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BOND RISK PREMIA AND THE MACROECONOMY
By Kasper Jørgensen
A PhD thesis submitted to
School of Business and Social Sciences, Aarhus University,
in partial fulfilment of the requirements of
the PhD degree in
Economics and Business Economics
May 2018
CREATESCenter for Research in Econometric Analysis of Time Series
This version: July 19, 2018 © Kasper Jørgensen
PREFACE
This dissertation is the outcome of my graduate studies at the Department of Eco-
nomics and Business Economics at Aarhus University during the period September
2013 through May 2018. I am grateful to the department and the Center for Research
in Econometric Analysis of Time Series (CREATES) funded by the National Research
Foundation (DNRF78) for providing an outstanding research environment and gen-
erous financial support for participation in numerous conferences and courses.
A number of people deserve special mention. First and foremost, I would like to
thank Martin M. Andreasen for his encouragement, valuable insights, and for always
being extremely helpful. It has been a privilege to work with and learn from you. The
first and third chapter of this dissertation is the result of our joint effort. I truly hope
that we can continue to collaborate in the years to come.
In the fall of 2016, I had the great pleasure to visit James D. Hamilton at the
Department of Economics at University of California, San Diego. I am indebted to
Jim for hosting me, his guidance, and for valuable comments on my work. I would
also like to thank the Department of Economics at UCSD for its hospitality.
The faculty at the Department of Economics and Business Economics at Aarhus
University deserves my gratitude for providing an inspiring research environment.
During my graduate studies, I have had the privilege of being surrounded by many
great colleagues, and I am grateful to all of them for creating an outstanding aca-
demic and social environment. I especially want to thank Alexander, Bo, Carsten,
Christian, Jakob, Johan, Jonas, Niels, Mikkel, and Thomas for contributing to making
my graduate studies a memorable time.
Finally and most importantly, I would like to thank my family for always being
supportive and understanding. My most heartfelt thank you goes to my girlfriend
Anna. Thank you for your encouragement, endless support, patience, and for bearing
over with me being absent-minded at times. Also thank you for our many trips and
experiences in California and elsewhere. It means everything to me.
Kasper Jørgensen
Aarhus, May 2018
i
UPDATED PREFACE
The pre-defense took place on June 26, 2018 in Aarhus. I am grateful to the members
of the assessment committee, Professor Joachim Grammig (University of Tuebingen),
Professor Claus Munk (Copenhagen Business School), and Professor Stig Vinther
Møller (Aarhus University) for their careful reading of the dissertation and their
many insightful comments and suggestions. Some of the suggestions have been
incorporated into the present version of the dissertation while others remain for
future work.
Kasper Jørgensen
Aarhus, July 2018
iii
CONTENTS
Summary vii
Danish summary xi
1 The Importance of Timing Attitudes in Consumption-Based Asset Pric-ing Models 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 A Long-Run Risk Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Estimation Results: The Long-Run Risk Model . . . . . . . . . . . . . 9
1.4 A New Keynesian Model . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2 How Learning from Macroeconomic Experiences Shapes the Yield Curve 432.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2 An Illustrative Consumption-Based Model . . . . . . . . . . . . . . . 46
2.3 Bond Return Predictability . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Term Premia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.5 The Decline in the Equilibrium Real Rate . . . . . . . . . . . . . . . . 69
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3 Bond Risk Premia at the Zero Lower Bound 1073.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.2 Bond Return Predictability at the ZLB . . . . . . . . . . . . . . . . . . 110
3.3 A Shadow Rate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.4 Regime-Dependent Market Prices of Risk . . . . . . . . . . . . . . . . 121
3.5 Economic Implications . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
v
SUMMARY
This dissertation is comprised by three self-contained chapters that all are concerned
with understanding the macroeconomics of risk premia in bond markets and, po-
tentially, financial markets more generally. Yields on long-term bonds consists of
two components: (i) average expected short term interest rates over the lifetime of
the bond, (ii) risk premia. Low long-term yields could then signal (i) low expected
short term interest rates, perhaps because of weak growth prospects, or (ii) low risk
premiums, perhaps because of low overall economic uncertainty. Clearly, the policy
implications are very different. For this reason, accurate decompositions of yields are
of critical importance for monetary policy makers.
Chapter 1, "The Importance of Timing Attitudes in Consumption-Based Asset Pric-
ing Models" (joint work with Martin M. Andreasen), studies a new utility kernel within
the Epstein and Zin (1989) and Weil (1990) framework.1 Epstein-Zin-Weil preferences
are widely used because they separate the intertemporal elasticity of substitution
(IES) and relative risk aversion (RRA) which otherwise have a perfect inverse rela-
tionship when using standard expected utility. It is well-known that Epstein-Zin-Weil
preferences achieve this separation by imposing a timing attitude on the household.
This embedded constraint implies that the standard implementation of Epstein-
Zin-Weil preferences determine (i) the IES, (ii) the RRA, and (iii) the timing attitude
using only two parameters. This raises the question; do Epstein-Zin-Weil preferences
perform well because they separate the IES from RRA or because they imply a timing
attitude? We augment the standard power-utility kernel in Epstein-Zin-Weil with a
constant which allows a more flexible specification of the timing attitude. We then
show that the mechanism enabling Epstein-Zin-Weil preferences to explain asset
prices, is not to separate the IES from RRA, but to introduce a strong timing attitude.
These new preferences resolve a puzzle in the long-run risk model, where consump-
tion growth is too strongly correlated with the price-dividend ratio and the risk-free
rate. The proposed preferences also enable a New Keynesian model to match equity
and bond premia with a low RRA of 5.
In chapter 2, "How Learning from Macroeconomic Experiences Shapes the Yield
Curve", I link the shape of the yield curve to macroeconomic fundamentals. I show
1Chapter 1 has a revise and resubmit invitation from Journal of Monetary Economics.
vii
viii SUMMARY
that constant-gain learning measures of inflation and consumption growth expec-
tations capture the long-run variation in the level and slope of the yield curve, re-
spectively. Controlling for the macroeconomic expectation factors, I extract cyclical
level and slope yield curve factors. The four factors decompose the usual level and
slope factors into trend and cycle components. This dynamic distinction is important
for extracting accurate measures of risk premia in long-term bonds. The four factors
predict excess returns with R2’s up to 56%, and subsume and add to the predictive
information in the most popular bond return predictors. The macroeconomic ex-
pectation factors predominantly capture variation in the expectation hypothesis
component of long-term yields, that is the long-run short rate expectations. The
cyclical level and slope factors capture risk premium variation. As a result, my de-
composition of long-term yields imply cyclical term premia. Cyclical term premia is
in line with macro-finance priors and risk premia in other asset classes (Fama and
French, 1989), but in contrast to the popular affine term structure models.
Finally, in chapter 3, "Bond Risk Premia at the Zero Lower Bound" (joint with
Martin M. Andreasen and Andrew C. Meldrum), we study the dynamics of bond risk
premia at the zero lower bound (ZLB). The classical studies by Fama and Bliss (1987)
and Campbell and Shiller (1991) relate the slope of the yield curve to risk premia in
bonds. However, the recent episodes with prolonged periods of short-term interest
rates being restricted by their ZLB poses a challenge to this linear relation. As the
short end of the yield curve becomes constrained from below, this in turn generates
a "slope compression effect", meaning that a given slope of the yield curve carries
a stronger signal at the ZLB. Furthermore, the recent low interest rate environment
has called for unconventional monetary policies. This is likely to affect the required
compensations for risk by bond investors, meaning that we also may have a "price of
risk effect". In predictive regressions of excess bond returns onto yield spreads, we
document a structural break in regression coefficients over the recent low interest
rate regime. The standard three-factor shadow rate model fails to account for this
empirical pattern. Instead, we propose a shadow rate model with market prices of
risk that switch across non-binding and binding zero lower bound regimes. Our
shadow rate model with regime-dependent market prices of risk is consistent with
the provided regression evidence. The regime-switching shadow rate model suggests
that markets expected monetary policy lift-off to occur later than otherwise thought.
ix
References
Campbell, J. Y., Shiller, R. J., 1991. Yield spreads and interest rate movements: A bird’s
eye view. Review of Economic Studies Vol. 58(3), 495–514.
Epstein, L., Zin, S., 1989. Substitution, risk aversion and the temporal behavior of
consumption and asset returns: A theoretical framework. Econometrica Vol. 57,
937–969.
Fama, E. F., Bliss, R. R., 1987. The information in long-maturity forward rates. Ameri-
can Economic Review Vol. 77(4), 680–692.
Fama, E. F., French, K. R., 1989. Business conditions and expected returns on stocks
and bonds. Journal of Financial Economics Vol. 25, 23–49.
Weil, P., 1990. Non-expected utility in macroeconomics. Quarterly Journal of Eco-
nomics Vol. 1, 29–42.
DANISH SUMMARY
Denne afhandling består af tre uafhængige kapitler, som alle omhandler forståel-
sen af makroøkonomien bag risikopræmier i obligationsmarkeder og, potentielt,
finansielle markeder mere generelt. Renten på langsigtede obligationer består af to
komponenter: (i) den gennemsnitlige forventede kortsigtede rente over levetiden
på obligationen og (ii) en risikopræmie. Lave renter kan således være et signal om
(i) lave forventede kortsigtede renter, for eksempel på grund af svage vækstprogno-
ser, eller (ii) lave risikopræmier, for eksempel på grund af lav generel økonomisk
usikkerhed. Implikationerne for økonomisk politik er åbenlyst forskellige. Af den-
ne grund er præcise dekomponeringer af renter utroligt vigtige for pengepolitiske
beslutningstagere.
Kapitel 1, "The Importance of Timing Attitudes in Consumption-Based Asset Pri-
cing Models"(i samarbejde med Martin M. Andreasen), studerer en ny nyttekerne
inden for rammerne af Epstein og Zin (1989) og Weil (1990).1 Epstein-Zin-Weil-
præferencer er vidt udbredte, fordi de separerer den intertemporale substitutionsela-
sticitet (IES) og relative risikoaversion (RRA), som ellers har et perfekt inversforhold
under forventet nytte. Det er velkendt, at Epstein-Zin-Weil præferencer opnår denne
separation ved at pålægge husholdningen en timing attitude. Denne integrerede
begrænsning betyder, at Epstein-Zin-Weil præferencer bestemmer (i) IES, (ii) RRA og
(iii) timing attituden ved hjælp af kun to parametre. Det rejser spørgsmålet: Virker
Epstein-Zin-Weil præferencer godt, fordi de separerer IES og RRA eller fordi de med-
fører en timing attitude? Vi tilføjer en konstant til potensnyttefunktionen i standard
Epstein-Zin-Weil-præferencer, som dermed tillader en mere fleksibel specifikation
af timing attituden. Vi viser dernæst, at mekanismen, som muliggør at Epstein-Zin-
Weil-præferencer kan forklare aktivpriser, ikke er separationen af IES og RRA, men
derimod fordi de introducerer en stærk timing attitude. De nye præferencer løser et
problem i long-run risk modellen, hvor forbrugsvækst er for stærkt korreleret med
pris-dividende ratioen og den risikofrie rente. De foreslåede præferencer muliggør
også, at en Ny Keynesiansk model kan matche aktie- og obligationspræmier med en
lav RRA på 5.
I kapitel 2, "How Learning from Macroeconomic Experiences Shapes the Yield
1Kapitel 1 har en revise og resubmit invitation ved Journal of Monetary Economics.
xi
xii DANISH SUMMARY
Curve", forbinder jeg formen på rentekurven med underliggende makroøkonomisk
forhold. Jeg viser, at constant-gain learning mål for forbrugsvækst- og inflations-
forventninger fanger langsigtet variation i henholdsvis niveauet og hældningen på
rentekurven. Efter at have kontrolleret for de makroøkonomiske forventningsfaktorer
udtrækker jeg cykliske niveau- og hældningsfaktorer. De fire faktorer dekomponerer
de typiske niveau- og hældningsfaktorer i trend- og cyklus-faktorer. Denne dynami-
ske sondring er vigtig for at udtrykke præcise mål for risikopræmien i langsigtede
obligationer. De fire faktorer forudsiger det overskydende afkast med op til 56%
forklaringsgrad, og inkluderer samt tilføjer til informationen i de mest populære
obligationsafkastsprædiktorer. De makroøkonomiske forventningsfaktorer fanger
overvejende forventningshypotese-komponenten i langsigtede renter, dvs. forvent-
ninger over lange horisonter til renten på kortsigtede obligationer. De cykliske niveau-
og hældningsfaktorer fanger variation i risikopræmien. Som et resultat heraf er de-
komponeringen af langsigtede renter ensbetydende med cykliske risikopræmier på
obligationer. Cykliske risikopræmier på obligationer er i overensstemmelse med
makro-finansielle intuition og risikopræmier i andre typer aktiver (Fama og French,
1989), men i kontrast til resultaterne fra de populære affine rentekurvemodeller.
Endeligt, studerer vi i kapitel 3, "Bond Risk Premia at the Zero Lower Bound"(i
samarbejde med Martin M. Andreasen og Andrew C. Meldrum), dynamikken i risiko-
præmierne på obligationer ved den nedre grænse på nominelle renter. De klassiske
studier af Fama og Bliss (1987) og Campbell og Shiller (1991) relaterer hældningen
på rentekurven til risikopræmierne på obligationer. De seneste episoder med vedva-
rende nominelle renter, som er restringeret af deres nedre grænse, udgør imidlertid
en udfordring for denne lineære relation. Når den korte ende af rentekurven bliver
begrænset nedenfra, så genererer dette en "hældningskompressionseffekt", hvilket
medfører, at en given hældning på rentekurven bærer et stærkere signal ved den
nedre grænse på nominelle renter. Derudover har det nylige lave rentemiljø kræ-
vet ukonventionelle pengepolitikker. Dette vil sandsynligvis have en effekt på de
kompensationer som obligationsinvestorer kræver for at påtage sig risiko, hvilket er
ensbetydende med, at vi kan have en "risikopris-effekt". I prædiktive regressioner
af overskydende obligationsafkast på rentespænd dokumenterer vi et strukturelt
brud i regressionskoefficienterne over det nylige lav-rente regime. Tre-faktor skyg-
gerentemodellen kan ikke forklare dette empiriske mønster. I stedet foreslår vi en
skyggerentemodel med risikopriser, som skifter over bindende og ikke-bindende
nedre grænse regimer. Vores skyggerentemodel med regime-afhængige risikopriser
er konsistent med de dokumenterede regressionsevidens. Regime-skifts skyggerente-
modellen antyder, at obligationsmarkedet forventede pengepolitiske rentestigninger
ville ske senere end tidligere troet.
xiii
Litteratur
Campbell, J. Y., Shiller, R. J., 1991. Yield spreads and interest rate movements: A bird’s
eye view. Review of Economic Studies Vol. 58(3), 495–514.
Epstein, L., Zin, S., 1989. Substitution, risk aversion and the temporal behavior of
consumption and asset returns: A theoretical framework. Econometrica Vol. 57,
937–969.
Fama, E. F., Bliss, R. R., 1987. The information in long-maturity forward rates. Ameri-
can Economic Review Vol. 77(4), 680–692.
Fama, E. F., French, K. R., 1989. Business conditions and expected returns on stocks
and bonds. Journal of Financial Economics Vol. 25, 23–49.
Weil, P., 1990. Non-expected utility in macroeconomics. Quarterly Journal of Econo-
mics Vol. 1, 29–42.
C H A P T E R 1THE IMPORTANCE OF TIMING ATTITUDES IN
CONSUMPTION-BASED ASSET PRICING MODELS
REVISE & RESUBMIT INVITATION FROM JOURNAL OF MONETARY ECONOMICS
Martin M. AndreasenAarhus University, CREATES, and the Danish Finance Institute
Kasper JørgensenAarhus University and CREATES
Abstract
A new utility kernel for Epstein-Zin-Weil preferences is proposed to disentangle
the intertemporal elasticity of substitution (IES), the relative risk aversion (RRA),
and the timing attitude. We then show that the mechanism enabling Epstein-Zin-
Weil preferences to explain asset prices, is not to separate the IES from RRA, but to
introduce a strong timing attitude. These new preferences resolve a puzzle in the
long-run risk model, where consumption growth is too strongly correlated with the
price-dividend ratio and the risk-free rate. The proposed preferences also enable a
New Keynesian model to match equity and bond premia with a low RRA of 5.
1
2 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
1.1 Introduction
Following the seminal work of Epstein and Zin (1989) and Weil (1990), a large number
of consumption-based models use so-called Epstein-Zin-Weil preferences to explain
asset prices (see Bansal and Yaron, 2004; Gourio, 2012, to name just a few). An
important property of these preferences is to disentangle relative risk aversion (RRA)
and the intertemporal elasticity of substitution (IES) which otherwise have an inverse
relationship when using expected utility. It is also well-known that the separation of
the IES and RRA in Epstein-Zin-Weil preferences is achieved by imposing a timing
attitude on the household, which either prefers early or late resolution of uncertainty.
This embedded constraint implies that Epstein-Zin-Weil preferences determine i)
the IES, ii) the RRA, and iii) the timing attitude using only two parameters. However,
experimental evidence suggests that the timing attitude has an independent effect
on decision making beyond what is implied by RRA, and that the timing attitude is
unrelated to RRA (see, for instance, Chew and Ho, 1994; van Winden, Krawczyk, and
Hopfensitz, 2011). This raises the question; do Epstein-Zin-Weil preferences perform
well because they separate the IES from RRA or because they imply a timing attitude?
We address this question in the present paper and explore whether a more flexible
specification of the timing attitude helps to explain asset prices. We study these
questions by augmenting the power-utility kernel adopted in Epstein and Zin (1989)
and Weil (1990) with a constant u0 to account for other aspects than consumption
Ct when modeling the household’s contemporaneous utility level. The benefit of
this extension of the utility kernel u(Ct
)is to obtain greater flexibility in setting
u′′ (Ct)
Ct /u′ (Ct)
and u′ (Ct)
Ct /u(Ct
)compared to the traditional specification of
Epstein-Zin-Weil preferences, where one parameter determines both ratios. Much
attention in the literature has been devoted to u′′ (Ct)
Ct /u′ (Ct), because it controls
the IES. The ratio u′ (Ct)
Ct /u(Ct
), on the other hand, is often ignored but is the main
focus of the present paper, because it determines how the household’s timing attitude
affects RRA. Thus, adding a constant to the utility kernel allows us to disentangle the
IES, the RRA, and the timing attitude.
We start by studying the asset pricing implications of our new utility kernel in the
long-run risk model of Bansal and Yaron (2004). Using an analytical second-order
perturbation approximation, we first show that the household’s timing attitude has
a separate effect on asset prices beyond the IES and RRA, which is consistent with
the experimental evidence cited above. Estimation results for the standard long-run
risk model confirm the finding in Beeler and Campbell (2012) that consumption
growth in the model is too highly correlated with the price-dividend ratio due to its
strong reliance on long-run risk. We further show that this property of the model
also makes the contemporaneous correlation between consumption growth and the
risk-free rate too high, and these findings therefore question the empirical support
for the required degree of long-run risk in the model of Bansal and Yaron (2004). An
important empirical finding in the present paper is to show that our utility kernel
1.1. INTRODUCTION 3
resolves these puzzles, because it reduces the reliance on long-run risk and instead
makes the household display strong preferences for early resolution of uncertainty.
The ability of our extended model to match means, standard deviations, and auto-
correlations is nearly identical to the standard long-run risk model, suggesting that
our extension is identified from contemporaneous correlations, which the literature
mostly ignores when taking the long-run risk model to the data. Another important
finding is that the satisfying performance of the long-run risk model is hardly affected
by lowering RRA from 10 to 5 once u0 is included in the utility kernel. In contrast, the
fit of the standard long-run risk model deteriorates with a RRA of 5. However, our
results also show that the timing premium of Epstein, Farhi, and Strzalecki (2014)
is very high for this model (even with our extension) and it easily implies that the
household is willing to give up 80% of lifetime consumption to have all uncertainty
resolved in the following period.
