the case,
because interest rates enter the production funcF E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Private sector parameters
b

0.7739
(0.0013)

λf

2.2681
(0.0323)

ξp

0.9371
(0.0004)

ξw

0.0000
(0.0083)

κ

16.4333
(0.1750)

σa


(—)

Monetary policy rule parameters

ρ

0.8720
(0.0907)

rπ

1.2813
(0.4977)

ry

0.3517
(0.6065)

r∆π

–0.5129
(0.6971)

r∆y

0.4259
(0.2759)

σR

0.1564
(0.0001)

NOTE: The number of impulse-response steps used, which
ideally should be determined statistically (see Hall et al., 2007),
is 25. A diagonal matrix is used to weight the responses.

tion, implying that potential output depends on
the nominal interest rate (see Ravenna and Walsh,
2006, for further discussion). Holding constant its
other effects, a cut in the interest rate stimulates
potential output and helps inflation stabilization in
the face of upward pressure on aggregate demand.
Therefore, in the wake of a monetary policy shock,
policymaker stabilization of the output gap and
inflation takes a three-pronged approach: a large
response to output to rein in incipient excessive
aggregate demand (ry > 0, r∆y > 0); a short-run cut
in the interest rate as inflation rises to stimulate
potential output (r∆π < 0); and a sizable and durable
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DiCecio and Nelson

Figure 2
Estimated and Model Responses to a Monetary Policy Shock, 1962:Q3–2005:Q4
Inflation

Output
1

0
0.5

−0.5

0

−1
0

5

10

15

20

0

5

Consumption

10

15

20

Investment
2

1

1.5
1

0.5

0.5
0

0
0

5

10

15

20

0

5

Productivity

10

15

20

Interest Rate

0.4
−0.2
0.2

−0.4
−0.6

0

−0.8
−0.2

−1
0

5

10

15

20

positive response of the interest rate to the level
of inflation relative to the target (rπ > 1).
The estimates imply large investment adjustment costs, mainly driven by the matching of the
smoother investment responses after the initial
period; the model does not match the apparent
initial spike in investment observed in the data.
Although the model allows for both wage
stickiness and indexation of wages to price inflation, our parameter estimates imply that both
these features are absent.14 On the other hand,
price stickiness is substantial. Because full indexation of prices is superimposed on this price
adjustment, it is not appropriate to infer from
14

That is, equation (2) collapses to the usual static labor-supply
condition, equating real wages to the marginal rate of substitution
between consumption and leisure.

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2007

0

5

10

15

20

the low implied Calvo probability that prices are
implausibly rigid; rather, the indexation implies
substantial price movements every period, even
with the underlying price stickiness. Empirical
support for lagged inflation terms in the Phillips
curve, when this parameter is estimated freely, is
not universal (see, e.g., Ireland, 2001), so our
assumption of full indexation may be restrictive.
A lagged inflation term in the Phillips curve is,
however, in line with the specification advocated
by Blake and Westaway (1996) for U.K. monetary
policy analysis.
The bottom line is that the estimates suggest
that an emphasis on price stickiness as opposed
to wage stickiness is appropriate in analyzing the
United Kingdom. This emphasis is consistent
with evidence for other European countries, such
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

DiCecio and Nelson

Figure 3
Estimated and Model Responses to a Monetary Policy Shock, 1962:Q3–1979:Q1
Inflation

Output

1

0.4

0.5

0.2

0

0

−0.5

−0.2

−1

−0.4
0

5

10

15

20

0

5

Consumption

10

15

20

Investment
0.5

0.5

0

0

−0.5

−0.5
0

5

10

15

20

0

5

Productivity

10

15

20

Interest Rate
0.4
0.2

0

0
−0.2

−0.2

−0.4
−0.6

−0.4

−0.8
0

5

10

15

20

as Coenen, Levin, and Christoffel’s (2007) study
of nominal rigidities in Germany.
Factor utilization is not found to be important,
the relevant parameter being driven to the boundary of its admissible region. Our VAR productivity
responses are not very precisely estimated. Consequently, the model can explain output variation
in terms of input responses and therefore has
little need to rely on the intensive margin to
explain the data responses.

ESTIMATES INCLUDING
PRE-1979 DATA
In this section we present results for the long
sample 1962-2005 as well as a sample using only
pre-1979 data. The long-sample impulse responses
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

0

5

10

15

20

and their matches are given in Figure 2, and those
for the pre-1979 sample are given in Figure 3.
Parameter estimates for each sample are given in
Table 5.
Turning to the policy rule first, the estimates
deliver substantially lower inflation responses
in the interest rate rule pre-1979, consistent with
the assignment of inflation control to nonmonetary devices in the United Kingdom before 1979.
But the response is large enough even in this sample period to deliver determinacy (i.e., a singlemodel equilibrium).15 The output response is
“wrongly” (negatively) signed pre-1979. This may
15

Our estimation routine considers only parameter combinations
that deliver a single solution. An alternative procedure, which we
have not pursued here, would be to consider both determinacy
and indeterminacy regions and select a solution in the latter case
using the minimal state-variable procedure.

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Table 5
Model Estimates for Samples that Include Pre-1970 Data
1962:Q3–2005:Q4

1962:Q3–1979:Q1

b

0.9410
(0.0003)

0.5806
(0.0015)

λf

1.3904
(0.0038)

1.2039
(0.0035)

ξp

0.4478
(0.0024)

0.1703
(0.0025)

ξw

0.9897
(0.0003)

0.6069
(0.0008)

κ

49.9261
(0.3288)


(—)

σa

0.3900
(0.0062)

0
(—)

ρ

0.9433
(0.0068)

0.7018
(0.0017)

rπ

1.2657
(0.1118)

0.9642
(0.0473)

ry

0.0321
(0.0125)

–0.1980
(0.0220)

r∆π

0.4128
(0.0034)

0.6066
(0.0163)

r∆y

0.0879
(0.0104)

0.1618
(0.0182)

σR

0.1959
(0.0002)

0.1719
(0.0002)

Private sector parameters

Monetary policy rule parameters

be another reflection of the lack of monetary policy response to inflationary pressure because, as
noted earlier, the output coefficient captures
responses to the specific type of output increases
that are likely to raise inflation. An additional
departure from our baseline estimates is that both
sets of estimates that include pre-1979 data have
a more standard (i.e., positive) interest rate
response to the change in inflation.
A major feature of the structural parameters
when we move away from our baseline sample is
that there is now sizable nominal wage rigidity.
Another difference from our baseline structural
parameter estimates pertains to investment behavior. In the pre-1979 sample, the model cannot
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2007

match the empirical investment impulse response;
the best the model can do is to suppress investment altogether (and so generate a flat investment
model response in Figure 3). Accordingly, the
investment adjustment-cost parameter estimate
becomes arbitrarily large.
It is tempting to suggest that our anomalous
results for investment occur because the estimates
including pre-1979 data are distorted by the existence of unmodeled breaks in monetary policy
regime. But this does not really provide a satisfactory answer why we get these particular results.
It is not obvious that estimated impulse responses
over a sample that includes multiple regimes will
be perverse in their shape; they are, more or less,
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

DiCecio and Nelson

an average of the responses observed across each
regime, we should expect them to be of standard
shape. Instead of this, we get model estimates
that appear to extinguish the investment portion
of aggregate demand.
It is likely instead that U.K. government policy
is indeed the culprit for the anomalies in the pre1979 results, but the policy actions responsible
were microeconomic interventions in the economy and not monetary policy. Before the 1980s,
many large industries (e.g., steel and telecommunications) were principally government-owned.
What is more, in a misguided effort to control
inflation by nonmonetary means, governments
frequently intervened in the pricing decisions of
their enterprises. For example, George Brown,
then Secretary of State for Economic Affairs, in
1965 said that the government was operating a
price-control policy “in the field of government
responsibility so far as charges for which they
are responsible, prices which are their responsibility…”(Glasgow Herald, 1965). Ted Heath,
prime minister 1970-74, said shortly before being
elected that “we are going to see to it that the
State does not put up its prices and charges with
gay abandon” (Russell, 1970). The attempt to
enforce this policy led to considerable interference in government enterprises’ operations, so
much so that Anthony Crosland, a leading Labour
Party figure, cited 1970-71 as a period that displayed a poor “balance…between Ministerial
control and entrepreneurial freedom” (Crosland,
1974, p. 39).
From around 1978, however, it became much
more standard for government-owned enterprises
to base their pricing and investment decisions
on market signals, with a government report on
the subject in 1978 stating that the “Government
intends that the nationalized industries will not
be forced into deficits by restraints on their
prices”(House of Commons, 1979). The stepping away by government from management of
investment decisions was cemented by the privatization of many government enterprises in
the 1980s.
Because the pre-1978 government interventions blocked investment from responding to
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

market signals, including those from monetary
policy shocks, one can understand why investment responses might deviate greatly from those
predicted by our model, in which investment
behavior is based on optimal firm choices. Government prohibitions on a firm’s ability to raise
prices might cut off funds to the firm, thus distorting investment decisions. On the other hand,
for given monetary policy, government intervention in investment decisions might merely transfer aggregate demand pressure from investment
to other categories of spending, rather than affect
total demand. So impulse responses other than
those for investment might still be compatible
with the model, which is essentially what we find.

EXTENSIONS
In this section, we report further results regarding the robustness of our results to alternative
data definitions (under “Alternative Investment
Series”) and our choice of regime dates (under
“VAR Stability and Regime Breaks”).

Alternative Investment Series
The BEQM model and such sources as the
Bank of England Inflation Report use a slightly
narrower definition of investment than we use
in our analysis. This narrower definition is
known as “business investment” (though, like
our series, it includes investment by government
enterprises).
We use this alternative investment series and
examine the effect on our results. In Figure 4,
we show that using this series in our VAR makes
little difference in the empirical responses to a
monetary policy shock.16 The parameter estimates
using this series are given in Table 6. These are
little changed from the baseline parameter estimates, with some important exceptions. Most
notably, the policy rule estimates are much less
precisely estimated and feature a much lower
16

The sample period we now use, 1980:Q1–2005:Q4, is slightly
shorter than the 1979:Q2–2005:Q4 sample we used for our baseline results, owing to difficulty obtaining parameter estimates for
1979-2005 with the business investment series.

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Figure 4
Estimated and Model Responses to a Monetary Policy Shock, with Alternative Investment Series,
1980:Q1–2005:Q4
Inflation

Output

0.4

0.3
0.2

0.2

0.1
0

0

−0.1
0

5

10

15

20

0

5

Consumption

10

15

20

Investment

0.4

1

0.2

0.5

0

0
−0.5
0

5

10

15

20

0

5

Productivity
0

0.1

−0.2

0

−0.4

−0.1

−0.6
5

10

15

20

estimated response to output; and the alternative
estimates give roughly equal importance to price
and wage stickiness, whereas in the baseline estimates prices were much stickier than wages.

VAR Stability and Regime Breaks
As noted above, our data cover the whole
1962-2005 period, but our baseline structural
estimates are based on a sample covering only
the period since the late 1970s, reflecting the
changes in U.K. industrial and monetary policies
that took place around that time. To investigate
further the issue of regime break dates, in Table 7
we follow Boivin and Giannoni (2002, p. 99) by
investigating the stability of the VAR when it is
estimated for the long sample 1962:Q1–2005:Q4.
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15

20

Interest Rate

0.2

0

10

0

5

10

15

20

We report the p-values for the constancy of the
coefficients associated with each group of regressors in the VAR. The break dates suggested by
the test are also reported, and those highlighted
in bold achieve statistical significance at the 10
percent level or better.
The results for the baseline VAR specification—that is, the VAR specification underlying
the Figure 2 impulse responses—occupy the top
half of the table. The results suggest a significant
break in the inflation equation around 1975:Q2.
This date, however, does not constitute a monetary policy regime break; instead, the mid-1975
instability reflects a one-time shock to industrial
policy. The previously described U.K. government
policy of holding down nationalized industries’
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

DiCecio and Nelson

prices underwent an adjustment in this period,
with prices allowed to rise to eliminate the accumulated discrepancy with costs. The large effect
on consumer prices that resulted has sometimes
been categorized as tantamount to a substantial
increase in indirect taxes (Wilson, 1984, p. 50).
The test statistics mechanically take this large
shock as evidence of regime change, though,
economically, it does not amount to the sort of
change in systematic policy responses that truly
qualify as a policy regime shift. The remaining
stability rejections are spread over 1977-81 and
so are roughly in line with our assumption of a
1979 break date.
Recall that our VAR does not use detrended
real data, nor does it include a trend as a regressor.
Some work on U.S. data—e.g., Rotemberg and
Woodford (1997), Boivin and Giannoni (2002),
and Giannoni and Woodford (2005)—detrends
real variables before putting them in the VAR. We
report in the bottom half of Table 7 the stability
results for our VAR when our specification is
modified by replacing the four real variables with
their detrended counterparts. The detrending
assumes that the log real variables are driven by
a broken linear trend, with constant and trend
breaks in both 1973:Q4 and 1981:Q4.
Besides continuing to show a break in 1975 in
some of the inflation coefficients, these stability
results largely reaffirm a focus on a regime break
around the early 1980s (specifically, 1980 or 1981).
Two of the significant rejections of stability do
suggest a break in 1984 in GDP and productivity
behavior, but these rejections can be discounted
as reflecting the temporary disturbances to output from the coal-mining strike of that year.
One puzzling aspect of the results with
detrended variables is that the interest rate equation no longer registers any significant regime
break. This, however, is not decisive evidence
against the importance of monetary policy regime
change. For one thing, relatively minor and statistically insignificant changes in the VAR coefficients can imply large changes in the implied
“long-run response” of the interest rate to endogenous variables. This is the case here because,
despite the lack of rejection of stability, the VAR
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Table 6
Estimates Using Alternative Investment
Series, 1980:Q1–2005:Q4
Private sector parameters
b

0.6621
(0.0015)

λf

2.4922
(0.0198)

ξp

0.8465
(0.0020)

ξw

0.9409
(0.0008)

κ

14.2684
(0.2340)

σa


(—)

Monetary policy rule parameters

ρ

0.8426
(1.2144)

rπ

2.0747
(21.0751)

ry

0.0546
(1.0372)

r∆π

–0.2935
(0.0368)

r∆y

0.4140
(2.1327)

σR

0.1509
(0.0001)

equation for the interest rate underlying the final
row of the table has a long-run solution with an
interest rate response to (annualized) inflation of
about 0.3; but this response rises to 1.0 on restricting the sample to 1979-2005.17 Furthermore, the
inflation VAR equation now exhibits a significant
early-1980s break, which is indirect support for a
monetary policy regime change around that time.
17

In some contexts (see, e.g., Rotemberg and Woodford, 1997, and
Rudebusch, 1998) the VAR equation for the interest rate coincides
with the interest rate policy rule. This is not the case in our analysis,
as the policy rule that we use in estimation differs from the VAR
equation. But solving the reduced-form VAR interest rate equation
for its long-run solution nevertheless provides a means of crosschecking the stability test results.

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Table 7
VAR Stability Tests, 1962:Q3–2005:Q4
Regressor
Dependent variable

π

y

c

i

y–h

r

π

0.002
1975:Q2

0.001
1977:Q1

0.713
1975:Q3

0.153
1971:Q1

0.510
1970:Q4

0.574
1973:Q2

y

0.127
1972:Q3

0.682
1975:Q3

0.373
1990:Q2

0.358
1976:Q1

0.948
1975:Q3

0.347
1977:Q4

c

0.469
1990:Q2

0.444
1990:Q2

0.001
1977:Q1

0.012
1980:Q2

0.001
1977:Q1

0.008
1980:Q2

i

0.553
1973:Q1

0.844
1968:Q4

0.788
1976:Q1

0.195
1979:Q2

0.524
1975:Q3

0.711
1975:Q3

y–h

0.382
1973:Q2

0.943
1975:Q3

0.922
1973:Q2

0.869
1973:Q2

0.361
1990:Q2

0.199
1974:Q1

r

0.275
1977:Q4

0.000
1981:Q2

0.000
1981:Q2

0.006
1977:Q4

0.000
1981:Q2

0.471
1973:Q1

1. Baseline VAR specification

2. Baseline VAR specification with y, c, i, y – h detrended

π

0.002
1975:Q2

0.072
1975:Q2

0.040
1981:Q4

0.004
1976:Q1

0.567
1978:Q4

0.741
1988:Q4

y

0.110
1972:Q3

0.036
1984:Q1

0.121
1981:Q2

0.224
1987:Q2

0.296
1981:Q4

0.867
1977:Q4

c

0.738
1970:Q1

0.063
1981:Q3

0.065
1980:Q2

0.095
1980:Q2

0.105
1975:Q2

0.001
1980:Q2

i

0.812
1968:Q4

0.269
1987:Q1

0.239
1985:Q1

0.014
1976:Q4

0.049
1980:Q1

0.662
1990:Q2

y–h

0.325
1972:Q3

0.147
1984:Q1

0.078
1984:Q1

0.187
1976:Q4

0.476
1981:Q4

0.665
1970:Q1

r

0.425
1977:Q4

0.813
1980:Q1

0.714
1986:Q1

0.801
1990:Q3

0.865
1998:Q3

0.460
1976:Q3

NOTE: Values reported are the p-values for the Andrews (1993) sup-Wald test, computed using the procedure of Diebold and Chen
(1996). The null hypothesis assumes no structural breaks, whereas the alternative hypothesis has breaks in the constant and group of
lag coefficients on the indicated regressor. Each panel also gives the break-date associated with the p-value.

CONCLUSIONS
In this paper, we have estimated the
Christiano, Eichenbaum, and Evans (2005) model
on U.K. data. Although CEE found plausible estimates on U.S. data when treating the period since
the 1960s as a single regime, for the United
Kingdom it appears that more satisfactory estimates emerge if pre-1979 data are excluded; otherwise, the estimates imply degenerate behavior of
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2007

investment. This result is consistent with policy
regime changes being an important factor in the
postwar U.K. economy. These regime changes
include not only changes in the role assigned to
monetary policy but also shifts toward making
investment decisions more closely related to
market forces. Another important implication of
our estimates is that price stickiness, rather than
wage stickiness, is the major source of nominal
rigidity in the United Kingdom.
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

DiCecio and Nelson

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England Quarterly Bulletin, Summer 2005, 45(2),
pp. 190-93.

Haskel, Jonathan; Martin, Christopher and Small,
Ian. “Price, Marginal Cost and the Business Cycle.”
Oxford Bulletin of Economics and Statistics,
February 1995, 57(1), pp. 25-41.

Ravenna, Federico and Walsh, Carl E. “Optimal
Monetary Policy with the Cost Channel.” Journal of
Monetary Economics, May 2006, 53(2), pp. 199-216.

House of Commons, Parliament of the United
Kingdom. House of Commons Debates, December
5, 1979, p. 563 (quoting U.K. Government, White
Papers on Nationalized Industries, London: HMSO,
1978).
Ireland, Peter N. “Sticky-Price Models of the Business
Cycle: Specification and Stability.” Journal of
Monetary Economics, February 2001, 47(1), pp. 3-18.
King, Mervyn A. Public Policy and the Corporation.
London: Chapman and Hall, 1977.
Levin, Andrew T.; Onatski, Alexei; Williams, John C.
and Williams, Noah. “Monetary Policy under
Uncertainty in Micro-Founded Macroeconometric
Models.” NBER Macroeconomics Annual, 2005,
20(1), pp. 229-87.
Neiss, Katharine S. and Nelson, Edward. “The RealInterest-Rate Gap as an Inflation Indicator.”
Macroeconomic Dynamics, April 2003, 7(2),
pp. 239-62.
Nelson, Edward and Nikolov, Kalin. “Monetary
Policy and Stagflation in the U.K.” Journal of
Money, Credit, and Banking, June 2004, 36(3),
pp. 293-318.
Newey, Whitney K. and McFadden, Daniel. “Large

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Ravn, Morten O. “Permanent and Transitory Shocks
and the U.K. Business Cycle.” Journal of Applied
Econometrics, January/February 1997, 12(1),
pp. 27-48.
Rotemberg, Julio J. and Woodford, Michael. “An
Optimization-Based Econometric Framework for
the Evaluation of Monetary Policy.” NBER
Macroeconomics Annual, 1997, 12(1), pp. 297-346.
Rudebusch, Glenn D. “Do Measures of Monetary
Policy Shocks in a VAR Make Sense?” International
Economic Review, November 1998, 39(4), pp. 907-31.
Russell, William. “‘Pushover’ Jenkins Surrendered to
Unions, Says Heath.” Glasgow Herald, June 3, 1970,
p. 8.
Sims, Christopher A. and Zha, Tao. “Were There
Regime Switches in U.S. Monetary Policy?”
American Economic Review, May 2006, 96(1),
pp. 54-81.
Smets, Frank and Wouters, Raf. “An Estimated
Stochastic Dynamic General Equilibrium Model of
the Euro Area.” Journal of the European Economic
Association, 2003, 1(5), pp. 1123-75.
Wilson, Thomas. “Monetarism in Britain,” in
T. Wilson, ed., Inflation, Unemployment, and the
Market. Oxford: Clarendon Press, 1984, pp. 41-78.

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DiCecio and Nelson

APPENDIX
Data Sources and Definitions
Nominal interest rate: U.K. Treasury bill rate, quarterly average. Source: Haver-IFS, quarterly average
series.
Output: real GDP, seasonally adjusted, quarterly, series abmi.q.
Source: U.K. Office of National Statistics (ons.gov.uk), downloaded May 2006.
Private household consumption: seasonally adjusted, quarterly, series abjr.q.
Source: ons.gov.uk, downloaded May 2006.
Investment: gross fixed capital formation, seasonally adjusted, quarterly, series npqt.q.
Source: ons.gov.uk, downloaded May 2006.
Alternative investment series: business investment at 2003 prices, seasonally adjusted, quarterly,
series npel.q.
Source: ons.gov.uk, downloaded August 2006.
Productivity: Y/H, where H = hours worked.
Source for H: series ybus.q (source: U.K. Office of National Statistics [ONS]), with splice into Ravn
(1997) U.K. hours worked series to obtain pre-1971:Q1 data.
Inflation: log difference of P, where P is a seasonally adjusted consumer price series. P was constructed
as follows: A quarterly average of the retail price index (RPI) was spliced into a quarterly average of
RPIX (RPI excluding mortgage interest payment) after 1973, the series was seasonally adjusted, then
tax-related spikes of 4 percent (in 1979:Q3) and 2 percent (1990:Q2) were removed from the series.
The seasonal regressions underlying the seasonal adjustment used the log-change as the dependent
variable, and seasonal patterns were allowed to differ across 1955-76, 1976-86, and 1987-2005.
Source for the monthly RPI underlying the quarterly averages: ONS (ons.gov.uk). The ONS, however,
provides RPIX data only from January 1987. An unofficial RPIX series starting in 1974 has, however,
been constructed at the Bank of England, and this series underlies studies such as Nelson and Nikolov
(2004) and Benati (2004). The OECD-Haver service also provides an RPIX series (though beginning
only in 1975) that closely matches this series. We used the quarterly average of the unofficial RPIX
series for 1974-87 and spliced it into the quarterly average of the official RPIX series that begins in
1987. Splicing this RPIX series at 1974:Q1 with RPI delivered the RPI/RPIX quarterly average series
underpinning P.

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Commentary
Martin Fukac̆ and Adrian Pagan

T

he paper by DiCecio and Nelson (2007;
DCN) considers the estimation of the
parameters of a dynamic stochastic
general equilibrium (DSGE) model for
the United Kingdom that is virtually the same
as that which Christiano, Eichenbaum, and Evans
(2005; CEE) estimated for the United States.
The CEE model is much larger than existing
academic DSGE models of the United Kingdom,
such as Lubik and Schorfheide (2007). It is not
as large as the Bank of England Quarterly Model
(BEQM), which has both DSGE elements and
data-imposed dynamics; however, because the
BEQM is used for policymaking, there is a much
greater imperative to match the data than found
in most academic work. There are a few other
DSGE models that have been applied to the
United Kingdom—for example, Leitemo (2006)—
but, in general, these are often used to examine
some particular question and are also rather
restricted in their mode of operation. Often they
use a standard open-economy New Keynesian
model rather than a straight DSGE model like
CEE’s. Moreover, the authors of these models are
often not that familiar with the U.K. context and
data; the current authors, however, are experts
in this area, and it certainly shows in their discussion of alternative data sources. So, given
the paucity of studies, any new one would be
welcome.
Now, as the Chinese proverb says, a journey
of a thousand miles starts with a single step. What
we have here is more than single step but well

short of a thousand miles. Reading it, one longed
for a fully fleshed-out model along the lines of
Smets and Wouters’s (2003) work on the United
States and the euro area (which is very similar to
CEE’s), where a complete set of shocks is described
and estimated. Because the DCN model identifies
only a money shock, there are few questions one
can ask about the model. So it was disappointing
that the authors were not a bit more adventuresome. But we presume that this will be part of a
broader piece of research and look forward to
seeing a more complex model that recognizes
the open-economy characteristics of the United
Kingdom. Of course, one does have to acknowledge that DSGE models have not had a good
record of producing useful models of the open
economy. One reason DCN point to is the prediction of stronger exchange effects than seen in the
data. We agree with this, and it was a central conclusion about the mini-BEQM model that was
calibrated to the U.K. economy in Kapetanios,
Pagan, and Scott (2007). Moreover, as Justiniano
and Preston (2006) argue, it has been very hard to
find much of an influence of the foreign economy
on a small open economy, and this is contrary to
evidence we have from vector autoregressions
(VAR). So there is quite a bit to be done both on
the broader front of developing useful openeconomy models and in getting a U.K. model that
is in a more complete state than this one. Because
the model is not fleshed out that much, we will
restrict comments to what DCN do rather than
alternatives that might have been tried.

Martin Fukac̆ is an economist at the Reserve Bank of New Zealand. Adrian Pagan is a professor of economics at Queensland University of
Technology and the University of New South Wales. This research was supported by ARC Grant DP0449659.
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 233-40.

© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
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Fukac̆ and Pagan

ESTIMATION STRATEGY
The methodology used for estimation is that
of CEE. It has four steps:
1. Identify monetary policy shocks using a
structural equation for the interest rate and
a VAR(2) to represent the rest of the system.
The identification condition used is that
monetary shocks have no contemporaneous
effect on any variables of the system except
the interest rate. The monetary policy rule
depends on all variables in the VAR, and
these enter the rule both with lags and
contemporaneously.
2. Compute the monetary impulse responses
C jD from this structural VAR (SVAR)
( j indexes the j th impulse response).
3. Choose some values for the DSGE model
parameters θ and use them to compute the
DSGE model monetary impulse responses
C jM.

is 77: Of these 77, 72 are from the 2 lags of the
six variables in the six equations plus the 5 possible coefficients attached to contemporaneous
coefficients in the interest rate rule. Because M
seems to be 25, that would mean that 150 impulses
are used. This is much larger than the number of
parameters determining them. Hence there are
many redundant impulse responses and the
D
must be singular.
covariance matrix of C1D,…, C 25
This might be a problem when one uses the δ
method to compute standard errors. Indeed, the
standard errors of θˆ found by moment matching
in DCN seem to be incredibly small. Thus they
have an estimate of the markup λ f parameter of
2.27 with a standard error of 0.03. It’s hard to
believe that one could ever get that degree of
precision with just 26 years of quarterly observations. Because the SVAR coefficients have a t ratio
above 5 in only one case (lagged productivity),
it’s hard to see how we can end up with t ratios
above 400 for θˆ, which are fundamentally derived
from the SVAR coefficients.

4. Find the value of θ that minimizes

∑ j =1(C jD − C jM )′ W (C jD − C jM ),
M

where W is a diagonal matrix of weights.
They apply this to U.K. data. The original
DSGE model they employ has 15 variables,
whereas the SVAR(2) has 6. Estimates of the
parameters are presented, and some standard
errors are given along with plots of the monetary
impulses implied by the SVAR and the DSGE
model calibrated with the estimated θ.

ESTIMATION PROBLEMS
What could go wrong with this methodology?
We discuss three issues in the following
subsections.

How Many Impulses To Use?
There is a maximum useful choice for M
because the C jD are simply functions of the SVAR
coefficients. Let there be n variables in the SVAR
(in DCN n = 6 and it is an SVAR(2)). Then the
total number of coefficients in the DCN SVAR(2)
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Approximating the DSGE Model with
an SVAR
There is a generic problem here in that the
DSGE model often determines m variables and
m > n; that is, the SVAR is fitted to a smaller number of variables than appear in the DSGE model.
This is true of the DCN model, where it appears
that n = 6, m = 15. Now the DSGE model will be
an SVAR in the m variables but is unlikely to be
an SVAR in the n variables. An old literature, due
to Zellner and Palm (1974) and Wallis (1977),
has noted that, when a system that is a VAR(p) in
n variables is reduced to a smaller system with
m < n variables, the smaller system will generally be a vector autoregression moving-average
(VARMA) process. Because the CEE procedure
involves such compression of variables, it might
be expected that a VARMA process is needed
rather than a VAR; so, the use of a VAR could lead
to specification bias. It might be expected that a
VAR of very high order could compensate for this
misspecification—and this is generally true—
but the order of VAR needed to deliver a good
approximation may in fact be far too high for the
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Fukac̆ and Pagan

data sets one is normally faced with. For example,
Kapetanios, Pagan, and Scott (2007) find that
reducing a model that is BEQM-like (but half
the size) would require a VAR(50) to capture the
effects of some shocks (and this with 30,000
observations). The problem has been analyzed in
a DSGE context by Ravenna (forthcoming) and
Fry and Pagan (2005). We adopt the exposition
of the latter.
Suppose that the DSGE model followed a
VAR(1) solution (assuming that ut is i.i.d.):

zt = Pzt −1 + Gut .
Now consider what happens if we model only a
subset of the variables. We will call the modeled
subset z1t and the omitted variables z2t .We can
decompose the VAR above as
(1)

z1t = P11z1t −1 + P12z2t −1 + G1ut ,

and we will assume that the following relation
holds between z1t and z2t :
z2t = D0 z1t + D1z2t −1 + D2ut .

Substituting this in we get

z1t =

( P11 + P12D0 ) z1t −1 + P12D1z2t −2 + G1ut + P12D2ut −1,
so that the sufficient conditions for there to be a
finite-order VAR in z1t will be that either P12 = 0
(i.e., z2t does not Granger cause z1t ; see Lütkepohl,
1993, p. 55, and Quenouille, 1957, p. 43-44) or
–
–
D 1 = 0, D 2 = 0 (i.e., the variables to be eliminated
must be connected to the retained variables
through an identity and there can be no “own lag”
in the omitted variables in the relation connecting
z1t and z2t ).
This observation looks trivial, but in fact many
of the problems that have arisen where a finiteorder VAR does not obtain occur because the
omitted variables are connected with the retained
variables through an identity, but one that contains an “own lag.” The classic example is in the
basic real business cycle (RBC) model where, after
log linearization around the steady state, we
would get
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

(2)

lt = y t − ct
C ∗ct + K ∗ kt = Y ∗ y t + (1 − δ ) K ∗kt −1

(3)

ct = Et (ct +1 − αγ ( y t +1 − kt +1 ))

y t = at + α kt −1 + (1 − α ) lt ,
where ct is the log of consumption, at is the log of
the technology shock, kt is the log of the capital
stock, lt is the log of labor input, and yt is the log
of output. An asterisk denotes steady-state values,
and α is the steady-state share of capital in output.
When at is an AR(1), the solution to this system
can be made a VAR(1) in ct , lt , yt , and kt . It’s clear
that we could eliminate any of ct , lt , or yt because
none appear as a lagged variable in the system.
Equally clearly, kt cannot be eliminated unless we
can find an identity that relates it to other variables but does not involve kt –1. Thus, the identity
(3) shows that this is not possible. Most of the
literature that seeks to establish that a SVAR cannot approximate a DSGE model (Chari, Kehoe,
and McGrattan, 2004; Erceg, Guerrieri, and Gust,
2005; Cooley and Dwyer, 1995) substitute out kt
and so end up with a non-finite-order VAR.
The implication of this for DCN’s work is that
the reduction of the system from 15 to 6 variables
might necessitate a very long VAR and not the
VAR(2) they adopted. They used statistical criteria
to determine the order of the VAR. Kapetanios,
Pagan, and Scott (2007) did this as well, and the
tests produced a VAR of order six, far below what
was needed (50th order) to produce the correct
impulse responses. The reason is that the tests
proceed on the assumption that the number of
variables in the VAR is correct and it is only the
order that needs to be found. So it seems that DCN
might be matching impulses that are not strictly
comparable. The appropriate procedure would
be to (i) simulate a long history of data from the
15-variable DSGE model, incorporating just monetary shocks; (ii) fit a VAR(2) in just 6 of the variables; and then (iii) find the impulse responses
from such an approximating VAR, being careful
to note that some of the lagged values will be
perfectly correlated with others and that it will
be necessary to combine variables together to
overcome that problem. These are then matched
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Fukac̆ and Pagan

up with the empirically observed VAR(2) impulse
responses in the six variables. We have assessed
this by examining a variant of their model, where
information is dated at t rather than the combination of t and t – 1 that is in their paper. However,
we used the same model parameters as DCN.
Although there are some differences between the
true impulse responses and those delivered by a
VAR(2), it seems that the approximations are quite
good, except at longer horizons. So there does not
seem to be an approximation issue here, although
in any application one should check that there is
no problem, as it is not very difficult to do. However, the δ method used by CEE to compute standard errors is correct only if the approximation is
satisfactory. Basically, the estimator of the DSGEmodel parameters is an indirect estimator, being
derived from functions of the SVAR coefficients
represented by the impulse responses. The covariance matrix of such an estimator requires that
derivatives of the model-implied VAR impulse
responses be computed with respect to the θ
parameters, and not the derivatives with respect
to the model impulses, as done by CEE. These are
only the same if there is no approximation error.

Multiple Solutions
Ignoring the problem identified in the previous section, estimators such as the maximum
likelihood estimator basically attempt to match
the VAR coefficients from the data with those
from the model, rather than attempting to match
impulse responses. To see the problems you might
encounter with the latter, let us look at the simple
model

y t = β E t −1 ( x t +1 ) + ε yt ,
x t = ρ x t −1 + ε xt .

If we would try to find β and ρ by matching the
first two impulse responses, we would be minimizing (assuming that the weights in W are equal)

(c
+ (c

D
1, y ε x

D
1,x ε x

2

) + (c
− ρ ) + (c

− βρ2

2

D
2, y ε x

D
2, x ε x

− βρ3

2

)

2

)

− ρ2 .

Clearly, such an approach has the problem of
producing an order-six polynomial in ρ, so that
we may get multiple solutions. This would not
arise if we were matching to the VAR coefficients,
because then ρ̂ = â 2, βˆ = â 1/ρ̂ 2. Bearing in mind
the first point as well, it seems better to match
the VAR to get estimates of θ and then to show
the impulse response correspondence.

LOOKING AT SOME OF THE
EULER EQUATIONS
Now it would seem useful to develop a
method that uses the same information as
impulse-response matching but that is a bit simpler, provides ready ways of computing standard
errors, emphasizes the economics, and can be
used to tell us something about the ability of the
DSGE model to match the data. Basically the
proposal is to work with the Euler equations and
estimate the model parameters directly from them
with a single-equation estimator. Of course this
is an old idea, but it has fallen out of favor, possibly because of the literature claiming that systems
estimators of parameters of the New Keynesian
system performed better than the single-equation
estimators because of weak instruments. But, in
many DSGE models, enough parameters are prescribed that weak instrument problems are not
present, and we will see this in the DCN context.
The Euler equations of DSGE models have
the generic form1

The VAR will be
Et −1zt = η1zt −1 + η2 Et −1 ( zt +1 ) + η3 Et −1w t .

y t = a1 x t −1 + ε yt
x t = a2 x t −1 + ε xt ,

where a1 = βρ 2, a2 = ρ, and the impulse responses
are

In this equation, zt is the endogenous variable
whose coefficient is normalized to unity, wt are
1

c1M,y ε x
236

= βρ

2

,c2M,y ε x

J U LY / A U G U S T

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2

= βρ ,c1,x ε x = ρ,c2,x ε x = ρ .
2007

The dating of expectations here comes from DCN and reflects the
assumption that interest rates have no effect on contemporaneous
variables.

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Fukac̆ and Pagan

either exogenous or other endogenous variables,
and the parameters ηj are functions of some of
the DSGE model parameters θ. Now this can be
written as
zt = η1zt −1 + η2 E t −1 ( zt +1 ) + η3 Et −1w t + ε t ,

and the right-hand-side regressors are uncorrelated with the error εt . If we had these conditional
expectations we could run a regression. We note
that this equation holds for any subset of information used by the economic agents. Hence, let us
define the information used in the estimation as
that of the DCN VAR(2), that is, two lagged values
of yt ,ct ,it ,yt – ht ,rt , and ∆pt .2 Call these the vector
ζ t–1. Then, if we can estimate Et–1共zt+1兲 and Et –1wt ,
we could simply fit a regression to this equation
and thereby measure ηj . Because the model is
linear, we can indeed estimate Et–1共zt+1兲 and Et –1wt
as the predictions from the regression of zt+1 and
wt against ζ t–1. Basically, this estimation method
uses the same information as moment matching,
that is, the information contained in the VAR.
Notice that standard errors are easily found from
this by estimating the Euler equation parameters
with an instrumental variables estimator. As we
will see later, in most cases the instruments are
very good and so there is no reason to doubt the
standard errors of ηj found in this way.
This is a relatively simple way to estimate the
ηj . Whether the DSGE model parameters θ can be
estimated is a different question, because there
may be a nonlinear mapping between the η and θ
and so we may not be able to recover θ uniquely.
This shouldn’t concern us unduly because, fundamentally, the impact of monetary policy depends
on the ηj ; but there may be some cases where we
want to think about changing θ and so would then
need to identify it. Ma (2002) pointed out that
there was an identification problem like this in
strictly forward-looking New Keynesian Phillips
curves, and we will see that it comes up in the
CEE model as well.
Let us look at the above principles in the
context of some of the equations in DCN. First
2

We work with data that are not deviations from steady-state values
and so will have to include intercepts in any equation we estimate.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

we look at the Phillips curve. After normalizing
on πt , the Euler equation becomes
− Et −1π t +

(4)
+

β
1
π t −1 +
E π
1+ β
1 + β t −1 t +1

(1 − βξ ) (1 − ξ ) E s .
t −1 t
(1 + β ) ξ

We can write this as an equation of the form
πt =
+

1
β
π +
π
1 + β t −1 1 + β t + 1

(1 − βξ )(1 − ξ ) E  α r k + 1 − α R + w  + ε
(
) ( t t )) t
t −1 (
t
(1 + β ) ξ

or

πt −

β
1
π
π t −1 −
1+ β
1 + β t +1

(

)

= η1 Et −1  α rtk + (1 − α )( Rt + w t )  + ε t ,
where Et–1共εt 兲 = 0. We note that, because β = 0.99
is imposed, we are not trying to estimate the
coefficients attached to πt and πt+1. Now, using
data, one can form α rtk + 共1 – α兲 共Rt + wt 兲. Because
DCN pre-set α to 0.36, this can then be regressed
against the information represented by the VAR
lagged variables to get

(

)

E t −1  α rtk + (1 − α ) ( Rt + w t )  .3

The regression of this variable against ζ t–1 (the
VAR(2) lagged variables) gives an R 2 of 0.99, so it
is a very good instrument for α rtk + 共1 – α兲 共Rt + wt 兲.
If we fit a nonlinear regression to this equation,
we get an estimated coefficient for ξp of 0.988,
which is reasonably close to what is reported in
the paper from impulse response matching. But
the standard deviation is 0.092, which is nowhere
near the 0.0004 given in the paper—although, if
one makes it robust to serial correlation, it halves.
Clearly, the estimate here implies very low frequency of price adjustment, as does DCN’s.
3

Because DCN conclude that σa = ⬁, there is no difference between
the capital stock and services. We compute the capital stock recursively, but this means the estimates are inaccurate until the initial
condition disappears. Because we start the recursion in 1955:Q2,
but use data only after 1979:Q2, we feel that the effects of the initial condition have died away, as it will be multiplied by the term
(0.975)104.

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Fukac̆ and Pagan

Although this estimate seems implausible, the
interpretation would seem to be that there are
some problems in the specification of the Phillips
curve (indeed the serial correlation in the residuals is consistent with that).
Another equation in DCN we consider estimating is the production function with the form
y t = λ f (α kt + (1 − α ) lt ).

Given that DCN prescribe α, we can form
ζ t = α kt + 共1 – α兲lt and treat this as a regressor to
estimate λf . There will of course be technology
in this relation and, by treating it as an AR(1)
process, the equation will have an AR(1) error
term. Because the regressor will generally be correlated with the white noise shock driving the AR
in technology, we need instruments to estimate
λf ; for this we use yt –1 and ζ t –1. We also include a
constant to reflect the fact that we are not using
variables that are deviations from a constant
steady state and that technology should have a
constant mean. Then we get an estimate of λf of
1.25, with a standard error of 0.05. This seems
more reasonable than the value of 2.27 that DCN
obtain, although they give a defense of it. Again,
the standard errors are very different.
The interest rate rule parameter values are
somewhat puzzling. Under the assumptions in
force here, one should be able to fit this rule by
ordinary least squares (OLS) regression because
it is assumed that the regressors are all uncorrelated with the interest rate shock. If we run the
regression, the fit we would get is
R t = 0.883Rt −1 + (1 − 0.833) (0.0001y t + 1.28π t −1 )
+ 0.05∆y t + 0.10∆π t ,

versus the estimated equation of the paper,

there is not much difference if we add on extra
lags. Notice also that the negative sign on ∆πt that
perturbed them has gone. Because this seems a
logical way to estimate the money rule, given the
assumptions made about the structure of the
model, one is puzzled about the results that come
from matching impulse responses.
What explains this? One possibility is that
the DSGE model implies a particular value for
the intercept of the equation, whereas we have
just subsumed this into a constant term that is
freely estimated. However, the steady-state values
used in the model for variables seemed quite
close to the sample means over the estimation
horizon; so, it would seem that one would get
much the same intercepts (provided of course
the slope coefficients were correct).
There are some problems with multiple
parameter values in both the Phillips curve and
the wage equation. Because

πt −

β
1
π t −1 −
π t +1 =
1+ β
1+ β

)

(

η1 Et −1  α rtk + (1 − α ) ( Rt + w t )  + ε t
and

η1 =

(1 − βξ )(1 − ξ ) ,
p

p

(1 + β ) ξp

we see that the solution for ξp involves a quadratic. There are two estimates of ξp that produce
exactly the same likelihood—the value of 0.988
given above and 1.02. A similar situation exists
for the wage equation. Perhaps this is one reason
why Bayesian methods might work better in these
models—they would impose the restriction that
ξp and ξw lie between 0 and 1.

Rt = 0.872Rt −1 + (1 − 0.872) (0.352y t + 1.28π t −1 )
+ 0.43∆y t − 0.62∆π t .
The standard deviation on yt from the OLS regression is very small, so these estimates are very
different. DCN note that the rule they fit is not
the one in the VAR(2), because that would include
other lags in the variables. But if we just fit a
VAR(1), then it should be comparable to what
they claim the estimated money rule is. In fact,
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2007

STRUCTURAL CHANGE IN THE
MODEL
The authors look at structural change in the
SVAR and conclude that there was some back in
the 1970s, but this was due to industrial issues
and not monetary policy regime changes. But it’s
always difficult to learn something about the staF E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Fukac̆ and Pagan

Figure 1
Coefficients of Lagged Inflation with Two Standard Errors on Each Side from a Rolling Regression
(68 periods)
1.4
Coefficient
2*SE+
2*SE–

1.2

1

0.8

0.6

0.4

0.2

19
72
19 :Q2
73
19 :Q
74 3
19 : Q
76 4
19 :Q
77 1
19 :Q
78 2
19 :Q
79 3
19 :Q4
81
19 :Q
82 1
19 :Q
83 2
19 :Q
84 3
19 :Q4
86
19 :Q
87 1
19 :Q2
88
19 :Q
89 3
19 :Q4
91
19 :Q
92 1
19 :Q
93 2
19 :Q
94 3
19 :Q
96 4
19 :Q
97 1
19 :Q
98 2
19 :Q
99 3
20 :Q
01 4
20 :Q
02 1
20 :Q2
03
20 :Q
04 3
:Q
4

0

bility of the parameters in a VAR. One might also
want to ask where to place a possible monetary
policy regime change. Is it when Thatcher came
in, when inflation targeting was adopted, or when
there was a formal change to the institution with
the formation of the Monetary Policy Committee
(MPC)? Pagan (2003) argued that there had been
a change in the level of persistence in inflation
in the United Kingdom after the formation of the
MPC. This is still evident in the data: See Figure 1,
which gives estimates of the coefficient of πt –1
using a rolling horizon of 68 quarters.
So this looks like structural change in the
dynamics, and perhaps the VAR stability tests
should have focused more around the post-1997
point, although this means a very short postbreak sample. At the end of the day, graphs like
this have to make one wonder about applying a
constant-parameter DSGE model to such data. It
would seem one might need to use only the post1997 period to estimate the DSGE model, although
with such small samples one might need to use
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

some sort of Bayesian approach. Perhaps one
could use the estimated values of this paper to
produce priors.

REFERENCES
Chari, V.V.; Kehoe, Patrick J. and McGrattan, Ellen R.
“A Critique of Structural VARs Using Real Business
Cycle Theory.” Working Paper No. 631, Federal
Reserve Bank of Minneapolis, 2004.
Christiano, Lawrence J.; Eichenbaum, Martin and
Evans, Charles L. “Nominal Rigidities and the
Dynamic Effects of a Shock to Monetary Policy.”
Journal of Political Economy, February 2005, 113(1),
pp. 1-45.
Cooley, Thomas F. and Dwyer, Mark. “Business
Cycles Without Much Theory: A Look at Structural
VARs.” Journal of Econometrics, March/April 1995,
83(1-2), pp. 57-88.

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Fukac̆ and Pagan

DiCecio, Riccardo and Nelson, Edward. “An
Estimated DSGE Model for the United Kingdom.”
Federal Reserve Bank of St. Louis Review, July/
August 2007, 89(4), pp. 215-31.
Erceg, Christopher J.; Guerrieri, Luca and Gust,
Christopher. “Can Long-Run Restrictions Identify
Technology Shocks?” Journal of the European
Economic Association, December 2005, 3(6), pp.
1237-78.
Fry, Renee and Pagan, Adrian R. “Some Issues in
Using VARs for Macroeconometric Research.”
Centre for Applied Macroeconomic Analyses,
CAMA Working Paper 2005/18, Australian National
University, 2005.
Justiniano, Alejandro and Preston, Bruce. “Can
Structural Small Open Economy Models Account
for the Influence of Foreign Disturbances.” Centre
for Applied Macroeconomic Analyses, CAMA
Working Paper, 2006/12, Australian National
University, 2006.

Lütkepohl, Helmut. An Introduction to Multiple
Time Series. Berlin: Springer-Verlag, 1993.
Ma, Adrian. “GMM Estimation of the New Phillips
Curve.” Economics Letters, August 2002, 76(3),
pp. 411-17.
Pagan, Adrian R. “Report on Modelling and
Forecasting at the Bank of England.” Bank of
England Quarterly Bulletin, Spring 2003, 43(1),
pp. 1-29.
Quenouille, M.H. The Analysis of Multiple Time
Series. Griffin’s Statistical Monographs and Courses
No. 1. London: Griffin, 1957.
Ravenna, Frederico. “Vector Autoregressions and
Reduced Form Representations of Dynamic
Stochastic General Equilibrium Models.” Journal
of Monetary Economics (forthcoming).
Smets, Frank and Wouters, Raf. “An Estimated
Stochastic Dynamic General Equilibrium Model of
the Euro Area.” Journal of the European Economic
Association, 2003, 1(5), pp. 1123-75.

Kapetanios, George; Pagan, Adrian R. and Scott,
Alasdair. “Making a Match: Combining Theory and
Evidence in Policy-Oriented Macroeconomic
Modeling.” Journal of Econometrics, February 2007,
136(2), pp. 505-94.

Wallis, Kenneth F. “Multiple Time Series and the
Final Form of Econometric Models.” Econometrica,
September 1977, 45(6), pp. 1481-97.

Leitemo, Kai. “Targeting Inflation by Forecast
Feedback Rules in Small Open Economies.” Journal
of Economic Dynamics and Control, March 2006,
30(3), pp. 393-413.

