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Federal Reserve Bank of Chicago

The Term Structure and
Inflation Uncertainty
Tomas Breach, Stefania D’Amico, and
Athanasios Orphanides

December 2016
WP 2016-22

The Term Structure and In‡ation Uncertainty
Tomas Breach, Stefania D’Amico, and Athanasios Orphanides
December 12, 2016

Abstract
This paper develops and estimates a Quadratic-Gaussian model of the U.S.
term structure that can accommodate the rich dynamics of in‡ation risk premia over the 1983-2013 period by allowing for time-varying market prices of
in‡ation risk and incorporating survey information on in‡ation uncertainty in
the estimation. The model captures changes in premia over very diverse periods, from the in‡ation scare episodes of the 1980s, when perceived in‡ation
uncertainty was high, to the more recent episodes of negative premia, when
perceived in‡ation uncertainty has been considerably smaller. A decomposition of the nominal ten-year yield suggests a decline in the estimated in‡ation
risk premium of 1.7 percentage points from the early 1980s to mid-1990s. Subsequently, its predicted value has ‡uctuated around zero and turned negative
at times, reaching its lowest values (about -0.6 percentage points) before the
latest …nancial crisis, in 2005-2007, and during the subsequent weak recovery,
in 2010-2012. The model’s ability to generate sensible estimates of the in‡ation risk premium has important implications for the other components of the
nominal yield: expected real rates, expected in‡ation, and real risk premia.

Keywords: Quadratic-Gaussian Term Structure Models, In‡ation Risk Premium,
Survey Forecasts, Hidden Factors.
JEL Classi…cation: G12, E43, E44, C58
Breach: Federal Reserve Bank of Chicago. E-mail: tbreach@frbchi.org. D’Amico: Federal
Reserve Bank of Chicago. E-mail: sdamico@frbchi.org. Orphanides: MIT Sloan School of Management. E-mail: athanasios.orphanides@mit.edu. For helpful comments, discussions, and suggestions we thank Bob Barsky, Alejandro Justiniano, Don Kim, Thomas King, Anna Paulson, Hiro
Tanaka, Min Wei, and seminar participants at the Federal Reserve Board and the Federal Reserve
Bank of Chicago. The views expressed here do not re‡ect o¢ cial positions of the Federal Reserve.

Introduction

Longer-term nominal yields contain rich information about real interest rates and
in‡ation rates that market participants expect to prevail in the future. Extracting
this valuable information, however, is complicated by the presence of unobservable
in‡ation risk premia (IRP) and real risk premia (RRP) that are widely acknowledged
to vary over time.
Monetary policymakers are keenly interested in understanding these premia for
multiple reasons. The IRP embedded in nominal yields may re‡ect factors such
as uncertainty about in‡ation and the credibility of the monetary authority (e.g.,
Argov et al. 2007; Palomino, 2012; Du et al., 2016), which may evolve over time
with the ability of the central bank to successfully communicate its policy strategy
and deliver on its in‡ation objective. Changes in perceptions of in‡ation risks, such
as the in‡ation scare episodes of the 1980s (Goodfriend, 1993) or the risk of de‡ation following the last …nancial crisis (Kitsul and Wright, 2013), may cause abrupt
changes in long-term yields not necessarily associated with shifts in expectations of
future interest rates or in‡ation. At times, signi…cant changes in risk premia may
complicate the transmission of monetary policy to longer-term rates, particularly
when premia move in the direction opposite to expectations of short-term rates, as
has apparently been the case in the "conundrum" period (2004-06) and the "taper
tantrum" episode (May-June 2013). Understanding in‡ation uncertainty and associated risk premia is important for the appropriate risk management of monetary
policy. (Evans et al, 2015; and Feldman et al, 2016).
In recent years, a new generation of dynamic term structure models has been
developed to estimate the various components of the term structure by allowing
‡exible speci…cations of risk premia while maintaining analytical tractability (e.g.,
Dai and Singleton, 2000; Du¤ee, 2002). Despite considerable progress in modelling
the term structure, estimation of the IRP has proven challenging. Alternative speci…cations estimated over di¤erent periods have resulted in a broad range of results
(surveyed in Bekaert and Wong, 2010). Term structure models estimated using data
prior to the last …nancial crisis (e.g., Ang, Bekaert, and Wei, 2008; Buraschi and
Jiltsov, 2005; and Chernov and Mueller, 2012) report estimates of the IRP that
are larger in magnitude and mainly positive. In contrast, models focusing on more
recent data (e.g., Abrahams et al, 2013; Grishchenko and Huang, 2013; and Fleckenstein, Longsta¤, and Lustig, 2014), deliver values of the IRP that are smaller in
magnitude and often negative, especially at shorter maturities.
One factor that could explain ‡uctuations in the IRP over time is the variation in
the level of actual and perceived in‡ation uncertainty. In‡ation uncertainty makes
nominal bonds risky, as their real value is eroded by surprise in‡ation, and thus
is expected to a¤ect the associated risk premium. Although the speci…c channels
may di¤er, a relation between in‡ation uncertainty and the IRP emerges in numerous models, as highlighted for example in the survey by Gurkaynak and Wright
2

(2012). In the data, both actual in‡ation volatility and survey-based in‡ation uncertainty have declined notably since the 1980s (D’Amico and Orphanides, 2008), as
the Federal Reserve adopted policies that gradually reestablished its credibility to
keep in‡ation low and stable, following a period of monetary neglect. And as shown
by D’Amico and Orphanides (2014), real-time measures of perceived in‡ation uncertainty contain meaningful information about future nominal bond excess returns
that is not contained in current yields or forward spreads. Another potentially critical factor for the evolution of the IRP is the changing covariance between bond and
stock returns, which a¤ects the hedging characteristics of Treasury nominal bonds.
As indicated in Campbell, Sunderam, and Viceira (2016), for example, the stockbond covariance was high and positive in the early 1980s but became negative in
the 2000s, which in their model mainly re‡ects time-variation in in‡ation volatility
and in the covariance between in‡ation and the real economy, with both accounting
for signi…cant changes in the sign and size of nominal bond risk premia.
We develop a Quadratic-Gaussian term structure model that is ‡exible enough
to encompass very diverse dynamic behaviors of the IRP over extreme episodes
like the early 1980s, characterized by high actual and expected in‡ation as well as
high in‡ation uncertainty, and the post-2008 period, characterized by low in‡ation
(and mild de‡ation) as well as very low expected in‡ation and in‡ation uncertainty.
The richer dynamic of the IRP is achieved by having time-varying market prices of
in‡ation risk in the model, which translates into time-varying in‡ation volatility and
time-varying covariances between yield curve factors and in‡ation. To obtain reliable
estimates of the parameters governing these sources of in‡ation risk and the IRP, we
incorporate information from survey-based in‡ation uncertainty in the estimation.
Using this new data input, which captures real-time perceptions of in‡ation risk,
proves quite valuable for pinning down the dynamics of the IRP which, in turn,
has important implications for the other components of nominal yields: expected
real short-term rates, expected in‡ation, and the RRP. The key novelty of this
approach is to tackle the di¢ culties in the estimation of the IRP by allowing for
very ‡exible market prices of in‡ation risk and by using a real-time measure of
in‡ation uncertainty.
Introducing survey-based information about second moments may also be seen as
an extension of the approach developed in Kim and Orphanides (2012) and Chernov
and Mueller (2012) who augment term structure models with information from
survey …rst moments. Similarly to the …rst of these studies, we include survey
forecasts of short-term interest rates to guard against estimation imprecision and
bias due to the highly persistent nature of interest rates; and, similarly to the second
study, we include survey forecasts of in‡ation that help pin down expected in‡ation
and real rates over a long sample period for which TIPS yields are not available.
To account for the possibility that survey forecasts provide noisy information and
to let the data determine the extent of this noise, following Kim and Orphanides
(2012) we allow for unconstrained variances of the measurement errors around all
3

forecasts in the estimation. Acknowledging the presence of measurement errors is
particularly important for survey-based in‡ation uncertainty, which is an imputed
variable derived from the subjective probability distributions in the Survey of Professional Forecasters (SPF) using the methodology in D’Amico and Orphanides (2008).
In addition, following the intuition in Du¤ee (2011) and motivated by preliminary
regressions similar to those conducted by Joslin, Priebsch, and Singleton (2014), in
our model in‡ation-related variables are not fully spanned by the current nominal
yield curve. This is achieved through two modelling choices. First, we introduce a
shock capturing short-run variations in CPI in‡ation that do not require a monetarypolicy response, such as, short-lived changes in energy and food prices. Second,
one of the factors is hidden in the nominal yield curve and can only in‡uence the
…rst and second moments of in‡ation. In principle, both of these features can be
important because, while the conditional volatility of the Brownian shock speci…c
to CPI a¤ects in‡ation uncertainty and risk premia but not the forecast of in‡ation
(i.e., its expected value is zero), the hidden factor can in‡uence expected in‡ation.
Moreover, the conditional volatility of the innovations to the hidden factor can also
contribute to ‡uctuations in in‡ation uncertainty and risk premia.
The resulting model produces IRP estimates at the 10-year horizon that are
larger and positive in the 1980s, then decline by about 1.7 percentage points by
the mid 1990s, and subsequently become negative at times, for example during the
conundrum period (2005-07) and during the de‡ation scare of 2010-2012. The estimates also capture episodes of sharp increases in the IRP, as for instance during
the taper tantrum of May-June 2013. Further, despite being estimated without
the use of TIPS yields, the model generates real yields that closely resemble those
on TIPS, except for a residual very similar to the liquidity premium estimated in
D’Amico, Kim, and Wei (2016). The model also does a good job at …tting the
survey-based one-year expected in‡ation and in‡ation uncertainty, and it produces
expected short-term rates that closely match those from survey forecasts. With
regard to longer-term trends, the model captures the decline in long-term in‡ation expectations from the 1980s to the 2000s that is associated with the Federal
Reserve’s overall disin‡ationary policies over this period as well as the decline in
long-term expectations of the short-term real interest rate re‡ecting the decline in
the equilibrium real interest rate.
The rest of the paper is organized as follows. Section 2 sets up the QuadraticGaussian model. Section 3 compares our model to other key studies in the literature.
Section 4 presents the state-space form of the model and the data. Section 5 provides
details about the identi…cation and estimation methodology. Section 6 presents
the main model’s empirical results and comparisons with alternative speci…cations,
which help assess the contribution of key elements to its improved performance.
Section 7 o¤ers concluding remarks.

