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Federal Reserve Bank of Chicago

Economic Determinants of the Nominal
Treasury Yield Curve
Charles L. Evans and David Marshall

REVISED November, 2006
WP 2001-16

Economic Determinants of the Nominal Treasury Yield
Curve
a

Charles L. Evansa, , David A. Marshallb∗†
Federal Reserve Bank of Chicago; b Federal Reserve Bank of Chicago
Received Date; Received in Revised Form Date; Accepted Date

Abstract
Macroeconomic shocks account for most of the variability of nominal Treasury yields,
inducing parallel shifts in the level of the yield curve. We develop a new approach to
identifying macroeconomic shocks that exploits model-based empirical shock measures.
Technology shocks shift yields through their effect on expected inflation and the term
premium. Shocks to preferences for current consumption affect yields through their impact
on real rates and expected inflation. For both shocks, the systematic reaction of monetary
policy is an important transmission pathway. We find little evidence that fiscal policy
shocks are an important source of interest rate variability.
K eyw ords: Term Structure, M onetary Policy, Vector Autoregression, identification
JE L classifi cation: E52, C32, E43

∗ Corresponding

author: david.marshall@chi.frb.org
paper represents the views of the authors and should not be interpreted as reflecting the
views of the Federal Reserve Bank of Chicago or the Federal Reserve System. We have benefitted
from helpful comments from Monika Piazzesi, Tao Zha, and an anonymous referee.
† The

Economic Determinants of the Nominal Treasury Yield Curve

1.

1

Introduction

Treasury yields assimilate vast amounts of information about economic activity.
The Treasury yield curve is often cited as providing information on the current
stance of monetary and fiscal policy, as well as expectations of future economic
activity, real interest rates, and inflation.1 For these reasons one would expect to
find links between movements in the nominal Treasury yields and observed macroeconomic shocks.
While a major theme of finance research is to understand the factors that move
the term structure, little work to date has focused on observable macroeconomic
factors. Rather, most recent work on the term structure assumes that interest rate
changes are driven by unobserved factors. Notable examples include Litterman and
Scheinkman (1991), Knez, Litterman and Scheinkman (1994), Backus, Foresi, and
Telmer (1998), and the empirical affine term structure literature.2 An important
exception is Ang and Piazzesi (2003). They introduce two observable macroeconomic factors into a Dai and Singleton (2000)-type affine model of the yield curve.
The first factor is the first principal component extracted from several measures of
real economic activity; the second factor is similarly extracted from several price
level indices. They find that macro factors explain up to 85% of the long-horizon
variance of shorter-term yields, but have a much smaller effect on long yields.
In this paper, we ask how different macroeconomic impulses affect the nominal
yield curve. Our first exercise confirms Ang and Piazzesi’s (2003) result that most of
the variability of short- and medium-term yields is driven by macroeconomic factors.
However, our results for the long-term yield are rather different from these authors.
In particular, we find that macro impulses account for almost 85% of the 5-year
yield variance. The key source of this difference is that, unlike Ang and Piazzesi
(2003), our model incorporates interest rate smoothing, in that interest rates depend
on their own lagged values. We show that when interest rate smoothing is omitted,
the importance of macroeconomic shocks for interest rates is severely attenuated.
Our second set of results provides evidence on how specific types of shocks
affect the yield curve. To identify economic shocks, we develop an approach that is
new to the VAR literature. Instead of imposing a priori covariance restrictions on
the relation between the VAR innovations and shocks, we infer these relationships
from empirical measures of economic shocks that economists have proposed, often
based on dynamic general equilibrium models. Our model-based measures include:
Basu, Fernald, and Shapiro’s (2001a,b) measure of technology shocks; Blanchard
and Perotti’s (2000) measure of fiscal policy shocks; and a measure of marginalrate-of-substitution (MRS) shocks similar to that studied by Hall (1997). We show
how this information is easily incorporated into the analysis of impulse responses.
We find that our MRS shock moves output and inflation in the same direction.
Many empirical macroeconomists refer to this sort of impulse as an aggregate de1 For example, see Bernanke and Blinder (1992), Estrella and Hardouvelis (1991), Blanchard
(1985), Mishkin (1990).
2 See Duffie and Kan (1996), Dai and Singleton (2000) and Backus, Foresi, Mozumdar, and Wu
(2001).

Economic Determinants of the Nominal Treasury Yield Curve

2

mand shock (e.g., see Blanchard (1989)). An expansionary MRS shock increases
both expected inflation and real interest rates, inducing a large, significant, and
persistent response in all nominal rates and shifting the yield curve level. In contrast, the technology shock moves output and inflation in opposite directions. An
expansionary technology shock drives real interest rates up and expected inflation
down, so its effect on nominal interest rates is, in principle, ambiguous. However, we
find that the expected inflation response dominates, so the expansionary technology
shock tends to reduce interest rates of all maturities. Our model-based measure of
fiscal shocks does not have a significant impact on interest rates.
Our third set of results relates to the transmission mechanisms by which these
shocks move the yield curve. We find that the systematic response of monetary
policy is an important pathway whereby macroeconomic shocks affect interest rates.
Monetary policy generally reacts to these shocks in the manner predicted by the
Taylor (1993) principle: shocks that increase expected inflation or the gap between
actual and potential output tend to increase the Federal funds rate. Longer-term
interest rates are affected by expectations of changes in the funds rate. In addition,
macroeconomic shocks can directly affect term premiums.
The remainder of this paper is structured as follows: In section 2 we describe
our basic statistical framework. In section 3 we conduct a preliminary empirical
exploration on the effect of macroeconomic factors on the yield curve. This section
uses an eigenvalue decomposition of impulse responses to examine the implications
of interest rate smoothing for the transmission of macroeconomic shocks. Section
4develops our identification methodology that uses model-based shock measures,
and section 5 explains how we implement this methodology empirically. Section 6
presents our empirical findings. Section 7 concludes the paper.
2.

Basic statistical framework

We use the following vector autoregression (VAR) framework throughout our
empirical analysis. Let Zt be an (n − 1) × 1 vector of nonfinancial macroeconomic
variables at time t; let F Ft denote the federal funds rate (included as the instrument
0
of monetary policy); let Yt ≡ (Zt0 , F Ft ) ; and let Rt denote an m × 1 vector of zerocoupon Treasury yields of different maturities. We estimate versions of the following
structural VAR:
·
¸·
¸ ·
¸·
¸ ·
¸
A 0
Yt
Ã(L)
0
Yt−1
εt
=
+
(1)
G H
Rt
Rt−1
γt
C̃(L) D̃(L)
where A is an n × n nonsingular matrix; H is an m × m nonsingular matrix; G
is a rectangular matrix, 0 is the zero matrix with appropriate dimensions; and
Ã(L), C̃(L), and D̃(L) are matrix polynomials in the lag operator L. The process
0
[εt , γ t ] is an i.i.d. vector of mutually and serially uncorrelated shocks whose variance
¡
¢
FF 0
is the identity matrix. It is useful to partition εt ≡ εZ0
, where εZ
t , εt
t is a
FF
(n − 1) × 1 vector of macroeconomic shocks and εt is a monetary policy shock.
One can think of γ t as yield shocks analogous to Ang and Piazzesi’s (2003) vector
of latent financial variables. The zero restrictions on the upper right-hand blocks

Economic Determinants of the Nominal Treasury Yield Curve

3

of the coefficient matrices in (1) imply that neither current nor lagged yields nor
the yield shocks γ t enter the law of motion for Yt . Thus, current and lagged Yt are
a sufficient state vector for spanning the space of all macroeconomic driving shocks.
We estimate system (1) via ordinary least squares using the following reduced
form:
·
¸ ·
¸ ·
¸·
¸ ·
¸
Yt
0
a(L)
0
Yt−1
ut
=
+
+
(2)
αYt
c(L) d(L)
Rt
Rt−1
vt
0

where α ≡ −H −1 G and [u0t vt0 ] is the vector of OLS residuals. If A is identified, then
the structural shocks εt can be recovered from the OLS residuals via the relation
Aut = εt

(3)

Once εt is identified, variance decompositions and impulse responses can be computed.
3.