Zellner, Arnold and Palm, Franz. “Time Series
Analysis and Simultaneous Equation Econometric
Models.” Journal of Econometrics, May 1974, 2(1),
pp. 17-54.

Lubik, Thomas A. and Schorfheide, Frank. “Do Central
Banks Respond to Exchange Rate Movements? A
Structural Investigation.” Journal of Monetary
Economics, May 2007, 54(4), pp. 1069-87.

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F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Macroeconomic Implications of
Changes in the Term Premium
Glenn D. Rudebusch, Brian P. Sack, and Eric T. Swanson
Linearized New Keynesian models and empirical no-arbitrage macro-finance models offer little
insight regarding the implications of changes in bond term premiums for economic activity. This
paper investigates these implications using both a structural model and a reduced-form framework.
The authors show that there is no structural relationship running from the term premium to economic activity, but a reduced-form empirical analysis does suggest that a decline in the term premium has typically been associated with stimulus to real economic activity, which contradicts
earlier results in the literature. (JEL E43, E44, E52, G12)
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 241-69.

F

rom June 2004 through June 2006, the
Federal Reserve gradually raised the
federal funds rate from 1 percent to 5¼
percent. Despite this 425-basis-point
increase in the short-term rate, long-term interest
rates remained at remarkably low levels, with the
10-year Treasury yield averaging 4¼ percent in
both 2004 and 2005 and ending September 2006
at just a little above 4½ percent. The apparent
lack of sensitivity of long-term interest rates to
the large rise in short rates surprised many
observers, as such behavior contrasted sharply
with interest rate movements during past policytightening cycles.1 Perhaps the most famous
expression of this surprise was provided by
then-Chairman of the Federal Reserve Alan
Greenspan in monetary policy testimony before
Congress in February 2005, in which he noted
that “the broadly unanticipated behavior of world
bond markets remains a conundrum.”
1

For example, from January 1994 to February 1995, the Federal
Reserve raised the federal funds rate by 3 percentage points and
the 10-year rate rose by 1.7 percentage points.

The puzzlement over the recent low and relatively stable levels of long-term interest rates has
generated much interest in trying to understand
both the source of these low rates and their economic implications. In addressing these issues,
it is useful to divide the yield on a long-term bond
into an expected-rate component that reflects the
anticipated average future short rate for the maturity of the bond and a term-premium component
that reflects the compensation that investors
require for bearing the interest rate risk from
holding long-term instead of short-term debt.
Chairman Greenspan’s later July 2005 monetary
policy testimony suggested that the conundrum
likely involved movements in the latter component, noting that “a significant portion of the
sharp decline in the ten-year forward one-year
rate over the past year appears to have resulted
from a fall in term premiums.” This interpretation has been supported by estimates from various finance and macro-finance models that
indicate that the recent relatively stable 10-year
Treasury yield reflects that the upward revisions

Glenn D. Rudebusch is a senior vice president and associate director of research and Eric T. Swanson is a research advisor at the Federal
Reserve Bank of San Francisco. Brian P. Sack is a vice president at Macroeconomic Advisers. The views expressed in this paper are the authors’
and do not necessarily reflect official positions of the Federal Reserve System. The authors thank John Cochrane, Monika Piazzesi, and
Jessica Wachter for helpful comments and suggestions. Michael McMorrow, Vuong Nguyen, and David Thipphavong provided excellent
research assistance.

© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
only with prior written permission of the Federal Reserve Bank of St. Louis.

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Rudebusch, Sack, Swanson

to expected future short rates that accompanied
the monetary policy tightening were offset, on
balance, by a decline in the term premium (e.g.,
Kim and Wright, 2005, and Rudebusch, Swanson,
and Wu, 2006).2
It is this recent experience of a declining term
premium in long-term rates that motivates our
paper. We examine what is known—both in theory and from the data—about the macroeconomic
implications of changes in the term premium.
This topic is especially timely and important
because of the practical implications of the recent
low term premium for the conduct of monetary
policy. Specifically, as noted by Federal Reserve
Governor Donald Kohn (2005), “the decline in
term premiums in the Treasury market of late may
have contributed to keeping long-term interest
rates relatively low and, consequently, may have
supported the housing sector and consumer
spending more generally.” Furthermore, any such
macroeconomic impetus would alter the appropriate setting of the stance of monetary policy, as
described by Federal Reserve Chairman Ben
Bernanke (2006):
To the extent that the decline in forward rates
can be traced to a decline in the term premium…the effect is financially stimulative
and argues for greater monetary policy
restraint, all else being equal. Specifically, if
spending depends on long-term interest rates,
special factors that lower the spread between
short-term and long-term rates will stimulate
aggregate demand. Thus, when the term premium declines, a higher short-term rate is
required to obtain the long-term rate and the
overall mix of financial conditions consistent
with maximum sustainable employment and
stable prices.

Under this “practitioner” view, which is also
prevalent among market analysts and private
sector macroeconomic forecasters, the recent fall
in the term premium provided a boost to real
economic activity and, therefore, optimal mone-

tary policy should have followed a relatively
more restrictive path as a counterbalance.3
Unfortunately, this practitioner view of the
macroeconomic and monetary policy implications
of a drop in the term premium is not supported
by the simple linearized New Keynesian model
of aggregate output that is currently so popular
among economic researchers. In that model, output is determined by a forward-looking IS curve:
(1)

where yt denotes aggregate output and it – Et πt +1
is the one-period ex ante real interest rate. Solving
this equation forward, output can be expressed as
a function of short-term real interest rates alone:
(2)

Of course, as we discuss in detail below, such decompositions of
the long rate into expected rates and a term premium are subject
to considerable uncertainty.

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2007

`
1
y t = − Et ∑ β j (it + j − π t +1+ j ) + et .
γ j =0

According to this equation, it is the expected
path of the short-term real interest rate that determines the extent of intertemporal substitution
and hence current output. Long-term interest rates
matter only because they embed expectations of
future short-term interest rates (as in McGough,
Rudebusch, and Williams, 2005). Taken literally,
this simple analytic framework does not allow
shifts in the term premium to affect output; therefore, according to this model, the recent decline
in the term premium should be ignored when
constructing optimal monetary policy, and the
only important consideration should be the
restraining influence of the rising expected-rate
component.
Given these contradictory practitioner and
New Keynesian views about the macroeconomic
implications of changes in the term premium,
this paper considers what economic theory more
generally implies about this relationship as well
as what the data have to say. We start in the next
section by examining a structural dynamic stochastic general equilibrium (DSGE) framework
3

2

1
y t = β Et y t +1 − (it − Et π t +1 ) + et ,
γ

For example, in a January 2005 commentary, the private forecasting
firm Macroeconomic Advisers argued that the low term premium
was keeping financial conditions accommodative and “would
require the Fed to ‘do more’ with the federal funds rate to achieve
the desired rate of growth.”

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Rudebusch, Sack, Swanson

that can completely characterize the relationship between the term premium and the economy.
In this framework, unlike its linearized New
Keynesian descendant, there are important connections between term premiums and the economy. Unfortunately, given theoretical uncertainties
and computational complexities, the model cannot be taken directly to the data, so it provides
only qualitative insights about the macroeconomic
implications of changes in term premiums, not
quantitative empirical assessments.
To uncover such empirical assessments, the
third section surveys the recent empirical macrofinance literature, which links the behavior of
long-term interest rates to the economy, with
varying degrees of economic structure (e.g., Ang
and Piazzesi, 2003, and Rudebusch and Wu, 2003,
denoted RW). However, although this new literature has made interesting advances in understanding how macroeconomic conditions affect the term
premium, it has made surprisingly little progress
toward understanding the reverse relationship.
Indeed, restrictions are typically imposed in these
models that either eliminate any effects of the
term premium on the economy or require the
term premium to affect the economy in the same
way as other sources of long-rate movements.
Accordingly, as yet, this literature is not very
useful for investigating whether there are important macroeconomic implications of movements
in the term premium.
In contrast, as reviewed in the fourth section,
several papers have directly investigated the predictive power of movements in the term premium
on subsequent gross domestic product (GDP)
growth (e.g., Favero, Kaminska, and Söderström,
2005, and Hamilton and Kim, 2002), but because
these analyses rely on simple reduced-form regressions, their structural interpretation is unclear.
Nevertheless, taken at face value, the bulk of the
evidence suggests that decreases in the term premium are followed by slower output growth—
clearly contradicting the practitioner view (as well
as the simple New Keynesian view). However, we
reconsider such regressions and provide some
new empirical evidence that supports the view
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

taken by many central bankers and market analysts
that a decline in the term premium typically has
been associated with stimulus to the economy.
The final section concludes by describing
some practical lessons for monetary policymakers
when confronted with a sizable movement in the
term premium.

A STRUCTURAL MODEL OF THE
TERM PREMIUM AND THE
ECONOMY
In this section, we use a standard structural
macroeconomic DSGE framework to study the
relationship between the term premium and the
economy. In principle, such a framework can
completely characterize this relationship; however, in practice the DSGE asset-pricing framework
has a number of well-known computational and
practical limitations that keep it from being a
useful empirical workhorse. Nevertheless, the
framework can provide interesting qualitative
insights, as we will now show.

An Asset-Pricing Representation of
the Term Premium
As in essentially all asset pricing, the fundamental equation that we assume prices assets in
the economy is the stochastic discounting
relationship:
(3)

pt = Et [mt +1 pt +1 ],

where pt denotes the price of a given asset at
time t and mt +1 denotes the stochastic discount
factor that is used to value the possible statecontingent payoffs of the asset in period t+1
(where pt +1 implicitly includes any dividend or
coupon payouts).4 Specifically, the price of a
default-free n-period zero-coupon bond that
pays one dollar at maturity, pt(n), satisfies
4

Cochrane (2001) provides a comprehensive treatment of this
asset-pricing framework. As Cochrane discusses, a stochastic discount factor that prices all assets in the economy can be shown to
exist under very weak assumptions; for example, the assumptions
of free portfolio formation and the law of one price are sufficient,
although these do require that investors are small with respect to
the market.

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Rudebusch, Sack, Swanson

pt(n) = E t [mt +1 pt(n+−11) ],

(4)

where pt(0) = 1 (the price of one dollar delivered
at time t is one dollar).
We can use this framework to formalize the
decomposition of bond yields described in the
introduction, with the term premium defined as
the difference between the yield on an n-period
bond and the expected average short-term yield
over the same n periods.5 Let i t(n) denote the continuously compounded n-period bond yield
(with it  i t(1) ); then the term premium can be
computed from the stochastic discount factor in
a straightforward manner:

it(n) −

1 n −1
1
1 n −1
Et ∑ it + j = − log pt(n) + Et ∑ log pt(1+)j
n j =0
n
n j =0

(5)
n
n
 1
1
= − log Et  ∏ mt + j  + Et ∑ log E t + j −1mt + j .
n
 j =1
 n j =1

Of course, equation (5) does not have an easy
interpretation without imposing additional structure on the stochastic discount factor, such as
conditional log-normality. Nonetheless, even in
this general form, equation (5) highlights an important point: The term premium is not exogenous,
as a change in the term premium can only be
due to changes in the stochastic discount factor.
Thus, to investigate the relationship between the
term premium and the economy in a structural
model, we must first specify why the stochastic
discount factor in the model is changing.
In general, the stochastic discount factor will
respond to all of the various shocks affecting the
economy, including innovations to monetary
policy, technology, and government purchases.
Of course, these different types of shocks also
have implications for the determination of output
and other economic variables. Thus, we would
expect the correlation between the term premium
and output to depend on which structural shock
5

This definition of the term premium (given by the left-hand side
of equation (5)) differs from the one used in the theoretical finance
literature by a convexity term, which arises because the expected
log price of a long-term bond is not equal to the log of the expected
price. Our analysis is not sensitive to this adjustment; indeed,
some of our empirical term-premium measures are convexityadjusted and some are not, and they are all highly correlated over
our sample.

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2007

was driving the change in the term premium. We
next elaborate on this point using a simple structural model.

A Benchmark DSGE Structural Model
The expression for the term premium
described by equation (5) is quite general but
not completely transparent, because it does not
impose any structure on the stochastic discount
factor. Thus, to illuminate the structural relationship between the term premium and the macroeconomy, we introduce a simple benchmark New
Keynesian DSGE model.
The basic features of the model are as follows.
Households are representative and have preferences over consumption and labor streams given
by
(6)

`
 (c − bht )1−γ
l 1+ χ 
− χ0 t  ,
max Et ∑ β t  t
1−γ
1+ χ

t =0

where β denotes the household’s discount factor, ct
denotes consumption in period t, lt denotes labor,
ht denotes a predetermined stock of consumption
habits, and γ, χ, χ0, and b are parameters. We set
ht = C t –1, the level of aggregate consumption in
the previous period, so that the habit stock is
external to the household. There is no investment in physical capital in the model, but there
is a one-period nominal risk-free bond and a
long-term default-free nominal consol that pays
one dollar every period in perpetuity (under our
baseline parameterization, the duration of the
consol is about 25 years). The economy also contains a continuum of monopolistically competitive firms with fixed, firm-specific capital stocks
that set prices according to Calvo contracts and
hire labor competitively from households. The
firms’ output is subject to an aggregate technology shock. Furthermore, we assume there is a
government that levies stochastic, lump-sum
taxes on households and destroys the resources
it collects. Finally, there is a monetary authority
that sets the one-period nominal interest rate
according to a Taylor-type policy rule:
(7)
it = ρi it −1 + (1 − ρi )  i ∗ + g y (y t − y t −1 ) + gπ π t  + ε ti ,

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Rudebusch, Sack, Swanson

where i* denotes the steady-state nominal interest
rate, yt denotes output, πt denotes the inflation rate
(equal to Pt /Pt –1 –1), ε ti denotes a stochastic monetary policy shock, and ρi , gy , and gπ are parameters.6 This basic structure is very common in the
macroeconomics literature, so details of the specification are presented in the appendix.
In equilibrium, the representative household’s
optimal consumption choice satisfies the Euler
equation:
(8) (ct − bct −1 )−γ = β exp(it )Et (ct +1 − bct )− γ Pt / Pt +1 ,
where Pt denotes the dollar price of one unit of
consumption in period t. The stochastic discount
factor is given by
(9)

mt +1 =

β(ct +1 − bct )− γ Pt
.
(ct − bct −1 )− γ Pt +1

The nominal consol’s price, pt(⬁), thus satisfies
(10)

pt(` ) = 1 + E t mt +1 pt(`+1) .

We define the risk-neutral consol price, pt(⬁)rn, to be
(11)

pt(` )rn = 1 + exp( − it )E t pt(`+1)rn .

The implied term premium is then given by7
(12)

 p(` ) 
 pt(` )rn 
log  (` t)
log
−
 p(` )rn − 1  .
 pt − 1 
t

Having specified the benchmark model, we
can now solve the model and compute the
responses of the term premium and the other variables of the model to economic shocks. Parameters

of the model are given in the appendix. We solve
the model by the standard procedure of approximation around the nonstochastic steady state, but
because the term premium is zero in a first-order
approximation and constant in a second-order
approximation, we compute a third-order approximation to the solution of the model using the
nth-order approximation package described in
Swanson, Anderson, and Levin (2006), called
perturbation AIM.
In Figures 1, 2, and 3, we present the impulse
response functions of the term premium and
output to a 1-percentage-point monetary policy
shock, a 1 percent aggregate technology shock,
and a 1 percent government purchases shock,
respectively. These impulse responses demonstrate that the relationship between the term
premium and output depends on the type of
structural shock. For monetary policy and technology shocks, a rise in the term premium is
associated with current and future weakness in
output. By contrast, for a shock to government
purchases, a rise in the term premium is associated
with current and future output strength. Thus,
even the sign of the correlation between the term
premium and output depends on the nature of
the underlying shock that is hitting the economy.
A second observation to draw from Figures 1,
2, and 3 is that, in each case, the response of the
term premium is quite small, amounting to less
than one-third of 1 basis point, even at the peak
of the response! Indeed, the average level of the
term premium for the consol in this model is only
15.7 basis points.8 This finding foreshadows
8

6

Note that the interest rate rule we use here is a function of output
growth rather than the output gap. We chose to use output growth
in the rule because definitions of potential output (and hence the
output gap) can sometimes be controversial. In any case, our results
are not very sensitive to the inclusion of output growth in the policy
rule. For example, if we set the coefficient on output growth to
zero, all of our results are essentially unchanged. We also follow
much of the literature in assuming an “inertial” policy rule with
gradual adjustment and i.i.d. policy shocks. However, Rudebusch
(2002 and 2006) argues for an alternative specification with serially correlated policy shocks and little such gradualism.

7

The continuously compounded yield to maturity of the consol is
given by log[p/(p –1)]. To express the term premium in annualized
basis points rather than in logs, equation (12) must be multiplied
by 40,000. We obtained qualitatively similar results using alternative term-premium measures in the model, such as the term premium on a two-period zero-coupon bond.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

From the point of view of a second- or third-order approximation,
this result is not surprising, because only under extreme curvature
or large stochastic variances do second- or third-order terms matter
much in a macroeconomic model. Some research has arguably
employed such model modifications to account for the term premium. For example, Hördahl, Tristani, and Vestin (2006b) assume
that the technology shock has a quarterly standard deviation of
2.5 percent and a persistence of 0.986. Adopting these two parameter values in our model causes the term premium to rise to 141
basis points. Ravenna and Seppälä (2006) assume a shock to the
marginal utility of consumption, with a persistence of 0.95 and a
quarterly standard deviation of 8 percent. A similar shock in our
model boosts the term premium to 41 basis points. Wachter (2006)
assumes a habit parameter (b) of 0.961, which in our model boosts
the term premium to 22.3 basis points. Thus, we are largely able to
replicate some of these authors’ findings; nonetheless, we believe
that our benchmark parameter values are the most standard ones
in the macroeconomics literature (e.g., Christiano, Eichenbaum,
and Evans, 2005).

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Rudebusch, Sack, Swanson

Figure 1
Impulse Responses to a 1-Percentage-Point Federal Funds Rate Shock
Term Premium

Basis Points
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
2

0

4

6

8

10

12

14

16

18

20

12

14

16

18

20

Quarters

Output

Percent
0.00

–0.05

–0.10

–0.15

–0.20
0

2

4

6

8

10
Quarters

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Figure 2
Impulse Responses to a 1 Percent Technology Shock
Term Premium

Basis Points
0.00
–0.05
–0.10
–0.15
–0.20
–0.25
–0.30
–0.35
0

2

4

6

8

10

12

14

16

18

20

12

14

16

18

20

Quarters

Output

Percent
0.20

0.15

0.10

0.05

0.00
0

2

4

6

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8

10
Quarters

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Rudebusch, Sack, Swanson

Figure 3
Impulse Responses to a 1 Percent Government Purchases Shock
Term Premium

Basis Points
0.25

0.20

0.15

0.10

0.05

0.00
2

0

4

6

8

10

12

14

16

18

20

12

14

16

18

20

Quarters

Output

Percent
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0

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4

6

8

10
Quarters

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Rudebusch, Sack, Swanson

one of the primary limitations of the structural
approach to modeling term premiums, which we
will discuss in more detail below.
Finally, we note that, although this structural
model is very simple, in principle there is no reason why the same analysis cannot be performed
using larger and more realistic DSGE models,
such as Smets and Wouters (2003), Christiano,
Eichenbaum, and Evans (2005), or the extensions
of these in use at a number of central banks and
international policy institutions.9 Even with these
larger models, we can describe the term-premium
response to any given structural shock and the
broader implications of the shock for the economy
and optimal monetary policy.

Limitations of the DSGE Model of the
Term Premium
Using a structural DSGE model to investigate
the relationship between the term premium and
the economy has advantages in terms of conceptual clarity, but there are also a number of limitations that prevent the structural-modeling
approach from being useful at present as an
empirical workhorse for studying the term premium. This remains true despite the increasing
use of structural macroeconomic models at policymaking institutions for the study of other macroeconomic variables, such as output and inflation.
These limitations generally fall into two categories:
theoretical uncertainties and computational
intractabilities.
Regarding the former, even though some
DSGE models—sometimes crucially augmented
with highly persistent structural shocks—appear
to match the empirical impulse responses of
macroeconomic variables, such as output and
inflation, researchers do not agree on how to
specify these models to match asset prices. For
example, a variety of proposals to explain the
equity-premium puzzle include habit formation
9

Some notable extensions include Altig et al. (2005) to the case of
firm-specific capital, Adolfson et al. (2007) to the case of a small
open economy, and Pesenti (2002) and Erceg, Guerrieri, and Gust
(2006) to a large-scale (several hundred equations) multicountryblock context for use at the International Monetary Fund and the
Federal Reserve Board, respectively.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

in consumption (Campbell and Cochrane, 1999),
time-inseparable preferences (Epstein and Zin,
1989), and heterogeneous agents (Constantinides
and Duffie, 1996, and Alvarez and Jermann, 2001).
This lack of consensus implies that there is much
uncertainty about the appropriate DSGE specification for analyzing the term premium.
The possibility that a heterogeneous-agent
model is necessary to understand risk premiums
poses perhaps the most daunting challenge for
structural modelers of the term premium. In the
case of heterogeneous agents with limited participation in financial markets, different households’
valuations of state-contingent claims are not equalized, so determining equilibrium asset prices can
become much more complicated than in the representative-household case. Although a stochastic
discount factor still exists under weak assumptions even in the heterogeneous-household case,
it need not conform to the typical utility functions that are in use in current structural macroeconomic models.10
The structural approach to asset pricing also
faces substantial computational challenges, particularly for the larger-scale models that are
becoming popular for the analysis of macroeconomic variables. Closed-form solutions do not
exist in general, and full numerical solutions are
computationally intractable except for the simplest
possible models.11 The standard approach of
log-linearization around a steady state that has
proved so useful in macroeconomics is unfortunately not applicable to asset pricing, because by
construction it eliminates all risk premiums in
the model. Some extensions of this procedure to
a hybrid log-linear log-normal approximation
(Wu, 2006, and Bekaert, Cho, and Moreno, 2005)
10

One might even question the assumptions required for a stochastic
discount factor to exist. For example, if there are large traders and
some financial markets are thin, then it is no longer the case that
all investors can purchase any amount of a security at a constant
price, contrary to the standard assumptions.

11

See Backus, Gregory, and Zin (1989), Donaldson, Johnsen, and
Mehra (1990), Den Haan (1995), and Chapman (1997) for examples
of numerical solutions for bond prices in very simple real business
cycle models. Gallmeyer, Hollifield, and Zin (2005) provide a
closed-form solution for bond prices in a simple New Keynesian
model, under the assumption of a very special reaction function
for monetary policy.

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Rudebusch, Sack, Swanson

and to a full second-order approximation around
the steady state (Hördahl, Tristani, and Vestin,
2006b) are only moderately more successful,
because they imply that all risk premiums in the
model are constant (in other words, these authors
all assume the weak form of the expectations
hypothesis). Obtaining a local approximation
that actually produces time-varying risk or term
premiums requires a full third-order approximation, as in our analysis above and in Ravenna and
Seppälä (2006). Even then, the implied time variation in the term premium is very small, due to
the inherently small size of third-order terms,
unless one is willing to assume very large values
for the curvature of agents’ utility functions, very
large stochastic shock variances, and/or very high
degrees of habit persistence (which goes back to
the theoretical limitations discussed above). Thus,
the challenges in computing the asset-pricing
implications of DSGE models, while becoming
less daunting over time, remain quite substantial.

MACRO-FINANCE MODELS OF
THE TERM PREMIUM
Because of the significant limitations in
applying the structural model discussed above,
researchers interested in modeling the term premium in a way that can be taken to the data have
had no choice but to pursue a less-structural
approach. Although one can model “yields with
yields” using a completely reduced-form, latentfactor, no-arbitrage asset-pricing model, as in
Duffie and Kan (1996) and Dai and Singleton
(2000), recent research has focused increasingly
on hybrid macro-finance models of the term
structure, in which some connections between
macroeconomic variables and risk premiums are
drawn, albeit not within the framework of a fully
structural DSGE model (see Diebold, Piazzesi, and
Rudebusch, 2005). The approaches employed in
this macro-finance literature have generally fallen
into two categories: vector autoregression (VAR)
macro-finance models and New Keynesian macrofinance models. We consider each in turn.
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VAR-Based Macro-Finance Models
The first paper in the no-arbitrage macrofinance literature was Ang and Piazzesi (2003).12
They assume that the economy follows a VAR:
(13)

X t = µ + ΦX t −1 + Σ εt ,

where the state vector, Xt, contains output, inflation, the one-period nominal interest rate, and two
latent factors (discussed below). The stochastic
shock, εt , is i.i.d. over time. In this model, the oneperiod nominal interest rate, it , is determined by
a Taylor-type monetary policy rule based on Xt ,
so that the model-implied expected path of the
short-term interest rate is known at any point in
time.
The VAR, however, does not contain any
information about the stochastic discount factor.
Ang and Piazzesi simply assume that the stochastic discount factor falls into the essentially
affine class, as in standard latent-factor finance
models, so it has the functional form
(14)

1


mt +1 = exp  − it − λt′ λt − λt′ε t +1  ,


2

where εt is assumed to be conditionally lognormally distributed and the prices of risk, λt ,
are assumed to be affine in the state vector, Xt :
(15)

λt = λ 0 + λ1X t .

Estimation of this model is complicated by
the inclusion of two unobserved, latent factors
in the state vector, Xt , which are typical of noarbitrage models in the finance literature. To make
estimation tractable, Ang and Piazzesi impose
the restriction that the unobserved factors do not
interact at all with the observed macroeconomic
variables (output and inflation) in the VAR.
Because of this very strong restriction, the macro12

A number of papers before Ang and Piazzesi (2003) investigated
the dynamic interactions between yields and macroeconomic
variables in the context of unrestricted VARs, including Evans
and Marshall (2001) and Kozicki and Tinsley (2001). Diebold,
Rudebusch, and Aruoba (2006) and Kozicki and Tinsley (2005)
provide follow-up analysis. As with the no-arbitrage papers discussed below, however, none of these papers has explored
whether the term premium implied by their models feeds back to
the macroeconomy, the question of interest in the present paper.

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Rudebusch, Sack, Swanson

economic variables in the model are determined
by a VAR that essentially excludes all interest
rates (both short-term and long-term rates). Thus,
while the Ang-Piazzesi model can effectively
capture the extent to which changes in macroeconomic conditions affect the term premium, it
cannot capture any aspects of that relationship
running in the reverse direction.13 In this regard,
their model falls short of addressing the topic of
interest in the present paper.14
Bernanke, Reinhart, and Sack (2004, denoted
BRS), employ a similar model but assume that the
state vector, Xt , consists entirely of observable
macroeconomic variables, which determine both
short-rate expectations (through the VAR) and
the prices of risk (15). By eliminating the use of
latent variables, the empirical implementation of
the model is simplified tremendously. Of course,
as in Ang and Piazzesi, the BRS framework will
capture effects of movements in the term premium
driven by observable factors included in the VAR,
but it does not empirically separate the role of
the term premium from that of lagged macroeconomic variables. Note that the BRS specification,
as in the Ang and Piazzesi model, does not include
longer-term interest rates in the VAR (but in this
case does include the short-term interest rate),
implying that movements in the term premium
not captured by the included variables are
assumed to have no effect on the dynamics of
the economy.
13

14

Even when movements in the term premium are driven by the
observed macroeconomic variables (output and inflation) rather
than the latent factors, the Ang-Piazzesi model fails to identify
effects of the term premium on the macroeconomy. For example,
suppose higher inflation is estimated to raise the term premium
and lead to slower growth in the future. We cannot ascribe the
slower growth to the term premium, because the higher inflation
may also predict tighter monetary policy or other factors that would
be expected to slow the economy. Note that the VAR does at least
partially address the issue that not all movements in the term premium are created equal, because the predictive power of a change
in the term premium will depend on the specific combination of
economic factors driving it.
Cochrane and Piazzesi (2006) also focus on the interaction between
macroeconomic conditions and the term premium. They use the
predictable component of the ex post returns from holding longerterm securities as a measure of the term premium. Their findings
support the case that the term premium varies importantly over
time, and they link those movements to macroeconomic conditions.
However, they do not address whether the term premium itself
affects economic activity.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Ang, Piazzesi, and Wei (2006, denoted APW),
also estimate a no-arbitrage macro-finance model
based on a VAR of observed state variables. However, in contrast to BRS, APW explicitly include
the five-year Treasury yield as an element of the
state vector. Thus, to a very limited extent, their
model begins to address the types of effects that
are the focus of the present paper. However,
their VAR does not distinguish between the riskneutral and term-premium components of the
five-year yield, so it is only able to capture distinct effects from these two components if they
are correlated (in different ways) with the other
variables in the VAR (which are, specifically, the
short-term interest rate and GDP growth). Even
then, it would not be possible in their model to
disentangle the direct effects of the short-term
interest rate and GDP growth on future output
from the indirect effects that changes in those
variables have on the term premium; it is in this
respect that the APW model cannot help answer
the question we are interested in, even though it
allows a separate role for longer-term yields in
the VAR.15
Finally, Dewachter and Lyrio (2006a,b) consider a model that is very similar in spirit to APW
and BRS, only they work in continuous time and
allow for a time-varying long-run inflation objective of the central bank, as argued for by Kozicki
and Tinsley (2001) and Gürkaynak, Sack, and
Swanson (2005). However, just as with the other
papers discussed above, Dewachter and Lyrio do
not allow changes in the term premium to feed
back to the macroeconomic variables of the model.

New Keynesian Macro-Finance Models
A separate strand of the macro-finance literature has attempted to bridge the gulf between
DSGE models and VAR-based macro-finance
models by incorporating more economic structure
into the latter. Specifically, these papers replace
the reduced-form VAR in the macro-finance
models with a structural New Keynesian macro15

APW also present some related reduced-form results on the forecasting power of the term premium for future GDP growth, which
we discuss in more detail in the next section.

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economic model that governs the dynamics of
the macroeconomic variables.
An early and representative paper in this literature was written by Hördahl, Tristani, and
Vestin (2006a, denoted HTV). They begin with a
basic New Keynesian structural model in which
output, inflation, and the short-term nominal
interest rate are governed by the equations
(16)
y t = µ y E t y t +1 + (1 − µ y )y t −1 − ζ i (it − E t π t +1 ) + εty ,
(17)
π t = µπ Et π t +1 + (1 − µπ )π t −1 + δ y y t − ε tπ ,
(18)
it = ρi it −1 + (1 − ρi )  gπ (E t π t +1 − π t∗ ) + g y y t  + ε ti .

Equation (16) describes a New Keynesian curve
that allows for some degree of habit formation on
the part of households through the lagged output
term; equation (17) describes a New Keynesian
Phillips curve that allows for some rule-of-thumb
price setters through the lagged inflation term;
and equation (18) describes the monetary authority’s Taylor-type short-term interest rate reaction
function. Equations (16) and (17) are structural
in the sense that they can be derived from a loglinearization of household and firm optimality
conditions in a simple structural New Keynesian
DSGE model along the lines of our benchmark
model (although HTV modify this structure by
allowing the long-run inflation objective, π t*, to
vary over time).
In contrast to a DSGE asset-pricing model,
however, HTV model the term premium using
an ad hoc affine structure for the stochastic discount factor, as in the VAR-based models above.
Although this approach is not completely structural, it makes the model computationally tractable and provides a good fit to the data while
allowing the term premium to vary over time in
a manner determined by macroeconomic conditions that are determined structurally (to first
order). The true appeal of this type of model is
that it is parsimonious and simple while allowing
for expectations to influence macroeconomic
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dynamics and for the term premium to vary nontrivially to macroeconomic developments.
However, as was the case in the VAR-based
models, the HTV model does not allow the term
premium to feed back to macroeconomic variables.
As discussed in the introduction, the structure
of the IS curve in the HTV model assumes that
economic activity depends only on expectations
of the short-term real interest rate and not on the
term premium. Thus, this approach is also unable
to address the issue considered in the current
paper.
RW develop a New Keynesian macro-finance
model that comes a step closer to addressing the
topic of this paper by allowing for feedback from
the term structure to the macroeconomic variables
of the model. In particular, RW incorporate two
latent term-structure factors into the model and
give those latent factors macroeconomic interpretations, with a level factor that is tied to the
long-run inflation objective of the central bank
and a slope factor that is tied to the cyclical stance
of monetary policy. Thus, the latent factors in
the RW model can affect economic activity, and
the term structure does provide information about
the current values of those latent factors. However, RW make no effort to decompose the effects
of long-term interest rates on the economy into
an expectations component and a term-premium
component, so there is no sense in which the
term premium itself affects macroeconomic
variables.
Wu (2006) and Bekaert, Cho, and Moreno
(2005) come closer to a true structural New
Keynesian macro-finance model by deriving the
stochastic discount factor directly from the utility
function of the representative household in the
underlying structural model. Thus, like a DSGE
model, their papers impose the cross-equation
restrictions between the macroeconomy and the
stochastic pricing kernel that are ignored when
the kernel is specified in an ad hoc affine manner.
However, these analyses also suffer from the computational limitations of working within the
DSGE framework (discussed above), because both
papers are unable to solve the model as specified.
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Rudebusch, Sack, Swanson

Instead, those authors use a log-linear, log-normal
approximation, which implies that the term premium in the model is time-invariant.16 Thus,
their papers do not address the question we have
posed in this paper.17

REDUCED-FORM EVIDENCE
ON THE EFFECTS OF THE
TERM PREMIUM
Because of the limitations discussed above,
the models in the previous two sections do not
provide us with much insight into the empirical
economic implications of changes in the term
premium. The benchmark structural model is
largely unable to reproduce the magnitude and
variation of the term premium that is observed
in the bond market, and, although the macrofinance models are more successful at capturing
the observed behavior of term premiums, they
typically impose very restrictive assumptions that
eliminate any macroeconomic implications of
changes in term premiums. A separate literature
that has provided a direct examination of these
implications is based on reduced-form empirical
evidence. Specifically, in the large literature that
uses the slope of the yield curve to forecast subsequent GDP growth, several recent papers have
tried to estimate separately the predictive power
of the term premium. In this section, we summarize these papers and contribute some new evidence on this issue.
An important caveat worth repeating is that
there is only a reduced-form relationship—not a
structural one—between the term premium and
future output growth, so even the sign of their pair16

Indeed, the term premium would be zero except for the fact that Wu
(2006) and Bekaert, Cho, and Moreno (2005) allow some secondand higher-order terms to remain in these models. In particular,
they leave the log-normality of the stochastic pricing kernel in its
nonlinear form, which implies a nonzero, albeit constant, risk
premium. A drawback of this approach is that it treats some secondorder terms as important, while dropping other terms of similar
magnitude.

17

A related paper by Gallmeyer, Hollifield, and Zin (2005) provides
a full nonlinear solution to a very similar model. However, they are
able to solve the model only under the assumption of an extremely
special reaction function for monetary policy; thus, their method
has no generality and is invalid in cases in which that policy
reaction function is not precisely followed.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

wise correlation over a given sample will depend
on which types of shocks are most influential.
Nevertheless, it may be of interest to consider
the average correlation between future output
growth and changes in the term premium over
some recent history. If the mixture of shocks is
expected to remain relatively stable, then the
average estimated reduced-form relationship
between the term premium and future economic
growth could be useful for forecasting. For this
reason, the historical relationship may provide
useful information to a policymaker who has to
decide whether and how to respond to a given
change in the term premium.

Evidence in the Literature
Recent research relating the term premium to
subsequent GDP growth has been part of a much
larger literature on the predictive power of the
slope of the yield curve. A common approach in
this literature is to investigate whether the spread
between short-term and long-term interest rates
has significant predictive power for future GDP
growth by estimating a regression of the form
(19) y t + 4 − y t = β0 + β1(y t − y t − 4 ) + β2 (it(n) − it ) + ε t ,
where yt is the log of real GDP at time t and i t(n) is
the n-quarter interest rate (usually a longer-term
rate such as the 10-year Treasury yield).18 The
standard finding is that the estimated coefficient
β2 is significant and positive, indicating that the
yield-curve slope helps predict growth.
Note that equation (19) is a reduced-form
specification that has no economic structure.
However, it can be motivated by thinking of the
long-term interest rate as a proxy for the neutral
level of the nominal funds rate, so that the yieldcurve slope captures the current stance of monetary policy relative to its long-run level. For
example, a steep yield-curve slope (with short
rates unusually low relative to long rates) would
indicate that policy is accommodative and would
18

This equation assumes that the dependent variable is future GDP
growth (a continuous variable). Other papers in this literature use
a dummy variable for recessions (a discrete variable). In either
case, the motivation for the approach is the same and the results
are qualitatively similar.

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be associated with faster subsequent growth, thus
accounting for the positive coefficient.
In this respect, the use of the long-term interest
rate in the regression (19) is motivated entirely
by the component related to the expected longrun level of the short rate. But the long-term rate
also includes a term premium; hence, any variation in this premium will affect the performance
of the equation. Indeed, it is useful to decompose
the yield-curve slope into these two components,
as follows:
 1 n −1
 

1 n −1
(20) it(n) − it =  ∑ Et it + j − it  +  it(n) − ∑ Et it + j  .
n j =0
 n j =0
 


The first term captures the expectations component, or the proximity of the short rate to its
expected long-run level. The second component
is the term premium, or the amount by which the
long rate exceeds the expected return from investing in a series of short-term instruments. For
notational simplicity, we will denote the first
component in (20) as exspt , the expected-rate
component of the yield spread, and the second
component as tpt , the term premium.
With this decomposition, the prediction
equation (19) can be generalized as follows:
(21)
y t + 4 − y t = β0 + β1 (y t − y t − 4 ) + β2exspt + β3 tpt + εt .
The standard equation (19) imposes the coefficient restriction β2 = β3. Loosening that restriction allows the term premium to have a different
implication for subsequent growth than the
expected-rate component.19 Several recent papers
have considered this issue, as we will briefly
summarize.
The first paper to examine the importance of
the above decomposition for forecasting was
19

Because this equation is intended to capture the effects on output
from changes in interest rates, it is not far removed from the literature on estimating IS curves. Most empirical implementations of
the IS curve, however, assume that output is related to short-term
interest rates rather than long-term interest rates. Or, as seen in
Fuhrer and Rudebusch (2004), these papers focus on the component
of long rates tied to short-rate expectations, following the New
Keynesian output equation very closely. As a result, even this literature is more closely tied to estimating the parameter β2 than the
parameter β3.

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Hamilton and Kim (2002), which forecasts future
GDP growth using a spread between the 10-year
and 3-month Treasury yields in equation (19). The
innovation of their paper is that it then separates
the yield spread into the expectations and termpremium components considered in equation (21).
The authors achieve this separation by considering the ex post realizations of short rates, using
instruments known ex ante to isolate the expectations component. They find that the coefficients
β2 and β3 are indeed statistically significantly
different from one another, although both coefficients are estimated to be positive. Note that a
positive value for β3 implies that a decline in the
term premium is associated with slower future
growth.
A second paper that decomposes the predictive power of the yield spread into its expectations
and term-premium components is Favero,
Kaminska, and Söderström (2005). These authors
differ from Hamilton and Kim (2002) by using a
real-time VAR to compute short-rate expectations
rather than a regression of ex post realizations of
short rates on ex ante instruments. As in Hamilton
and Kim (2002), they find a positive sign for the
coefficient β3, so that a lower term premium again
predicts slower GDP growth.
A third relevant paper is by Wright (2006),
who touches on this issue in the context of a
probit model for forecasting recessions. Wright
considers the predictive power of the yield slope,
and then he investigates whether the return forecasting factor from Cochrane and Piazzesi (2005)
also enters those regressions significantly. Since
this factor is correlated with the term premium,
he is implicitly controlling for the term premium,
as in equation (21). He finds that this factor is
insignificant for predicting recessions over horizons of two or four quarters but has a significant
negative coefficient for predicting recessions over
a six-quarter horizon; that is, a lower term premium raises the odds of a recession, consistent
with the findings of the other papers that it would
predict slower growth.
A final reference is APW. As noted above,
they use a VAR that includes long rates, GDP
growth, and a short rate, but they cannot separate
out the effects of the term premium from other
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Rudebusch, Sack, Swanson

variable macroeconomic VAR comprising
four lags each of the unemployment rate,
quarterly inflation in the consumer price
index, and the 3-month Treasury bill rate.
At each date the VAR can be used to forecast the short rate over a given horizon, and
the average expected future short rate can
be used as an estimate of the risk-neutral
long-term rate of that maturity.21 The difference between the observed long-term rate
and the risk-neutral long-term rate then
provides a simple estimate of the term
premium. This approach has been used
by Evans and Marshall (2001), Favero,
Kaminska, and Söderström (2005), Diebold,
Rudebusch, and Aruoba (2006), and
Cochrane and Piazzesi (2006).

movements in long-term interest rates. However,
the authors perform an additional exercise in
which they calculate the expected-rate and termpremium components of the long rate as implied
by the VAR and then estimate the forecasting
equation (21), allowing for different effects from
these two components. In contrast to the previously discussed papers, APW find that the term
premium has no predictive power for future GDP
growth; that is, the coefficient β3 is zero.
Overall, the handful of papers that have
directly tackled the predictive power of the term
premium have produced results that starkly contrast with the intuition that Chairman Bernanke
expressed in his March 2006 speech (see the introduction). The empirical studies to date suggest
that, if anything, the relationship has the opposite
sign from the practitioner view. According to these
results, policymakers had no basis for worrying
that the decline in the term premium might be
stimulating the economy and instead should have
worried that it was a precursor to lower GDP
growth.

2. Bernanke-Reinhart-Sack measure: A potential shortcoming of using a VAR to estimate
the term premium is that it does not impose
any consistency between the yield curve
at a given point in time and the VAR’s projected evolution of those yields. Such pricing consistency can be imposed by using a
no-arbitrage model of the term structure.
As discussed in the previous section, a noarbitrage structure can be laid on top of a
VAR to estimate the behavior of the term
premium, as in BRS. Here, we consider
the term-premium estimate from that paper,
as updated by Rudebusch, Swanson, and
Wu (2006).

Empirical Estimates of the Term
Premium
Estimation of equation (21) requires a measure
of the term premium, and there are a variety of
possibilities in the literature. We begin our empirical analysis by collecting a number of the prominent term-premium measures and examining
some of the similarities and differences among
them.
Specifically, we consider five measures of
the term premium on a zero-coupon nominal 10year Treasury security20:

3. Rudebusch-Wu measure: No-arbitrage
restrictions can also be imposed on top of
a New Keynesian macroeconomic model.
Here we take the term premium estimated
from one such model, Rudebusch and Wu
(2003 and 2007), discussed earlier. As with
the Bernanke-Reinhart-Sack measure, this
term-premium measure was extended to a

1. VAR measure: The first of these measures,
which we label the “VAR” measure, is based
on a straightforward projection of the short
rate from a simple but standard three20

Note that some of these term-premium measures are adjusted
for convexity (e.g., Kim-Wright, Bernanke-Reinhart-Sack, and
Rudebusch-Wu), and some are not (e.g., our VAR-based measure
and our extension of the Cochrane-Piazzesi measure). The adjustment for convexity has little or no impact on our results, however;
for example, the correlation between the VAR-based term-premium
measure and the Kim-Wright and Bernanke-Reinhart-Sack measures
are 0.94 and 0.96, respectively.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

21

Of course, there are several reasons for not taking these VAR projections too seriously as good measures of the actual interest rate
expectations of bond traders at the time. Rudebusch (1998) describes
three important limitations of such VAR representations: (i) the
use of a time-invariant, linear structure, (ii) the use of final revised
data and full-sample estimates, and (iii) the limited number of
information variables. We examined several rolling-sample estimated VARs as well and obtained similar results.

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Rudebusch, Sack, Swanson

longer sample by Rudebusch, Swanson,
and Wu (2006), and we use this extended
version below.
4. Kim-Wright measure: One can also estimate
the term premium using a standard noarbitrage dynamic latent-factor model from
finance (with no macroeconomic structure
underlying the factors). In these models,
risk-neutral yields and the term premium
are determined by latent factors that are
themselves linear functions of the observed
bond-yield data. We use the term-premium
measure from a three-factor model discussed by Kim and Wright (2005), which
we extend back to 1961.22
5. Cochrane-Piazzesi measure: Cochrane and
Piazzesi (2005) analyze excess returns for
a range of securities over a one-year holding period. Their primary finding is that a
single factor—a particular combination of
current forward rates—predicts a considerable portion of the excess returns from
a one-year holding period for Treasury
securities. For our purposes, however, we
are interested in the term premium on a
10-year security, or the (annualized) excess
return expected over the 10-year period.
Sack (2006a) provides a straightforward
approach for converting the CochranePiazzesi one-year holding-period results
into a measure of the term premium. Specifically, the expected one-period excess
returns implied by the Cochrane-Piazzesi
estimates, together with the one-year riskfree rate, imply an expected set of zerocoupon yields one year ahead (because the
only way to generate expected returns on
zero-coupon securities is through changes
22

We extend the Kim-Wright measure back to 1961 by regressing
the three Kim-Wright latent factors on the first three principal
components of the yield curve and using these coefficients to estimate the Kim-Wright factors in prior years. Because the term premium in the model is a linear function of observed yields, and
because the Kim-Wright model fits the yield-curve data very well,
this exercise should come very close to deriving the same factors
that would be implied if we extended their model back to 1961.
Over the period where our proxy and the actual Kim-Wright term
premium overlap, the correlation between the two measures is
0.998 and the average absolute difference between them is less
than 4 basis points.

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2007

in yield). Those expected future yields can
then be used to compute the expected
Cochrane-Piazzesi factor one year ahead
and, hence, the expected excess returns
over the one-year period beginning one
year ahead. By iterating forward, one can
compute the expected excess return for
each of the next 10 years, thereby yielding
a measure of the term premium on the 10year security.
As is clear from the above descriptions, the
approaches used to derive the five term-premium
measures differ considerably in the variables
included and the theoretical restrictions incorporated. Nevertheless, the measures show many
similar movements over time, as can be seen in
Figure 4, which plots the five measures of the
term premium for the 10-year zero-coupon
Treasury yield back to 1984.
Three of the measures, in particular—the VAR,
Bernanke-Reinhart-Sack, and Kim-Wright—are
remarkably highly correlated over this period.23
As shown in Table 1, the correlation coefficients
among these measures range from 0.94 to 0.98.
The other two measures—Rudebusch-Wu and
Cochrane-Piazzesi—are less correlated with the
others. For example, the correlation coefficients
with the VAR measure are 0.68 for RudebuschWu and 0.88 for Cochrane-Piazzesi. These lower
correlations largely reflect that the RudebuschWu measure is more stable than the others and
that the Cochrane-Piazzesi measure is more
volatile.
The greater stability of the Rudebusch-Wu
measure can be easily understood. Their underlying model attributes much of the variation in
the 10-year Treasury yield to changes in the
expected future path of short rates, reflecting, in
their framework, variation in the perceived inflation target of the central bank. That assumption
is supported by other research. For example,
Gürkaynak, Sack, and Swanson (2005) found
significant systematic variation in far-ahead for23

These correlations are very high in comparison with, say, the zero
correlations exhibited by various authors’ measures of monetary
policy shocks, as noted in Rudebusch (1998).

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Rudebusch, Sack, Swanson

Figure 4
Five Measures of the 10-Year Term Premium
Percent
8
Bernanke-Reinhart-Sack
Cochrane-Piazzesi
Kim-Wright
Rudebusch-Wu
VAR

7
6
5
4
3
2
1
0
–1
–2
–3
1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

Table 1
Correlations Among the Five Measures of the Term Premium
BRS

RW

KW

CP

BRS

1.00

RW

0.76

1.00

KW

0.98

0.81

1.00

CP

0.92

0.87

0.96

1.00

VAR

0.96

0.68

0.94

0.88

ward nominal interest rates in response to macroeconomic news in a way that suggested changes
in inflation expectations rather than changes in
term premiums. Similarly, Kozicki and Tinsley
(2001) found that statistical models that allow
for a “moving endpoint” are able to fit interest
rate and inflation time series much better than
standard stationary or difference-stationary VARs.
By attributing more of the movement in long rates
to short-rate expectations, the Rudebusch-Wu
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

VAR

1.00

analysis does not need as much variation in the
term premium to explain the observed variation
in yields.24
The behavior of the measure based on
Cochrane and Piazzesi (2005) is harder to understand. This measure is well below the other meas24

One could argue that a weakness of the other term-premium estimates is that they are based on models that assume that the longrun features of the economy, such as the steady-state real interest
rate and rate of inflation, are completely anchored.