A Quadratic Gaussian Model

In this section we develop a Quadratic Gaussian model of the term structure of interest rates that accommodates nominal yields, CPI in‡ation, survey-based expected
in‡ation, expected interest rates, and in‡ation volatility.

2.1

The basic building blocks

We start with specifying the state factor dynamics under the physical measure, P:
dxt
=
dzt

x
2 2

z
1 1

(
(

xt )2
zt )1

x
z

dt +

x
2 2
x;z

1
z

dBtx
dBtz

where zt is a factor hidden in the nominal yield curve in the sense of Du¤ee (2011)
but its shocks can be correlated with shocks relevant for nominal interest rates as
indicated by the unconstrained x;z , Bt denotes a 3-dimensional standard Brownian
motion, and thus all the factors are Gaussian.
The nominal pricing kernel under P is given by:
dMtN
=
MtN

N0
x
t dBt

rtN dt

where the nominal short rate is an a¢ ne function of only two state variables:
rtN =

N
0

N0
1 xt ;

and the 2-dimensional vector of the market price of nominal risk is given by:
N
t

N
0

xt ;

with N being a 2 2 constant matrix that will be left unrestricted to allow for a
‡exible speci…cation of the market price of nominal risk. However, by preventing N
t
and rtN from loading on zt , we make sure that this factor is unspanned by nominal
yields.1
The log price level follows the process
d log Qt =

t dt

q0
x
t dBt

+ ! t dBt?

and is governed by xt = [xt ; zt ] rather than xt , as the factors underlying the nominal
yields are not su¢ cient to span in‡ation and expected in‡ation, which is a¢ ne in
xt :
0
t = 0 + 1 xt ;
1

This can be also achieved by imposing restrictions under the risk-neutral measure as in Du¤ee
(2011), and we veri…ed that results are not very sensitive to the way we impose the unspanning
restrictions.

and the two conditional volatility processes are given by
q
t

= q0 + q xt ;
! t = ! 0 + ! 01 xt ;
where 0 and ! 0 are scalars, 1 and q0 are 3 1 vectors, ! 01 is a 1 3 vector, q
is a 3 3 matrix, and dBtx dBt? = 0. The orthogonal shock speci…c to the in‡ation
process is supposed to capture, for instance, short-run variations in in‡ation that
do not require a monetary-policy response and thus do not a¤ect the nominal short
rate (Kim, 2008). In particular, since we use total CPI in our estimation, not only it
is important to have a separate shock for CPI innovations driven by changes in food
and energy prices, but since these components are usually more volatile it is key
to allow for time variation in the conditional volatility of this shock. Overall, this
implies that we treat much of high-frequency variation in in‡ation as unspanned by
interest rates.
The real pricing kernel is given by MtR = MtN Qt , which by Ito’s Lemma follows
the dynamics:
dMtR
dMtN
dQt dMtN dQt
=
+
+
=
Qt
MtR
MtN
MtN Qt

R0
x
t dBt

rtR dt

! t dBt?

where the real short rate becomes a quadratic function of the state variables because
dM N
t
, as each of these elements contains a state-dependent
of the interaction term M Nt dQ
Qt
t

market price of risk, that is,

(xt ) and

rtR =

R
0

(xt ), respectively:

R0
1 xt

+ xt 0

xt ;

all parameters are linked by the no-arbitrage conditions:

2.2

R
0

N
0

R
1
R
t

=
=

N
1
N
t

1
q
t

N0 q
0
0

q0 N
0

1
2

1
(
2

q0 q
0 0

N0 q
0

+ ! 20 )
q0 q
0

!0!1

1
! 1 ! 01 :
2

Bond Pricing

Under the risk-neutral measure, Q, xt follows the dynamics:
dxt =

(

=
= e (e

dBtx +

xt ) dt +
xt

i
0

xt ) dt + dBt
6

i
t dt

xt dt +

i
t dt

dBtx +

i
t dt

where
e=

i
0
i
t dt

ee =

dBt = dBtx +

and i = N; R indicating either the nominal or real risk neutral measure.
The price of a nominal and real zero-coupon bond with maturity is:
Pt;i

i
Et (Mt+
)
=
= EtQ exp
i
Mt
i

= exp A + B xt +

t+
R

rsi ds

x0t C i xt

;

i = N; R

with the solution satisfying the following di¤erential equations:
dAi
=
d
dB i
=
d
dC i
=
d

i
0
i
1
i

1
+ B i0 ee + B i0
2

B i + tr

e0 B i + 2C i ee + 2C i
Ci e

e0 C i + 2C i

0
0

C i0 :

In the case of nominal bonds (i.e., i = N ), C N = 0, as in this model we start with
specifying an a¢ ne nominal short rate and the real short rate inherits the quadratic
component through the no arbitrage condition MtR = MtN Qt , and therefore nominal
bonds’prices preserve the same functional form usually obtained in a¢ ne Gaussian
models. It follows that since yt;i = 1 log(Pt;i ); nominal and real yields are equal
to:
yt;N = aN + bN 0 xt
yt;R = aR + bR0 xt + xt 0 cR xt ;
where ai =

2.3

A i , bi =

B i ; and ci =

Ci .

In‡ation: Expected and Unexpected

In‡ation between t and t +
it; ,

Qt+
1
log
=
Qt

is de…ned as:

(xt+s )ds +

(xt+s ) dBsx
0

and annual average expected in‡ation over horizon
7

!(xt+s )0 dBs?

is given by:

Et [it+ ] =

Et [

t+s ] ds

therefore unexpected in‡ation can be expressed as follows:
Z
1
it;
Et [it+ ] =
( t+s Et [ t+s ]) ds+
0

q
0

xt+s

dBsx

s ds

)

+ (xt

(

)

dBsx

+ (! 0 +

! 01

)

! 1 dBs? +

q
0

! 0 + ! 1 xt+s dBs? =

dBs?

0
s

dBsx

0
?
s ! 1 dBs :

It is easy to note that for the unexpected in‡ation to be time varying, that is, to be
function of the factors xt , it is su¢ cient that either q or ! 1 are di¤erent from zero,
meaning that the time-varying market prices of in‡ation risk are the key features of
the model permitting time variation in in‡ation volatility, which we derive below.
We can re-write the unexpected in‡ation in matrix form:
it;
2

6 q0 + q
6
6
6
1
C=6
6
6! 0 + ! 01
6
4
1

Et [it+ ] =
2
03
603
7
6
7
6I3
7
6
7
7 ; D = 601
6
7
601
7
6
7
4I3
5
01
3

(C + Dxt )0

7
7
37
7
37 ;
7
37
5
3
37

R
R0

s ds
dB x
0 0 s
s q0

7
6
7
6R
x 7
6 e
dB
s
6 0R 0 q0 x 7
7
=6
s
7
6 0 R s dB
?
7
6
dB
0 0 s
7
6R
4 e s ! 1 dBs? 5
0R
0
! dBs?
0 s 1

R
R
0
where 0 s ds = 1 0 I3 3 e s
dBsx : By observing the elements in ,
it is easy to note that the unexpected in‡ation is driven by all four shocks in the
model: the innovations to the yield factors, the innovations to the hidden factor,
and the shock speci…c to CPI, as well as the conditional volatility of these shocks.
The in‡ation variance is a quadratic function of the state variables:
var(it; ) =

1
2

Et (C + Dxt )0

(C + Dxt ) :

0
, the
In the Appendix A, we provide a detailed derivation of all the elements in
block matrix whose expected value delivers the variances of and covariances between
the shocks that drive unexpected in‡ation (and thus uncertainty).
As it will become clear later, having a survey-based measure of in‡ation uncertainty allows us to better pin down some of the parameters in C and : More
importantly, the vector of parameters ! 1 can be identi…ed only if we incorporate
survey data on this second moment.