We also study the asset pricing implications of our new utility kernel in a New Key-
nesian dynamic stochastic general equilibrium (DSGE) model, where consumption
and dividends are determined endogenously. Our estimates reveal that the proposed
utility kernel in this setting resolves the puzzlingly high RRA required in many DSGE
models to explain asset prices. More precisely, the model matches the equity pre-
mium and the bond premium (i.e. the mean and variability of the 10-year nominal
term premium) with a low RRA of 5. The mechanism explaining this substantial im-
provement of the New Keynesian model is similar to the one offered in the long-run
risk model, namely that our new utility kernel allows strong preferences for early
resolution of uncertainty to coincide with low RRA. We also find that changing RRA
has a very small effect on the model’s ability to match the data when using our new
utility kernel. As in the long-run risk model, this suggest that it is not the high RRA in
the traditional formulation of Epstein-Zin-Weil preferences that helps to match asset
prices, but instead the strong timing attitude that is induced by high RRA. We also
find that the timing premium in the New Keynesian model is in the order of 5% to
10% due to the endogenous labor supply, consumption habits, and a low IES. Our
extension preserves this property of the New Keynesian model and hence matches
asset prices with a low RRA and a low timing premium.
Conducting a number of counterfactual experiments, we study the asset pricing
implications of the timing attitude and long-run risk in the two considered models.
To examine the effects of the timing attitude, we set the Epstein-Zin-Weil parameter
to zero in both models such that the RRA is tightly linked to the IES. This modification
generates a small reduction in RRA for the two models, but both models are now un-
able to explain asset prices. A second counterfactual re-introduces strong preferences
for early resolution of uncertainty but omits long-run risk. Here, we also find that the
two models cannot match asset prices, although the IES, the RRA, and the timing
attitude are identical to their estimated values in both models. These experiments,
and our remaining analysis, therefore suggest that the mechanism enabling Epstein-
4 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
Zin-Weil preferences to explain asset prices, is not to separate the IES from RRA, but
to introduce strong preferences for early resolution of uncertainty to amplify effects
of long-run risk.
The remainder of this paper is organized as follows. Section 1.2 introduces our new
utility kernel within the long-run risk model. Section 1.3 estimates this extension of
the long-run risk model and studies its empirical performance. Section 1.4 considers
a New Keynesian model with the proposed utility kernel and explores its empirical
performance. Concluding comments are provided in Section 1.5.1
1.2 A Long-Run Risk Model
The representative household is introduced in Section 1.2.1, and the exogenous
processes for consumption and dividends are specified in Section 1.2.2. We present
the new utility kernel in Section 1.2.3 and derive the IES and RRA. The asset pricing
properties of the proposed utility kernel are explored analytically in Section 1.2.4.
1.2.1 The Representative Household
Consider a household with recursive preferences as in Epstein and Zin (1989) and
Weil (1990). Using the formulation in Rudebusch and Swanson (2012), the value
function Vt is given by
Vt = ut +βEt [V 1−αt+1 ]
11−α (1.1)
for ut > 0, where Et [·] is the conditional expectation given information in period t .2
Here, β ∈ (0,1) and ut ≡ u(Ct
)denotes the utility kernel as a function of consumption
Ct . For higher values of α ∈ R \ {1}, these preferences generate higher risk aversion
when ut > 0 for a given IES, and vice versa for ut < 0.
Another important property of (1.1) is to embed the household with preferences
for resolution of uncertainty. This behavioral property is determined by the aggrega-
tion function in (1.1), i.e. by f
(ut ,Et
[V 1−α
t+1
])≡ ut +β
(Et
[(Vt+1
)1−α]) 11−α
, where the
household displays preferences for early (late) resolution of uncertainty if f(·, ·) is
convex (concave) in its second argument (see Weil, 1990). The formulation in (1.1)
therefore implies preferences for early (late) resolution of uncertainty ifα> 0 (α< 0).3
Given that α controls the degree of curvature in f(·, ·) with respect to Et
[V 1−α
t+1
], it
seems natural to consider α as measuring the strength of the household’s timing
attitude. Another and slightly more intuitive measure for temporal resolution of un-
certainty is the timing premium Πt of Epstein et al. (2014), which is the fraction of
lifetime consumption that the household is willing to give up to have all uncertainty
1All technical derivations and proofs are deferred to an online appendix available.2When ut < 0, we define Vt = ut −βEt [(−Vt+1)1−α]
11−α as in Rudebusch and Swanson (2012).
3The opposite sign restrictions apply when ut < 0.
1.2. A LONG-RUN RISK MODEL 5
resolved in the following period. Epstein et al. (2014) show that Πt depends on the
strength of the timing attitude α and the amount of consumption uncertainty. Thus,
it may be useful to think of α as controlling the ’price of timing risk’, whereas the
law of motion for consumption controls the ’quantity of timing risk’. However, the
timing premium is generally not available in closed form, and we will therefore rely
on the household’s timing attitude α when studying the analytical properties of the
proposed preferences.
The household has access to a complete market for state contingent claims At+1.
Resources are spent on Ct and At+1, and we therefore have the budget restriction
Ct +Et[Mt ,t+1 At+1
]= At , where Mt ,t+1 denotes the real stochastic discount factor.
1.2.2 Consumption and Dividends
The process for consumption is specified to be compatible with production economies
displaying balanced growth. Hence, we let Ct ≡ Zt ×Ct , where Zt > 0 is the balanced
growth path of technology, or simply the productivity level. The variable Ct intro-
duces cyclical consumption risk, which in production economies originates from
demand-related shocks, monetary policy shocks, or short-lived supply shocks (see,
for instance, Justiniano and Primiceri, 2008).
Inspired by the work of Bansal and Yaron (2004), we let
log Zt+1 = log Zt + logµz +xt +σzσtεz,t+1
xt+1 = ρx xt +σxσtεx,t+1
σ2t+1 = 1−ρσ+ρσσ2
t +σσεσ,t+1
(1.2)
where σ2t introduces stochastic volatility. Here, εi ,t+1 ∼NID
(0,1
)for i ∈ (
z, x,σ)
with∣∣ρx∣∣< 1 and
∣∣ρσ∣∣< 1.4 Thus, xt introduces persistent changes in the growth rate of
Zt and captures long-run productivity risk. The innovation εz,t does not generate
any persistence in the growth rate of Zt and is therefore referred to as short-run
productivity risk.5 Variation in consumption around Zt is specified as in Bansal et al.
(2010) by letting logCt+1 = ρc logCt +σcσtεc,t+1, where εc,t ∼NID(0,1
)and
∣∣ρc∣∣< 1.
The process for dividends D t is given by∆dt+1 = logµd +φx xt +φc ct +σdσtεd ,t+1,
where dt+1 ≡ logD t+1 and εd ,t ∼NID(0,1
). Here, φx and φc capture firm leverage
in relation to long-run and cyclical risk, respectively, as in Bansal et al. (2010). For
completeness, all innovations are assumed to be mutually uncorrelated at all leads
and lags.
4Although (1.2) does not enforce σ2t ≥ 0, we nevertheless maintain this specification for comparison
with Bansal and Yaron (2004) and Bansal, Kiku, and Yaron (2010).5Hence, we follow the terminology from the long-run risk model (see for instance Bansal et al. (2010)),
although variation in εz,t has a permanent effect on the level of Zt .
6 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
1.2.3 The Utility Kernel
To motivate our new utility kernel for disentangling the IES, the RRA, and the timing
attitude, it is useful to start with the general expression for RRA. Recall, that RRA
measures the amount that the household is willing to pay to avoid a risky gamble
over wealth. With recursive preferences as formulated in (1.1), the general expression
for RRA in the steady state (ss) is given by (see Swanson, 2018)
RRA =− u′′ (Ct)
Ct
u′ (Ct) ∣∣∣∣∣
ss
+α u′ (Ct)
Ct
u(Ct
) ∣∣∣∣∣ss
. (1.3)
Hence, the RRA depends on the timing attitudeα and the two ratios u′′ (Ct)
Ct /u′ (Ct)
and u′ (Ct)
Ct /u(Ct
). The first term in (1.3) is the familiar expression for the inverse
of the IES, where the IES measures the percentage change in consumption growth
from a one percent change in the real interest rate under the absence of uncer-
tainty. The second term in (1.3) is controlled by the timing attitude α and the ratio
u′ (Ct)
Ct /u(Ct
). The presence of the ratio u′ (Ct
)Ct /u
(Ct
)in this second term is
rarely mentioned, but this ratio plays a key role for RRA because it determines how
the household’s timing attitude α affects risk aversion. That is, for a given IES and
a given timing attitude α, the ratio u′ (Ct)
Ct /u(Ct
)determines the RRA. This prop-
erty of u′ (Ct)
Ct /u(Ct
)appears to have been largely overlooked in the literature,
because much focus has been devoted to the power utility kernel 11−1/ψC 1−1/ψ
t , where
ψ determines both u′′ (Ct)
Ct /u′ (Ct)
and u′ (Ct)
Ct /u(Ct
).
This observation suggest that the IES, the RRA, and the timing attitude may
be disentangled by considering a utility kernel, where the ratios u′′ (Ct)
Ct /u′ (Ct)
and u′ (Ct)
Ct /u(Ct
)can be determined separately. A simple way to achieve this
separation is to let
u(Ct ) = u0Z 1−1/ψt + 1
1−1/ψC 1−1/ψ
t , (1.4)
where the constant u0 ∈ R augments the standard power kernel. To avoid that this
constant diminishes relative to the utility from consumption as the economy grows,
it is necessary to scale u0 by Z 1−1/ψt to ensure a balanced growth path in the model.6
In this modified utility kernel, the constant u0 determines u′ (Ct)
Ct /u(Ct
), whereas
the ratio u′′ (Ct)
Ct /u′ (Ct)
and the IES are controlled by ψ as in the conventional
power kernel.
The presence of u0 in (1.4) may be motivated by accounting for other aspects
than consumption when modeling household utility. We provide two examples. First,
the household may enjoy utility from government spending Gt on roads, public parks,
6The kernel in (1.4) is obviously not the only way to separately determine u′′ (Ct)
Ct /u′ (Ct)
andu′ (Ct
)Ct /u
(Ct
). A previous version of this paper studied a utility kernel that modifies the standard power
utility kernel by changing u′ (·) and u′′ (·) as opposed to the level of u (·) as in (1.4). However, this alternativespecification is slightly more complicated than (1.4), and we therefore prefer the specification in (1.4),which we are grateful to the associate editor, Eric Swanson, for proposing.
1.2. A LONG-RUN RISK MODEL 7
law and order, etc. When these spendings grow with the size of the economy, i.e. Gt =gss Zt where gss ∈R+, and the utility from Gt is separable from Ct , then conditions
for balanced growth imply a utility kernel of the form g1−1/ψss
1−1/ψ Z 1−1/ψt + 1
1−1/ψC 1−1/ψt as
captured by (1.4). Second, the household may also consume home-produced goods
Ch,t that are made using the technology Lss Zt , where Lss denotes a fixed supply of
labor. When utility from home-produced goods is separable from Ct , conditions
for balanced growth dictates a utility kernel of the form L1−1/ψss
1−1/ψ Z 1−1/ψt + 1
1−1/ψC 1−1/ψt ,
which also has the structure captured by (1.4).
It is straightforward to show that RRA with (1.4) is given by
RRA = 1
ψ+α
1− 1ψ
1+u0
(1− 1
ψ
) , (1.5)
which reduces to the familiar expression 1ψ +α
(1− 1
ψ
)when u0 = 0. Thus, a high
value of u0 reduces RRA, and vice versa. To understand the intuition behind this
effect, consider the case where u0 is high, such that u′ (Ct)
Ct /u(Ct
)is low, and hence
variation in Ct has only a small effect on the overall utility level across the business
cycle. This implies that the value function attains a high and stable level even when
faced with a risky gamble, and the household is therefore only willing to pay a small
amount to avoid this gamble, i.e. it has a low RRA. Thus, by varying u0, we can
separately set RRA, for a given IES and timing attitude α.
1.2.4 Understanding Asset Prices
To explain how the IES, the RRA, and the timing attitude affect asset prices, we follow
Bansal and Yaron (2004) and consider a simplified version of the long-run risk model
without stochastic volatility, i.e. σσ = 0. The presence of u0 in (1.4) implies that
we cannot obtain the household’s wealth in closed form and hence eliminate the
value function from the stochastic discount factor using the procedure in Epstein
and Zin (1989). We are therefore unable to obtain an analytical expression for asset
prices by the log-normal method as in Bansal and Yaron (2004). Instead, we use the
perturbation method to derive an analytical second-order approximation to the long-
run risk model around the steady state. In the interest of space, we only provide the
solution for the value function vt ≡ logVt , the mean of the risk-free rate r ft ≡ logR f
t ,
and the mean of equity return r mt ≡ logRm
t in excess of the risk-free rate.
Proposition 1. The second-order approximation to vt around the steady state is given
by
vt = vss + v c ct + vx xt + 1
2v c c c2
t +1
2vxx x2
t + v cx ct xt + 1
2vσσ
8 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
where
vss = log
(∣∣∣∣∣u0 + 11− 1
ψ
∣∣∣∣∣)− log
(1−κ0
)v c = 1−κ0
1−κ0ρc
1− 1ψ
1+u0
(1− 1
ψ
)vx = κ0
1−κ0ρx
(1− 1
ψ
)v c c = 1−κ0
1−κ0ρ2c
(1− 1
ψ
)2
1+u0
(1− 1
ψ
) −[
1− 1ψ
1+u0
(1− 1
ψ
) 1−κ01−κ0ρc
]2
vxx = κ0
1−κ0ρ2x
1−κ0
(1−κ0ρx )2
(1− 1
ψ
)2
v cx =(1− 1
ψ
)2
1+u0
(1− 1
ψ
) 1−κ01−κ0ρc
[ρcκ0
1−κ0ρcρx− κ0
1−κ0ρx
]vσσ = κ0
1−κ0
[v c cσ
2c + (1−α) v2
cσ2c + vxxσ
2x + (1−α) v2
xσ2x + (1−α)
(1− 1
ψ
)2σ2
z
]
with κ0 ≡βµ1− 1
ψ
z .
The steady state of the value function vss is obviously increasing in u0, whereas
the loadings v c and v cx are decreasing in u0. That is, a higher value of u0 raises the
level of the value function and makes it less responsive as argued above. The lower
value of v c is further seen to reduce the contribution from cyclical consumption in
the risk correction vσσ. A key determinant for the size of vσσ is the timing attitude
α, which has a negative impact on vσσ through cyclical, short- and long-run risk,
because α> 1 for plausible levels of RRA with uss ≡ u(Css ) > 0.
Proposition 2. The unconditional mean of the risk-free rate E[
r ft
]and the ex ante
equity premium E[
r mt+1 − r f
t
]in a second-order approximation around the steady state
are given by
E[
r ft
]= rss − 1
2αv2
xσ2x −
1
2
1+ (α−1)
(1− 1
ψ2
)σ2z −
1
2
(1
ψ2 + 1
ψ2αv c +αv2
c
)σ2
c
and
E[
r mt+1 − r f
t
]=ακ1vx
φx − 1ψ
1−κ1ρxσ2
x +(αv c + 1
ψ
)φc +
(1−ρc
) 1ψ
1−κ1ρcκ1σ
2c .
Proposition 2 shows that the mean risk-free rate is given by its steady state level
rss =− logβ+ 1ψ logµz minus uncertainty corrections for each of the shocks affecting
consumption. The first term − 12αv2
xσ2x corrects for long-run risk and is negative
and increasing in the timing attitude α. The second uncertainty correction in E[
r ft
]relates to short-run risk and is also negative ifψ> 1 andα> 1. The final term in E
[r f
t
]corrects for cyclical risk and is also negative and becomes larger (in absolute terms)
1.3. ESTIMATION RESULTS: THE LONG-RUN RISK MODEL 9
when ψ falls and α increases. The effect of u0 enters in the uncertainty correction for
ct through v c , where a lower value of u0 gives a high RRA and a high v c that results in
a large uncertainty correction from cyclical risk.
The equity premium depends positively on long-run risk if φx > 1ψ and ψ > 1,
where the latter requirement is needed to ensure that vx > 0. We also note that this
uncertainty correction is increasing in i) the persistence of xt as determined by ρx ,
ii) the timing attitude α, and iii) firm leverage φx . The second term in E[
r mt+1 − r f
t
]is also positive and corrects for cyclical risk. The size of this term increases in i) the
persistence of ct as determined by ρc , ii) the timing attitude α, iii) firm leverage φc ,
and iv) the loading v c . The latter implies that a lower value of u0 (to increase the RRA
and v c ) also increases the contribution of cyclical risk in the equity premium.
To summarize our insights from these analytical expressions, recall that existing
models tend to generate too low equity premia and too high risk-free rates. Given
identical returns for equity and the risk-free rate under certainty equivalence, we thus
require a positive uncertainty correction in E[
r mt+1 − r f
t
]and a negative uncertainty
correction in E[
r ft
]to resolve the equity premium and risk-free rate puzzles. The
long-run risk model does exactly so for a high timing attitude α and a high RRA,
provided the IES is larger than one. The proposed utility kernel also shows that the
household’s timing attitude α has a separate effect on asset prices beyond the IES
and RRA consistent with the evidence in Chew and Ho (1994) and van Winden et al.
(2011).
1.3 Estimation Results: The Long-Run Risk Model
This section studies the ability of the long-run risk model to explain key features
of the post-war U.S. economy. We first describe the model solution and estimation
methodology in Section 1.3.1. The estimation results for the standard long-run risk
model are provided in Section 1.3.2 as a natural benchmark. Section 1.3.3 considers
our extension of the long-run risk model, while Section 1.3.4 studies the performance
of the model on moments that are not included in the estimation. We finally consider
a number of counterfactuals in Section 1.3.5.
1.3.1 Model Solution and Estimation Methodology
Pohl, Schmedders, and Wilms (2018) show that the widely used log-normal method
to approximate the solution to the long-run risk model may not always be sufficiently
accurate. Our extension allows α to take on even larger values than traditionally
considered, and this may generate even stronger nonlinearities in the long-run risk
model than reported in Pohl et al. (2018). We address this challenge by using a second-
order projection solution, where we exploit properties of quadratic systems with
Gaussian innovations to analytically carry out the required integration. Avoiding
10 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
numerical integration allows us to greatly reduce the executing time of this projection
solution to a few seconds, which makes the approximation sufficiently fast to be
used inside an estimation routine. Appendix A.2 provides further details on this
approximation, which constitutes a new numerical contribution to the literature. We
also show in Appendix A.3 that this second-order projection solution is more accurate
than the widely used log-normal method, and that it generally performs as well as a
highly accurate fifth-order projection solution.
The estimation is carried out on quarterly data, as this data frequency strikes
a good balance between getting a reasonably long sample and providing reliable
measures of consumption and dividend growth. Consistent with the common cal-
ibration procedure for the long-run risk model, we let one period in the model
correspond to one month and time-aggregate the theoretical moments to a quarterly
frequency. When simulating model moments, Bansal and Yaron (2004) enforce the
non-negativity of σ2t by replacing negative draws with a small positive number. We
follow their procedure and set this small number to σ2σ.
Our quarterly data set is from 1947Q1 to 2014Q4, where we use the same five
variables as in Bansal and Yaron (2004): i) the log-transformed price dividend ratio
pdt , ii) the real risk-free rate r ft , iii) the market return r m
t , iv) consumption growth
∆ct , and v) dividend growth ∆dt . All variables are stored in this order in datat with
dimension 5×1. We explore whether the model can match the means, variances,
contemporaneous covariances, and persistence in these five variables, as well as
the ability of pdt to forecast excess market return ext ≡ r mt − r f
t and the inability
of pdt to forecast dividend growth. To ease the estimation, the values of µz and µd
are calibrated to match the sample mean of consumption growth and dividends,
respectively. Hence, for the estimation we let
qt≡
�data′t
vec(datat data′
t
)′di ag
(datat data′
t−1
)′(ext −ex
)×pdt−1
∆dt ×pdt−1
,
where �datat contains the first three elements of datat , di ag (·) denotes the diagonal
elements of a matrix, and ex t is the sample average of ext . The model is estimated by
simulated method of moments (SMM), where the model-implied moments 1S
∑Ss=1 qs
are computed by simulation using S = 250,000 monthly observations. We adopt the
conventional two-step implementation of SMM and use a diagonal weighting matrix
in a preliminary first step, where moments related to consumption and dividend
growth have a relatively high weight to ensure that the model does not match as-
set prices at the expense of a distorted fit to macro fundamentals. Based on these
estimates, we then obtain our final estimates using the optimal weighting matrix
computed by the Newey-West estimator with 15 lags.