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Rudebusch, Sack, Swanson

Figure 5
Kim-Wright Decomposition of the 10-Year Zero-Coupon Yield
Percent
14

10-Year Zero-Coupon Yield

12

10
Risk-Neutral
10-Year Zero-Coupon Yield
8

6

4
10-Year Term Premium
2

0
1961 1964 1967 1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006

NOTE: The shaded bars indicate recessions as dated by the National Bureau of Economic Research.

ures and is much more volatile. To a large extent,
this behavior simply mimics the one-period
expected excess returns computed by Cochrane
and Piazzesi. Indeed, Sack (2006b) and Wright
(2006) have pointed out that the implied oneperiod expected excess returns are surprisingly
volatile and are currently very negative. This
behavior partly shows through to the implied
term-premium measure.
Overall, Figure 4 provides us with a menu of
choices for the analysis that follows.25 Even with
the differences noted above, the five measures
show considerable similarities in their variation
over this sample. Indeed, the first principal component captures 95 percent of the variation in
these five term-premium estimates. In the analysis
in the next section, we focus our attention on the

Kim-Wright measure. This measure appears to be
representative of the other measures considered.
In fact, it is very highly correlated (0.99) with the
first principal component of all five measures.
Moreover, it has the advantage that it can be
extended back to the early 1960s, allowing us to
conduct our analysis over a longer sample.
The 10-year zero-coupon yield is shown in
Figure 5 along with the two components based on
the Kim-Wright term-premium estimate.26 As can
be seen, both short-rate expectations and the term
premium contributed to the run-up in yields
through the early 1980s and, since then, to the
decline in yields. As noted by Kim and Wright
(2005), the term premium recently has fallen to
very low levels, a pattern consistent with the
26

25

In contrast to the measures shown in Figure 4, Ludvigson and Ng
(2006) provide one that has considerable high-frequency variation
and little persistence or predictive power for economic activity.
However, we have some reservations about their identification of
the term premium and exclude it from our analysis.

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2007

The yield data considered here are from the Gürkaynak, Sack,
and Wright (2006) database. Those authors do not recommend
using the 10-year Treasury yield before 1971, as there are very few
maturities at that horizon for estimating the yield curve. However,
their 10-year yield is highly correlated with the Treasury constantmaturity 10-year yield over that period, which justified its use. All
results that follow are robust to beginning the sample in 1971.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Rudebusch, Sack, Swanson

Figure 6
Kim-Wright Term Premium and the CBO Output Gap
Percent
8
6
10-Year Term Premium

4
2
0
–2
–4
–6

Output Gap

–8
–10
1960 1963 1966 1969 1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005

NOTE: The shaded bars indicate recessions as dated by the National Bureau of Economic Research.

conundrum discussed by former Chairman Alan
Greenspan.
Figure 6 plots this term-premium measure
along with the Congressional Budget Office (CBO)
output gap and provides the first hint of a negative
relationship between the two. It is this relationship that we now explore in more detail.

New Evidence on the Implications of
the Term Premium
We begin by estimating the standard relationship between the slope of the yield curve and
subsequent GDP growth, using the specification
in equation (19). The long rate is a 10-year zerocoupon Treasury yield, taken from the Gürkaynak,
Sack, and Wright (2006) database. The short rate
is the 3-month Treasury bill rate from the Federal
Reserve’s H.15 data release. All data are quarterly
averages, and the sample ranges from 1961:Q3 to
2005:Q4. We examine both this full sample and
a shorter subsample beginning in 1984, which
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

arguably has a more consistent monetary policy
regime (e.g., Rudebusch and Wu, 2007).
Results are presented in the first column of
Table 2. Over the full sample, we find that the
coefficient for the yield-curve slope is highly statistically significant and has a positive sign. This
estimate implies that a flatter yield curve predicts
slower GDP growth, the standard finding in the
academic literature. Over the shorter sample, the
estimated coefficient loses its significance, reflecting another fact that is well-appreciated among
researchers—that the predictive power of the
yield-curve slope for growth appears to have
diminished in recent decades.
As discussed above, this approach is purely
a reduced-form exercise that is not explicitly tied
to a theoretical structure. However, a common
motivation for using the yield-curve slope as a
predictor is that it serves as a proxy for the stance
of monetary policy relative to its neutral level.
Given this motivation, one would prefer to measure the yield-curve slope based strictly on the
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Rudebusch, Sack, Swanson

Table 2
Prediction Equations for GDP Growth
Dependent Variable: yt+4 – yt
(1)

(2)

(3)

yt – yt–4

0.15 (1.57)

0.12 (1.18)

0.32 (3.04)

it(n)

0.64 (3.64)
0.68 (4.03)

1.03 (5.64)

(4)

1962-2005 Sample
– it

exspt
exspt–4

0.38 (4.22)

–0.79 (–3.49)

tpt

0.30 (0.92)

tpt–4

–0.61 (–1.34)
0.54 (1.24)

exspt – exspt–4

0.96 (5.62)

tpt – tpt–4

–0.77 (–1.95)

1985-2005 Sample
yt – yt–4

0.26 (2.54)

it(n) – it

0.28 (1.29)

exspt

0.32 (2.31)
0.35 (1.59)

exspt–4

0.36 (2.30)

0.36 (2.68)

0.46 (1.92)
–0.07 (–0.32)

tpt

0.07 (0.25)

tpt–4

–0.46 (–1.15)
0.61 (2.18)

exspt – exspt–4

0.30 (1.37)

tpt – tpt–4

–0.59 (–1.93)

NOTE: Coefficient estimates are shown with their t-statistics in parentheses (t-statistics have been corrected for residual heteroskedasticity and autocorrelation). Each regression includes a constant that is not reported.

portion of the long-term interest rate associated
with expectations of the short-term rate. In that
context, we can also ask how the other component
of the long rate—the term premium—affects
growth. This consideration leads to specification
(21) above, in which the two components of the
yield-curve slope are allowed to have different
predictive effects for subsequent GDP growth.
We can implement this approach using the
term-premium measure described above.27 The
results are shown in column 2. For both samples,
the expectations-based component of the yield
slope has slightly stronger predictive power than
the pure yield-curve slope (that is, the coefficient
27

In our analysis, we ignore any potential issues associated with
generated regressors.

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on this component is slightly larger and more
significant than the coefficient on the overall
slope reported in column 1), and the coefficient
on the term premium, β3, is not significantly
different from zero. However, we are unable to
reject the hypothesis that β2 = β3 at even the 10
percent level over either the post-1962 or post1985 sample.
Our findings are similar in spirit to the existing empirical evidence that the term premium
has a different effect on subsequent growth than
the expectations-related component of the yield
curve. Note that the only purpose of having a termpremium measure, according to these results, is
to determine the expectations component of the
yield slope more accurately. The term premium
itself has no predictive power for future growth.
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Rudebusch, Sack, Swanson

However, the specification of these regression equations seems somewhat at odds with the
models we presented earlier. For example, the
New Keynesian IS curve (2) could be used to
motivate the use of the yield-curve slope, as it
assumes that output is determined by the deviation of the real short-term interest rate from its
equilibrium level. The expectations component
of the yield-curve slope might capture this variable, but it should then be related to the level of
the output gap. In contrast, the reduced-form
specifications (19) and (21) relate the slope of
the yield curve to the growth rate of output. Thus,
this specification seems to differ from the more
structural models by a derivative. Moreover, the
term premium in Figures 5 and 6 appears to be
nonstationary or nearly nonstationary, while GDP
growth is much closer to being stationary. Thus,
from a statistical point of view, specifications
(19) and (21) are also highly suspect.
If we difference equation (2) to arrive at a
specification in growth rates, it would suggest
that it is changes in the stance of monetary policy
that predict future GDP growth.28 This suggests
investigating whether GDP growth is tied to
changes in the stance of policy and changes in
the term premium, as opposed to the levels of
those variables.
As an exploratory step in this direction, we
re-estimate equation (21) with an additional oneyear lag of the right-hand-side variables included
in the regression. The results, shown in column
3 of Table 2, strongly hint that there is greater
predictive power associated with the changes in
these variables than with their levels. Indeed, one
can reject the hypothesis that the coefficients on
the lagged variables are zero (at the 1 percent significance level). Moreover, one cannot reject that
the right-hand-side variables enter the regression
only as changes. That is, the hypothesis that the
coefficients on the lag of these components equal
the negative of the coefficients on their current
levels cannot be rejected even at the 10 percent
28

Some might argue that the dependent variable here should be the
growth of the output gap rather than GDP. As discussed below, we
obtained similar results using the change in the CBO output gap
as the dependent variable.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

significance level. A similar (though less striking)
pattern is found in the shorter sample.
Because both the theory and the hypothesis
tests in the preceding paragraph suggest that only
differences should matter, column (4) of the table
presents results from estimating the baseline forecasting regression equation in differences, namely,
(22)
y t + 4 − y t = β0 + β1 (y t − y t − 4 ) + β2 (exspt − exspt − 4 )
+ β3 (tpt − tpt −4 ) + ε t .
The full-sample results indicate that both components of the yield-curve slope matter for future
growth. The coefficient on the risk-neutral expectations component of the yield-curve slope is now
larger and more statistically significant than in
any of the earlier specifications. We can also overwhelmingly reject the hypothesis that β2 = β3
(with p-values less than 10–4). This finding indicates that GDP growth is expected to be higher
not when the short-term interest rate is merely
low relative to its long-run level, but when it has
fallen relative to that level.
More importantly for this paper, we find that
the estimated coefficient on the term premium is
now negative and (marginally) statistically significant. According to these results, a decline in
the term premium tends to be followed by faster
GDP growth—the opposite sign of the relationship
uncovered by previous empirical studies. (In the
shorter sample, all of the coefficients are again less
significant. However, we still reject the hypothesis that β2 = β3 [with a p-value of 0.0395] in column 4, and the coefficient on the change in the
term premium is again negative and borderline
statistically significant.29)
Our findings line up with the intuition
expressed by Chairman Bernanke when he suggested that the declining term premium signaled
additional stimulus to the economy. Our results
29

Furthermore, using the year-on-year change in the CBO output
gap as the predicted variable rather than the year-on-year change
in output itself gave similar results. Specifically, the coefficient
on the term premium remains negative, with a p-value just less
than 0.05. These results suggest that a decline in the term premium
predicts a higher future value of the output gap and that policymakers might want to take that prediction into account when formulating the optimal policy response.

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Rudebusch, Sack, Swanson

are the first piece of evidence (that we are aware
of) to support this hypothesis, and they stand in
sharp contrast to the previous empirical evidence
presented by Hamilton and Kim (2002), Favero,
Kaminska, and Söderström (2005), and Wright
(2006).

nomic models. Understanding and incorporating
this correlation within the framework of a model
would appear to be a useful addition to the
research agenda. In this regard, we only speculate
that our empirical findings may reflect a heterogeneous population in which a decline in the term
premium makes financial market conditions more
accommodative for certain classes of borrowers.

CONCLUSIONS
Our results can be usefully summarized from
the perspective of advising monetary policymakers. Specifically, policymakers may wonder
how they should respond when confronted with
a substantial change in the term premium, such
as the recent decline that appears to have taken
place during 2004 and 2005.
The first, and perhaps most important, conclusion from our analysis is that policymakers
should always try to determine the source of the
change in the term premium. If that source can
be identified, then policymakers are advised to
consider the repercussions of that underlying
driving force more broadly rather than focusing
exclusively on the change in the term premium.
In this way, policymakers can take into account
the macroeconomic implications of the structural
shifts or disturbances that are affecting the term
premium.
Of course, policymakers often may be uncertain about the reasons for changes in the term
premium. Indeed, during the past few years, a
variety of only tentative explanations have been
offered for the seemingly low term premium. In
such a situation, policymakers may find our
reduced-form analysis of the implications of the
term premium for future economic activity to be
a useful baseline. Our results suggest that a decline
in the term premium has typically been associated
with higher future GDP growth, which appears
consistent with the practitioner view. Indeed,
according to our reduced-form analysis, the attention that Federal Reserve officials paid to the
seemingly large decline in the term premium in
2004 and 2005 may have been justified.
Finally, our finding that changes in the term
premium have a significant correlation with future
GDP growth is not captured by many macroeco262

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2007

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Favero, Carlo A.; Kaminska, Iryna and Söderström,
Ulf. “The Predictive Power of the Yield Spread:
Further Evidence and a Structural Interpretation.”
Unpublished manuscript, Università Bocconi, 2005.
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Gallmeyer, Michael F.; Hollifield, Burton and Zin,
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Gürkaynak, Refet S.; Sack, Brian and Swanson, Eric.
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Economic News: Evidence and Implications for
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Gürkaynak, Refet S.; Sack, Brian and Wright,
Jonathan. “The U.S. Treasury Yield Curve: 1961 to
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APPENDIX
Benchmark New Keynesian Model
To better understand the structural relationship between the term premium and the macroeconomy,
we define a simple New Keynesian DSGE model to use as a benchmark. This appendix provides a
detailed description of the model, the benchmark parameter values we used in computing the impulse
responses in Figures 1 to 3, and our solution algorithm.
The economy contains a continuum of households with a total mass of unity. Households are representative and seek to maximize utility over consumption and labor streams given by
`
 (c − bht )1−γ
l 1+ χ 
− χ0 t  ,
max Et ∑ β t  t
1−γ
1+ χ

t =0

(23)

where β denotes the household’s discount factor, ct denotes consumption in period t, lt denotes labor,
ht denotes a predetermined stock of consumption habits, and γ, χ, χ0, and b are parameters. We will set
ht = Ct –1, the level of aggregate consumption in the previous period, so that the habit stock is external
to the household.30 The household’s stochastic discount factor from period t to t + j thus satisfies

mt ,t + j ; β j

(ct + j − bCt + j −1 )−γ Pt
.
(ct − bCt −1 )− γ Pt + j

The economy also consists of a continuum of monopolistically competitive intermediate goods
firms indexed by f ∈[0,1]. Firms have Cobb-Douglas production functions:
y t (f ) = At k α lt (f )1−α ,

(24)

–
where k is a fixed, firm-specific capital stock (identical across firms) and At denotes an aggregate
technology shock that affects all firms. The level of aggregate technology follows an exogenous AR(1)
process:
log At = ρA log At −1 + εtA ,

(25)

where εtA denotes an i.i.d. aggregate technology shock with mean zero and variance σA2. Intermediate
goods are purchased by a perfectly competitive final goods sector that produces the final good with a
constant elasticity of substitution production technology:
1
Yt =  ∫ y t (f )1/ (1+θ )df 
 0


(26)

1+θ

.

Each intermediate goods firm f thus faces a downward-sloping demand curve for its product given by

 p (f ) 
y t (f ) =  t 
 Pt 

(27)

30

−(1+θ )/ θ

Yt ,

Campbell and Cochrane (1999) consider instead a habit stock, which is an infinite sum of past aggregate consumption with geometrically
decaying weights, and a slightly different specification of the utility kernel. They argue that this specification fits asset prices better than the
one-period habits used here. However, Lettau and Uhlig (2000) argue that the Campbell-Cochrane specification significantly worsens the
model’s ability to fit consumption and labor data.

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where
1
Pt ;  ∫ pt (f )−1/ θ df 
 0


(28)

−θ

is the constant elasticity of substitution aggregate price of a unit of the final good.
Each firm sets its price, pt (f ), according to a Calvo contract that expires with probability 1 – ξ each
period. There is no indexation, so the price, pt (f ), is fixed over the life of the contract. When a contract
expires, the firm is free to reset its price as it chooses. In each period t, firms must supply whatever
output is demanded at the posted price, pt (f ). Firms hire labor, lt (f ), from households in a competitive
labor market, paying the nominal market wage, wt . Marginal cost for firm f at time t is thus given by

mct (f ) =

(29)

w t lt (f )
.
(1 − α )y t (f )

Firms are collectively owned by households and distribute profits and losses back to the households.
When a firm’s price contract expires and it is able to set a new contract price, the firm maximizes the
expected present discounted value of profits over the lifetime of the contract:
`

(30)

Et ∑ ξ j mt ,t + j  pt (f )y t + j (f ) − w t + j lt + j (f ) ,
j =0

where mt,t+j is the representative household’s stochastic discount factor from period t to t + j. The firm’s
optimal contract price, pt*( f ), thus satisfies
`

(31)

pt∗ (f ) =

(1 + θ )Et ∑ j = 0 ξ j mt,t + j mct + j (f )y t + j (f )
`

Et ∑ j = 0 ξ j mt,t + j y t + j (f )

.

To aggregate up from firm-level variables to aggregate variables, it is useful to define the crosssectional price dispersion, ∆t :
(32)

`

∆1t / (1−α ) ; (1 − ξ ) ∑ ξ j pt∗− j (f )−(1+θ )/ (θ (1−α )),
j =0

where the exponent 1/(1 – α) arises from the firm-specificity of capital.31 We can then write
(33)

Yt = ∆ t−1 At K α L1t −α ,

– –
where K = k and
1

(34)

Lt ; ∫ lt (f )df
0

and equilibrium in the labor market requires Lt = lt , where lt is the labor supplied by households.
Optimizing behavior by households gives rise to the intratemporal condition,
(35)

31

wt
χ 0 ltχ
=
,
Pt (ct − bCt −1 )− γ

Allowing a competitive capital market with free mobility of capital across sectors or considering industry-specific labor markets as well as
firm-specific capital does not alter our basic findings presented in the benchmark model section.

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and the intertemporal Euler equation,

(ct − bCt −1 )− γ = β exp(it )E t (ct +1 − bCt )− γ Pt / Pt +1 ,

(36)

where it denotes the continuously compounded interest rate on the riskless one-period nominal bond.
There is no investment in physical capital in the model.
There is a government in the economy, which levies lump-sum taxes, Gt, on households and
destroys the resources it collects. The aggregate resource constraint implies that
(37)

Yt = Ct + Gt ,

where Ct = ct , with ct denoting the consumption of the representative household. Government consumption follows an exogenous AR(1) process:

log Gt = ρG log Gt −1 + εtG ,

(38)

where εtG denotes an i.i.d. government consumption shock with mean zero and variance σG2.
Finally, there is a monetary authority in the economy that sets the one-period nominal interest rate
according to a Taylor-type policy rule:
it = ρi it −1 + (1 − ρi )  i ∗ + g y (Yt − Yt −1 ) + gπ π t  + ε ti ,

(39)

where i* denotes the steady-state nominal interest rate, πt denotes the inflation rate (equal to Pt /Pt–1 – 1),
εti denotes an i.i.d. stochastic monetary policy shock with mean zero and variance σi2, and ρi , gy , and
gπ are parameters. Of course, the steady-state inflation rate, π*, in this economy must satisfy
1 + π* = β exp(i*).
As noted above, households have access to a long-term default-free nominal consol that pays one
dollar every period in perpetuity. The nominal consol’s price, pt(⬁), thus satisfies

pt(` ) = 1 + Et mt +1 pt(`+1) ,

(40)

where mt +1  mt,t +1 is the representative household’s stochastic discount factor. We define the riskneutral consol price, pt(⬁)rn, to be
pt(` )rn = 1 + exp( − it(1) )Et pt(`+1)rn,

(41)

and the implied term premium is then given by
 pt(` )rn 
 p(` ) 
−
log  (` t)
log
 p(` )rn − 1  .
 pt − 1
t

(42)

Note that under our baseline parameterization, the consol in our model has a duration of about 25 years.
This completes the specification of the benchmark model referred to in the text. In computing
impulse response functions, we use the parameter values as specified in Table A1. A technical issue
that arises in solving the model above is the relatively large number of state variables, eight in all: Ct –1,
At –1, Gt –1, it –1, ∆t –1, plus the three shocks εtA, εtG, εti.32 Because of dauntingly high dimensionality, valuefunction iteration-based methods, such as projection methods (or, even worse, discretization methods),
are computationally intractable. We instead solve the model above using a standard macroeconomic
32

The number of state variables can be reduced a bit by noting that Gt and At are sufficient to incorporate all of the information from Gt –1, At –1,
εtG, and εtA, but the basic point remains valid—namely, that the number of state variables in the model is large from a computational point of
view.

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technique that approximates the model’s soluTable A1
tion around the nonstochastic steady state—a socalled perturbation method.
Benchmark Model Parameter Values
As discussed in the text, a first-order approxα
0.3
ρA
0.9
imation (i.e., a linearization or log-linearization)
β
0.99
ρ
0.9
G
of the model around the steady state eliminates
θ
0.2
ρi
0.7
the term premium from the model entirely,
ξ
0.75
σA2
0.012
because equations (40) and (41) are identical in
γ
2
σG2
0.0042
the first-order approximation. A second-order
approximation produces a nonzero but constant
χ0
(1 – b)–γ
σi2
0.0042
–
2
2
term premium (a sum of the variances σA , σG ,
χ
1.5
K
1
and σi2). Because our interest in this paper is not
π*
0
b
0.66
just in the level of the term premium but also in
2
gπ
its variation over time, we must compute a thirdgy
0.5
order approximation to the solution of the model
around the nonstochastic steady state. We do so
using the nth-order perturbation AIM algorithm of
Swanson, Anderson, and Levin (2006). This algorithm requires that the equations of the model be put
into a recursive form, which for the model above is fairly standard. The most difficult equation is (31),
which can be written in recursive form as

(43)

 pt∗(f ) 
 P 

1+

α 1+θ
1−α θ

=

t

(44)

zn,t
zd ,t

zd ,t = Yt (Ct − bCt −1 )−γ + βξ E t π t1+/1θ zd ,t +1
1+θ 1

(45)

zn,t

χ
= (1 + θ ) 0 L1t + χ ∆ t−1/ (1−α ) + βξ E t π t +θ1 1−α zn,t +1.
1−α

The computational time required to solve our model to the third order is minimal—no more than about
10 seconds on a standard laptop computer.
Computing impulse responses for this model is actually simpler than the use of a third-order
approximation might suggest. We are interested in the responses of output and the term premium to
an exogenous shock to εtA, εtG, or εti. For output, we plot the standard first-order (i.e., log-linear) responses
of output to each shock. For small shocks, such as those of the size we are considering here (1 percent),
these responses are highly accurate. For the term premium, of course, the first- and second-order
responses of that variable to each shock would be identically zero, so we plot the third-order responses
of that variable. These third-order terms are all of the form σZ2X, where Z ∈{A,G,i } and X is one of the
state variables of the model,33 so if we plug in the values of σA2, σG2, and σi2 given in Table A1, these
terms are linear as well, which makes them easy to plot.

33

In perturbation analysis, stochastic shocks of the model are given an auxiliary “scaling” parameter, so these shocks are third-order in a rigorous
sense. See Swanson, Anderson, and Levin (2006) for details.

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Commentary
John H. Cochrane

T

he paper by Glenn Rudebusch, Brian
Sack, and Eric Swanson (2007) is an
impressive survey of several literatures
concerned with monetary economics
and interest rates. It is well done, so I think the
best thing for me to do is to highlight what I
think are the central points and to give my views
on those points.

WHAT “CONUNDRUM”?
Figure 1 presents the federal funds rate and
1- to 15-year forward rates through the past two
recessions. This comparison lets us easily consider
to what extent the recent behavior of long-term
forward rates represents an unusual experience
or not.
My first reaction to Figure 1 is that the patterns are strikingly similar. Short-term yields and
forwards decline, spreads widen, and then yields
and forwards recover as spreads tighten again. In
both episodes there is a little blip on the way down
in which long-term yields and forwards rise much
more than short-term ones, despite no movement
in the funds rate (late 1992 and 2002). In both
episodes there is an event on the way up in which
all yields and forwards increase sharply (late 1994
and 2004).
The main difference between the two episodes
is that the rise in the federal funds rate in 2004-06
is much smoother and more predictable and longterm forward rates, in particular the 10-year rate,

falls while the funds rate is rising. Though longterm forwards decline overall in both recoveries
(1994-96 and 2004-06), the earlier experience
includes a blip up in all rates through 1995, which
is later reversed. This experience is missing in
the second period. This remaining difference is
Greenspan’s “conundrum.”
The difference is already small. Furthermore,
because the rise in the funds rate was much steadier in the later episode, the behavior of market
rates relative to the funds rate (which reflects
different behavior by the Fed) is even less different
across the two episodes than the overall behavior
of interest rates. If one regards long-term rates as
dynamically driven by the federal funds rate, it’s
not obvious that there is any difference in the
behavior of markets.
Even if the later period is different, why is it
puzzling? First, long forwards should fall when
the Fed tightens. This is exactly how the world
is supposed to work. Tighter policy now means
lower inflation later, and thus lower nominal rates
in 10 years. There is no model or estimate anywhere in which the Fed can raise real rates for
10 years without reducing inflation. Prices are not
that sticky! In 1994, the opposite nearly one-forone rise of long forwards with rises in the federal
funds rate was viewed as a conundrum for just
this reason. The main, somewhat convoluted,
story used to explain the 1994 events is that an
interest rate rise communicates bad news about
inflation from the Fed to the markets—information

John H. Cochrane is a professor of finance at the Graduate School of Business at the University of Chicago and a research associate at the
National Bureau of Economic Research. The author acknowledges research support from the Center for Research and Security Prices and
from a National Science Foundation grant administered by the National Bureau of Economic Research. A draft of this paper with color
graphics is available at http://faculty.chicagogsb.edu/john.cochrane/research/Papers/.
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 271-82.
© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
only with prior written permission of the Federal Reserve Bank of St. Louis.

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Cochrane

Figure 1
Federal Funds Rate and Selected 1- to 15-Year Forward Rates Through Two Recessions
10

8

6
15
10
4

5
1
Federal Funds

2
1990

1991

1992

1993

1994

1995

1996

2002

2003

2004

2005

2006

8

6

4
15
10
2

5
1
Federal Funds

0
2000

2001

SOURCE: Forward-rate data are from Gürkaynak, Sack, and Wright (2006).

that for some reason the markets did not already
have, of course. Greenspan himself echoed this
view in 19941:
In early February, we thought long-term rates
would move a little higher as we tightened.
The sharp jump in [long] rates that occurred
appeared to reflect the dramatic rise in market
expectations of economic growth and associated
concerns about possible inflation pressures.

Of course, this is a simplistic discussion. A
tightening has to be unanticipated in order for it
1

I owe the quote to Gallmeyer et al. (2007).

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to lower forward rates through this channel, and
evidence from other sources, such as the Treasury
inflation-protected securities mentioned by Chairman Greenspan or foreign interest rates, also bears
on the issue. Still, where did anyone get the idea
that monetary policy should control long-term
rates and that it is puzzling if long-term rates do
not “respond” positively to tightening? The natural benchmark predicts exactly the opposite, if
any, effect.
Second, to the extent that the decline in forward rates represents a cyclical or secular decline
in term premia, that decline also is perfectly natural. Term premia, like all risk premia, should
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Cochrane

decline as we come out of recessions, and have
done so in every past recession. Even negative
term premia are not a puzzle—they should be negative. In a world with stable inflation, interest rate
variation comes from variation in real rates; and,
in such a world, long-term bonds are safer investments for long-term investors. Rolling over shortterm bonds runs the “reinvestment risk” that
short-term (real) rates will change, so short-term
bonds should bear the burden of any bond risk
premia. We expect only a positive term premium
in a world with unstable inflation and relatively
constant real rates, such as the 1970s. Because
short-term rates adapt quickly to inflation changes,
rolling over short-term bonds has less risk to a
long-term investor than does buying only longterm bonds in this environment.
In sum, were I a Fed Chairman testifying to
Congress with the plots of Figure 1 in hand, I
would be tempted to point out that, far from a
“conundrum,” the world is finally behaving
exactly the way it should—and so is the central
bank. The increased transparency and predictability of operating procedures, seen in the steadiness
of the rise in funds rates in 2004-06 verses the less
predictable rise in 1994-96, has communicated
to the markets the Fed’s steadfastness in controlling inflation. We are moving to the sensible world
of negative risk premia, which is exactly what we
should see once markets understand that inflation
is vanquished forever. The conquest of inflation
has removed an unnecessary risk premium for
long-run investors and issuers of long-dated nominal bonds. I don’t necessarily believe all this, of
course, but it would be awfully tempting to make
this argument were I defending the Fed’s actions
before a congressional committee. The “conundrum” is Greenspan: Why did he say anything
else?
Finally, it is academics’ job to remind policy
debaters of basic economics, so I think we should
pounce anytime somebody says something like
“[the] decline in the term premium...is financially
stimulative and argues for greater monetary policy
restraint.” Every price reflects both supply and
demand. Low interest rates can reflect a lack of
good investment projects as easily as they can
reflect an abundance of savings. To take a local
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

example, low housing prices in East St. Louis do
not seem to be particularly “stimulative.”

DECOMPOSING THE YIELD
CURVE
My grumpy comments about “conundrum”
and the “stimulative” effects of low prices notwithstanding, this episode does highlight the
importance of splitting the yield curve into
expected future rates and risk premia and of
understanding the dynamic structure of risk premia and their macroeconomic underpinnings.
Here Rudebusch, Sack, and Swanson provide a
very nice summary of the state of the art.
I think the bottom line is that we know less
than we think about this decomposition and far
less than the pronouncements in policymakers’
quotes imply. The paper can be read as a comprehensive survey of one failure after another. Here,
let me give two quick, and I hope memorable,
points in this litany of ignorance.

Levels, Differences, and Standard Errors
I learned two important lessons while Monika
Piazzesi and I (2006) investigated this kind of
decomposition. First, how you specify trends,
cointegration, etc.—which the data say very little
about—is overwhelmingly the most important
issue in driving the decomposition of the longmaturity end of the yield curve. Second, the standard errors are very large. For these reasons alone,
any statements decomposing the recent experience of forward rates into changes in expected
interest rates versus declining term premia are
subject to huge uncertainty.
To see this point, let’s try the simplest
approach to decomposing the yield curve. I run a
vector autoregression (VAR) of five forward rates
on their lags (I use the Fama-Bliss data available
from the Center for Research and Security Prices,
and I use a three-month moving average of forward rates on the right-hand side, which Piazzesi
and I (2005) find improves forecasts by mitigating
measurement error):
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Figure 2
March 2006 Forward Curve and Expected
One-Year Rates from a VAR in Levels and
from a Cointegrated VAR
Rate
8

7
VAR Expected Rate
6

Negative Term Premium
March 2006 Forward
Curve

5

Positive Term Premium
∆ VAR Expected Rate

4

3

2
0

5

10

15

Maturity

NOTE: The one-standard-error bars are computed from a direct
regression forecast, Xt′cov(βˆ )Xt , using Hansen-Hodrick correction for serial correlation due to overlap.

 y t(1+)1 
 y t(1) 




 ft(+21) 
 ft(2) 

 = A+ B
 + ε t + 1,
 A 
 A 
 f (5) 
 f (5) 
 t +1 
 t 
where
ft共n兲 = forward at time t for loans from
t + n – 1 to t + n
yt共1兲 = one-year rate at time t.
We can use this VAR to generate forecasts at each
date of future one-year rates, leaving (“estimating”)
the term premium as a residual,

(

)

ft(n ) = Et y t(1+)n + rpft(n ) .

You don’t have to estimate fancy term-structure
models to decompose the yield curve into
expected interest rates and a risk premium.
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Figure 2 presents the results, evaluated on
March 2006 (the last point in my data sample),
in the line labeled “VAR expected rate.” The line
captures a lot of common opinion: It says that
interest rates are expected to rise gently over the
next few years, leaving a negative term premium,
which is puzzling until you think through the
economics of long-term bond investing in a lowinflation world. This kind of decomposition also
says that much of the recent decline in forward
rates comes from the term premium rather than
changes in expected long-term rates.
This all seems very sensible. However,
Figure 3 examines the same calculation over a
longer time interval. The lines represent, at each
date, expected one-year rates one, two, three,
(1)
etc., years in the future: that is, for Et 共y t+k
兲 for
k = 1,2,3,... at each t. The graph dramatically
makes the point that long-horizon expected oneyear rates calculated by this method simply reflect
reversion to the mean. The 6.25 percent asymptote
in Figure 2 represented no specially sophisticated
regression forecast; it was simply the sample
interest rate.
There is nothing logically or econometrically
wrong with this conclusion, but do we really
believe it? For example, in 1980, this decomposition says that everyone knew interest rates would
decline from 16 percent back to an unconditional
mean of a bit over 6 percent, and rather rapidly,
so the then-flat yield curves represented very
large risk premia for holding long-term bonds.
But did people really believe inflation would be
tamed, or did perhaps the flat yield curves of the
time really represent a good chance that inflation
would re-emerge? Similarly, perhaps the sample
mean is now too high an estimate. Our data come
from inflation and its conquest. Perhaps it is
sensible now to think a “structural shift” has
happened, so the long-run mean should be a
good deal less than 6.25 percent.
As an alternative, let us try a forecast that
ignores this “level” information. On a statistical
basis, forward rates are clearly best modeled by a
single common trend that has a root that is near
if not equal to 1 and stationary spreads around
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

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Figure 3
Multiperiod One-Year-Rate Forecasts from a VAR in Levels
16

14

12

10

8

6

4

2

0

1975

1980

1985

1990

1995

2000

2005

NOTE: Current one-year rate and expectations of one-year rates one, two, three, etc., years in the future, calculated by a simple VAR.

that trend. I estimate a VAR imposing that
restriction:
 y t(1+)1 − y t(1) 

 ft(2) − y t(1) 


(2)
(1)


 ft +1 − y t +1 
A
 + ε t +1 .

 =A+B
A
 ( 5)


(1) 
 ft − y t 
 f (5) − y (1) 
t +1 
 t +1

This is equivalent to simply running forecasting
regressions that set to zero a coefficient on the
level of interest rates:
Before : ft(+n1) = a(n) + Bn,1 × y t(1) + Bn,2  ft(2) − y t(1) 
+ Bn,3  ft(3) − y t(1)  +;
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

now : ft(+n1) = a(n) + 0 × y t(1) + Bn,2  ft(2) − y t(1) 
+ Bn,3  ft(3) − y t(1)  + ...

Intuitively, we still allow information such as
“the yield curve is upward sloping” to forecast
interest rate changes. We ignore information such
as “interest rates are low” to tell us interest rates
will rise.
Figure 4 presents expected one-year rates over
time by this method. You can see the huge difference. One-year rates are certainly not being forecast to revert to a constant unconditional mean!
In particular, the flat yield curves of the 1980s are
not now interpreted to reveal huge risk premia
plus expected declines in interest rates. This is
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Figure 4
Multiperiod One-Year-Rate Forecasts from a VAR in Levels

15

10

5

0
1975

1980

1985

1990

1995

2000

2005

NOTE: Current one-year rate and expectations of one-year rates one, two, three, etc., years in the future, calculated by a VAR that
imposes a single common trend in forward rates.

not a pure random-walk model, and there is still
some forecastability left. For example, the steeply
upward-sloping yield curves of 2003-04 do forecast substantial rises in short rates.
Figure 2 includes the March 2006 one-year
rate forecast from this method, in the line labeled
“∆ VAR Expected Rate.” This is also a sensible
forecast. Because we no longer use the information that the current one-year rate is slightly below
its sample mean, we are left only with slope information. The unusually flat slope of the forward
curve means, in this forecast, that interest rates
will decline somewhat, so that the term premium
is still somewhat positive. However, Figure 2
shows that long-term interest rate forecasts by
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this method have been rising in recent years; so,
the decline in forward rates since 2004 is attributed even more to declining term premia by this
method than by the VAR in levels.
Can statistics help us? Alas, no. Testing for
unit roots, cointegration, etc., and imposing the
resulting structure on the analysis is not fruitful.
One naturally wants to think about “structural
shifts,” changing means, and so forth, and these
will be even more imprecisely estimated in nowshorter samples. It is certainly true that the
dominant root of a persistent set of variables is
estimated with downward bias, so the actual
reversion to the mean is slower than the VAR in
levels indicates, but whether that mean makes
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any sense in the first place is not something statistics can really help us with.
Can fancier models help us? In particular,
most of the term-structure literature does not look
at simple VAR forecasts. Instead, it estimates the
parameters of “affine models.” To think about
what these can do, it’s useful to have a specific
example in front of us, so here is the Ang and
Piazzesi (2003) model that we use in Cochrane
and Piazzesi (2005 and 2006). A vector of state
variables Xt follows an AR(1) process; the stochastic discount factor is an exponential function of the state variables, with “market prices
of risk” (loadings of M on shocks to X) that also
depend on the state variables,
(1)

(2)

X t = µ + φ X t −1 + Σε t
1


M t = exp  −δ 0′ X t − λt′λt − λt′εt 


2
λt = λ0 + λ1′ X t .

Assuming the shocks ε are i.i.d. normal with unit
variance, we can find this model’s prediction for
bond prices,
(3)

Pt(n) = E t ( M t +1 M t +2 ...M t + n )

(4)

pt(n) = An + Bn X t ,

and then yields and forward rates. Inverting (4),
we can reveal the “state variables” from bond
prices, yields, or forward rates. Thus, this model
becomes a structured factor model in which a
large collection of prices, yields, or forward rates
are described in terms of a few linear combinations
of those same prices, yields, or forward rates.
But, underlying the whole thing, we see a
VAR(1) in yields, prices, or forward rates—just
as we have been estimating all along! Thus the
only way the affine model can give us any different
answers from those of the ordinary least squares–
estimated VARs above is if the structure of market
prices of risk means that we use information in
the cross-section of bond prices to infer something about the dynamics. In general, this is not
the case. In Cochrane and Piazzesi (2005) we show
how to construct market prices of risk, λ, from a
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

given discrete-time VAR(1) to turn it into an affine
model. Thus, in general, the affine model lives
on top of a VAR estimate of long-term forward
rates and adds nothing to it. (There remains the
possibility that by restricting or modeling market
prices of risk, λ, in sensible ways, one obtains
information about the VAR (1), and this is the
point of our 2006 paper. But this is not [yet] a
common idea, and its success lies in the believability of a priori restrictions on λ .)
In addition, once we have settled on a specification, we have to wonder how much sampling
uncertainty in estimating the parameters translates
into uncertainty about the forecasts. To address
this question in a simple and transparent way, I
run direct forecasting regressions,

y t(1+)k − y t(1) =
1 ft(2) − y t(1)

ft(3) − y t(1) ... ft(5) − y t(1)  βk + ε t + k ,

y t(1+)k − y t(1) = X t′βk + εt + k ,
where the second equation defines notation. I find
the covariance matrix of βˆk , including a HansenHodrick correction for serial correlation due to
overlapping data, and then I calculate the error as

σ t2  Êt y t(1+)k − y t(1)  = X t ′cov (βˆ k ,βˆ k ′ )X t .

(

)

This is the error in the measurement of expected
interest rates due to sampling uncertainty in the
coefficients that comprise the regression forecast.
It is not the forecast error—that is, it is not a measure of how large σ 2(ε t+k) is. The one-standard-error
bars in Figure 2 present this calculation. The term
premium is not statistically significant, and the
large difference between the two specifications is
barely two standard errors. The Hansen-Hodrick
correction for serial correlation is undoubtedly
optimistic—at the right-hand end of the graph
we’re forecasting interest rates 10 years ahead in
45 years of data—so the true sampling uncertainty
is undoubtedly a good deal larger.
Now, understanding that large roots and
common trends, which often must be specified a
priori, are crucial to long-term forecasts and that
long-run forecasts are subject to enormous sampling uncertainty is not news. However, as I read
it, this sensitivity is not at all considered by the
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literature that uses affine models to compute
long-term yield-curve decompositions. We are
usually treated only to one estimate based on
one a priori specification, usually in levels, and
usually with no measure of the huge sampling
uncertainty. Needless to say, the usual habit of
estimating 10-year interest rate forecasts by
extrapolating models fit to weekly or monthly
data is no help, and possibly a hindrance. The
520th power of a matrix is a difficult object to
estimate.
In sum, when a policymaker says something
that sounds definite, such as “long-run forward
rates have declined, while interest rate expectations have remained constant, so risk premia
have declined,” he is really guessing, and we
really have no idea whether this is a fact.

Measuring Risk Premia
We also know a good deal less about long-term
risk premia than we think we do. Quotes such as
those at the beginning of the paper suggest that
risk premia are well measured if perhaps poorly
understood. Nothing of the sort is true. We may
have a decent handle on one-year risk premia, as
surveyed in the paper and the subject of my next
set of comments, but the 10-year forward-rate
premium reflects not only this year’s expected
excess bond returns, but this year’s expectations
of next year’s expected returns, and so on and so
forth. If you like equations, an easy one in which
to see this point is
(5)
1  (1)
y t + Et y t(1+)1 + Et y t(1+)2 + ... + Et y t(1+)n −1 
n
+  Et rx t(n+)1 + Et rxt(n+−21) + ... + Et rxt(2+)n −1  ,

)

(

y t(n) =

(

)

(

( )
(
)
(
)
)

(n)
(n)
= r t+1
– y t(1) =
where y (1) = one-year yield and rx t+1
excess returns. The first term is the expectations
hypothesis. The second term is the risk premium,
and you see that the risk premium depends on
future expected excess returns, not just on current
expected excess returns.
Now, if expected excess returns lived off in
their own space, moving away in response to
shocks and then recovering without relation to
the rest of the yield curve, then, yes, there would

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be one “risk premium” that accounts for expected
excess returns, as well as long-horizon forward
and yield-curve risk premia. Alas, this is not the
case. Today’s level, slope, and curvature have
strong power to forecast next year’s expected
excess returns. (Characterizing these dynamics
is a major point of Cochrane and Piazzesi, 2006.)
We can easily be in a situation that this year’s
expected excess return, Et rxt +1, is large and positive, while future expected excess returns,
Et (rxt +k ), are strongly negative, so the risk premium in the yield curve can be negative as well.
The one-year expected excess return can be positive while the 10-year forward rate is below its
corresponding expected one-year rate. It is precisely by such differences in expected future risk
premia that the two decompositions shown in
Figure 2 can produce forward rate premia of different signs, despite the same initial return risk
premium.
In sum, there is no single “risk premium.”
There is a full-term structure of return risk premia, which moves over time in interesting and
still poorly measured ways. Sure statements that
risk premia have moved down over time do not
reflect any solid and independent measurement.

FORECASTING, TERM PREMIA,
AND MACROECONOMICS
One of the major contributions of the
Rudebusch, Sack, and Swanson paper is the
empirical work linking bond risk premia and
macroeconomics. By restating the points in my
own way and slightly disagreeing with some
conclusions, I think I can usefully highlight this
important part of the paper.
Naturally, I like the Cochrane and Piazzesi
(2005) measurement of the risk premium, so I’ll
focus my comments on that paper. Briefly, we
noticed that regressions of excess returns on ex
ante forward rates follow a nearly exact one-factor
structure: That is, that regressions

rx t(n+)1 = α n + βn,1 y t(1) + βn,2 ft(2) + ... + βn,5 ft(5) + εt(n+)1
almost exactly follow
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Cochrane

)

(

rx t(n+)1 = bn γ 0 + γ 1 y t(1) + γ 2 ft(2) + ... + γ 5 ft(5) + εt(n+)1
= bn (γ ′ ft ) + εt(n+1) ,

where the last equality defines notation. A single
“return-forecasting factor,” γ ′ft, describes expected
excess returns of all maturities. Longer-maturity
bonds’ expected excess returns move more, and
shorter-maturity bonds’ expected excess returns
move less, but they all move in lockstep. Thus,
we estimate the common “return-forecasting factor” by running a single regression of average
(across maturity) returns on all forward rates:
rx t +1 =

1 5
∑ rx (n+)
4 n =2 t 1

= γ 0 + γ 1 y t(1) + γ 2 ft(2) + .... + γ 5 ft(5) + ε t +1,

where the first equality defines notation. Sensitive to “levels” issues, we obtain nearly identical
results by ruling out a level effect:

rx tn+1 = γ 0 + γ 2 (ft(2) − y t(1) )

(

)

+ γ 3(ft(3) − y t(1) ) + .... + γ 5 ft(5) − y t(1) + εt +1 .
The coefficients γ have a pretty tent shape. This
measure of bond risk premia values curvature in
the forward curve, not slope in the forward curve.
Figure 5 shows how this works and the connection between macroeconomics and bond risk
premia: In January 2002 (shown by the first vertical lines in panels A, B, and D), the recession
and interest rates have just finished their stage of
steep decline, as seen also in the unemployment
rate (panel D). The forward curve is upward sloping, but it is also very curved (panel C). The curved
forward rate, interacting with the tent-shaped γ,
is the sign of risk premia. This means (statistically)
that the upward slope will not be soon matched
by rises in interest rates, so the greater yields on
long-term bonds are (risky) profit for investors.
The risk premium (panel B) is very high. In fact,
this prediction is borne out: Interest rates do not
rise for several years, so investors who bought
long-term bonds in January 2002 profited handsomely for a few years.
By contrast, consider January 2004. Now, the
forward curve still slopes up substantially (panel
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

C), but it is no longer particularly curved, so the
tent-shaped γ coefficients no longer predict much
of a risk premium (panel B). Now, the upwardsloping yield curve does signal rises in interest
rates; the expectations hypothesis is working;
returns on long-term bonds will be no higher
(on average) than those on short-term bonds.
Again, this prediction is borne out. This time,
interest rates do rise. This is a repeated statistical
pattern, working the same way in many previous
recessions.
Having digested what term-premium forecasts
are and how they work, we see that the graphs
show several patterns seen in more-formal regressions. First, the term premium ( γ ′f here, as well
as other measures in the paper) drives out slope
variables for forecasting bond excess returns.
Previously, Fama and Bliss (1987), Campbell and
Shiller (1991), and others found that measures of
the term-structure slope forecast excess returns.
Yes, we see the slope is high in 2002, when longterm bond holders turn out to make money. But
it is also high in 2004 when they don’t. When you
put the slope and the curvature of the forward rate
together in a multiple regression, the curvature
measured by γ wins out. The slope seemed to forecast bond returns because it was correlated with
the curve measure. (See, for example, Table A3
of the appendix to Cochrane and Piazzesi, 2005.)
Second, the term premium is high in the
depths of a recession. In Figure 5, this is measured
by the association of the term premium (panel B)
with unemployment (panel D). The association
is even stronger in previous recessions. In macroeconomic terms (that’s why we’re here), this is
natural. The risk premium is high at the early
stage of a recession, a time in which investors
don’t want to hold risk of any kind. Stock prices
are low, predicting higher-than-average stock
returns; interest rates are low relative to foreign
interest rates, predicting high returns for holding
exchange-rate risk. By January 2004, however,
the recession is over, the period of growth and
rising interest rates has set in, and everybody
knows it. It’s not a surprise that the premium for
holding risk during recessions has vanished.
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Figure 5
Macroeconomics and the Yield Curve
A. 1- to 5-, 10-, and 15-Year Forward Rates and
Federal Funds

B. Cochrane and Piazzesi Risk Premium

8

80

All Forward Rates
No Level

60

6

40
4
20
2

0
−20

0
2000

2002

2004

2006

2000

C. Forward Curve

2002

2004

2006

2004

2006

D. Unemployment
7

6
2002

5

6

4

2004
5

3
4

2

1
1

2

3

4

5

3
2000

2002

NOTE: Vertical lines mark interesting dates. Panel A: federal funds rate and 1- to 15-year forward rates through the previous recession.
Panel B: Cochrane and Piazzesi (2005) measures of the bond risk premium. Panel C: forward curves on the two indicated dates.
Panel D: unemployment rate.
SOURCE: Data for yields and forward rates past a five-year maturity are from Gürkaynak, Sack, and Wright (2006).