2.4

In‡ation Risk Premium

We now turn to the main object of interest in this study, that is, the IRP, which is
de…ned as follows:

IRPt = rtN

rtR

q0 N
0

xt 0

N0 q
0

N0 q
0
0

1
2

q0 q
0
q0

1
2

q0 q
0 0

1 2
! +
2 0

! 0 ! 1 xt +

1
! 1 ! 01 xt :
2

Our IRP has a richer dynamic behavior than permitted by previous studies in the
literature, for example, Chernov and Mueller (2012) and D’Amico, Kim and Wei
(2016), who already allowed for quite ‡exible dynamics. Particularly, in D’Amico,
Kim and Wei (2016), the IRP is linear in the state variables and is time varying
because of the state-dependent market price of nominal risk–i.e., the time variation
is obtained by having just the term N 0 q0 di¤erent from zero in the expression
above.
In this model, the resulting speci…cation of the IRP has two additional sources
of ‡exibility. First, as shown in the last term of the above equation, it is a quadratic
function of the state variables because of q and ! 1 . Second, the linear portion can
vary because either the market price of nominal risk or the market price of in‡ation
risk changes over time, as N , q , and ! 1 multiply xt .
This extremely adaptable functional form should allow our model to accommodate very di¤erent dynamic behaviors of the IRP over a long and diverse sample
period including the in‡ation scare episodes of the 1980s when, in principle, perceptions of heightened in‡ation risk would have commanded large and positive values
of the IRP, and the de‡ation scare episode of 2009-2012, when disin‡ation and low
growth made nominal bonds a very good hedge against adverse outcomes possibly
pushing the IRP into negative territory.
To provide a simple intuition for why the data on real-time in‡ation variance
can improve the estimation of the IRP, we rewrite the IRP in the following way:

IP Rt;

1
1

log 41 +

Cov
Et

R
Mt+
MtR

; QQt+t
Qt
Qt+

log 1 + Cov rt;R ; it;

5 + Jt;

=Et rt;R Et (it; )

log 1 + Cov rt;N ; it;

var(it; ) =Et rt;R Et (it; ) :

where for simplicity we are assuming that the real pricing kernel is mainly driven
by the real yield and we are ignoring the Jensen’s inequality term, which in practice
is fairly small.2
Based on this simpli…cation, it is easy to see that to the extent that variations
in the covariance between the real economy and in‡ation arises from ‡uctuations in
the variance of in‡ation, accurate measurement of these speci…c ‡uctuations would
be important. Survey data on real-time in‡ation uncertainty serve this purpose,
that is, they help identifying ‡uctuations in the variance of in‡ation and thus in
the time-varying IRP. Further, as we will explain shortly in Section 4.1 where we
describe the covariances of the state variables, having a time-varying market price
on in‡ation risk qt also allows time variation in the covariance between nominal
interest rates and in‡ation, thus having more data to pin down qt also helps in the
estimation of that covariance.

Comparison to previous studies

This paper draws on contributions from several streams of the term-structural literature. First of all, to achieve time-varying second moments, we favor the use
of Quadratic Gaussian (QG) models because a¢ ne term-structure models with stochastic volatility typically fail to produce reasonable risk premia (Dai and Singleton,
2002 and Du¤ee, 2002) and …tted yield volatilities that resemble the time-varying
volatilities estimated from semi-parametric time-series models (Ahn, Dittmar, and
Gallant, 2002; Collin-Dufresne, Goldstein, and Jones, 2009). For example, Haubrich,
Pennacchi, and Ritchken (2012) develop a completely a¢ ne model that has four stochastic drivers and seven factors, but it still generates IRP that do not seem very
sensible up to the two-year horizon, as it is mostly negative even in the early 1980s,
when most other studies …nd that IRP estimates reach their highest peak.
In contrast, as shown in Kim (2004), QG models do not seem to exhibit a tradeo¤ between …tting yield volatility and risk premia, therefore, we build on these type
of models (e.g., Kim, 2004; and Kim and Singleton, 2012) and expand on them
2

Jt;

( 1 )[log(Et (Qt =Qt; ))

Et (log(Qt =Qt; ))]:

by adding ‡exibility to market prices of in‡ation risk and allowing for unspanned
in‡ation risk. Particularly, we decided to expand in this direction because Le and
Singleton (2013) show that substantial variation in risk premia is unspanned by
nominal bond yields and seems to arise from a time-varying market price of in‡ation
risk; and, D’Amico and Orphanides (2014) show that perceived in‡ation risk is an
important driver of excess bond returns beyond and above the information contained
in nominal yields.
To allow for unspanned in‡ation risk, our model includes some of the unspanning restrictions emphasized in Du¤ee (2011) and Joslin, Priebsch, and Singleton
(2014), and similarly to the latter, we also run preliminary regressions to motivate
our hidden factor. Table 1 reports the percentage of variation (R2 ) in in‡ation related variables explained by the 3 latent factors of an a¢ ne term-structure model
estimated using only nominal yields and short-term rate forecasts. We …nd that
although more than 80% of variation in expected in‡ation is explained by these
factors, only half of the variation in in‡ation uncertainty is explained by those same
factors. In line with this observation, our unspanning restrictions permit our third
factor to drive both expected in‡ation and in‡ation uncertainty while remaining
hidden from the nominal yield curve.
Our paper is also closely related to studies emphasizing the size and nature of
the IRP. For example, similarly to Chernov and Mueller (2012), we use surveybased in‡ation expectations at various horizons, but while their preferred model
uses TIPS yields in the estimation, we use short- and long-horizon survey forecasts
of nominal interest rates that together with surveys forecasts of in‡ation help to
pin down the term-structure of expected real rates over a longer sample. A more
important di¤erence is that in this paper, we focus on modeling time variation in
the market price of in‡ation risk and incorporate information from survey-based
in‡ation variance, which in turn permits us to identify a more ‡exible dynamic of
the IRP. Another relevant study that, however, uses a quite di¤erent approach is
Buraschi and Jiltsov (2005). Speci…cally, these authors develop a structural model
that can identify the underlying nominal and real factors driving the IRP, but also
su¤ers from the shortcoming that the market price of risk, even if state dependent,
is not as ‡exible as ours, which is based on a more reduced-form approach. Further,
their dataset consists only of interest rates, CPI, and money supply, and thus does
not include any information from survey forecasts. In addition, di¤erently from our
work, in both of these studies, the sample period stops before 2008.
Finally, our study is also related to equilibrium term-structure models implying
that time-variation in expected excess returns of nominal risk-free bonds is driven
by changes in variances of real and in‡ation risks (e.g., Bansal and Shaliastovich,
2012); however, in most of these models, the market price of risk is assumed to
be constant and macro risk is fully spanned by nominal yields. This is also true
for Campbell, Sunderam, and Viceira (2016), who assume that all time variation
in bond risk premia is driven by variation in bond risk and not by variation in the
11

aggregate price of risk. Importantly, their estimates of the variables governing bond
risk are informed by realized second moments of high-frequency returns, while our
estimates are informed by a real-time measure of perceived in‡ation risk. Moreover,
we are more focused on modeling and estimating the IRP, while they emphasize
the importance of the time-varying stock-bond covariance for the term structure of
interest rates.

State-space form and data

In this section, we …rst present the state equation and emphasize the role of timevarying market prices of in‡ation risk in generating time variation in the covariances
of the state variables, then we turn to the observation equations and highlight how
they link the data to our state variables.

4.1

State variables and their covariances

We rewrite the model in a state-space form and estimate it by quasi maximum likelihood (QML) using the Augmented State Space Extended Kalman Filter method
developed in Kim (2004). The basic idea of his approach is to augment the state
vector st with the quadratic term vech(xt xt 0 ), st = [xt ; vech(xt xt 0 ); qt ]0 ; such that
the state equations can be written in the usual linear matrix form:
st = Gh +

h st h

s
t;

where st = [ t ; vt ; q (xt h )0 t + !(xt h )0 ?
t ] is the vector of innovations to xt ,
vech(xt xt 0 ), and qt , respectively, and vt , Gh and h are de…ned in the Appendix B.
The conditional variance of the state variables, st h = V ar(st jIt h ) = Et h ( st st );
is given by:
3
2
0
0 q
0 0
0
t t
t vt
t t (xt h )
5=
vt 0t 0
vt vt0
vt 0t q (xt h )0
E4
q
q
q
0 0
0
0 q
0
2 ?2
(xt h ) t t
(xt h ) t vt
(xt h ) t t (xt h ) + !(xt h ) t
2
3
V art h (xt )
Covt h (xt ; vech(xt xt 0 ))
Covt h (xt ; qt )
4Covt h (xt ; vech(xt xt 0 ))
V art h (vech(xt xt 0 ))
Covt h (vech(xt xt 0 ); qt )5
0
Covt h (xt ; qt )
Covt h (vech(xt xt ); qt )
V art h (qt )
It is worth noting the di¤erent roles played by q (xt ) and !(xt ): q (xt ) allows
covariances between all latent factors and the log price level qt to be time-varying
and also contributes to the time variation in the variance of qt ; in contrast, !(xt )
governs only the variance of qt . This suggests that, in principle, the estimated values
of !(xt ) should be strongly in‡uenced by data on in‡ation uncertainty, which will
also help identifying ‡uctuations in the variable q (xt ).
12

4.2

Observation equations and data

From January 1983 to December 2013, we observe seven nominal yields YtN =
fyt;N i g7i=1 , the 6-month, 12-month, and 6-to-11 years ahead forecasts of the nominal short rate ft6m ; ft12m ; and ftlong respectively, the survey in‡ation expectations
at one- and 11-year horizons EIt1y and EIt11y ; as well as the one-year real-time
in‡ation uncertainty IUt1y : We collect all the observable variables in the vector
ot = [YtN ; ft6m ; ft12m ; ftlong ; EIt1y ; EIt11y ; IUt1y ]0 and write also the observation equations in a matrix form:
ot = a + F st + "t
where "t denotes the vector of measurement errors, assumed to be i.i.d., with freely
N
N (0; 2EI; i ); and
N (0; 2N; i ); "ft; i
N (0; 2f; i ); "EI
estimated variances: "Yt; i
t; i
"IU
N (0; 2IU ):
t
More details about the functional form of the observation equations and thus of
a and F are provided in Appendix C. However, we stress here how each observation
equation links speci…c data to all or some of the state variables. Further, it should
be noted that in‡ation and survey-based variables are not available for all dates,
which introduces missing data in the observation equation and are handled in the
standard way by allowing the dimensions of a and F to be time-dependent (see, for
example, Harvey 1989).
The …rst seven measurement equations relate observable Treasury nominal yields
only to the two state variables xt ; due to the unspanning restrictions. Speci…cally,
we use the 3- and 6-month Treasury bill rates from the Federal Reserve Board’s
H.15 release and converted them to continuously compounded basis. The 1-, 2-, 4-,
7-, and 10-year nominal yields are based on zero-coupon yield curves …tted at the
Federal Reserve Board (see Gurkaynak, Sack, and Wright, 2007; Gurkaynak, Sack,
and Wright, 2010 for details). We sample yields at the weekly frequency and assume
that the monthly CPI-U data is observed on the last week of the current month.34
Similarly, our eighth and ninth measurement equations also link the 6- and 12month-ahead forecasts of the 3-month Treasury bill rate from Blue Chip Financial
Forecasts (BCFF), which are available monthly, only to xt : We complement these
measurement equations with another one that uses the long-range forecast (6-to-11
years ahead) of the same rate. In BCFF, this forecast is provided only semiannually,
but we follow the procedure in D’Amico and King (2015) to convert them to a
consistent quarterly frequency, as we think that information from longer-term survey
forecasts is very important to correctly estimate the persistency of the yield factors
under the physical measure. The basic idea consists of combining the long-range
3