Macroeconomic shocks as drivers of interest rates

Our first exercise using framework (1) is our “baseline model”. It explores the
fraction of interest rate variability that can be attributed to macroeconomic shocks.
The data vector is given by Z ≡ (IP, P, P COM )03 , whereIP denotes the logarithm
of industrial production, P denotes the logarithm of the personal consumption expenditure chain-weight price index, and P COM denotes the smoothed change in an
updated version of the index of sensitive materials prices originally published in the
index of leading indicators. The yields we use here, and throughout the paper, are
the 1-month, 12-month, and 60-month zero coupon bond yields from the Fama-Bliss
files of the CRSP data base. The data are monthly, from January 1959 through
December 2000. The VAR incorporates 12 lags.
For this exercise, we posit a lower-triangular structure for matrix A in system (1).
This is equivalent to a simple recursive orthogonalization of the VAR residual vector
ut . We give no structural interpretation to the elements of εt thus constructed,
except that its first three elements, εZ
t , span the impulses driving the nonfinancial
macro variables Z. Panel A of Table 1 displays the fraction of the 5-year ahead
conditional variance of each yield attributable to each of the orthogonalized macro
residuals. (Numbers in parentheses are lower and upper 90% error bands, computed
using 500 Monte Carlo draws from the Bayesian posterior distribution of the model
parameters.) As can be seen from the Table, 75% to 84% of the 5-year ahead
variance of the yields is due to nonfinancial macro factors. That is, macroeconomic
impulses account for the vast preponderance of interest rate variability.
Our estimates for the shorter yields are similar to those reported by Ang and
Piazzesi (2003). However, our estimate for the 60-month yield is much higher than
that reported by those authors. In particular, they find that only 48% of the 5-year
ahead variance of this long bond yield is explained by macroeconomic factors.
3 We

thank John Fernald for providing us with this time series on technology shocks.

Economic Determinants of the Nominal Treasury Yield Curve

4

What accounts for this discrepancy? One important reason is that our model
incorporates interest rate smoothing. This term often refers to models in which
monetary policy is a function of lagged interest rates. For example, Christiano,
Eichenbaum and Evans (1999) study monetary policy rules in VARs that include
substantial lags of interest rates. Similarly, the literature on Taylor rules tends to
find that specifications with partial-adjustment match the data better (Orphanides,
1999; Evans, 1998).4 From a theoretical perspective, Woodford (2003) argues that
interest-rate smoothing is more effective at imparting persistence in long-term interest rates and influencing the interest-sensitive sectors of the economy with smaller
policy movements. Building on the empirical Taylor rule literature, we generalize
the term “interest rate smoothing” to refer to any model in which lagged interest rates enter the interest rate equations. The model of Ang and Piazzesi (2003)
excludes interest rate smoothing, since they model interest rates as functions exclusively of current and lagged macro factors, along with current and lagged latent
exogenous factors that are orthogonal to the macro factors.
We construct a “no-smoothing” version of model (1) by setting all elements of
D̃(L) as well as the fourth row of Ã(L) equal to zero. By doing so, the regression
equations for F F and R exclude lagged values of these interest rates. Note that
lagged values of the federal funds rate still indirectly affect interest rates because Zt
is still be a function of F Ft−1 , F Ft−2 , etc. In this no-smoothing version of equation
F
(1) the interest rate residuals εF
and γ t display pronounced serial correlation (a
t
point noted by Rudebusch, 2002, and Ang and Piazzesi, 2003). To account for
this serial persistence, our no-smoothing model assumes that these residuals are
autoregressive processes.5
The implications of the no-smoothing model are given in Panel B of Table 1.
Compared to the baseline model, the no-smoothing model has a much smaller fraction of interest rate variance accounted for by nonfinancial macro shocks. Specifically, only 47% - 53% of the 5-year ahead conditional interest rate variance is
attributable to macro shocks when interest rate smoothing is excluded.
There appear to be two reasons for the dramatic differences between these two
models. First, in the baseline model the macro shocks εZ
t are propagated to future interest rates Rt+h both through their direct affect on future macro variables Zt , Zt+1, ...,Zt+h , and through their effect on the interest rates (F Ft , Rt ) ,
(F Ft+1 , Rt+1 ) , ... , (F Ft+h−1 , Rt+h−1 ). These interest rate channels are missing
in the no-smoothing case. Rather, when interest rate smoothing is ruled out, the
dynamic propagation of macro shocks εZ
t to Rt+1 , Rt+2 etc. proceeds exclusively
through its effect on Zt , Zt+1 , Zt+2 , etc. One might conjecture that, by restricting
the propagation of macro shocks to this single pathway, the magnitude of the interest rate responses (and therefore the fraction of interest rate variation explained
by macro shocks) would be attenuated.
Second, as we noted above, the no-smoothing model implies a yield residual γ t
that is highly serially correlated. This implies that a good deal of the dynamics of
4 In contrast, Rudebusch (2002) argues that serial persistence in Taylor rules comes from serially
correlated shocks and not interest-rate smoothing.
5 In particular, εF F and each of the elements of γ are modelled as AR(8) processes.
t
t

Economic Determinants of the Nominal Treasury Yield Curve

5

Rt in the no-smoothing model is accounted for by the dynamic properties of γ t which,
by construction, is orthogonal to the macro shocks and the current and lagged macro
variables. As a result, the estimator of the no-smoothing model relies less on the
propagation of macro shocks in explaining interest rate persistence, and places a
greater reliance on exogenous persistence that is orthogonal to the macroeconomy.
These two explanations should be reflected in an eigenvalue decomposition of impulse responses. In particular, in any linear dynamical
system the h-step response
Pq h (k,j) i h
of variable k to shock j can be written as i=1 wi
λi , where q is the number

of eigenvalues characterizing the dynamical system, λi is the ith such eigenvalue,
(k,j)
and wi
denotes the weight put on the ith eigenvalue λi in
impulse
o
n this particular
response. (See the Appendix for a detailed derivation of

(k,j)

wi

.)

Note that

(k,j)
wi

the weights
do not depend on h, the number of steps ahead in the impulse
response. Note also that the eigenvalues can be complex, in which case the corresponding weight will also be complex. (Complex eigenvalues characterize oscillatory
impulse responses.)
The 5-year ahead variance decompositions, displayed in Table 1, are proportional
to the squared sum of the impulse responses for the first 60 steps, so they are largely
captured by studying the largest eigenvalues. In the baseline model with interest
rate smoothing, the medium- to long-run variation in interest rates is well-explained
by focusing on only three large eigenvalues:
1. A complex conjugate pair with modulus = .9903:
λ = .9900 + .0242i
¡ ¢
conj λ = .9900 − .0242i

(4)

This eigenvalue pair receives substantial weight in the responses of both macro
variables and yields to macro shocks. It has the potential to impart longterm dynamics in macro and interest rate responses. The periodicity of this
complex pair is about 21 years, so we tentatively associate it with the long
swings in productivity growth that characterize the post-war data.
2. A real eigenvalue with modulus .9606. This eigenvalue has a half-life of
approximately 17.2 months. It seems to be associated with the medium-term
dynamics induced by having lagged F Ft−h in the federal funds equation. It
has small but non-zero weight for the Zt responses, and a large weight for the
responses of both F Ft and Rt .
3. A real eigenvalue with modulus .9590. This eigenvalue has a half-life of
approximately 16.6 months. It appears to be associated with the additional
medium-term interest rate dynamics induced by having lagged Rt−h in the Rt
equations, since it receives a large weight for the Rt responses, but it receives
a weight of zero for both Zt and F Ft .

Economic Determinants of the Nominal Treasury Yield Curve

6

In contrast, in the no-smoothing model the medium- to long-run variation in
interest rates is well-explained using only a single complex eigenvalue pair that is
almost identical to λ in equation (4) and its conjugate.6 The subsidiary eigenvalues
0.96 and 0.959 that we found to be important in the baseline system are completely
absent in the no-smoothing system. This is not surprising, since these subsidiary
eigenvalues capture dynamics associated with the lagged fed funds rate and lagged
yields, dynamics which have been excluded from the no-smoothing case. Thus,
the absence of these subsidiary eigenvalues from the no-smoothing case is a formalization of the first explanation, discussed above. In particular, the no-smoothing
model omits propagation paths for the macro shocks that are responsible for the
medium-term dynamics of interest rates. In contrast, this medium-term propagation mechanism is present in the baseline model.
The eigenvalue decomposition of impulse responses also sheds light
© on the¡sec¢ª
ond explanation. As mentioned above, the complex eigenvalue pair λ, conj λ
given in equation (4) is important for both the baseline and no-smoothing systems.
(k,j)
However, the weights wi associated with this eigenvalue pair tend to be substantially larger for the baseline smoothing model than for the no-smoothing model.
This is especially important for the interest rate responses to the P COM shock,
which is the most important macro shock for interest rate variability in Table 1.
For example, the weight associated with λ for the response of the one-month rate
to this shock has a modulus of 0.1849 in the baseline case, but only 0.1548 in the
no-smoothing case. This increased weight on λ is the main reason why, in Table
1, the P COM shock accounts for substantially more interest rate variation in the
baseline model, as compared to the no-smoothing model.
What is the economics underlying this larger weight on λ in the baseline case?
Recall that λ captures the low-frequency component of interest rate movements.
The smoothing model ascribes essentially all of this component of interest rate
variation to macro movements at the same frequency. In contrast, the no-smoothing
model ascribes a portion of this component to low-frequency movements in the
serially-correlated residual γ t . In fact, there appears to be a pronounced peak
in the spectral density of γ t at a periodicity between 20 and 30 years (although
estimates of spectra at very low frequencies are necessarily imprecise, for obvious
reasons). With a process γ t that is orthogonal to the macro shocks explaining a good
deal of the low-frequency variation in interest rates, there is less remaining interest
rate variability to be explained by the macro variability at this periodicity. The
reduced weight on λ in the no-smoothing model formalizes this reduced dependency
on macro fluctuations to explain these low frequency interest rate swings.
4.