Third, and the major point of the authors’
paper (as I see it), the slope of the yield curve
drives out the risk premium for forecasting 1-year
gross domestic product (GDP) growth. We see this
in panel B of Figure 5: The risk premium is high
in 2002, when GDP is not about to grow. The risk
premium is low in 2004, when GDP is about to
grow. The slope is high in both times. Thus, the
slope carries GDP forecast power, and the slope,
purged of its correlation with the risk premium,
forecasts GDP even better.
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This point is made in their paper in the
regression of Table 2, last column, which I take
to be the central result:
y t + 4 − y t = 0.38(4.22) + 0.96(5.62)(exspt − exspt − 4 )
− 0.59( − 1.93)(tpt − tpt − 4 ) + ε t + 4 ,

where y = GDP; exsp is the expectationshypothesis component of the 10-year rate; tp is
the term premium component of the 10-year rate,
as in (5); t-statistics are in parentheses; and the
sample is from 1962 to 2005.
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Cochrane

The authors (p. 261) say of this regression,
“The coefficient on the risk-neutral expectations
component of the yield curve slope [0.96] is now
larger and more statistically significant than in
any of the earlier specifications,” which is the
same interpretation I gave in discussing the figure.
The authors also say (p. 261), “More importantly for this paper, we find that the estimated
coefficient on the term premium is now negative
and (marginally) statistically significant. According to these results, a decline in the term premium
tends to be followed by faster GDP growth—the
opposite sign of the relationship uncovered by
previous empirical studies.” I read the evidence
differently: Rather than accept a marginally significant coefficient with the wrong sign, it seems
to me the right lesson is that the second coefficient
is zero. The slope of the yield curve forecasts GDP
growth, but not risk premia. The curvature of the
forward curve measures risk premia, but not GDP
growth. Risk premia are high precisely when we
are not sure whether the recession is over.
Table 8 of Ang, Piazzesi, and Wei (2006) runs
the same sort of regression. At a four-quarter
horizon, they find
y t + 4 − y t = a + 1.15(5.00)EH t − 0.47(0.30)RPt + εt + 4 ,

where EH = expectations hypothesis; RP = risk
premium in the 20-quarter term spread (i.e., the
terms in (5), estimated from a macro-affine model);
and t-statistics are in parentheses. In this slightly
different specification, they confirm the huge
significance of the expectations-hypothesis term,
but find an insignificant contribution due to the
risk-premium term.
This view dovetails with the other side of risk
premia that Monika Piazzesi and I (2006) have
recently started investigating. From the basic
asset-pricing relation, 1 = Et (Mt +1Rt +1), and (2),
we can write
(6)

1
Et (rx t(n+)1 ) + σ t2 (rx t(n+)1 ) = cov (rxt(n+)1,εt′+1 )λt .
2

Note the absence of an n index on λ . The point
of this equation is that expected excess returns
on each bond must be earned in compensation
for, and in proportion to, the covariance of that
bond’s return with macroeconomic shocks, ε . So
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

far, we have been talking about the left-hand side:
What models or state variables drive variation
over time in expected excess returns? Now, it’s
time to start working on the right-hand side: What
are the shocks? Piazzesi and I find that the term
premium is almost exactly earned entirely in
compensation for shocks to the level of the term
structure. The prices of risk, λ , corresponding to
other term-structure shocks are essentially zero.
In particular, expected returns seem not to be
earned in compensation for “slope” shocks. This
finding lets us start to think about macroeconomics. Whatever the macroeconomic determinants of bond risk premia, they must be variables
with “level” effects on the term structure. This
observation quickly rules out monetary policy,
whose shocks typically raise short rates while
lowering or leaving unchanged long rates—
a “slope” shock.

CONCLUDING COMMENT
In sum, I think we are headed to a view that
slope movements in the yield curve, which are
related to monetary policy, are also related to
expected GDP growth, as seen in the usual
impulse-response functions. But slope movements do not signal risk premia (left-hand side
of (6)), nor does covariance with monetary policy
shocks generate real risk premia (right-hand side
of (6)). Term premia are large in the early phases
of recessions, when it’s not clear how long the
recession will last; they are revealed by the curvature of the forward rate, and they are earned in
compensation for macroeconomic risks that correspond to shocks in the level of the yield curve.
Of course, we have no economic models of
any of these fascinating statistical regularities.
This point is made clear in the brilliant survey
of total failures that occupies a large part of the
paper. First, what are the macroeconomic state
variables that drive variation in expected returns?
What exactly are the times and states of nature
in which expected returns are high? Second,
expected returns are generated by the covariance
of returns with macroeconomic shocks. What are
these macroeconomic shocks? These questions
are, as ever, the Holy Grail of macro-finance.
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REFERENCES
Ang, Andrew and Piazzesi, Monika. “A No-Arbitrage
Vector Autoregression of Term Structure Dynamics
with Macroeconomic and Latent Variables.”
Journal of Monetary Economics, July 2003, 50(5),
pp. 745-87.
Ang, Andrew; Piazzesi, Monika and Wei, Min.
“What Does the Yield Curve Tell Us about GDP
Growth?” Journal of Econometrics, March/April
2006, 131(1-2), pp. 359-403.
Campbell, John Y. and Shiller, Robert J. “Yield Spreads
and Interest Rate Movements: A Bird’s Eye View.”
Review of Economic Studies, May 1991, 58(3),
pp. 495-514.
Cochrane, John H. and Piazzesi, Monika. “The Fed
and Interest Rates—A High Frequency
Identification.” American Economic Review, May
2002, 92(2), pp. 90-95.
Cochrane, John H. and Piazzesi, Monika. “Bond Risk
Premia.” American Economic Review, March 2005,
95(1), pp. 138-60.

Fama, Eugene F. and Bliss, Robert R. “The Information
in Long-Maturity Forward Rates.” American
Economic Review, September 1987, 77(4), pp. 680-92.
Gallmeyer, Michael F.; Hollifield, Burton; Palomino,
Francisco and Zin, Stanley F. “Arbitrage-Free
Pricing with Dynamic Macroeconomics Models.”
Federal Reserve Bank of St. Louis Review, July/
August 2007, 89(4), pp. 305-26.
Greenspan, Alan. “Statement by Alan Greenspan,
Chairman, Board of Governors of the Federal
Reserve System, before the Committee on Banking,
Housing, and Urban Affairs, May 27, 1994.” Federal
Reserve Bulletin, July 1994.
Gürkaynak, Refet S.; Sack, Brian and Wright,
Jonathan H. “The U.S. Treasury Yield Curve: 1961
to the Present.” Unpublished manuscript, Board of
Governors of the Federal Reserve System, 2006.
Rudebusch, Glenn D.; Sack, Brian P. and Swanson,
Eric T. “Macroeconomic Implications of Changes
in the Term Premium.” Federal Reserve Bank of
St. Louis Review, July/August 2007, 89(4),
pp. 241-69.

Cochrane, John H. and Piazzesi, Monika.
“Decomposing the Yield Curve.” Unpublished
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F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Long-Run Risks and Financial Markets
Ravi Bansal
The recently developed long-run risks asset pricing model shows that concerns about long-run
expected growth and time-varying uncertainty (i.e., volatility) about future economic prospects
drive asset prices. These two channels of economic risks can account for the risk premia and
asset price fluctuations. In addition, the model can empirically account for the cross-sectional
differences in asset returns. Hence, the long-run risks model provides a coherent and systematic
framework for analyzing financial markets. (JEL G0, G00, G1, G10, G12)
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 283-99.

F

rom the perspective of theoretical
models, several key features of asset
markets are puzzling. Among others,
these puzzling features include the level
of equity premium (see Mehra and Prescott, 1985),
asset price volatility (see Shiller, 1981), and the
large cross-sectional differences in average returns
across equity portfolios such as value and growth
portfolios. In bond and foreign exchange markets,
the violations of the expectations hypothesis
(see Fama and Bliss, 1987; Fama, 1984) and the
ensuing return predictability is quantitatively
difficult to explain. What risks and investor concerns can provide a unified explanation for these
asset market facts? One potential explanation of
all these anomalies is the long-run risks model
developed in Bansal and Yaron (2004) (henceforth
BY). In this model, fluctuations in the long-run
growth prospects of the economy and the timevarying level of economic uncertainty (consumption or output volatility) drive financial markets.
Recent work indicates that many of the asset
prices anomalies are a natural outcome of these
channels developed in BY. In this article I explain
the key mechanisms in the BY model that enable
it to account for the asset market puzzles.

In the BY model, the first economic channel
relates to expected growth: Consumption and
dividend growth rates contain a small long-run
component in the mean. That is, current shocks
to expected growth alter expectations about future
economic growth not only for short horizons but
also for the very long run. The second channel
pertains to varying economic uncertainty: Conditional volatility of consumption is time varying.
Fluctuations in consumption volatility lead to
time variation in risk premia. Agents fear adverse
movements in the long-run growth and volatility
components because they lower equilibrium consumption, wealth, and asset prices. This makes
holding equity quite risky, leading to high risk
compensation in equity markets.
The preferences developed in Epstein and Zin
(1989) play an important role in the long-run risks
model. These preferences allow for a separation
between risk aversion and intertemporal elasticity
of substitution (IES) of investors: The magnitude
of the risk aversion relative to the reciprocal of
the IES determines whether agents prefer early
or late resolution of uncertainty regarding the
consumption path. In the BY model, agents prefer
early resolution of uncertainty; that is, risk aver-

Ravi Bansal is a professor of finance at the Fuqua School of Business, Duke University. He thanks Dana Kiku, Tom Sargent, Ivan Shaliastovich,
and Amir Yaron for comments.

© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
only with prior written permission of the Federal Reserve Bank of St. Louis.

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283

Bansal

sion is larger than the reciprocal of the IES. This
ensures that the compensation for long-run
expected growth risk is positive. The resulting
model has three distinct sources of risks that determine the risk premia: short-run, long-run, and
consumption volatility risks. In the traditional
power utility model, only the first risk source
carries a distinct risk price and the other two risks
have zero risk compensation. Separate risk compensation for shocks to consumption volatility
and expected consumption growth is a novel
feature of the BY model relative to earlier asset
pricing models.
To derive model implications for asset prices,
the preference parameters are calibrated. The
calibrated magnitude of the risk aversion and the
IES is an empirical issue. Hansen and Singleton
(1982), Attanasio and Weber (1989), and VissingJorgensen and Attanasio (2003) estimate the IES
to be well in excess of 1. Hall (1988) and Campbell
(1999), on the other hand, estimate its value to be
well below 1. BY show that even if the population
value of the IES is larger than 1, the estimation
methods used by Hall would measure the IES to
be close to zero. That is, in the presence of timevarying consumption volatility, there is a severe
downward bias in the point estimates of the IES.
Using data from financial markets, Bansal,
Khatchatrian, and Yaron (2005) and Bansal and
Shaliastovich (2007) provide further evidence
on the magnitude of the IES.
Different techniques are employed to provide
empirical and theoretical support for the existence
of long-run components in consumption and
dividends. Whereas BY calibrate parameters to
match the annual moments of consumption and
dividend growth rates, Bansal, Gallant, and
Tauchen (2007) and Bansal, Kiku, and Yaron
(2006) formally test the model using the efficient
and generalized method of moments, respectively.
Using multivariate analysis, Hansen, Heaton, and
Li (2005) and Bansal, Kiku, and Yaron (2006)
present evidence for long-run components in
consumption growth. Colacito and Croce (2006)
also provide statistical support for the long-run
components in consumption data for the United
States and other developed economies. Lochstoer
and Kaltenbrunner (2006) provide a production284

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based motivation for long-run risks in consumption. They show that in a standard production
economy, where consumption is endogenous,
the consumption growth process contains a predictable long-run component similar to that in the
BY model. There is considerable support for the
volatility channel as well. Bansal, Khatchatrian,
and Yaron (2005) show that consumption volatility is time varying and that its current level predicts future asset valuations (price-dividend ratio)
with a significantly negative projection coefficient; this implies that asset markets dislike economic uncertainty. Exploiting the BY uncertainty
channel, Lettau, Ludvigson, and Wachter (2007)
provide interesting market premium implications
of the low-frequency decline in consumption
volatility.
BY show that their long-run risks model can
explain the risk-free rate, the level of the equity
premium, asset price volatility, and many of the
return and dividend growth predictability dimensions that have been characterized in earlier work.
The time-varying volatility in consumption is
important to capture some of the economic outcomes that relate to time-varying risk premia.
The arguments presented in their work also
have immediate implications for the crosssectional differences in mean returns across assets.
Bansal, Dittmar, and Lundblad (2002 and 2005)
show that the systematic risks across firms should
be related to the systematic long-run risks in firms’
cash flows that investors receive. Firms whose
expected cash-flow (profits) growth rates move
with the economy are more exposed to long-run
risks and hence should carry higher risk compensation. These authors develop methods to measure
the exposure of cash flows to long-run risks, and
show that these cash flow betas can account for
the differences in risk premia across assets. They
show that the high book-to-market portfolio (i.e.,
value portfolio) has a larger long-run risks beta
relative to the low book-to-market portfolio (i.e.,
growth portfolio). Hence, the high mean return
of value firms relative to growth firms is not puzzling. The Bansal, Dittmar, and Lundblad (2002
and 2005) evidence supports a long-run risks
explanation for the cross-sectional differences in
mean returns.
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Bansal

Several recent papers use the BY long-run
risks model to address a rich array of asset market
questions; among others, these include Kiku
(2006), Colacito and Croce (2006), Lochstoer and
Kaltenbrunner (2006), Chen, Collin-Dufresne,
and Goldstein (2006), Chen (2006), Eraker (2006),
Piazzesi and Schneider (2005), and Bansal and
Shaliastovich (2007). Kiku (2006) shows that the
long-run risks model can account for the violations of the capital asset pricing model (CAPM)
and consumption CAPM (C-CAPM) in explaining
the cross-sectional differences in mean returns.
Further, the model can capture the entire transition density of value or growth returns, which
underscores the importance of long-run risks in
accounting for equity markets’ behavior. Eraker
(2006) and Piazzesi and Schneider (2005) consider
the implications of the model for the risk premia
on U.S. Treasury bonds and show how to account
for some of the average premium puzzles in the
term structure literature. Colacito and Croce (2006)
extend the long-run risks model to a two-country
setup and explain the issues about international
risk sharing and exchange rate volatility. Bansal
and Shaliastovich (2007) show that the long-run
risks model can simultaneously account for the
behavior of equity markets, yields, and foreign
exchange and explain the nature of predictability and violations of the expectations hypothesis
in foreign exchange and Treasury markets. Chen,
Collin-Dufresne, and Goldstein (2006) and Chen
(2006) analyze the ability of the long-run risks
model to explain the credit spread and leverage
puzzles of the corporate sector.
Hansen, Heaton, and Li (2005) consider a
long-run risks model with a unit IES specification.
Using different methods to measure long-run risks
exposures of portfolios sorted by book-to-market
ratio, they find, as in Bansal, Dittmar, and
Lundblad (2005), that these alternative long-run
risk measures do line-up in the cross-section with
the average returns. They further show that the
measurement of long-run risks can be sensitive
to the econometric methods used. Hansen and
Sargent (2006) highlight the interesting implications of robust decisionmaking for risks in financial markets when the representative agent
entertains the long-run risks model as a baseline
description of the economy.
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

The above results indicate that the long-run
risks model can go a long way toward providing
an explanation for many of the key features of
asset markets.
The remainder of the article has three sections.
The next section reviews the long-run risks model
of Bansal and Yaron (2004). The third section
discusses the empirical evidence of the model
and, in particular, its implications for the equity,
bond, and currency markets. The final section
concludes.

LONG-RUN RISKS MODEL
Preferences and the Environment
Consider a representative agent with the
following Epstein and Zin (1989) recursive
preferences,
θ
1− γ

U t = (1 − δ )Ct θ + δ E t U t1+−1γ 


(

1/ θ 
 1− γ

)




,

where the rate of time preference is determined
by δ, with 0 < δ < 1. The parameter θ is determined
by the risk aversion and the IES—specifically,

θ;

1−γ
,
1
1−
ψ

where γ ≥ 0 is the risk aversion parameter and
ψ ≥ 0 is the IES. The sign of θ is determined by
the magnitudes of the risk aversion and the elasticity of substitution. In particular, if ψ >1 and
γ > 1, then θ will be negative. Note that, when
θ = 1 (that is γ = 1/ψ ,), one obtains the standard
case of expected utility.
As is pointed out in Epstein and Zin (1989),
when risk aversion equals the reciprocal of IES
(expected utility), the agent is indifferent to the
timing of the resolution of the uncertainty of the
consumption path. When risk aversion exceeds
(is less than) the reciprocal of IES, the agent
prefers early (late) resolution of uncertainty of
consumption path. In the long-run risks model,
agents prefer early resolution of the uncertainty
of the consumption path.
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Bansal

The period budget constraint for the agent
with wealth, Wt , and consumption, Ct , at date t is
(1)

Wt +1 = (Wt − Ct ) Ra , t +1 .

Ra, t +1 =

Wt +1
W
( t − Ct )

is the return on the aggregate portfolio held by
the agent. As in Lucas (1978), we normalize the
supply of all equity claims to be 1 and the riskfree asset to be in zero net supply. In equilibrium,
aggregate dividends in the economy (which also
include any claims to labor income) equals aggregate consumption of the representative agent.
Given a process for aggregate consumption, the
return on the aggregate portfolio corresponds to
the return on an asset that delivers aggregate
consumption as its dividends each time period.
The logarithm of the intertemporal marginal
rate of substitution (IMRS), mt +1, for these preferences (Epstein and Zin, 1989) is
(2)

mt +1 = θ log δ −

θ
g + (θ − 1) ra, t +1,
ψ t +1

and the asset pricing restriction on any continuous return, ri,t +1, is



θ
(3) Et exp θ log δ − g t +1 + (θ − 1) ra, t +1 + ri , t +1  = 1,
ψ




where g t +1 equals log 共Ct +1/Ct 兲—the log growth
rate of aggregate consumption. The return, ra,t +1,
is the log of the return (i.e., continuous return)
on an asset that delivers aggregate consumption
as its dividends each time period.
The return on the aggregate consumption
claim, ra,t +1, is not observed in the data, whereas
the return on the dividend claim corresponds to
the observed return on the market portfolio, rm,t +1.
The levels of market dividends and consumption
are not equal; aggregate consumption is much
larger than aggregate dividends. The difference
is financed by labor income. In the BY model,
aggregate consumption and aggregate dividends
are treated as two separate processes and the difference between them defines the agent’s labor
income process.
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The key ideas of the model are developed,
and the intuition is provided by means of approximate analytical solutions. However, for the key
qualitative results, the model is solved numerically. To derive the approximate solutions for
the model, we use the standard Campbell and
Shiller (1988) return approximation,
(4)

ra , t +1 = κ 0 + κ 1zt +1 − zt + gt +1 ,

where lowercase letters refer to variables in logs,
in particular, ra,t +1 = log 共Ra,t +1 兲 is the continuous
return on the consumption claim and the price-toconsumption ratio is zt = log 共Pt /Ct 兲. Analogously,
rm,t+1 and zm,t correspond to the continuous return
on the dividend claim and the log of the priceto-dividend ratio. The approximating constants,
κ 0 and κ 1, are specific to the asset under consideration and depend only on the average level of
zt , as shown in Campbell and Shiller (1988). It is
important to keep in mind that the average value
of zt for any asset is endogenous to the model
and depends on all its parameters and the
dynamics of the asset’s dividends.
From equation (2), it follows that the innovation in IMRS, mt +1, is driven by the innovations
in g t +1 and ra,t +1. Covariation with the innovation
in mt+1 determines the risk premium for any asset.
We characterize the nature of risk sources and
their compensation in the next section.

Long-Run Growth and Economic
Uncertainty Risks
The agent’s IMRS depends on the endogenous
consumption return, ra,t+1. The risk compensation
on all assets depends on this return, which itself
is determined by the process for consumption
growth. The dividend process is needed for determining the return on the market portfolio. To capture long-run risks, consumption and dividend
growth rates, g t +1 and gd,t +1, respectively, are
modeled to contain a small persistent predictable
component, xt , while fluctuating economic uncertainty is introduced through the time-varying
volatility of the cash flows:

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Bansal

x t +1 = ρx t + ϕeσ t et +1
gt +1 = µ + x t + σ t ηt +1

(5)

gd , t +1 = µd + φ x t + ϕdσ tut +1

(

)

σ t2+1 = σ 2 + ν1 σ t2 − σ 2 + σ w w t +1
et +1 , ut +1 , ηt +1 , w t +1 ~ N .i.i.d. (0, 1),

with the shocks et +1, ut +1, ηt +1, and wt +1 assumed
to be mutually independent. The parameter ρ
determines the persistence of the expected growth
rate process. First, note that when ϕe = 0, the
processes gt and gd,t +1 have zero autocorrelation.
Second, if et+1 = ηt+1, the process for consumption
is ARMA(1,1) used in Campbell (1999), Cecchetti,
Lam, and Mark (1993), and Bansal and Lundblad
(2002). If in addition ϕe = ρ, then consumption
growth corresponds to an AR(1) process used in
Mehra and Prescott (1985). The variable σt +1 represents the time-varying volatility of consumption
and captures the intuition that there are fluctuations in the level of uncertainty in the economy.
The unconditional volatility of consumption is
σ 2. To maintain parsimony, it is assumed that
the shocks are uncorrelated and that there is only
one source of time-varying economic uncertainty
that affects consumption and dividends.
Two parameters, φ > 1 and φd > 1, calibrate the
overall volatility of dividends and its correlation
with consumption. The parameter φ is larger than
1 because corporate profits are more sensitive to
changing economic conditions relative to consumption. Note that consumption and dividends
are not cointegrated in the above specification;
Bansal, Gallant, and Tauchen (2007) develop a
specification that does allow for cointegration
between consumption and dividends.
The better understand the role of long-run
risks, consider the scaled long-run variance (or
variance ratio) of consumption for horizon J,

σ c2, J

J
Var  ∑ j =1 gt + j 

.
=
J Var  gt 

The magnitude of this consumption growth
volatility is the same for all J if consumption is
uncorrelated across time. This scaled variance
increases with the horizon when the expected
growth is persistent. Hence, agents face larger
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

aggregate consumption volatility at longer horizons. As the persistence in x and/or its variance
increases, the magnitude of long-run volatility
will rise. In equilibrium, this increase in magnitude of aggregate consumption volatility will
require a sizeable compensation if the agents
prefer early resolution of uncertainty about the
consumption path.
Using multivariate statistical analysis,
Hansen, Heaton, and Li (2005) and Bansal, Kiku,
and Yaron (2006) provide evidence on the existence of the long-run component in observed
consumption and dividends. Using simulation
methods, Bansal and Yaron (2005) document the
presence of the long-run component in U.S. consumption data, whereas Colacito and Croce (2006)
estimate this component in consumption for
many developed economies. Note that there can
be considerable persistence in the time-varying
consumption volatility as well; hence, the longrun variance of the conditional volatility of consumption can be very large as well.
To see the importance of the small lowfrequency movements for asset prices, consider
the quantity

`

Et  ∑ κ 1j g t + j  .
 j =1

With κ 1 < 1, this expectation equals

κ 1xt
.
1 − κ 1ρ
Even though the variance of x is tiny, while ρ is
fairly high, shocks to xt (the expected growth rate
component) can still alter growth rate expectations for the long run, leading to volatile asset
prices. Hence, investor concerns about expected
long-run growth rates can alter asset prices quite
significantly.

Equilibrium and Asset Prices
The consumption and dividend growth rate
processes are exogenous in this endowment
economy. Further, the IMRS depends on an
endogenous return, ra,t+1. To characterize the
IMRS and the behavior of asset returns, a solution for the log price-consumption ratio, zt , and
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Bansal

the log price-dividend ratio, zm,t , is needed. The
relevant state variables for zt and zm,t are the
expected growth rate of consumption, xt , and
the conditional consumption volatility, σt2.
Exploiting the Euler equation (3), the approximate solution for the log price-consumption
ratio, zt , has the form zt = A0 + A1xt + A2σt2. The
solution for A1 is

1
ψ
A1 =
.
1 − κ 1ρ
1−

(6)

This coefficient is positive if the IES, ψ, is greater
than 1. In this case, the intertemporal substitution
effect dominates the wealth effect. In response to
higher expected growth, agents buy more assets
and consequently the wealth-to-consumption
ratio rises. In the standard power utility model
with risk aversion larger than 1, the IES is less
than 1 and therefore A1 is negative—a rise in
expected growth potentially lowers asset valuations. That is, the wealth effect dominates the
substitution effect.1
Corporate payouts (i.e., dividends), with φ > 1,
are more sensitive to long-run risks and changes
in the expected growth rate lead to a larger reaction in the price of the dividend claim than in
the price of the consumption claim. Hence, for
the dividend asset,

φ−
A1, m =

An analogous coefficient for the market pricedividend ratio, A2,m , is provided in BY.
The expression for A2 provides two valuable
insights. First, if the IES and risk aversion are
larger than 1, then θ and consequently A2 are
negative. In this case, a rise in consumption
volatility lowers asset valuations and increases
the risk premia on all assets. Second, an increase
in the permanence of volatility shocks—that is,
an increase in ν1—magnifies the effects of volatility shocks on valuation ratios as investors perceive changes in economic uncertainty as very
long lasting.

Pricing of Short-Run, Long-Run, and
Volatility Risks
Substituting the solutions for the price-consumption ratio, zt , into the expression for equilibrium return for ra,t+1 in equation (4), one can
now characterize the solution for the IMRS that
can be used to value all assets. The log of the
IMRS mt +1 can always be stated as the sum of
its conditional mean and its one-step-ahead
innovation. The conditional mean is affine in
expected mean and conditional variance of consumption growth and can be expressed as
Et ( m t + 1) =

(8)

1
ψ

1 − κ 1, m ρ

and is larger in absolute value than the consumption asset.
The solution coefficient, A2, for measuring
the sensitivity of the price-consumption ratio to
volatility fluctuations is

where m0 is a constant determined by the preference and consumption dynamics parameters.
The innovation in the IMRS is very important
for thinking about risk compensation (risk premia)
in various markets. Specifically, it is equal to
(9)

2

(7)

1



θ
2
0.5   θ −  + (θ A1κ 1ϕe ) 
ψ
 

A2 =
.
θ (1 − κ 1ν1 )

An alternative interpretation with the power utility model is that
higher expected growth rates increase the risk-free rate to an extent
that discounting dominates the effects of higher expected growth
rates. This leads to a fall in asset prices.

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2007

1

− γ  (γ − 1) 
2

 κ 1ϕe   2
ψ

1
1 + 
m 0 − xt +
 σt ,
ψ
2
  1 − κ 1ρ  

m t + 1 − E t ( m t + 1) =
− λm,ησ t ηt +1 − λm,e σ t et +1 − λm,w σ w w t +1,

where λm,η , λm,e , and λm,w are the market prices
for the short-run, long-run, and volatility risks,
respectively. The market prices of systematic
risks, including the compensation for stochastic
volatility risk in consumption, can be expressed
in terms of the underlying preferences parameters
that govern the evolution of consumption growth:
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Bansal

λ m, η = γ

1 κ ϕ 
λm,e =  γ −   1 e 
ψ   1 − κ 1ρ 

   κ ϕ 2 
(10)
1 e
κ 1  1 +

 1 − κ ρ   


1



1
.
λm,w =  γ −  (1 − γ ) 
ψ
1
−
κ
ν
2

(
)


1 1





The risk compensation for the ηt +1 shocks is
very standard, and λm,η equals the risk aversion
parameter, γ . In the special case of power utility,
γ = 1/ψ, the risk compensation parameters λm,e
and λm,w are zero. Long-run risks and volatility
are priced only when the agent is not indifferent
to the timing of the uncertainty resolution for the
consumption path—that is, when risk aversion
is different from the reciprocal of the IES. For this
to be the case, γ should be larger than 1/ψ. The
market prices of long-run and volatility risks are
sensitive to the magnitude of the permanence
parameter, ρ, as well. The risk compensation for
long-run risks and volatility risks rises as the
permanence parameter, ρ, rises.
The equity premium in the presence of timevarying economic uncertainty is
(11)

(

)

E t rm, t +1 − rf, t =

βm,η λm,ησ t2 + βm,e λm,eσ t2 + βm,w λm,w σ w2 − 0.5Vart ( rm, t +1 ).

The first beta corresponds to the exposure to
short-run risks and the second to long-run risks.
The third beta (that is, βm,w ) captures the return’s
exposure to volatility risks. It is important to note
that all the betas in this general equilibrium framework are endogenous. They are completely pinned
down by the dynamics of the asset’s dividends
and the preferences parameters of the agent. The
quantitative magnitude of the betas and the risk
premium for the consumption claim is discussed
below.
The risk premium on the market portfolio is
time varying as σt fluctuates. The ratio of the
conditional risk premium to the conditional
volatility of the market portfolio fluctuates with
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

σt and therefore the Sharpe ratio is time varying.
The maximal Sharpe ratio in this model, which
approximately equals the conditional volatility of
the log IMRS, also varies with σt. During periods
of high economic uncertainty (i.e., consumption
volatility), all risk premia rise.
The first-order effects on the level of the riskfree rate, as discussed in Bansal and Yaron (2005),
are the rate of time preference and the average
consumption growth rate divided by the IES.
Increasing the IES keeps the level low. The variance of the risk-free rate is determined by the
volatility of the expected consumption growth
rate and the IES. Increasing the IES lowers the
volatility of the risk-free rate. In addition, incorporating economic uncertainty leads to an interesting channel for interpreting fluctuations in the
real risk-free rate. In addition, this has serious
implications for the measurement of the IES in
the data, which heavily relies on the link between
the risk-free rate and expected consumption
growth. In the presence of varying volatility, the
estimates of the IES based on the projections
considered in Hall (1988) and Campbell (1999)
are seriously biased downward.
Hansen, Heaton, and Li (2005) also consider
a long-run risks model specification where the IES
is pinned at 1. This specific case affords considerable simplicity, as the wealth-to-consumption
ratio is constant. To solve the model at values of
the IES that differ from 1, the authors provide
approximations around the case where the IES is
1. Bansal, Kiku, and Yaron (2006) provide an
alternative approximate solution that relies on
equation (4); they show how to derive the return
ra,t along with the endogenous approximating
constants, κ 1 and κ 0 , for any configuration of
preferences parameters.

DATA AND MODEL IMPLICATIONS
Data and Growth Rate Dynamics
BY calibrate the model described in (5) at the
monthly frequency. From this monthly model
they derive time-aggregated annual growth rates
of consumption and dividends to match key
aspects of annual aggregate consumption and
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Bansal

Table 1
Time-Series Properties of Data
Estimate

Standard error

σ (g)

2.93

(0.69)

AC(1)

0.49

(0.14)

AC(2)

0.15

(0.22)

AC(5)

–0.08

(0.10)

AC(10)

0.05

(0.09)

VR(2)

1.61

(0.34)

VR(5)

2.01

(1.23)

VR(10)

1.57

(2.07)

σ (gd)

11.49

(1.98)

AC(1)

0.21

(0.13)

corr(g,gd)

0.55

(0.34)

Variable

NOTE: This table displays the time-series properties of aggregate
consumption and dividend growth rates: g and gd , respectively.
The statistics are based on annual observations from 1929 to
1998. Consumption is real per capita consumption of nondurables and services; dividends are the sum of real dividends
across all CRSP firms. AC( j ) is the jth autocorrelation, VR( j ) is
the jth variance ratio, σ is the volatility, and corr denotes the
correlation. Standard errors are Newey and West (1987)–
corrected using 10 lags.

dividends data. For consumption, Bureau of
Economic Analysis data on real per capita annual
consumption growth of non-durables and services
for the period 1929-98 is used. Dividends and
the value-weighted market return data are taken
from the Center for Research in Security Prices
(CRSP). All nominal quantities are deflated using
the consumer price index.
The mean annual real per capita consumption
growth rate is 1.8 percent, and its standard deviation is about 2.9 percent. Table 1, adapted from
BY, shows that, in the data, consumption growth
has a large first-order autocorrelation coefficient
and a small second-order coefficient. The standard
errors in the data for these autocorrelations are
sizeable. An alternative way to view the longhorizon property of the consumption and dividend
growth rates is to use variance ratios, which are
themselves determined by the autocorrelations
(Cochrane, 1988). In the data, the variance ratios
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first rise significantly and at a horizon of about
seven years start to decline. The standard errors
on these variance ratios, not surprisingly, are
quite substantial.
In terms of the specific parameters for the
consumption dynamics, BY calibrate ρ at 0.979,
which determines the persistence in the long-run
component in growth rates. Their choice of ϕe and
σ ensures that the model matches the unconditional variance and the autocorrelation function
of annual consumption growth. The standard deviation of the innovation in consumption equals
0.0078. This parameter configuration implies that
the predictable variation in monthly consumption
growth is very small—the implied R 2 is only 4.4
percent. The exposure of the corporate sector to
long-run risks is governed by φ, and its magnitude
is similar to that in Abel (1999). The standard
deviation of the monthly innovation in dividends,
ϕd σ , is 0.0351. The parameters of the volatility
process are chosen to capture the persistence in
consumption volatility. Based on the evidence of
slow decay in volatility shocks, BY calibrate ν1,
the parameter governing the persistence of conditional volatility, at 0.987. The shocks to the
volatility process have very small volatility; σw is
calibrated at 0.23 × 10–5. At the calibrated parameters, the modeled consumption and dividend
growth rates very closely match the key consumption and dividends data features reported in
Table 1. Bansal, Gallant, and Tauchen (2007)
provide simulation-based estimation evidence
that supports this configuration as well.
Table 2 presents the targeted asset market
data for 1929-98. The equity risk premium is 6.33
percent per annum, and the real risk-free rate is
0.86 percent. The annual market return volatility
is 19.42 percent, and that of the real risk-free rate
is quite small, about 1 percent per annum. The
volatility of the price-dividend ratio is quite high,
and it is a very persistent series. In addition to
these data dimensions, BY also evaluate the ability
of the model to capture predictability of returns.
Bansal, Khatchatrian, and Yaron (2005) show
that, consistent with the implications of the BY
model, price-dividend ratios are negatively correlated with consumption volatility at long leads
and lags.
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Bansal

It is often argued that, in the data, consumption and dividend growth are close to being i.i.d.
BY show that their model of consumption and
dividends is also consistent with the observed
data on consumption and dividends growth rates.
However, although the financial market data are
hard to interpret from the perspective of the i.i.d.
growth rate dynamics, BY show that it is interpretable from the perspective of the growth rate
dynamics that incorporate long-run risks. Given
these difficulties in discrimination across these
two models, Hansen and Sargent (2006) use features of the long-run model for developing economic models where agents update their model
beliefs in a manner that incorporates robustness
against model misspecification.

Preference Parameters
The preference parameters take account of
economic considerations. The time preference
parameter, δ < 1, and the risk aversion parameter,
γ , is either 7.5 or 10. Mehra and Prescott (1985)
argue that a reasonable upper bound for risk
aversion is around 10. The IES is set at 1.5: An
IES value that is not less than 1 is important for
the quantitative results.
There is considerable debate about the magnitude of the IES. Hansen and Singleton (1982)
and Attanasio and Weber (1989) estimate the IES
to be well in excess of 1. More recently, Guvenen
(2001) and Vissing-Jorgensen and Attanasio (2003)
also estimate the IES over 1; they show that their
estimates are close to that used in BY. However,
Hall (1988) and Campbell (1999) estimate the IES
to be well below 1. BY argue that the low IES
estimates of Hall and Campbell are based on a
model without time-varying volatility. They show
that ignoring the effects of time-varying consumption volatility leads to a serious downward bias
in the estimates of the IES. If the population value
of the IES in the BY model is 1.5, then the estimated value of the IES using Hall estimation
methods will be less than 0.3. With fluctuating
consumption volatility, the projection of consumption growth on the level of the risk-free rate does
not equal the IES, leading to the downward bias.
This suggests that Hall’s and Campbell’s estimates
may not be a robust guide for calibrating the IES.
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Table 2
Asset Market Data
Estimate

Standard error

E(rm – rf )

6.33

(2.15)

E(rf )

0.86

(0.42)

σ (rm )

19.42

(3.07)

σ (rf )

0.97

(0.28)

Variable
Returns

Price-dividend ratio
E(exp(p – d))

26.56

(2.53)

σ (p – d)

0.29

(0.04)

AC1(p – d)

0.81

(0.09)

AC2(p – d)

0.64

(0.15)

NOTE: This table presents descriptive statistics of asset market
data. The moments are calculated using annual observations
from 1929 through 1998. E(rm – rf ) and E(rf ) are, respectively,
the annualized equity premium and mean risk-free-rate; σ (rm),
σ (rf ), and σ ( p – d ) are, respectively, the annualized volatilities
of the market return, risk-free rate, and log price-dividend ratio;
AC1 and AC2 denote the first and second autocorrelations.
Standard errors are Newey and West (1987)–corrected using
10 lags.

In addition to the above arguments, the empirical evidence in Bansal, Khatchatrian, and Yaron
(2005) shows that a rise in consumption volatility
sharply lowers asset prices at long leads and lags,
and high current asset valuations forecast higher
future corporate earnings growth. Figures 1
through 4 use data from the United States, United
Kingdom, Germany, and Japan to evaluate the
volatility channel. The asset valuation measure
is the price-to-earnings ratio, and the consumption volatility measure is constructed by averaging
eight lags of the absolute value of consumption
residuals. It is evident from the figures that a rise
in consumption volatility lowers asset valuations
for all countries under consideration; this highlights the volatility channel and motivates the
specification of an IES larger than 1. In a twocountry extension of the model, Bansal and
Shaliastovich (2007) show that dollar prices of
foreign currency and forward premia co-move
negatively with the consumption volatility differential, whereas the ex ante currency returns
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Bansal

Figure 1

Figure 2

P/E Ratio and Consumption Volatility:
United States

P/E Ratio and Consumption Volatility:
United Kingdom

3

3

Consumption Volatility (12 quarters)

Consumption Volatility (12 quarters)

Log Price-Earnings Ratio

2

Log Price-Earnings Ratio

2

1

1

0

0

–1

–1

–2

–2

–3

–3

55

60

65

70

75

80

85

90

95

76

78

80

82

84

86

88

90

92

94

96

98

NOTE: This figure plots consumption volatility along with the
logarithm of the price-earnings ratio for the United States. Both
series are standardized.

NOTE: This figure plots consumption volatility along with the
logarithm of the price-earnings ratio for the United Kingdom.
Both series are standardized.

Figure 3

Figure 4

P/E Ratio and Consumption Volatility:
Germany

P/E Ratio and Consumption Volatility:
Japan

3

3

Consumption Volatility (12 quarters)

Consumption Volatility (12 quarters)

Log Price-Earnings Ratio

2

Log Price-Earnings Ratio
2

1
1
0
0
–1
–1

–2
–3

–2
76

78

80

82

84

86

88

90

92

94

96

98

NOTE: This figure plots consumption volatility along with the
logarithm of the price-earnings ratio for Germany. Both series
are standardized.

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76

78

80

82

84

86

88

90

92

94

96

98

NOTE: This figure plots consumption volatility along with the
logarithm of the price-earnings ratio for Japan. Both series are
standardized.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Bansal

Table 3
Risk Components and Risk Compensation
ψ = 0.1

ψ = 0.5

ψ = 1.5

mprη

93.60

93.60

93.60

mpre

0.00

137.23

160.05

mprw

0.00

–27.05

–31.56

βη

1.00

1.00

1.00

βe

–16.49

–1.83

0.61

βw

11,026.45

1,225.16

–408.39

prmη

0.73

0.73

0.73

prme

0.00

–1.96

0.76

prmw

0.00

–0.08

0.03

NOTE: This table presents model-implied components of the risk premium on the consumption asset for different values of the
intertemporal elasticity of substitution parameter, ψ. All entries are based on γ = 10. The parameters that govern the dynamics of the
consumption process in equation (5) are identical to those in Bansal and Yaron (2004): ρ = 0.979, σ = 0.0078, ϕe = 0.044, ν1 = 0.987,
σw = 0.23 × 10–5, and κ1 = 0.997. The first three rows report the annualized percentage prices of risk for innovations in consumption,
the expected growth risks, and the consumption volatility risks—mprη , mpre , and mprw , respectively. These prices of risks correspond
to annualized percentage values for λm,η σ, λm,e σ, λm,wσw in equation (9). The exposures of the consumption asset to the three systematic
risks, βη , βe , and βw , are presented in the middle part of the table. Total risk compensation in annual percentage terms for each risk
is reported as prm* and equals the product of the price of risk, the standard deviation of the shock, and the beta for the specific risk,
respectively.

have positive correlations with it. This provides
further empirical support for a magnitude of the
IES. In terms of growth rate predictability, Ang
and Bekaert (2007) and Bansal, Khatchatrian, and
Yaron (2005) report a positive relation between
asset valuations and expected earnings growth.
These data features, as discussed in the theory
sections above, again require an IES larger than 1.

Asset Pricing Implications
To underscore the importance of two key
aspects of the model, preferences and long-run
risks, first consider the genesis of the risk premium on ra,t+1—the return on the asset that delivers aggregate consumption as its dividends. The
determination of risk premia for other dividend
claims follows the same logic.
Table 3 shows the market price of risk and
the breakdown of the risk premium from various
risk sources. Column 1 considers the case of
power utility where the IES equals the reciprocal
of the risk aversion parameter. As discussed earF E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

lier, the prices of long-run and volatility risks are
zero. In the power utility case, the main risk is
the short-run risk and the risk premium on the
consumption asset equals γ σ 2, which is 0.7 percent per annum.
Column 2 of Table 3 considers the case with
an IES less than 1 (set at 0.5). For long-run growth
rate risks, the price of risk is positive; for volatility
risks, the price of risk is negative, as γ is larger
than the reciprocal of the IES. However, the consumption asset’s beta for long-run risks (beta with
regard to the innovations in xt+1) is negative. This,
as discussed earlier, is because A1 is negative
(see equation (6)), implying that a rise in expected
growth lowers the wealth-to-consumption ratio.
Consequently, long-run risks in this case contribute a negative risk premium of –1.96 percent
per annum. The market price of volatility risks is
negative and small; however, the asset’s beta for
this risk source is large and positive, reflecting
the fact that asset prices rise when economic
uncertainty rises (see equation (7)). In all, when
the IES is less than 1, the risk premium on the
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consumption asset is negative, which is highly
counterintuitive, and highlights the implausibility
of this parameter configuration.
Column 3 of Table 3 shows that when the
IES is larger than 1, the price of long-run growth
risks rises. More importantly, the asset’s beta for
long-run growth risks is positive and that for
volatility risks is negative. Both these risk sources
contribute toward a positive risk premium. The
risk premium from long-run growth is 0.76 percent and that for the short-run consumption
shock is 0.73 percent. The overall risk premia for
this consumption asset is 1.52 percent. This evidence shows that an IES larger than 1 is required
for the long-run and volatility risks to carry a
positive risk premium.
It is clear from Table 3 that the price of risk is
highest for the long-run risks (see columns 2 and
3) and smallest for the volatility risks. A comparison of columns 2 and 3 also shows that raising the
IES increases the prices of long-run and volatility
risks in absolute value. The magnitudes reported
in Table 3 are with ρ = 0.979—lowering this persistence parameter also lowers the prices of longrun and volatility risks (in absolute value).
Increasing the risk aversion parameter increases
the prices of all consumption risks. Hansen and
Jagannathan (1991) document the importance of
the maximal Sharpe ratio, determined by the
volatility of the IMRS, in assessing asset pricing
models. Incorporating long-run risks increases
the maximal Sharpe ratio in the model, which
easily satisfies the non-parametric bounds of
Hansen and Jagannathan (1991).
The risk premium on the market portfolio
(i.e., the dividend asset) is also affected by the
presence of long-run risks. To underscore their
importance, assume that consumption and dividend growth rates are i.i.d. This shuts off the
long-run risks channel. The market risk premium
in this case is
(12)

(

)

E t rm, t +1 − rf, t =

γ cov ( gt +1, gd , t +1) − 0.5 Var ( gd , t +1),
and market return volatility equals the dividend
growth rate volatility. If shocks to consumption
and dividends are uncorrelated, then the geo294

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metric risk premium is negative and equals
–0.5Var共gd,t+1兲. If the correlation between monthly
consumption and dividend growth is 0.25, then
the equity premium is 0.08 percent per annum,
which is similar to the evidence documented in
Mehra and Prescott (1985) and Weil (1989).
BY show that their model, which incorporates
long-run growth rate risks and fluctuating economic uncertainty, provides a very close match
to the asset market data reported in Table 2. That
is, the model can account for the low risk-free rate,
high equity premium, high asset price volatility,
and low risk-free rate volatility. The BY model
matches additional data features, such as (i) predictability of returns at short and long horizons
using dividend yield as a predictive variable, (ii)
time-varying and persistent market return volatility, (iii) negative correlation between market return
and volatility shocks, i.e., the volatility feedback
effect, and the (iv) negative relation between
consumption volatility and asset prices at long
leads and lags. (Also see Bansal, Khatchatrian,
and Yaron, 2005.)

Cross-Sectional Implications
Table 4, shows that there are sizable differences in mean real returns across portfolios sorted
by book-to-market ratio, size, and momentum for
quarterly data from 1967 to 2001. The size and
book-to-market sorts place firms into different
deciles once per year, and the subsequent return
on these portfolios is used for empirical work.
For momentum assets, CRSP-covered New York
Stock Exchange and American Stock and Options
Exchange stocks are sorted on the basis of their
cumulative return over months t –12 through
t –1. The loser portfolio (M1) includes firms with
the worst performance over the past year, and the
winner portfolio (M10) includes firms with the
best performance. The data show that subsequent
returns on these portfolios have a large spread
(i.e., M10 return – M1 return), about 4.62 percent
per quarter: This is the momentum spread puzzle.
Similarly, the highest book-to-market firms (B10)
earn average real quarterly returns of 3.27 percent,
whereas the lowest book-to-market (B1) firms
average 1.54 percent per quarter. The value spread
(return on B10 – return on B1) is about 2 percent
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Bansal

Table 4
Portfolio Returns
Mean

Standard
deviation

Mean

Standard
deviation

Mean

Standard
deviation

S1

0.0230

0.1370

B1

0.0154

0.1058

M1

–0.0104

0.1541

S2

0.0231

0.1265

B2

0.0199

0.0956

M2

0.0070

0.1192

S3

0.0233

0.1200

B3

0.0211

0.0921

M3

0.0122

0.1089

S4

0.0233

0.1174

B4

0.0218

0.0915

M4

0.0197

0.0943

S5

0.0242

0.1112

B5

0.0200

0.0798

M5

0.0135

0.0869

S6

0.0207

0.1050

B6

0.0234

0.0813

M6

0.0160

0.0876

S7

0.0224

0.1041

B7

0.0237

0.0839

M7

0.0200

0.0886

S8

0.0219

0.1001

B8

0.0259

0.0837

M8

0.0237

0.0825

S9

0.0207

0.0913

B9

0.0273

0.0892

M9

0.0283

0.0931

S10

0.0181

0.0827

B10

0.0327

0.1034

M10

0.0358

0.1139

NOTE: This table presents descriptive statistics for the returns on the 30 characteristic-sorted decile portfolios. Value-weighted returns
are presented for portfolios formed on market capitalization (S), book-to-market ratio (B), and momentum (M). M1 represents the
lowest momentum (loser) decile, S1 the lowest size (small firms) decile, and B1 the lowest book-to-market decile. Data are converted to
real values using the personal consumption expenditure deflator, are sampled at the quarterly frequency, and cover 1967:Q1–2001:Q4.

per quarter: This is the value spread puzzle. What
explains these big differences in mean returns
across portfolios?
Bansal, Dittmar, and Lundblad (2002 and
2005) connect systematic risks to cash-flow risks.
They show that an asset’s risk measure (i.e., its
beta) is determined by its cash-flow properties. In
particular, their paper shows that cross-sectional
differences in asset betas mostly reflect differences
in systematic risks in cash flows. Hence, systematic risks in cash flows ought to explain differences
in mean returns across assets. They develop two
ways to measure the long-run risks in cash flows.
First they model dividend and consumption
growth rates as a VAR and measure the discounted
impulse response of the dividend growth rates to
consumption innovations. This is one measure
of risks in cash flows. Their second measure is
based on stochastic cointegration, which is estimated by regressing the log level of dividends
for each portfolio on a time trend and the log
level of consumption. Specifically, consider the
projection

dt = τ (0) + τ (1)t + τ (2)ct + ζt ,
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

where the projection coefficient, τ 共2兲, measures
the long-run consumption risk in the asset’s dividends. The coefficient τ 共2兲 will be different for
different assets.
Bansal, Dittmar, and Lundblad (2002 and
2005) show that the exposure of dividend growth
rates to the long-run component in consumption
has considerable cross-sectional explanatory
power. That is, dividends’ exposure to long-run
consumption risks is an important explanatory
variable in accounting for differences in mean
returns across portfolios. Portfolios with high
mean returns also have higher dividend exposure
to consumption risks. The cointegration-based
measure of risk, τ2, also provides very valuable
information about mean returns on assets. The
cross-sectional R 2 from regressing the mean
returns on the dividend-based risk measures is
well over 65 percent. In contrast, other approaches
find it quite hard to explain the differences in
mean returns for the 30-asset menu used in
Bansal, Dittmar, and Lundblad (2005). The standard consumption betas (i.e., C-CAPM) and the
market-based CAPM asset betas have close to
zero explanatory power. The R 2 for the C-CAPM
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Bansal

is 2.7 percent, and that for the market CAPM is
6.5 percent, with an implausible negative slope
coefficient. The Fama and French three-factor
empirical specification also generates point estimates with negative, and difficult to interpret,
prices of risk for the market and size factors; the
cross-sectional R 2 is about 36 percent. Compared
with all these models, the cash-flow risks model
of Bansal, Dittmar, and Lundblad (2005) is able
to capture a significant portion of the differences
in risk premia across assets. Hansen, Heaton, and
Li (2005) inquire about the robustness of the stochastic cointegration-based risk measures considered in Bansal, Dittmar, and Lundblad (2002).
They argue that the dividend-based consumption
betas—particularly, the cointegration-based risk
measures—are imprecisely estimated in the time
series. Interestingly, across the different estimation procedures, the cash-flow beta risk measures
across portfolios line-up closely with the average
returns across assets. That is, in the cross-section
of assets (as opposed to the time series), the price
of risk associated with the long-run risks measures
is reliably significant.
Bansal, Dittmar, and Kiku (2006) derive new
results that link this cointegration parameter to
consumption betas by investment horizon and
evaluate the ability of their model to explain differences in mean returns for different horizons.
They provide new evidence regarding the robustness of the stochastic cointegration-based measures of permanent risks in equity markets. Parker
and Julliard (2005) evaluate whether long-run
risks in aggregate consumption can account for
the cross-section of expected returns. Malloy,
Moskowitz, and Vissing-Jorgensen (2005) evaluate whether long-run risks in stockholders’ consumption relative to aggregate consumption has
greater ability to explain the cross-section of
equity returns, relative to aggregate consumption
measures.