Here we abstract from the real-time data issue by assuming that investors correctly infer the
current in‡ation rate in a timely fashion.
4
The data source for the nominal yields and CPI-U is Haver.

forecasts from BCFF with those from Blue Chip Economic Indicators (BCEI). This
is because BCFF provide these long-range projections in June and December, while
the BCEI report them in March and October, these values can then be interpolated
to obtain the September value and have a regularly-spaced quarterly time series.5
The eleventh and twelfth equations relate the observed measures of expected
in‡ation at the 1- and 11-year horizon to all state variables xt , as in‡ation-related
variable are allowed to load on the hidden factor. Speci…cally, we use the median
forecast of average in‡ation over the following year from the Survey of Professional
Forecasters (SPF) because it is reported at a consistent quarterly frequency and
therefore does not require interpolation. However, since the longest available forecasting horizon in these data is one-year ahead, to measure longer-term in‡ation
expectations we turn again to the BCS, which has been providing semiannual longrange (2-to-6 and 7-to-11 years ahead) consensus forecasts of CPI since 1983. Once
we have converted them to a consistent quarterly frequency using the same methodology described for interest rate forecasts, we can compute the expected average
value over the next 11 years— by taking the weighted average of the one-year, 2-6year, and 7-11-year expectations, respectively.
Finally, the last observation equation relates the real-time measure of in‡ation
variance at one-year horizon to all state variables xt as well as to vech(xt xt 0 ). The
real-time measure of in‡ation variance is derived from the subjective probability
distributions in the SPF using the methodology of D’Amico and Orphanides (2008),
therefore it should capture ex-ante in‡ation risk perceived by investors rather than
ex-post realized volatility.

Identi…cation and estimation methodology

Except for the unspanning restrictions already described in Section 2.1, for all other
parameters in the model, we only impose restrictions that are necessary for achieving
identi…cation to allow a maximally ‡exible correlation structure between the factors,
which has shown to be critical in …tting the rich behavior of risk premia observed
in the data. In particular:
2
3
2
3
0
0
1
0 0
11
0 5 ; = 4 21 1 05
= 03 1 ; = 4 0
22
0
0
33
31
32 1
and N is unrestricted.
Regarding the set of parameters that allow for time variation in the variance of
in‡ation and covariances of in‡ation with the other state variables, we have that q
is lower triangular and ! 1 is left unrestricted:
5

For more details see the Appendix in D’Amico and King (2015).

q
11
q
21
q
31

0
q
22
q
32

3
0
0 5 and ! 1= ! 11 ! 12 ! 13
q
33

This implies that the market price of in‡ation risk can be a¤ected by all three
factors xt and their interactions, and that the conditional volatility of the shock
speci…c to CPI is also a¤ected by the same three factors xt .
To facilitate the estimation by starting with reasonable initial values of the parameters and to make the results easily replicable, we break the estimation in a
few easier steps: We …rst perform a “pre”-estimation where a set of preliminary
parameter estimates governing the nominal term structure is obtained using YtN
and survey forecasts of 3-month TBill rate alone;6 second, based on these estimates
and data on YtN , we can obtain a preliminary estimate of the state variables, xt
and dBt ; third, a regression of monthly in‡ation onto estimates of xt and dBt gives
preliminary estimates of 0 ; 1 , q0 ; q , ! 0 ; fourth, a regression of quarterly in‡ation uncertainty on xt and x2t gives preliminary estimates of ! 1 ; and …nally, these
preliminary estimates are used as starting values in the full, one-step estimation of
all model parameters by QML.

Empirical Findings

In this section, we …rst provide a summary description of the results based on
our "full" model speci…cation, which includes all the features described above and
incorporates in the estimation all the information from surveys. Then, we dissect
the results to highlight the contribution of key elements of our approach separately,
by presenting comparisons with simpler speci…cations and with the estimation that
does not make use of survey information on the second moment of in‡ation.

6.1

Full model speci…cation

A visual description of our main …ndings is presented in Figures 1, 2, and 3. Specifically, Figure 1 shows the decomposition of the 10-year nominal yield into three
components: The real yield (including the RRP), the expected in‡ation at the pertinent horizon, and the corresponding IRP. Figure 2 focuses on the four components
of the 10-year nominal yield, as in addition to the expected in‡ation rate and IRP
(also shown in Figure 1), it shows the expected future short real rate and the RRP
separately. Finally, Figure 3 summarizes the overall …t of the full model, as it
compares the model-implied one-year in‡ation variance, 5-year real yield, one-year
6

It is important to keep in mind that in this preliminary estimation we do not impose unspanning
and therefore derive 3 latent factors from the nominal term structure. This implies that especially
the third factor will have a dynamic quite di¤erent from that one of the hidden factor obtained in
the …nal step of the estimation.

expected in‡ation, and one-year expectation of the nominal short-term rate to their
counterparts in the data (shown in orange).
As it can be seen in Figure 1, the model estimation over the 1983 to 2013 period
captures the main characteristics of the time variations in longer-term nominal yields
that have been discussed in the earlier literature. Overall, in‡ation expectations, real
interest rate expectations, the IRP as well as the RRP all trended down during the
1980s and 1990s. Real yields dominate the other components in accounting for the
‡uctuations in nominal yields. However, the major sources of variation di¤er at low
and high frequencies. While the expectation component of the yield— the expected
real interest rate and expected in‡ation— dominate at business cycle frequencies,
the risk premia largely drive higher-frequency ‡uctuations.
Focusing on the estimates of the 10-year IRP in Figure 2, our …ndings suggest
that it was consistently positive in the …rst part of the sample, reaching its highest
peak (about 1:7 percentage points) in the spring of 1984, and then spiked again in
May-October 1987. Since the mid 1990s, it has ‡uctuated around zero, reaching its
most negative values (about 0:6 percentage points) in 2005-2007, just before the
most recent …nancial crisis, and during the subsequent weak recovery, in 2011-2012.
The largest ‡uctuations in the estimated IRP capture notable episodes documented over this period that re‡ected changes in perceptions of in‡ation risks. The
spikes in 1984 and 1987, for example, coincide with the narrative of the in‡ation
scares of the 1980s documented by Goodfriend (1993). Similarly, the substantial
decline over the 2010-2012 period largely coincides with the de‡ation scare episode
described in Kitsul and Wright (2013). Our estimates of the IRP also capture
episodes that have occupied discussions relating to monetary policy. One notable
example is the "conundrum" period in the mid-2000s when, as shown in Figure 2,
risk premia started declining sharply in 2004 while the Federal Reserve was raising
short-term nominal interest rates. Another example is the "taper-tantrum" in the
summer of 2013, when longer-term Treasury yields rose dramatically following Fed
Chairman Ben Bernanke’s remarks about the possibility of moderating the pace of
asset purchases later that year, implying a lower degree of expected monetary policy
accommodation.
Interestingly, our …ndings also illustrate the time-varying nature of the covariance
of yield components. While in much of the 1980s, all four components broadly
move in the same direction, after 1987 expectations and risk premia start moving
in opposite directions. This pattern is particularly evident in 1987-1992, 2001-02,
2004-08, and 2011-2013. These are periods highlighting the presence of a hidden
factor: Changes in the hidden factor would move the IRP and RRP in the same
amount of but opposite to the expected future short real rates and the expected
in‡ation. This could explain the conundrum period and also indicates that the
entire increase in the nominal yields observed during the taper tantrum was indeed
due to increases in risk premia.
Turning attention to the expected in‡ation and expected short-term real interest
16

rates, Figure 2 also shows that the model captures their secular decline since the
1980s. With respect to in‡ation expectations, this decline is consistent with the
Federal Reserve’s successful disin‡ation e¤orts over the 1980s and 1990s and its
strategy of maintaining mostly stable in‡ation since then. With respect to the
decline in long-term expectations of the short-term real interest rate, the model’s
…ndings are consistent with studies suggesting a notable decline in the equilibrium
real interest rate over this period (Holtson, Laubach and Williams, 2016).
Moving to the overall …t of the full model, Figure 3 suggests that the modelimplied variables match their data counterparts quite well. Starting from the top left
panel, it can be noted that the ‡uctuations in the model-implied one-year in‡ation
variance track quite closely those in the survey-based in‡ation variance. Further,
despite being estimated without the use of TIPS yields, as shown in the top right
panel, the full model generates a 5-year real yield that closely resembles that one on
TIPS (when available), except for a residual very similar to the liquidity premium
estimated in D’Amico, Kim, and Wei (2016). In their study, the estimated TIPS
liquidity premium is fairly high and positive in the early years of TIPS, then declines
steadily and stays close to zero from 2004 until the height of the 2007-08 …nancial
crisis, when it surges to its highest level, to then turn negative around 2011. The
two bottom panels indicate that the model can match pretty well one-year survey
forecasts of in‡ation and of the short-term rate. This also illustrates that the survey
information about …rst moments of key variables like in‡ation and the short rate
help the model capturing the slow moving trend in those expectations as well as the
ZLB period.