Identifying structural shocks using model-based measures

According to the evidence of section 3, a large fraction of the interest rate
variance for all maturities is accounted for by macroeconomic impulses. However,
6 In the no-smoothing case, this eigenvalue is estimated to be 0.9890 ± .0213i, almost identical
to λ.

Economic Determinants of the Nominal Treasury Yield Curve

7

unless substantially more structure is imposed on the VAR innovations in equation
(19) this description of the data’s conditional second moment properties represents
an incomplete characterization of the economic determinants of the nominal yield
curve. According to equation (19), identification of the structural shock vector
εt requires restricting matrix A. We propose in this section an approach that
closely ties the identifying restrictions to specific economic theories. In particular,
as in Prescott (1986) and Hall (1997), we exploit the ability of economic models
to guide directly the construction of empirical measures of fundamental economic
impulses, such as technology shocks, fiscal policy shocks, and shocks to households’
marginal rate of substitution (MRS) between consumption and leisure. As a result,
few prior restrictions are placed on the covariance structure of the VAR innovations
ut . As much as possible, we allow the model-based measures to dictate the VAR
identification of macroeconomic shocks εt .
Let ηt denote the vector of observable model-based measures. (In section 5
we describe in detail how these measures are constructed from data.) We assume
that these measures represent noisy measures of the true underlying shocks εt .
Specifically,
η t = D εt + wt

(5)

where D is a non-singular (n × n) matrix and wt is a vector of measurement errors
independent of εt (and therefore of ut ). To identify the model, we must uniquely
determine the matrices A and D. To that end, substitute equation (19) into equation
(21) to get
η t = Cut + wt

(6)

where
C ≡ DA

(7)

or, equivalently,
A = D−1 C

(8)

This condition is important: Since wt is uncorrelated with ut , the matrix C can
readily be estimated from equation (22) by ordinary least squares. Therefore, A
could be identified with no a priori restrictions if the n2 elements of D were known.
In effect, this shifts identifying restrictions from the matrix A to matrix D.
Using equations (19), (24), and the fact that E [εε0 ] = I, one obtains
DD0 = CΣu C 0 .

(9)

The C and Σu matrices on the right-hand side of equation (25) can be estimated
directly, so equation (25) imposes n(n + 1)/2 restrictions on D. Identification of D
then requires an additional n(n − 1)/2 a priori restrictions. Arguably, restrictions
on D are easier to justify than restrictions on A, since the former maps underlying
structural shocks into their empirical counterparts, while the latter maps the underlying shocks to the VAR residuals. For example, D may be diagonal, in which

8

Economic Determinants of the Nominal Treasury Yield Curve

case the η measures are contaminated only by classical measurement error. Alternatively, theory and measurement limitations may indicate that some η measures
are linear combinations of the underlying shocks. In that case, D would have some
non-zero off-diagonal elements. We discuss the specific identifying assumptions we
impose on D in section 5, below.
When the system is exactly identified, D can be computed directly from equation
(25) as the unique factorization of CΣu C 0 satisfying the identifying restrictions.
When the system is overidentified, neither equation (20) nor equation (25) will
hold exactly in finite samples. Nevertheless, one can still estimate D by using the
maximum likelihood procedure described in Hamilton (1994, pp.331-332). Once D
is determined, matrix A can be computed using equation (24).
5.

Model-based measures of structural shocks

To implement the model-based identification strategy described above in section
4, we must obtain model-based measures of macroeconomic driving shocks. In this
section we describe four quarterly model-based measures that we use: technology,
preference, fiscal policy and monetary policy.
5.1.

Technology Shocks

Since Prescott (1986), the driving process for aggregate technology shocks in
real business cycle models has been calibrated to empirical measures of Solow residuals. A large literature, including Prescott (1986), has noted that a portion of
the fluctuations in standard Solow residual measures is endogenous, responding to
macro shocks.7 Basu, Fernald, and Shapiro (2001b) provide a recent estimate of
technology innovations that attempts to reduce these influences. Ignoring industry
composition effects, their aggregate analysis specifies production as follows:
Yt = zt gt F (vt Kt , et Nt )
ln zt = µ + ln zt−1 + εT ech,t

(10)

where Y , z, v, K, e, and N are the levels of output, technology, capital utilization
rate, capital stock, labor effort, and labor hours. The object gt represents costs of
adjusting employment and the capital stock. It is is an explicit function of observable data, and is calibrated from econometric estimates in the literature (see
Shapiro (1986) and Basu, Fernald, and Shapiro (2001a,b)). F is a production function that is homogeneous of degree γ ≥ 1, allowing for the possibility of increasing
returns. Basu, Fernald, and Shapiro specify an economic environment where the
unobserved variables v and e can be measured as proportional to the workweek of
labor and capital. Assuming γ = 1 – constant-returns-to-scale – Basu, Fernald,
and Shapiro (2001b) use time-varying cost shares to compute a quarterly, aggregate
measure of the technology innovation.
7 For

example, see Burnside, Eichenbaum and Rebelo (1993) and Braun and Evans (1998))

Economic Determinants of the Nominal Treasury Yield Curve

9

We use Basu, Fernald, and Shapiro’s (2001b) quarterly, aggregate measure of
technology for our model-based empirical measure ηT ech 8 of the aggregate technology shock εT ech . Although this quarterly measure includes controls for many latent,
endogenous features, data limitations prevent controlling for industry compositional
effects. This potentially introduces measurement error into this series. The data
begin in 1965:II and end in 2000:IV.
5.2.

Marginal-Rate-Of-Substitution Shocks

A shock to the marginal rate of substitution between consumption and leisure
can potentially shift aggregate demand for goods and services. Hall (1997), Shapiro
and Watson (1988) and Baxter and King (1990) find substantial business cycle
effects from empirical measures of intratemporal marginal rates of substitution between consumption and leisure. To generate a model-based empirical measure of an
MRS shock, we generalize Hall’s (1997) procedure to allow for time-nonseparable
preferences.9 Consider a representative consumer with the following utility specification that includes external habit persistence
¡
¢1−γ
Ct − bC t−1
N 1+φ
U (Ct , Nt ) = ξ t
−
1−γ
1+φ
ln ξ t = ρ(L) ln ξ t−1 + ηMRS,t

(11)

where C is consumption of the representative agent, C represents the per-capita
aggregate consumption level, N is labor hours, ξ is a serially correlated preference
shifter, and εMRS is a serially independent shock. The first-order conditions for
consumption and labor hours lead to the following intratemporal Euler equation
(or MRS relationship)
¡
¢−γ
ξ t Ct − bC t−1
= 1/Wt
(12)
Ntφ
where W is the real wage. Taking logs, one obtains
£
¤
ln ξ t = φ ln Nt − lnWt + γln Ct − bC t−1 .

(13)

In equilibrium, the per-capita aggregate consumption equals the consumption levels
of the representative agent, so C = C.
We use equation (13) to obtain an empirical measure of ln ξ t . We then estimate autoregression (11) to obtain our model-based measure ηMRS,t of the MRS
shock. Our data are quarterly and extend from 1964:I to 2000:IV. Consumption is
measured by per capita nondurables and services expenditures in chain-weighted
1996 dollars. Labor hours correspond to hours worked in the business sector per
capita. The real wage corresponds to nominal compensation per labor hour worked
in the business sector deflated by the personal consumption expenditure chain price
8 Throughout
9 Holland

this paper, we omit the time subscript t if no ambiguity is implied.
and Scott (1998) study a similar MRS shock for the United Kingdom economy.

Economic Determinants of the Nominal Treasury Yield Curve

10

index. The hours and compensation data are reported in the BLS productivity
release. The utility function parameters are taken from previous studies. First, to
ensure balanced growth we set γ = 1, corresponding to log utility for consumption services. Second, we use Hall’s (1997) value for φ = 1.7, corresponding to a
compensated elasticity of labor supply of 0.6. Finally, we set the habit persistence
parameter b = 0.73 as estimated by Boldrin, Christiano and Fisher (2001).
We measure ηMRS as the residual in equation (11). We estimate a sixth-order
polynomial for ρ(L). In addition, the M RS measure ξ exhibits noticeable low
frequency variation, so we also include a linear time trend in the regression to
account for demographic factors that are beyond the scope of this analysis. In order
to allow for serially-correlated measurement errors in ξ t , we use an instrumental
variables estimator to estimate ρ(L). 10
Macroeconomic researchers have offered differing interpretations for the random
marginal rate of substitution shifter in ξ t in equation (12).11 First, the home production literature due to Benhabib, Rogerson, and Wright (1991) and Greenwood
and Hercowitz (1991), among others, suggests that ξ t could be a productivity shock
to the production of home goods. Second, inertial wage and price contracts will
distort the simple intratemporal Euler equation as it is specified in (12) . In particular, in the Calvo pricing environments considered by Christiano, Eichenbaum, and
Evans (2001) and Galí, Gertler, Lopez-Salido (2001), alternative versions of (12)
hold. Third, Chari, Kehoe, and McGrattan (2002) and Mulligan (2002) interpret ξ t
as reflecting wedges or distortions, such as changes in tax rates or union bargaining
power. To the extent that these alternative explanations have different theoretical
implications for impulse response functions, an empirical analysis of our MRS shock
can help shed light on which explanation seems to be consistent with the aggregate
data.
5.3.