1.3. ESTIMATION RESULTS: THE LONG-RUN RISK MODEL 11
A preliminary analysis reveals that σσ is badly identified. Given that the long-run
risk model requires high persistence inσ2t , we occasionally find that large estimates of
σσ generate a fairly low probability of σ2t being non-negative (e.g., Pr
(σ2
t ≥ 0)≈ 60%),
making (1.2) a poor approximation for the evolution of σ2t . Therefore, we impose an
upper bound of 0.999 on ρσ as in Bansal, Kiku, and Yaron (2012) and set the value of
σσ to 0.05. This value of σσ ensures that Pr(σ2
t ≥ 0)
is at least 83% with ρσ ≤ 0.999.7
1.3.2 The Benchmark Model
As a natural benchmark, we first consider the standard long-run risk model by letting
u0 = 0 in (1.4). For comparability with nearly all calibrations of this model, we let
the IES = 1.5 and RRA = 10 by setting α appropriately using (1.5). The estimates in
the second column of Table 1.1 show that xt generates a small but very persistent
component in consumption growth with σx = 1.16×10−4 and ρx = 0.990. As in the
calibration of Bansal et al. (2012), σ2t displays high persistence with ρσ = 0.9983.
Cyclical consumption risk is mean-reverting with ρc = 0.975 and fairly volatile with
σc = 0.0027. We also note that the constraint on the effective discount factor β∗ ≡βµ
1−1/ψz < 1 is binding, because a high value of β is needed to generate a low risk-free
rate.
Table 1.1 also reports the timing premium Πt of Epstein et al. (2014). We find
thatΠss = 70%, meaning that the household is willing to give up 70% of its lifetime
consumption to know all future realizations of consumption in the following period.
This level of the timing premium is somewhat higher than the reported 31% for the
long-run risk model in Epstein et al. (2014), but lower than 77% as implied by the
calibrated version of the long-run risk model in Bansal et al. (2012).8
Column three in Table 1.2 verifies the common finding in the literature that the
standard long-run risk model with IES = 1.5 and RRA = 10 is able to explain sev-
eral asset pricing moments. In particular, the model provides a very satisfying fit to
the means and standard deviations of the price-dividend ratio and market return.
However, the risk-free rate has an elevated mean (1.96% vs. 0.83%) and displays insuf-
ficient variability with a standard deviation of 0.75% compared to 2.22% in the data.
Table 1.2 also shows that our estimated version of the long-run risk model matches
the standard deviation and persistence in consumption and dividend growth, al-
though the auto-correlation for dividend growth is somewhat higher than in the data
(0.52 vs. 0.40). It is, however, within the 95% confidence interval[0.27,0.52
], which is
7For comparison, Bansal et al. (2012) let σσ = 0.0378, and our calibration is thus very similar to theirpreferred value of σσ.
8The difference in the timing premium reported in Epstein et al. (2014) and the implied value fromthe calibration in Bansal et al. (2012) is mainly explained by the considered values of β and ρσ. Epsteinet al. (2014) use ρσ = 0.987 and β= 0.9980, but increasing ρσ to 0.999 as in Bansal et al. (2012) raises Πssfrom 31% to 50%. If we also increase β to 0.9989 as in Bansal et al. (2012), then Πss = 82% and hence closeto the 77% in Bansal et al. (2012). Slightly different values of σz and σx in Bansal et al. (2012) and Epsteinet al. (2014) account for the remaining difference.
12 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
Table
1.1:Th
eLo
ng-R
un
Risk
Mo
del:T
he
Structu
ralParameters
Estim
ation
results
usin
gd
atafro
m1947Q
1to
2014Q4
and
aseco
nd
-ord
erp
rojectio
nap
pro
ximatio
n.T
he
mo
delh
asa
mo
nth
lytim
efreq
uen
cyw
ithm
od
el-imp
lied
mom
ents
time-aggregated
toa
qu
arterlytim
efreq
uen
cyb
asedon
asim
ulated
samp
leof250,000
mon
thly
observation
s.Th
erep
ortedestim
atesare
fromth
esecon
dstep
inSM
Mw
ithth
eop
timalw
eigthin
gm
atrixestim
atedby
the
New
ey-Westestim
atoru
sing
15lags.Stan
dard
errors
arerep
ortedin
paren
thesis,excep
twh
enan
estimate
is
onth
eb
oun
dary
and
itsstan
dard
erroris
notavailab
le(n
.a.).Th
evalu
esofµ
zan
dµ
dare
calibrated
tom
atchth
esam
ple
mom
ents
ofconsu
mp
tionan
dd
ividen
dgrow
th,
respectively,im
plyin
gµ
z =1.0016
andµ
d =1.0020.T
he
value
ofσσ
issetto
0.05.Th
etim
ing
prem
ium
atthe
steady
state(Π
ss )isd
efin
edas
in(1.8)an
dcom
pu
tedb
ased
on
aseco
nd
-ord
erp
rojectio
no
fthe
value
fun
ction
and
the
utility
levelwh
enu
ncertain
tyis
resolved
inth
efo
llowin
gp
eriod
isco
mp
uted
bysim
ulatio
nu
sing
anti-th
etic
samp
ling
with
10,000d
raws
and
15,000term
sto
app
roximate
the
lifetime
utility
stream.
Ben
chm
arkM
od
elE
xtend
edM
od
el
(1)(2)
(3)(4)
(5)(6)
(7)(8)
RR
A=
5R
RA
=10
RR
A=
5R
RA
=10
IES
=1.5
IES
=1.5
IES
=1.1
IES
=1.5
IES
=2.0
IES
=1.1
IES
=1.5
IES
=2.0
u0
−−
71.37( 3.36)
24.72( 3.09)
9.91( 0.64)
33.22( 4.12)
9.87( 0.90)
2.56( 0.30)
β0.9991
( n.a
.)0.9991
( n.a
.)0.9995
( n.a
.)0.9991
( n.a
.)0.9988
( n.a
.)0.9995
( n.a
.)0.9991
( n.a
.)0.9988
( n.a
.)
ρc
0.7577( 0.3681)
0.9748( 0.0209)
0.9810( 0.0075)
0.9831( 0.0027)
0.9828( 0.0086)
0.9805( 0.0048)
0.9832( 0.0071)
0.9809( 0.0104)
ρx
0.9926( 0.0024)
0.9899( 0.0041)
0.9822( 0.0254)
0.9684( 0.0003)
0.9774( 0.0100)
0.9928( 0.0017)
0.9675( 0.0003)
0.9849( 0.0157)
ρσ
0.9986( 0.0011)
0.9983( 0.0025)
0.9974( 0.0081)
0.9990( n
.a.)
0.9990( n
.a.)
0.9990( n
.a.)
0.9990( n
.a.)
0.9986( 0.0047)
φx
3.2053( 0.2223)
4.3843( 0.0621)
3.5511( 3.5511)
4.595( 0.2558)
4.3246( 1.0974)
3.3772( 2.4230)
4.5664( 0.7024)
4.0767( 0.8778)
φc
2.4172( 0.0751)
0.2396( 0.1219)
0.2745( 0.0620)
0.2737( 0.0028)
0.2716( 0.0976)
0.3263( 0.0537)
0.2630( 0.0839)
0.2763( 0.0786)
σc
0.00001( n
.a.)
0.0027( 0.0008)
0.0030( 0.0006)
0.0027( 0.0003)
0.0027( 0.0005)
0.0026( 0.0003)
0.0027( 0.0004)
0.0028( 0.0008)
σz
0.0020( 0.0003)
0.0014( 0.0012)
0.0013( 0.0011)
0.0016( 0.0004)
0.0016( 0.0006)
0.0020( 0.0002)
0.0016( 0.0003)
0.0015( 0.0010)
σd
0.0125( 0.0004)
0.0116( 0.0010)
0.0116( 0.0009)
0.0107( 0.0001)
0.0106( 0.0008)
0.0108( 0.0007)
0.0107( 0.0011)
0.0108( 0.0018)
σx
1.57×10 −
4(2.34×
10 −5 )
1.16×10 −
4(2.30×
10 −5 )
1.03×10 −
4(6.56×
10 −5 )
1.20×10 −
4(0.70×
10 −5 )
1.20×10 −
4(4.17×
10 −5 )
0.46×10 −
4(4.56×
10 −5 )
1.23×10 −
4(1.35×
10 −5 )
1.02×10 −
4(5.04×
10 −5 )
Mem
o
Pr (σ
2t ≥0 )
86.9%89.2%
93.1%82.6%
82.6%82.6%
82.6%86.9%
u ′( Ct ) C
t
u( Ct ) ∣∣∣∣ss
0.3330.333
0.0120.036
0.0840.023
0.0780.219
Πss
72%70%
93%86%
75%99%
86%73%
α13.00
28.00336.96
120.1053.59
402.04120.08
43.36
1.3. ESTIMATION RESULTS: THE LONG-RUN RISK MODEL 13
Tab
le1.
2:T
he
Lon
g-R
un
Ris
kM
od
el:F
ito
fMo
men
tsT
he
mo
del
has
am
on
thly
tim
efr
equ
ency
wit
hm
od
el-i
mp
lied
mo
men
tsti
me-
aggr
egat
edto
aq
uar
terl
yti
me
freq
uen
cyu
sin
gth
esa
me
pro
ced
ure
asin
Ban
sala
nd
Yaro
n(2
004)
.All
mea
ns
and
stan
dar
dd
evia
tio
ns
are
exp
ress
edin
ann
ual
ized
per
cen
t,ex
cep
tfo
rth
ep
rice
-div
iden
dra
tio.
Th
atis
,th
ere
leva
ntm
om
ents
are
mu
ltip
lied
by
400,
exce
ptf
orth
est
and
ard
dev
iati
onof
the
mar
ketr
etu
rnth
atis
mu
ltip
lied
by20
0.A
llm
odel
-im
plie
dm
omen
tsin
colu
mn
s(2
)to
(9)
are
from
the
un
con
dit
ion
ald
istr
ibu
tion
com
pu
ted
usi
ng
asi
mu
late
dsa
mp
leof
250,
000
mon
thly
obse
rvat
ion
s,w
her
eas
the
emp
iric
ald
ata
mom
ents
inco
lum
n(1
)are
the
emp
iric
alsa
mp
lem
omen
ts.I
nco
lum
n(1
),fi
gure
sin
par
ente
sis
refe
rto
the
stan
dar
der
ror
oft
he
emp
iric
alm
om
ent,
com
pu
ted
bas
edo
na
blo
ckb
oo
tstr
apu
sin
g5,
000
dra
ws
and
ab
lock
len
gth
of3
2q
uar
ters
.
Dat
aB
ench
mar
kM
od
elE
xten
ded
Mo
del
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
RR
A=
5R
RA
=10
RR
A=
5R
RA
=10
IES
=1.
5IE
S=
1.5
IES
=1.
1IE
S=
1.5
IES
=2.
0IE
S=
1.1
IES
=1.
5IE
S=
2.0
Mea
ns
pd
t3.
495
(0.1
22)
3.49
13.
297
3.27
73.
290
3.28
43.
294
3.28
63.
278
rf t
0.83
1(0
.547
)1.
839
1.95
92.
162
1.69
31.
591
1.67
61.
706
1.62
8
rm t6.
919
(1.8
79)
5.70
36.
320
6.31
86.
262
6.30
06.
215
6.27
66.
330
∆c t
1.90
5(0
.244
)1.
894
1.90
21.
905
1.89
71.
896
1.89
41.
897
1.90
0
∆d
t2.
391
(0.9
75)
2.35
42.
377
2.39
82.
363
2.35
72.
358
2.36
32.
370
Std
sp
dt
0.42
1(0
.068
)0.
419
0.34
20.
262
0.28
40.
302
0.26
30.
285
0.29
7
rf t
2.22
4(0
.397
)1.
142
0.75
00.
698
0.49
50.
451
0.58
80.
496
0.46
6
rm t16
.45
(1.1
38)
14.1
014
.48
14.8
314
.80
14.7
314
.77
14.7
814
.694
∆c t
2.03
5(0
.172
)2.
054
2.06
22.
033
2.01
22.
022
2.07
62.
013
2.03
4
∆d
t9.
391
(1.5
31)
9.22
29.
045
8.99
58.
807
8.80
88.
785
8.80
18.
779
Per
sist
ence
corr
(pd
t,p
dt−
1)
0.98
2(0
.056
)0.
985
0.97
60.
957
0.96
30.
968
0.95
70.
964
0.96
7
corr
( rf t
,rf t−
1
)0.
866
(0.0
35)
0.98
70.
978
0.96
40.
951
0.96
60.
981
0.94
90.
975
corr
( rm t,r
m t−1
)0.
084
(0.0
48)
0.01
70.
012
0.00
30.
003
0.00
60.
000
0.00
30.
006
corr
( ∆c t,∆
c t−1
)0.
306
(0.1
18)
0.71
80.
378
0.26
90.
257
0.28
60.
240
0.25
80.
289
corr
( ∆d t,∆
dt−
1) 0.
396
(0.0
63)
0.46
70.
523
0.52
90.
552
0.55
50.
544
0.55
20.
553
14 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
Table
1.2:Lon
g-Ru
nR
iskM
od
el:FitofM
om
ents
(con
tinu
ed)
Data
Ben
chm
arkM
od
elE
xtend
edM
od
el
(1)(2)
(3)(4)
(5)(6)
(7)(8)
(9)
RR
A=
5R
RA
=10
RR
A=
5R
RA
=10
IES
=1.5
IES
=1.5
IES
=1.1
IES
=1.5
IES
=2
IES
=1.1
IES
=1.5
IES
=2
Co
rrelation
s
corr (p
dt ,r
ft )0.035( 0.212)
0.9130.668
-0.0840.040
0.303-0.052
0.0330.367
corr (p
dt ,r
mt )0.058( 0.062)
0.1850.212
0.2840.256
0.2360.288
0.2550.243
corr (p
dt ,∆
ct )
0.025( 0.080)
0.6520.366
0.1390.107
0.1480.118
0.1060.187
corr (p
dt ,∆
dt )
−0.017
( 0.095)0.499
0.5350.635
0.6050.573
0.6630.604
0.586
corr (r
ft,r
mt )0.023( 0.044)
0.1640.083
-0.021-0.006
0.013-0.072
-0.0050.004
corr (r
ft,∆
ct )
0.161( 0.080)
0.7890.468
0.3050.253
0.2890.200
0.2560.230
corr (r
ft,∆
dt )
−0.168
( 0.093)0.565
0.336-0.035
0.0090.088
-0.1630.011
0.072
corr (r
mt,∆
ct )
0.233( 0.054)
0.1350.395
0.6230.592
0.5560.558
0.5970.554
corr (r
mt,∆
dt )
0.104( 0.050)
0.2960.294
0.2960.290
0.2890.292
0.2890.289
corr (∆
ct ,∆
dt )
0.062( 0.0496)
0.4650.236
0.0690.075
0.1070.028
0.0760.107
corr (r
mt−
rft
,pd
t−1 )
−0.134
( 0.048)-0.017
-0.0140.006
-0.002-0.009
0.011-0.002
-0.007
corr (∆
dt ,p
dt−
1 )−
0.0163( 0.104)
0.4670.498
0.5860.562
0.5330.616
0.5600.545
Go
od
ness
offi
tQ
step2
-0.0632
0.06210.0624
0.05910.0592
0.06160.0592
0.0593J-test:P-valu
e-
10.93%26.44%
20.20%24.78%
24.59%21.24%
24.5524.49%
Qsca
led-
3.352.26
1.891.54
1.531.62
1.541.61
1.3. ESTIMATION RESULTS: THE LONG-RUN RISK MODEL 15
derived from the reported standard error for each of the sample moments in Table
1.2 shown in parenthesis and computed using a block bootstrap.
The last part of Table 1.2 shows the contemporaneous correlations. We find that
consumption growth is too highly correlated with the price-dividend ratio (0.37
vs. 0.03). This is similar to the finding reported in Beeler and Campbell (2012). We
also find that consumption growth is too strongly correlated with the risk-free rate
(0.47 vs. 0.16). Conventional two-sided t-tests further show that the differences in
cor r(pdt ,∆ct
)and cor r
(pdt ,rt
)have t-statistics of 4.26 and 3.84, respectively.9
To understand why consumption growth is too highly correlated with pdt and
r ft , recall that the standard long-run risk model relies on the power utility kernel
with an IES = 1.5 and RRA = 10. Equation (1.5) then implies a relatively low timing
attitude with α= 28. To explain the market return, the model therefore requires high
persistence in xt to amplify the long-run risk channel (see Section 1.2.4). But, such a
high level of persistence in xt makes consumption growth too highly correlated with
the price-dividend ratio and the risk-free rate. To realize this, consider the analytical
approximation in Section 1.2.4 which implies
cov(∆ct , pdt ) =φ− 1
ψ
1−κ1ρxρx
σ2x
1−ρ2x+
(1−ρc
)2
1−κ1ρc
1
ψ
σ2c
1−ρ2c
(1.6)
and
cov(∆ct ,r ft ) = 1
ψ
[ρx
σ2x
1−ρ2x− (
1−ρc)2 σ2
c
1−ρ2c
], (1.7)
which both are increasing in ρx for the parameter values in Table 1.1. Hence, an
undesirable effect of the high persistence in xt is to amplify the comovement of
consumption growth with pdt and r ft .
The tight link between the timing attitudeα and the degree of long-run risk is seen
clearly when estimating the model with RRA = 5, as shown in the first column of Table
1.1. This lower level of RRA weakens the effect from the timing attitude, asα falls from
28 to 13. To match asset prices, we therefore find an increase in the degree of long-
run risk compared to the benchmark specification with RRA = 10, as σx increases
from 1.16× 10−4 to 1.57× 10−4 and ρx increases from 0.990 to 0.993. The second
column in Table 1.1 shows that this increase in long-run risk produces too much
auto-correlation in consumption growth (0.72 vs. 0.31) and amplifies cor r (∆ct , pdt )
and cor r (∆ct ,r ft ) further.
9Using the log-normal method and the calibration in Bansal and Yaron (2004), the long-run risk model
implies cor r(pdt ,∆ct
) = 0.547 and cor r
(r
ft ,∆ct
)= 0.581. The corresponding empirical moments on
annual data are 0.061 and 0.356, respectively. The slightly modified calibration in Bansal et al. (2012) with
less long-run risk gives cor r(pdt ,∆ct
)= 0.368 and cor r
(r
ft ,∆ct
)= 0.473. Thus, the elevated correlations
for cor r(pdt ,∆ct
)and cor r
(r
ft ,∆ct
)also appear in calibrated versions of the long-run risk model using
annual data.
16 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
1.3.3 The Extended Model
We next introduce u0 in the utility kernel and re-estimate the long-run risk model
when conditioning on the familiar values of RRA = 10 and IES = 1.5. Column seven in
Table 1.1 shows that we find u0 = 9.87 with a standard error of 0.90, meaning that u0 is
statistically different from zero at all conventional significance levels. With u0 = 9.87,
the key ratio u′ (Ct)
Ct /u(Ct
)∣∣∣ss
is much lower than in the benchmark version of the
model (0.078 vs. 0.333), and this allows the timing attitude α to increase from 28 to
120 while keeping RRA at 10. Less long-run risk is therefore needed to match asset
prices and this explains the fall in ρx from 0.990 to 0.968. As a result, cor r(pdt ,∆ct
)falls from 0.37 to 0.10 and cor r
(r f
t ,∆ct
)falls from 0.47 to 0.26, implying that both
moments are no longer significantly different from their empirical moments. We also
see improvements in the ability of the model to match cor r(pdt ,r f
t
), cor r
(r f
t ,∆dt
),
cor r(∆ct ,∆dt
), and the mean of r f
t . On the other hand, the fit to cor r(r m
t ,∆ct
),
cor r(r m
t − r ft , pdt−1
), cor r
(∆dt , pdt−1
), and the standard deviations of pdt and r f
t
worsen slightly when including u0.
To evaluate the overall goodness of fit for the long-run risk model, Table 1.2 also
reports the value of the objective function Q step2 in step 2 of our SMM estimation
and the related p-value for the J-test for model misspecification. The benchmark
model and our extension are not rejected by the data, but we note that the J-test
has low power given our short sample (T = 271). The values of Q step2 are unfortu-
nately not comparable across models, because they are computed for model-specific
weighting matrices. To facilitate model comparison, we therefore introduce the fol-
lowing measure for goodness of fit Q scaled =∑ni=1
((md at a
i −mmodeli
)/(1+md at a
i
))2
,
where md at ai and mmodel
i refer to the scaled moments in the data and the model,
respectively, as reported in Table 1.2.10 Although the moments in Q scale are weighted
differently than in the estimation, Q scaled may nevertheless serve as a natural sum-
mary statistic for model comparison from an economic perspective. We find that
the benchmark model implies Q scaled = 2.26, but allowing for u0 in the utility kernel
gives Q scaled = 1.54. This corresponds to an 32% improvement in model fit from
disentangling the timing attitude α from the IES and RRA.