6.2

Dissecting the model’s key features

The main empirical contributions of our study can be more easily illustrated and
understood by comparing the empirical performance of the full model to the results
derived from di¤erent model speci…cations, with each speci…cation obtained by removing from the full model one of its key ingredients. We consider three simpli…cations: 1) the model without time-varying in‡ation volatility, called Model No_TVV
(No time-varying volatility, i.e., ! 1 = 03 1 and q = 03 3 ); 2) the model estimated
without data on in‡ation uncertainty (called No_IU) and thus without ! 1 ; which
cannot be correctly identi…ed without those data; and 3) the model estimated letting also zt to be spanned by nominal yields, called No_Unsp (No unspanning, i.e.,
N
N
a 3 3 unrestricted matrix). Table 2 summarizes those
1 (3) unrestricted and
model speci…cations and associated parameters restrictions.
The …rst exercise quanti…es the contribution of time-varying market prices of
in‡ation risk to the overall model performance. The second exercise aims at understanding the value added by survey information about perceived in‡ation uncertainty, and as a consequence the role played by the time-varying conditional volatility of the orthogonal shock speci…c to CPI, which should capture high-frequency
17

variations in in‡ation. Finally, the third experiment is meant to shed light on the
importance of the hidden factor for capturing variations in the …rst and second
moments of in‡ation.
Figure 4 summarizes the comparison between the model with homoskedastic
in‡ation shocks (Model No_TVV), whose results are plotted in the left panels, and
the full model, whose results are plotted in the right panels. For each model, we show
in blue the estimated values of the one-year in‡ation variance, the two-year IRP,
the 5-year real yield, and the one-year expected in‡ation, and in orange their data
counterparts. For brevity, we do not report the estimates for longer-term variables
as they provide the same message and, for the full model, have been highlighted in
the Figure 1 and 2.
The panel’s …rst row shows the implications of restricting the model to homoskedastic in‡ation shocks. While the full model is able to match quite closely the
‡uctuations in the survey-based in‡ation variance, the Model No_TVV estimates
the in‡ation variance to be constant at 0:8 percent which is too low to capture the
1980s and too high to capture the more recent period of relative stability. As shown
in the second and third rows, this has important implications for the estimated
IRP and real yields. The homoskedastic model generates IRP estimates that are
implausible: They are extremely large (in absolute value), as they vary between
10 and +11 percent, and are trending upward over the sample period, with the
lowest values in the early 1980s and the highest peak in 2013. In contrast, the full
model estimates the 2-year IRP to reach its highest value of about 50 basis points
in the early 1980s, then to decline quite consistently through the mid 1990s when it
turns negative, particularly in 2001-02, 2004-06, and 2010-12, but also to increase
sharply at the height of the recent …nancial crisis in 2008-09 and in the summer of
2013 during the so-called taper tantrum. As shown in the third row, the 5-year real
yield implied by Model No_TVV also ‡uctuates within an unreasonable range, as it
reaches almost 20 percent in the early 1980s and about 15 percent in 2012; while,
on the other hand, the full model generates a 5-year real yield that reaches at most
about 7 percent in the early 1980s and closely resembles that one on TIPS (when
available), as already noted in the discussion of Figure 3. However, the homoskedastic model …ts the one-year survey expected in‡ation slightly better, indicating that,
if survey forecasts of in‡ation are used in the estimation, having time-varying in‡ation volatility does not add much along this dimension. This may be due to the
hidden factor, which is responsible solely for variations in in‡ation-related variables.
Since the hidden factor has to capture only ‡uctuations in in‡ation expectations
but not in the in‡ation variance, it is possible that it is doing a much better job in
…tting the survey-based …rst moments.
The next four …gures compare results from the full model to the other two simpli…cations we consider, that is, Model No_IU and Model No_Unsp. Figure 5 shows
the estimates of the IRP at 2- and 10-year maturity and of the one-year in‡ation
variance together with the SPF counterpart across the three model speci…cations.
18

Looking at the third column, it is evident that only the full model is successful in
capturing the ‡uctuations in the survey-based in‡ation variance. Of course, this is
not that surprising relative to the model estimated without data on in‡ation uncertainty, but is interesting to note the deterioration in the …t when the data on
in‡ation uncertainty is used in the estimation of the model without unspanning.
Indeed, as illustrated by the contrast between the top and bottom right panels,
it seems that allowing for a factor that does not in‡uence nominal interest rates
but does in‡uence in‡ation-related variables is important to capture ‡uctuations in
perceived in‡ation risk. The regression analysis reported in Table 3 con…rms this
observation. The table shows the R2 from regressions of the in‡ation-related concepts onto the three factors implied by the full model. As shown in the last column
of the table, the hidden factor explains a large portion of variations in the surveybased in‡ation variance that is not explained by the other two factors: Including the
hidden factor in the regression raises the R2 from 47% to 83%. In turn, since Model
No_IU and Model No_Unsp do not …t in‡ation variance well, they do not generate
very sensible IRP especially in the 1980s. For the Model No_IU, the estimated
2-year IRP is implausibly small and even negative in the early 1980s. This is not
consistent with most estimates available in the literature, which tend to be sizable
and positive across maturities during those years. In contrast, Model No_Unsp estimates values of the IRP that are as high as 7:3 percent in the early 1980s and are
always positive, which is implausibly high. Indeed, most studies obtain estimates of
the IRP that hardly reach 2 percent, even at longer maturities, and often turn negative starting in the 2000s (e.g., Buraschi and Jiltsov, 2005; Chernov and Mueller,
2012; Haubrich et al. 2012, Ajello et al., 2012). Based on those previous …ndings, it
seems that the estimated IRP from the full model, reported in the left and middle
top panels, is much more sensible. In addition to the dynamic behavior of the IRP
already described in Figure 2, it is worth noting that the average term structure
of the IRP is upward sloping, as it is usually more di¢ cult to predict in‡ation at
longer horizons and thus uncertainty about in‡ation is larger. Further, the greater
duration of longer-term bonds ampli…es the impact of a given amount of in‡ation
uncertainty.
Figure 6 illustrates the implications of the IRP estimates for the model-implied
real yields and RRP. The bottom row shows quite starkly that, in the case of the
Model No_Unsp, the ‡ip side of extremely large and positive IRP is extremely low
and ‡at real yields and RRP, which at the 10-year horizon reaches 1 percent in
1984. To a much lesser extent there is a similar trade-o¤ also in the case of the
Model No_IU, but only in the early 1980s, which at the 10-year maturity is less
evident than at the 2-year maturity (not shown for brevity). In particular, since in
the absence of survey data on in‡ation uncertainty, this model produces IRP that
are too low or even negative in the early 1980s, it generates real yields and RRP
that seem a bit too high in the same period, with the 2-year real yield as high as the
10-year real yield, and the 2-year RRP reaching a peak of about 2 percent in 1984 to
19

counterbalance the negative values of the IRP in the same period. Finally, the full
model, similarly to the results for the 5-year real yield already described in Figure
3, generates a 10-year real yield that closely resembles that one on TIPS, again
except for a residual very similar to the 10-year TIPS liquidity premium estimated
in D’Amico, Kim, and Wei (2016). This model also delivers a 10-year RRP that is
mostly positive over the sample period, displaying a marked downward trend as it
declines from a level of about 3:5 percent in the early 1980s to almost 0:5 percent
at the end of 2013.
Figure 7 makes a very simple point: the …t of survey in‡ation expectations across
the three models is very similar. This suggests that the data on in‡ation variance and
the hidden factor have almost no e¤ect on the model-implied estimates of in‡ation
expectations at short and long horizons, when their survey counterparts are included
in the estimation. It also implies that these estimates are mainly governed by the
two latent factors that extract information mostly from nominal yields and the
survey forecasts of the short-term rate. Table 3 con…rms this observation: R2 from
regressions of the in‡ation-related concepts onto the …rst two factors (the yield
factors) are as high as 84 percent, and the R2 does not increase much once we
include the hidden factor in the regression speci…cation. Using long-range survey
forecasts of the short-term interest rate in the estimation produces a level yield
factor that is quite persistent and is therefore able to capture the gradual downward
trend in in‡ation expectations.
Finally, …gure 8 clearly illustrates that also the …t of survey forecasts of the
nominal short-term rate, at short and long horizons, is very similar across the three
models. This, together with the evidence presented in Figure 7, in turn, suggests
that expected real rates are well pinned down simply by the di¤erence between survey forecasts of nominal interest rates and in‡ation. Thus, if a term-structure model
allows for a ‡exible speci…cation of the IRP, whose richer dynamics are better identi…ed using survey information on in‡ation variance, as it is the case in the full model,
then the di¤erence between the observed nominal yields and the sum of expected
real rates, expected in‡ation and IRP (all of which are extracting information from
survey data), will be su¢ cient to inform the estimates of the RRP. This is the basic
intuition to understand the ability of the full model to generate more sensible IRP
and RRP over this long sample period.
Finally, it is also worth observing that, since all the models …t survey forecasts
of the short-term rate very closely, even during the ZLB period, and since these
forecasts do not violate the ZLB, then also the model-implied estimates of nominal
short rates obey the ZLB at these maturities. In other words, information from
surveys is extremely helpful for the estimation of our model also at the ZLB.