Fiscal Policy Shocks

The modern business cycle literature that includes fiscal policy effects has focused primarily on exogenous specifications of government spending and tax rates.12
To relate these theoretical studies to aggregate data requires distinguishing between
the exogenous and endogenous components of fiscal policy. Blanchard and Perotti
(2000) construct a quarterly series of exogenous fiscal shocks by using regression
methods to control for the systematic response of fiscal policy. We use their measures to identify fiscal policy shocks.
1 0 Our shock identification strategy assumes that the measurement errors in our model-based
shocks are independent of the VAR innovations. Consequently, we use real GDP, the GDP price
index and commodity prices as instruments.
1 1 As Hall (1997) pointed out, the greatest amount of evidence against Eichenbaum, Hansen,
and Singleton’s (1988) preference specifications surrounded the intratemporal Euler equation for
consumption and leisure.
1 2 Baxter and King (1993) and Christiano and Eichenbaum (1992) study permanent and transitory changes in exogenous government purchases. Braun (1994) and McGrattan (1994) study
transitory changes in exogenous tax rates. An alternative approach is taken by Leeper and Sims
(1994): they allow tax rates to respond systematically to the state of the economy.

Economic Determinants of the Nominal Treasury Yield Curve

11

Blanchard and Perotti (2000) start with measures of GDP, government spending excluding transfers, and tax receipts net of transfers. The latter two variables
include federal, state, and local measurements. Blanchard and Perotti control for
the automatic responses of spending and taxes to changes in GDP, using measures
of the elasticity of different types of taxes, transfers, and spending to output. Additional restrictions are imposed to identify exogenous shocks to taxes and government
spending.13 We construct our model-based empirical measure η F iscal as a shock to
the government deficit, defined as the difference between Blanchard-Perotti’s government spending and tax shocks.14 We treat ηF iscal as a noisy measure of the
underlying fiscal policy shock εF iscal .
5.4.

Accounting for Monetary Policy Shocks

The effects of monetary policy shocks on the term structure have been studied
elsewhere,15 and are not the focus of this paper. However, to isolate the effects
of technology, MRS, and fiscal shocks, we control for monetary policy impulses
so that the effects of monetary policy shocks are not incorrectly ascribed to these
other shocks. To do so, we introduce an empirical measure of monetary policy
shocks, denoted η MP . We use an updated version of the monetary policy shock
measure in Christiano, Eichenbaum, and Evans (1996). This measure is derived
from an identified VAR using the following variables: the logarithm of real GDP;
the logarithm of the GDP chain-weighted price index; the smoothed change the
index of sensitive materials prices used in section 3; the Federal funds rate; the
logarithm of nonborrowed reserves; and the logarithm of total reserves. The data
run from 1959:I through 2000:IV.
5.5.

Correlation Structure of the Model-Based Measures

Table 2 displays the contemporaneous correlation matrix for the model-based
0
shock measures η ≡ (η MP , η MRS , η T ech , η F iscal ) described in sections 5.1 - 5.4. .
Note that the correlations are fairly low, with the exception of corr (ηT ech , ηF iscal ),
which exceeds 0.30.
According to equation (22), the model-based measures only provide useful information for identifying A if they are correlated with the VAR residuals ut . Table
3 provides evidence on these correlations for the data we use. It displays the R2 s for
the OLS regressions in system (22) using the measures of η = (η MP , η MRS , η T ech , η F iscal )0
described in sections 5.1 - 5.3.. The variables in our macro VAR block are quarterly analogues to the monthly measures used in section 3: real GDP, the GDP
price deflator, the commodity price index P COM , and the Federal funds rate. The
1 3 Blanchard and Perotti (2001) estimate their VAR under two different trend assumptions.
First, they incorporate deterministic time trends; second, they allow for stochastic trends. We
have done our analysis with fiscal shocks computed both ways. The results are very similar, so
we only display the results for the model with deterministic time trends.
1 4 We have also performed the analysis with the individual tax and spending shocks. The results
are qualitatively unchanged, although the impulse responses to these individual shocks are smaller.
1 5 See Gordon and Leeper (1994), Bernanke, Gertler, and Watson (1997), and Evans and Marshall
(1998).

Economic Determinants of the Nominal Treasury Yield Curve

12

only problematic shock measure in Table 3 is the fiscal shock, whose R2 is only
8.7%.16 This suggests that our fiscal shock measure η F iscal may not provide strong
identification for an underlying fiscal shock in the context of our VAR system. As a
result, caution should be exercised in interpreting the responses to the fiscal shock
implied by this exercise.
5.6.

Identifying restrictions

Given the empirical estimates of ηt , the key step in the identification is to
specify restrictions on D, the mapping from the model-based measures η to the
true underlying shocks ε. A straightforward approach would be to assume that
each element of ηt equals the corresponding element of εt plus measurement error.
In this case, D is diagonal. We find that the data strongly reject the overidentifying
restrictions implied by this model.17 Alternatively, some η measures may be linear
combinations of the underlying shocks, perhaps due to mismeasurement in the way
the series in η were computed. This could account for the correlation structure
among η it elements, described in Table 2, and would imply non-zero off-diagonal
elements of D.
As we noted in section 5.1, there is a large literature on possible mismeasurement
of technology shocks. Evans (1992) points to possible contamination of technology
shocks by monetary policy; Burnside, Eichenbaum, and Rebelo (1993) discuss the
problem of unobserved labor hoarding, and Burnside and Eichenbaum (1993) note
the problem of variable capital utilization. While the Basu-Fernald-Shapiro technology measure that we use in this paper attempts to correct for many of these
sources of mismeasurement, it may do so imperfectly. Consequently, we wish to
allow for the possibility that η T ech may be a linear combination of several shocks.
To this end, we selected the following specification of system (21):

 
 

ηMP
d11 0
εMP
0
0
 ηMRS   0 d22 0


0 

 
  εMRS 
(14)
 ηT ech  =  d31 d32 d33 0   εT ech  + w.
d41 d42 d43 d44
ηF iscal
εF iscal

Specification (14) is overidentified: it imposes seven zero restrictions, whereas exact
identification requires only six restrictions. Specification (14) assumes that our
monetary policy and MRS measures equal the true underlying shock plus classical
measurement error. In contrast, η T ech is allowed to incorporate the influence of the
true underlying monetary policy and MRS shocks. Note that we do not permit any
of these three measures to be contaminated by the underlying fiscal shocks. The
reason for this assumption is that the fiscal policy shock measure has the smallest
correlation with the VAR innovations ut of all of our η it elements. (See Table 3.)
1 6 Our measure of η
F iscal is the difference between the government spending shock and the tax
shock, both estimated by Blanchard and Perotti (2000). When we estimate regression (22) using
the spending shock or the tax shock individually, the R2 s are all below 6%.
1 7 The likelihood ratio statistic comparing the diagonal-D model to the model where D is exactly
identified is χ2 (6). We obtain a value of 718.9 for this statistic.

Economic Determinants of the Nominal Treasury Yield Curve

13

Consequently, this row of the matrix C is likely to be estimated imprecisely, so
we wish to limit the influence of the fiscal policy measure on the other analyses.18
Furthermore, Blanchard and Perotti’s (2000) approach uses only three variables:
GDP, government spending, and government taxes. So this measure has not been
projected onto innovations from omitted variables that would be included in larger
systems. These considerations motivate us to restrict the potential influence of the
fiscal policy measure on the identification of the other shocks.
6.

Empirical Results

In this section we explore how macroeconomic shocks affect the term structure
using the identification strategy described in sections 4 and 5. The results are displayed in Figures 1 and 3. These figures display the responses of macroeconomic
variables and yields to the MRS and technology shocks. In addition, we plot the responses of the one-month real rate19 and the 60-month term premium.20 The dashed
lines give 90% probability error bands for the impulse responses, computed using
500 Monte Carlo draws from the posterior distribution of the model’s parameters.
Table 4 displays the decomposition of the variance of 5-year ahead macroeconomic
forecast errors implied by the ε vector of shocks, also with 90% probability error
bands. We compute the posterior distribution using the approach described in
Evans and Marshall (2002), which extends the Bayesian methods described in Sims
and Zha (1999), Zha (1999), and Waggoner and Zha (2003) in a natural way to
account for uncertainty in regression (22).21
6.1.