A natural way to extend the timing premium of Epstein et al. (2014) to the utility
10The difference md at ai −mmodel
i in Qscale is standardized by 1+md at ai , as oppose to just md at a
i , toensure that moments close to zero do not get very large weights.
1.3. ESTIMATION RESULTS: THE LONG-RUN RISK MODEL 17
kernel in (1.4) is to define Πt implicitly as
Vt =u0Z 1−1/ψ
t + 1
1− 1ψ
C1− 1
ψ
t
(1−Πt
)1− 1ψ (1.8)
+β
Et
∞∑
i=1βi−1
u0Z 1−1/ψt+i + C
1− 1ψ
t+i
1− 1ψ
(1−Πt
)1− 1ψ
1−α
1/(1−α)
.
That is, we combine Z1− 1
ψ
t u0 and the utility from Ct when computing Πt , because
Z1− 1
ψ
t u0 is a reduced-form term that captures other aspects of consumption than
included in Ct (see Section 1.2.3). This implies thatΠt measures the fraction of overall
lifetime consumption that the household is willing to pay to have all uncertainty
resolved in the following period. Clearly, equation (1.8) reduces to the definition of
Πt in Epstein et al. (2014) when u0 = 0. Table 1.1 shows that Πss increases from 70%
to 86% when introducing u0 in the utility kernel when RRA = 10 and IES = 1.5. That is,
the pronounced increase in the timing attitudeα from 28 to 120 more than outweighs
the effects from less long-run risk and leads to an even higher timing premium.
The remaining columns in Table 1.1 and 1.2 explore the robustness of these
findings to lowering the IES to 1.1, increasing the IES to 2, and reducing RRA to 5.
We emphasize the following two results. First, lowering RRA from 10 to 5 does hardly
affect the model’s ability to match asset prices once u0 is included in the utility kernel.
For instance, we find Q scaled = 1.54 when the IES = 1.5 for both levels of RRA. In
contrast, when using the traditional utility kernel with a RRA of 5, the model’s ability
to match the data deteriorates as Q scaled increases from 2.26 to 3.35. Second, the
effects of changing the IES are generally also small, in particular for RRA = 10. Thus,
we find that the satisfying ability of the long-run risk model to match asset prices
extends to the case of a lower IES of 1.1 and a lower RRA of 5, once u0 is included in
the utility kernel. However, separating these three behavioral characteristics in the
utility function does not alleviate the problem of seemingly implausible high levels
of the timing premium, which remains very high (i.e. above 70%) for all considered
specifications of the IES and RRA.
1.3.4 Additional Model Implications
In addition to the moments used in the estimation, the long-run risk model is also
frequently evaluated based on its ability to reproduce several stylized relationships for
the U.S. stock market. Following Beeler and Campbell (2012), we first study the ability
of the price-dividend ratio to explain past and future consumption growth. Figure 1.1
shows that past and future consumption growth are too highly correlated with the
18 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
price-dividend ratio compared to empirical evidence in the standard long-run risk
model. A similar finding is reported in Beeler and Campbell (2012) for two calibrated
versions of this model. In contrast, our extension of the long-run risk model implies
that past and future consumption growth display the same low correlations with
the price-dividend ratio as seen in the data. Figure 1.1 considers the case where the
IES =1.5 and RRA = 5 in our extension of the long-run risk model, but the results are
robust to using any of the other specifications for the IES and RRA reported in Table
1.1. Thus, disentangling the timing attitude from the IES and RRA is also supported
by these stylized regressions, because a higher timing attitude reduces the amount of
long-run risk and hence the degree of predictability in consumption growth.
Figure 1.1: Properties of Consumption Growth and Volatility
All model-implied moments are computed given the estimated parameters in Table 1.1 using a simulatedsample path of 1,000,000 observations. The conditional volatility σt is estimated by
∣∣ut∣∣, where ut is the
residual from the OLS regression ∆ct =α+∑5j=1β
(j)∆ct− j +ut . All the 95 percent confidence bands are
computed using a block bootstrap applied jointly to the regressant and the regressor with a block length of2× j lags.
-5 -4 -3 -2 -1 0 1 2 3 4 5
Forecast horizon j in quarters
0
0.05
0.1
0.15
0.2
0.25
0.3
-5 -4 -3 -2 -1 0 1 2 3 4 5
Forecast horizon j in quarters
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1 2 3 4 5 6 7 8 9 10
Forecast horizon j in quarters
0.1
0.2
0.3
0.4
0.5
1 2 3 4 5 6 7 8 9 10
Forecast horizon j in quarters
0
0.05
0.1
0.15
0.2
The last two charts in Figure 1.1 explore the relationship between consumption
volatility and the price-dividend ratio. We find that our extension of the long-run risk
model preserves the good performance of the benchmark model and implies that
i) a high price-dividend ratio predicts future low volatility and ii) high uncertainty
forecasts a low price-dividend ratio.
1.3. ESTIMATION RESULTS: THE LONG-RUN RISK MODEL 19
1.3.5 The Key Mechanisms
We next consider a number of experiments to illustrate some of the key mechanisms
in the model. Here, we apply the estimated version of the model in column four of
Table 1.1 with an IES of 1.5 and a RRA of 5.
Table 1.3: The Long-Run Risk Model: Analyzing the Extended ModelThe model has a monthly time frequency with model-implied moments time-aggregated to a quarterlytime frequency using the same procedure as in Bansal and Yaron (2004). All means and standard deviationsare expressed in annualized percent by multiplying by 400, except for the standard deviation of the marketreturn that is multiplied by 200. The moments are from the unconditional distribution computed usinga simulated sample of 250,000 monthly observations. Unless stated otherwise, all parameters attain theestimated values from column (4) in Table 1.1, meaning that the IES = 1.5 and RRA = 5.
(1) (2) (3) (4) (5) (6) (7)u0 = 10 u0 = 20 u0 = u0 σx = 0 σσ = 0 β= 0.998 α= 0
Meanspdt 66.31 15.71 3.29 14.04 4.46 3.16 8.50
r ft 2.18 1.86 1.69 2.24 2.17 3.21 2.39
r mt 2.36 2.36 6.26 2.38 3.60 6.80 2.40
Stdspdt 0.31 0.31 0.28 0.30 0.27 0.28 0.82
r ft 0.47 0.48 0.50 0.24 0.43 0.50 0.45
r mt 17.99 18.02 14.80 17.02 16.18 14.55 22.22
MemoRRA 5 5 5 5 5 5 0.67Πss 46% 79% 86% 18% 33% 34% 0%α 56.33 99.67 120.10 120.10 120.10 120.10 0
The first experiment we consider is to gradually increase u0 to its estimated value
of 24.72. Table 1.3 shows that a higher value of u0 generates a substantial increase in
the required timing attitude α to ensure a constant RRA. This in turn has desirable
effects on the level of asset prices because a higher value of α reduces E[pdt ] as well
as E[r ft ] and increases E[r m
t ]. To understand these effects of increasing α for a given
level of RRA, recall that the household is indifferent to resolution of uncertainty when
α= 0. Now suppose we increase α to make the household prefer early resolution of
uncertainty, but without affecting the RRA. This modification increases the variability
of the value function and hence increases the precautionary motive. The one-period
risk-free bond therefore becomes more attractive, and this reduces the risk-free rate
as shown in Proposition 2. On the other hand, uncertain future dividends from equity
become less attractive for higher values of α due to the presence of long-run risk.
A household with strong preferences for early resolution of uncertainty therefore
requires a larger compensation for holding equity compared to the case of α= 0 and
20 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
this explains the increase in E[r mt ] for higher values of α.
The second experiment we consider is to omit long-run risk by letting σx = 0. The
fourth column in Table 1.3 shows that this modification has profound implications,
as the model now generates a too high level for the the price-dividend ratio (14.04
vs. 3.50 in the data) and the risk-free rate (2.24% vs. 0.83% in the data), whereas the
average market return is too low (2.38% vs. 6.92% in the data). Omitting long-run
risk also has a large effect on the timing premium, which falls from 86% to 18%.
Thus, disentangling the timing attitude α from the IES and RRA does not alleviate the
reliance on long-run risk in the model.
Our third experiment imposes σσ = 0 to evaluate the importance of stochastic
volatility. The fifth column in Table 1.3 shows that the mean of the price-dividend
ratio increases to 4.46 and the mean market return falls to 3.66%. We also find that
the timing premium decreases from 86% to 33%. This shows that stochastic volatility
may have a much larger impact on the timing premium in long-run risk models than
suggested by the results in Epstein et al. (2014). Thus, stochastic volatility remains an
important feature of the long-run risk model, even when the timing attitude is set
independently of the IES and RRA.
The fourth experiment explores whether the high subjective discount factor β
may help to explain the high timing premium in the long-run risk model. We address
this question in the sixth column of Table 1.3 by reducing β from its estimated value
of 0.9991 to 0.9980 as considered in Epstein et al. (2014). This small change in β
reduces the timing premium from 86% to 34%, which is in the neighborhood of the
31% reported in Epstein et al. (2014). However, a β of 0.9980 gives a too high mean for
the risk-free rate (3.21% vs. 0.83% in the data), and hence makes the model unable
to resolve the risk-free rate puzzle. This result explains why our estimation prefers a
high β, although it implies high timing premia.
Our final experiment studies the effect of the timing attitude by letting α = 0,
implying that the household is indifferent between early and late resolution of uncer-
tainty. The seventh column of Table 1.3 shows that this modification only lowers the
RRA from 5 to 0.67, but it nevertheless has a profound impact on the model despite
the presence of long-run risk. That is, the model is simply unable to match asset
prices without strong preferences for early resolution of uncertainty.
1.4 A New Keynesian Model
To provide further support for the considered Epstein-Zin-Weil preferences, we next
show that they also help explain asset prices in an otherwise standard New Keynesian
model. The processes for consumption and dividends are here determined within
the model, whereas they are assumed to be exogenously given in the long-run risk
model. We proceed by presenting our New Keynesian model in Section 1.4.1, the
adopted estimation routine in Section 1.4.2, and the estimation results in Section
1.4. A NEW KEYNESIAN MODEL 21
1.4.3. We finally examine the key mechanisms in our extended New Keynesian model
in Section 1.4.4.
1.4.1 Model Description
1.4.1.1 Household
The household is similar to the one considered in Section 1.2 except for a variable
labor supply Lt . To match the persistence in consumption growth, we follow much
of the New Keynesian tradition and allow for exogenous consumption habits of the
form bCt−1. These modifications are included in the new utility kernel by letting
u(Ct ,Lt ) = u0Z 1−1/ψt +
(Ct −bCt−1
)1−1/ψ
1−1/ψ
+ϕ0Z 1−1/ψt
(1−Lt
)1− 1ϕ
1− 1ϕ
(1.9)
with ϕ0 > 0 and ϕ ∈ R \{1}, which reduces to the specification in Rudebusch and
Swanson (2012) when u0 = b = 0. The constant u0 does not affect the IES at the steady
state ψ(1− b
µZ ,ss
), where consumption habits reduce the IES compared to the value
implied by ψ. The expression for the RRA is slightly more involved than the one
provided in (1.5) due to consumption habits and the variable labor supply, where the
latter gives the household an additional margin to absorb shocks. For the Epstein-
Zin-Weil preferences in (1.1), Swanson (2018) shows that RRA in the steady state is
given by RRA= 1IES
(1+ Wt
ZtΛt
)−1∣∣∣∣
ss+α uC (Ct ,Lt )Ct
u(Ct ,Lt )
∣∣∣ss
, where Λt ≡ −uL (Ct ,Lt )uCC (Ct ,Lt )uC (Ct ,Lt )uLL (Ct ,Lt )
accounts for the labor margin. When inserting for the utility kernel in (1.9) we get
RRA = 1
IES+ ϕWss (1−Lss )Css
+α(1− 1
ψ
)1− b
µZ ,ss+ 1− 1
ψ
Css
[u0C
1ψ
ss
(1− b
µZ ,ss
) 1ψ + (1−Lss )Wss
1− 1ϕ
] .(1.10)
Here, Css and Wss refer to the steady state of consumption and the real wage in
the normalized economy without trending variables, and µZ ,ss denotes the deter-
ministic trend in consumption and productivity, which we specify below in (1.11).
Equation (1.10) shows that u0 also with consumption habits and a variable labor
supply controls RRA through the ratio uC(Ct
)Ct /u
(Ct
).
The real budget constraint for the household is given by Et
[Mt ,t+1
X t+1πt+1
]+Ct =
X tπt
+Wt Lt +D t , where Mt ,t+1 is the nominal stochastic discount factor, X t is nominal
state-contingent claims, πt denotes gross inflation, Wt is the real wage, and D t is real
dividend payments from firms.
22 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
1.4.1.2 Firms
Final output Yt is produced by a perfectly competitive representative firm, which
combines differentiated intermediate goods Yt(i)
using Yt =(∫ 1
0 Yt(i) η−1
η di
) ηη−1
with
η > 1. This implies that the demand for the i th good is Yt(i) = (
Pt (i)Pt
)−ηYt , where
Pt ≡(∫ 1
0 Pt(i)1−ηdi
) 11−η
denotes the aggregate price level and Pt(i)
is the price of the
i th good.
Intermediate firms produce the differentiated goods using Yt(i)= Zt At K θ
ss Lt(i)1−θ ,
where Kss and Lt(i)
denote capital and labor services at the i th firm, respectively.
Productivity shocks are allowed to have the traditional stationary component At ,
but also a non-stationary component Zt to generate long-run risk in the model.
For the stationary shocks, we let log At+1 = ρA log At +σAεA,t+1, where∣∣ρA
∣∣ < 1,
σA > 0, and εA,t+1 ∼ NID(0,1
). Similarly for the non-stationary shocks, we intro-
duce µZ ,t+1 = Zt+1/Z and let
log
(µZ ,t+1
µZ ,ss
)= ρZ log
(µZ ,t
µZ ,ss
)+σZ εZ ,t+1, (1.11)
where∣∣ρZ
∣∣< 1, σZ > 0, and εZ ,t+1 ∼NID(0,1
).11
Intermediate firms can freely adjust their labor demand at the given market wage
Wt and are therefore able to meet demand in every period. Similar to Andreasen
(2012), price stickiness is introduced as in Rotemberg (1982), where ξ≥ 0 controls
the size of firms’ real cost ξ2
(Pt
(i)
/(Pt−1
(i)πss
)−1
)2
Yt when changing the optimal
nominal price Pt(i)
of the good they produce.12
1.4.1.3 The Central Bank and Aggregation
The central bank sets the one-period nominal interest rate it according to it = iss +βπ log
(πtπss
)+βy log
(Yt
Zt Yss
), based on a desire to close the inflation and output gap.
Note that the inflation gap accounts for steady-state inflation πss , and that the output
gap is expressed in deviation from the steady state level of output in the normalized
economy Yss without trending variables.
11The specification of long-run productivity risk adopted in the endowment model, i.e. (1.2), couldalso be used in the New Keynesian model, but we prefer the more parsimonious specification in (1.11) forcomparability with the existing DSGE literature (see, for instance, Justiniano and Primiceri, 2008). Thisdifference explains the slightly different notation used in (1.11) for µZ ,t , µZ ,ss , σZ , and εZ ,t+1 comparedto the corresponding parameters in (1.2).
12Specifying nominal regidities by Calvo pricing as in Rudebusch and Swanson (2012) gives largelysimilar results to those reported below. The considered specification is chosen because the solution tothe New Keynesian model with Rotemberg pricing is approximated more accurately by the perturbationmethod than with Calvo pricing. The reason seems to be that Calvo (unlike Rotemberg) pricing induces aprice dispersion index as an extra state variable that makes the New Keynesian model very nonlinear incertain areas of the state space, as shown in Andreasen and Kronborg (2018).
1.4. A NEW KEYNESIAN MODEL 23
Summing across all firms and assuming that δKss Zt units of output are used
to maintain the constant capital stock as in Rudebusch and Swanson (2012), the
resource constraint becomes Ct +ZtδKss =(1− ξ
2
(πtπss
−1)2
)Yt .
1.4.1.4 Equity and Bond Prices
Equity is defined as a claim on aggregate dividends from firms, i.e. D t = Yt −Wt Lt ,
and its real price is therefore 1 = Et
[Mt ,t+1Rm
t+1
]where Rm
t+1 =(D t+1 +P m
t+1
)/P m
t .
The price in period t of a default-free zero-coupon bond B (n)t maturing in n
periods with a face value of one dollar is B (n)t = Et
[Mt ,t+1πt+1
B (n−1)t+1
]for n = 1, ..., N with
B (0)t = 1. Its yield to maturity is i (n)
t =− 1n logB (n)
t . Following Rudebusch and Swanson
(2012), we define term premia as Ψ(n)t = i (n)
t − i (n)t , where i (n)
t is the yield to maturity
on a zero-coupon bond B (n)t under risk-neutral valuation, i.e. B (n)
t = e−it Et
[B (n−1)
t+1
]with B (0)
t = 1.
1.4.2 Model Solution and Estimation Methodology
We approximate the model solution by a third-order perturbation solution. The model
is estimated by GMM using unconditional first and second moments computed as
in Andreasen, Fernandes-Villaverde, and Rubio-Ramirez (2018). The selected series
describing the macro economy and the bond market are given by ∆ct , πt , it , i (40)t ,
Ψ(40)t , and logLt , where one period in the model corresponds to one quarter. The 10-
year nominal interest rate and its term premium (obtained from Adrian, Crump, and
Moench, 2013) are available from 1961Q3, leaving us with quarterly data from 1961Q3
to 2014Q4. We include all means, variances, and first-order auto-covariances of these
six variables for the estimation, in addition to five contemporaneous covariances
related to the correlations reported at the end of Table 1.5. To examine whether
our New Keynesian model is able to match the equity premium, we also include
the mean of the net market return r mt = logRm
t in the set of moments. Finally, the
GMM estimation is implemented using the conventional two-step procedure for
moment-based estimators as outlined in Section 1.3.1.
We estimate all structural parameters in the model except for a few badly identi-
fied parameters. That is, we let δ= 0.025 and θ = 1/3 as typically considered for the
U.S. economy. We also let η= 6 to get an average markup of 20% and impose ϕ= 0.25
to match a Frisch labor supply elasticity in the neighborhood of 0.5. The ratio of
capital to output in the steady state is set to 2.5 as in Rudebusch and Swanson (2012).
We follow Andreasen (2012) and set ξ based on a linearized version of the model to
match a Calvo parameter of αp = 0.75, giving an average duration for prices of four
24 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
quarters.13 Finally, the estimates of the subjective discount factor for all considered
specifications of the New Keynesian model hit the upper bound for this parameter
and we therefore simply let β= 0.9995.
1.4.3 Estimation Results
1.4.3.1 A Standard Power Utility Kernel
We first consider the standard implementation of Epstein-Zin-Weil preferences with
u0 = 0 and condition the estimation of the New Keynesian model on different values
of RRA. Table 1.4 shows that we get fairly standard estimates when RRA = 5. That
is, we find strong habits (b = 0.72), very persistent technology shocks (ρA = 0.99
and ρZ = 0.33), and a central bank that assigns more weight to stabilizing inflation
than output (βπ = 1.46 and βy = 0.02). Table 1.5 shows that the model does well in
matching the mean and variability of inflation, the short rate, the 10-year interest rate,
and the 10-year term premium. The model-implied level of the market return is 3.61%
and reasonably close to the empirical value of 5.53%, when accounting for its large
standard error of 2.01% computed by a block bootstrap. However, the model also
generates too much variability in consumption growth (2.35% vs. 1.80%) and labor
supply (2.85% vs. 1.62%), predicts too strong autocorrelation in consumption growth
(0.73 vs. 0.53), and is unable to match the negative correlation between consumption
growth and inflation (0.19 vs. −0.18). Table 1.4 and 1.5 also show that increasing RRA
to 10 does not materially affect the estimates and performance of the New Keynesian
model. Thus, these results just iterate the finding in Rudebusch and Swanson (2008)
that the standard New Keynesian model with low RRA struggles to match key asset
pricing moments without distorting the fit to the macro economy.