Concluding remarks

We show that a Quadratic-Gaussian model of the term structure resulting from a
‡exible speci…cation of the market prices of in‡ation risk and estimated using surveybased in‡ation uncertainty can capture the rich dynamics of in‡ation and real risk
premia over the 1983-2013 period. It can also provide guidance on expected real
interest rates and expected in‡ation embedded in longer-term yields.
In addition to a very ‡exible market price of in‡ation risk, two other features
of the model appear particularly useful to capture correctly the dynamics of the
in‡ation risk premia over time in our long sample. First, the introduction of timevarying volatility of the shock speci…c to CPI, which mainly captures short-run
in‡ation ‡uctuations. Second, the presence of a hidden factor, which is supposed
to govern the component of the in‡ation-related variables not spanned by nominal
yields. Both of these elements improve the reliability of the estimated in‡ation
risk and associated premium, and thus of the decomposition of nominal yields.
Interestingly, our results suggest that the hidden factor is important mainly for the
in‡ation variance. In contrast, in‡ation expectations load mostly on the level-yield
factor, the most persistent state variable implied by our model.
With regard to the key novelty in the estimation, the use of real-time data on
in‡ation uncertainty proves crucial for pinning down the dynamics of the in‡ation
risk premium over our sample that includes both the 1980s, when perceived in‡ation
uncertainty was high, and the 2000s and 2010s, when perceived in‡ation uncertainty
was low. Use of this information would be much less important if attention were
restricted to the more recent period of greater in‡ation stability.
The estimated model captures both the decline in in‡ation expectations from the
1980s to the 2000s that is associated with the Federal Reserve’s disin‡ationary e¤orts
and the notable decline in the equilibrium real interest rate, as measured by the
long-term expectations of the short-term real interest rate. A decomposition of the
10-year nominal yield suggests that the expectation components— the expected real
interest rate and expected in‡ation— dominate at business cycle frequencies, while
the risk premia largely drive higher-frequency ‡uctuations. Focusing on the IRP, the
results con…rm that it was considerably higher in the 1980s than over later periods.
The estimates also identify episodes of notably negative IRP, such as the 20052007 period, just before the most recent …nancial crisis, and during the subsequent
weak recovery, in 2011-2012. Overall, incorporating available survey information
regarding …rst and second moments, allows for the estimation of a ‡exible term
structure model that can capture the rich dynamics of risk premia and expectations.

Appendix A: Components of the in‡ation variance

For the interested reader we provide details on the in‡ation variance computation:
V art (it; ) = Et [

1
2

where

(C + Dxt )0
2

Vaa
6 Vab0
6 0
6 Vac
6
V =6
6 01 n
6 01 n
6
40n n
01 n

Vab
Vbb
Vbc0
01 n
01 n
0n n
01 n

(C + Dxt )] =

Vac
Vbc
Vcc
01 n
01 n
0n n
01 n

1
2

(C + Dxt )0 V (C + Dxt )

0n 1 0n 1
0n 1 0n 1
0n 1 0n 1
Vdd
0
0
Vee
0n n Vef0
0
0

3
0n 1
0n 1 7
7
0n 1 7
7
0 7
7
0 7
7
0n 1 5
Vgg

0n n
0n n
0n n
0n n
Vef
Vf f
01 n

This can be written as:

V art (it; ) = aiu + biu xt + xt 0 Ciu xt

1
2

[

q
0

(Vaa

+ Vab (

q
0

+ Vbb ( +
) Vbc )
0
0
+ (! 0 + ! 1 ) (Vee (! 0 + ! 01 )
2
biu = 2 [ 1 0 Vac + ( q0 + q )0 Vbc
ciu =

1
2

)

Vac ) + (

(Vac0 1
Vef
0

+
)

Vbc0 ( q0

q
0

+
q

(Vef0 (! 0

)0 (Vab0

) Vcc ) + Vdd
+ ! 01 ) Vf f ) + Vgg ]

Vcc + (! 0 + ! 01 )0 Vcf

[Vcc + Vf f ]

(2)

(3)

where:
aiu =

(1)

Vf f ]

with elements given by:
Z
1
(Inxn e (
Vaa =
Z0
0
0
1
e
=

(

)

(Inxn

s)0

(

)

(

s) 0
0

(4)

ds
s)

(

s)0

Vab =

[
Z

(Inxn

(Inxn
(

e
s)

)

(Inxn

)

+ F0; ( ; 0 ;

)]

) ds

1
[ Inxn
(Inxn e )]
Z
1
(Inxn e ( s) ) q e s ds
Z0
1
q
e s e ( s) q e s ds

=
Vac =
=

(5)

=
Vbb =

[

(Inxn

)

F0; (

; ;

)]
(6)

Inxn ds = Inxn

Vbc =
Vcc =

e
Z

e
s0

ds =

(Inxn

ds = F0; ( 0 ; ;

(7)

)
q0

;

(8)

)

0q0 q
q0 q
Vdd = Et [
)
s ds] = G0; (
s
0
Z
Vee =
ds =
0
Z
! 01 s ds = ! 01 1 (In n e )
Vef =
Z0
e s0 ! 1 ! 01 s ds = F0; ( 0 ; ; ! 1 ! 01 )
Vf f =
0
Z
0
0
0
Vgg = Et [
s ! 1 ! 1 s ds] = G0; (! 1 ! 1 )

(9)
(10)
(11)
(12)
(13)

Appendix B: The discrete state equation

To estimate the model, we need to discretize it and derive its state-space form. Let
h be a very small time interval, it follows that:
xt =

h + (Inxn

h)xt

= K + Hxt

(14)

where
is:

N (0; hInxn ). This means the discretized expression for the log price level
qt = q t

(xt

0
h) t

+ !(xt

?
h) t

(15)

In order to capture the quadratic dynamics of real bond prices, we will have to
augment our state vector with the term vech(xt x0t ). We introduce the operator
Dn such that vec() = Dn vech(). We also introduce Dn+ = (Dn0 Dn ) 1 Dn0 so that
Dn+ vec(xt xt 0 ) = vech(xt xt 0 ). Thus, applying properties of vec(),
vech(xt xt 0 ) = Dn+ vec(KK 0 + h 0 ) + Dn+ (H K + K H)xt
+ Dn+ (H H)vec(xt h xt 0 h ) + Dn+ vec( t 0t 0
h 0 + (K + Hxt h ) 0t 0 + t (K 0 + xt 0 h H 0 ))
= vech(KK 0 + h 0 ) + Dn+ (H K + K H)xt h
+ Dn+ (H H)Dn vech(xt h xt 0 h ) + t

where the errors terms are collected in
t

= Dn+ (

)vec(

0
t t)

vech(h

) + M (xt

(16)

Inxn )

(17)

and
M (xt

= Dn+ (Inxn

(K + Hxt

+ (K + Hxt

This permits us to de…ne a linear state space equation:
st = Gh +

h st h

in which:
2

3
3
2
K
xt
st = 4vech(xt xt 0 )5 ; Gh = 4vech(KK 0 + h 0 )5 ;
qt
0h
2
3
H
0
0
4 +
K + K H) Dn+ (H H)Dn 05
h = Dn (H
0
0
1
1 h

s
t

(18)
2

t
t
q

(xt

+ !(xt

?
t

3
5

To perform Kalman …ltering, we will need the conditional moments of the state
variables. We can easily see that
E[st jFt
The conditional variance

s
t h

= Gh +

= V ar(st jFt

(19)

h st h

= V ar( st ) = E[

s s0
t t ].

Appendix C: Observation equations

Observed variables ot are linear in the underlying state vector:
(20)

ot = a + F st + "t

Using the expression for the bond prices derived earlier (and solving the di¤erential
equations), the continuously compounded yields can be expressed as:
YtN = aN + bN 0 xt + "N
t ;
N 0
where aN = [aN
3m :::a10y ] is the stacked vector of coe¢ cients for each maturity, and
aN = 1 AN , where AN are the pricing parameters.
The short rate forecasts are ft = Et [rt+ ;3m ] for = 6m; 12m. The 6-to-11
R 11y
years ahead forecast is ftlong = 15 6y Et [rt+ ;3m ]d . We can solve for these using our
expressions for the yields to …nd:
0
ft = an3m + bN
3m (Inxn
N0
ftlong = aN
3m + b3m (Inxn

xt + "ft

0
) + bN
3m e

e
1
5

(21)

0
)) + bN
3m

1
5

)xt + "ft l
(22)

for

= 6m; 12m.
The in‡ation expectations can be expressed as:

EIt =
for

1
1

(Inxn

) +

1
1

(Inxn

)xt + "EI
t

(23)

= 1; 11.
Lastly, the one-year in‡ation uncertainty can be expressed:
IUt1y = C 0 V C + (2C 0 V D)xt + vec(D0 V D)0 Dn vech(xt xt 0 )) + "IU
t

Thus, collecting all the coe¢ cients of the constant terms in a and the coe¢ cients
multiplying the states in F , we have:
2
3
2
3
0
0
0
1
6 aN 7
6 bN 0
0
07
6 f;6m 7
6 f 6m
7
6a
7
6b
0
07
6 f;12m 7
6 f 12m
7
6a
7
6b
7
0
0
7
6
7
a=6
(24)
6 af;l 7 F = 6 bf l
0
07
6 i;1yr 7
6 i;1yr
7
6a
7
6b
0
07
6 i;10yr 7
6 i;10yr
7
4a
5
4b
0
05
au;1yr
bu;1yr vec(D0 V D)0 Dn 0
The error vector "t will have a diagonal covariance matrix.
25

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Table 1: R2 from regressions with nominal yield factors
Dependent Variable 1st factor 1st and 2nd factor 1st , 2nd ,
.05
.06
E[ 1y ]
.83
.84
E[ 11y ]
.86
.86
V ar( 1y )
.49
.49

and 3rd factor
.06
.84
.87
.51

Note: Entries show the R2 of regressions of each of the in‡ation variables (in the
…rst column) on the estimated factors from an a¢ ne term structure model.