Responses to MRS shock

Figure 1 gives the responses to the εMRS shock. Upon impact, real GDP rises
immediately with a persistent effect that lasts several years. The fraction of output
variance accounted for by εM RS is 39% at the 5-year horizon in Table 4. Inflation
rises, peaking after one year, although the estimates are imprecise. The transitory
nature of the inflation response implies that the MRS shock contributes only a
small portion of the total price level variation. According to Table 4, the fraction
of inflation variation accounted for by εMRS is only 9% at the 5-year horizon.
Consider now the systematic response of monetary policy to εMRS . The nominal
federal funds rate responds to an εMRS impulse with a persistent and significant
increase. The Taylor (1993) principle is evident in this response: the real funds rate
1 8 With specification (14), the coefficients in the regression of η
F iscal on ut only affect the
identification of εF iscal , not the other elements of ε.
1 9 In the figures, the one-month real rate denotes the real return to the one-month nominal bond.
£ 1 ¤
P59
2 0 If y k denotes the k-month yield, the 60-month term premium equals y 60 − 1
E yt+s
.
t
t
60
s=0 t
2 1 Because this system is overidentified, the standard Bayesian procedure (Doan, 2000) is inappropriate. (See Sims and Zha, 1999). In our analysis, matrix D in equation (14) is estimated by
maximum likelihood for each Monte Carlo draw. Sims and Zha (1999) refer to this as the “naive
Bayesian procedure”. However, when we implement similar, exactly identified systems, the error
bands are virtually identical to those displayed in the figures.

Economic Determinants of the Nominal Treasury Yield Curve

14

rises in response to a shock that increases both deviations of output and inflation
from their target levels. Table 4 indicates that εMRS is an important driver of
systematic monetary policy, accounting for 34% of the 5-year ahead variance of the
Federal funds rate. Together, these results depict shocks that shift the aggregate
economy’s demand for goods and services; the Fed responds by “leaning against the
wind.”
Turning to the responses of yields to the MRS shock, note that εMRS has substantial, persistent and significant effects on individual nominal yields. In particular, the three yields respond to a one-standard deviation positive εMRS shock by
increasing between 25 and 38 basis points on impact. These responses are longlived, remaining well above zero over four years after the initial impulse. These
responses indicate that aggregate demand shocks can lead to substantial variation
in nominal yields. Table 4 confirms this conclusion: the εMRS accounts for 37%,
37%, and 26% of the 5-year ahead forecast variance of the one-month, 12-month,
and 60—month yields, respectively.
Since the responses of the three yields to εMRS are similar, the MRS shock
induces a parallel shift in the level of the yield curve. The reason for this pronounced
response of the yield curve is that εMRS shifts inflation and real rates in the same
direction. We discussed above the positive response of one-month inflation. The
positive response of the one-month real rate is displayed in the lower left-hand graph
of Figure 1. It peaks at 48 basis points in the quarter following the impulse, decaying
gradually. The pronounced real rate response following an MRS shock is consistent
with our interpretation of these shocks as transitory impulses to the marginal utility
of consumption. The responses (not displayed) of the longer-term inflation and real
rates are quantitatively similar to their short-term counterparts. These positive
inflation and real-rate responses at all maturities result in a significant upward shift
in the level of the yield curve .
The lower right-hand graph in Figure 1 gives the response of the 5-year term
premium. As can be seen, there is no evidence that this term premium responds
to εMRS . Thus, movements in the long yield due to the MRS shock appear to be
well-described by the expectations hypothesis of the term structure.
6.2.

Responses to the technology shock

Figure 2 shows responses to εT ech , the technology shock. This shock induces an
increase in output, a fall in the inflation rate, and a decline in all three bond yields.
The response of output is pronounced and long-lived after about 5 quarters, but the
initial output response is negligible. This delayed response to a technology shock is
consistent with several recent empirical and theoretical analyses.22 The εT ech shock
accounts for 20% of the five-year ahead output variance. (See Table 4.)
The εT ech shock induces a pronounced short-run decline in the one-month-ahead
2 2 Galí (1999) and Basu, Fernald and Kimball(2000) interpret the delayed response of output to
technology shocks as evidence of inertial aggregate demand due to price stickiness. In Boldrin,
Christiano, and Fisher (2001) and Francis and Ramey (2001), this delay is consistent with inertial
aggregate demand due to habit persistence in consumption and investment adjustment costs.

Economic Determinants of the Nominal Treasury Yield Curve

15

inflation rate: the response peaks at -55 basis points three quarters after the impulse.
This inflation response is transitory, dissipating over the next three to four years.
According to Table 4, the εT ech shock accounts for 55% of the 5-year ahead variance
of the inflation rate. The inflation responses are consistent with an economy in
which monetary policy allows falling real marginal costs to show through to smaller
price increases.
The technology shock induces pronounced, persistent, and significant declines
in nominal yields of all maturities. For εT ech , the responses to a one-standard
deviation shock bottom out in three to four quarters at -43, -47, and -38 basis
points for the 1-, 12-, and 60-month yields respectively. Overall, εT ech accounts for
39%, 46%, and 57% of the one-month, 12-month, and 60-month yield variance at
the 5-year horizon. Since these negative responses of the three yields are similar in
magnitude, the level of the yield curve falls significantly. This decline is economically important: a one-standard deviation shock to εT ech induces approximately
a 30 basis point decline in Cochrane’s (2001) measure of the yield curve level (not
displayed).
Again, it is useful to decompose these responses of nominal yields into their
expected inflation and real-rate components. The initial responses of the real rate
to these shocks are positive, as one would expect from a positive impulse to the
marginal product of capital. However, the large deflationary impact of these shocks
overwhelms the contribution of the real rate, hence the negative initial responses
of nominal interest rates. Interestingly, the initial positive response of the real rate
turns negative in about seven quarters, so the real rate response actually serves to
prolong the negative response of nominal rates. A key factor driving this reversal
is the systematic response of monetary policy. In response to εT ech , the monetary
authority reduces the nominal Federal funds rate by a total of nearly 70 basis points
over the next three quarters. This policy response is quite persistent. According to
Table 4, εT ech accounts for 46% of the 5-year ahead variance of the Federal funds
rate. This response is consistent with the literature on Taylor (1993) rules: εT ech
moves inflation below the target inflation rate and reduces (or leaves unchanged)
the output gap.23 Note that after the first two quarters the funds rate response
exceeds the inflation response. Consequently, the real Federal funds rate falls. This
decline in the real funds rate is consistent with the stability condition of the Taylor
rule literature, that the nominal interest rate respond more than one-for-one with
inflation.
An additional factor that shifts the level of the yield curve is the response of
term premiums. According to last graph in Figure 2, the εT ech shocks tend to
induce an increase in the five year term premium over the first six months, with
a subsequent decline after three years. So, conditional on a technology shock, the
expectations hypothesis fails and we see time-varying term premia. Notice that
the real rate level response mimics the response of the five year term premium. If
2 3 The sign of the output gap turns on whether potential output rises immediately with the
expansionary technology shock (as in Galí, 1999 and Basu, Fernald, and Kimball, 2000) or is
delayed due to adjustment costs and habit persistence (as in Boldrin, Christiano, and Fisher,
2001, and Francis and Ramey, 2001).

Economic Determinants of the Nominal Treasury Yield Curve

16

this term premium response were flat, the real-rate response would be shorter-lived.
Together, the monetary policy reduction in the Federal funds rate and the negative
term premiums three to four years after impact tend to pull the level of the real
yield curve down.
6.3.

Responses to the fiscal shock

Another fundamental macroeconomic impulse comes from exogenous shifts in
fiscal policy. The responses to εF iscal are displayed in Figure 3. Output displays a
delayed response to εF iscal , peaking about seven quarters after the shock impact.
This delayed response of economic activity to fiscal shocks was also noted by Ramey
and Shapiro (1998). Somewhat surprisingly, we find little evidence of significant
yield responses to the fiscal shock. In particular, the error bands for all impulse
responses are quite wide. Perhaps this reflects the low R2 in the regression of the
VAR innovations ut onto the model-based measures η F iscal,t .
7.

Conclusion

This paper presents empirical evidence that macroeconomic factors account for
most of the movement in nominal Treasury yields of maturities ranging from one
month through five years. Technology shocks and shocks to the marginal rate of
substitution between consumption and leisure strongly influence the level of the
yield curve. In particular, our MRS shock acts like an aggregate demand shock, in
that it simultaneously increases real GDP and prices. This shock increases both real
interest rates and inflation, which together serve to raise the level of the nominal
yield curve. Our measures of technology shocks also lead to nominal yield curve
level effects. These shocks produce competing influences: positive shocks increase
real GDP and real interest rates but lower inflation. The overall transmission to
nominal yields is attenuated by these contrasting influences, but the effect is still
quite large, with the inflation effect playing the dominant role.
Our results differ from those of Ang and Piazzesi (2003), who find a smaller
response of the five-year yield to macroeconomic impulses. A key difference between
our approach and that of Ang and Piazzesi (2003) is that we allow interest rates to
depend on their lagged values, providing additional pathways for the transmission
of macroeconomic shocks.
The role of systematic monetary policy is critical for understanding the way
macro impulses jointly affect interest rates and the real economy. The Taylor principle seems to be a feature of these empirical responses: if inflationary expectations
rise above the inflation target, the Federal funds rate increases by more than the
inflation gap. In addition, the funds rate rises in response to real GDP above its
potential level. Our macroeconomic shocks induce changes in output and inflation
gaps, and systematic monetary policy adjusts the funds rate accordingly. Long term
interest rates move in anticipation of these systematic policy responses.
We also found evidence that term premiums respond to the technology shock.
Changes in term premiums are associated with time-variation in the market price

Economic Determinants of the Nominal Treasury Yield Curve

17

of risk. Consistent implications for term premium responses may help macroeconomists and financial economists further integrate macroeconomic facts into assetpricing models. More generally, by matching our economic factors with the latent
factors that have been the focus of much of the term structure literature in empirical
finance, it should be possible to further integrate this literature into the analysis of
dynamic general equilibrium models.