We next increase RRA to 60, although such an extreme level of risk aversion is
hard to justify based on micro-evidence. Table 1.5 shows that the New Keynesian
model now reproduces all means without generating too much variability in the
macro economy, except for a slightly elevated standard deviation in labor supply
(2.45% vs. 1.62%). Thus, a high RRA of 60 implies that the model delivers a better
overall fit to the data with Q scaled = 0.34 compared to Q scaled = 0.76 when RRA =
5. To compute the timing premium in our New Keynesian model we must extend
the definition in Epstein et al. (2014) to account for an endogenous labor supply.
The labor margin gives the household an extra dimension to absorb shocks and
this affects its willingness to pay for getting uncertainty resolved in the following
period. The problem is thus very similar to the one considered in Swanson (2018) for
extending expressions of RRA to account for a variable labor supply, and we therefore
follow his approach and use the equilibrium condition for the consumption-leisure
13The mapping is ξ=(1−θ+ηθ)(
η−1)αp(
1−αp
)(1−θ)
1−αpβµ1− 1
ψZ ,ss
as derived in the online appendix.
1.4. A NEW KEYNESIAN MODEL 25
Table 1.4: The New Keynesian Model: The Structural ParametersEstimation results using data from 1961Q3 to 2014Q4 using a third-order perturbation approximation with
model-implied moments computed as in Andreasen et al. (2018). The reported estimates are from the
second step of GMM with the optimal weigthing matrix estimated by the Newey-West estimator with 15
lags. The estimates of β are for all specifications on the boundary 0.9995 and therefore not reported below.
The timing premium at the steady state (Πss ) is computed based on (1.12) and a third-order perturbation
approximation, where the utility level when uncertainty is resolved in the followingt period is computed
by simulation using anti-thetic sampling with 5,000 draws and 10,000 terms to approximate the lifetime
utility stream.
Benchmark Model Extended Model(1) (2) (3) (4) (5) (6)
RRA=5 RRA=10 RRA=60 RRA=5 RRA = 10 RRA = 60
u0 - - - −938.67(215.71)
−294.23(36.214)
−24.503(5.4683)
ψ 0.1084(0.0109)
0.2088(0.0216)
0.4040(0.1440)
0.1835(0.0375)
0.3039(0.0470)
0.5169(0.0527)
b 0.7248(0.0165)
0.7588(0.0186)
0.7912(0.0302)
0.4867(0.0255)
0.5575(0.0308)
0.5785(0.0485)
βπ 1.4588(0.0568)
1.4326(0.0734)
1.4597(0.2267)
1.4229(0.0381)
1.3814(0.0576)
1.3263(0.0563)
βy 0.0209(0.0036)
0.0294(0.0053)
0.0565(0.0242)
0.0192(0.0042)
0.0228(0.0106)
0.0563(0.0184)
µZ ,ss 1.0029(0.0002)
1.0038(0.0003)
1.0052(0.0003)
1.0049(0.0004)
1.0050(0.0004)
1.0051(0.0004)
πss 1.0635(0.0053)
1.0458(0.0033)
1.0311(0.0026)
1.0683(0.0119)
1.0458(0.0047)
1.0290(0.0023)
Lss 0.3375(0.0009)
0.3371(0.0007)
0.3368(0.0007)
0.3381(0.0014)
0.3369(0.0009)
0.3378(0.0015)
ρA 0.9910(0.0009)
0.9885(0.0011)
0.9867(0.0012)
0.9927(0.0011)
0.9896(0.0013)
0.9818(0.0011)
ρZ 0.3254(0.0817)
0.5185(0.0718)
0.7883(0.0783)
0.2084(0.1196)
0.3653(0.1780)
0.4434(0.5371)
σA 0.0168(0.0008)
0.0143(0.0010)
0.0116(0.0023)
0.0214(0.0017)
0.0177(0.0013)
0.0098(0.0012)
σZ 0.0098(0.0010)
0.0070(0.0010)
0.0017(0.0010)
0.0053(0.0024)
0.0028(0.0008)
0.0019(0.0019)
MemoIESss 0.030 0.051 0.086 0.095 0.135 0.219u′C
u
∣∣∣ss
−1.93 −1.82 −1.59 −0.10 −0.08 −0.21
Πss 0.1% 4% 10% 6% 11% 17%α −1.27 −4.17 −36.43 −28.43 −103.6 −275.9
26 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
Table 1.5: The New Keynesian Model: Fit of MomentsAll variables are expressed in annualized terms in percent, except for the mean of log(Lt ). All model-
implied moments in columns (2) to (7) are from the unconditional distribution, whereas the empirical
data moments in column (1) are given by the sample means. In column (1), figures in parenthesis refer to
the standard error of the empirical moment, computed based on a block bootstrap using 5,000 draws and
a block length of 32 quarters.
Benchmark Model Extended Model(1) (2) (3) (4) (5) (6) (7)
Data RRA=5 RRA=10 RRA=60 RRA=5 RRA=10 RRA=60
Means (in pct)∆ct 1.975
(0.276)1.142 1.497 2.069 1.970 1.984 2.048
πt 3.890(0.793)
3.856 3.789 3.672 3.792 3.781 3.474
it 4.999(0.994)
5.090 5.104 5.161 5.178 5.164 5.115
i (40)t 6.497
(0.904)6.509 6.510 6.551 6.512 6.516 6.513
Ψ(40)t 1.663
(0.355)1.745 1.775 1.768 1.672 1.755 1.777
logLt −1.081(0.004)
−1.080 −1.080 −1.081 −1.080 −1.080 −1.080
r mt 5.527
(2.012)3.607 3.829 3.907 4.669 4.166 3.515
Stds (in pct)∆ct 1.802
(0.122)2.352 2.259 1.444 2.146 1.877 1.400
πt 2.716(0.612)
2.493 2.601 2.899 2.273 2.536 2.997
it 3.173(0.579)
3.045 2.944 2.935 2.374 2.651 2.848
i (40)t 2.621
(0.532)2.635 2.618 2.592 2.360 2.542 2.573
Ψ(40)t 1.165
(0.170)0.967 0.864 0.874 1.000 0.894 0.870
logLt 1.619(0.163)
2.853 2.697 2.450 2.506 2.509 2.082
Persistencecor r
(∆ct ,∆ct−1
)0.529(0.083)
0.727 0.757 0.764 0.479 0.538 0.527
cor r(πt ,πt−1
)0.953(0.056)
0.943 0.958 0.960 0.977 0.972 0.972
cor r(it , it−1
)0.949(0.031)
0.913 0.926 0.911 0.954 0.952 0.955
cor r(i (40)
t , i (40)t−1
)0.976(0.031)
0.989 0.987 0.985 0.989 0.987 0.980
cor r(Ψ(40)
t ,Ψ(40)t−1
)0.937(0.032)
0.991 0.988 0.986 0.993 0.989 0.982
cor r(logLt , logLt−1
)0.932(0.476)
0.751 0.767 0.800 0.875 0.868 0.871
1.4. A NEW KEYNESIAN MODEL 27
Table 1.5: The New Keynesian Model: Fit of Moments (continued)
Benchmark Model Extended Model(1) (2) (3) (4) (5) (6) (7)
Data RRA=5 RRA=10 RRA=60 RRA=5 RRA=10 RRA=60
Correlationscor r
(∆ct ,πt
) −0.184(0.150)
0.193 0.017 −0.167 −0.104 −0.180 −0.185
cor r(∆ct , it
)0.021(0.199)
0.239 0.020 −0.241 −0.110 −0.203 −0.255
cor r(πt , it
)0.703(0.074)
0.966 0.969 0.959 0.925 0.970 0.977
cor r(it , i (40)
t
)0.900(0.048)
0.809 0.854 0.878 0.912 0.939 0.961
cor r(i (40)
t ,Ψ(40)t
)0.757(0.148)
0.900 0.958 0.988 0.815 0.921 0.976
Goodness of fitQStep2 - 0.061 0.062 0.060 0.050 0.059 0.061J-test: P-value - 0.453 0.437 0.467 0.552 0.399 0.373Q scaled - 0.758 0.445 0.344 0.258 0.280 0.305
trade-off. This implies that the value function can be expressed in consumption units
as
Vt = Z1− 1
ψ
t u0 + 1
1− 1ψ
(Ct −bCt−1
)1− 1ψ (1.12)
+Z1− 1
ψ
t
ϕϕ0
1− 1ϕ
Z
(1− 1
ψ
)(ϕ−1)
t
W (ϕ−1)t
(Ct −bCt−1
) 1ψ (ϕ−1) +β
(Et
[V 1−α
t+1
]) 11−α
,
and it is then straightforward to compute the timing premium. Table 1.4 shows that
the timing premium at the steady stateΠss is 0.1% with RRA = 5, 4% with RRA = 10,
and only 10% with RRA = 60. Note also that this increase in Πss coincides with higher
levels of the timing attitude, as the absolute value of α increases gradually for higher
RRA. Importantly, the timing premium in the New Keynesian model is substantially
lower than in the long-run risk model, even when considering an extreme RRA of 60.
To explore whether the labor margin helps to account for the low timing premium
in the New Keynesian model, we next condition on the reported estimates in Table 1.4
with RRA = 60 and changethe Frisch labor supply elasticityϕ(1/Lt −1
)by considering
different values of ϕ. It is a priori not obvious how the timing premium should be
affected by changing the variability of the labor supply. As argued by Swanson (2018)
in the context of RRA, a higher labor supply elasticity allows the household to better
self-insure against bad productivity shocks to reduce the variability in consumption.
This effect should therefore reduce the timing premium for higher values of ϕ. But, a
more volatile labor supply also makes the household’s value function more uncertain
through the direct effect of leisure in the utility kernel, and this effect should therefore
28 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
increase the timing premium for higher values of ϕ. Panel A in Table 1.6 shows that
the second effect dominates, as the timing premium is 10% for ϕ = 0.25, 35% for
ϕ= 0.50, and 94% for ϕ= 0.75. These computations are conditioned on a RRA of 60
by appropriately changing the timing attitude α, which increases substantially in
absolute terms for higher values of ϕ. Panel B of Table 1.6 adopts another approach
by conditioning on α = −36 and instead let RRA vary as we change the value of ϕ.
When using this alternative benchmark, we find a much more gradual increase in the
timing premium when increasing ϕ, showing that the main effect of the labor margin
operates through the timing attitude α. Two other features of the New Keynesian
model that also may have a sizable impact on the timing premium are consumption
habits and the low estimate of ψ. Both features help to generate a low IES, which
reduces the timing premium as shown in Epstein et al. (2014). Panel C in Table
1.6 shows that low consumption habits and higher values of ψ increase the timing
premium. For instance, we find that the timing premium is 45% with b = 0 and
ψ= 0.75.
Thus, the labor margin, consumption habits, and a low estimate of ψ help to
generate a low timing premium in the New Keynesian model.
1.4.3.2 The Extended Utility Kernel
We next let u0 be a free parameter and estimate the New Keynesian model when
conditioning on a RRA of 5. The fourth column in Table 1.4 shows that u0 = −939
and with a standard error of 216. Hence, we clearly reject the null hypothesis of
u0 = 0 (t-statistic = −4.35) and therefore the standard utility kernel. This means
that accounting for other aspects than consumption and leisure when modeling
household utility also helps the New Keynesian model to explain postwar U.S. data.
The estimate of u0 is clearly larger (in absolute terms) than any of the estimates of u0
in the long-run risk model, but such a direct comparison is not particularly useful
because of the structural differences between the two models. For instance, the New
Keynesian model implies Css = 0.80, includes habits, and gives a substantial utility
contribution from leisure (as ϕ0 = 41.49 to match Lss ), whereas the long-run risk
model has Css = 1 and abstracts from both habits and leisure. Instead, it is much more
informative to study the value of uC(Ct ,Lt
)Ct /u
(Ct ,Lt
)∣∣∣ss
, because both models
determine u0 from this ratio to attain a given level of RRA. Table 1.4 shows that
our large estimate of u0 gives a fairly low value of uC(Ct ,Lt
)Ct /u
(Ct ,Lt
)∣∣∣ss=−0.10,
which is remarkably close to the corresponding ratio in the long run risk model,
which is 0.08 with IES = 1.5 and RRA = 10. Thus, the large estimate of u0 in the New
Keynesian model is in this sense in line with our results for the long-run risk model.
We generally find small effects on most of the structural parameters from includ-
ing u0. The main exceptions are smaller consumption habits (b = 0.49), a reduction
in the amount of long-run productivity risk (ρZ and σZ fall), and more risk related to
stationary productivity shocks (ρA and σA increase). We see also find a large increase
1.4. A NEW KEYNESIAN MODEL 29
Table 1.6: The New Keynesian Model: Analysis of Timing PremiumIn Panel A, the timing premium is computed for different values of ϕ and a RRA = 60, while the remainingparameters are as reported in column (3) of Table 1.4. In Panel B, the timing premium is computed fordifferent values ofϕ and with a constant timing attitude ofα=−36.42, while the remaining parameters areas reported in column (3) of Table 1.4. In Panel C, the timing premium is computed for different values ofψand b, while all the remaining parameters are as reported in column (3) of Table 1.4. The timing premiumis computed based on (1.12) and a third-order perturbation approximation, while the utility level whenuncertainty is resolved in the following period is computed by simulation using anti-thetic sampling with5,000 draws and 10,000 terms to approximate the lifetime utility stream.
Panel A: RRA = 60
ϕ= 0.75 ϕ= 0.50 ϕ= 0.25 ϕ= 0.10Πss 94% 35% 10% 0.0%α −267.34 −94.20 −36.42 −17.06std
(∆ct
)1.615 1.25 1.44 2.80
std(logLt
)14.52 3.95 2.45 7.84
φ0 81.475 61.96 27.25 2.32
Panel B:α=-36.42
ϕ= 0.75 ϕ= 0.50 ϕ= 0.25 ϕ= 0.10Πss 20% 11% 10% 0.0%RRA 8.91 23.96 60 123.21std
(∆ct
)1.386 1.41 1.44 2.31
std(logLt
)3.247 2.62 2.45 7.98
φ0 81.475 61.96 27.25 2.32
Panel C:
ψ= 0.25 ψ= ψ ψ= 0.5 ψ= 0.75b = 0 15% 23% 27% 45%b = 0.25 13% 18% 21% 34%b = 0.5 11% 14% 15% 22%b = b 9% 10% 10% 9%
in the timing attitude, as α increases from −1.3 to −28.4 when RRA = 5. However, this
increase does not generate a substantially higher timing premium, which remains
low at 6% with RRA = 5.
Table 1.5 shows that including u0 in the New Keynesian model enables the model
to match all means and standard deviations, except for the labor supply that displays
the same degree of variability as in the standard New Keynesian model with RRA =
60. Subject to this qualification, the New Keynesian model now explains the equity
premium with a low RRA = 5 and a low timing premium of 6%. The model also
matches the mean and the standard deviation of the 10-year nominal term premium,
implying that we also explain the bond premium puzzle with low RRA and low timing
premium. The auto- and contemporaneous correlations are also well matched, and
the proposed extension of the New Keynesian model therefore has better overall
30 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
fit with Q scaled = 0.26 compared to Q scaled = 0.34 for the standard New Keynesian
model with RRA = 60.
The final two columns of Table 1.4 and 1.5 study the effects of higher RRA when
allowing for an unrestricted timing attitude α through u0. We find that higher RRA
does not improve the performance of the New Keynesian model. Actually, its perfor-
mance worsens slightly with Q scaled increasing from 0.26 to 0.31 when changing RRA
from 5 to 60. This suggests that it is not the high RRA in the traditional formulation
of Epstein-Zin-Weil preferences that helps the New Keynesian model match asset
prices, but instead the high timing attitude α that is induced by the high RRA.14
1.4.4 The Key Mechanisms
We next run three experiments to explore some of the key mechanisms in the New
Keynesian model with the extended utility kernel in (1.9). The first experiment con-
sidered in Table 1.7 illustrates the implications of gradually increasing u0. As for the
long-run risk model, a numerically larger value of u0 lowers u′ (Ct)
Ct /u(Ct
)and
allows for strong preferences for early resolution of uncertainty through a high α
without affecting RRA. The large value ofα then amplifies the existing risk corrections
and enables the model to explain asset prices with low RRA.
Our second experiment abstracts from long-run productivity risk by lettingσZ = 0.
The fourth column in Table 1.7 shows that this modification has very large effects as
the model now is unable to explain both the level and variability of πt , it , i (40)t , and
Ψ(40)t . Thus, long-run risk is also an essential feature of the New Keynesian model.
Our final experiment omits Epstein-Zin-Weil preferences by letting α= 0 to make
the household indifferent between early and late resolution of uncertainty. Although
this modification only has a small effect on RRA (reducing it from 5 to 2.2) it neverthe-
less has a profound impact on the model, which largely displays the same properties
as when omitting long-run productivity risk. In other words, the New Keynesian
model is unable to explain asset prices without Epstein-Zin-Weil preferences, and
hence strong preferences for early resolution of uncertainty.
Thus, we confirm the result from the long-run risk model, namely that the main
effect of Epstein-Zin-Weil preferences with our extended utility kernel is not to sepa-
rate the IES from RRA but instead to introduce strong preferences for early resolution
of uncertainty. This finding also helps to clarify why consumption habits may struggle
to match asset prices in DSGE models, although they allow for additional flexibility
in setting the IES and RRA (see Rudebusch and Swanson, 2008). The reason being
that consumption habits do not introduce preferences for early resolution of uncer-
tainty, which we find are essential to explain asset prices in a standard New Keynesian
model.
14The accuracy of the third-order perturbation solution used to estimate the New Keynesian model isdiscussed in Appendix A.4.
1.5. CONCLUSION 31
Table 1.7: The New Keynesian Model: Analyzing the Key MechanismsAll moments are computed using a third-order perturbation and represented as in Table 1.5. Unless statedotherwise, all parameters attain the estimated values from column (4) in Table 1.4.
(1) (2) (3) (4) (5)u0 = 0 u0 =−450 u0 = u0 σZ = 0 α= 0
Means∆ct 1.970 1.970 1.970 1.970 1.970πt 20.674 12.355 3.792 21.856 23.751it 29.230 17.378 5.178 30.913 33.617i (40)
t 30.441 18.649 6.512 32.111 34.198Ψ(40)
t 1.549 1.608 1.672 1.535 0.918logLt -1.074 -1.077 -1.080 -1.074 -1.073r m
t 9.681 7.183 4.669 9.956 10.444
Stds∆ct 3.170 2.622 2.146 2.849 3.372πt 3.407 2.535 2.273 2.746 4.494it 4.236 2.852 2.374 3.390 5.847i (40)
t 4.040 2.856 2.360 3.315 5.138Ψ(40)
t 0.831 0.963 1.000 0.936 0.466logLt 12.475 6.632 2.506 13.269 14.751
MemoRRA 5 5 5 5 2.2Πss 1% 3% 6% 0.0% 0.0%α −1.685 −14.51 −28.43 −28.43 0
1.5 Conclusion
The present paper highlights the importance of the timing attitude for consumption-
based asset pricing. To isolate the effects of the timing attitude, we propose a slightly
more general formulation of Epstein-Zin-Weil preferences than considered previously
to disentangle the timing attitude from the IES and RRA. We then show that this
extension enables us to explain several asset pricing puzzles in both endowment and
production economies. In particularly, we resolve a puzzle in the long-run risk model
where consumption growth is too highly correlated with the price-dividend ratio and
the risk-free rate. We also resolve the need for high RRA in DSGE models by enabling
an otherwise standard New Keynesian model to match the equity premium and the
bond premium with a low RRA of 5. Our analysis also reveals that the reason Epstein-
Zin-Weil preferences help to explain asset prices, is not because they separate the
IES from RRA, but because they introduce strong preferences for early resolution of
uncertainty in the presence of long-run risk.
32 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
Acknowledgements
We thank Ravi Bansal, John Cochrane, Mette Trier Damgaard, Wouter den Haan,
James D. Hamilton, Alexander Meyer-Gohde, Claus Munk, Olaf Posch, Morten Ravn,
and Eric Swanson for useful comments and discussions. We acknowledge access to
computer facilities provided by the Danish Center for Scientific Computing (DCSC).
We acknowledge support from CREATES - Center for Research in Econometric Analy-
sis of Time Series (DNRF78), funded by the Danish National Research Foundation.
1.6. REFERENCES 33
1.6 References
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state space system for non-linear dsge models: Theory and empirical applications
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34 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
Pohl, W., Schmedders, K., Wilms, O., 2018. Higher-order effects in asset pricing models
with long-run risks. Journal of Finance Forthcoming.