Table 2: Summary of four alternative model speci…cations
Model
Model
Model
Model
Model

Restrictions and Identi…cations
N
Full
! 1 unrestricted, q lower-triangular, N
1 (3) = 0;
2 2
N
(3)
=
0;
! 1 = 03 1 ; q = 03 3 ; N
No_TVV
2 2
1
N
No_IU
1, ! 1 = 03 1 ; q lower-triangular, N
iu
2 2,
1 (3) = 0;
q
N
No_Unsp ! 1 unrestricted,
lower-triangular, 1 unrestricted, N
3 3.

Table 3: R2 from regressing with full model’s factors
Dependent Variable 1st factor 1st and 2nd factor 1st , 2nd , and hidden factor
.06
.06
.07
E[ 1y ]
.80
.83
.90
E[ 11y ]
.70
.84
.89
V ar( 1y )
.44
.47
.83
Note: Entries show the R2 of regressions of each of the in‡ation variables (in the
…rst column) on the estimated factors xt from the full model.

14
Nominal Yield

Real Yield

ExPi

IRP

0
1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014

Figure 1: Decomposition of the 10-year zero-coupon nominal yield. Chart decomposes the 10-year nominal yield into the 10-year real yield (including the RRP), the
expected in‡ation over the next 10 years (ExPi) and the corresponding IRP implied
by the full model.

Expected Real Rate

ExPi

IRP

RRP

-1
1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014

Figure 2: Expectation components and risk premia in the 10-year zero-coupon nominal yield. Chart shows the expected average short-term real interest rate over the
next 10 years and the corresponding RRP together with the expected in‡ation over
the next 10 years (ExPi) and the corresponding IRP, implied by the full model.

1 Yr Inflation Uncertainty

5 Yr Real Yield

6
1.5
4
1

2
0

0.5

-2
1990

2000

2010

1990

1 Yr Expected Inflation

2000

2010

1 Yr TBill Fcast

4
5
2
0

0
1990

2000

2010

1990

2000

2010

Figure 3: Overall Fit of the Full Model. Model estimates (in blue) compared with
actual data and surveys (in orange). The top left panel plots the one-year surveybased in‡ation variance versus the model-implied in‡ation variance; the top right
panel plots the 5-year actual TIPS yield versus the 5-year model-implied real yield;
the bottom left panel plots the survey-based one-year expected in‡ation versus the
one-year model-implied expected in‡ation; and the bottom right panel plots the
one-year ahead survey forecast of the 3-month T-Bill rate versus the model-implied
one-year ahead expectation of the 3-month rate.

Infl Uncer (Pct)

1.5

1.5
1
0.5

1
0.5

2 Yr IRP (Pct)

1990

2000

2010

0.5

2000

2010

2000

2010

1990

2000

2010

1990

2000

2010

1990

2000

2010

6
4
2
0

10
0
-10
1990
1 Yr Ex Inf (Pct)

1990

-0.5

-10
1990

5 Yr R Yld (Pct)

Full Model

Homoskedastic Model

2000

2010

1990

2000

2010

Figure 4: Comparison of homoskedastic and full models. The left panels show results
from the model with homoskedastic in‡ation shocks and the right panels show the
corresponding results from the full model. The …rst row plots the estimated value of
the one-year in‡ation variance and the one-year survey-based in‡ation uncertainty
(orange). The second row plots the estimated two-year IRP. The third row plots
the model-implied 5-year real yield and the actual 5-year TIPS yield (orange). The
last row plots the estimated one-year expected in‡ation versus the one-year ahead
survey forecast of in‡ation (orange).

Full Model

2 Yr IRP (Pct)

10 Yr IRP (Pct)
1.5

1
0

1
0
0.5

-1
1990 2000 2010

No IU Model

1990 2000 2010

2
1.5
1

1
0
0.5

-1
1990 2000 2010

No UnSp Model

Infl Uncer (Pct)

1990 2000 2010

1.5

4
1

0.5

0
1990 2000 2010

1990 2000 2010

Figure 5: In‡ation Risk Premium Estimates and Model Fit of In‡ation Uncertainty.
Each row plot results from the Full Model, No IU Model, and No Unsp Model,
respectively. The left and middle panels plot the estimated values of the 2-year and
10-year IRP. The right panels plot the one-year model-implied in‡ation variance
together its survey counterpart (orange).

Full Model

10 Yr R Yield (Pct)

No IU Model

1990

2000

2010

1990

2000

2010

1990

2000

2010

1990

2000

2010

0
1990

No UnSp Model

10 Yr RRP (Pct)

2000

2010

0
-2
1990

2000

2010

Figure 6: Model-Implied Real Yields versus TIPS yields and RRP Estimates. Each
row plots results from the Full Model, No IU Model, and No Unsp Model, respectively. The left panels plot the model-implied 10-year real yield together with the
actual 10-year TIPS yield (orange). The right panels plot the estimated values of
the 10-year RRP.

Full Model

Exp Inflation 1 Yr (Pct)

No IU Model

1990

2000

2010

2
1990

No UnSp Model

Exp Inflation Over Next 11 Yr (Pct)

2000

2010

2
1990

2000

2010

1990

2000

2010

1990

2000

2010

1990

2000

2010

Figure 7: Model Fit of Survey In‡ation Expectations at one-year and 11-year horizons. Each row plots results from the Full Model, No IU Model, and No Unsp
Model, respectively. The left panels plot one-year model-implied in‡ation expectation together with the corresponding survey forecast (orange). The right panels plot
average model-implied in‡ation expectation over the next 11 years together with the
corresponding survey forecast (orange).

Full Model

TBill Fcast 6 Mnth (Pct)

Fwd Rate Fcast 6 to 11 Yr (Pct)

6
5

No IU Model

1990

2000

2010

2000

2010

1990

2000

2010

1990

2000

2010

6
5

2
1990

No UnSp Model

1990

2000

2010
8

6
5

2
1990

2000

2010

Figure 8: Model Fit of Survey Forecasts of the Short Rate at short and long horizons.
Each row plots results from the Full Model, No IU Model, and No Unsp Model,
respectively. The left panels plot the 6-month-ahead model-implied expectation of
the 3-month rate together with the 6-month-ahead survey forecast of the 3-month
T-Bill rate (orange). The right panels plot the 6-to-11 years ahead model-implied
expectation of the 3-month rate together with the 6-to-11 years ahead survey forecast
of the 3-month T-Bill rate (orange).

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.
The Urban Density Premium across Establishments
R. Jason Faberman and Matthew Freedman

WP-13-01

Why Do Borrowers Make Mortgage Refinancing Mistakes?
Sumit Agarwal, Richard J. Rosen, and Vincent Yao

WP-13-02

Bank Panics, Government Guarantees, and the Long-Run Size of the Financial Sector:
Evidence from Free-Banking America
Benjamin Chabot and Charles C. Moul

WP-13-03

Fiscal Consequences of Paying Interest on Reserves
Marco Bassetto and Todd Messer

WP-13-04

Properties of the Vacancy Statistic in the Discrete Circle Covering Problem
Gadi Barlevy and H. N. Nagaraja

WP-13-05

Credit Crunches and Credit Allocation in a Model of Entrepreneurship
Marco Bassetto, Marco Cagetti, and Mariacristina De Nardi

WP-13-06

Financial Incentives and Educational Investment:
The Impact of Performance-Based Scholarships on Student Time Use
Lisa Barrow and Cecilia Elena Rouse

WP-13-07

The Global Welfare Impact of China: Trade Integration and Technological Change
Julian di Giovanni, Andrei A. Levchenko, and Jing Zhang

WP-13-08

Structural Change in an Open Economy
Timothy Uy, Kei-Mu Yi, and Jing Zhang

WP-13-09

The Global Labor Market Impact of Emerging Giants: a Quantitative Assessment
Andrei A. Levchenko and Jing Zhang

WP-13-10

Size-Dependent Regulations, Firm Size Distribution, and Reallocation
François Gourio and Nicolas Roys

WP-13-11

Modeling the Evolution of Expectations and Uncertainty in General Equilibrium
Francesco Bianchi and Leonardo Melosi

WP-13-12

Rushing into the American Dream? House Prices, the Timing of Homeownership,
and the Adjustment of Consumer Credit
Sumit Agarwal, Luojia Hu, and Xing Huang

WP-13-13

Working Paper Series (continued)
The Earned Income Tax Credit and Food Consumption Patterns
Leslie McGranahan and Diane W. Schanzenbach

WP-13-14

Agglomeration in the European automobile supplier industry
Thomas Klier and Dan McMillen

WP-13-15

Human Capital and Long-Run Labor Income Risk
Luca Benzoni and Olena Chyruk

WP-13-16

The Effects of the Saving and Banking Glut on the U.S. Economy
Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti

WP-13-17

A Portfolio-Balance Approach to the Nominal Term Structure
Thomas B. King

WP-13-18

Gross Migration, Housing and Urban Population Dynamics
Morris A. Davis, Jonas D.M. Fisher, and Marcelo Veracierto