Term Structure and Currency Markets
Colacito and Croce (2006) consider a twocountry version of the BY model. They show that
this model can account for the low correlation in
consumption growth across countries but high
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correlation in marginal utilities across countries
(high risk sharing despite a low measured crosscountry consumption correlation). This feature
of international data is highlighted in Brandt,
Cochrane, and Santa-Clara (2006). The key idea
that Colacito and Croche pursue is that the longrun risks component is very similar across countries, but in the short-run consumption growth
can be very different. That is, countries share very
similar long-run prospects, but in the short-run
they can look very different. This dimension, they
show, is sufficient to induce high correlation in
marginal utilities across countries. It also accounts
for high real exchange volatility.
BY derive implications for the real term
structure of interest rates for the long-run risks
model. More recent papers by Eraker (2006) and
Piazzesi and Schneider (2005) also consider the
quantitative implications for the nominal term
structure using the long-run risks model. Bansal
and Shaliastovich (2007) show that the BY model
can simultaneously account for the upwardsloping terms structure, the violations of the
expectations hypothesis in the bond markets,
the violations in the foreign currency markets,
and the equity returns. This evidence indicates
that the long-run risks model provides a solid
baseline model for understanding financial markets. With simple modifications the model can
be used to analyze the impact of changing shortterm interest rates on financial markets; that is,
it can help in designing policy.

CONCLUSION
The work of Bansal and Lundblad (2002),
Bansal and Yaron (2004), and Bansal, Dittmar,
and Lundblad (2005) shows that the long-run
risks model can help interpret several features
of financial markets. These papers argue that
investors care about the long-run growth prospects
and the uncertainty (time-varying consumption
volatility) surrounding the growth rate. Risks
associated with changing long-run growth
prospects and varying economic uncertainty
drive the level of equity returns, asset price volatility, risk premia across assets, and predictability
of returns in financial markets.
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Bansal

Recent papers indicate that the channel in
this model can account for nominal yield curve
features, such as the violation the expectations
hypothesis and the average upward-sloping nominal yield curve. Evidence presented in Colacito
and Croce (2006) and Bansal and Shaliastovich
(2007) shows that the model also accounts for key
aspects of foreign exchange markets.
Growing evidence suggests that the long-run
risks model can explain a rich array of financial
market facts. This suggests that the model can be
used to analyze the impact of economic policy
on financial markets.

REFERENCES
Abel, Andrew. “Risk Premia and Term Premia in
General Equilibrium.” Journal of Monetary
Economics, 1999, 43(1), pp. 3-33.
Ang, Andrew and Bekaert, Geert. “Stock Return
Predictability: Is It There?” Review of Financial
Studies, May 2007, 20(3), pp. 651-707.
Attanasio, Orazio P. and Weber, Guglielmo.
“Intertemporal Substitution, Risk Aversion, and
the Euler Equation for Consumption.” Economic
Journal, March 1989, 99(Suppl 395), pp. 59-73.
Bansal, Ravi; Dittmar, Robert F. and Kiku, Dana.
“Cointegration and Consumption Risks in Asset
Returns.” Forthcoming in Review of Financial
Studies, 2007.

Bansal, Ravi; Khatchatrian, Varoujan and Yaron,
Amir. “Interpretable Asset Markets.” European
Economic Review, April 2005, 49(3), pp. 531-60.
Bansal, Ravi; Kiku, Dana and Yaron, Amir. “Risks for
the Long-Run: Estimation and Inference.” Working
paper, Duke University, 2006.
Bansal, Ravi and Lundblad, Christian T. “Market
Efficiency, Asset Returns, and the Size of the Risk
Premium in Global Equity Markets.” Journal of
Econometrics, August 2002, 109(2), pp. 195-237.
Bansal, Ravi and Shaliastovich, Ivan. “Risk and
Return in Bond, Currency and Equity Markets:
A Unified Approach.” Working paper, Duke
University, 2007.
Bansal, Ravi and Yaron, Amir. “Risks for the LongRun: A Potential Resolution of Asset Pricing
Puzzles.” Journal of Finance, August 2004, 59(4),
pp. 1481-509.
Bansal, Ravi and Yaron, Amir. “The Asset Pricing
Macro Nexus and Return Cash Flow Predictability.”
Working paper, Duke University, 2005.
Brandt, Michael W.; Cochrane, John H. and SantaClara, Pedro. “International Risk Sharing Is Better
Than You Think (or Exchange Rates Are Much Too
Smooth).” Journal of Monetary Economics, May
2006, 53(4), pp 671-98.

Bansal, Ravi; Dittmar, Robert F. and Lundblad,
Christian T. “Interpreting Risk Premia Across Size,
Value and Industry Portfolios.” Working paper,
Duke University, 2002.

Campbell, John Y. “Asset Prices, Consumption and
the Business Cycle,” in John B. Taylor and Michael
Woodford, eds., Handbook of Macroeconomics.
Volume 1. Amsterdam: Elsevier Science, North
Holland, 1999.

Bansal, Ravi; Dittmar, Robert F. and Lundblad,
Christian T. “Consumption, Dividends, and the
Cross-Section of Equity Returns.” Journal of Finance,
August 2005, 60(4), pp. 1639-72.

Campbell, John Y. and Shiller, Robert J. “The
Dividend-Price Ratio and Expectations of Future
Dividends and Discount Factors.” Review of
Financial Studies, 1988, 1(3), pp. 195-228.

Bansal, Ravi; Gallant, A. Ronald and Tauchen,
George. “Rational Pessimism, Rational Exuberance,
and Markets for Macro Risks.” Forthcoming in
Review of Economic Studies, 2007.

Cecchetti, Stephen G.; Lam, Pok-Sang and Mark,
Nelson C. “The Equity Premium and the Risk-Free
Rate: Matching the Moments.” Journal of Monetary
Economics, February 1993, 31(1), pp. 21-46.

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Chen, Hui. “Macroeconomic Conditions and the
Puzzles of Credit Spreads and Capital Structure.”
Working paper, University of Chicago, 2006.
Chen, Long; Collin-Dufresne, Pierre and Goldstein,
Robert S. “On the Relation Between the Credit
Spread Puzzle and the Equity Premium Puzzle.”
Working paper, University of California–Berkeley,
2006.
Cochrane, John H. “How Big Is the Random Walk in
GNP?” Journal of Political Economy, October 1988,
96(5), pp. 893-920.
Colacito, Riccardo and Croce, Mariano M. “Risks for
the Long-Run and the Real Exchange Rate.”
Working paper, University of North Carolina, 2006.
Epstein, Larry G. and Zin, Stanley E. “Substitution,
Risk Aversion and the Temporal Behavior of
Consumption and Asset Returns: A Theoretical
Framework.” Econometrica, July 1989, 57(4),
pp. 937-69.
Eraker, Bjorn. “Affine General Equilibrium Models.”
Working paper, Duke University, 2006.
Fama, Eugene F. “Forward and Spot Exchange
Rates.” Journal of Monetary Economics, November
1984, 14(3), pp. 319-38.
Fama, Eugene F. and Bliss, Robert R. “The Information
in Long-Maturity Forward Rates.” American
Economic Review, September 1987, 77(4), pp. 680-92.
Guvenen, Fatih. “Mismeasurement of the Elasticity
of Intertemporal Substitution: The Role of Limited
Stock Market Participation.” Working paper,
University of Rochester, 2001.
Hall, Robert E. “Intertemporal Substitution in
Consumption.” Journal of Political Economy, April
1988, 96(2), pp. 339-57.
Hansen, Lars P.; Heaton, John C. and Li, Nan.
“Consumption Strikes Back? Measuring Long-Run
Risk.” Working paper, University of Chicago, 2005.
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Economies.” Journal of Political Economy, April
1991, 99(2), pp. 225-62.
Hansen, Lars P. and Sargent, Thomas. “Fragile Beliefs
and the Price of Model Uncertainty.” Working
paper, University of Chicago, 2006.
Hansen, Lars P. and Singleton, Kenneth. “Generalized
Instrumental Variables Estimation of Nonlinear
Rational Expectations Models.” Econometrica,
September 1982, 50(5), pp. 1269-86.
Kiku, Dana. “Is the Value Premium a Puzzle?”
Working paper, University of Pennsylvania, 2006.
Lettau, Martin; Ludvigson, Sydney and Wachter,
Jessica. “The Declining Equity Premium: What
Role Does Macroeconomic Risk Play?” Forthcoming
in Review of Financial Studies, 2007.
Lochstoer, Lars A. and Kaltenbrunner, Georg. “LongRun Risk through Consumption Smoothing.”
Working paper, London Business School, 2006.
Lucas, Robert E. Jr. “Asset Prices in an Exchange
Economy.” Econometrica, November 1978, 46(6),
pp. 1429-46.
Malloy, Christopher J.; Moskowitz, Tobias J. and
Vissing-Jorgensen, Annette. “Long-Run Stockholder
Consumption Risk and Asset Returns.” Working
paper, University of Chicago, 2005.
Mehra, Rajnish and Prescott, Edward C. “The Equity
Premium: A Puzzle.” Journal of Monetary
Economics, March 1985, 15(2), pp. 145-61.
Parker, Jonathan and Julliard, Christian. “Consumption
Risk and the Cross-Section of Asset Returns.”
Journal of Political Economy, February 2005,
113(1), pp. 185-222.
Piazzesi, Monika and Schneider, Martin. “Equilibrium
Yield Curves.” NBER Working Paper No. 12609,
National Bureau of Economic Research, 2005.
Shiller, Robert J. “Do Stock Prices Move Too Much
To Be Justified by Subsequent Changes in
Dividends?” American Economic Review, July
1981, 71(3), pp. 421-36.

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Bansal

Vissing-Jorgensen, Annette and Attanasio, Orazio P.
“Stock Market Participation, Intertemporal
Substitution, and Risk-Aversion.” American
Economic Review (Papers and Proceedings), May
2003, 93(2), pp. 383-91.
Weil, Philippe. “The Equity Premium Puzzle and the
Risk-Free Rate Puzzle.” Journal of Monetary
Economics, November 1989, 24(3), pp. 401-21.

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Commentary
Thomas J. Sargent

I

n several recent papers—including the
paper from this conference, Bansal (2007)—
Ravi Bansal and his coauthors have constructed an interpretation of some asset
pricing puzzles that I think macroeconomists
should pay attention to. Why? Because a representative agent’s consumption Euler equation
that links a one-period real interest rate to the
consumption growth rate is the “IS curve” that
is central to the policy transmission mechanism
in today’s New Keynesian models. A long list
of empirical failures called puzzles come from
applying the stochastic discount factor implied
by that Euler equation. Until we succeed in getting a consumption-based asset pricing model
that works well, the New Keynesian IS curve is
built on sand.
In several exciting papers, Bansal and his
coauthors propose a way to explain some of those
asset pricing puzzles by (i) specifying the intertemporal structure of risks to put long-run risks
into consumption and assets’ cash flows and (ii)
altering preferences to make the representative
consumer care more about those long-run risks.

LONG-RUN RISK
Let ct be the logarithm of aggregate consumption. A workhorse model that does a good job of
fitting per capita U.S. consumption of nondurables
and services makes ct a random walk with constant drift and i.i.d. Gaussian innovations. Bansal

and his coauthors begin from the observation that
it is difficult to distinguish that specification
from an alternative one in which the drift in log
consumption growth is itself a highly persistent
covariance stationary process with low conditional volatility but high unconditional volatility.
Thus, the drift itself is almost but not quite a random walk. The high unconditional volatility of
the drift confronts the representative consumer
with what Bansal and his coauthors call long-run
risk because the conditional mean of consumption
growth is not constant but wanders.
Bansal also posits that cash flows on particular portfolios differ in the extent to which they are
subject to long-run risks that are more or less correlated with the long-run risk in aggregate consumption. For example, Bansal and coauthors as
well as Hansen, Heaton, and Li (2006) have offered
evidence that the cash flows from Fama and
French’s high book-to-market portfolios have longrun components that are more highly correlated
with long-run components of consumption than
are the cash flows from low book-to-market portfolios. Can the need to compensate the representative consumer for that higher long-run correlation
with consumption explain why those high bookto-market portfolios have higher returns?

PREFERENCES
The answer is no if the representative consumer’s preferences are usual ones assumed by

Thomas J. Sargent is a professor of economics and business at New York University, a senior fellow at the Hoover Institution, and the president
of the American Economic Association.
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 301-03.
© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
only with prior written permission of the Federal Reserve Bank of St. Louis.

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Sargent

macroeconomists—for example, time-separable
logarithmic preferences with discount factor
β 僆 (0, 1). Why? Those preferences lead to the
usual stochastic discount factor whose logarithm is

st +1, t = log β – ∆ ct +1,
which has the property that the representative
consumer just doesn’t care enough about those
long-run risks to pump up the returns on those
high book-to-market portfolios. Therefore, Bansal
and his coauthors adopt a preference specification of Epstein and Zin (1989) that separates the
intertemporal elasticity of substitution (IES) from
the reciprocal of the coefficient of relative risk
aversion, both of which are unity for the log
preference specification mentioned above. With
the IES held fixed at 1, those preferences imply a
stochastic discount factor whose logarithm is

st +1, t = log β – ∆ ct +1
`

– (γ –1) ∑ ( E t +1 – Et )∆ ct + j +1

risk aversion γ > 1 in terms of a representative
consumer who has IES and risk aversion both
equal to 1, but where now γ > 1 measures his
doubts about the stochastic specification of his
model for consumption growth and cash flows.
This reinterpretation is achieved by noting that
the continuation value in Epstein and Zin’s formulas equals the indirect utility function for a
robust valuation problem in which a malevolent
nature helps the decisionmaker construct valuations that are robust to misspecification by choosing a worst-case model from a set of models
surrounding the decisionmaker’s approximation
model. Now γ acquires the interpretation of a
penalty parameter on the relative entropy between
the approximating and the distorted model. The
additional terms in the log stochastic discount
factor that appear when γ > 1 encode the likelihood ratio of the worst case to the approximating
model.1 This reinterpretation is of special interest
for the work of Bansal and his coauthors, who
reason as follows:

j =0

1. Statistically, it is difficult to distinguish a
stochastic specification with long run-risk
from one without it.


 `
1
2
– (γ – 1) vart  ∑ β j ( Et +1 − E t ) ∆ ct + j +1 ,
2
 j =0

where γ is the coefficient of relative risk aversion
and Et denotes mathematical expectation conditioned on time-t information. Setting γ > 1 adds
forward-looking terms to the stochastic discount
factor that make the representative consumer care
today about rates of log consumption growth far
in the future. For a γ high enough, the representative consumer does have to be compensated for
long-run cash flow risk correlated with long-run
consumption growth risk in amounts that are big
enough to explain how the market prices those
Fama and French portfolios. Furthermore,
Tallarini (2000), Bansal and Yaron (2004) and
others have shown that with a high enough γ
this kind of preference specification can provide
a neat explanation for both the risk-free rate and
the equity premium puzzles.

It is possible to reinterpret the above stochastic discount factor with IES = 1 and atemporal
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3. Besides, by using the rational expectations
cross-equation restrictions associated with
the consumption Euler equation, we can
infer that the representative consumer has
to believe the long-run risk specification
to explain the asset pricing data.
There is more to say here. Although long-run
risks are difficult to detect (assertion 1), EpsteinZin preferences or concerns about robustness
make it vital for the representative consumer to
1

ASSESSMENT

302

2. Therefore, without attributing wacky ideas
to their representative consumer, Bansal
and coauthors can assume that the representative consumer assigns probability 1
to the long-run risk model and takes our
original i.i.d. log consumption growth
model off the table.

2007

Barillas, Hansen, and Sargent (2007) show that by interpreting γ
as measuring fear of model misspecification rather than riskaversion, a moderate fear of model misspecification can do most
of the job of the large risk-aversion parameters of Tallarini (2000)
and Bansal and Yaron (2004) in explaining the equity premium.

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Sargent

care about them. The tenuous part of this argument is how the representative consumer can
come to be sure about the presence of these longrun risks when they are so difficult to detect statistically. Assertion 3 differs from arguments in
the least-squares learning literature that typically
have an agent learn about a forcing process by
way of a least-squares learning algorithm on that
process itself, not by using prices and the rational
expectations cross-equation restrictions to reverse
engineer what that process must be.
The robustness interpretation can help with
these learning issues. Hansen and Sargent (2007)
address these in the context of a model with a
representative consumer who responds to assertion 1 by leaving both the i.i.d. and long-run risk
models for log consumption growth on the table,
attaching a prior initialized at the equal ignorance
value of 0.5 to the long-run risk model, then
updating by Bayes’ law. We show that a consumer
who distrusts both submodels and the posterior
over submodels that emerges from Bayes’ law
will behave in a way that supports much of what
Bansal and his coauthors do. In particular, because
the long-run risk model is worse for the representative consumer, his worst-case probabilities
become slanted toward that model and possibly
put almost all of the mass on that model. This is
nice because it provides an alternative defense
of Bansal’s assumption that the representative
consumer acts as if he puts probability 1 on the
long-run risk model.
But the structure of Hansen and Sargent (2007)
yields other interesting outcomes, too. Even when
the robust investor slants his worst-case probability to put probability 1 on the long-run risk model,
the gap between the ordinary Bayesian probability and this worst-case probability contributes a
potential source of time-varying countercyclical
risk premia.

easy to care about is worth taking seriously. When
many macroeconomists are now busy attaching
loosely interpreted shocks or wedges to agents’
first-order condition to make our dynamic stochastic general equilibrium (DSGE) models fit
better, I welcome Bansal’s new approach.

REFERENCES
Bansal, Ravi. “Long-Run Risks and Financial
Markets.” Federal Reserve Bank of St. Louis
Review, July/August 2007, 89(4), pp. 283-99.
Bansal, Ravi and Yaron, Amir. “Risks for the Long
Run: A Potential Resolution of Asset Pricing
Puzzles.” Journal of Finance, August 2004, 59(4),
pp. 1481-509.
Barillas, Francisco; Hansen, Lars Peter and Sargent,
Thomas J. “Doubts or Variability?” Working paper,
University of Chicago and New York University,
2007.
Epstein, Larry G. and Zin, Stanley E. “Substitution,
Risk Aversion, and the Temporal Behavior of
Consumption and Asset Returns: A Theoretical
Framework.” Econometrica, July 1989, 57(4),
pp. 937-69.
Hansen, Lars Peter; Heaton, John and Li, Nan.
“Consumption Strikes Back: Measuring Long Run
Risk.” Working paper, University of Chicago, 2006.
Hansen, Lars Peter and Sargent, Thomas J. “Fragile
Beliefs and the Price of Model Uncertainty.”
Unpublished manuscript, 2007.
Tallarini, Thomas D. Jr. “Risk-Sensitive Real Business
Cycles.” Journal of Monetary Economics, June
2000, 45(3), pp. 507-32.

CONCLUSION
Bansal and Yaron’s idea of stressing long-run
risks that are difficult to detect but, with EpsteinZin preferences or fear of model misspecification,

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Arbitrage-Free Bond Pricing with
Dynamic Macroeconomic Models
Michael F. Gallmeyer, Burton Hollifield, Francisco J. Palomino, and Stanley E. Zin
The authors examine the relationship between changes in short-term interest rates induced by
monetary policy and the yields on long-maturity default-free bonds. The volatility of the long end
of the term structure and its relationship with monetary policy are puzzling from the perspective
of simple structural macroeconomic models. The authors explore whether richer models of risk
premiums, specifically stochastic volatility models combined with Epstein-Zin recursive utility,
can account for such patterns. They study the properties of the yield curve when inflation is an
exogenous process and compare this with the yield curve when inflation is endogenous and
determined through an interest rate (Taylor) rule. When inflation is exogenous, it is difficult to
match the shape of the historical average yield curve. Capturing its upward slope is especially
difficult because the nominal pricing kernel with exogenous inflation does not exhibit any negative
autocorrelation—a necessary condition for an upward-sloping yield curve, as shown in Backus
and Zin. Endogenizing inflation provides a substantially better fit of the historical yield curve
because the Taylor rule provides additional flexibility in introducing negative autocorrelation
into the nominal pricing kernel. Additionally, endogenous inflation provides for a flatter term
structure of yield volatilities, which better fits historical bond data. (JEL G0, G1, E4)
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 305-26.

T

he response of long-term interest rates
to changes in short-term interest rates
is a feature of the economy that often
puzzles policymakers. For example, in
remarks made on May 27, 1994, Alan Greenspan
(1994, p. 5) expressed concern that long rates
moved too much in response to an increase in
short rates:
In early February, we had thought long-term
rates would move a little higher temporarily
as we tightened…The sharp jump in [long]
rates that occurred appeared to reflect the
dramatic rise in market expectations of economic growth and associated concerns about
possible inflation pressures.

Then in his February 16, 2005, testimony,
Chairman Greenspan (2005) expressed a completely different concern about long rates:
[L]ong-term interest rates have trended lower
in recent months even as the Federal Reserve
has raised the level of the target federal funds
rate by 150 basis points…Historically, though,
even these distant forward rates have tended
to rise in association with monetary policy
tightening...For the moment, the broadly
unanticipated behavior of world bond markets
remains a conundrum.

Chairman Greenspan’s comments reflect the
fact that we do not yet have a satisfactory under-

Michael F. Gallmeyer is an assistant professor of finance at the Mays Business School, Texas A&M University; Burton Hollifield is an associate
professor of financial economics at the David A. Tepper School of Business, Carnegie Mellon University; Francisco J. Palomino is a PhD candidate
at the David A. Tepper School of Business, Carnegie Mellon University; and Stanley E. Zin is a professor of economics at the David A. Tepper
School of Business, Carnegie Mellon University, and a research fellow at the National Bureau of Economic Research. The authors thank David
Backus and Pamela Labadie for valuable comments and suggestions, Bill Gavin for suggesting the topic, Monika Piazzesi and Chris Telmer
for providing data, and Zhanhui Chen for excellent research assistance.

© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
only with prior written permission of the Federal Reserve Bank of St. Louis.

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Gallmeyer, Hollifield, Palomino, Zin

Figure 1
Time-Series Properties of the Yield Curve, 1970:Q1–2005:Q4

Yields (% Continuously Compounded Annual Return [CCAR])

Nominal Yields

14

12

10

8

6

4

2

40
30
20
Maturity (Quarters)

10
2005

2000

standing of how the yield curve is related to the
structural features of the macroeconomy, such as
investors’ preferences, the fundamental sources
of economic risk, and monetary policy.
Figure 1 plots the nominal yield curve for a
variety of maturities from 1 quarter—which we
refer to as the short rate—up to 40 quarters for
U.S. Treasuries, starting in the first quarter of 1970
and ending in the last quarter of 2005.1 Figure 2
plots the average yield curve for the entire sample
and for two subsamples. Figure 3 plots the standard deviation of yields against their maturities.
Two basic patterns of yields are clear from these
figures: (i) On average, the yield curve is upward
1

Yields up to 1991 are from McCulloch and Kwon (1993) then
Datastream from 1991 to 2005.

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1995

1990

1985

1980

1975

1970

sloping and (ii) there is substantial volatility in
yields at all maturities. Chairman Greenspan’s
comments, therefore, must be framed by the fact
that long yields are almost as volatile as short
rates. The issue, however, is the relationship of
the volatility at the long end to the volatility at
the short end, and the correlation between changes
in short-term interest rates and changes in longterm interest rates.
We can decompose forward interest rates into
expectations of future short-term interest rates and
interest rate risk premia. Because long-term interest rates are averages of forward rates, long-run
interest rates depend on expectations of future
short-term interest rates and interest rate risk premiums. A significant component of long rates is
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Gallmeyer, Hollifield, Palomino, Zin

Figure 2
Average Yield-Curve Behavior
Mean Nominal Yields (% CCAR)
9
8.5
1970:Q1–1989:Q4

8
7.5
7

1970:Q1–2005:Q4

6.5
6
5.5
5

1990:Q1–2005:Q4
4.5
4
0

5

10

15

20

25

30

35

40

Maturity (Quarters)

the risk premium, and there is now a great deal
of empirical evidence documenting that the risk
premiums are time-varying and stochastic. Movements in long rates can therefore be attributed to
movements in expectations of future nominal
short rates, movements in risk premiums, or some
combination of movements in both.
Moreover, if monetary policy is implemented
using a short-term interest rate feedback rule—for
example, a Taylor rule—then inflation rates must
adjust so that the bond market clears. The resulting endogenous equilibrium inflation rate will
then depend on the same risk factors that drive
risk premiums in long rates. Monetary policy
itself, therefore, could be a source of fluctuations
in the yield curve in equilibrium.
We explore such possibilities in a model of
time-varying risk premiums generated by the
recursive utility model of Epstein and Zin (1989)
combined with stochastic volatility of endowment
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

growth. We show how the model can be easily
solved using now-standard affine term-structure
methods. Affine term-structure models have the
convenient property that yields are maturitydependent linear functions of state variables. We
examine some general properties of multi-period
default-free bonds in our model, assuming first
that inflation is an exogenous process and by
allowing inflation to be endogenous and determined by an interest rate feedback rule. We show
that the interest rate feedback rule—the form of
monetary policy—can have significant impacts on
properties of the term structure of interest rates.

THE DUFFIE-KAN AFFINE TERMSTRUCTURE MODEL
The Duffie and Kan (1996) class of affine termstructure models, translated into discrete time
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Gallmeyer, Hollifield, Palomino, Zin

Figure 3
Volatility of Yields of Various Maturities
Standard Deviation of Nominal Yields (% CCAR)
4

3.5
1970:Q1–2005:Q4
3

2.5

1970:Q1–1989:Q4

2

1.5

1990:Q1–2005:Q4

1

0.5

0
0

5

10

15

20
25
Maturity (Quarters)

by Backus, Foresi, and Telmer (2001), is based
on a k-dimensional vector of state variables z
that follows a “square root” model:
1/ 2

zt +1 = ( I − Φ )θ + Φzt + Σ ( zt )

ε t +1 ,

where {ε t } ~ NID共0, 1兲, Σ共z兲 is a diagonal matrix
with a typical element given by σi 共z兲 = ai + bi′z,
where bi has nonnegative elements, and Φ is stable
with positive diagonal elements. The process for
z requires that the volatility functions, σi 共z兲, be
positive, which places additional restrictions on
the parameters.
The asset-pricing implications of the model
are given by the pricing kernel, mt+1, a positive
random variable that prices all financial assets.
That is, if a security has a random payoff, ht+1, at
date t + 1, then its date-t price is Et [mt+1ht+1]. The
pricing kernel in the affine model takes the form
308

J U LY / A U G U S T

2007

30

35

40

1/ 2

−log mt +1 = δ + γ ′zt + λ ′ Σ ( zt )

εt + 1,

where the k × 1 vector γ is referred to as the
“factor loadings” for the pricing kernel, the k × 1
vector λ is referred to as the “price of risk” vector
because it controls the size of the conditional
correlation of the pricing kernel and the underlying sources of risk, and the k × k matrix Σ共zt 兲 is
the stochastic variance-covariance matrix of the
unforecastable shock.
Let bt共n兲 be the price at date t of a default-free
pure-discount bond that pays 1 at date t + n, with
bt共0兲 = 1. Multi-period default-free discount bond
prices are built up using the arbitrage-free pricing
restriction,
(1)

bt(n ) = Et  mt +1bt(+n1−1)  .

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Gallmeyer, Hollifield, Palomino, Zin

Bond prices of all maturities are log-linear functions of the state:

1/ ρ

−log bt(n ) = A(n ) + B (n ) zt ,
共n兲

the intertemporal utility function, Ut , is the solution to the recursive equation,
(3)

ρ
U t = (1 − β ) ctρ + βµt (U t +1 ) 



,

共n兲

where A is a scalar and B is a 1 × k row vector.
The intercept and slope parameters, which we
often refer to as “yield-factor loadings,” of these
bond prices can be found recursively according to
(2)
A( n+1) = A( n) + δ + B ( n ) ( I − Φ )θ −

2
1 k
λ j + B j( n) a j ,
∑
2 j =1

(

)

where 0 < β < 1 characterizes impatience (the
marginal rate of time preference is 1 – 1/β ), ρ ≤ 1
measures the preference for intertemporal substitution (the elasticity of intertemporal substitution for deterministic consumption paths is
1/共1 – ρ兲), and the certainty equivalent of random
future utility is
1

2
1 k
n
B ( n+1) = γ ′ + B ( n )Φ − ∑ λ j + B j( ) bj′ ,

(

)

2 j =1

(

(4)

)

Bj共n兲 is
共0兲

共n兲

where
the j th element of the vector B .
Because b = 1, we can start these recursions
using A共0兲 = 0 and Bj共0兲 = 0, j = 1,2,…,k.
Continuously compounded yields, yt共n兲, are
defined by bt共n兲 = exp共–nyt共n兲兲, which implies
yt共n兲 = –共log bt共n兲兲/n. We refer to the short rate, it , as
the one-period yield: it ⬅ yt共1兲.
This is an arbitrage-free model of bond pricing
because it satisfies equation (1) for a given pricing
kernel mt . It is not yet a structural equilibrium
model, because the mapping of the parameters of
the pricing model to deeper structural parameters
of investors’ preferences and opportunities has
not yet been specified. The equilibrium structural
models we consider will all lie within this general
class, hence, can be easily solved using these
pricing equations.

where α ≤ 1 measures static risk aversion (the
coefficient of relative risk aversion for static
gambles is 1 – α ). The marginal rate of intertemporal substitution, mt +1, is

c 
mt +1 = β  t +1 
 c 
t

We begin our analysis of structural models
of the yield curve by solving for equilibrium
real yields in a representative-agent exchange
economy. Following Backus and Zin (2006), we
consider a representative agent who chooses
consumption to maximize the recursive utility
function given in Epstein and Zin (1989). Given
a sequence of consumption, {ct,ct+1,ct+2,…}, where
future consumptions can be random outcomes,
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

ρ −1

 U t +1 
 µ (U ) 
 t t +1 

α −ρ

.

Time-additive expected utility corresponds to
the parameter restriction ρ = α .
In equilibrium, the representative agent consumes the stochastic endowment, et , so that,
log 共ct +1/ct 兲 = log 共et +1/et 兲 = xt +1, where xt +1 is the
log of the ratio of endowments in t + 1 relative to t .
The log of the equilibrium marginal rate of substitution, referred to as the real pricing kernel, is
therefore given by
(5)

A TWO-FACTOR MODEL WITH
EPSTEIN-ZIN PREFERENCES

µt (U t +1 ) ; Et U tα+1  α ,

log mt +1 = log β + ( ρ − 1) x t +1
+ (α − ρ )  log Wt +1 − log µt (Wt +1 ) ,

where Wt is the value of utility in equilibrium.
The first two terms in the marginal rate of
substitution are standard expected utility terms:
the pure time preference parameter, β, and a consumption growth term times the inverse of the
negative of the intertemporal elasticity of substitution. The third term in the pricing kernel is a new
term coming from the Epstein-Zin preferences.
The endowment-growth process evolves stochastically according to

x t +1 = (1 − φx )θ x + φx x t + v t1/ 2ε tx+1,

J U LY / A U G U S T

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Gallmeyer, Hollifield, Palomino, Zin

where

Given the state variables and the log-linear
structure of the model, we conjecture a solution
for the log value function of the form

v t +1 = (1 − φv )θv + φv v t + σ v εtv+1

is the process for the conditional volatility of
endowment growth. We will refer to vt as stochastic volatility. The innovations εtx and εtv are
distributed NID共0,1兲.
Note that the state vector in this model conforms with the setup of the Duffie-Kan model
above. Define the state vector zt ⬅ [xt vt ]′, which
implies parameters for the Duffie-Kan model:

θ = [θ x θv ]′

w t +1 + x t +1 = ω + (ω x + 1) x t +1 + ωv v t +1 .
Because xt +1 and vt +1 are jointly normally distributed, the properties of normal random variables
can be used to solve for ut :

( (

a1 = 0, b1 = [0 1]′, a2 = σ v2 , b2 = [ 0 0]′ .

Following the analysis in Hansen, Heaton, and
Li (2005), we will work with the logarithm of the
value function scaled by the endowment:
Wt / et = (1 − β ) + β µt (Wt +1 ) / et


(

)

))

1


α
= log  E t exp (w t +1 + x t +1 )  α 
 
 


Σ ( zt ) = diag {a1 + b1′zt , a2 + b2′zt }

ρ 1/ ρ

α
Vart w t +1 + x t +1 
2
= ω + (ω x + 1) (1 − φx )θ x + ωv (1 − φv )θv

= Et w t +1 + x t +1  +

+ (ω x + 1) φ x x t + ωv φv v t
+



α
(ω x + 1)2 v t + α ωv2σ v2 .
2
2

1// ρ

ρ

 W
et +1   
t +1

= (1 – β ) + β  µt 
×

  et +1
et   




,

where we have used the linear homogeneity of µt
(see equation (4)). Take logarithms of (6) to obtain
w t = ρ −1log (1 − β ) + β exp ( ρu t )  ,

where wt ⬅ log共Wt /et 兲 and ut ⬅ log 共 µt 共exp共wt +1 +
xt +1 兲兲兲. Consider a linear approximation of the
right-hand side of this equation as a function of
–:
ut around the point m

w t ≈ ρ −1log (1 − β ) + β exp ( ρm )

β exp ( ρm ) 
+
 (ut − m )
 1 − β + β exp ( ρm )
; κ + κ ut ,
where κ < 1. For the special case with ρ = 0, that
is, a log time aggregator, the linear approximation
is exact, implying κ– = 1 – β and κ = β (see Hansen,
Heaton, and Li, 2005). Similarly, approximating
– = 0, results in κ– = 0 and κ = β.
around m
310

where ω–, ωx , and ωv are constants to be determined. By substituting,

ut ; log µt exp (w t +1 + x t +1 )

Φ = diag {φx ,φv }

(6)

w t = ω + ω x x t + ωv v t ,

J U LY / A U G U S T

2007

We can use the above expression to solve for the
value-function parameters and verify its loglinear solution:

ω x = κ (ω x + 1)φx
 κ 
φx
⇒ ωx = 
 1 − κφx 

α
2

ωv = κ ωv φv + (ω x + 1) 
2


2
 κ  α  1  


⇒ ωv = 
 1 − κφv   2  1 − κφx  


(ω x + 1) (1 − φx )θ x

κ
κ 
.
+
ω =
1 − κ 1 − κ + ωv (1 − φv )θv + α ωv2σv2 

2

The solution allows us to simplify the term
[logWt +1 – log µt 共Wt +1兲] in the real pricing kernel
in equation (5):

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Gallmeyer, Hollifield, Palomino, Zin

log Wt +1 − log µt (Wt +1 ) = w t +1 + x t +1 − log µt (w t +1 + xt + 1 )
= (ω x + 1)  x t +1 − Et xt + 1  + ωv v t +1 − Et v t +1 
α
α
2
− (ω x + 1) Vart  x t +1  − ωv2 Vart v t +1 
2
2
α
α
2
v
1/ 2 x
= (ω x + 1) v t εt + 1 + ωv σ v ε t +1 − (ω x + 1) v t − ωv2σ v2 .
2
2

The real pricing kernel, therefore, is a member
of the Duffie-Kan class with two factors and
parameters:

δ = −log ( β ) + (1 − ρ ) (1 − φx )θ x +

α
(α − ρ )ωv2σ v2
2

that there is a frictionless conversion of money
for goods in this economy, the nominal kernel is
given by
(8)

)

(

log m$t +1 = log ( m t +1) − pt +1,

where pt +1 is the log of the money price of goods
at time t +1 relative to the money price of goods
at time t, that is, the inflation rate between t and
t +1. Clearly then, the source of inflation, its random properties, and its relationship to the real
pricing kernel is of central interest for nominal
bond pricing. We next consider two different
specifications for equilibrium inflation.

γ = [γ x γ v ]′

(7)

2 ′

 1  
α

= (1 − ρ )φx
(α − ρ ) 
2
 1 − κφx  



λ = [ λx λv ]′

 κφx  
(1 − α ) − (α − ρ ) 

 1 − κφx  

=
.
2
 −  α   κ (α − ρ )   1  
  2   1 − κφv   1 − κφx  



We can now use the recursive formulas in equation (2) to solve for real discount bond prices and
the real yield curve.
Note how the factor loadings and prices of
risk depend on the deeper structural parameters
and the greatly reduced dimensionality of the
parameter space relative to the general affine
model. Also, for the time-additive expected utility
special case, α = ρ, the volatility factor does not
enter the conditional mean of the pricing kernel
because γ v = 0; and the price of risk for the volatility factor is zero because λv = 0. Finally, we can
see from the expressions for bond prices that the
two key preference parameters, ρ and α , provide
freedom in controlling both the factor loadings
and the prices of risk in the real pricing kernel.

NOMINAL BOND PRICING
To understand the price of nominal bonds,
we need a nominal pricing kernel. If we assume
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Exogenous Inflation
If we expand the state space to include an
exogenous inflation process, pt , the state vector
becomes zt = [xt vt pt ]′. The stochastic process
for exogenous inflation is

(

)

pt +1 = 1 − φ p θ p + φp pt + σ pε tp+1 ,
P
where ε t+1
is also normally distributed independently of the other two shocks. In this case, the
parameters for the affine nominal pricing kernel
are

(

)

δ $ = δ + 1 − φp θ p
γ $ = γ x γ v φ p  ′
λ $ = [ λx λv 1]′ .
In the exogenous inflation model, the price of
inflation risk is always exactly 1 and does not
change with the values of any of the other structural parameters in the model. In addition, the
factor loadings and prices of risk for output growth
and stochastic volatility are the same as in the
real pricing kernel. We will refer to this nominal
pricing kernel specification as the exogenous
inflation economy.

Monetary Policy and Endogenous
Inflation
We begin by assuming that monetary policy
follows a simple nominal interest rate rule. We
will abuse conventional terminology and often
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Gallmeyer, Hollifield, Palomino, Zin

refer to the interest rate rule as a Taylor rule.
Although there are a variety of ways to specify a
Taylor rule—see Ang, Dong, and Piazzesi (2004)—
we will consider a rule in which the short-term
interest rate depends on contemporaneous output,
inflation, and a policy shock:

From these dynamics, the nominal one-period
interest rate, it = –log 共Et [m $t+1]兲, is

it = δ + π + π x (1 − φx )θ x + π v (1 − φv )θv
+ (γ x + π x φx ) x t + (γ v + π v φv )v t + π s φs st
−

it = τ + τ x x t + τ p pt + st ,

(9)

where the monetary policy shock satisfies

Comparing this with the interest rate rule,
it = τ + τ x x + τ p (π + π x x t + π v v t + π s st ) + st ,

st = φs st −1 + σ s ε ts
and where εts ~ NID共0,1兲 is independent of the
other two real shocks.
Because this nominal interest rate rule must
also be consistent with equilibrium in the bond
market, that is, it must be consistent with the
nominal pricing kernel in equation (8) as well as
equation (9), we can use these two equations to
find the equilibrium process for inflation. Conjecture a log-linear solution for pt ,
(10)

pt = π + π x xt + π v v t + π s st ,

with π–, πx , and πs constants to be solved.
To solve for a rational expectations solution
to the model, we substitute the guess for the
inflation rate into both the Taylor rule and the
nominal pricing kernel and solve for the parameters π–, πx , πv , and πs that equate the short rate
determined by the pricing kernel with the short
rate determined by the Taylor rule.
From the dynamics of xt +1, vt +1, and st +1,
inflation, pt +1, is given by
pt +1 = π + π x x t +1 + π v v t +1 + π s st +1
= π + π x (1 − φ x ) θ x + π v (1 − φ v ) θ v
+ π x φx x t + π v φv v t + π s φs st
+ π x v t1/ 2εtx+1 + π v σ v εtv+1 + π s σ s ε ts+1 .

Substituting into the nominal pricing kernel,

(

)

− log m$t +1 = −log ( mt +1 ) + pt +1
= δ + γ x xt + γ v v t + λ x v t1/ 2ε tx+1 + λ v σ v ε tv+1 + pt +1
= δ + π + π x (1 − φ x )θ x + π v (1 − φv )θv
+ (γ x + π x φx ) xt + (γ v + π v φv ) v t + π s φs st
+ ( λ x + π x ) v t1/ 2ε tx+1 + ( λv + π v )σ v εtv+1 + π s σ s ε ts+1 .

312

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1
( λ x + π x )2 v t − 1 ( λv + π v )2 σ v2 − 1 π s2σ s2.
2
2
2

gives the parameter restrictions consistent with
equilibrium:
(11)

πx =

γ x − τx
τ p − φx
γv −

πv =
πs = −

π=

1
2
λx + π x )
(
2
τ p − φv

1
τ p − φs

 δ − τ + π x (1 − φx )θ x + π v (1 − φv )θv 
1 
.

τ p − 1  − 1 ( λv + π v )2 σ v2 − 1 π s2σ s2

 2
2

These expressions form a recursive system we
use to solve for the equilibrium parameters of
the inflation process. See Cochrane (2006) for a
more detailed account of this rational expectations solution method.
It is clear from these expressions that the
equilibrium inflation process will depend on the
preference parameters of the household generally
and attitudes toward risk specifically.
In a similar fashion, we can extend the analysis to any Taylor-type rule that is linear in the
state variables, including (i) lagged short rates,
(ii) other contemporaneous yields at any maturity,
and (iii) forward-looking rules, such as those in
Clarida, Galí, and Gertler (2000). Such extensions
are possible because, in the affine framework,
interest rates are all simply linear functions of
the current state variables. See Ang, Dong, and
Piazzesi (2004) and Gallmeyer, Hollifield, and
Zin (2005) for some concrete examples.
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Gallmeyer, Hollifield, Palomino, Zin

A Monetary Policy–Consistent Pricing
Kernel
Substituting the equilibrium inflation process
from equations (10) and (11) into the nominal
pricing kernel, we obtain an equilibrium threefactor affine term-structure model that is consistent with the nominal interest rate rule. The
state space is

zt ;  x t v t st  ′
Φ = diag {φx , φv , φs}

θ = [θ x θv 0 ]′
Σ ( zt ) = diag {a1 + b1′zt , a2 + b2′ zt , a3 + b3′ zt }
a1 = 0, b1 = [0 1 0]′
a2 = σ v2 , b2 = [0 0 0]′
a3 = σ s2 , b3 = [0 0 0]′ ,
and the parameters of the pricing kernel are

δ $ = δ + π + π x (1 − φx )θ x + π v (1 − φv )θv
γ $ = γ x + φx π x γ v + φv π v φs π s  ′
λ $ =  λx + π x λv + π v π s  ′ .
We will often refer to this nominal pricing kernel
specification as the endogenous inflation
economy.
The Taylor rule parameters, through their
determination of the equilibrium inflation process,
affect both the factor loadings on the real factors
as well as their prices of risk. Monetary policy
through its effects on endogenous inflation, therefore, can result in risk premiums in the term structure that are significantly different from those in
the exogenous inflation model. We explore such
a possibility through numerical examples.

QUANTITATIVE EXERCISES
We calibrate the exogenous processes in our
model to quarterly postwar U.S. data as follows:
1. Endowment growth: φx = 0.36, θx = 0.006,
σx = 0.0048共1 – φx2 兲1/2
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

2. Inflation: φp = 0.8471, θp = 0.0093,
σp = 0.0063共1 – φp2 兲1/2
3. Stochastic volatility: φv = 0.973,
θv = 0.0001825, σv = 0.9884 × 10–5
4. Policy shock: φs = 0.922,
σs = 共0.023 × 10–4兲1/2
The endowment growth process is calibrated
to quarterly per capita consumption of durable
goods and services, and inflation is calibrated to
the nondurables and services deflator, similarly
to Piazzesi and Schneider (2007). The volatility
process is taken from Bansal and Yaron (2004),
who calibrate their model to monthly data. We
adjust their parameters to deal with quarterly
time-aggregation. We take the parameters for the
policy shock from Ang, Dong, and Piazzesi (2004),
who estimate a Taylor rule using an affine termstructure model with macroeconomic factors
and an unobserved policy shock.
Figures 4 though 7 depict the average yield
curves and yield volatilities for different preference parameters for the exogenous and endogenous inflation models. In the top panel of each
figure, asterisks denote the empirical average
nominal yield curve, a blue dashed-dotted line
denotes the average real yield curve common
across both inflation models, a dashed line
denotes the average nominal yield curve in the
exogenous inflation economy, and a solid line
denotes the average nominal yield curve in the
endogenous inflation economy. The bottom panel
depicts yield volatilities for the same cases as the
average yield curve in the top panel. (Asterisks
in Figures 4 through 9 are the moments—means
and standard deviations—of the data in Figure 1.)
Each figure is computed using a different set
of preference parameters. We fix a level of the
intertemporal elasticity parameter, ρ, for each
panel and pick the remaining preference parameters—the risk aversion coefficient, α , and the rate
of time preference, β—to minimize the distance
between the average nominal yields and yield
volatilities in the data and those implied by the
exogenous inflation economy. We pick the
Taylor rule parameters to minimize the distance
between the average nominal yields and yield
volatilities in the data and those implied by the
endogenous inflation economy. Table 1 reports
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Gallmeyer, Hollifield, Palomino, Zin

Figure 4
Average Yield Curve and Volatilities for the Epstein-Zin Model with Stochastic Volatility
Average Term Structures

Annual %
8

7

6

5

Endogenous Inflation

4

Exogenous Inflation
Nominal Data
Real Yield Curve

3

2

0

5

10

Annual %

15

20
Quarters

25

30

35

40

25

30

35

40

Volatility of Yields

4.5
4
3.5
3
2.5
2
1.5
1
0.5

0

5

10

15

20
Quarters

NOTE: The parameters are ρ = –0.5, α = –4.835, β = 0.999, τ– = 0.003, τx = 1.2475, and τp = 1.000. The empirical moments for the full
sample (1970:Q1–2005:Q4) are plotted with asterisks, properties of the real yield curve are plotted with a dashed-dotted blue line,
properties of the yield curve in the exogenous inflation economy are plotted with a dashed black line, and properties of the yield
curve in the economy with endogenous inflation are plotted with a solid black line.

314

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F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Gallmeyer, Hollifield, Palomino, Zin

Figure 5
Average Yield Curve and Volatilities for the Epstein-Zin Model with Stochastic Volatility
Average Term Structures

Annual %
8

7

6

5

Endogenous Inflation

4

Exogenous Inflation
Nominal Data
Real Yield Curve

3

2
0

5

10

15

20
Quarters

25

30

35

40

30

35

40

Volatility of Yields

Annual %
3.5

3

2.5

2

1.5

1

0

5

10

15

20
Quarters

25

NOTE: The parameters are ρ = 0.0, α = –4.061, β = 0.998, τ– = 0.003, τx = 0.973, and τp = 0.973. The empirical moments for the full
sample (1970:Q1–2005:Q4) are plotted with asterisks, properties of the real yield curve are plotted with a dashed-dotted blue line,
properties of the yield curve in the exogenous inflation economy are plotted with a dashed black line, and properties of the yield
curve in the economy with endogenous inflation are plotted with a solid black line.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

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Gallmeyer, Hollifield, Palomino, Zin

Figure 6
Average Yield Curve and Volatilities for the Epstein-Zin Model with Stochastic Volatility
Average Term Structures

Annual %
8

7

6

5
Endogenous Inflation
4

Exogenous Inflation
Nominal Data
Real Yield Curve

3

2
0

5

10

15

20

25

30

35

40

30

35

40

Quarters

Volatility of Yields

Annual %
3.5

3

2.5

2

1.5

1
0

5

10

15

20
Quarters

25

NOTE: The parameters are ρ = 0.5, α = –4.911, β = 0.994, τ– = –0.015, τx = 3.064, and τp = 2.006. The empirical moments for the full
sample (1970:Q1–2005:Q4) are plotted with asterisks, properties of the real yield curve are plotted with a dashed-dotted blue line,
properties of the yield curve in the exogenous inflation economy are plotted with a dashed black line, and properties of the yield
curve in the economy with endogenous inflation are plotted with a solid black line.