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nomics Vol. 1, 29–42.
A.1. THE LONG-RUN RISK MODEL: A PERTURBATION APPROXIMATION 35
Appendix
A.1 The Long-Run Risk Model: A Perturbation Approximation
Proposition A.1.1. The second-order approximation to evt ≡ logEt
e(1−α)
(vt+1+
(1− 1
ψ
)logµz,t+1
)with µz,t ≡ Zt /Zt−1 around the steady state is given by
evt = evss +ev c ct +evx xt + 1
2ev c c c2
t +1
2evxx x2
t +ev cx ct xt + 1
2evσσ,
where
evss = (1−α)
log
(∣∣∣∣∣u0 + 11− 1
ψ
∣∣∣∣∣)− log
(1−κ0
)+ (1− 1
ψ
)logµz
ev c = (1−α)ρc
1−κ01−κ0ρc
1− 1ψ
1+u0
(1− 1
ψ
)evx = 1−α
1−κ0ρx
(1− 1
ψ
)ev c c = (1−α)ρ2
c1−κ0
1−κ0ρ2c
(1− 1
ψ
)2
1+u0
(1− 1
ψ
) − (1−α)ρ2c
[1− 1
ψ
1+u0
(1− 1
ψ
) 1−κ01−κ0ρc
]2
evxx = (1−α)ρ2x
κ0
1−κ0ρ2x
1−κ0
(1−κ0ρx )2
(1− 1
ψ
)2
ev cx = (1−α)ρxρc vxc
evσσ = 1−α1−κ0
[v c cσ
2c + (1−α) v2
cσ2c + vxxσ
2x + (1−α) v2
xσ2x + (1−α)
(1− 1
ψ
)2σ2
z
]Proposition A.1.2. The second-order approximation to the risk-free rate r f
t and the
expected equity return r m,et around the steady state are given by
r ft = rss + r c ct + rx xt + 1
2 r fσσ
r m,et = rss + r c ct + rx xt + 1
2 r m,eσσ
where
rss = − logβ+ 1ψ logµz
r c = −(1−ρc
) 1ψ
rx = 1ψ
r fσσ = −αv2
xσ2x −
[1− (1−α)
(1− 1
ψ
)(1+ 1
ψ
)]σ2
z −(
1ψ2 + 1
ψ2αv c +αv2c
)σ2
c
r m,eσσ = −(
1−κ1)
pdσσ+κ1
(pdc c +pd 2
c
)σ2
c +κ1
(pdxx +pd 2
x
)σ2
x +σ2d
with κ1 ≡ epdss
1+epdss.
Proposition A.1.3. The second-order approximation to the log-transformed price-
dividend ratio pdt around the steady state is given by
pdt = pdss +pdc ct +pdx xt + 1
2pdc c c2
t +1
2pdxx x2
t +pdcx ct xt + 1
2pdσσ,
36 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
where
pdss = log κ11−κ1
pdc = φc+(1−ρc ) 1ψ
1−κ1ρc
pdx = φx− 1ψ
1−κ1ρx
pdc c = −pd 2c +
2κ1ρc (1−ρc ) 1ψ+φc+φcκ1ρc
1−κ1ρcpdc −
(1−ρ2
c
)1ψ2 −φc (1−ρc ) 1
ψ−2(1−ρc ) 1ψ2
1−κ1ρ2c
pdxx = −pd 2x +
(φx− 1
ψ
)2
1−κ1ρ2x+2κ1ρx
φx− 1ψ
1−κ1ρ2x
pdx
pdcx = −pdc pdx +κ1ρc pdc
(φx− 1
ψ
)1−κ1ρcρx
+(
1ψ (1−ρc )+φc
)(φx− 1
ψ+ρxκ1pdx
)1−κ1ρcρx
pdσσ = σ2d
1−κ1+ σ2
z1−κ1
[α+ (1−α) 1
ψ2
]+ σ2
c1−κ1
[αv2
c −2ακ1pdc v c ++κ1pdc c +κ1pd 2c +2α 1
ψv c −2κ1pdc1ψ + 1
ψ2
]+ σ2
x1−κ1
[αv2
x −2ακ1pdx vx +κ1pdxx +κ1pd 2x
]with κ1 ≡ epdss
1+epdss.
A.2 The Long-Run Risk Model: Second-Order ProjectionApproximation
The long-run risk model may be summarized by the following four equilibrium
equations:
Vt = u0 + 1
1− 1ψ
C1− 1
ψ
t +βEV1
1−αt
EV t = Et
[V 1−α
t+1 µ
(1− 1
ψ
)(1−α)
z,t+1
]
1 = Et
β
(EV t
) 11−α
Vt+1µ−
(1− 1
ψ
)z,t+1
α (
Ct+1
Ct
)− 1ψ
µ− 1ψ
z,t+1R ft
(P/D
)t = Et
β
(EV t
) 11−α
Vt+1µ−
(1− 1
ψ
)z,t+1
α (
Ct+1
Ct
)− 1ψ
µ− 1ψ
z,t+1
((P/D
)t+1µd ,t+1 +µd ,t+1
)
as the market return is given by Rmt =
((P/D
)t +1
)1
(P/D)t−1µd ,t . Here, EVt ≡ Et
[V 1−α
t+1
],
Vt ≡Vt /Z1− 1
ψ
t , and EV t ≡ EVt /Z
(1− 1
ψ
)(1−α)
t . We consider a second-order log-approximation
to the four control variables in the model, i.e. vt = g v0 +gv
s st + 12 s′t gv
ssst , ev t = g ev0 +
A.2. THE LONG-RUN RISK MODEL: SECOND-ORDER PROJECTION APPROXIMATION 37
gevs st + 1
2 s′t gevss st , rt = g r
0 +grs st + 1
2 s′t grssst , and pdt = g pd
0 +gpds st + 1
2 s′t gpdss st , where
vt ≡ log Vt , ev t ≡ log EV t , rt ≡ logR ft , and pdt ≡ log
(P/D
)t . The law of motion for
the states is known and given by ct+1
xt+1
σ2t+1
︸ ︷︷ ︸
st+1
=
0
0
1−ρσ
︸ ︷︷ ︸
h0
+
ρc 0 0
0 ρx 0
0 0 ρσ
︸ ︷︷ ︸
hs
ct
xt
σ2t
︸ ︷︷ ︸
st
+
σcσ+t 0 0
0 σxσ+t 0
0 0 σσ
︸ ︷︷ ︸
ηt
εc,t+1
εx,t+1
εσ,t+1
︸ ︷︷ ︸
εt+1
m
st+1 = h0 +hsst +ηtεt+1, (A.1)
where st is a matrix of size ns ×1 and σ+t ≡
√max
(σ2
t ,0). Below, we use the notation[
g vs
(1, c
)g v
s
(1, x
)g v
s
(1,σ2
) ]to index the elements in gv
s and similar for gevs , gr
s ,
and gpds . Also, gv
ss(c, c) denotes the element on the first row and first column of the
matrix gvss, and so forth. To derive the approximation, we exploit the following result
which we prove in the online appendix:
Proposition A.2.1. Let a ∈R, b be an 1×ns matrix, and C a symmetric ns ×ns matrix.
Given (A.1), we then have that
Et
[exp
{a +bst+1 +s′t+1Cst+1
}]= exp
{a +bh0 +h′
0Ch0 +(2h′
0Chs +bhs)
st +s′t h′sChsst
}×exp
{1
2
(bηt +2h′
0Cηt +2s′th′sCηt
)(I−2η′t Cηt
)−1 (bηt +2h′
0Cηt +2s′th′sCηt
)′}×
∣∣∣(I−2η′t Cηt
)∣∣∣− 12
The projection approximation can be implemented sequentially by first obtaining
vt and then ev t , afterwhich rt and pdt are easily computed using the expressions for
vt and ev t . To conserve space, we only show how to solve for vt , as the remaining
three controls variables are obtained in a similar way. We first note that the expression
for the scaled value function reads
Vt = u0 + 1
1− 1ψ
C1− 1
ψ
t +βEt
[V 1−α
t+1 µ
(1− 1
ψ
)(1−α)
z,t+1
] 11−α
m
38 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
exp{
v(st
)} = u0 + 1
1− 1ψ
exp
(
1− 1
ψ
)ct
+βEt
exp{
(1−α) v(st+1
)}exp
(
1− 1
ψ
)(1−α) logµz,t+1
11−α
,
where v(st
) ≡ logV(st
). Due to the independence of the shocks, it is possible to
integrate out εz,t+1 manually as we have
exp{
v(st
)} = u0 + 1
1− 1ψ
exp
(
1− 1
ψ
)ct
+β{Et
[exp
{v
(st+1
)}(1−α)]
exp
(
1− 1
ψ
)(logµz +xt
)1−α
×exp
1
2
(1− 1
ψ
)2
(1−α)2σ2z
(σ+
t
)2
}1
1−α ,
mexp
{v
(st
)}= u0 + 11− 1
ψ
exp
{(1− 1
ψ
)ct
}
+βEt
[exp
{v
(st+1
)+ (1− 1
ψ
)(logµz +xt
)+ 12
(1− 1
ψ
)2(1−α)σ2
z
(σ+
t
)2}1−α] 1
1−α.
To avoid numerical overflow of exp{·}(1−α), given the large values of v(st+1
)and
α, we scale this term by Vt . That is,
exp{
v(st
)}= u0 + 11− 1
ψ
exp
{(1− 1
ψ
)ct
}
+VtβEt
exp
{v(st+1)+
(1− 1
ψ
)(logµz+xt )+ 1
2
(1− 1
ψ
)2(1−α)σ2
z
(σ+
t
)2}
Vt
1−α
11−α
.
Focusing on the last term we have
βEt
(exp
{−vt + vt+1 +
(1− 1
ψ
)(logµz +xt
)+ 12
(1− 1
ψ
)2(1−α)σ2
z
(σ+
t
)2})1−α 1
1−α
=βEt [exp{−v(st
)(1−α)+
(g v
0 +gvs st+1 + 1
2 s′t+1gvssst+1
)(1−α)
+(1− 1
ψ
)(1−α)
(logµz +xt
)+ 12
(1− 1
ψ
)2(1−α)2σ2
z
(σ+
t
)2}]1
1−α
=βexp
{(1− 1
ψ
)(logµz +xt
)+ 12
(1− 1
ψ
)2(1−α)σ2
z
(σ+
t
)2}
×Et
exp
{((1−α)
(g v
0 − v(st
))+ (1−α)gvs st+1 +s′t+1
(1−α)gvss
2 st+1
)}1/(1−α)
A.2. THE LONG-RUN RISK MODEL: SECOND-ORDER PROJECTION APPROXIMATION 39
To apply Proposition A.2.1, let a ≡ (1−α)(g v
0 − v(st
)), b ≡ (1−α)gv
s , and C ≡(1−α)gv
ss2 . This implies
Et
exp
{((1−α)
(g v
0 − v(st
))+ (1−α)gvs st+1 +s′t+1
(1−α)gvss
2 st+1
)}= exp
{(1−α)
(g v
0 − v(st
)+gvs h0 +h′
0gv
ss2 h0 +
(h′
0gvsshs +gv
s hs
)st +s′t h′
sgv
ss2 hsst
)}×exp{ 1
2 (1−α)2(gv
sηt +h′0gv
ssηt +s′th′sgv
ssηt
)(I−η′t (1−α)gv
ssηt
)−1
×(gv
sηt +h′0gv
ssηt +s′th′sgv
ssηt
)′}
×∣∣∣∣(I−η′t (1−α)gv
ssηt
)∣∣∣∣− 12
.
Hence, the Euler residuals for the log-transform value function R v(st
)reads
R v(st
)=−exp{
g v0 +gv
s st + 12 s′t gv
ssst
}+u0 + 1
1− 1ψ
exp
{(1− 1
ψ
)ct
}+βVt exp
{(1− 1
ψ
)(logµz +xt
)+ 12
(1− 1
ψ
)2(1−α)σ2
z
(σ+
t
)2}
×exp
{(g v
0 − v(st
))+gvs h0 +h′
0gv
ss2 h0 +
(h′
0gvsshs +gv
s hs
)st +s′t h′
sgv
ss2 hsst
}×exp{ (1−α)
2
(gv
sηt +h′0gv
ssηt +s′th′sgv
ssηt
)×
(I−η′t (1−α)gv
ssηt
)−1 (gv
sηt +h′0gv
ssηt +s′th′sgv
ssηt
)′}
×∣∣∣∣(I−η′t (1−α)gv
ssηt
)∣∣∣∣− 12(1−α)
.
We then determine g v0 , gv
s , and gvss as follows:
• Construct a multi-dimensional grid for the states based on the Cartesian set
Ss ≡ Sc ×Sx ×Sσ2t.
• Generate Ns points{
sit
}Ns
i=1from the set Ss.
• Determine g v0 , gv
s , and gvss by solving the nonlinear least squares problem,(
g v0 ,gv
s ,gvss
)= argmin
Ns∑i=1
(R v
(si
t
))2
.
The grid for the state variables Ss is constructed using 10 points uniformly distributed
along each dimension, implying Ns = 1,000. The upper and lower bounds along each
dimension is determined following a simulation of the states to cover the maximum
and minimum levels. We evaluate R v(st
)across all Ns points simultaneously by
using a vectorized implementation in MATLAB, where the symbolic toolbox is used to
analytically compute the matrix products, matrix inversions, and determinants in
the expression for R v(st
).
40 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
A.3 The Long-Run Risk Model: Accuracy of Solution
This section evaluates the accuracy of the adopted second-order projection approxi-
mation for each of the eight estimated versions of the long-run risk model in Table
1.1. The performance of this approximation is benchmarked to the widely used log-
normal method, a first-order projection solution, and a highly accurate fifth-order
projection solution. As in Pohl et al. (2018), we focus on means and standard devia-
tions for pdt , r ft , and r m
t , because these moments are most sensitive to the adopted
approximation method. The results are summarized in Table A.1, where we highlight
the following results. First, the log-normal method generally underpredicts E[pdt
],
generates too high values of E[
r mt
], and overpredicts the variability in pdt . Hence,
we reproduce the key findings of Pohl et al. (2018) on our estimated models. Second,
a first-order projection solution generally implies that these errors go in the opposite
direction, as it overpredicts E[pdt
]and underpredicts E
[r m
t
]. Third, the proposed
second-order projection solution displays no systematic biases and produces mo-
ments that are nearly identical to those from the fifth-order projection solution. The
main exception is for the extended model with IES = 1.1 and RRA = 10, where we see
somewhat larger deviations.
A.4 The New Keynesian Model: Accuracy of Solution
We evaluate the accuracy of the adopted third-order perturbation approximation by
computing unit-free Euler-equation errors on a grid of 1,000 points. The accuracy
of this solution is benchmarked to a standard first-order approximation and a fifth-
order approximation using the codes of Levintal (2017). Table A.2 reports the root
mean squared Euler-equation errors (RMSEs) for the six estimated versions of the
New Keynesian model in Table 1.4. We generally find that a third-order approxima-
tion improves the accuracy of the linearized solution, both for the Euler-equations
relating to the macro part of the model and for the 40 Euler-equations describing
bond prices. This improvement is particularly evident for bond prices. Increasing
the approximation order from three to five provides only a small improvement to
the macro part of the model when RRA equals 10 and 60, while accuracy actually
deteriorates slightly for RRA = 5. We find even smaller effects on bond prices of going
from third to fifth order, where accuracy only increases for the benchmark model
with RRA = 60 and the extended model with RRA = 10. Thus, these results indicate
that little would be gained by considering a fifth-order approximation. However,
going to fifth order is computationally much more demanding than the adopted
third-order approximation and would therefore not make a formal estimation of the
New Keynesian model feasible.
A.4. THE NEW KEYNESIAN MODEL: ACCURACY OF SOLUTION 41
Table A.1: The Long-Run Risk Model: Accuracy of MomentsThis table reports unconditional moments for the eight estimated versions of the long-run risk model
in Table 1.1 when using the log-normal method as well as a first-, second-, and fifth-order projection
solution with log-transformed variables. The projection approximations are computed by minimizing
the squared Euler-equation errors on a grid of 1,000 points, with 10 points uniformally distributed along
each dimension between its maximum and minimum level in a simulated sample of 250,000 observations.
The fifth-order projection solution is computed using complete Chebyshev polynomials. The log-normal
method is implemented using a first-order projection approximation of the value function and the tradi-
tional log-linear approximation of the price-dividend ratio at the unconditional mean of the price-dividend
level, which is obtained by iterating on the approximated loadings.
IES RRA = 5 RRA = 10
Means Stds Means Stds
pdt r ft r m
t pdt r ft r m
t pdt r ft r m
t pdt r ft r m
tBenchmark Model: 1.5
Log-normal method 3.27 1.83 6.18 0.52 1.15 15.87 3.12 1.95 6.81 0.42 0.75 15.83
1st order 3.76 1.83 4.87 0.38 1.15 13.61 3.58 1.95 5.32 0.32 0.75 14.13
2nd order 3.49 1.84 5.70 0.42 1.14 14.10 3.30 1.96 6.32 0.34 0.75 14.48
5th order 3.49 1.84 5.72 0.44 1.14 14.47 3.31 1.96 6.28 0.36 0.75 14.89
Extended Model: 1.1
Log-normal method 3.12 2.16 6.87 0.28 0.70 15.30 2.74 1.56 8.86 0.28 0.59 14.35
1st order 3.80 2.16 4.72 0.26 0.70 15.39 3.59 1.56 5.23 0.27 0.59 15.38
2nd order 3.28 2.16 6.32 0.26 0.70 14.83 3.29 1.68 6.22 0.26 0.59 14.77
5th order 3.27 2.16 6.36 0.27 0.70 15.08 4.26 1.67 3.83 0.29 0.59 16.99
Extended Model: 1.5
Log-normal method 3.05 1.63 7.14 0.32 0.50 15.20 3.05 1.64 7.12 0.32 0.50 15.21
1st order 3.81 1.63 4.68 0.28 0.50 15.69 3.81 1.64 4.68 0.28 0.50 15.69
2nd order 3.29 1.69 6.26 0.28 0.50 14.80 3.29 1.71 6.28 0.29 0.50 14.78
5th order 3.28 1.69 6.33 0.29 0.49 15.00 3.28 1.71 6.33 0.29 0.50 15.00
Extended Model: 2.0
Log-normal method 3.06 1.52 7.08 0.36 0.47 15.42 3.06 1.58 7.11 0.35 0.48 15.50
1st order 3.66 1.52 5.06 0.29 0.47 15.16 3.64 1.58 5.10 0.28 0.48 14.89
2nd order 3.28 1.59 6.30 0.30 0.45 14.73 3.28 1.63 6.33 0.30 0.47 14.69
5th order 3.29 1.59 6.30 0.31 0.45 15.00 3.28 1.63 6.32 0.31 0.47 15.02
42 CHAPTER 1. TIMING ATTITUDES IN ASSET PRICING MODELS
Table A.2: The New Keynesian Model: Euler-Equation ErrorsThis table reports the root mean squared unit-free Euler-equation errors (RMSEs) on a grid of 1,000 pointsfor a first-, third-, and fifth-order perturbation approximation. The grid is constructed by considering 10points uniformly between −2×σx,i and 2×σx,i for each state dimension, whereσx,i denotes the standarddeviation of the i ’th state in a log-linearized solution. Conditional expectations in the Euler-equationsare evaluated by Gauss-Hermite quadratures using 7 points. The considered model parameters are thosereported in Table 1.4. The RMSEs to the 12 equations describing the model without bond prices aresummarized under the label ’Macro Part’, while the RMSEs the 40 equations describing all bond prices aresummarized under the label ’Bond Prices’. The label ’Total’ refers to the RMSEs for the entire model.
Benchmark Model Extended Model(1) (2) (3) (4) (5) (6)
RRA=5 RRA=10 RRA=60 RRA=5 RRA = 10 RRA = 60
Macro Part:1st order 0.0253 0.0275 0.0703 0.2756 0.4170 0.28893rd order 0.1163 0.0565 0.0215 0.1375 0.0579 0.01965th order 0.1274 0.0474 0.0182 0.1525 0.0508 0.0149
Bonds Prices:1st order 0.0426 0.0466 0.1197 0.4717 0.7125 0.49213rd order 0.0013 0.0014 0.0046 0.0014 0.0016 0.00215th order 0.0076 0.0020 0.0033 0.0056 0.0014 0.0038
Total:1st order 0.0382 0.0418 0.1072 0.4224 0.6381 0.44083rd order 0.0543 0.0264 0.0108 0.0642 0.0271 0.00935th order 0.0599 0.0222 0.0089 0.0714 0.0237 0.0077
C H A P T E R 2HOW LEARNING FROM MACROECONOMIC
EXPERIENCES SHAPES THE YIELD CURVE
Kasper JørgensenAarhus University and CREATES
Abstract
I link constant-gain learning expectations of inflation and consumption growth to the
long-run variation in the level and slope of the U.S. Treasury yield curve, respectively.