WP-13-19

Very Simple Markov-Perfect Industry Dynamics
Jaap H. Abbring, Jeffrey R. Campbell, Jan Tilly, and Nan Yang

WP-13-20

Bubbles and Leverage: A Simple and Unified Approach
Robert Barsky and Theodore Bogusz

WP-13-21

The scarcity value of Treasury collateral:
Repo market effects of security-specific supply and demand factors
Stefania D'Amico, Roger Fan, and Yuriy Kitsul
Gambling for Dollars: Strategic Hedge Fund Manager Investment
Dan Bernhardt and Ed Nosal
Cash-in-the-Market Pricing in a Model with Money and
Over-the-Counter Financial Markets
Fabrizio Mattesini and Ed Nosal

WP-13-22

WP-13-23

WP-13-24

An Interview with Neil Wallace
David Altig and Ed Nosal

WP-13-25

Firm Dynamics and the Minimum Wage: A Putty-Clay Approach
Daniel Aaronson, Eric French, and Isaac Sorkin

WP-13-26

Policy Intervention in Debt Renegotiation:
Evidence from the Home Affordable Modification Program
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
Tomasz Piskorski, and Amit Seru

WP-13-27

Working Paper Series (continued)
The Effects of the Massachusetts Health Reform on Financial Distress
Bhashkar Mazumder and Sarah Miller

WP-14-01

Can Intangible Capital Explain Cyclical Movements in the Labor Wedge?
François Gourio and Leena Rudanko

WP-14-02

Early Public Banks
William Roberds and François R. Velde

WP-14-03

Mandatory Disclosure and Financial Contagion
Fernando Alvarez and Gadi Barlevy

WP-14-04

The Stock of External Sovereign Debt: Can We Take the Data at ‘Face Value’?
Daniel A. Dias, Christine Richmond, and Mark L. J. Wright

WP-14-05

Interpreting the Pari Passu Clause in Sovereign Bond Contracts:
It’s All Hebrew (and Aramaic) to Me
Mark L. J. Wright

WP-14-06

AIG in Hindsight
Robert McDonald and Anna Paulson

WP-14-07

On the Structural Interpretation of the Smets-Wouters “Risk Premium” Shock
Jonas D.M. Fisher

WP-14-08

Human Capital Risk, Contract Enforcement, and the Macroeconomy
Tom Krebs, Moritz Kuhn, and Mark L. J. Wright

WP-14-09

Adverse Selection, Risk Sharing and Business Cycles
Marcelo Veracierto

WP-14-10

Core and ‘Crust’: Consumer Prices and the Term Structure of Interest Rates
Andrea Ajello, Luca Benzoni, and Olena Chyruk

WP-14-11

The Evolution of Comparative Advantage: Measurement and Implications
Andrei A. Levchenko and Jing Zhang

WP-14-12

Saving Europe?: The Unpleasant Arithmetic of Fiscal Austerity in Integrated Economies
Enrique G. Mendoza, Linda L. Tesar, and Jing Zhang

WP-14-13

Liquidity Traps and Monetary Policy: Managing a Credit Crunch
Francisco Buera and Juan Pablo Nicolini

WP-14-14

Quantitative Easing in Joseph’s Egypt with Keynesian Producers
Jeffrey R. Campbell

WP-14-15

Working Paper Series (continued)
Constrained Discretion and Central Bank Transparency
Francesco Bianchi and Leonardo Melosi

WP-14-16

Escaping the Great Recession
Francesco Bianchi and Leonardo Melosi

WP-14-17

More on Middlemen: Equilibrium Entry and Efficiency in Intermediated Markets
Ed Nosal, Yuet-Yee Wong, and Randall Wright

WP-14-18

Preventing Bank Runs
David Andolfatto, Ed Nosal, and Bruno Sultanum

WP-14-19

The Impact of Chicago’s Small High School Initiative
Lisa Barrow, Diane Whitmore Schanzenbach, and Amy Claessens

WP-14-20

Credit Supply and the Housing Boom
Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti

WP-14-21

The Effect of Vehicle Fuel Economy Standards on Technology Adoption
Thomas Klier and Joshua Linn

WP-14-22

What Drives Bank Funding Spreads?
Thomas B. King and Kurt F. Lewis

WP-14-23

Inflation Uncertainty and Disagreement in Bond Risk Premia
Stefania D’Amico and Athanasios Orphanides

WP-14-24

Access to Refinancing and Mortgage Interest Rates:
HARPing on the Importance of Competition
Gene Amromin and Caitlin Kearns

WP-14-25

Private Takings
Alessandro Marchesiani and Ed Nosal

WP-14-26

Momentum Trading, Return Chasing, and Predictable Crashes
Benjamin Chabot, Eric Ghysels, and Ravi Jagannathan

WP-14-27

Early Life Environment and Racial Inequality in Education and Earnings
in the United States
Kenneth Y. Chay, Jonathan Guryan, and Bhashkar Mazumder

WP-14-28

Poor (Wo)man’s Bootstrap
Bo E. Honoré and Luojia Hu

WP-15-01

Revisiting the Role of Home Production in Life-Cycle Labor Supply
R. Jason Faberman

WP-15-02

Working Paper Series (continued)
Risk Management for Monetary Policy Near the Zero Lower Bound
Charles Evans, Jonas Fisher, François Gourio, and Spencer Krane
Estimating the Intergenerational Elasticity and Rank Association in the US:
Overcoming the Current Limitations of Tax Data
Bhashkar Mazumder

WP-15-03

WP-15-04

External and Public Debt Crises
Cristina Arellano, Andrew Atkeson, and Mark Wright

WP-15-05

The Value and Risk of Human Capital
Luca Benzoni and Olena Chyruk

WP-15-06

Simpler Bootstrap Estimation of the Asymptotic Variance of U-statistic Based Estimators
Bo E. Honoré and Luojia Hu

WP-15-07

Bad Investments and Missed Opportunities?
Postwar Capital Flows to Asia and Latin America
Lee E. Ohanian, Paulina Restrepo-Echavarria, and Mark L. J. Wright

WP-15-08

Backtesting Systemic Risk Measures During Historical Bank Runs
Christian Brownlees, Ben Chabot, Eric Ghysels, and Christopher Kurz

WP-15-09

What Does Anticipated Monetary Policy Do?
Stefania D’Amico and Thomas B. King

WP-15-10

Firm Entry and Macroeconomic Dynamics: A State-level Analysis
François Gourio, Todd Messer, and Michael Siemer

WP-16-01

Measuring Interest Rate Risk in the Life Insurance Sector: the U.S. and the U.K.
Daniel Hartley, Anna Paulson, and Richard J. Rosen

WP-16-02

Allocating Effort and Talent in Professional Labor Markets
Gadi Barlevy and Derek Neal

WP-16-03

The Life Insurance Industry and Systemic Risk: A Bond Market Perspective
Anna Paulson and Richard Rosen

WP-16-04

Forecasting Economic Activity with Mixed Frequency Bayesian VARs
Scott A. Brave, R. Andrew Butters, and Alejandro Justiniano

WP-16-05

Optimal Monetary Policy in an Open Emerging Market Economy
Tara Iyer

WP-16-06

Forward Guidance and Macroeconomic Outcomes Since the Financial Crisis
Jeffrey R. Campbell, Jonas D. M. Fisher, Alejandro Justiniano, and Leonardo Melosi

WP-16-07

Working Paper Series (continued)
Insurance in Human Capital Models with Limited Enforcement
Tom Krebs, Moritz Kuhn, and Mark Wright

WP-16-08

Accounting for Central Neighborhood Change, 1980-2010
Nathaniel Baum-Snow and Daniel Hartley

WP-16-09

The Effect of the Patient Protection and Affordable Care Act Medicaid Expansions
on Financial Wellbeing
Luojia Hu, Robert Kaestner, Bhashkar Mazumder, Sarah Miller, and Ashley Wong

WP-16-10

The Interplay Between Financial Conditions and Monetary Policy Shock
Marco Bassetto, Luca Benzoni, and Trevor Serrao

WP-16-11

Tax Credits and the Debt Position of US Households
Leslie McGranahan

WP-16-12

The Global Diffusion of Ideas
Francisco J. Buera and Ezra Oberfield

WP-16-13

Signaling Effects of Monetary Policy
Leonardo Melosi

WP-16-14

Constrained Discretion and Central Bank Transparency
Francesco Bianchi and Leonardo Melosi

WP-16-15

Escaping the Great Recession
Francesco Bianchi and Leonardo Melosi

WP-16-16

The Role of Selective High Schools in Equalizing Educational Outcomes:
Heterogeneous Effects by Neighborhood Socioeconomic Status
Lisa Barrow, Lauren Sartain, and Marisa de la Torre
Monetary Policy and Durable Goods
Robert B. Barsky, Christoph E. Boehm, Christopher L. House, and Miles S. Kimball

WP-16-17

WP-16-18

Interest Rates or Haircuts?
Prices Versus Quantities in the Market for Collateralized Risky Loans
Robert Barsky, Theodore Bogusz, and Matthew Easton

WP-16-19

Evidence on the within-industry agglomeration of R&D,
production, and administrative occupations
Benjamin Goldman, Thomas Klier, and Thomas Walstrum

WP-16-20

Expectation and Duration at the Effective Lower Bound
Thomas B. King

WP-16-21

Working Paper Series (continued)
The Term Structure and Inflation Uncertainty
Tomas Breach, Stefania D’Amico, and Athanasios Orphanides

WP-16-22

Full text of Working Papers (Federal Reserve Bank of Chicago) : The Term Structure and Inflation Uncertainty, Working Paper 2016-22

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