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F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Gallmeyer, Hollifield, Palomino, Zin

Figure 7
Average Yield Curve and Volatilities for the Epstein-Zin Model with Stochastic Volatility
Average Term Structures

Annual %
8

7

6

5

Endogenous Inflation

4

Exogenous Inflation
Nominal Data
Real Yield Curve

3

2

0

5

10

Annual %

15

20
Quarters

25

30

35

40

25

30

35

40

Volatility of Yields

3.5

3

2.5

2

1.5

1

0.5
0

5

10

15

20
Quarters

NOTE: The parameters are ρ = 1.0, α = –6.079, β = 0.990, τ– = –0.004, τx = 1.534, and τp = 1.607. The empirical moments for the full
sample (1970:Q1–2005:Q4) are plotted with asterisks, properties of the real yield curve are plotted with a dashed-dotted blue line,
properties of the yield curve in the exogenous inflation economy are plotted with a dashed black line, and properties of the yield
curve in the economy with endogenous inflation are plotted with a solid black line.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

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Gallmeyer, Hollifield, Palomino, Zin

Table 1
Factor Loadings and Prices of Risk
Factor loadings (γ s)
Constant
xt
vt
pt
–
A. ρ = –0.5, α = –4.835, β = 0.999, τ = 0.003, τx = 1.2475, τp = 1.000
Real kernel

0.01

0.54

25.45

—

Prices of risk (λ s)
st

x
ε t +1

ε t +1

v

ε t +1

p

ε t +1

s

—

8.25

–902.98

—

—

Exogenous inflation

0.01

0.54

25.45

0.85

—

8.25

–902.98

1.00

—

Endogenous inflation

0.02

0.14

21.63

—

–1.44

7.15

–906.90

—

–1.56

—

7.34

–677.11

—

—

–
B. ρ = 0.0, α = –4.061, β = 0.998, τ = 0.003, τx = 0.973, τp = 0.973
Real kernel

0.01

0.36

20.07

—

Exogenous inflation

0.01

0.36

20.07

0.85

—

7.34

–677.11

1.00

—

Endogenous inflation

0.02

0.00

33.56

—

–1.51

6.34

–663.24

—

–1.63

—

8.93

–972.61

—

—

–
C. ρ = 0.5, α = –4.911, β = 0.994, τ = –0.015, τx = 3.064, τp = 2.006
Real kernel

0.01

0.18

32.23

—

Exogenous inflation
Endogenous inflation

0.01

0.18

32.23

0.85

—

8.93

–972.61

1.00

—

0.02

–0.45

38.34

—

–0.56

7.18

–966.33

—

–0.61

—

10.99

–1,398.00

—

—

–
D. ρ = 1.0, α = –6.079, β = 0.990, τ = 0.004, τx = 1.534, τp = 1.607
Real kernel

0.02

0.00

51.82

—

Exogenous inflation
Endogenous inflation

0.02

0.00

51.82

0.85

—

10.99

–1,398.00

1.00

—

0.03

–0.44

58.30

—

–0.74

9.76

–1,391.30

—

–0.80

NOTE: The table reports the affine term-structure parameters for the real term structure, the nominal term structure in the exogenous
inflation economy, and the nominal term structure in the endogenous inflation economy. The parameters in each panel are computed
using a different set of preference parameters. We fix a level of the intertemporal elasticity parameter, ρ, and choose the remaining
preference parameters—the risk aversion coefficient, α , and the rate-of-time preference, β, to minimize the distance between the
average nominal yields and yield volatilities in the data and those implied by the exogenous inflation economy. We pick the Taylor
rule parameters to minimize the distance between the average nominal yields and yield volatilities in the data and the those implied
by the endogenous inflation economy.

the factor loadings and the prices of risk for each
economy corresponding to the figures. Table 2
reports the coefficients on the equilibrium inflation rate and properties of the equilibrium inflation rate in the endogenous inflation economy.
Figure 4 reports the results with ρ = –0.5; here
the representative agent has a low intertemporal
elasticity of substitution. The remaining preference
parameters are α = –4.835 and β = 0.999. With
this choice of parameters, the average real term
structure is slightly downward sloping.
Backus and Zin (1994) show that a necessary
condition for the average yield curve to be upward
sloping is negative autocorrelation in the pricing
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kernel.2 Consider an affine model with independent factors zt1, zt2,…, ztk with an innovation εtj on
the j th factor, a factor loading γ j on the j th factor,
and a price of risk λj on the j th factor. In such a
model, the j th factor contributes
(12)

( )

(

γ 2j Autocov ztj + γ j λ j Cov ztj, ε tj

)

to the autocovariance of the pricing kernel.
In our calibration, the exogenous factors in
2

Piazzesi and Schneider (2007) argue that an upward-sloping
nominal yield curve can be generated if inflation is bad news for
consumption growth. Such a structure leads to negative autocorrelation in the nominal pricing kernel.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Gallmeyer, Hollifield, Palomino, Zin

Table 2
Properties of pt in the Endogenous Inflation Economy
π–

πs

E(pt )

σ (pt )

AR(1)

–3.92

–1.56

0.01

0.02

0.37

13.87

–1.63

0.01

0.02

0.44

6.28

–0.61

0.01

0.03

0.37

6.66

–0.80

0.01

0.02

0.37

πx
πv
–
A. ρ = –0.5, α = –4.835, β = 0.999, τ = 0.003, τx = 1.2475, τp = 1.000
0.01

–1.11

–
B. ρ = 0.0, α = –4.061, β = 0.998, τ = 0.003, τx = 0.973, τp = 0.973
0.01

–1.00

–
C. ρ = 0.5, α = –4.911, β = 0.994, τ = –0.015, τx = 3.064, τp = 2.006
0.02

–1.75

–
D. ρ = 1.0, α = –6.079, β = 0.990, τ = 0.004, τx = 1.534, τp = 1.607
0.01

–1.23

NOTE: The first four columns are coefficients of the inflation rate: the equilibrium inflation rate coefficients on a constant, output,
stochastic volatility, and the monetary policy shock, respectively. The last two columns are properties of inflation: the unconditional
standard deviation and the first-order autocorrelation of inflation, respectively.

the real economy—output growth and stochastic
volatility—all have positive autocovariances and
the factor innovations have positive covariances to
the factor levels. This implies that γ j2Autocov共ztj 兲
and Cov共ztj , εtj 兲 are both positive. For a factor to
contribute negatively to the autocorrelation of the
pricing kernel, the factor loading γ j and the price
of risk λj must have opposite signs. Additionally,
the price of risk λj must be large enough relative
to the factor loading γ j to counteract the positive
autocovariance term γ j2Autocov共ztj 兲.
Output growth has a lower autocorrelation
coefficient than stochastic volatility in our calibration, but because output growth has a much
higher unconditional volatility, it has a much
higher autocovariance than stochastic volatility.
In the real economy, the factor loading γ x on the
level of output growth is equal to 共1 – ρ兲φx , which
is nonnegative for all ρ ≤ 1. Also, the price of risk
for output growth, λ x , is positive at the parameter
values used in Figure 4 because a sufficient condition for it to be positive is α ≤ 0 and |ρ|≤|α|.
From (12), output growth contributes positively
to the autocovariance of the pricing kernel.
From the real pricing kernel parameters given
in (7), the price of risk for volatility is related to
the factor loading on the level of volatility by
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

λv = −

β
γv.
1 − βφv

Because 1 – βφv > 0, the volatility price of risk,
λ v , and the volatility factor loading, γ v , have opposite signs, implying that the volatility factor can
contribute a negative autocovariance to the pricing kernel. But output growth has the strongest
effect on the autocovariance of the pricing kernel,
leading to positive autocovariance in the pricing
kernel. As a consequence, the average real yield
curve is downward sloping. The numerical values
for the real pricing kernel’s factor loadings and
prices of risk from Figure 4 are reported in panel A
of Table 1.
In the exogenous inflation economy, shocks
to inflation are uncorrelated to output growth and
stochastic volatility—the factor loadings and
prices of risk on output growth and stochastic
volatility in the nominal pricing kernel are the
same as in the real pricing kernel. Average nominal yields in the exogenous inflation economy are
equal to the real yields plus expected inflation
and inflation volatility with an adjustment for
properties of the inflation process. The inflation
shocks are positively autocorrelated, with a factor
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Gallmeyer, Hollifield, Palomino, Zin

loading and a price of risk that are both positive.
The average nominal yield curve has approximately the same shape as the real yield curve—
it is downward sloping.
In the endogenous inflation economy, inflation is a linear combination of output growth,
stochastic volatility, and the monetary policy
shock. As shown in panel A of Table 2, endogenous inflation’s loading on output, πx , is negative.
This implies that the nominal pricing kernel’s
output-growth factor loading and price of risk are
lower than in the exogenous inflation economy.
As a consequence, output growth contributes
much less to the autocovariance of the pricing
kernel with endogenous inflation. The factor loading and price of risk for stochastic volatility are
also lower in the endogenous inflation economy.
The policy shocks are positively autocorrelated,
but the factor loading and the price of risk for the
policy shock are of opposite sign. The average
nominal yield curve in the endogenous inflation
economy is therefore flatter than both the real
yield curve and the nominal yield curve with
exogenous inflation.
Turning to the volatilities in the bottom panel
of Figure 4, the exogenous inflation economy
exhibits more volatility in short rates and less
volatility in long rates than found in the data. This
is a fairly standard finding for term-structure
models with stationary dynamics (see Backus and
Zin, 1994). The volatility of long rates is mainly
driven by the loading on the factor with the largest
innovation variance and that factor’s autocorrelation. The closer that autocorrelation is to zero,
the faster that yield volatility decreases as bond
maturity increases. In our calibration, output
growth has the largest innovation variance and a
fast rate of mean reversion, equal to 0.36. Yield
volatility drops quite quickly as bond maturity
increases. In general, the lower the loading on output growth, the slower that yield volatility drops
as bond maturity increases. Because endogenous
inflation is negatively related to output growth,
the factor loading on output growth is lower. Yield
volatility drops at a slower rate (relative to maturity) in the endogenous inflation economy than
in the exogenous inflation economy.
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Figure 5, panel B of Table 1, and panel B of
Table 2 report yield-curve properties with a higher
intertemporal elasticity of substitution 共ρ = 0兲 or
a log time aggregator. Piazzesi and Schneider
(2007) study a model with the same preferences,
but without stochastic volatility. The factor loading on output growth in the real economy is higher
than in the economy with ρ = –0.5 reported in
Figure 4 (compare panel A with panel B of
Table 1). The average real yield curve and the
average nominal yield curve with exogenous
inflation are less downward sloping when ρ = 0
than when ρ = –0.5. Similarly, increasing ρ further
to 0.5 (see Figure 6) or 1.0 (see Figure 7) leads to
a real yield curve that is less downward-sloping.
Because increasing ρ decreases the factor loading
on output growth, it also decreases the volatility
of real yields: See the bottom panels in Figures 4
through 7.
As ρ increases, the representative agent’s
intertemporal elasticity of substitution increases,
implying less demand for smoothing consumption
over time. Increasing ρ decreases the representative agent’s demand for long-term bonds for the
purpose of intertemporal consumption smoothing
and leads to lower equilibrium prices and higher
yields for real long-term bonds. The average real
yield curve therefore is less downward-sloping
as ρ increases. Increasing ρ also reduces the sensitivity of long-term real yields to output growth,
leading to less volatile long-term yields: See the
bottom panels in Figures 4 through 7.
Nominal yields in the economies with exogenous inflation are approximately equal to the real
yields plus a maturity-independent constant.
But in the economies with endogenous inflation,
inflation and output growth have a negative
covariance, leading to a decrease in the factor
loading on output growth: See panels C and D of
Tables 1 and 2. For ρ ≥ 0.5 (see Figures 6 and 7),
the average nominal yield curve is upward sloping and the shape of the volatility term structure
decays similarly to that observed in the data.
The final three columns of Table 2 report
unconditional moments of inflation in the economy with endogenous inflation. There are a few
notable features. First, the unconditional moments
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Gallmeyer, Hollifield, Palomino, Zin

Figure 8
The Effects of Increasing

τx
Average Term Structure, τ x , Increased by 10%

Annual %
8

7.5

7

6.5

Benchmark
Increased τ x
Nominal Data

6

5.5

0

5

10

15

20

25

30

35

40

35

40

Quarters

Volatility of Yields, τ x , Increased by 10%

Annual %
4

3.5

3

2.5

2

1.5

0

5

10

15

20
Quarters

25

30

NOTE: The baseline parameters are ρ = 1.0, α = –6.079, β = 0.990, τ– = –0.004, τx = 1.534, and τp = 1.607. Empirical moments for the full
sample (1970:Q1–2005:Q4) are plotted with asterisks, results from the baseline parameters are plotted with a solid black line, and results
when the feedback from output growth to short-term interest rates is increased by 10 percent are plotted with a dashed black line.

are not particularly sensitive to the intertemporal
elasticity of substitution. Second, the unconditional variance of inflation in the calibrated economy is an order of magnitude higher than that in
the data: 0.0033 in empirical data and about 0.02
in these economies. Finally, inflation is much
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

more autocorrelated in the data—the AR(1) coefficient is 0.85 in the data and about 0.4 in the
model economies.
Figure 8, Figure 9, and Table 3 show results
from changing the Taylor rule parameters. We
keep the remaining parameters fixed at the values
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Gallmeyer, Hollifield, Palomino, Zin

Figure 9
The Effects of Increasing

τp
Average Term Structure, τ p , Increased by 10%

Annual %
8

7.5

7

6.5
Benchmark

6

Increased τ p
Nominal Data

5.5

5

0

5

10

15

20

25

30

35

40

35

40

Quarters

Volatility of Yields, τ p , Increased by 10%

Annual %
3.5

3

2.5

2

1.5

1

0

5

10

15

20

25

30

Quarters

NOTE: The baseline parameters are ρ = 1.0, α = –6.079, β = 0.990, τ– = –0.004, τx = 1.534, and τp = 1.607. Empirical moments for the
full sample (1970:Q1–2005:Q4) are plotted with asterisks, results from the baseline parameters are plotted with a solid black line, and
results when the feedback from inflation to short-term interest rates is increased by 10 percent are plotted with a dashed black line.

used to generate Figure 7. Figure 8 shows that
increasing τx , the interest rate’s responsiveness
to output growth shocks, leads to a reduction in
average nominal yields and a steepening in the
average yield curve (top panel), as well as an
increase in yield volatility (bottom panel).
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Panel A of Table 3 shows that increasing τx
decreases the constant in the nominal pricing
kernel, decreases the factor loading on output
growth, decreases the price of risk for output
growth, and also increases the factor loading on
stochastic volatility. The loading on output growth
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Gallmeyer, Hollifield, Palomino, Zin

Table 3
Comparative Statics for the Taylor Rule Parameters
Nominal pricing kernel
Factor loadings (γ s)
Constant

xt

Prices of risk (λ s)

vt

st

x
ε t +1

v
ε t +1

s
ε t +1

Equilibrium inflation loadings
π–

πx

πv

πs

A. τx increased by 10%, from 1.53 to 1.69
Baseline

0.03

–0.44

58.30

–0.74

9.76

–1,391.30 –0.80

0.01

–1.23

6.66

–0.80

Increased τx

0.02

–0.49

60.13

–0.74

9.63

–1,389.40 –0.80

0.01

–1.35

8.55

–0.80

B. τp increased by 10%, from 1.61 to 1.77
Baseline

0.03

–0.44

58.30

–0.74

9.76

–1,391.30 –0.80

0.01

–1.23

6.66

–0.80

Increased τp

0.02

–0.39

55.30

–0.66

9.90

–1,394.40 –0.71

0.01

–1.09

3.58

–0.71

NOTE: The table reports the effect of changing the Taylor rule parameter τx or τp on the affine term-structure parameters as well as
properties of pt in the endogenous inflation economy. The equilibrium inflation rate coefficients on output, stochastic volatility, and
the monetary policy shock are reported. The baseline parameters are ρ = 1.0, α = –6.08, β = 0.990, τ– = –0.004, τx = 1.53, and τp = 1.61.

in the pricing kernel drops because the sensitivity
of the inflation rate to output growth drops; in
turn, the sensitivity of inflation to stochastic
volatility increases by a large amount—from 6.66
to 8.55.
Figure 9 shows that increasing τp , the interest
rate responsiveness to inflation, leads to a reduction in average nominal yields and a flattening in
the average yield curve (top panel) and a decrease
in yield volatility (bottom panel).
Panel B of Table 3 shows that increasing τp
decreases the constant in the nominal pricing
kernel, increases the factor loading on output
growth, increases the price of risk for output
growth, decreases the factor loading on stochastic volatility, and also drops the factor loading on
the monetary policy shock. The constant in the
pricing kernel drops because the constant in the
inflation rate drops, the factor loading on output
growth increases because the sensitivity of the
inflation rate to output growth increases; in turn,
the sensitivity of inflation to stochastic volatility
decreases by a large amount—from 6.66 to 3.58.
Overall, the experiments reported in Figure 8
and Figure 9 show that properties of the term
structure depend on the form of the monetary
authorities’ interest rate feedback rule. In particular, the factor loading on stochastic volatility is
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

quite sensitive to the interest rate rule. In this
economy, because stochastic volatility is driving
time-variation in interest rate risk premiums,
monetary policy can have large impacts on interest rate risk premiums.

RELATED RESEARCH
The model we develop is similar to a version
of Bansal and Yaron’s (2004), which includes
stochastic volatility; however, our simple autoregressive state-variable process does not capture
their richer ARMA specification. Our work is also
related to Piazzesi and Schneider (2007), who
emphasize that, for a structural model to generate an upward-sloping nominal yield curve, it
requires joint assumptions on preferences and
the distribution of fundamentals. Our work highlights how an upward-sloping yield curve can also
be generated through the monetary authority’s
interest rate feedback rule.
Our paper adds to a large and growing literature combining structural macroeconomic models
that include Taylor rules with arbitrage-free termstructure models. Ang and Piazzesi (2003), following work by Piazzesi (2005), have shown that a
factor model of the term structure that imposes
arbitrage-free conditions can provide a better
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Gallmeyer, Hollifield, Palomino, Zin

empirical model of the term structure than a
model based on unobserved factors or latent
variables alone. Estrella and Mishkin (1997),
Evans and Marshall (1998 and 2001), Hördahl,
Tristani, and Vestin (2004), Bekaert, Cho, and
Moreno (2005), and Ravenna and Seppala (2006)
also provide evidence of the benefits of building
arbitrage-free term-structure models with macroeconomic fundamentals. Rudebusch and Wu
(2004) and Ang, Dong, and Piazzesi (2004) investigate the empirical consequences of imposing a
Taylor rule on the performance of arbitrage-free
term-structure models.
For an alternative linkage between short- and
long-maturity bond yields, see Vayanos and Vila
(2006), who show how the shape of the term structure is determined in the presence of risk-averse
arbitrageurs, investor clienteles for specific bond
maturities, and an exogenous short rate that could
be driven by the central bank’s monetary policy.

serious econometric exercise. Further research
will explore the trade-offs between shock specifications, preference parameters, and monetary
policy rules for empirical yield-curve models
that more closely match historical evidence.
Finally, it would be instructive to compare
and contrast the recursive utility model with
stochastic volatility with other preference specifications that are capable of generating realistic
risk premiums. The leading candidate on this
dimension is the external habits models of
Campbell and Cochrane (1999). We are currently
pursuing an extension of the external habits
model in Gallmeyer, Hollifield, and Zin (2005)
to include an endogenous, Taylor rule–driven
inflation process.

REFERENCES
Ang, Andrew; Dong, Sen and Piazzesi, Monika.
“No-Arbitrage Taylor Rules.” Working paper,
Columbia University, 2004.

CONCLUSIONS
We demonstrate that an endogenous monetary
policy that involves an interest rate feedback rule
can contribute to the riskiness of multi-period
bonds by creating an endogenous inflation process
that exhibits significant covariance risk with the
pricing kernel. We explore this through a recursive utility model with stochastic volatility that
generates sizable average risk premiums. Our
results point to a number of additional questions.
First, the Taylor rule that we work with is arbitrary, so how would the predictions of the model
change with alternative specification of the rule?
In particular, how would adding monetary nonneutralities along the lines of a New Keynesian
Phillips curve as in Clarida, Galí, and Gertler
(2000) and Gallmeyer, Hollifield, and Zin (2005)
alter the monetary policy–consistent pricing kernel? Second, what Taylor rule would implement
an optimal monetary policy in this context?
Because preferences have changed relative to the
models in the literature, this is a nontrivial theoretical question.
In addition, the simple calibration exercise in
this paper is not a very good substitute for a more
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2007

Ang, Andrew and Piazzesi, Monika. “A No-Arbitrage
Vector Autoregression of Term Structure Dynamics
with Macroeconomic and Latent Variables.” Journal
of Monetary Economics, May 2003, 50(4), pp. 745-87.
Backus, David K.; Foresi, Silverio and Telmer, Chris I.
“Affine Term Structure Models and the Forward
Premium Anomaly.” Journal of Finance, February
2001, 56(1), pp. 279-304.
Backus, David K. and Zin, Stanley E. “Reverse
Engineering the Yield Curve.” NBER Working
Paper No. 4676, National Bureau of Economic
Research, 1994.
Backus, David K. and Zin, Stanley E. “Bond Pricing
with Recursive Utility and Stochastic Volatility.”
Working paper, New York University, 2006.
Bansal, Ravi and Yaron, Amir. “Risks for the Long
Run: A Potential Resolution of Asset Pricing
Puzzles.” Journal of Finance, August 2004, 59(4),
pp. 1481-510.
Bekaert, Geert; Cho, Seonghoon and Moreno, Antonio.
“New-Keynesian Macroeconomics and the Term

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Gallmeyer, Hollifield, Palomino, Zin

Structure.” Working paper, Columbia University,
2005.
Campbell, John Y. and Cochrane, John H. “By Force
of Habit: A Consumption-Based Explanation of
Aggregate Stock Market Behavior.” Journal of
Political Economy, April 1999, 107(2), pp. 205-51.
Clarida, Richard; Galí, Jordi and Gertler, Mark.
“Monetary Policy Rules and Macroeconomic
Stability: Evidence and Some Theory.” Quarterly
Journal of Economics, February 2000, 115(1),
pp. 147-80.
Cochrane, John H. “Identification and Price
Determination with Taylor Rules: A Critical Review.”
Working paper, University of Chicago, 2006.
Duffie, Darrell and Kan, Rui. “A Yield-Factor Model
of Interest Rates.” Mathematical Finance, October
1996, 6(4), pp. 379-406.
Epstein, Larry G. and Zin, Stanley E. “Substitution,
Risk Aversion, and the Temporal Behavior of
Consumption and Asset Returns: A Theoretical
Framework.” Econometrica, July 1989, 57(4),
pp. 937-69.
Estrella, Arturo and Mishkin, Frederic S. “The
Predictive Power of the Term Structure of Interest
Rates in Europe and the United States: Implications
for the European Central Bank.” European Economic
Review, July 1997, 41(7), pp. 1375-401.
Evans, Charles L. and Marshall, David A. “Monetary
Policy and the Term Structure of Nominal Interest
Rates: Evidence and Theory.” Carnegie-Rochester
Conference Series on Public Policy, December 1998,
49, pp. 53-111.
Evans, Charles L. and Marshall, David A. “Economic
Determinants of the Term Structure of Nominal
Interest Rates.” Working Paper WP-01-16, Federal
Reserve Bank of Chicago, 2001.
Gallmeyer, Michael F.; Hollifield, Burton and Zin,
Stanley E. “Taylor Rules, McCallum Rules, and the
Term Structure of Interest Rates.” Journal of
Monetary Economics, July 2005, 52(5), pp. 921-50.

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Greenspan, Alan. Testimony before the U.S. Senate
Committee on Banking, Housing, and Urban
Affairs. Board of Governors of the Federal Reserve
System, May 27, 1994;
http://fraser.stlouisfed.org/historicaldocs/ag94/
download/27982/Greenspan_19940527.pdf.
Greenspan, Alan. Testimony before the U.S. Senate
Committee on Banking, Housing, and Urban
Affairs. Board of Governors of the Federal Reserve
System, February 16, 2005;
www.federalreserve.gov/boarddocs/hh/2005/
february/testimony.htm.
Hansen, Lars Peter; Heaton, John and Li, Nan.
“Consumption Strikes Back? Measuring Long-Run
Risk.” NBER Working Paper No. 11476, National
Bureau of Economic Research, 2005.
Hördahl, Peter; Tristani, Oreste and Vestin, David.
“A Joint Econometric Model of Macroeconomic
and Term Structure Dynamics.” Working Paper
Series No. 405, European Central Bank, 2004.
McCulloch, J. Huston and Kwon, Heon-Chul. “U.S.
Term Structure Data, 1947-1991.” Working Paper
No. 93-6, The Ohio State University, 1993.
Piazzesi, Monika. “Bond Yields and the Federal
Reserve.” Journal of Political Economy, April 2005,
113(2), pp. 311-44.
Piazzesi, Monika and Schneider, Martin.
“Equilibrium Yield Curves,” in Daron Acemoglu,
Kenneth Rogoff, and Michael Woodford, eds.,
NBER Macroeconomics Annual 2006. Cambridge,
MA: MIT Press, 2007.
Ravenna, Frederico and Seppala, Juha. “Monetary
Policy and Rejections of the Expectations
Hypothesis.” Working paper, University of Illinois,
2006.
Rudebusch, Glenn D. and Wu, Tao. “A Macro-Finance
Model of the Term Structure, Monetary Policy, and
the Economy.” Presented at the Federal Reserve
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Gallmeyer, Hollifield, Palomino, Zin

Vayanos, Dimitri and Vila, Jean-Luc. “A PreferredHabitat Model of the Term Structure of Interest
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Commentary
Pamela A. Labadie

T

he questions addressed in the Gallmeyer
et al. (2007) paper are important ones:
How does monetary policy affect longterm interest rates? How can we explain
the volatility of the long end of the yield curve
and its relationship with monetary policy? A
successful model is one that answers these questions and is useful to policymakers. To be useful
to policymakers, the model should yield quantitative answers to questions such as, What will
happen to long-term rates if the central bank
raises the federal funds rate by 25 basis points?
Since money is not introduced in a way that is
essential,1 this model is not designed to answer
questions about the mechanism by which monetary policy affects the yield curve. This is not
necessarily a deficiency, because models where
money is essential have yet to prove useful in
policy discussions and empirical results.2 The
strength of this model is that it provides quantitative answers to the questions. Hence, it is reasonable to set a standard of success where the
model-generated time series must, in some sense,
“look like” actual time series.
As in any paper where hard modeling decisions are made, there are both strengths and weaknesses in the choices made by the authors. To
describe these choices concisely, I’ll provide a
brief overview of the model.
1

Wallace (2001) says that money is “essential” in an economy if it
permits allocations that would otherwise not be achieved.

2

See Kocherlakota (2002).

OVERVIEW OF THE MODEL
There is a single, exogenous, stochastic, and
perishable endowment good. The endowment
grows at the rate xt , with stochastic volatility vt :
1
1 − φx θ x + φx xt + v t2 ε tx+1

xt +1 = (

)
v t +1 = (1 − φv )θv + φv v t + σ v ε tv+1 .
Hence, the endowment process is characterized
by a parameter vector 共φx , φv , θx , θv , σv 兲.
There is a representative agent with EpsteinZin preferences. The advantages of this preference
structure, with its property of separating relative
risk aversion from the elasticity of intertemporal
substitution, are well known. The preference
parameter vector is 共β , ρ , α 兲. The pricing kernel
is the intertemporal marginal rate of substitution
in consumption, denoted log共mt +1兲. The price of
default-free discount bonds can be determined
recursively through an arbitrage-free restriction
of the form

bt(n ) = Et m t +1bt(+n1−1) .
With this structure for the endowment process
and the preferences, the model can be mapped
into the Duffie-Kan affine term-structure model
with two factors. In particular, the authors guess
a form of the value function and then verify that
this guess is a solution. The value function then
implies the form of the real pricing kernel. The
process by which the authors relate the deeper
preference and endowment parameters to the

Pamela A. Labadie is a professor of economics at The George Washington University.
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 327-29.
© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
only with prior written permission of the Federal Reserve Bank of St. Louis.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

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Labadie

discrete-time affine term-structure model is interesting and elegant.
To introduce money and inflation into the
model, the authors add a stochastic process to
the real pricing kernel:

)

(

log mt$+1 = log ( mt +1 ) − pt +1 ,

where pt +1 is inflation. Hence, money, prices, and
inflation in the model are merely noise, creating
a wedge between the real and nominal pricing
kernels. Two specifications of the inflation process
are studied: exogenous and endogenous.

Exogenous Inflation
Inflation is conjectured to take the form

(

)

pt +1 = 1 − φ p θ p + φ p pt + σ p εtp+1 .
This specification does not link inflation to the
endowment process or to any money-growth
process. The state space is expanded to xt , vt , pt .
They calibrate the model to data and set the
parameters values at

φp = 0.8471

leads to my second comment. The model, as
specified, can be estimated using formal econometric techniques. The endowment process and
inflation processes, along with the real and nominal pricing kernels, form a system of equations.
Using data on nominal interest rates, one can
estimate inflation and consumption growth, the
parameters of the model, and in particular the
preference parameters. The preference-parameter
estimates could then be used to generate a real
pricing kernel. What are the estimated preference
parameters and does economic intuition suggest
that they are sensible? How does the real pricing
kernel behave? It seems a missed opportunity.
Finally, the model, as posed, severely restricts
the price of inflation risk by fixing the price of
inflation risk at unity. A very simple cash-inadvance model with fixed velocity has the property that inflation is a function of both money
growth and output growth, so there is a statevarying inflation risk premium. Even with a
zero mean, the inflation-risk premium may be
an additional source of variability and may help
to remedy the lack of volatility at the long end of
the yield curve in the model.

θ p = 0.0093
σ p = 0.0063

1
1 − φp2 2

(

)

Endogenous Inflation
.

The preference parameters are set at ρ = 0 and
α = –2.91, where ρ is the elasticity of intertemporal substitution and α is relative risk aversion,3
and there is little discussion of this choice. With
this set of parameter values, the model is used
to derive a yield curve. The result is a modelgenerated yield curve that matches the shape of
the historical yield curve, but exhibits less volatility in long rates. Since an explicit goal of the
paper is to explain volatility at the long end of
the yield curve, this answer is not satisfactory.
Three comments arise: The first is why are these
values for the preference parameters chosen?
How does varying the preference parameters
change the results? Should we just aimlessly
search the parameter space for a better fit? This
3

These are the values chosen in the version of the paper presented
at the conference.

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2007

To make inflation endogenous, the authors
assume that monetary policy follows a nominal
interest rate rule (Taylor rule) of the form
it = τ + τ x x t + τ p pt + st .

This rule raises short-term rates aggressively in
response to inflation. There are many other
specifications that could be considered, and it
would be helpful to have a discussion on why
this rule is chosen over other specifications. Is
this the type of rule the authors believe monetary
authorities are using? Is this the rule that gives
the best results in the sense that the modelgenerated yield curve matches the data? Is it
chosen for tractability?
This process must be consistent with the
other equations in the model, which requires the
derivation of an inflation process consistent with
the interest rate rule and other equations. To link
the rule to the nominal pricing kernel and bondF E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Labadie

market equilibrium, they use a guess-and-verify
method to derive a consistent inflation process
of the form
pt = π + π x x t + π v v t + π s st .

The state space is now zt = 共xt , vt , st 兲, and there are
additional restrictions on the means and conditional variances. Once again they calibrate the
model, choosing parameter values taken from the
data. The calibrated model fits the average yield
curve and has a volatility pattern closer to the
data, especially at the long end.
This is a major goal of the paper and, in that
sense, it is successful; but the question arises as
to whether the endogenous inflation process in
any way resembles the inflation process in the
data. My conjecture is that it does not—and in
some important ways—and the differences need
to be made explicit. How do the endogenous and
exogenous inflation processes compare? If the
calibrated model fits the average yield curve and
closely matches the volatility, but is based on an
inflation process that differs significantly from
the actual inflation process, how useful is this to
policymakers? How useful is a model that fits
the yield curve and its volatility but is greatly at
odds with the actual inflation process? Common
sense would suggest it is of limited usefulness.

CONCLUSION
The authors are to be commended for devising
a model with such a rich potential for explaining
yield curves and their volatility. Linking the
Epstein-Zin preferences to a discrete-time affine

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

term-structure model is no easy task, although
they seem to do so effortlessly. The model is
devised to be estimated with standard econometric
methods, using data on bond prices, consumption
growth, and inflation. Such an exercise would
provide useful insights into the real pricing kernel
and the parameters of the Epstein-Zin preferences.
The exogenous inflation specification is too
restrictive and should permit a variable inflationrisk premium. Finally, alternative interest rate
rules should be examined with the explicit goal
of generating an inflation process using the model
that matches the actual inflation process, according to some explicit criterion. While much of this
may sound negative, I want to emphasize that
this model has the potential to be very useful to
policymakers and the steps needed to make it so
are straightforward ones to take.

REFERENCES
Gallmeyer, Michael F.; and Hollifield, Burton;
Palomino, Francisco and Zin, Stanley. “AritrageFree Bond Pricing with Dynamic Macroeconomic
Models.” Federal Reserve Bank of St. Louis Review,
July/August 2007, 89(4), pp. 305-26.
Kocherlakota, Narayana. “Money: What’s the Question
and Why Should We Care about the Answer?”
American Economic Review, May 2002, 92(2),
pp. 58-61.
Wallace, Neil. “Lawrence R. Klein Lecture 2000:
Whither Monetary Economics?” International
Economic Review, November 2001, 42(4), pp. 847-69.

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Monetary Policy as Equilibrium Selection
Gaetano Antinolfi, Costas Azariadis, and James B. Bullard
Can monetary policy guide expectations toward desirable outcomes when equilibrium and welfare
are sensitive to alternative, commonly held rational beliefs? This paper studies this question in
an exchange economy with endogenous debt limits in which dynamic complementarities between
dated debt limits support two Pareto-ranked steady states: a suboptimal, locally stable autarkic state
and a constrained optimal, locally unstable trading state. The authors identify feedback policies
that reverse the stability properties of the two steady states and ensure rapid convergence to the
constrained optimal state. (JEL E31, E42, E58)
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 331-41.

INTRODUCTION
Overview

I

ndeterminacy, or non-uniqueness, of rational
expectations equilibrium has been a prominent feature of monetary policy analysis since
Sargent and Wallace (1975) found that passive
interest rate policies cause indeterminacy in an
IS-LM framework with rational expectations.
Generally speaking, policy choices influence
equilibrium outcomes, and passive choices can
support multiple equilibria. This situation has
been viewed as one to be avoided if at all possible; the prospect of the economy coordinating
on the “wrong” set of self-confirming beliefs is
unnecessary at best and detrimental to welfare
at worst. In the standard New Keynesian model,
for example, the monetary policymaker must
follow a sufficiently active policy to avoid
indeterminacy. “Active” means that the policy
instrument cannot be held fixed, or allowed to
fluctuate randomly, but instead must adjust to
the state of the economy according to a specific,
widely understood rule. A policy that is too

passive—say, too close to a nominal interest rate
peg—allows indeterminacy.
Results with this flavor depend critically on
the expectations of the private sector regarding
future monetary policy actions, and this has led
many to describe the problem of monetary policy
as one of managing or shaping expectations to
rule out private sector beliefs that may send the
economy toward a suboptimal course. How can
policy be designed to stop this process? Can policy
somehow strengthen rational beliefs in the desired
inflation target and in moderate inflationary
expectations?
This paper considers indeterminacy and
monetary policy from a dynamic general equilibrium perspective in order to study the robustness
of activist monetary policy advice, like that coming from the large literature on Taylor-type rules.1
We find these results to be quite robust. In fact,
multiple Pareto-ranked dynamic equilibria turn
out to occur whenever the monetary instrument
1

See the discussion in Woodford (2003) and Bullard and Mitra
(2002). For a discussion of the Taylor principle, see Woodford
(2001).

Gaetano Antinolfi is an associate professor of economics and Costas Azariadis is a professor of economics at Washington University in St. Louis.
Azariadis is a visiting scholar and James B. Bullard is a vice president and deputy director of monetary analysis at the Federal Reserve Bank
of St. Louis. The authors thank Peter Ireland for extensive comments on an earlier draft.

© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
only with prior written permission of the Federal Reserve Bank of St. Louis.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

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Antinolfi, Azariadis, Bullard

is used passively, without regard to the state of
the economy. In contrast, some types of informed
active policies eliminate most of the indeterminacy and force the economy onto a constrained
optimal path. Our framework also begins to
address interesting questions concerning the
nature of the interaction between monetary policy
and the smooth operation of credit markets.

What We Do
We study a dynamic general equilibrium
model of pure exchange that is a simplified version of Azariadis and Kaas (2007). Following
Eaton and Gersovitz (1981) and Kehoe and Levine
(1993), endogenous debt limits deter default by
households that cannot be forced to repay debts.
These households live forever and have variable
incomes. To keep the analysis tractable, we focus
on an economy with just two types of agents who
share a constant flow of total income. Income
shares fluctuate between low and high levels in
alternating periods. To smooth consumption perfectly, high-income agents could in principle lend
a large enough amount to low-income agents each
period to ensure that every household’s share of
total consumption remains constant. We show
that, under certain reasonable assumptions, this
first-best outcome cannot be achieved as an equilibrium with endogenous debt limits. Instead,
there are two steady states: a constrained efficient
outcome at a high interest rate in which credit markets work as well as possible and an inefficient
autarkic outcome at a low interest rate in which
credit markets break down and agents are unable
to smooth consumption at all.2 A continuum of
dynamical equilibria indexed by initial conditions
all tend toward the suboptimal steady state.
We introduce policy into this environment.
We discuss the possibility of fiscal tax-transfer
schemes that would in principle work well,
but which also require the policymaker to use
detailed information concerning household
incomes to make the correct resource realloca2

In the model, credit markets break down completely, but we think
of this as representing poorly functioning credit markets in which
the volume of borrowing and lending is less than it could be.

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tions. Passive monetary policy, which we think
of as a constant rate of growth of the money stock,
is always associated with indeterminacy and
particularly poor dynamics. We then turn to active
monetary policy, in which the policy instrument
is adjusted in a particular way in reaction to the
current state of the economy. We show that credible commitment to a certain active policy can
converge to the constrained efficient outcome
immediately if the policymaker reacts to the entire
state of the economy and gradually if the policy
rule responds only to prices. We regard this as a
version of the policy advice coming from related
literature on monetary policy in the face of important frictions in the economy, even though the
friction in this paper is quite different. We also
think this result suggests that good monetary
policy is partly responsible for the smooth functioning of credit markets, a sentiment that is often
stated in monetary policy circles.

Recent Related Literature
It is a typical result from the literature that
models with a role for fiat money tend to have a
nonmonetary steady state and an associated
indeterminacy. This is true in models of overlapping generations; but the demand for money
depends on beliefs in the search-theoretic monetary literature as well.3 The model here is more
closely related to Bewley-type economies.4
In the New Keynesian literature, such as
Woodford (2003), credit markets are complete
and work perfectly, even though there are other
frictions in that model. We also have complete
markets, but the friction in our setting directly
affects the incentives of households to lend appropriately. Thus, monetary policy in our framework
improves the operation of credit markets, whereas
in Woodford (2003) it has no particular effect on
the operation of these markets.
3

See, for example, the discussion in Wright (2005). Other examples
of indeterminacy include older Keynesian models with rational
expectations, and dynamic general equilibrium structures with
bubbles, complementarities, and increasing returns, such as those
reviewed in Boldrin and Woodford (1990), Cooper (1999), and
Benhabib and Farmer (1999).

4

See Bewley (1980) and Townsend (1980).

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Antinolfi, Azariadis, Bullard

Our results have a certain global flavor. In part
that is because the model is simple enough that
we can characterize the entire set of equilibria in
a fairly straightforward way. Other authors have
focused on a global perspective in models of
monetary policy, perhaps most prominently
Benhabib, Schmitt-Grohé, and Uribe (2001) and
Benhabib and Eusepi (2005). They emphasize
that active policies may be associated with local
determinacy but global indeterminacy. In
Benhabib, Schmitt-Grohé, and Uribe (2001), the
second steady state (the one not associated with
the inflation target of the monetary authorities)
is close to the Friedman rule, whereas the second
steady state in our framework is associated with
high inflation. Benhabib and his collaborators
emphasize how the design of policy may or may
not be able to avoid too low an inflation rate relative to the target, whereas we stress how the
design of monetary policy can avoid inflation
rates that exceed any reasonable target. In particular, Benhabib and Eusepi (2005) show that, in a
model with sticky prices, a feedback rule can
eliminate global indeterminacy if the monetary
instrument responds to the output gap.

A NONMONETARY MODEL
The economy we have in mind, but do not
analyze here, consists of a large number of agents,
possibly a continuum, with a common utility
function and a large variety of income processes.
Aggregate income can be thought of as constant so
that we may focus on fluctuations in the distribution of income among households and on the asset
trades they will conduct as they attempt to smooth
consumption. Individual consumption shares will
be constant if asset markets are perfect, but will
necessarily fluctuate if endogenous debt limits
constrain household borrowing.
To simplify matters and maintain tractability,
we analyze an economy with deterministic individual incomes populated by two agents indexed
by i = 0,1. Time is discrete and denoted by
t = 0,1,2,… Each agent i has preferences given by
`

(1)

∑

( )

β tu cti ,

t =0

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

with 0 < β < 1. The aggregate endowment is constant at two units, but its distribution over agents
changes deterministically over time. In particular,
individual endowments are periodic5; that is,
(2)

(1 + α , 1 − α ) if t = 0,2,...

(ω ,ω ) = (1 − α , 1 + α )
0
t

1
t



if t = 1,3,... ,

with α ∈共0,1兲. In addition, agent zero owes an
initial debt, B = α /共1+ β 兲, to agent one. This debt
makes the initial wealth of the two agents identical when incomes are discounted at the common
rate of time preference. In a more complicated
economy, agents would be indexed by α ∈共0,1兲;
some individual incomes would fluctuate only a
little, others would fluctuate quite a bit.

Perfect Enforcement
To fix ideas and notation, we start with a
standard dynamic general equilibrium model
with perfect enforcement of loan contracts. In
this setting, an equilibrium is an infinite sequence
共ctH, ctL, Rt 兲 that describes for each period t consumption for the high- and low-income agents
and the gross yield on loans. This sequence satisfies consumption Euler equations for each person, two intertemporal budget constraints, and
market clearing. Based on our assumptions concerning the initial distribution of wealth, it is obvious that the unique equilibrium is 共ctH, ctL, Rt 兲 =
共1,1,1/β 兲 for all t, and it is Pareto optimal. Individual consumption is a constant fraction of
aggregate consumption at all times.
Commitment to repay debts is essential in
achieving this allocation of resources. If borrowers
can in principle default on their loan obligations
at the cost of perpetual exclusion from both sides
of the asset market, as suggested by Kehoe and
Levine (1993), then the Pareto-optimal allocation
cannot be decentralized as a competitive equilibrium with limited enforcement unless it is weakly
5

In a growing economy, individual incomes need not be negatively
correlated but income shares must be. This simple deterministic
endowment process is the degenerate case of a stochastic economy
with two Markovian states with a zero probability of remaining in
the same state. Markovian endowments with two states are a
straightforward extension. The assumption of two states or dates
has obvious geometric advantages, but it is not innocuous where
policy is concerned. We discuss this point further in the conclusion.

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Antinolfi, Azariadis, Bullard

preferred to autarky by all agents at all times. It
is easy to check that the current autarky payoff is
(3)

u (1 + α ) + βu (1 − α )
1 − β2

for a high-income agent and
(4)

u (1 − α ) + βu (1 + α )
1 − β2

for a low-income one. These are dominated by
market participation and perpetual consumption
of one unit if, and only if,
(5)

u (1 + α ) + βu (1 − α ) ≤ (1 + β )u (1) .

This inequality holds under conditions similar
to those enumerated in Alvarez and Jermann
(2000, Proposition 4.9), which require that all
individuals have a strong need for consumption
smoothing. In particular, inequality (5) holds if
all individuals have a low intertemporal elasticity
of substitution, or a low rate of time preference,
or are subject to large individual income shocks.
Reasonable as they might seem for an economy
with two agents, these conditions are difficult to
achieve in an environment with a large variety of
agent types, some of whom will necessarily experience small income shocks. In what follows we
assume that inequality (5) fails6 and that autarky
is a state with a low implied rate of interest. Specifically, we assume

These relations are shown in Figure 1, where
the first-best allocation is on the diagonal and
point A represents autarky. An implied interest
factor of unity corresponds to point M.

Limited Enforcement
In environments where loan contracts are
enforced by perpetual exclusion of defaulters
from asset markets, equilibria are defined somewhat differently from standard models. In particular, an equilibrium is an infinite sequence,
共ctH, ctL, Rt , bt 兲, where bt is the debt limit assigned
to the low-income person at t. Agents maximize
taking Rt and bt as given, markets clear, and bt is
the largest possible debt limit that will keep borrowers at t from defaulting at date t + 1. These
limits must be binding by inequality (6), which
states that the first-best allocation 共ctH,ctL兲 = 共1,1兲᭙t
is ruled out by debt limits. In particular, (i) the
consumption Euler equation holds for the highincome agent and fails for the low-income agent;
that is,

β Rt =

(8)

( ) < u ′ (c ) .
u ′ (c ) u ′ ( c )
u ′ ctH

L
t +1

L
t

H
t +1

(ii) Budget constraints apply, with the low-income
agent borrowing at the debt limit from the highincome agent; that is,

ctH = 1 + α − R t −1bt −1 − bt

(9)
and

(6)

u (1 + α ) + βu (1 − α ) > (1 + β )u (1)
ctL = 1 − α + Rt −1bt −1 + bt .

(10)
and

(iii) Markets clear; that is,
(7)

u ′ (1 + α ) < βu′ (1 − α ) .
ctH + ctL = 2.

(11)
In a more complicated model with a continuum
of agents indexed by α, inequality (7) would have
to hold for some interval of α, in particular for
the highest values of α .7

And (iv) debt limits equate the autarkic and market
payoffs for a high-income consumer who is about
to repay last period’s debt; specifically,

6

(12)

7

If inequality (5) fails, then it is straightforward to show that highincome agents will prefer autarky to the perfect enforcement
allocation for any initial distribution of debt, not just for the distribution assumed in this paper.
If the utility function were logarithmic, inequality (7) would
require that the maximal value of α should exceed 共1 – β 兲/共1 + β 兲,
which implies that the maximal annual fluctuation in individual

334

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2007

( )

( )

u ctH + βu ctL+1 = u (1 + α ) + βu (1 − α )

for all t.
income should be no less than approximately 2 percent. Hence, it
seems quite plausible that the first-best allocation will be prevented
by endogenous debt limits.

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Antinolfi, Azariadis, Bullard

Figure 1
The Fundamental Diagram
c Lt+1 = 2 – xt+1

45˚

2

u(1+ α– ) + β u(1 – α– )
A′ = (1 – α , 1+ α )
u(1+ α ) + β u(1 – α )
(1+ β )u(1)

T ′ = (2 – x̂, x̂ )
(1,1)

T = ( x̂ , 2 – x̂)

M

A = (1+ α ,1 – α )

1+ α–

If we define ctH = xt ∈ 关1,1+ α 兴, then it is clear
that equilibria are solution sequences to equation
(12), that is, to
(13) u ( x t ) + βu (2 − x t +1 ) = u (1 + α ) + βu (1 − α ) .
These sequences are shown in Figure 1.