8.

Appendix A: Eigenvalue decomposition of impulse responses
System (2) with l lags can be written as a companion-form VAR system:
Xt = ΦXt−1 + Ωvt ,

(15)

where Xt is an q × 1 vector (q ≡ (n + m)l) that stacks {Zt , F Ft , Rt } and their lags,
vt is the companion-form disturbance vector whose first seven elements are {ε0t , γ 0t }0 ,
and Φ is a q × q matrix of companion-form coefficients. Consider the eigenvalue
decomposition of Φ:
Φ = P ΛP −1

(16)

where Λ is a diagonal matrix with the q eigenvalues {λ1 , ..., λq } of Φ along its
diagonal, and P is the associated eigenvector matrix. Use the following notation:
Pi ≡ ith column of P ;

Pei ≡ ith row of P −1 ;

Ωj ≡ j th column of Ω;

(17)

and let ek ≡ kth column of the q ×q identity matrix. The hth -period-ahead response
of variable k to the j th shock is e0k Φh Ωj , which can be written
≡

q h
i
X
(k,j)
wi
λhi

(18)

i=1

(k,j)
≡ e0k Pi Pei Ωj ;
where wi
The impulse responses in the no-smoothing case are not quite captured by equations (15) and (18), since, with no lagged interest rates in the system, Φ is singular.
However, an expression analogous to equation (18) still holds. Let X̂t denote a
q̂ × 1 vector (where q̂ ≡ nl) that stacks {Zt , F Ft } and their lags. The no-smoothing
model can then be written in the following companion form:

b X̂t−1 + Ωb
b vt ,
X̂t = Φ

Rt = α
b 0 X̂t + α
b 1 X̂t−1 + H −1 γ t

where vbt is the companion-form disturbance vector whose first 4 elements are εt ,
b is a companion-form coefficient matrix for the dynamics of X̂t , and both α
Φ
b 0 and
b 1:n denote the first n rows
α
b 1 are m × q̂ companion form coefficient matrices. Let Φ
b and use definitions (16) and (17) with Φ replaced by Φ.
b Then for h ≥ 1,
of Φ,

Economic Determinants of the Nominal Treasury Yield Curve

18

the hth -period-ahead response of the kth yield Rk,t+h to the j th macro shock εj,t is
Pq̂ h (k,j) i h−1
bi
λi ,where now
i=1 w
(k,j)

w
bi

h
i
b 1:n + α1 Pi Pei Ωj
≡ e0k α0 Φ

The contemporaneous response, i.e. for h = 0, is e0k α0 Ωj .

Economic Determinants of the Nominal Treasury Yield Curve

9.

19

Appendix B: Bayesian inference when identification uses modelbased measures of fundamental shocks

This Appendix is a self-contained derivation of the Bayesian posterior distribution of the impulse responses when the VAR is identified using the model-based
approach of section 4.. As is typical in VARs, it is assumed that there is a linear
mapping from fundamental shocks to VAR residuals. That is, if ut denotes the
(n × 1) vector of VAR residuals and εt denotes the (n × 1) vector of fundamental
shocks then there exists a matrix of constants A such that
A ut = εt .

(19)

In addition, it is assumed that the elements of εt are mutually uncorrelated. If the
variance-covariance matrix of εt is normalized to the identity matrix, then equation
(19) implies
AΣu A0 = I

(20)

where Σu ≡ Eut u0t . Identification then requires imposing sufficient restrictions
such that there is a unique matrix A satisfying equation (20).
As described in section 4., one can use model-based measures of the fundamental
shocks to achieve identification. Suppose the econometrician has an (n × 1) time
series ηt consisting of noisy measures of the true shock vector εt . We shall assume
that the noisy measures are related to the fundamental shocks εt by
η t = D εt + wt

(21)

where D is a non-singular (n × n) matrix and wt is a vector of measurement errors
independent of εt (and therefore of ut ). Under this structure, identification of A
can be achieved by imposing restrictions on D, rather than restricting A directly.
Substitute equation (19) into equation (21) to get
η t = Cut + wt

(22)

where
C ≡ DA

(23)

or, equivalently,
A = D−1 C.

(24)

Since wt is uncorrelated with ut , the matrix C can readily be estimated from equation (22) by ordinary least squares. Given this estimate of C, A could be identified
from equation (24) with no a priori restrictions if D were known. To estimate D,
equation 20 can now be restated as
DD0 = CΣu C 0

(25)

As in the typical identified VAR, at least n(n − 1)/2 a priori identifying restrictions
on D are required.

20

Economic Determinants of the Nominal Treasury Yield Curve

In this Appendix, we derive an algorithm to conduct Bayesian inference when
this sort of model-based identification is used. In subsection 9.1., we review the
basic Zellner (1971) approach to Bayesian inference in vector autoregressions with
a diffuse prior. Subsection 9.2. then extends the Zellner approach to encompass
identification using model-based measures.
9.1.

A review of Zellner’s (1971) approach to Bayesian inference in multivariate
time series

In this section, we review Zellner’s (1971) derivation of the posterior distribution for the VAR slope coefficients and the covariance matrix of the reduced-form
residuals. Let Ye ([T + l] × n) denote the data used in the VAR. (Here, T denotes
the number of usable observations, l denotes the number of lags in the VAR, and n
denotes the number of series in the VAR.) To write the VAR in regression notation,
let ν ≡ nl + 1, the number of regressors per equation, let the (T × n)matrix of
dependent variables in the VAR be denoted Y, 24


Yel+1,1 · · · Yel+1,n


·
·




·
·
Y ≡



·
·
Yel+T,1
Yel+T,n
and let the (T × ν) matrix of VAR regressors be denoted X,

1
Yel,1
· · · Ye1,1
Yel−1,1
Yel,2
 1
e
e
e
e
Yl+1,1
· · · Y2,1
Yl,1
Yl+1,2

 ·
·
X≡
 ·
·

 ·
·
1 Yel+T −1,1 Yel+T −2,1 · · · YeT,1 Yel+T −1,2

Yel−1,2
Yel,2
Yel+T −2,2

The reduced form of the VAR is given by the regression equation

···
···

Ye1,2
Ye2,2

· · · YeT,2

Y = XB + U

(26)

where U stacks the n × 1 i.i.d. error process ut as U = (u1 , u2 , · · ·, uT )0 , and it is
assumed that
ut ∼ N (0, Σu ) .

(27)

In equation (26), the coefficient matrix B has dimension (ν × n). The rows of B
correspond to the regressors X; the columns correspond to the n equations.
The parameters of model (26) are {B, Σu }. Zellner (1971) derives the joint
posterior distribution of {B, Σu }, given the data {X, Y }. For the exactly identified
case, A is an exact function of {Σu } and the impulse responses are exact functions
2 4 Note

that our usage of the notation Y differs from that used above in equation (1).

Yel,3
e
Yl+1,3
Yel+T −1,3

···
···

Ye1,n
Ye2,n
·
·
·
YeT,N










21

Economic Determinants of the Nominal Treasury Yield Curve

of {A, B}, so we can take Monte Carlo draws from this joint posterior to derive
Monte Carlo estimates of the posterior distribution of the impulse responses.
Zellner (1971, pp. 224-227) shows that under the Jeffreys priors25
prior(B) = constant

(28)

−(n+1)/2

prior (Σu ) ∝ |Σu |

(29)

the joint posterior for {B, Σu } is given by
−(n+T +1)/2

p (B, Σu |Y, X) ∝ |Σu |

½
¸¾
·
³
´0
³
´
1
−1
0
−1
b
b
exp − trace SΣu + B − B X X B − B Σu
(30)
2

b denotes the matrix of OLS estimates of the VAR slope coefficients
where B
b ≡ (X 0 X)−1 X 0 Y
B

(31)

and S denotes T times the sample covariance matrix of the VAR disturbances
´
³
´0 ³
b .
b
Y − XB
S ≡ Y − XB

b and S are functions of the data only, not of any model parameters.
Note that both B
Equations (26) and (27) imply that the likelihood of Y conditional on {B, Σu , X}
is simply the multivariate normal density, which is proportional to
½
¾
¤
£
1
|Σu |−T /2 exp − trace (Y − XB)0 (Y − XB) Σ−1
(32)
u
2

The likelihood of {B, Σu } conditional on {Y, X} is proportional to the object in
(32), treated as a function of {B, Σu } .The posterior p (B, Σu |Y, X) is the likelihood
multiplied by the joint prior (the product of prior(B) and prior (Σu )) :
½
¾
¤
£
1
−(n+T +1)/2
0
p (B, Σu |Y, X) ∝ |Σu |
exp − trace (Y − XB) (Y − XB) Σ−1
(33)
u
2

To derive (30) from (33), note that
³
´0 ³
´
b + XB
b − XB
b + XB
b − XB (34)
(Y − XB)0 (Y − XB) =
Y − XB
Y − XB
³
´0 ³
´ ³
´0
³
´
b
b + B
b − B X 0X B
b−B
=
Y − XB
Y − XB
³
´0 ³
´ ³
´0
³
´
b X B
b−B + B
b − B X0 Y − XB
b .
+ Y − XB
Equation (30) follows from equation (34) by applying the identity
³
´0
b X≡0
Y − XB

(35)

2 5 Here, and throughout this paper, we express densities using the “∝” sign. This simply means
that we have omitted the normalizing constant that ensures that the density integrates to unity.