The variation in yields that is orthogonal to the consumption-based equilibrium
factors has a two-factor structure with cyclical level and slope factor interpretation.
The four factors predict excess returns with R2’s up to 56%, and subsume and add to
the predictive information in the most popular bond return predictors. My four-factor
model implies cyclical term premia, because the macroeconomic expectations drive
time-variation in long-run short rate expectations that captures the trend component
of long-term yields. The cyclicality of term premia contrasts the implications of the
workhorse affine term structure model.
43
44 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
2.1 Introduction
Accurate decompositions of long-term nominal yields into expected short rates and
term premia are crucial for understanding the expected returns in bond markets, the
conventional-, and unconventional monetary policy pass-through. The central ques-
tion is what information should go into this decomposition? I address this question
in the present paper.
I uncover a novel factor structure in U.S. Treasury bonds. Two key term structure
factors have a clear equilibrium-based interpretation. Consumption-based asset
pricing models generically imply that time-varying consumption growth and inflation
expectations are nominal yield curve factors. My main contribution is to show that
expected inflation and consumption growth capture the most persistent component
— the trend component — of yield curve level and slope, respectively. Importantly,
the macroeconomic expectations drive time-variation in long-horizon expected
short rates. Long-horizon short rate expectations capture crucial variation in the
expectation hypothesis component of the yield curve. Controlling for long-horizon
short rate expectations, I extract yield curve level and slope cycles. The two cycle
factors capture the least persistent variation in the level and slope of yields, and are
important sources of risk premium variation. The dynamic trend and cycle distinction
in yield curve level and slope greatly improves the measure of bond risk premia.
My work builds on the macro-finance literature studying bond risk premia. A
recent literature has challenged the conventional wisdom that the current yield curve
spans all relevant information for estimating risk premia.1 This paper is most closely
related to the work of Cieslak and Povala (2015) and Bauer and Rudebusch (2017a).
Cieslak and Povala (2015) control for expected inflation and extract a measure of
bond risk premia. I show that further controlling for expected consumption growth
improves upon the bond risk premia measure, because it captures additional vari-
ation in long-run real rate expectations. Bauer and Rudebusch (2017a) control for
expected inflation and extract additional variation in the long-run real rate — the
equilibrium real rate — from the yield curve. I relate this equilibrium real rate varia-
tion to underlying macroeconomic fundamentals and provide an interpretation of
the apparent decline in the equilibrium real rate.
I start from a simple consumption-based argument. Underlying a bond investor’s
behavior is the economics of the fundamental consumption-savings trade-off. High
expected consumption tomorrow disincentivizes real savings, since the expected
marginal utility from consumption tomorrow is comparably low. Equilibrium bond
prices exactly balances this trade-off. Thus, expected consumption growth is an
important factor of equilibrium bond prices. As bonds are nominal, expected inflation
1The macroeconomic variables that have been found to have additional information about bondrisk include the output gap (Cooper and Priestley, 2008), factors from a large macroeconomic data set(Ludvigson and Ng, 2009), Treasury bond supply (Greenwood and Vayanos, 2014), economic activity andinflation (Joslin, Priebsch, and Singleton, 2014), and trend inflation (Cieslak and Povala, 2015).
2.1. INTRODUCTION 45
is another important factor. Beyond the macroeconomic expectations, other factors
affecting the marginal utility of consumption are potentially priced into the term
structure.
Macroeconomic expectations are not directly observable, thus a modeling choice
is needed. One approach is to use survey responses. Piazzesi, Salomao, and Schneider
(2015), Buraschi, Piatti, and Whelan (2017), Cieslak (2017) document that consensus
survey responses are biased forecasts of bond market variables. This is consistent
with the interpretation that the consensus survey response is unrepresentative of
the marginal bond investor’s expectations. Therefore, I pursue a different approach.
Nagel and Malmendier (2016) provide a natural micro-foundation for constant-gain
learning expectations based on a learning from experiences argument. People are un-
certain about the true data generating process, but learn from their experiences. That
is, people overweight macroeconomic data experienced over their lifetime compared
to the macroeconomic history they did not experience. This implies that macroe-
conomic history is down-weighted as new generations emerge and old generations
pass.
Empirically, constant-gain learning expectations for consumption growth and
inflation are significant factors for 1- through 10-year U.S. Treasury bonds. Across
the maturity spectrum, the macroeconomic expectations capture 80-89% of the total
variation in yields from 1971-2015. Expected inflation loads almost equally on short-
and long-maturity yields. This is consistent with the common interpretation of a level
factor. Expected consumption growth loads heavily on short-maturity yields and less
on longer-maturity yields. This is consistent with the common interpretation of a
slope factor. I follow Cieslak and Povala (2015) and extract maturity-specific yield
curve cycles as the variation in yields orthogonal to the macroeconomic expectations.
The maturity-specific cycle factors have a two-factor structure. These two yield curve
cycle factors also admit an interpretation of level and slope factors, respectively. In
fact, the four factors decompose the first two principal components of the yield
curve — that is, the usual measures of level and slope factors. Expected inflation
and consumption growth capture the trend component of the level and slope of the
yield curve, respectively. The two yield curve cycle factors capture less persistent
components of the level and slope of the yield curve. The four factors fully span the
conventional level and slope factors of the term structure, but provide an important
dynamic distinction. The macro-founded level and slope trends predominantly cap-
ture the expectation hypothesis component of long-term yields, whereas the cycle
factors capture risk premium variation.
Following Cochrane and Piazzesi (2005), I combine these four factors into a
single bond return predictor. Across maturities, I find that the predictor holds vast
information about future bond returns with R2’s as high as 56%. The identified bond
return predictor subsumes and improves upon the information in the first three
principal components of yields, the Cochrane and Piazzesi (2005) factor formed
46 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
from five forward rates, and the Cieslak and Povala (2015) interest cycle factor. This
added predictability is a robust feature of the data, as it holds (i) in- and out-of-
sample, (ii) when controlling for small-sample distortions, (iii) across sub-samples,
(iv) for different holding periods, (v) across data sets that differ in how zero-coupon
bonds are constructed, and (vi) for macroeconomic expectations computed from an
optimal-gain learning algorithm using the Kalman filter recursions.
Finally, I use the model to decompose the 10-year yield into term premia and
expected short rates. I benchmark the results against the workhorse affine term
structure model (Vasicek, 1977; Duffie and Kan, 1996; Dai and Singleton, 2000; Duffee,
2002). My model implies average expected short rates that move at a lower frequency,
whereas term premia move at a higher frequency. This is because — in my model
— the short rate reverts towards a moving long-run mean. The long-run nominal
short rate reverts to the sum of the equilibrium real rate and long-run inflation
expectations. Both the equilibrium real rate and long-run inflation expectations have
time-variation driven by the two macroeconomic expectations factors. I find that
the equilibrium real rate has declined in recent years. This is consistent with recent
evidence (Hamilton, Harris, Hatzius, and West, 2016; Holston, Laubach, and Williams,
2017; Bauer and Rudebusch, 2017a). In contrast, the affine term structure model
implies a fixed long-run nominal short rate. As a result, long-horizon expectations
display stronger mean reversion. Term premia, as the residual component, then
captures some of the trend component in the 10-year yield. A cyclical term premium
is consistent with the behaviour of risk premia in other asset classes (Fama and
French, 1989).
The paper proceeds as follows. Section 2.2 outlines an illustrative consumption-
based model of the nominal term structure. Section 2.3 shows that the identified
factors predict bond returns both in- and out-of-sample, and that they subsume the
information in the most popular predictors in the literature. Section 2.4 decomposes
long-maturity yields into short rate expectations and term premia, while Section
2.5 discusses the model-implied equilibrium real rate. Finally, Section 2.6 provides
concluding remarks.
2.2 An Illustrative Consumption-Based Model
I argue that expected consumption growth and inflation are two key factors that
help explain variation in the nominal term structure. The economic intuition comes
from a simple consumption-based equilibrium model, where the key ingredient is a
marginal bond investor that uses constant-gain learning to update macroeconomic
expectations.
2.2. AN ILLUSTRATIVE CONSUMPTION-BASED MODEL 47
2.2.1 The Consumption-Savings Trade-Off
The standard consumption-based asset pricing model with time-separable log utility
implies a nominal stochastic discount factor m$t ,t+1 given by
m$t ,t+1 = logδ−∆ct+1 −πt+1, (2.1)
where δ denotes a subjective discount factor, ∆ct+1 denotes consumption growth
and πt+1 denotes the net inflation rate. Consider the decomposition of consumption
growth and inflation into conditionally expected and unexpected components,
∆ct+1 = τc,t +εc,t+1
πt+1 = τπ,t +επ,t+1,(2.2)
where τc,t = Et[∆ct+1
], τπ,t = Et
[πt+1
]are expectations conditional on information
available to the marginal investor at time t and εc,t+1, επ,t+1 are forecast errors.
Equations (2.1) and (2.2) summarize the consumption-savings trade-off. In times of
high expected consumption growth postponing current consumption is less desirable,
since the marginal utility from consuming tomorrow is expected to be comparably
low. That is, saving in bonds is unattractive in times of high expected consumption
growth, and vice versa. Similarly, high expected inflation discourages savings in
nominally denoted bonds, since the real purchasing power of the promised payments
is expected to be low. These arguments imply that expected consumption growth τc,t
and τπ,t are important factors of equilibrium yield curves.
Of course, marginal utility from consumption might exhibit time-variation due
to other factors as well. Popular reasons are external habits as in Campbell and
Cochrane (1999) and Wachter (2006) or demand shocks as in Albuquerque, Eichen-
baum, and Rebelo (2016), Creal and Wu (2016), and Schorfheide, Song, and Yaron
(2017). Incorporating this feature implies a nominal stochastic discount factor given
by
m$t ,t+1 = logδ−τc,t −εc,t+1 −τπ,t −επ,t+1 +∆γt+1, (2.3)
where ∆γt+1 denotes demand shock growth, which in general can depend on a set of
additional factors Pt as well as expected macroeconomic conditions τt =[τc,t τπ,t
]′.The nominal stochastic discount factor implies that, for demand shock processes that
do not depend on expected consumption growth and inflation in an exactly offsetting
manner, the main consumption-savings intuition continues to hold true. Outside
of such knife-edge restrictions on demand shock growth, expected macroeconomic
conditions continue to be factors of equilibrium yields — and any other nominal
asset. Furthermore, the set of factors Pt reflecting additional time-variation in the
marginal utility from consumption is potentially priced into the term structure of
yields. That is, the equilibrium arguments suggest a yield curve factor structure given
by
y (n)t =An +Bτ,nτt +BP,nPt (2.4)
48 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
where y (n)t denotes the yield-to-maturity on an n-year bond.2
2.2.2 Learning from Macroeconomic Experiences
The marginal bond investor’s expectations of future consumption growth and in-
flation are not directly observable, and a modelling choice is therefore needed to
obtain these expectations. Here, I follow a large literature in macroeconomics and
finance that emphasizes informational frictions (Mankiw and Reis, 2002; Sims, 2003;
Woodford, 2003, among others). I construct the marginal investor’s macroeconomic
expectations using the popular constant-gain learning algorithm and let
τc,t+1 = τc,t +νcεc,t+1
τπ,t+1 = τπ,t +νπεπ,t+1,(2.5)
where εc,t+1 and επ,t+1 are the realized forecast errors from (2.2). The two constant-
gain parameters νc and νπ summarize how fast forecast errors are incorporated
into expected consumption growth and expected inflation, respectively. In this way,
the constant-gain learning algorithm captures the idea that the bond investor is
uncertain about the true data generating process and learns from his experiences, in
which case the algorithm has been shown to provide a robust optimal prediction rule
(Evans, Honkapohja, and Williams, 2010).
Nagel and Malmendier (2016) provide a natural micro-foundation for the constant-
gain learning algorithm. Individuals overweight consumption growth and inflation
experienced during their lifetimes compared to the macroeconomic history they did
not experience when forming expectations. This behavior implies that macroeco-
nomic history is down-weighted as new generations emerge and old generations
pass. Further, Nagel and Malmendier (2016) show that the average learning from
experience forecast matches closely that of the constant-gain learning algorithm.
Usually, constant-gain learning is motivated with structural shifts or other forms of
parameter instability. Instead, the learning from experiences motivation builds on
the psychological evidence on personal experience and availability bias. The psycho-
logical aspects of personal experiences and availability are discussed in e.g. Tversky
and Kahneman (1973).
2.3 Bond Return Predictability
In this section, I discuss the identification of the term structure factors and present
evidence on their information for current yields and expected bond returns. In partic-
ular, I examine how learning from macroeconomic experiences help measure bond
2I derive the equilibrium factor structure from the marginal bond investor’s optimization problemin the online appendix accompanying this paper. Further, I elaborate on the restrictions on the factorloadings An , Bτ,n , and BP,n that rule out arbitrage opportunities. The online appendix is available on mywebsite or upon request.
2.3. BOND RETURN PREDICTABILITY 49
risk premia. I use data on consumption of non-durables and service goods, core
CPI, and end-of-month unsmoothed Fama and Bliss (1987) yields for 1- through
10-year maturities. The data is for November 1971 through December 2014, giving a
total of T = 518 observations. I provide details on the data construction in the online
appendix.
2.3.1 Identification: Macroeconomic Expectations Factors
As described in Section 2.2, expected consumption growth and inflation are identified
solely from the macroeconomic data using the constant-gain learning algorithm. By
recursive substitution of (2.2) and (2.5), the constant-gain learning algorithm implies
τc,t = νc
t−1∑i=0
(1−νc
)i∆ct−i
τπ,t = νπt−1∑i=0
(1−νπ
)iπt−i .
(2.6)
In principle, the macroeconomic expectations thus depend on their entire history.
However, I follow Cieslak and Povala (2015) and truncate the sums in (2.6) after
K = 120 terms, i.e. only the most recent 10 years of data are used. As the distant
past is heavily downweighted, truncating the sums has no meaningful effect on the
measurement of τc,t and τπ,t . Varying K between 100 and 150 terms verifies that the
results are not sensitive to this choice.
For now, I fix the constant-gain parameters νc = νπ = 0.016, which is consistent
with typical parameter values considered in the literature. Piazzesi et al. (2015) use a
constant-gain parameter of 0.026, Kozicki and Tinsley (2001), Faust and Wright (2013),
and Cieslak and Povala (2015) use 0.013, and Orphanides and Williams (2005) and
Nagel and Malmendier (2016) use 0.005. Hence, my choice is well within the range
of often considered values. Further, Branch and Evans (2006) find that a common
constant-gain parameter close to zero provides the best forecasting rule for inflation
and real activity measured by GDP growth. The results I present here are not sensitive
to the equality constraint, nor to varying the constant-gain parameters between 0.005
and 0.05. Figure 2.1 plots the computed expectations in blue, the cross-section of
expectations from the survey of professional forecasters (SPF) in red, along with the
realized series in black.3 Consumption growth series are in the leftmost graph, and
inflation series are in the rightmost graph. For both series, the constant-gain learning
expectations capture the low-frequency movements in the underlying series.
The constant-gain learning expectations also align well with the cross-section
of expectations from the survey of professional forecasters. However, Figure 2.1 also
3For Figure 2.1, I have computed the constant-gain learning expectations using real personal con-sumption expenditures and all-item CPI inflation for comparison with the SPF responses. All remaininganalysis in this text is done using real per capita consumption of non-durables and service goods and coreCPI inflation.
50 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
Figure 2.1: Consumption Growth, Inflation, and Expectations
Consumption growth is the log growth rate of real personal consumption expenditures and inflationis the log growth rate of all-item CPI for the period 1971:11 - 2014:12. Both are sampled with monthlyfrequency. Survey expectations are from the survey of professional forecasters (SPF). Constant-gain
learning expectations for consumption growth and inflation are computed as τc,t = νc∑K
i=0
(1−νc
)i∆ct−1
and τπ,t = νπ∑Ki=0
(1−νπ
)iπt−i , respectively. Parameters are νc = νπ = 0.016 and K = 120. Shaded areas
indicate NBER recession dates.
highlights some limitations to using survey data directly. In particular, it is unclear
which moment of the cross-sectional distribution of survey responses that provides
a good proxy for the marginal bond investor’s expectations. Piazzesi et al. (2015),
Buraschi et al. (2017), and Cieslak (2017) document that the consensus survey re-
sponse implies repeated forecast errors for bond market variables, which is consistent
with the interpretation that the consensus survey response is unrepresentative for
the marginal bond investor’s expectations. Greenwood and Shleifer (2014) document
a similar result for stock market variables.
Although the marginal bond investor may have irrational conditional expecta-
tions, in the sense that the constant-gain learning forecasts for consumption growth
and inflation could be systematically wrong, I find that this is in fact not the case.
Regressions of consumption growth and inflation onto their perceived conditional
expectations, i.e.
∆ct+h = ρ(h)0,c +ρ(h)
1,c τc,t +εc,t+h
πt+h = ρ(h)0,π +ρ(h)
1,πτπ,t +επ,t+h ,(2.7)
show that the constant-gain learning algorithm is not significantly misspecified.
Across different forecasting horizons, the constants ρ(h)0,c and ρ(h)
0,π are never signifi-
2.3. BOND RETURN PREDICTABILITY 51
Table 2.1: Mincer and Zarnowitz (1969) Regressions
Results are for the regressions∆ct+h/12 = ρ(h)0,c +ρ(h)
1,c τc,t +εc,t+h/12 andπt+h/12 = ρ(h)0,π+ρ
(h)1,πτπ,t +επ,t+h/12.
Newey and West (1987) corrected t-statistics in absolute values using cei l(1.5×h
)lags for the null hypoth-
esisH0 : ρ(h)0,i = 0 andH0 : ρ(h)
1,i = 1 for i ∈ {c,π} are in parenthesis.
Panel A : ∆ct+h/12
h = 1 h = 3 h = 6 h = 12
ρ(h)0,c −0.0009
(0.3057)0.0014(0.3401)
0.0048(0.9303)
0.0104(1.6535)
ρ(h)1,c 1.1214
(0.6973)0.9761(0.0974)
0.7610(0.7828)
0.3966(1.7093)
R2 0.20 0.15 0.09 0.02
Panel B : πt+h/12
h = 1 h = 3 h = 6 h = 12
ρ(h)0,π −0.0022
(0.8651)−0.0012
(0.3370)0.0002(0.0368)
0.0028(0.4207)
ρ(h)1,π 1.1674
(1.7841)1.1394(1.0392)
1.0979(0.5602)
1.0225(0.0955)
R2 0.59 0.56 0.52 0.44
cantly different from zero, and the slope coefficients ρ(h)1,c and ρ(h)
1,π are never signif-
icantly different from one. Thus, in the sense of Mincer and Zarnowitz (1969), the
constant-gain learning forecasts are rational forecasts of consumption growth and
inflation. And so, the marginal bond investor has a weak form of rational expectations.
Further, the two macroeconomic expectations series combine to capture the
low-frequency movements in the yield curve very well. The top-left panel of Figure
2.2 depicts the level of the yield curve measured by a maturity-averaged yield, y t =1
10
∑10i=1 y(i)
t . Fitted values from the simple regression of the maturity-averaged yield
onto a constant and expected inflation are plotted in blue. Expected inflation captures
well the low-frequency variation in the level of the yield curve. The expected inflation
factor is statistically significant with a Newey and West (1987) t-statistic (corrected
using 18 lags) of 15.06 and explains 85% of the total variation.
The top-right panel of Figure 2.2 depicts the slope of the yield curve measured by
the 10-year yield spread, y (10)t − y (1)
t . Fitted values from the simple regression of the
10-year yield spread onto a constant and expected consumption growth are plotted
in blue. Expected consumption growth captures well the low-frequency variation in
the slope of the yield curve. The expected consumption growth factor is negatively
related to the slope of the yield curve, and is statistically significant with a Newey and
West (1987) t-statistic of -5.01. Expected consumption growth explains 22% of the
total variation in the 10-year yield spread.