Real Indeterminacy
If inequalities (6) and (7) hold, Figure 1 shows
that there are two steady states. This first is a sta–
ble autarkic state, 共ctH, ctL, Rt , bt 兲 = 共1+ α , 1 – α , R ,0兲
for all t, where
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

(14)

2

cH
t = xt

R=

u ′ (1 + α )
< 1.
βu ′ (1 − α )

This state corresponds to point A in Figure 1.
The loan market is shut down in this state. The
second is an unstable trade state, 共ctH, ctL, Rt , bt 兲 =
共x̂,2 – x̂, R̂,b̂ 兲 for all t, where x̂ ∈ 共1,1+ α 兲 is the
unique solution to
(15) u ( x ) + βu (2 − x ) = u (1 + α ) + βu (1 − α ),

(16)

Rˆ =

u ′ ( xˆ )
,
βu ′ (2 − xˆ )

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Antinolfi, Azariadis, Bullard

and
(17)

1 + α − xˆ
bˆ =
.
1 + Rˆ

This state corresponds to point T in Figure 1. The
loan market is active in this state. Because T lies
between point M and the diagonal, we have
(18)

Rˆ ∈(1, 1 / β ).

PASSIVE FISCAL AND MONETARY
POLICIES
Fiscal Policy with Zero Debt Limits
We explore here the possibility of achieving
the constrained efficient allocation by a passive
fiscal or monetary policy, that is, by choosing
policy instruments that are invariant to the history
of economic events. We start with a constant
lump-sum tax, τ , on the high-income agent and
an equal subsidy to the low-income agent. Any
tax we choose must support an equilibrium allocation that is weakly preferred to autarky by all
agents at all times. This feasibility requirement
excludes tax and transfer schemes that would
equalize post-transfer endowments in all periods,
thus implementing the first-best allocation xt = 1᭙t
as an autarkic equilibrium. However, the policy
τ = 1+ α – x̂ shifts the endowment point from
point A to point T in Figure 1 and implements
the constrained optimal allocation as a unique
post-transfer autarkic equilibrium at the highinterest yield, R̂ ∈ 共1,1/β 兲. All agents weakly
prefer this outcome to the pre-transfer autarkic
–
equilibrium at the low-interest yield, R < 1.
The only problem with this policy is that it
relies on precise information about individual
incomes, especially if there were a large variety
of income types. Policy in this setting must be
able to tailor individual transfers to individual
incomes. Are there simpler ways to achieve
desirable outcomes with a blunter policy instrument that requires less information—that is, that
does not discriminate between individuals?

Because autarky is associated with an interest
factor below 1, and the trading state with an interest factor above 1, it follows from Alvarez and
Jermann (2000, Proposition 4.6) that the trading
state is constrained optimal8 and the autarkic
state is not. Individual consumption shares fluctuate less in the constrained optimal state than
they do in the autarkic state.
In addition to the two steady states, there is a
continuum of equilibrium sequences 共xt 兲 indexed
on x0 ∈ 共x̂,1+ α 兲, which converge to autarky. See
again Figure 1. All of these sequences can be
Pareto ranked by the initial consumption, x0.
Equilibrium outcomes are indeterminate in
this nonmonetary economy for reasons that have
nothing to do with the intertemporal elasticity of
substitution in consumption or the lack of gross
substitutes as commonly understood. Instead,
indeterminacy in this environment comes from
dynamic complementarities between current and
expected future debt limits. In particular, low
future debt limits reduce gains from future asset
trading and lower the current payoff to solvency.
This, in turn, raises the incentive to default, which
must be deterred by tighter debt limits now.
We conclude that the constrained optimal
allocation of consumption 共x̂,2–x̂兲 can be achieved
only if all future debt limits are expected to stay
exactly at b̂ . Any other expectations will lead
inevitably to autarky or to the nonexistence of
equilibrium. In the remainder of the paper, we
will explore whether, and how, policies can guide
individual expectations in a manner that leads
away from autarky and, perhaps, toward the
constrained optimal allocation.

One completely anonymous instrument is
fiat money printed to pay an equal lump-sum
transfer to all agents. Positive lump-sum transfers
flatten the distribution of current resources among
households, and negative transfers skew that
distribution in favor of high-income persons.
This, in turn, enables monetary policy to control
the real yield on money,9 which is the reciprocal

8

9

An allocation is constrained optimal if it satisfies the usual resource
constraints, is weakly preferred to autarky by all agents at all times,
and cannot be dominated by another feasible allocation.

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2007

Monetary Policy with Zero Debt Limits

Identical outcomes can be achieved by changes in the stock of
public debt because money and debt are perfect substitutes in
our economy. We use the term “monetary policy” advisedly here

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Antinolfi, Azariadis, Bullard

of the inflation factor, along any equilibrium path.
To see this, we let Mt be the stock of money per
agent, µt be the gross rate of money growth, pt be
the price level, τ t be the real value of the transfer,
and mt = Mt /pt . Policymakers choose the sequence
共µt 兲 under the restriction that the resulting monetary equilibrium is weakly preferred to autarky
by all agents at all times. We assume that agents
have the option of rejecting monetary transfers
and taxes in favor of autarky.
Assuming for the moment that debt limits
are zero (we relax this assumption in the next
subsection), budget constraints are
(19)

ctH = 1 + α + τ t − mdt ,

where mtd is the demand for money by highincome agents and
(20)

ctL+1 = 1 − α + τ t +1 + R t mdt ,

where Rt = pt /pt +1 is the real rate of return on
money. Low-income agents are assumed to be
rationed and to spend their entire money balances
to raise current consumption.
Equilibrium in this economy satisfies the
consumption Euler equation (8) for the highincome agent, rewritten here as
(21)

( )

( )

u′ ctH = β Rt u′ ctL+1 ,

as well as equilibrium in the goods and money
markets; that is,
(22)

ctH + ctL = 2,

(23)

mtd = 2mt .

In addition, individual budget constraints apply;
that is,
(24)


1
ctH ; x ( mt , µt ) = 1 + α −  1 +  mt .
µt 


The real return on money is
(25)

Rt =

pt
mt +1
=
.
pt +1 µt +1mt

because we expect that our results carry over to economies where
debt dominates money in rate of return.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

We conclude that, for a given policy sequence
共µt 兲, equilibria are bounded non-negative solution
sequences 共mt 兲 to the nonautonomous equation,
mt +1
(26) u′  x ( mt ,µt ) = β
u ′ 2 − x ( mt +1 ,µt +1 ) .
µ m 
t +1

t

Equilibria that converge to autarky are driven by
self-confirming inflationary expectations that
reduce the demand for real money balances and
diminish trading between high- and low-income
agents. One important drawback of passive policies is that they are unable to connect future
returns on money with the current state of the
economy and therefore cannot counter inflationary expectations with tighter monetary policy,
that is, by lowering µt .
Figure 2 shows the qualitative properties of
solutions that correspond to a passive monetary
–
policy, µt = µ ∈[1/Rˆ ,1/R ]᭙t, for an economy in
which dated consumption goods are gross substitutes. Each policy is associated with two steady
states: a stable autarkic state with m = 0 and an
unstable trading state with m *共µ 兲 > 0. In general,
higher values of µ correspond to lower steadystate returns on money and to a lower demand
for money, m*共µ 兲. For example, µ = 1/R̂ ∈共β,1兲
supports the constrained optimal trading state,
ctH = x̂, for all t by raising the value of real balances to
(27)

1 + α − xˆ
m∗ 1 / Rˆ =
.
1 + Rˆ

(

)

This value of µ, which involves a mild deflation at the steady state of an economy with zero
income growth, is the lowest feasible rate of
growth consistent with all agents preferring monetary equilibrium to autarky at all times. The
Friedman rule, µ = β , which would support the
first-best allocation ctH = 1 for all t is simply not
feasible: It imposes too large a tax on high-income
agents, causing them to choose autarky over the
use of money. As we raise the value of µ above
1/R̂, the amount of trading between the two
groups of agents shrinks, vanishing at µ = 1/R̂;
here the steady-state return on money is equal to
the autarkic rate of return.
Figure 2 reveals that feasible passive policies
cannot overcome the indeterminacy problem of
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Antinolfi, Azariadis, Bullard

Figure 2
Passive Policies
mt+1

µ=β
45˚

µ = 1/R̂
–
µ = 1/R

µ=1

Infeasible Policies

(0,0)

1+ α – x̂
1+ R̂

self-fulfilling inflationary expectations that leads
to reduced trade among households. To solve this
problem, policy should connect the current state
of the economy with future returns on money,
that is, with expectations of future inflation. We
explore active feedback policies in the “Active
Monetary Policy” section.

Monetary Policy with Positive Debt
Limits
Positive debt constraints add nothing essential to the equilibria described in the previous
subsection because holdings of private debt are a
perfect substitute for balances of fiat money. A
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2007

α
1+1/β

mt

high-income individual has the same payoff when
trading loans as he would if he held only money.
Loan default does not hurt the trading opportunities of any individual.
If debt limits are a positive sequence 共bt 兲, the
budget constraint for a high-income individual,

1
ctH = 1 + α −  1 +  mt − Rt −1bt −1 − bt
µt 

(28)
= 1 + α − ( mt + bt ) − Rt −1 ( mt −1 + bt −1 ),

indicates that only the size of the asset portfolio
matters for individual plans, not its division into
fiat money and debt. Money displaces private
loans at a one-to-one rate.
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Antinolfi, Azariadis, Bullard

ACTIVE MONETARY POLICY
Suppose next that monetary policy can control directly the real yield on money balances in
a manner that depends on the entire state of the
economy. We index that state by 共xt , Rt 兲, where xt
is twice the consumption share of the high-income
agent and Rt is the real yield on money and debt.
Equivalently, we may index the state by 共mt , Rt 兲,
where mt is real balances per capita. Then we
write the policy function as
(29)

Rt +1 = f ( x t , Rt ) .

R = f (1 + α , R ),

(31)

Rˆ = f xˆ , Rˆ .

( )

An example of this type of policy is

(


R
if ( x , R) = 1 + α , R

(32) f ( x , R) =  2 − x̂
 β 2 − β Rx otherwise.
)
 (

)

Given the policy f, an equilibrium is a
sequence 共xt , Rt 兲 that satisfies (29), plus the consumption Euler equation of the high-income
agent, rewritten here as
(33)

(34)

x t +1 = 2 − β R t x t ,

(35)

Rt +1 = f (x t ,Rt ) .

All we need do is study the characteristic polynomial p共λ 兲 in the neighborhood of any steady
state 共x,R兲. That polynomial is
(36) g ( λ ) = λ 2 + β R ( β R − ε R ) λ + β R ( ε x − ε R ),

The arbitrary function f maps the product space
–
–
[1,1+ α] × [R ,R̂] into [R ,R̂]. It should be consistent
with autarky (because autarky is an equilibrium
whenever households refuse to accept fiat money
in exchange for goods) and with the constrained
optimal state 共x̂, R̂兲 (because this state is a reasonable target for a benevolent policymaker). This
requires that
(30)

policies for a logarithmic utility function in the
neighborhood of each steady state. In this class
of economies, equilibria satisfy

u ′ ( x t ) = β Rt u ′ (2 − x t +1 ) .

By construction, the dynamical system (29) and
–
(33) has two steady states, 共x,R兲 = 共1+ α ,R 兲 and
共x̂,R̂兲. A sensible policy f共.兲 ensures that equilibria
starting at any point 共x0 ,R0 兲 move away from the
–
suboptimal autarkic state 共1+ α ,R 兲 and converge
rapidly to the constrained optimal state 共x̂,R̂兲.
We do not attempt a global characterization
of policies that achieve this objective for an arbitrary utility function u共.兲. We instead confine
ourselves to exploring the properties of such
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

where εx and εR are partial elasticities of the policy
function f with respect to x and R, and β R < 1 at
each steady state.
Desirable policies, as we have described them,
should turn the constrained optimal steady state
into an attractor, or sink, and the suboptimal state
into a source. The eigenvalues, or roots of the
polynomial g共λ 兲, should be inside the unit circle
at 共x,R兲 = 共x̂, R̂兲 and outside the unit circle at
–
共x–, R 兲. One way to choose is to focus on functions
f that raise future real yields whenever households with currently high incomes consume “too
much” (relative to the efficient outcome, x̂) and
demand “too little” money.
It is easy to check whether the policy function
in equation (32) has exactly this property, which
would furthermore guarantee immediate convergence to the constrained efficient state 共x̂,R̂兲 from
any initial condition other than autarky. Under
this policy, the dynamical system consisting of
equations (34) and (35) has a double real eigenvalue with modulus zero at 共x̂,R̂兲.
Exactly what does the monetary authority
have to do to control the real rate of interest in
the manner specified by the policy function in
(32)? This can be answered with a logarithmic
utility function: Combine the budget constraint
(24) with the Euler equation (26) to obtain an
expression that connects monetary policy at date
t +1 with monetary policy at t and with the state
of the economy at t. In particular, we find that

(37)

1 + µt +1


1−α 
β xt −

Rt 
1 + µt 
=
.
µt
(1 + α − x t )
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Antinolfi, Azariadis, Bullard

We conclude that monetary policy tightens
( µt +1 falls) subsequent to a rise in the rate of inflation and a drop in the rate of return on money.
Equation (37) says that the tightening appears to
be substantial. For example, if α is small relative
to unity, β is about 0.95, and the ratio of money
to income is about 1:7; then, each additional 1
percentage point of inflation near the constrained
efficient steady state causes the money growth
rate to drop by about 7 percentage points. To see
this, we calculate the response of the money
growth rate to changes to the past inflation rate
from equations (37) and (24) and obtain
d µt +1
1−α
.
=
d (1 / R t )
mt

Then, we set α = 0 and mt = 1/7.
This is the sort of strongly reactive policy
that guides inflationary expectations to just the
level needed to support the constrained efficient
outcome. In a similar but not quite as effective
way, outcomes can be achieved if the policy rule
simply maps current inflation into future inflation, ignoring current quantities such as xt and
mt . Specifically, if we employ the rule
Rt +1 = φ ( Rt ),
–
where φ maps the interval [R ,R̂] into itself, then
the dynamical system consisting of equations (34)
and (38) has two real eigenvalues, –β R and φ ′共R兲,
at any steady state 共x,R兲. Recall that β R ∈共0,1兲 at
–
both R and R̂.
–
Therefore, any policy rule such that 円φ ′共R 兲円 > 1
and 円φ ′共Rˆ 兲円 < 1 will convert the autarkic state A
into a saddle and the trading state T into a sink.
For most initial conditions 共x0 ,R0 兲, monetary
policy leads the economy to converge asymptotically, but not immediately, to the constrained
efficient state.

(38)

CONCLUSIONS AND EXTENSIONS
This paper provides general equilibrium
examples of how active monetary policy can be
used to select a desirable outcome in economies
where passive policies are associated with many
Pareto-ranked dynamic equilibria.
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2007

In our setting, monetary policy works directly
on rational beliefs about future values of the
inflation rate, debt limits, and other financial
variables. It does so by committing to a feedback
rule that connects current financial conditions
with future values of the policy instrument and,
in particular, to a shared belief that asset returns
will improve substantially when the volume of
asset trading falls below what is consistent with
an efficient allocation of resources.
When viewed as an exercise in equilibrium
selection, monetary policy is an attempt to foster
expectations that lead to socially desirable states
of the economy as rapidly as possible. This attempt
is completely successful in our simple setting
where money is a perfect substitute for private
debt, that is, a store of value for two agents trading in complete, albeit imperfect, asset markets.
Efficiency in this setting is achieved when the
volume of loans is as large as capital-market
imperfections will allow. If that volume is less
than it should be, properly valued money can
act as a substitute for private loans. The job of
the central bank is to defend the correct value of
money by connecting expectations of future inflation with current economic conditions and intervening aggressively to pin inflation expectations
to the right value.
We are not sure that a blunt policy instrument
such as anonymous monetary policy will be as
successful in selecting constrained optimal outcomes in a richer environment with many agents
and uncertainty. In particular, if money and debt
are imperfect substitutes because the former has
a liquidity advantage over the latter, then monetary policy has implications for debt limits and
for the participation of households in financial
markets. In addition, policy choices may not be
conditioned on the entire state of the economy if
that state includes detailed information about
individual incomes and trading plans. In that case,
the policymaker may have to settle for something
less than constrained efficiency, as in Benhabib
and Eusepi (2005). These implications need to be
carefully explored before we can design monetary
rules with any degree of confidence.
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Antinolfi, Azariadis, Bullard

REFERENCES
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Azariadis, Costas and Kaas, Leo. “Asset Price
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Monetary and Fiscal Policy: A Global Perspective.”
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Benhabib, Jess and Farmer, Roger E.A. “Indeterminacy
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Benhabib, Jess; Schmitt-Grohé, Stephanie and Uribe,
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Bewley, Truman. “The Optimum Quantity of Money,”
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Boldrin, Michele and Woodford, Michael. “Equilibrium
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Cooper, Russel W. Coordination Games:
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Sargent, Thomas J. and Wallace, Neil. “Rational
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Townsend, Robert M. “Models of Money with
Spatially Separated Agents,” in J. Kareken and
N. Wallace, eds., Models of Monetary Economies.
Federal Reserve Bank of Minneapolis, 1980.
Woodford, Michael. “The Taylor Rule and Optimal
Monetary Policy.” American Economic Review,
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Woodford, Michael. Interest and Prices: Foundations
of a Theory of Monetary Policy. Princeton, NJ:
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Wright, Randall. “Introduction to ‘Models of Monetary
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Bullard, James B. and Mitra, Kaushik. “Learning
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F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Commentary
Peter N. Ireland

I

n their conference paper, Gaetano Antinolfi,
Costas Azariadis, and James Bullard (2007)
develop and analyze a macroeconomic
model with heterogeneous agents in which
individual incomes fluctuate, aggregate income
remains constant, and frictions that inhibit the
strict enforcement of private contracts place
endogenous limits on agents’ ability to borrow
and lend and hence to engage in intertemporal
trade. Further, because the model features only
a single good, intertemporal trade is the only
trade that potentially takes place in equilibrium.
Bewley (1980), Townsend (1980), Kehoe and
Levine (1993), Kocherlakota (1996), and Alvarez
and Jermann (2000) previously and famously
considered similar models. Here, however,
Antinolfi, Azariadis, and Bullard go beyond all
of this previous work by highlighting that these
models typically feature multiple equilibria.
Here, in fact, the authors’ model has two
steady-state equilibria under laissez-faire. In one,
no trade takes place, so that equilibrium allocations are autarkic; in the other, agents trade
actively. Hence, the two steady states can be
Pareto-ranked: All agents prefer the good equilibrium with trade to the bad equilibrium without.
The authors’ policy problem then arises, because
the bad steady state is stable and the good steady
state is unstable, implying that even if the economy begins arbitrarily close to but not exactly in
the good steady state, it will converge over time
to the bad steady state. In Antinolfi, Azariadis,
and Bullard’s analysis, the government’s stabiliza-

tion policy aims at keeping the economy at or near
the good steady state.
Stabilization policy in this analysis therefore
plays an important but somewhat unfamiliar role.
Typically, in mainstream macroeconomic models,
stabilization policy calls for the monetary and
fiscal authorities to adjust their policy instruments
in response to shocks that buffet the economy
around a given steady state. In Antinolfi, Azariadis,
and Bullard’s model, by contrast, stabilization
policy works on a more fundamental level, to
actually pick out the steady state toward which
the economy gravitates. Hence their paper’s title,
“Monetary Policy as Equilibrium Selection.”
Here, monetary policy helps achieve this stabilization goal by reversing the properties of the
two steady states, rendering the good steady state
stable and the bad steady state unstable.
Specifically, the authors show that active
policies that call for the government to adjust its
policy instruments vigorously in response to
changes in the underlying state of the economy
succeed in achieving this goal. By contrast, passive policies—including constant money growth
rate rules—that call for little or no policy response
to changes in the economy fail by leaving the
bad steady state as the economy’s most likely
destination.
Antinolfi, Azariadis, and Bullard’s analysis,
results, and conclusions combine to make their
paper quite interesting and useful. The paper is
novel in its focus on active versus passive policy
rules in models of the type used in Bewley (1980)
and Townsend (1980). Ljungqvist and Sargent

Peter N. Ireland is a professor of economics at Boston College and a research associate at the National Bureau of Economic Research.
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 343-48.
© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
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Ireland

(2000), for example, take a far more limited
approach to policy analysis in a version of
Townsend’s (1980) turnpike model. Specifically,
Ljungqvist and Sargent simply assume that the
economy starts in its good steady state and then
ask what the optimal constant rate of money
growth in that good steady state is. Antinolfi,
Azariadis, and Bullard qualify and extend these
earlier results in an important way by making
clear that Ljungqvist and Sargent’s preferred constant money growth rate rule does not prevent
the economy from leaving a neighborhood of its
good steady state and converging to the bad
steady state instead.
By highlighting the importance of this activeversus-passive distinction for the design of
welfare-enhancing monetary policy, Antinolfi,
Azariadis, and Bullard’s paper also becomes quite
useful, as it draws previously unnoticed links
between the branch of the literature that works
with Bewley-Townsend-type models and another
branch of the literature in monetary economics
that works with a very different class of models.
In particular, recent work with New Keynesian
models featuring monopolistic competition and
staggered nominal price setting in goods markets
establishes what Woodford (2003) and others call
the “Taylor principle.” This Taylor principle indicates that the central bank can stabilize the inflation rate around a desired target value through the
use of an interest rate rule for monetary policy of
the kind proposed by Taylor (1993), provided that
rule is sufficiently active, calling for a vigorous
adjustment of the short-term nominal interest
rate instrument in response to shocks that push
the inflation rate away from target. Antinolfi,
Azariadis, and Bullard’s results favor the use of
active monetary policy rules as well, helping to
establish the generality and robustness of these
findings across two otherwise divergent branches
of inquiry.
This new paper by Antinolfi, Azariadis, and
Bullard thereby contributes importantly to the
literature. It extends, as the other conference
papers do, the “Frontiers in Monetary Policy
Research.” What’s more, like many other papers
that extend the frontiers of research—particularly
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in monetary economics, it seems—this new paper
raises a host of additional questions at the same
time that it provides answers to existing ones.
The remainder of my discussion focuses on
some of these additional questions, pointing as
Antinolfi, Azariadis, and Bullard’s paper itself
does to promising avenues for future research.

IS PUBLIC POLICY REALLY
NECESSARY?
This first and most basic question asks
whether public policy is really crucial in an
economic environment like the one described by
Antinolfi, Azariadis, and Bullard’s model. In their
paper, the authors themselves provide a partial
response to this question by indicating that the
answer is “no” if private credit markets work well
to begin with. In particular, the authors show that,
when contracts can be perfectly enforced, trading
in private credit markets supports an equilibrium
allocation that is Pareto optimal. In this special
case, government policy cannot help; laissez-faire
works best.
Yet one might go a step further, as I am
tempted to do, and note that even with limited
contractual enforcement, the scope for welfareenhancing public policy, though present, will
necessarily be limited to the extent that the
autarkic equilibrium is really not so bad. I raise
this possibility with a specific concern in mind.
The point is that all of these terms—“bad equilibrium,” “autarkic allocations,” “unstable steady
states,” and so on—have very specific meanings
when used in the context of a formal study in
macroeconomic theory like Antinolfi, Azariadis,
and Bullard’s. Of course, the authors very carefully and properly use these terms in their paper.
However, the risk remains that, when presented
to a broader audience of nonspecialists and policymakers, these words will unintentionally conjure
up images of disastrous outcomes under laissezfaire; in fact, though, a full, quantitative assessment of the welfare properties of equilibrium
outcomes with and without government intervention—perhaps along the same lines as that
presented by Krueger and Perri (2005) but applied
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Ireland

to the specific environment studied here—remains
a task for future research.
In his famous essay “The Role of Monetary
Policy,” Milton Friedman (1968, p. 14) cautions
against the tendency toward overconfidence in
economists offering policy advice: “[I]n this area
particularly,” he warns, “the best is likely to be the
enemy of the good.” It is almost surely true that,
in reality, as in Antinolfi, Azariadis, and Bullard’s
model, frictions prevent private markets—especially private credit markets—from operating with
total efficiency so as to bring equilibrium allocations in line with Pareto-optimal outcomes. Yet,
as Friedman emphasizes, it seems equally true
that, in reality, even the most carefully designed
government policies introduced into environments in which outcomes under laissez-faire are
clearly suboptimal have often made matters much
worse instead of much better. The inefficiencies
in private credit markets are usefully highlighted
in Antinolfi, Azariadis, and Bullard’s model. But,
before we lean too heavily on those inefficiencies
as the basis for justifying activist government
intervention in those same segments of the U.S.
economy, future research must more forcefully
establish that those inefficiencies are severe
enough, quantitatively, to also justify the risk
that a well-designed public policy will be poorly
implemented or will otherwise have unintended
and detrimental consequences. Many sad lessons
from history teach us that “reversion to autarky,”
in the vernacular as opposed to the language for
formal economic theory, most frequently occurs
precisely because of excessive government
involvement in private markets.

IS MONETARY POLICY REALLY
NECESSARY?
Although fiscal policy, in the form of a carefully designed system of income taxes and transfers, might seem to be the most direct and effective
way of helping private agents in Antinolfi,
Azariadis, and Bullard’s model stabilize their consumptions in the face of their fluctuating income
streams, the authors point out that the successful
implementation of such a policy requires the
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government to obtain and exploit detailed information about individual agents’ economic circumstances. On these grounds, they advocate the
search for monetary policy rules that help accomplish the same goal of income redistribution.
Along the same lines, however, one might
also note that the monetary policy rule that the
authors propose later, shown in their equation
(37), requires the central bank to adjust the rate
of money growth in response not just to movements in the aggregate variable R, which measures
the real return to money (or the inverse of the
inflation rate), but also to the variable x, which
measures not aggregate income or consumption
but rather the share of aggregate consumption
enjoyed by high-income agents. In a more complicated model with richer forms of heterogeneity,
the analog to the variable x would be a statistic
or set of statistics summarizing the cross-sectional
distribution of consumption. Successful implementation of this preferred monetary policy, therefore, also requires the government to collect and
process much of the same individual-specific
data needed to run an optimal tax-and-transfer
fiscal scheme.
For this reason, an alternative policy rule that
takes the form of the authors’ equation (38) and
therefore calls for a monetary response to changes
in the aggregate variable R alone may represent a
more appealing and realistic alternative to pure
laissez-faire or to a passive constant money growth
rate rule. In any case, working out the implications
of private information and the incentives that the
government can offer agents to truthfully reveal
that private information in settings like that
described by Antinolfi, Azariadis, and Bullard’s
model remains another important task for future
research; those implications may draw sharper
and more reliable distinctions between fiscal and
monetary policies as effective tools for income
redistribution.

IS TIME CONSISTENCY A
PROBLEM?
In Antinolfi, Azariadis, and Bullard’s model,
activist policy works to stabilize the economy
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around its good steady state by influencing private
expectations of future inflation under various
contingencies that arise both in and out of equilibrium. As Kydland and Prescott (1977) emphasize, however, public policymakers who make
announcements in an attempt to shape private
expectations often fall victim to the time-consistency problem. Once private expectations based
on a policymaker’s announcements have been
built into private decisions, that same policymaker may have an incentive to deviate from his
or her promised action. The problem then arises
because private agents recognize that the policymaker has this incentive to renege on any initial
promise. In equilibrium, a policymaker without
the ability to commit strongly to a preannounced
policy may be unable to influence expectations
in the desired way.
All of Antinolfi, Azariadis, and Bullard’s
analysis proceeds under the assumption that the
central bank has this ability to commit. At the
same time, however, their model builds directly
and importantly on the idea that private agents’
inability to precommit to their own future actions
is precisely what provides room for welfareenhancing public policy in the first place. What
justifies this assumption that the government faces
no similar commitment problem? And if the
optimal activist monetary policy rules shown in
equations (37) and (38) turn out to be time inconsistent, how do optimal policies under discretion
compare with these counterparts under commitment, both in terms of their implications for the
behavior of the money stock and inflation and in
terms of their ability to stabilize the economy
around the good steady state? These questions,
too, remain to be answered in future research.

IS CREDIBILITY A PROBLEM?
WHICH EQUILIBRIA ARE
EXPECTATIONALLY STABLE?
In addition to the time-consistency problem
described above, a second potential difficulty
may arise when the central bank tries to use the
optimal activist policies described by equations
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(37) and (38) to stabilize the economy around its
good steady state: Once the economy reaches the
good steady state—immediately under (37) and
eventually under (38)—these activist policies call
for constant money growth and inflation rates and
may therefore appear to private agents as being
observationally equivalent to passive policies,
such as a constant money growth rate rule. Hence,
once the economy reaches the good steady state,
either of these activist policies retains its power
to stabilize the economy only through the effects
that the central bank’s commitment to the policy
rule has on private expectations of what would
happen, out of equilibrium, if the economy begins
to slip away from that good steady state.
Given the potential tenuousness of the expectational forces keeping the economy in the good
steady state, even under an active monetary policy
rule, one might reasonably ask, What would happen if, instead of forming their expectations based
on how they believe the government would
behave out of equilibrium, private agents formed
their expectations based on how they actually
observe the government to behave in equilibrium?
Would the central bank have to act, periodically
at least, to maintain the credibility of its commitment to the optimal rule?
Often, in the literature following Kydland
and Prescott (1977), “credibility” is used synonymously with “time consistency.” In this case,
however, the term as I use it refers to ideas that
are closer in spirit to the concepts of “expectational stability” and “learnability” that, in previous work, Bullard (2006) uses to characterize the
government’s ability to keep the economy in or
around a desired steady state when private agents
form their expectations adaptively, based on historical data as opposed to full knowledge of the
economy’s true structure.
Examining the need and scope for activist
monetary policy to stabilize the economy
described by Antinolfi, Azariadis, and Bullard’s
model around the good steady state when expectations are formed through adaptive learning
also remains an important and useful task for
future research.
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ARE ACTIVE POLICIES
ROBUSTLY OPTIMAL?
This last question comes full circle, back to
Milton Friedman’s (1968) caveats about activist
public policymaking. Antinolfi, Azariadis, and
Bullard’s main result, concerning the optimality
of activist policy rules, seems quite sensible: If
the central bank wants to stabilize the economy
around a desirable steady state, then it certainly
stands to reason that its monetary policy ought
to react strongly whenever the economy begins
to deviate from that steady state. The authors’
main result shares the same powerful, intuitive
appeal as the Taylor principle from the literature
on New Keynesian economics.
However, their statement about robustness—
that, looking across many different macroeconomic models, optimal policy rules are all
activist—remains logically distinct from (and
therefore does not imply) another statement about
robustness: that any given activist policy rule,
fine-tuned to fit the special features of any given
model, will continue to work well across many
different macroeconomic models. Barnett and
He (2002) make this point quite forcefully, using
methods and arguments that are quite similar to
Antinolfi, Azariadis, and Bullard’s.
In this earlier paper, Barnett and He focus on
a macroeconomic model that is quite different
from the one studied here by Antinolfi, Azariadis,
and Bullard; specifically, Barnett and He work
with an older-style, medium-scale macroeconometric model developed originally by Bergstrom,
Nowman, and Wymer (1992). Nevertheless,
Barnett and He begin their analysis just as
Antinolfi, Azariadis, and Bullard do, by demonstrating that, although the Bergstrom-NowmanWymer model has an unstable steady state under
laissez-faire, it can be stabilized by an appropriately designed activist fiscal policy rule. At the
same time, however, Barnett and He also show
that this activist fiscal policy rule, when properly
calibrated to stabilize the economy under a given
configuration of the model’s nonpolicy parameters,
works counterproductively to destabilize the
economy still further when improperly calibrated
to a slightly different set of nonpolicy parameters.
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Barnett and He’s results thereby echo
Friedman’s caveat about the best being the enemy
of the good by confirming that an activist policy
that is fine-tuned to work well within one particular model may perform quite poorly when
applied to a very similar, but still slightly different,
economic environment. Barnett and He’s results
clearly indicate that additional careful and rigorous analyses like Antinolfi, Azariadis, and
Bullard’s are needed to establish the robustness
of optimal activist fiscal and monetary policies.

CONCLUSION
Antinolfi, Azariadis, and Bullard’s conference
paper contributes to scientific knowledge in several ways. It stands as the first paper to consider
the important distinction between active and
passive policy rules in a heterogeneous-agent
model with endogenously incomplete markets
that builds on Bewley’s (1980) and Townsend’s
(1980) early formulations. By considering this
distinction and by highlighting the stabilizing
powers of activist monetary policy rules, it also
draws useful and previously unnoticed links
between the branch of the literature in monetary
economics that studies the properties and implications of Bewley-Townsend-type models and
the until-now completely distinct branch of the
literature that studies New Keynesian models of
monopolistic competition and nominal price
rigidity. Finally, Antinolfi, Azariadis, and Bullard’s
paper contributes to scientific knowledge by raising a host of questions for future researchers who
share these authors’ technical sophistication, fine
attention to detail, and intellectual rigor.
Before closing, let me ask some of these questions again, phrasing them in a slightly different
way than they appear in my discussion above. The
optimal activist monetary policy characterized
by Antinolfi, Azariadis, and Bullard’s equation
(37) calls, in the authors’ own words (p. 340), for
the “money growth rate to drop by about 7 percentage points” in response to “each additional
1 percentage point of inflation.” Is this optimal
policy time consistent? Is this optimal policy
credible or expectationally stable? Is this optimal
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policy robust to various changes in the economy
environment? And is this optimal policy really
necessary? All of these questions await the same
type of careful and rigorous analysis contained
in Antinolfi, Azariadis, and Bullard’s very fine
conference paper.

REFERENCES

Kehoe, Timothy J. and Levine, David K. “DebtConstrained Asset Markets.” Review of Economic
Studies, October 1993, 60(205), pp. 865-88.
Kocherlakota, Narayana R. “Implications of Efficient
Risk Sharing without Commitment.” Review of
Economic Studies, October 1996, 63(217),
pp. 595-609.

Alvarez, Fernando and Jermann, Urban J. “Efficiency,
Equilibrium, and Asset Pricing with Risk of Default.”
Econometrica, July 2000, 68(4), pp. 775-97.

Krueger, Dirk and Perri, Fabrizio. “Public versus
Private Risk Sharing.” Unpublished manuscript.
Department of Economics of Goethe University,
Frankfurt, December 2005.

Antinolfi, Gaetano; Azariadis, Costas and Bullard,
James B. “Monetary Policy as Equilibrium
Selection.” Federal Reserve Bank of St. Louis
Review, July/August 2007, 89(4), pp. 331-41.

Kydland, Finn E. and Prescott, Edward C. “Rules
Rather than Discretion: The Inconsistency of Optimal
Plans.” Journal of Political Economy, June 1977,
85(3), pp. 473-92.

Barnett, William A. and He, Yijun. “Stabilization
Policy as Bifurcation Selection: Would Stabilization
Policy Work if the Economy Really Were Unstable?”
Macroeconomic Dynamics, November 2002, 6(5),
pp. 713-47.

Ljungqvist, Lars and Sargent, Thomas J. Recursive
Macroeconomic Theory. Cambridge: MIT Press,
2000.

Bergstrom, A.R.; Nowman, K.B. and Wymer, C.R.
“Gaussian Estimation of a Second Order Continuous
Time Macroeconometric Model of the UK.”
Economic Modelling, October 1992, 9(4), pp. 313-51.
Bewley, Truman. “The Optimum Quantity of Money,”
in John H. Kareken and Neil Wallace, eds., Models
of Monetary Economies. Federal Reserve Bank of
Minneapolis, 1980.
Bullard, James B. “The Learnability Criterion and
Monetary Policy.” Federal Reserve Bank of St. Louis
Review, May/June 2006, 88(3), pp. 203-17.

Taylor, John B. “Discretion versus Policy Rules in
Practice.” Carnegie-Rochester Conference Series on
Public Policy, December 1993, 39, pp. 195-214.
Townsend, Robert M. “Models of Money with
Spatially Separated Agents,” in John H. Kareken
and Neil Wallace, eds., Models of Monetary
Economies. Federal Reserve Bank of Minneapolis,
1980.
Woodford, Michael. Interest and Prices; Foundations
of a Theory of Monetary Policy. Princeton: Princeton
University Press, 2003.

Friedman, Milton. “The Role of Monetary Policy.”
American Economic Review, March 1968, 58(1),
pp. 1-17.

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Model Fit and Model Selection
Narayana R. Kocherlakota
This paper uses an example to show that a model that fits the available data perfectly may provide
worse answers to policy questions than an alternative, imperfectly fitting model. The author
argues that, in the context of Bayesian estimation, this result can be interpreted as being due to
the use of an inappropriate prior over the parameters of shock processes. He urges the use of priors
that are obtained from explicit auxiliary information, not from the desire to obtain identification.
(JEL C11, E40, E60)
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 349-60.

I

n an influential recent paper, Smets and
Wouters (2003) construct a dynamic stochastic general equilibrium (DSGE) model
with a large number of real and nominal
frictions and estimate the unknown parameters
of the model using sophisticated Bayesian techniques. They document that the estimated model
has out-of-sample forecasting performance superior to that of an unrestricted vector autoregression. They write of their findings (p. 1125), “This
suggests that the current generation of SDGE
[stochastic dynamic general equilibrium] models
with sticky prices and wages is sufficiently rich
to capture the stochastics and the dynamics in
the data, as long as a sufficient number of structural shocks is considered. These models can
therefore provide a useful tool for monetary
policy analysis” (italics added for emphasis).
The European Central Bank (ECB) agrees. They
are planning to begin using models with explicit
micro-foundations for the first time in their analyses of monetary policy. In doing so, they are
explicitly motivated by the Smets and Wouters
(2003) analysis.1

Smets and Wouters and the ECB are adherents
to what one might call the principle of fit. According to this principle, models that fit the available
data well should be used for policy analysis;
models that do not fit the data well should not
be. The principle underlies much of applied
economic analysis. It is certainly not special to
sophisticated users of econometrics: Even calibrators who use little or no econometrics in their
analyses believe in the principle of fit. Indeed,
there are literally dozens of calibration papers
concerned with figuring out what perturbation
in a given model will lead it to fit one or two more
extra moments (like the correlation between hours
and output or the equity premium).
In this paper, I demonstrate that the principle
of fit does not always work. I construct a simple
example economy that I treat as if it were the true
world. In this economy, I consider an investigator
who wants to answer a policy question of interest
and estimates two models to do so. I show that
1

See www.ecb.int/home/html/researcher_swm.en.html for details.

Narayana R. Kocherlakota is a professor of economics at the University of Minnesota, a consultant at the Federal Reserve Bank of Minneapolis,
and a research associate at the National Bureau of Economic Research. The author thanks Ricardo DiCecio, Lee Ohanian, Tom Sargent, Adam
Slawski, Hakki Yazici, Stan Zin, and especially Barbara McCutcheon for conversations about this paper. He learned much of what is in this
paper from joint work that he did in the 1990s with Beth Ingram and Gene Savin when they were colleagues at the University of Iowa. The
views expressed herein are the author’s and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.

© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
only with prior written permission of the Federal Reserve Bank of St. Louis.

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model 1, which has a perfect fit to the available
data, may actually provide worse answers than
model 2, which has an imperfect fit.
The intuition behind this result is quite
simple. The policy question of interest concerns
how labor responds to a change in the tax rate.
The answer depends on the elasticity of the
labor supply. In both models, the estimate of this
parameter hinges on a particular non-testable
assumption about how stochastic shocks to the
labor-supply curve covary with tax rates. When
model 2’s identification restriction is closer to
being correct than model 1’s, model 2 provides a
better answer to the policy question, even though
its fit is always worse.
In the second part of the paper, I consider a
potential fix. I enrich the class of possible models
by discarding the non-testable assumption mentioned above. The resultant class of models is, by
construction, only partially identified; there is a
continuum of possible parameter estimates that
are consistent with the observed data. I argue that,
from the Bayesian perspective, a user of model 1
essentially has an incorrect prior over the set of
parameters of this richer third model. As a solution, I suggest using a prior that is carefully motivated from auxiliary information, so that it does
not assign zero probability to positive-probability
events.
In general, there is much prior information
available about behavioral parameters, such as
those governing preferences and technology. However, there is much less prior information about
the parameters governing shock processes. One
possible response to this problem is to be what I
will term agnostic—that is, to be fully flexible
about the specification of the prior concerning
the shock-process parameters. I argue in the
context of the example that if one takes such an
agnostic approach, the data themselves reveal no
information about the behavioral parameters. I
interpret this result as an indirect argument for
the procedure commonly called calibration, in
which an investigator picks a plausible range for
technology and preference parameters based only
on prior auxiliary information.
In the final part of the paper, I return to Smets
and Wouters (2003). Using the above analysis, I
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offer a critique of their approach to estimation
and model evaluation. I suggest how one might
change their estimation and evaluation approach
to ensure more reliable policy analyses.
The second part of the paper is about Bayesian
estimation of models given limited a priori information, which is, as far as I know, novel. However, the first part is not new: It is well-known that
there are potential problems with the principle
of fit. In early contributions, Marschak (1950) and
Hurwicz (1950) emphasize that multiple structures (mappings between interventions and outcomes) may be consistent with a given reduced
form (probabilistic description of available data).
Liu (1960) argues that this potential problem is,
in fact, endemic: The available data never serve to
identify the true structure uniquely. In perhaps
the most related work, Sims (1980) argues explicitly that large-scale models may fit the data well
and yet provide misleading answers to (some)
questions because their estimates are based on
incredible identification restrictions.
Though it lacks novelty, my discussion about
the principle of fit serves three purposes. First,
the principle remains a dominant one among
policymakers and others (as my opening paragraphs indicate). Given the recent excitement
about Smets and Wouters (2003) and other related
papers, it is worthwhile (I believe) to remind
everyone of the principle’s limitations.
Second, I want to make absolutely clear that
we cannot resolve the problem by using structural
models. Most macroeconomists are highly cognizant of the Lucas critique (1976). It correctly
emphasizes that to assess a policy intervention a
model’s parameters should be structural—that is,
invariant to the intervention. In response, most
macroeconomists now use structural models to
analyze policy interventions. My paper demonstrates that this response is not a panacea. In particular, I show that even if it is structural and
well-fitting, a model may provide misleading
answers to policy questions.
Finally, my argument is not just that better fit
can lead to worse answers but that we should
expect that obtaining better fit will lead to worse
answers. Archetypal macroeconomic models are
usually under-shocked relative to the data under
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Kocherlakota

consideration. Hence, to get macroeconomic
models that fit well, we need to add shocks. But
we generally know little about these shocks. It
should not be at all surprising if adding them
were to create new, possibly substantial, sources
of error.

AN ARTIFICIAL WORLD AND
POLICY INTERVENTIONS
The basic structure of this paper is akin to a
Monte Carlo study. I first set up an artificial world
over which I have complete control. I introduce
an investigator (econometrician) into this artificial
world who does not know the structure of the
artificial world but is instead limited to using
one of two possible (classes of) models. Both are
false because the artificial world is not a special
case of either model; however, the investigator
does not know that they are false. Based on data
from the artificial world, the investigator uses a
variety of possible methods to determine which
model has superior fit.
In this section, I describe the artificial world
and a class of policy interventions under consideration in that world. In the artificial world, agents
decide how much to work at each date. Their decisions are influenced by shocks to labor productivity, taxes, and preferences. The means of these
random variables are hit by observable shocks in
each period. Preference shocks and tax rates
covary; it is this covariance that makes estimation
of the parameters of the model challenging.

The Artificial World
Time is discrete and continues forever. There
is a unit measure of agents who live forever, and
preferences are given by
`

∑
t =1

∗

δ t −1  ln ct − exp (ψ t ) ntγ / γ ∗ , 0 < δ < 1,



where ct is consumption in period t and nt is labor
in period t. Technology is given by
y t = exp (At ) nt ,

where yt is the amount of consumption produced
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in period t. Agents are taxed at rate τt , where τt is
governed by the policy rule
lnÄ (1 − τ t ) = − βψ t + ε t

β > 0.
The proceeds of the taxes are handed back lumpsum to the agents.
The random variables 共At ,ψt , εt 兲 are i.i.d.,
over time. There is another random variable λ t,
which is equally likely to be 0 or 1. Conditional
on λ t = i, the random variables 共A,ψ, ε 兲 are all
Gaussian and mutually independent, with
means 共µA共i 兲, µψ 共i 兲, µε 共i 兲兲1i = 0 and positive variances 共σA2, σψ2, σε2兲. Note that the means depend
on i, but the variances do not.
It is easy to prove that, in this economy, there
is a unique equilibrium of the form

lnÄ nt = ( ln (1 − τ t ) − ψ t ) / γ ∗
lnÄ y t = At + lnÄ nt .

Interventions
Consider the following class of interventions,
indexed by the real variable ∆. With intervention
∆ , the tax rate follows the rule

ln (1 − τ t ( ∆ )) = ∆ + ln (1 − τ t ).
The policy question is this: How much does
average logged output change in response to a
change in the tax rate? Mathematically, let yt 共∆兲
denote per capita output under intervention ∆.
What is E 共ln共yt 共∆*兲兲兲 – E 共ln共yt 兲兲, where ∆* is a
given intervention? The true answer to this
question is ∆*/γ *.

TWO (IDENTIFIED) MODELS
There is an investigator who wants to know
the answer to the given policy question. The
investigator does not know the structure of the
artificial world, but does observe the following
data:

(lnÄ y t , lnÄ nt , ln (1 − τ t ), λt )t`=1 .
The investigator has two possible models to use
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ments of the models are the same as that of the
artificial world itself. In each model, there is a
unit measure of identical agents who work to
produce output. The agents face a linear tax on
output, and the proceeds of this tax are handed
out lump-sum. However, the shock-generation
processes in the two models are different from
each other and from the artificial world.

Model 1
In model 1, preferences are of the form

δ

t −1

ψ 1t ntγ 1 /

lnÄ ct − exp (


)

γ 1 ;

t =1

technology is given by yt = exp共A1t 兲nt ; and agents
are taxed at rate τt , where ln共1 – τt 兲 = ε1t . The
random variables 共A1t ,ψ1t , ε1t 兲 are i.i.d. over time
and mutually independent. The random variable
λt has support {0, 1}; the probability that λt equals
1 is given by p1. Conditional on λt = i, the random
variables 共A1t ,ψ1t , ε 1t 兲 are Gaussian, with means
2
, σ12ψ , σ12ε 兲.
共µ1A共i兲, µ1ψ共i兲, µ1ε共i兲兲1i=0 and variances 共σ1A
The investigator does not know these means
and variances; they will have to be estimated in
some fashion from the data. Put another way,
this is actually a class of models indexed by the
11 parameters 共γ 1, 共µ1A共i 兲, µ1ψ 共i 兲, µ1ε 共i 兲兲1i = 0,
2
σ 12ε , σ 1A
, σ 12ψ , p1兲.
Model 1 implies that in equilibrium

lnÄ ( nt ) =  ln (1 − τ t ) − ψ 1t  / γ 1
ln ( y t ) = A1t + ln ( n1t ).
How does model 1 differ from the artificial
world? It is alike in all respects except one: In
model 1, the parameter β has been set to zero. As
we shall see, this additional restriction allows
the investigator to estimate γ1 from the available
data.

Model 2
In model 2, preferences are given by
`

∑

δ t −1 lnÄ ct − exp (ψ 2 ) ntγ 2 / γ 2

t =1

and technology is given by yt = exp[A2共1兲λt +
A2共0兲共1 – λt 兲]nt . Here, ψ2, A2共1兲, and A2共0兲 are all
352

ln (1 − τ t ) = ε2 (1) λt + ε2 (0) (1 − λt ).
The random variable λt is i.i.d. over time, with
support {0, 1}, and the probability that λt equals 1
is given by p2. The parameters ε2共1兲 and ε2共0兲 are
both constants. Hence, in model 2, there are seven
unknown parameters, 共γ 2, ψ2, A2共1兲, A2共0兲, ε2共1兲,
ε2共0兲, p2 兲. The model implies that
ln ( nt ) = ln (1 − τ t ) − ψ 2  / γ 2

`

∑

constants; λt is a random variable; and agents are
taxed at rate τt , where

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2007

ln ( y t ) = A2 (1) λt + A2 ( 0) (1 − λt ) + ln ( nt ).

How does model 2 differ from the artificial
world? In model 2, tastes do not vary at all. As
well, the variances of the other shocks around their
means are both set to zero. Like many modern
macroeconomic models, model 2 has relatively
few sources of uncertainty compared with what
is true of the (artificial) world.