Economic Determinants of the Nominal Treasury Yield Curve

22

which follows directly from equation (31) as a matter of algebra. Equation (35)
is simply the normal equation of the OLS estimator. It holds exactly because it
b were replaced by the true parameter B,
defines the OLS estimator. Note that if B
it would only hold in expectation.
The marginal posterior density p (Σu ) can be derived by integrating the righthand side of equation (30) with respect to B:
½
¸¾
·
Z
³
´0
³
´
1
0
−1
b
b
p (Σu ) ∝
|Σu |−(n+T +1)/2 exp − trace SΣ−1
+
B
−
B
X
X
B
−
B
Σ
dB
u
u
2
¾
½
¤
£
1
= |Σu |−(T −ν+n+1)/2 exp − trace SΣ−1
(36)
u
2
½
¸¾
·³
Z
³
´
´0
1
b Σ−1
b X 0X B − B
× |Σu |−ν/2 exp − trace B − B
dB
u
2
½
¾
¤
£
1
∝ |Σu |−(T −ν+n+1)/2 exp − trace SΣ−1
u
2
½
·³
¸¾
´0
³
´
b X 0X B − B
b Σ−1
because |Σu |−ν/2 exp − 12 trace B − B
, being proportional
u
to the normal density, integrates to a constant.26 The distribution in equation (36)
is inverse Wishart with parameter S. Equations (30) and (36) give us the conditional posterior for B
½
¸¾
·³
³
´
´0
p(B, Σu )
1
−ν/2
b Σ−1
b X 0X B − B
p (B|Σu ) =
exp − trace B − B
(37)
∝ |Σu |
u
p(Σu )
2

It is more conventional to write the slope coefficients in B as a column vector. Let
Bs denote the vector formed by stacking the columns of coefficient matrix B (with
b
Then the density in equation (37) can be
B̂s similarly denoting the stacked B).
written
½
´£
´i¾
h³
¤³
1
−ν/2
−1
0
c
c
p (Bs |Σu ) ∝ |Σu |
exp − trace Bs − Bs Σu ⊗ X X Bs − Bs
(38)
2
implying that the vector of slope coefficients Bs is multivariate normal with mean
cs and variance Σu ⊗ (X 0 X)−1 , precisely the distribution one would obtain from a
B
classical statistical analysis.
n
h¡
io
¢
¡
¢
2 6 There are two subtleties here. First, |Σ |−ν/2 exp − 1 trace
b 0 X0X B − B
b Σ−1
B−B
u
u
2

b and variance
is indeed proportional to the multivariate normal density for B with mean B
Σu ⊗ (X 0 X)−1 , but the constant of proportionality involves |X 0 X|. This is legitimate, since in
Bayesian statistics X is treated as a known constant, not a random variable. Second, Zellner’s
prior implies a “degrees of freedom” correction in the exponent for |Σu |. Sims and Zha (1999) and
Doan (2000) omit the degrees of freedom correction, which is equivalent to choosing a different
prior for |Σu |.

Economic Determinants of the Nominal Treasury Yield Curve

9.2.

23

Inference when identification uses noisy measures of the fundamental shocks

In this section, we apply the Zellner (1971) approach, described in section 9.1.,
above, to this model where exactly n(n − 1)/2 identifying restrictions are imposed
on D. In that case, there exists a unique D satisfying equation (25).
To put the notation in the same format as in section 9.1., let the (T ×n) matrix H
T
contain the model-based shocks measures {η t }t=1 . Note that we are treating H as
data, even though H may contain constructed variables that depends on estimated
parameters. We treat H as data for two reasons. First, a researcher may obtain
a model-based shock measure from some outside source without receiving all the
underlying data used to generate the measure. Second, some of these model-based
measures are generated by nonlinear models, which would make it very difficult to
derive the posterior distribution of the parameter estimates.
The model now consists of two regression equations: equation (26) and equation
(22), which can be written
H = U C + W.

(39)

In equation (39), W stacks the n × 1 i.i.d. measurement error process wt as W =
(w1 , w2 , · · ·, wT )0 , and it is assumed that
W ∼ N (0, Σw ) .

(40)

The parameters of the model given by equations (26) and (39) are {C, Σw , B, Σu }.
Analogously with the task in section 9.1., we wish to derive the joint posterior distribution of {C, Σw , B, Σu }, given the data {X, Y, H}. We then will to use this
posterior distribution to compute error bands for impulse responses. For the exactly identified case, A is an exact function of {C, Σu } and the impulse responses
are exact functions of {A, B}, so we can take Monte Carlo draws from this joint posterior to derive Monte Carlo estimates of the posterior distribution of the impulse
responses.
We can write the joint posterior density p (C, Σw , B, Σu ) as27
p (C, Σw , B, Σu ) = p (C|Σw , B, Σu ) p (Σw |B, Σu ) p (B|Σu ) p (Σu )

(41)

Under the Jeffreys priors (28) and (29), the densities p (Σu ) and p (B|Σu ) are as
derived in section 9.1. in equations (36) and (37). It remains to derive the final two
conditional densities: p (C|Σw , B, Σu ) and p (Σw |B, Σu ) . To do so, we follow exactly
the same steps as we used to derive p (Σu ) and p (B|Σu ), except that we condition
on B. (It turns out that Σu does not directly affect the conditional distribution of
C and Σw .) For a given B, let us write
U (B) ≡ Y − XB
b (B) ≡ (U (B)0 U (B))−1 U (B)0 H.
C

2 7 All densities in equation (41) are conditional on the data {Y, X, H}. This dependency is not
noted explicitly.

Economic Determinants of the Nominal Treasury Yield Curve

24

and
³
´0 ³
´
b (B)
b (B)
V (B) ≡ H − U (B)C
H − U (B)C

The interpretation of these objects is as follows: U (B) is the matrix of residuals
b (B) is the estimate of C that one
implied by equation (26) for the given B. C
would obtain from U (B) and H if one estimated equation (39) via OLS. V (B)
is the moment matrix of the residuals from this OLS estimation of equation (39).
b (B), and V (B) are functions of the data, so derivations
Conditional on B, U (B), C
analogous to those for equations (30), (36), and (37) can be performed in which we
b by C
b (B), and S by V (B). Note that we use the
replace X by U (B), Y by H, B
analogue to equation (35), which is
³
´0
b (B) U (B) ≡ 0
(42)
H − U (B)C
Noting that the number of estimated parameters per equation in the regression of
η t on ut is n, one obtains the following conditional posterior densities:

−(n+T +1)/2
p (C, Σw |B,
½ Σu , Y, X,
· H) ∝ |Σw | ³
¸¾
´0
³
´
(43)
0
−1
b
b
× exp − 12 trace V (B) Σ−1
+
C
−
C
(B)
U
(B)
U
(B)
C
−
C
(B)
Σ
w
w
−(T +1)/2

p (Σw |B, Σu , Y, X, H) ∝ |Σw |

½
¾
¤
£
1
exp − trace V (B) Σ−1
w
2

(44)

−n/2

, B, Σu , Y, ·X, H) ∝ |Σw |
p (Cs |Σw½
³
´0 £
´¸¾ (45)
¤³
0
bs (B) Σ−1
c
× exp − 12 trace Cs − C
⊗
U
(B)
U
(B)
C
−
C
(B)
s
w

where the subscript “s” denotes a matrix stacked columnwise, as above. Note
that the conditional distribution in (44) is of the inverse-Wishart form, and the
conditional distribution in (45) is Gaussian.
Together, equations (36), (37), (44), and (45) give the posterior distribution for
{C, Σw , B, Σu } . One can obtain a draw from this distribution using the following
Monte Carlo procedure:
1. Draw Σu from the inverted Wishart density given by equation (36);
2. Given this draw of Σu , draw B from the multivariate normal distribution
given by equation (37);
3. Given this draw of B, draw Σw from the inverted Wishart density given by
equation (44);
4. Given these draws of B and Σw , draw C from the multivariate normal distribution given by equation (45);

Economic Determinants of the Nominal Treasury Yield Curve

25

5. We now have one draw from the joint density given by equation (41). In the
exactly-identified case we can use this draw of {C, Σu } to compute D as the
unique matrix satisfying equation (25). A draw of A is then computed from
D and the draw of C using equation (24). The draws of A and B can then be
used to generate one draw of the impulse responses.