THE FALLACY OF FIT
The investigator has two models available.
He wants to use his infinitely long sample to
decide which model to use in order to answer
the policy question. The sample 共ln共yt 兲, ln共nt 兲,
ln共1– τ 兲兲⬁t = 1 is jointly Gaussian conditional on
λt = i, for i = 0, 1. The means of the conditional
distributions depend on λt; the conditional distributions have the same variance-covariance
matrix. Hence, the sample can be fully summarized by 13 moments: the probability p that λt
equals 1, the means 共 µy共i 兲, µn共i 兲, µτ 共i 兲兲 of 共ln共y兲,
ln共n兲, ln共1– τ 兲兲 conditional on λt = i, and the variance-covariance matrix Σ of 共ln共y兲, ln共n兲, ln共1– τ 兲兲
conditional on λ .
Note that in these data, there are two distinct
kinds of variation. The first kind is because of λ.
Movements in λ generate changes in 共 µy共i 兲, µn共i 兲,
µτ 共i 兲兲; these changes can provide information
about the unknown parameters of the two models.
At the same time, 共ln共y兲, ln共n兲, ln共1– τ 兲兲 vary
around these fluctuating means. This information
is summarized by the six moments of Σ. The goal

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of the investigator is to use these two sources of
variation to estimate the unknown parameter γ.
Given this information, the investigator has
available three methods of estimating/evaluating
the models.

Method 1: Maximum Likelihood
In this subsection, I suppose that the investigator estimates the unknown parameters of each
model by maximum likelihood, and then compares the models’ abilities to fit the 13 population
moments.
Model 2 implies that, conditional on λt = i, the
data is deterministic. In other words, according
to model 2, the conditional variance-covariance
matrix of 共ln共yt 兲, ln共nt 兲, ln共1– τ t 兲兲 contains only
zeros. It follows that the likelihood of the data,
conditional on any specification of model 2, is
zero.2
For model 1, the maximum-likelihood estimates of the 11 unknown parameters are given by

pˆ 1 = 1 / 2
γˆ 1 = Σττ / Σ nτ

σˆ 12A = Σ yy − Σ nn
(1)

σˆ 12ε = Σττ
2
σˆ 12ψ = (γˆ 1 ) Σ nn − Σττ

µˆ 1A ( i ) = µ y ( i ) − µn ( i ), i = 0, 1
µˆ 1ε ( i ) = µτ ( i )
µˆ 1ψ ( i ) = µτ ( i ) − µn ( i ) γˆ 1 , i = 0, 1.
Given the infinitely long sample, these estimates
are very precise; the likelihood of the sample is 1
given this parameter setting and zero given all
others. Note that under this parameter setting,
the model fits all 13 moments of the data exactly.
Hence, according to maximum likelihood,
only model 1 should be used to answer the policy
question (with the parameter estimates (1)); no
specification of model 2 should be used. The lack
of fit is because model 2 is “under shocked” rela2

It is worth noting that model 2 implies that the sample variancecovariance matrix, conditional on λt = i, is noninvertible in any
finite sample. Hence, the likelihood of any finite sample, conditional on model 2, is zero.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

tive to the data. The world has four distinct shocks
generating the data, but model 2 has only one.
Maximum likelihood punishes this kind of discrepancy severely; from a statistical point of
view, it is the most readily detectable form of
misspecification.

Method 2: Bayesian Estimation
In this subsection, I suppose that the investigator applies Bayesian estimation methods to the
available data from the artificial world. Obviously,
models 1 and 2 are nested—if model 2 is true,
model 1 is also true. Consider an econometrician
who has a prior over the 11 unknown parameters
of model 1. The prior is such that it puts probability q on the parameters being consistent with
model 2 and puts probability 共1 – q兲 on the parameters being inconsistent with model 2.
Now suppose the econometrician observes
an infinite sequence of data from the artificial
world. The data bleaches out the effect of the initial prior; the econometrician’s posterior will be
concentrated on the parameter estimates (1). A
Bayesian econometrician with an infinitely long
sample will reach the same policy conclusions
as does a classical econometrician using maximum likelihood.

Method 3: Method of Moments
As we have seen, maximum likelihood and
Bayesian estimation simply discard all undershocked models. Now, we consider a less severe
measure of fit: method of moments, by which I
mean the following. Consider the 13 population
moments that characterize the sample. Pick 13
positive weights that sum to 1. Estimate the
unknown parameters in each model by minimizing the weighted sum of squared deviations
between model-generated moments and sample
moments. Then compare model 1 and model 2
by the value of the minimized objective.
Note again that the models are nested. Because
we are minimizing the same objective for each
model, model 1 must do at least as well as model
2. By setting the parameters in model 1 according
to (1), the objective is set equal to zero. Because
model 2 (incorrectly) generates a non-invertible
variance-covariance matrix for any parameter
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Kocherlakota

setting, the objective must be strictly larger than 0.
Model 1 must fit the data better than model 2,
according to this measure of fit.
However, using method of moments, we can
now actually estimate parameters for model 2, as
opposed to simply discarding it as maximum
likelihood does. For model 2, regardless of the
weights, the estimated seven parameters are

ˆ ( i ) =  µ ( i ) − µ ( i ) , i = 0, 1
A
n
2
 y


εˆ 2 ( i ) = µτ ( i ), i = 0, 1
µτ (1) − µτ (0 )
µn (1) − µn ( 0)

The seven parameters are set so that the model
generates the values in the data for the moments
共p, µy共1兲, µy共0兲, µτ 共1兲, µτ 共0兲, µn共1兲, µn共0兲兲. Model 2
predicts that the other moments are zero for any
choice of parameters, so that part of the minimization problem is irrelevant for parameter estimation.

Using the Estimated Models to Answer
the Policy Question
Recall that the policy question is this: What
is the value of

( ))

E ln ( y t ) ∆ ∗ − E ( ln ( y t ))

when taxes are changed so that ln共1– τt 共∆*兲兲 –
ln共1– τt 兲 = ∆*? The true answer to this question is
given by ∆*/γ *.
Here is what the two models deliver. Under
model 1, the answer is ∆*/γˆ1 = ∆*Σnτ /Στ τ , where
Σnτ is the population covariance of ln共nt 兲 and
ln共1– τt 兲 in the artificial world, and Στ τ is the
population variance of ln共1– τt 兲 in the true world.
(This is ∆* multiplied by the population regression coefficient.) We can calculate these population moments to find that that answer in model 1
is given by

(

{

})

ANS1 = ∆ ∗ 1 / γ ∗ + βγ ∗−1σ ψ2 / β 2σ ψ2 + σ ε2 ,

which is too large in absolute value relative to
the true answer of ∆*/γ *.
354

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2007

= ∆∗

 µε (1) − µε ( 0) / γ ∗ − β  µψ (1) − µψ (0 ) / γ ∗ −  µψ (1) − µψ (0 ) / γ ∗
µε (1) − µε (0 ) − β  µψ (1) − µψ (0 )

(

= ∆∗ / γ ∗ − ∆∗ / γ ∗

 µψ (1) − µψ ( 0)

) µ (1) − µ (0) − β  µ
ε

ε



ψ

(1) − µψ ( 0)

.

Why Doesn’t Fit Work?

ψˆ 2 =  µτ (1) − γˆ 2 µn (1) =  µτ ( 0) − γˆ 2 µn (0) .

(

 µ (1) − µn (0 ) ∗
ANS2 =  n
∆
 µτ (1) − µτ ( 0)

Note that if 共 µ ψ 共1兲 – 共 µ ψ 共0兲兲 is sufficiently close
to zero in absolute value, ANS2 is nearer to 1/γ *
than is ANS1. Even though model 2’s fit is worse
than that of model 1, model 2 may still deliver a
superior answer to the policy question.

pˆ 2 = 1 / 2

γˆ 2 =

Under model 2, the answer is given by

In the above discussion, we have seen that
the model that fits better—indeed, the model
that fits the available data perfectly—may well
deliver a worse answer to the policy question.
What is going on here? The policy question is
this: What happens to hours worked if we increase
the tax rate on labor? The answer is wholly governed by the elasticity of labor, which is equal to
1/γ * in the artificial world. To answer the question the investigator has to estimate γ * well, but
there is a traditional difficulty associated with
doing so. If there are no shifts in the labor supply,
then the comovement in hours and tax rates will
pin down the elasticity of the labor supply. However, if the labor supply shifts (that is, movements
in ψ ) are correlated with the variation in tax rates,
then the investigator will achieve biased estimates
of 1/γ *.
How do the two models estimate γ *? In the
artificial world, there are two sorts of variation
in the data. The first is that the means of the distributions of 共A, ψ, ln共1– τ 兲兲 fluctuate over time.
The second is that the realizations of the random
variables fluctuate around their means. The
good news is that, because λ is observable, the
two kinds of variations are distinct. The bad
news is that both kinds of fluctuations feature
potential comovement between ψt andτt —
comovement that makes our task of estimating γ
more difficult.
The two models differ in their estimates of γ
because each one relies on a different type of
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Kocherlakota

fluctuation to pin down γ . Model 1 assumes
(incorrectly) that the fluctuations of ln共1– τt 兲 and
ψt around their means are independent. It then
exploits the fluctuations in tax rates and hours
around their means to estimate γ. Model 2 assumes
(incorrectly) that the mean of ψt does not fluctuate
at all. It then uses the shifts in the means of hours
and tax rates over time to estimate γ . Which one
works better depends on which incorrect assumption is a better approximation to reality. Nothing
in the data answers this question.
The key point is that the relative fit of the
models does not tell us which of these assumptions is closer to being right. More generally,
parameter estimation of any kind always relies
on two sources of information: the data and nontestable identification assumptions. The fit of a
model tells us nothing about the reliability of the
latter.3 Yet their reliability is essential if one is to
obtain accurate parameter estimates.

PRIOR CARE
The problem with model 1 is that it includes
a false restriction, which is included solely to
identify the unknown parameter. In this section,
I consider a richer model than model 1, in which
I dispense with the false identification restriction.
By construction, this model is only partially
identified. I argue that one way to interpret the
problem with model 1 is that the investigator is
using an incorrect prior over the larger parameter
space of this richer model. I suggest a simple fix
to these problems: estimate the larger, partially
identified model in a Bayesian fashion while
being meticulous in building the prior explicitly
from auxiliary information.
As before, assume that there is an investigator who has an infinite sample 共ln共yt 兲, ln共nt 兲,
ln共1– τt 兲,λt 兲⬁t = 1 from the artificial world described
3

Model 1 is a just-identified model; the number of identifying
restrictions is equal to the number of estimated paramaters. More
generally, there may be more identifying restrictions than unknown
parameters. It is commonplace to construct tests of the overidentifying restrictions in such models. However, it is important to keep
in mind that these are tests only of some linear combinations of
the restrictions. The other linear combinations are being used to
estimate the parameters and are, as in the just-identified case,
nontestable.

in the first section. The investigator does not use
model 1 or model 2 though. Instead, the investigator uses a new model, model 3.

Model 3
In model 3, preferences are of the form
`

∑

δ t −1 ln ct − exp (ψ 3t ) ntγ 3 / γ 3 ,γ 3 ≥ 0 ;

t =1

technology is given by yt = exp共A3t 兲nt ; and agents
are taxed at rate τt , where
ln (1 − τ t ) = − β3ψ 3t + ε3t , β3 ∈ R.

The random variables 共A3t ,ψ3t , ε3t 兲 are i.i.d. over
time and mutually independent. The collection
of random variables {λt }⬁t = 1 are i.i.d., with support
{0,1}; the probability that λt equals 1 is given by p3.
Conditional on λt = i, the random variables 共A3t ,
ψ3t , ε3t 兲 are Gaussian, with means 共µ 3A共i兲, µ 3ψ共i兲,
2
µ 3ε共i兲兲1i=0 and variances 共σ3A
, σ32ψ , σ32ε 兲. In this class
of models, there are 12 unknown parameters
2
共γ 3, β3, 共µ 3A共i 兲, µ 3ψ 共i 兲, µ 3ε 共i 兲兲1i = 0, σ3A
, σ32ψ , σ32ε , p3兲.
Model 3 implies that in equilibrium
ln ( nt ) = ln (1 − τ t ) − ψ 3t  / γ 1
ln ( y t ) = A3t + ln ( nt ).

Model 3 is exactly the same as model 1,
except that, in model 3, tax rates may be correlated with the preference shock ψ3. This change
means that model 3 is sufficiently rich—to nest
both model 1 and the artificial world.
Model 3 is only partially identified. Suppose
γ3 = γˆ3.Then, there is a unique specification of
the other 11 parameters so that model 3 fits the
available data exactly. In particular, let
pˆ 3 = 1 / 2

σˆ 32ψ = ( 0 γˆ 3 − 1)′ Σ (0 γˆ 3 − 1)
γˆ Σ − Σ
βˆ 3 = 3 nτ2 ττ
σˆ 3ψ

(2)

σˆ 32ε

σˆ 32ψ

σˆ 32A = Σ yy − Σ nn
ˆ

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

=

(γˆ 3 )2  Σ nn Σττ − Σ2nτ 

A

(
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Kocherlakota

=

µˆ 3 A ( i ) = µ y ( i ) − µn ( i ), i = 0, 1

A Mistaken Prior: The Case of Model 1

µˆ 3ψ ( i ) = µτ ( i ) − µn ( i ) γˆ 3

Suppose g is such that the prior puts probability 1 on the event β3 = 0. In this case, after seeing the available data, the investigator’s posterior
is concentrated on the vector (1). With this kind
of prior, model 3 is equivalent to model 1.
We have seen that using model 1 gives misleading answers to the policy question. A prior
like g implicitly contains a great deal of information, because no amount of data can shift the
investigator from his belief that β3 = 0. It should
not be used unless the investigator actually has
this information about the world.

µˆ 3ε ( i ) = µτ ( i ) + βˆ 3 µˆ 3ψ ( i ),
and then the 13 moments generated by model 3
correspond to the moments of the sample. (Note
that all parameter estimates that are supposed to
be non-negative [that is, variances] are in fact
non-negative.) Hence, for each specification of
the parameter γˆ3, there exists a specification of
the other 11 parameters so that the model exactly
fits the data.
Recently, there has been a great deal of work
on classical methods to estimate partially identified models (see Manski, forthcoming, for a useful survey). However, I believe it is most useful
to consider the estimation of model 3 from a
Bayesian perspective.4 Specifically, let θ = 共β3,
2
, σ32ψ , σ32ε , p3兲 represent
共µ 3A共i 兲, µ 3ψ 共i 兲, µ 3ε 共i 兲兲1i = 0, σ3A
the parameters of the model other than γ3. Suppose that the parameter space for 共γ3,θ3兲 is given
by R+ × Θ, where Θ = R 7 × R+3 × [0, 1]. This parameter space is a 12-dimensional manifold. I assume
that the investigator has a prior density over this
manifold such that γ3 is stochastically independent from θ. I will let the marginal prior density
over γ3 be denoted by f and the prior density over
θ be denoted by g.
A basic intuition in Bayesian estimation/
learning is that the prior is essentially irrelevant
if one has a large amount of data. Intuitively, the
impact of the data is sufficiently large to bleach
out the initial information in the prior. However,
this intuition applies only when the model is
identified. As we shall see, when one uses the
partially identified model 3, the prior over 共γ3,θ 兲
affects the posterior distribution over γ3, even
though the investigator has access to an infinite
sample.
4

Lubik and Schorfheide (2004) use a Bayesian procedure to estimate
a partially identified model. As Schorfheide (forthcoming) emphasizes, identification problems—that is, the presence of ridges or
multiple peaks in the likelihood—do not create any problems for
Bayesian estimation: “Regardless, the posterior provides a coherent summary of pre-sample and sample information and can be
used for inference and decision making.”

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An Arbitrary Prior
Suppose instead that g and f are such that
the support of the investigator’s prior is the
entire parameter space. Let h共γˆ3 ; Data兲 represent
the parameter estimates (2) when γ 3 = γˆ3 . Then,
after seeing the infinite sample, the investigator’s
posterior is concentrated on a one-dimensional
manifold 共γˆ3 ,h共γˆ3 ; Data兲兲 indexed by γˆ3 ∈ [γL , γH ].
His posterior over this one-dimensional manifold
is proportional to φ 共γˆ3 兲, where
∂hn (γˆ 3 ; Data )
.
∂γ 3
n=1
11

φ (γˆ 3 ) = f (γˆ 3 ) g ( h (γˆ 3 ; Data )) ∏

(Here, hn represents the nth component of the
function h.) Given this posterior uncertainty, the
investigator’s answer to the policy question is no
longer a single number. Instead, the investigator’s
answer is now a random variable, with support
equal to the interval [∆*/γH , ∆*/γL ] and density
proportional to

φ (1 / x ) / x 2 ,
where x represents the answer to the policy
question.
Because the model is only partially identified,
the investigator’s posterior over the answer to
the policy question is influenced by his marginal
prior f over the preference parameter γ3 and his
prior g over the other parameters. This dependence
exists even though he sees an infinite sample.5
5

2

0

1

Note that in (2), the estimates 共σ̂ 3A , µ̂ 3A , µ̂ 3A , p̂3兲 depend only on the

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This means that the investigator cannot count on
the available data to eventually correct all misinformation in his initial prior. Instead, he must be
sure that his prior truly represents information
about these parameters derived from auxiliary
sources.

No Prior Information About Shock
Processes
It is easy to see how to construct a prior f over
the preference parameters (or over technology
parameters in a more general context). We can
derive information about such behavioral parameters from other data sources, from introspection,
or from experiments. However, it is more difficult
to obtain this information about the joint shock
process (A, τ, ψ ). In at least some, and perhaps
most, cases, there will be no auxiliary information
available about these processes. What should be
done?6
In this subsection, I assume that the investigator has information that leads to a prior f with
support [γL , γH ], where γL > 0. The investigator
has no auxiliary information about the shock
processes.
The Bayesian Approach. One possible
response to this no-information situation is to
formulate a purely subjective prior belief over
the 11 parameters of the shock process and then
proceed in a standard Bayesian fashion. In doing
so, it is important to keep two issues in mind.
First, as we have seen above, when the model is
partially identified, the prior impacts the answer
to the policy question regardless of how large the
sample is. The subjective beliefs always matter.
Second, every prior—regardless of how neutral it may seem—has some information embedded in it. To appreciate this last point, suppose
there is a parameter α. All that an investigator
data and not on γˆ3. Hence, the posterior over these four parameters is concentrated on a single vector after the investigator sees
an infinite sample.
6

Recently, del Negro and Schorfheide (2006) have suggested using
prior beliefs about endogenous variables (such as output and inflation) as a way to construct legitimate priors about exogenous shocks.
This approach is potentially interesting. One concern is that usually our prior beliefs about endogenous variables come from the
macroeconomic data that will, in fact, be used for estimation.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

truly knows about α is that α lies in [0,1]; he wants
his prior over α to be neutral over its location
within that interval. It is tempting to conclude
that we can capture this neutrality by using a
uniform distribution over [0, 1]. But now consider y = α 1000. What does the investigator know
about y ? Presumably, all that the investigator
knows about y is that it lies in [0, 1]—if he knew
more, he would have known more about α. However, if the investigator has a uniform prior over α,
then the investigator’s prior over y is proportional
to y –999/1000. This density is far from uniform over
[0, 1]; it places a lot more weight on low values of
y than on high values of y. The uniform density
over α actually does smuggle information about
α into the analysis.
An Agnostic Approach. The Bayesian
approach weds the investigator to a single
prior g. As I suggest above, this prior contains
information that the investigator does not literally have. One response to this problem is to
use what I would call an agnostic approach: Be
flexible about the choice of g and compute a
posterior density for each possible prior g over
γ 3. By doing so, the investigator’s answer to the
policy question is no longer a single number, or
even a single posterior, but rather a collection of
posteriors generated by varying g. All of these
posteriors have support [∆*/γH , ∆*/γL ].
The resulting collection of posteriors is large.
In particular, let p be any continuous probability
density function over [∆*/γH , ∆*/γL ]. Let gp be a
continuous function mapping Θ into R+ such that
g p ( h (1 / x ; Data )) =

(

x 2 p ∆ ∗−1 x

)

N

f (1 / x ) ∏ hn (1 / x ; Data )
n =1

for all x in [1/γH ,1/γL ]. (This pins down the
behavior of gp only on a given line in Θ.). If the
investigator’s prior over Θ is given by gp, then his
posterior over [∆*/γH , ∆*/γL ] is given by p. Thus,
the agnostic approach imposes no discipline on
the question of interest beyond the upper or lower
bounds on γ 3 imposed by the prior f.
This kind of agnostic analysis is reminiscent
of calibration. Under calibration, an investigator
uses information from auxiliary sources to pin
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down a range of possible values for behavioral
parameters. He then reports answers to the policy
question for all of the parameter settings in this
range. It is exactly this information that the investigator ends up reporting under the agnostic
approach: a range of possible values for the policy
question given the range of possible values for
the behavioral parameters.
It is important to emphasize that this conclusion does not mean that the data is useless under
the agnostic approach. Estimation collapses the
support of the original joint prior from a 12dimensional manifold to the 1-dimensional support of the posterior. Hence, the prior information
about γ 3, combined with the data, does help the
investigator learn a great deal about the nature of
the shocks hitting the economy. It is true that
this information is irrelevant given the policy
question posed. For other potential questions,
though, this information may well be valuable.
The Agnostic Approach and Decisionmaking.
Of course, more generally, the investigator may
have some information about the underlying
shocks that restricts the possible specifications
of g. Then, the agnostic approach is not equivalent to calibration. In this general case, the agnostic approach implies that each policy intervention (each ∆) leads to a set of posterior probability
distributions over outcomes.
It is interesting to consider the problem of
choosing ∆ in this setting. Suppose there is a
social welfare W共p兲 associated with a given posterior p over the set of outcomes. Let Π共∆兲 represent the set of possible posteriors implied by a
given ∆. Then, choosing ∆ is akin to optimizing
under Knightian uncertainty, as opposed to risk.
It is standard in such settings to use a maximin
approach, under which the choice of ∆ solves
the problem:

max min W ( p ).
∆

p ∈Π( ∆ )

Hurwicz (1950, p. 257) provides a similar resolution to the problem of decisionmaking with partially identified models.7
7

See Gilboa and Schmeidler (1989) for an axiomatization of this
approach to uncertainty.

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RELATIONSHIP TO SMETS AND
WOUTERS (2003)
As reported in the introduction, Smets and
Wouters (2003) estimate a DSGE monetary model.
They note that their model is highly similar to
that of Christiano, Eichenbaum, and Evans (2005).
The big difference between the two specifications
is in the number of shocks: Smets and Wouters
allow for 10 different shock processes. None of
these shock processes represent measurement
error. Instead, they all play a substantive economic
role.
Smets and Wouters use a Bayesian procedure
to estimate their model. As argued above, the
prior plays an important role in this kind of estimation. Smets and Wouters correctly spend a
great deal of time in their paper discussing the
specification of the prior over the preference and
technology parameters. They motivate this part
of the prior thoroughly using explicit auxiliary
information.
The motivation for their choice of prior over
the 10 shock processes is quite different. They
write (p. 1140), “Identification is achieved by
assuming that each of the structural shocks [is]
uncorrelated and that four of the ten shocks...
follow a white noise process.” In other words,
they choose the prior over the 10 shock processes
in order to achieve identification, not because of
auxiliary information. The first example makes
clear the problems with this approach. Like Smets
and Wouters, the user of model 1 chooses the
prior over the shock processes to achieve identification. Because this prior does not truly reflect
auxiliary information, the resulting estimates are
severely biased, even though the model fits the
data exactly. Smets and Wouters give us no reason to believe why the same should not be true
of the estimates of their model.8
The second part of the current paper suggests
an alternative approach. The investigator should
not pick a prior that is designed to achieve iden8

In their recent discussion of identification of DSGE models, Canova
and Sala (2006, p. 40) write that “resisting the temptation to arbitrarily induce identifiability is the only way to make DSGE models
verifiable and knowledge about them accumulate on solid ground.”
I agree.

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Kocherlakota

tification. Instead, the prior—or collection of
priors—over the shock processes should reflect
the investigator’s actual beliefs about those
processes. The resulting set of posteriors will
naturally contain less information—but also be
more reliable. The key property of model 3 is
that it is sufficiently rich to include as a special
case the artificial world that is actually generating the data. It is certainly difficult to build such
a class of models in the real world. Nonetheless,
Bayesian estimation techniques (or any other for
that matter) are only reliable if one does so.9

CONCLUSIONS
A model-based analysis of a policy intervention has two steps. The first is to figure out the
key parameters that shape the quantitative impact
of the intervention. The second is to gather information about these parameters. This information
can come in two forms: prior information and
information derived from estimating the model
using a particular data set. The first part of this
paper argues that the fit of a model tells us little
about the quality of information coming from
either source. The second part of the paper argues
that the latter source of information (estimation)
is not useful unless the investigator has reliable
prior information about shock processes.
There is an important lesson for the analysis
of monetary policy. Simply adding shocks to
models in order to make them fit the data better
should not improve our confidence in those
models’ predictions for the impact of policy
changes. Instead, we need to find ways to improve
our information about the models’ key parameters
(for example, the costs and the frequency of price
adjustments). It is possible that this improved
information may come from estimation of model
parameters using macroeconomic data. However,
as we have seen, this kind of estimation is only
useful if we have reliable a priori evidence about
the shock processes. My own belief is that this
kind of a priori evidence is unlikely to be avail9

See Schorfheide (2000) for a discussion of how to augment
Bayesian techniques to allow for the possibility that no model
under consideration is true.

F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

able. Then, auxiliary data sources, such as the
microeconometric evidence set forth by Bils and
Klenow (2004), will serve as our best source of
reliable information about the key parameters in
monetary models.

REFERENCES
Bils, Mark and Klenow, Peter J. “Some Evidence on
the Importance of Sticky Prices.” Journal of
Political Economy, October 2004, 112(5), pp. 947-85.
Canova, Fabio and Sala, Luca. “Back to Square One:
Identification Issues in DSGE Models.” Working
Paper 583, European Central Bank, 2006.
Christiano, Lawrence J; Eichenbaum, Martin and
Evans, Charles. “Nominal Rigidities and the
Dynamic Effects of a Shock to Monetary Policy.”
Journal of Political Economy, February 2005,
113(1), pp. 1-45.
del Negro, Marco and Schorfheide, Frank. “Forming
Priors for DSGE Models and How It Affects the
Assessment of Nominal Rigidities.” Working paper,
University of Pennsylvania, 2006.
Gilboa, Itzhak and Schmeidler, David. “Maxmin
Expected Utility with a Non-unique Prior.” Journal
of Mathematical Economics, 1989, 18(2), pp. 141-53.
Hurwicz, Leonid. “Generalization of the Concept of
Identification,” in Tjalling Koopmans, ed.,
Statistical Inference in Dynamic Economic Models.
New York: John Wiley and Sons, 1950.
Liu, Ta-Chung. “Underidentification, Structural
Estimation, and Forecasting.” Econometrica,
October 1960, 28(4), pp. 855-65.
Lubik, Thomas A. and Schorfheide, Frank. “Testing
for Indeterminacy: An Application to U.S. Monetary
Policy.” American Economic Review, March 2004,
94(1), pp. 190-217.
Lucas, Robert E. “Econometric Policy Evaluation:
A Critique,” in K. Brunner and A. Meltzer, eds.,
The Phillips Curve and Labor Markets. Amsterdam:
North-Holland, 1976.

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Kocherlakota

Manski, Charles F. “Partial Identification in
Econometrics,” in Stephen Durlauf and Lawrence
Blume, eds., The New Palgrave Dictionary of
Economics. Second Edition (forthcoming).
Marschak, Jacob. “Statistical Inference in Economics,”
in T. Koopmans, ed., Statistical Inference in
Dynamic Economic Models. New York: John Wiley
and Sons, 1950.
Schorfheide, Frank. “Loss Function Based Evaluation
of DSGE Models.” Journal of Applied Econometrics,
November-December 2000, 15(6), pp. 645-70.
Schorfheide, Frank. “Bayesian Methods in
Macroeconometrics,” in Stephen Durlauf and
Lawrence Blume eds., The New Palgrave Dictionary
of Economics. Second Edition (forthcoming).
Sims, Christopher A. “Macroeconomics and Reality.”
Econometrica, January 1980, 48(1), pp. 1-48.
Smets, Frank and Wouters, Raf. “An Estimated
Stochastic Dynamic General Equilibrium Model of
the Euro Area.” Journal of the European Economic
Association, September 2003, 1(5), pp. 1123-75.

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F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Commentary
Lee E. Ohanian

I

n “Model Fit and Model Selection,”
Narayana Kocherlakota (2007) warns econometricians and the users of econometric
analyses that macroeconomic models that
fit the data well—as measured by a high R 2 and/or
low residual variance—may not be very useful
for policy advice because key parameters may
not be identified. As an alternative, Kocherlakota
provides a Bayesian approach that recognizes the
significant challenge of identifying all parameters
in a fully specified general equilibrium model
and that also treats uncertainty about parameter
values in a consistent fashion.
Kocherlakota’s agnostic approach means that
the range of uncertainty associated with the conditional forecasts (policy advice) generated by the
macroeconometric models used by central banks
and other policymaking agencies is probably
much larger than recognized by macroeconometric
practitioners. This also suggests that policymakers,
who use the forecasts from these models as an
input into policymaking, should also modify their
priors and recognize the considerable uncertainty
in conditional forecasts.
There is substantial evidence that supports
Kocherlakota’s recommendation, and there is also
substantial evidence that this practice—or other
practices that explicitly recognize the degree of
uncertainty in modeling the economy—is not
followed by macroeconometric model builders.
Nor is agnosticism followed by policymakers
regarding the structure of the economy. Instead,
current model-building practice focuses largely

on models that feature a Phillips curve, and
recent monetary policy decisions also appear to
focus on the Phillips curve. I first discuss
Kocherlakota’s analysis of fit and identification
and then discuss the broader issues of agnosticism in choosing among alternative theoretical
frameworks for evaluating policy and the role of
agnosticism in making monetary policy.

AGNOSTICISM IN MODELING
Kocherlakota reminds macroeconometric
modelers and the users of these models that a
model with a good fit may not be a useful tool
for conditional forecasting and policy analysis.
To summarize practitioners’ current view about
fit and its implications for policy analysis,
Kocherlakota quotes from recent influential
work by Smets and Wouters (2003), who suggest
that a useful model for policy analysis is one that
includes enough shocks to fit the data well. Smets
and Wouters’s view is quite representative of
macroeconometric modeling strategies used today.
To illustrate why a model that fits well may
be not be useful for conditional forecasting,
Kocherlakota constructs a model economy in
which a model fits the data perfectly. He then
shows that despite the perfect fit, the model cannot accurately forecast the impact of a tax cut on
the economy because the elasticity of labor supply
is unidentified. The reason that the perfect-fitting
model provides a poor conditional forecast is

Lee E. Ohanian is a professor of economics at the University of California at Los Angeles, a visiting scholar at the Federal Reserve Bank of
Minneapolis, and a research associate of the National Bureau of Economic Research.
Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 361-69.
© 2007, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
only with prior written permission of the Federal Reserve Bank of St. Louis.

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Ohanian

because it includes a non-testable identifying
restriction that is false.
Kocherlakota suggests an alternative procedure
that completely discards the non-testable identifying restriction. As an alternative, Kocherlakota
recommends using auxiliary information to construct a range of values that a parameter can take.
This is not standard Bayesian analysis, however,
in that the standard approach commits the investigator to a single prior. Instead, Kocherlakota’s
procedure considers many priors over the parameter of interest. This delivers a collection of posteriors, which are restricted only in that they have
support between the minimum and maximum
values specified for the parameter. In principle,
this approach is very sensible, as it explicitly and
systematically allows the researcher to conduct
a sensitivity analysis.
From a practical perspective, however, this
procedure may be difficult to apply. To see this,
note that in Kocherlakota’s example of the impact
of a tax cut, there is a single parameter. In this
case, the application of the agnostic procedure
yields a collection of posteriors over this single
parameter, with support over a minimum and
maximum value. This one-dimensional case is
fairly simple to implement and to investigate.
However, in a high-dimensional setting, the investigator must specify many priors over several
parameters. Specifying multidimensional priors
can be difficult; in practice, specifications are
often chosen for computational ease, but in this
case the prior is particularly important because
the effect of the prior does not wash out as the
sample size becomes arbitrarily large, as in standard analysis with full identification. Understanding how various multidimensional priors affect
the analysis is very much an open, and difficult,
question. Moreover, understanding how to distill
and interpret the information from a collection
of posteriors is also an open question. Making
progress on these fronts seems necessary to successfully apply the agnostic, Bayesian approach
in any rich model that includes many parameters
and/or shocks.
Kocherlakota’s agnostic approach presumes
that there is an inherent identification problem
in macroeconomics that is not easy to resolve. Is
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2007

the identification problem in macroeconomics
as difficult as suggested by Kocherlakota? Unfortunately, there is no easy answer to this question;
the profession has wrestled with identification in
aggregate economics since the work of Tinbergen
(1937), Haavelmo (1944), the Cowles Commission
(Koopmans, 1950), Liu (1960), Sims (1980), and
it continues today (Canova and Sala, 2006).
The identification debate in macroeconomics
has suggested many different resolutions to the
problem. One must understand the differences:
Sims (1980) viewed the identification challenge
in macroeconomics a sufficiently tall order to fill
as to recommend relatively unrestricted vector
autoregression (VAR) models that achieved identification by imposing a sufficient number of lags
in the VAR to generate white noise innovations
and then impose a Wold causal ordering on the
innovation covariance matrix. In contrast,
Kocherlakota recommends a very different
approach, in which the behavioral equations of
the model are tightly restricted by theory, but only
minimal restrictions are imposed on the structure
of the shock processes. Regarding the relative
merits of these two different approaches, identification achieved through restrictions on shock
processes are often difficult to justify because
economic theory typically does not shed much
light on the correct stochastic specification of
shocks. Moreover, evaluating the identification
of shocks is difficult, as identifying shocks almost
always requires strong non-testable restrictions.
Some economists argue that shocks should be
uncorrelated, and that this apparently innocuous
assumption can go a long ways toward achieving
identification. But we have several observations
that shocks can be correlated. For example, there
were several scientific, productivity breakthroughs
in World War II that were largely the consequence
of the large wartime government spending shock.
Similarly, World War II monetary shocks were
due to fiscal spending requirements that induce
the need for seignorage finance. The deregulation
of financial markets over the past 30 years has led
to significant technological change in financial
intermediation. The Great Depression led to
enormous changes in economic regulation and
government management of the economy. These
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Ohanian

are just a few examples that indicate that achieving identification through orthogonality assumptions can be at variance with the data.
Identification will always be a difficult issue
in macroeconomics, as maintained identifying
restrictions are by definition not testable and
are almost always open to debate. But what
researchers can do is not make the identification
problem any more difficult than it needs to be,
and here Kocherlakota’s recommendations are
particularly valuable. The focus on fit, as exemplified by Smets and Wouters, tends to increase
the difficulty of the identification problem. This
is because increasing the richness of the model—
by including more shocks—makes identification
harder by requiring more restrictions to be placed
on the shock process. From this perspective, relatively simple models may be easier to identify
than densely parameterized models.
It is puzzling that the profession needs to be
reminded of the “fallacy of fit” (Kocherlakota’s
words). The profession learned this dictum the
hard way in the 1970s, when the apparently wellfitting large-scale macroeconometric models
broke down, particularly the Phillips curve (the
inflation-unemployment relationship), which was
a central component of these models. Specifically,
the 1970s witnessed both unemployment and
inflation rising to levels far outside their fitted
historical relationship. At the same time Charles
Nelson (1972) showed that atheoretic, low-order
univariate ARMA models of macroeconomic time
series—that typically were characterized by a
worse fit than the large-scale models—dominated
the large-scale models in forecasting competitions.
Further improvements in forecasting were generated by pseudo-Bayesian VARs, which imposed
random-walk priors on time series to reduce the
problem of overparameterization that is inherent
in VARS. Bayesian VARs are used for forecasting
at several research agencies and commercial
banks, including the Minneapolis Fed and the
Richmond Fed. All of these events led to traditional large-scale econometric models playing a
much smaller role in central bank research and
in policymaking.
So why did central bank researchers and
policymakers return to macroeconometric models
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

after these failures? One reason stems from the
fact that current models feature a much deeper
structure than their large-scale predecessors.
Today’s models include dynamically maximizing
households, maximizing firms, an internally
consistent set of expectations, and a precise definition of equilibrium. All of these advances were
absent from earlier models, and it is believed by
many that these improvements would allow
macroeconometric models to successfully confront the Lucas critique. Nevertheless, it is critical
to distinguish between a model with a deep structure that in principle can be used for conditional
forecasting and a model in which the parameters
are reasonably identified. The first feature doesn’t
imply the second; ironically, specifying rich, fully
articulated general equilibrium models will tend
to make the identification problem even more
difficult.
Applied economists face a difficult trade-off
in specifying macroeconomic models. Simple
models may in principle be easier to identify,
conditional on correct specification, but simple
models will tend to be false, and thus may not be
useful for parameter estimation, at least from a
classical perspective. Richer models may be more
difficult to identify but, conditional on identification, may fare better in terms of parameter estimation. This trade-off is one reason why calibration,
which sidesteps these difficult issues, has been
so popular among applied macroeconomists.
Kocherlakota’s approach is another proposal in a
research program that has attempted to place
calibration into either an explicit classical or
Bayesian framework (see Watson, 1993; Diebold,
Ohanian, and Berkowitz, 1998; Schorfheide, 2000;
and Fernéndez-Villaverde and Rubio-Ramírez,
2004).

THE PHILLIPS CURVE PRIOR
IN MACROECONOMETRIC
MODELING
Central bank research staff have strong priors
on the class of models that are used in monetary
policy analysis. The dominant class of models
are those with a Phillips curve. Here, I define the
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Ohanian

Figure 1
Is the NAIRU Phillips Curve Still There? Unemployment vs. Future Changes in Inflation, 1984-2006
Change in Inflation
2

1.5

1

0.5

0

–0.5

–1

–1.5

–2
3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

Unemployment

Phillips curve as the view that during periods of
slack economic capacity—such as a period of
relatively high unemployment—rapid economic
growth will not raise inflation very much; but
during periods of low unemployment, economic
growth can raise inflation considerably. The
Phillips curve view has implications for monetary
policy. Specifically, it implies that monetary
stimulus during periods of high unemployment
will not raise inflation very much and that there
is scope for the Fed to moderate recessions (which
are periods of slack capacity) through expansionary monetary policy. It also implies that, as capacity becomes tight, the Fed controls inflation by
attempting to reduce the growth rate of the real
economy through monetary contraction.
It may be reasonable for model builders and
policymakers to narrowly focus on this class of
models if there is strong empirical support for
the Phillips curve. In contrast, if there is limited
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support for this class of models, then there is
scope to consider alternative theoretical channels
for the determination of inflation. Here, I present
U.S. time-series evidence that shows little support
for the view that inflationary risks are significantly higher during periods of rapid growth and
tight capacity, relative to rapid growth and slack
capacity.
Atkeson and Ohanian (2001), Stock and
Watson (2006), and others have recently analyzed
the Phillips curve in U.S. time series. Under the
Phillips curve view, low unemployment should
be associated with rising inflation. This is often
referred to as the NAIRU (non-accelerating inflation rate of unemployment) Phillips curve.
Figure 1 shows the NAIRU Phillips curve by presenting the change in inflation and the unemployment rate for the period 1960-2006. The figure
updates my earlier study with Atkeson to include
data from 2000-06. The heavy gray line is an ordiF E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Ohanian

nary least squares (OLS) regression line, which
shows a modest negative relationship between
the change in inflation and unemployment for
1960-83. The blue line is the OLS regression line
between these variables for 1984-2006. The slope
coefficient is very close to zero and is also statistically insignificantly different from zero. This
latter result indicates that there has been no systematic relationship between the change in the
inflation rate and unemployment since 1984. In
other words, there has been no simple NAIRU
Phillips curve in U.S. data for more than 20 years.
In Atkeson and Ohanian (2001), we extended
our analysis of the Phillips curve by examining
whether changes in unemployment, or other
measures of slack capacity, help forecast future
inflation relative to a naive forecasting model that
simply extrapolates the current inflation rate into
the future. Surprisingly, we found that inflation
was not forecasted well by measures of slack
capacity. In particular, the root mean-squared
forecast error (RMSE) for the core consumer
price index, which is a measure of inflation that
excludes volatile food and energy prices and a
key indicator of inflation for both financial markets and central banks, is as much as 94 percent
higher compared with the forecast from the naive
model that extrapolates the current inflation rate
into the future. More sophisticated forecasting
models did not fare much better: for example,
inflation forecasts from Stock and Watson’s macroeconomic activity index model, which forecasts
inflation from a much larger information set than
just unemployment. This model had an RMSE of
between 33 and 81 percent higher than the naive
forecast. These results indicate that there is no
significant, predictable relationship between
cyclical fluctuations in the real economy and
future inflation. Paradoxically, forecasts from
sophisticated models, which clearly fit much
better in sample, are deficient to those from very
simple models that do not fit so well in sample,
such as the naive model we used in Atkeson and
Ohanian (2001).
Tables 1 through 3 show the results of other
tests of the Phillips curve. These tests evaluate
whether economic growth generates inflation
differentially when unemployment is low (tight
F E D E R A L R E S E R V E B A N K O F S T. LO U I S R E V I E W

Table 1
Testing the Phillips Curve Correlation
Between GDP Growth and Inflation,
Controlling for Unemployment
Period

Deflator

Core CPI

1957:Q1–2006:Q2

–0.23

–0.20

1960s

–0.20

–0.23

1970s

–0.43

–0.40

1980s

–0.28

–0.11

1990s

–0.45

–0.44

2000:Q1–2006:Q2

–0.05

–0.22

capacity) relative to when unemployment is high
(slack capacity). Table 1 shows the correlation
between quarterly gross domestic product (GDP)
growth and inflation, measured by the core CPI
and by the GDP deflator for the period 1957-2006,
and also for each decade during the past 50 years.
The data are conditioned on the unemployment
rate as a measure of capacity. The most striking
finding is that the relationship between growth
and inflation is negative, not positive as suggested
by the Phillips curve. Tables 2 and 3 show correlations between GDP growth and inflation, conditioned on unemployment, for various leads and
lags up to 1 year (4 quarters). The correlations
are primarily negative or close to zero.
These tests indicate that there is not sufficient
evidence to support the strong prior for the
Phillips curve that characterizes current econometric practice in central banks. Instead, these
results suggest that models with alternative inflation mechanisms should be analyzed.

THE PHILLIPS CURVE PRIOR
AND MONETARY POLICY
The strong emphasis of the Phillips curve in
macroeconomic research at central banks suggests
that this view is also prominent among the policymakers who are the primary users of central bank
economic research. The importance of the Phillips
curve in policymaking appears to vary over time.
There does not appear to be a substantial Phillips
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Ohanian

Table 2
Testing the Phillips Curve Correlation Between GDP Growth and Inflation (Core CPI):
Leads and Lags, Controlling for Unemployment
1957:Q1–2006:Q2

i=4

i=3

i=2

i=1

∆ ln (yt+i ), ∆ ln (pt )

–0.23

–0.45

–0.41

–0.28

∆ ln (yt ), ∆ ln (pt+i )

–0.02

0.04

–0.10

–0.12

Table 3
Testing the Phillips Curve Correlation Between GDP Growth and Inflation (Deflator):
Leads and Lags, Controlling for Unemployment
1957:Q1–2006:Q2

i=4

i=3

i=2

i=1

∆ ln (yt+i ), ∆ ln (pt )

–0.27

–0.29

–0.26

–0.22

∆ ln (yt ), ∆ ln (pt+i )

0.04

–0.04

–0.09

–0.16

curve policy bias during the Volcker-Greenspan
disinflation of 1982-95. This period is certainly
one of the great triumphs of central banking. After
engineering the largest peacetime inflation in the
history of the United States, with inflation rising
from about 1 percent in the early 1960s to more
than 13 percent by 1980, the Fed lost credibility
with financial markets. By 1980, long-term interest
rates rose to 13 percent. A standard Fisher equation decomposition, which relates nominal interest rates to expected inflation over the horizon of
the security, clearly indicates that financial markets were systematically expecting permanent
high inflation. Beginning at this time, however,
Paul Volcker initiated a low-inflation monetary
policy, and inflation declined to less than 3 percent by the mid-1990s.
To analyze the potential impact of the Phillips
curve on monetary policy, I examine the relationship between the federal funds rate and the
unemployment rate. If there is a strong influence
of the Phillips curve on policy, then we should
observe a systematic inverse relationship between
the funds rate and the unemployment rate.
Figure 2 shows monthly data on these variables
between 1981 and 1995. Note that there is little
systematic relationship between the federal funds
rate and the unemployment rate (the correlation
is about 0.3), suggesting that policy was not par366

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2007

ticularly focused on the Phillips curve. This is not
surprising, as there is little disagreement among
economists or financial market participants that
monetary policy during this period was unconditionally committed to reducing inflation, without much reference to the business cycle. The
policy was indeed effective, as inflation fell and
long-term interest rates fell.
But the nature of policy seemed to change
considerably after inflation declined. Figure 3
shows the funds rate and the unemployment rate
between 1996 and 2006. This figure shows a distinct and systematic inverse relationship between
unemployment and the funds rate, with a correlation of –0.91. As the unemployment rate declined
to 4 percent in 1999 and 2000, Fed officials worried about tight labor markets and inflationary
pressures and raised the funds rate. Then, as the
unemployment rate rose from 4 percent to more
than 6 percent, the Fed pursued a more expansionary policy, driving the funds rate down from
6.5 percent in late 2000 to just 1 percent in late
2003. The policy record since 1995 is consistent
with a strong Phillips curve prior and is reminiscent of the “fine tuning” that policymakers pursued in the 1960s and 1970s. U.S. time-series data
provides little support that a fine-tuning policy
based on Phillips curves will be successful.
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Ohanian

Figure 2
Monetary Policy Without a Phillips Curve Focus
Rate
20
Federal Funds Effective Rate

18

Unemployment Rate

16

Correlation = 0.30
14
12
10
8
6
4
2
0
1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

Figure 3
Unemployment Rate–Driven Monetary Policy? The Return of Phillips Curve Policymaking
Rate
7

6
5
4

3
Federal Funds Effective Rate

2

Unemployment Rate
Correlation = –0.91

1

0
1996

1997

1998

1999

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2001

2002

2003

2004

2005

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Ohanian

CONCLUSION
During the 1960s, policymakers believed
they understood the U.S. economy so well that
they could achieve virtually any desired result
with the appropriate mix of fiscal and monetary
stimulus or contraction. Part of this belief
stemmed from the close fit macroeconometric
model builders were able to achieve with largescale macroeconometric models. This belief
ended abruptly in the stagflation of the 1970s, in
which the perceived stable and systematic tradeoff between unemployment and inflation broke
down and both unemployment and inflation rose
to unprecedented postwar levels. The belief that
tight-fitting models could generate accurate conditional forecasts also broke down and formed
the basis of Robert Lucas’s famous critique of
econometric models.
Macroeconomics and economic modeling
have advanced enormously since the large-scale
models of the 1960s, and these advances are
largely responsible for the return of macroeconometric model building to the forefront of
central bank research and policymaking. But as
Kocherlakota points out, identification of all
parameters in these models is tenuous, particularly
in models with many shock processes. Good
macroeconometric practice almost by necessity
requires sensitivity analysis that provides a systematic treatment of the uncertainty underlying
model parameters. And when there is considerable uncertainty in conditional forecasts, policymakers should recognize this uncertainty as well.
Current policy and the current menu of
models analyzed appear to be too responsive to
the Phillips curve, more so than is warranted by
the data. Model fit is a seductive property; it is
hard for model builders to resist modifying model
equations to achieve a better fit, even when the
modifications do not have strong theoretical
underpinnings. Fromm and Klein (1965) showcase how model builders of the 1960s focused
on fit and modified models to achieve low mean
square error, despite the fact there were few, if any,
economic foundations for these modifications.
Econometric practice today is in some ways
reminiscent of 1960s practice. Shocks are being
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added to various equations to achieve a close fit
to the data, but without necessarily understanding
deeply what the frictions or market imperfections
underlying these shocks are. And current policymaking seems far too responsive to a perceived
Phillips curve that is not present in the data. We
know all too well the outcome of the fitting exercises of the 1970s and the reliance on the Phillips
curve. Perhaps the best way to avoid the monetary policy mistakes of the past is to remember
that these mistakes were partly the consequence
of relying too much on an empirical relationship
that does not have strong theoretical underpinnings and that is not a robust feature of U.S. data.
Agnostic approaches to modeling, as suggested
by Kocherlakota, can significantly aid in the
process of quantifying macroeconomic uncertainty and understanding its implications for
monetary policy.

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