Economic Determinants of the Nominal Treasury Yield Curve

26

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30

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Economic Determinants of the Nominal Treasury Yield Curve

31

Table 1: Fraction of 5-Year Ahead Yield Variance Attributable to
Macro Shocks under Recursive Orthogonalization
Panel A: Baseline Case
One-month Yield
Shock to IP
0.263 (0.083,0.495)
Shock to P
0.066 (0.014,0.268)
Shock to PCOM
0.467 (0.240,0.647)
Total Macro Shocks .796

Panel B: No-Smoothing Case
One-month Yield
Shock to IP
.081
Shock to P
.140
Shock to PCOM
.249
Total Macro Shocks .470

12-month Yield
0.260 (0.077,0.522)
0.073 (0.011,0.282)
0.518 (0.273,0.684)
.751

12-month Yield
.071
.154
.261
.486

60-month Yield
0.189 (0.032,0.443)
0.119 (0.014,0.358)
0.535 (0.218,0.666)
.843

60-month Yield
.045
.202
.286
.533

Panel A gives the fraction of 5-year ahead forecast variance of the 1-, 12-, and 60-month
zero-coupon Treasury yields explained by each of the three orthogonalized shocks (listed
in the left-hand-most column) according to the baseline model. Numbers in parentheses
are lower and upper 90% error bands, computed using 500 Monte Carlo draws from the
Bayesian posterior distribution of the model parameters. Panel B gives the point estimates for the corresponding variance fractions implied by the model without interest rate
smoothing. The bottom row in each panel sums the point estimates in the first three
rows.

Economic Determinants of the Nominal Treasury Yield Curve

32

Table 2: Correlation Matrix of {ηMP , ηMRS , ηT ech , ηF iscal }
η MP
η MRS
η T ech
η f iscal

ηMP
1.0
0.11
-0.11
-0.02

η MRS

ηT ech

η F iscal

1.0
0.05
0.15

1.0
0.31

1.0

The model-based shocks to monetary policy, preferences, technology, and fiscal policy are denoted η MP , η MRS , η T ech , and η f iscal , respectively. The
derivation of these shocks is described in section ??.

Table 3: R2 Estimates in Regressions of Model-Based Shocks on VAR
Residuals
Shock
ηMP
η MRS
η T ech
ηf iscal

R2
66.0%
22.7%
36.7%
8.7%

This table displays the R2 s from the regression of each of the model-based
shocks {η MP , η MRS , η T ech , η F iscal } on the VAR residuals ut , as in equation
(22).

Economic Determinants of the Nominal Treasury Yield Curve

33

Table 4: Variance Decompositions at 5-Year Horizon Using Identified
Shocks
Real GDP
Inflation
PCOM
Fed Funds
1-month yield
12-month yield
60-month yield

εM RS
39
(18 - 62)
9
(2 - 36)
14
(3 -36)
34
(9 - 65)
37
(13 - 63)
37
(12 - 66)
26
(6 - 57)

εT ech
20
(2 - 46)
55
(10 - 71)
45
(4 - 67)
46
(5 - 57)
39
(4 - 50)
46
(4 - 57)
57
(4 - 64)

εF iscal
1
(0 - 17)
23
(7 - 62)
22
(2 - 73)
4
(2 - 51)
3
(1 - 44)
3
(1 - 52)
4
(1 - 55)

εMP
40
(15 - 59)
14
(4 - 43)
19
(6 - 39)
16
(9 - 36)
11
(6 - 31)
6
(3 - 25)
3
(2 - 22)

For each of the four macro variables {GDP, price, P COM, F ed F unds}
and each of the three yields, the table gives the percentage of the 5-year
ahead forecast error variance attributable to each of the four identified shocks
{εMRS , εT ech , εF iscal , εMP }. (Percentages for the yields do not add up to
unity, due to the effects of the yield shocks γ t in equation (1).) The shocks
use the identification described in section 5.6. Numbers in parentheses are
lower and upper 90% error bands, computed using 500 Monte Carlo draws
from the Bayesian posterior distribution of the model parameters.

34

Economic Determinants of the Nominal Treasury Yield Curve

Figure 1: Responses to MRS Shock
MRS Shock --> Real GDP

1.5

MRS Shock --> 1-mo. Yield

0.8

1.0
0.4

0.5
0.0

0.0

-0.5
-1.0

-0.4
0

5
quarters

10

15

MRS Shock --> 1-mo.-ahead Infl.

0.75

0

10

15

10

15

10

15

10

15

MRS Shock --> 12-mo. Yield

0.9

0.50

5
quarters

0.6

0.25
0.3
0.00
0.0

-0.25
-0.50

-0.3
0

5
quarters

10

15

MRS Shock --> RFF

1.2

0

MRS Shock --> 5-yr. Yield

0.8

0.8

5
quarters

0.4

0.4
0.0

0.0
-0.4

-0.4
0

5
quarters

10

15

MRS Shock --> 1-mo. Real Rate

1.0

0

5
quarters
MRS Shock --> 5-yr. Term Prem.

0.6
0.4

0.5

0.2
0.0

0.0

-0.2
-0.5

-0.4
0

5
quarters

10

15

0

5
quarters

Figure 1: This figure displays the responses of eight endogenous variables to a unit
shock to the marginal rate of substitution shock εMRS . The dashed lines give
90% probability error bands implied by the posterior distribution of the model’s
parameters.

Economic Determinants of the Nominal Treasury Yield Curve

35

Figure 2: Responses to Technology Shock
Tech shock --> Real GDP

1.8

Tech shock --> 1-mo. Yield

0.4

1.2

0.0

0.6
-0.4

0.0
-0.6

-0.8
0

5
quarters

10

15

Tech shock --> 1-mo.-ahead Infl.

0.5

0

0.0

-0.5

-0.3

-1.0

10

15

10

15

10

15

10

15

Tech shock --> 12-mo. Yield

0.3

0.0

5
quarters

-0.6
0

5
quarters

10

15

Tech shock --> RFF

0.5

0

5
quarters
Tech shock --> 5-yr. Yield

0.2
-0.0

0.0

-0.2
-0.5

-0.4

-1.0

-0.6
0

5
quarters

10

15

Tech shock --> 1-mo. Real Rate

0.8

0

Tech shock --> 5-yr. Term Prem.

0.3

0.4

0.0

0.0

-0.3

-0.4

5
quarters

-0.6
0

5
quarters

10

15

0

5
quarters

Figure 2: This figure displays the responses of eight endogenous variables to a unit
shock to the technology shock εT ech . The dashed lines give 90% probability error
bands implied by the posterior distribution of the model’s parameters.

Economic Determinants of the Nominal Treasury Yield Curve

36

Figure 3: Responses to Fiscal Shock
Fiscal shock --> Real GDP

1.0

Fiscal shock --> 1-mo. Yield

0.50
0.25

0.5

0.00
0.0

-0.25

-0.5

-0.50
0

5
quarters

10

15

Fiscal shock --> 1-mo.-ahead Infl.

0.6

0

0.25

0.0

0.00

-0.3

-0.25

-0.6

10

15

10

15

10

15

10

15

Fiscal shock --> 12-mo. Yield

0.50

0.3

5
quarters

-0.50
0

5
quarters

10

15

Fiscal shock --> RFF

0.6

0

Fiscal shock --> 5-yr. Yield

0.4

0.3

0.2

0.0

-0.0

-0.3

-0.2

-0.6

-0.4

-0.9

5
quarters

-0.6
0

5
quarters

10

15

Fiscal shock --> 1-mo. Real Rate

0.50

0

Fiscal shock --> 5-yr. Term Prem.

0.4

0.25

0.2

0.00

0.0

-0.25

-0.2

-0.50

5
quarters

-0.4
0

5
quarters

10

15

0

5
quarters

Figure 3: This figure displays the responses of eight endogenous variables to a unit
shock to the fiscal policy shock εF iscal . The dashed lines give 90% probability error
bands implied by the posterior distribution of the model’s parameters.

Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and economic topics.
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The Informational Advantage of Specialized Monitors:
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Stock Market and Investment Good Prices: Implications of Microeconomics
Lawrence J. Christiano and Jonas D. M. Fisher

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Understanding the Effects of a Shock to Government Purchases
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The Quest for the Natural Rate: Evidence from a Measure of Labor Market Turbulence
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Projected U.S. Demographics and Social Security
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1

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2

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WP-99-19

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WP-99-20

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6

Working Paper Series (continued)
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WP-01-07

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WP-01-15

Economic Determinants of the Nominal Treasury Yield Curve
Charles L. Evans and David Marshall

WP-01-16

7