The bottom panel of Figure 2.2 plots the factor loadings from the maturity-specfic
regressions
y (n)t =α(n) +β(n)
c τc,t +β(n)π τπ,t +e(n)
t , (2.8)
52 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
Figure 2.2: Fitted Expectations and Factor Loadings
Yields are unsmoothed Fama-Bliss yields for the period 1971:11 - 2014:12. The fitted expectations are
obtained as the fitted values from the regressions (i) yt =α+βτπ,t +et and (ii) y (10)t − y (1)
t =α+βτc,t +et , where yt = 1
10∑10
i=1 y(i)t . Shaded areas indicate NBER recession dates. Factor loadings are from the
regressions y (n)t =α(n) +β(n)
c τc,t +β(n)π τπ,t +e(n)
t .
as function of maturity. For n = 1,2, . . . ,10, the R2’s associated with the regressions
in (2.8) range between 80-89%. The loadings on the inflation expectation factor
range from 1.62 for the short maturities to 1.44 for the longest maturity. This is
consistent with the interpretation of inflation expectations as a level factor. Loadings
for consumption growth expectations are large at the short end (1.43) and smaller
at the long end (0.53), thus consistent with an interpretation of a slope factor. Both
macroeconomic expectation factors are statistically significant factors across all
maturities.
2.3.2 Identification: Yield Curve Cycle Factors
As the Pt factors are unobserved, I take them to be orthogonal to the macroeconomic
expectations. This implies that the factors can be extracted from the yield curve
information that is orthogonal to the macroeconomic expectations. This approach
is similar to Cieslak and Povala (2015). The cycle factors Pt are only identified up to
a rotation. I fix a particular rotation by assuming that the cycle factors are the first
nP principal components of the variation that is orthogonal to the macroeconomic
expectations. I simply define Pt =We et , where We is a nP×N matrix with weights
and et is a vector with the N yield residuals. That is, I extract maturity-specific cycle
2.3. BOND RETURN PREDICTABILITY 53
factors as the information in yields orthogonal to the macroeconomic expectations,
i.e.
e(n)t = y (n)
t − α(n) − β(n)c τc,t − β(n)
π τπ,t . (2.9)
I then perform a singular value decomposition to obtain the Pt ’s.4
Table 2.2 reveals that the maturity-specific residuals from (2.9) are highly cor-
related across maturities, have very similar first-order autoregressive coefficients,
and that the first two principal components, P(1)t and P(2)
t , explain 99% of the total
variation in e(1)t , e(2)
t , . . . , e(10)t . For this reason, I let nP = 2 and consider P(1)
t and
P(2)t as the yield curve cycle factors. Figure 2.3 plots the time series dynamics and
Table 2.2: Descriptive Statistics: Latent Factors
Latent factors Pt are obtained from the residuals e(n)t = y (n)
t − β(n)0 − β(n)
c τc,t − β(n)π τπ,t . P(i)
t is the i ’th
principal component of the vector of yield residuals e(n)t for n = 1,2, . . . ,10.
Panel A : Correlation matrix
e(1)t e(2)
t e(3)t e(4)
t e(5)t e(6)
t e(7)t e(8)
t e(9)t e(10)
t
e(1)t 1.00 · · · · · · · · ·
e(2)t 0.98 1.00 · · · · · · · ·
e(3)t 0.94 0.99 1.00 · · · · · · ·
e(4)t 0.90 0.96 0.99 1.00 · · · · · ·
e(5)t 0.85 0.93 0.97 0.99 1.00 · · · · ·
e(6)t 0.81 0.90 0.95 0.98 0.99 1.00 · · · ·
e(7)t 0.78 0.87 0.93 0.96 0.98 0.99 1.00 · · ·
e(8)t 0.74 0.84 0.91 0.95 0.97 0.98 0.99 1.00 · ·
e(9)t 0.73 0.83 0.90 0.93 0.96 0.98 0.98 0.99 1.00 ·
e(10)t 0.70 0.80 0.87 0.91 0.94 0.96 0.97 0.98 0.98 1.00
Panel B : Autocorrelation
e(1)t e(2)
t e(3)t e(4)
t e(5)t e(6)
t e(7)t e(8)
t e(9)t e(10)
t
AR(1) 0.95 0.94 0.94 0.94 0.93 0.93 0.93 0.93 0.93 0.93
Panel C : Variation explained by PCs
P(1)t P(2)
t P(3)t P(4)
t P(5)t P(6)
t P(7)t P(8)
t P(9)t P(10)
t
% 93.0 6.00 0.49 0.17 0.12 0.08 0.06 0.04 0.03 0.02
factor loadings related to P(1)t and P(2)
t , respectively. The factor loadings admit an
interpretation of P(1)t as a cycle level factor and P(2)
t as a cycle slope factor. However,
the time series dynamics of especially P(1)t is somewhat different from the typical
4I perform the singular value decomposition after normalizing the variance covariance matrix toobtain the correlation matrix.
54 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
Figure 2.3: Cycle Factors and Factor Loadings
This figure plots the time series of P(1)t and P(2)
t , where P(1)t and P(2)
t are obtained from the singularvalue decomposition Pt = We et . Here, et is the vector of residuals from the restricted regressions, i.e.
e(n)t = y (n)
t − α(n) − β(n)c τc,t − β(n)
π τπ,t . The factor loadings are obtained from the regressions y (n)t =α(n) +
β(n)c τc,t +β(n)
π τπ,t +β(n)1 P(1)
t +β(n)2 P(2)
t +e(n)t .
level factor in affine term structure models. Although the autocorrelation of P(1)t is
high at 0.94, it is smaller than the typical level factor in affine term structure models,
which is often close to unity (see Christensen, Diebold, and Rudebusch, 2011; Duffee,
2011a, among many others). The same is true for the P(2)t factor — the autocorrelation
is high at 0.946, but smaller than for the typical slope factor in affine term structure
models.
2.3.3 An Excess Return Predictor
Macroeconomic expectations and yield curve cycles provide a useful decomposition
of the yield curve for measuring bond risk premia. The standard practice for assessing
the usefulness is predictive regressions of the general form
r x(n)t+1 = Ztθ
(n) +u(n)t+1, (2.10)
where p(n)t denotes the n-year log bond price at time t and r x(n)
t+1 = p(n−1)t+1 −p(n)
t +p(1)
t denotes excess holding return — borrow in the 1-year bond, buy an n-year
bond and sell it one year later. Zt denotes a set of common explanatory variables.
The model-implied explanatory variables for the predictive regressions are Zt =[1 τc,t τπ,t P(1)
t P(2)t
]. I use these variables to construct a single excess return fore-
2.3. BOND RETURN PREDICTABILITY 55
casting factor by running regressions of the form
r x t+1 = Ztθ+ut+1, (2.11)
and obtain the fitted values as in Cochrane and Piazzesi (2005). The left hand side
variable is the maturity-averaged excess return, r x t = 19
∑10i=2 r x(i)
t . I consider four
Table 2.3: Excess Return Forecasting Factor Identifying Regression
This table reports regression coefficients for r x t+1 = θ0+θcτc,t +θπτπ,t +θ1P(1)t +θ2P
(2)t +ut+1. Forecast-
ing horizon is yearly, but observations are sampled monthly. Newey and West (1987) corrected t-statisticsusing 18 lags are in parentheses () and Hansen and Hodrick (1980) corrected t-statistics using 12 lags are inbrackets []. χ2
NW and χ2H H are Wald test statistics for the null hypothesisH0 : θc = θπ = θ1 = θ2 = 0, where
the variance-covariance matrix is Newey and West (1987) and Hansen and Hodrick (1980) corrected using18 and 12 lags, respectively. The 5-percent and 1-percent critical values for a χ2 distribution with 2 degreesof freedom are 5.99 and 9.21, respectively. The 5-percent and 1-percent critical values for a χ2 distributionwith 3 degrees of freedom are 7.81 and 11.35, respectively. The 5-percent and 1-percent critical values for aχ2 distribution with 4 degrees of freedom are 9.49 and 13.28, respectively.
Regression outputθ0 ×100 θc θπ θ1 θ2 χ2
NW χ2H H R2
(1) 6.3527 -2.3326 -0.1896 - - 3.90 3.07 0.04(2.6490) (-1.7133) (-0.2682) - -
[2.3333] [-1.5288] [-0.2347] - -
(2) 1.7529 - - 1.4820 -3.7313 61.2 51.3 0.48(2.5697) - - (7.7245) (-5.9825)
[2.2371] - - [7.1352] [-5.2526]
(3) 5.6881 -2.3739 - 1.4727 -3.7138 101.4 85.8 0.52(4.0576) (-2.9925) - (8.6278) (-6.6317)
[3.5824] [-2.6483] - [8.1041] [-5.8389]
(4) 6.0004 -2.1697 -0.1783 1.4723 -3.7130 97.6 81.2 0.52(3.7788) (-2.4534) (-0.3605) (8.2961) (-6.6890)
[3.3063] [-2.1822] [-0.3137] [7.7424] [-5.8920]
specifications. First, I report a restricted version using only the two factors capturing
macroeconomic expectations. Neither of the two factors are statistically significant
predictors. The t-statistics for τc,t and τπ,t are -1.71 (-1.53) and -0.27 (-0.23) using
Newey and West (1987) (Hansen and Hodrick, 1980) corrected standard errors with
18 (12) lags, respectively. The Wald test for joint insignificance cannot be rejected
at any of the usual significance levels. This result suggests that the macroeconomic
expectations predominantly capture the expectation hypothesis components in long-
term yields. In a second restricted version, I use only the two latent factors. These two
factors combine to capture an impressive 48% of the total variation in the maturity-
averaged excess returns. Both factors are highly statistically significant predictors.
The t-statistics for P(1)t and P(2)
t are 7.72 (7.14) and -5.98 (-5.25), respectively. The
56 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
Wald test for joint insignificance rejects at minuscule significance levels. This result
suggests that the cycle factors are important sources of risk premium variation. Next, I
include conditional expectations for consumption growth. The predictive coefficient
is -2.37 and it is significant with a t-statistic of -2.99 (-2.65). Augmenting the two
latent factors with expected consumption growth increases the explained variation
to 52% of the total variation in maturity-averaged excess returns. Finally, including
inflation expectations as well does not add any significant explanatory power. In fact,
the predictive coefficient for τπ,t is statistically insignificant for any conventional
significance level. For these reasons, I use the third specification and construct the
single excess return forecasting factors as RFt = θ0+ θcτc,t + θ1P(1)t + θ2P
(2)t . Although
τπ,t does not directly affect RFt it is still critical for identifying P(1)t and P(2)
t . Thus,
τπ,t still indirectly affects RFt .
I compute an additional return factor RF?t as the fitted values from the predictive
regression
r x t+1 = 0.06(3.82)
− 2.17(−2.45)
τ?c,t − 0.18(−0.37)
τ?π,t + 1.47(8.22)
P(1),?t − 3.69
(6.68)P(2),?
t +ut+1, (2.12)
where stars indicate that the constant-gain parameters νc , νπ are calibrated to mini-
mize the squared prediction errors. That is, the constant-gain parameters are cali-
brated to bond market data, which exactly reflect the expectations of the marginal
bond investor. The optimal constant-gain parameters are ν?c = 0.0158 and ν?π =0.0164, and these values are of course reflected in the choice of the common constant-
gain parameter νc = νπ = 0.016 used throughout. Restricting a common constant-
gain parameter of νc = νπ = 0.016 comes at a fairly low loss in predicted variation.
The regression in (2.12) achieves R2 = 0.52, and is only marginally higher than the R2
from specifications three and four in Table 2.3.
2.3.4 Other Excess Return Predictors
I consider three sets of prominent excess return predictors identified in the literature.
These predictors are based on the studies of Litterman and Scheinkman (1991),
Cochrane and Piazzesi (2005), and Cieslak and Povala (2015), respectively.
The first set of predictors comprises the contemporaneous information in the
yield curve by the first three principal components; commonly referred to as level,
slope, and curvature. This is a widely used model for decompositions of the yield
curve. That is, Zt =[1 PC1,t PC2,t PC3,t
].
The second set of predictors summarize the information in the current yield
curve by forward rates. I use five forward rates with maturities n = 1,2,3,4,5. That
is, Zt =[
1 f (1,1)t f (2,1)
t f (3,1)t f (4,1)
t f (5,1)t
], where f (n,h)
t denotes the forward rate in
period t for loans between period t +n −h and t +n.
The final set of predictors capture the information in the current yield curve that
is orthogonal to constant-gain inflation expectations. To this end, the predictors
2.3. BOND RETURN PREDICTABILITY 57
are constrained versions of the latent factors Pt identified in this paper by impos-
ing β(n)c = 0 in (2.8). Cieslak and Povala (2015) label these maturity-specific cycle
factors c(n)t , and summarize their information by c(1)
t and c t = 19
∑10i=2 c(i)
t . That is,
Zt =[
1 c(1)t c t
].
I obtain a single return predicting factor associated with each of the three sets of
yield curve factors Zt as the fitted values from regressions of the form in (2.11). These
three predictive factors are labelled PCt , C Pt , and C Ft , respectively. Three principal
components of the yield curve, five forward rates, and two maturity-specific cycle
factors capture 17%, 20%, and 43% of the total variation in maturity-averaged excess
returns, respectively.
2.3.5 Predictive Regressions
Table 2.4 and Table 2.5 shows the results from the predictive regressions in (2.10) with
a constant and PCt , C Pt , C Ft , RFt , and RF?t as predictors, respectively. Table 2.4
covers the full November 1971 - December 2014 sample. Table 2.5 takes the apparent
yield curve shift documented in Rudebusch and Wu (2007) into account by consid-
ering the shorter January 1990 - December 2014 sample. The results suggest that
all four excess return predictors have significant predictive power for all maturities
considered. However, judged from the reported t-statistics and predictive R2’s, the
RFt factor is the strongest predictor across all maturities. Over the full sample, ac-
counting for consumption growth expectations improves the predictive power of the
C Ft factor by 4-5% for the shortest maturities and by 7% for the longest maturities.
In relative terms, this amount to sizeable improvements in the range of 10−19%
depending on the maturity considered. Compared to the PCt and C Pt the predictive
power more than doubles for all maturities by considering the RFt factor. Over the
shorter sample, the improvements in predictive power over the C Ft factor is even
more pronounced. The predictive power of the RFt factor improves the C Ft factor by
5−7% for the shortest maturities and by as much as 20% for the longest maturities. In
relative terms, the improvements are as large as 32−57% depending on the maturity
considered. Compared to the PCt and C Pt the predictive power more than doubles
for all maturities except for the 2-year bond by considering the RFt factor.
The four excess return predictors naturally share some of the same predictive
information. This is the case, since they are all constructed using the same yield
curve and (potentially) macroeconomic data. However, it is possible that the different
predictors complement each other. For this reason, I run the multivariate regressions
r x(n)t+1 = θ(n)
0 +θ(n)RF RFt +θ(n)
F Ft +u(n)t+1, (2.13)
where Ft in turn refers to PCt , C Pt , and C Ft . I present the regression results based on
the longer sample, since this includes the Volcker inflation event and thus is the most
challenging sample. The results are similar or even stronger on the shorter sample.
58 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
Table 2.4: Predictive Regressions, 1971-2015
This table reports regression coefficients for r x(n)t+1 = Ztθ
(n) +u(n)t+1. Forecasting horizon is yearly, but
observations are sampled monthly. Newey and West (1987) corrected t-statistics using 18 lags are inparentheses () and Hansen and Hodrick (1980) corrected t-statistics using 12 lags are in brackets [].
Maturities
2 3 4 5 6 7 8 9 10
Panel A: r x(n)t+1 = θ(n)
0 +θ(n)PC
PCt +u(n)t+1.
θ(n)PC
0.2067 0.3964 0.6156 0.8267 1.0372 1.2052 1.4411 1.5794 1.6916(2.6865) (2.7124) (2.9553) (3.2749) (3.3838) (3.3864) (3.7655) (3.7299) (3.4650)[2.3667] [2.3760] [2.5854] [2.8684] [2.9710] [2.9726] [3.3034] [3.2835] [3.0464]
R2 0.10 0.11 0.13 0.15 0.17 0.17 0.19 0.19 0.17
Panel B: r x(n)t+1 = θ(n)
0 +θ(n)C P C Pt +u(n)
t+1.
θ(n)C P 0.2157 0.4295 0.6719 0.8366 1.0404 1.2053 1.4036 1.5201 1.6771
(3.7464) (3.8851) (4.3975) (4.4450) (4.5814) (4.6946) (4.9545) (4.7644) (4.7869)[3.3321] [3.4343] [3.8833] [3.9252] [4.0813] [4.1865] [4.4253] [4.2785] [4.2726]
R2 0.13 0.15 0.19 0.19 0.20 0.20 0.21 0.21 0.20
Panel C: r x(n)t+1 = θ(n)
0 +θ(n)C F C Ft +u(n)
t+1.
θ(n)C F 0.2345 0.4510 0.6588 0.8412 1.0301 1.2066 1.3906 1.5213 1.6660
(5.8566) (6.3891) (6.8434) (7.2674) (7.6226) (8.0163) (8.0546) (7.8907) (7.8089)[5.2505] [5.7265] [6.1187] [6.5173] [6.8757] [7.2428] [7.3064] [7.1624] [7.0939]
R2 0.33 0.37 0.40 0.42 0.44 0.46 0.47 0.46 0.44
Panel D: r x(n)t+1 = θ(n)
0 +θ(n)RF RFt +u(n)
t+1.
θ(n)RF 0.2302 0.4450 0.6486 0.8336 1.0236 1.2015 1.4005 1.5293 1.6876
(6.2104) (7.3131) (8.1573) (8.8618) (9.4483) (10.1541) (10.3266) (9.9930) (10.3324)[5.5887] [6.6121] [7.3809] [8.0615] [8.6629] [9.3382] [9.5838] [9.2297] [9.6175]
R2 0.37 0.41 0.44 0.47 0.50 0.52 0.54 0.53 0.51
Panel E: r x(n)t+1 = θ(n)
0 +θ(n)RF?
RF?t +u(n)t+1.
θ(n)RF?
0.2278 0.4426 0.6476 0.8328 1.0221 1.2043 1.3989 1.5269 1.6972(5.9877) (7.0617) (7.8005) (8.4570) (9.0050) (9.6335) (9.8434) (9.5450) (9.8765)[5.3840] [6.3777] [7.0408] [7.6625] [8.2178] [8.8096] [9.0782] [8.7721] [9.1315]
R2 0.36 0.41 0.44 0.47 0.50 0.52 0.54 0.53 0.52
2.3. BOND RETURN PREDICTABILITY 59
Table 2.5: Predictive Regressions, 1990-2015
This table reports regression coefficients for r x(n)t+1 = Ztθ
(n) +u(n)t+1. Forecasting horizon is yearly, but
observations are sampled monthly. Newey and West (1987) corrected t-statistics using 18 lags are inparentheses () and Hansen and Hodrick (1980) corrected t-statistics using 12 lags are in brackets [].
Maturities
2 3 4 5 6 7 8 9 10
Panel A: r x(n)t+1 = θ(n)
0 +θ(n)PC
PCt +u(n)t+1.
θ(n)PC
0.2232 0.4038 0.6259 0.8311 1.0109 1.2145 1.4076 1.5636 1.7194(2.0039) (2.0695) (2.4189) (2.7856) (2.9673) (3.2068) (3.5625) (3.7857) (3.8269)[1.7360] [1.7964] [2.1087] [2.4482] [2.6307] [2.8697] [3.2340] [3.4892] [3.5287]
R2 0.11 0.11 0.13 0.15 0.15 0.17 0.18 0.18 0.19
Panel B: r x(n)t+1 = θ(n)
0 +θ(n)C P C Pt +u(n)
t+1.
θ(n)C P 0.2227 0.4078 0.6407 0.8535 1.0287 1.2306 1.4060 1.5548 1.6552
(2.3551) (2.5280) (3.0302) (3.5498) (3.7550) (4.0599) (4.4149) (4.6013) (4.4917)[2.0375] [2.1933] [2.6406] [3.1313] [3.3537] [3.6730] [4.0787] [4.3331] [4.2470]
R2 0.13 0.12 0.15 0.17 0.18 0.19 0.20 0.20 0.19
Panel C: r x(n)t+1 = θ(n)
0 +θ(n)C F C Ft +u(n)
t+1.
θ(n)C F 0.1572 0.3715 0.6015 0.8346 1.0439 1.2551 1.4281 1.5839 1.6552
(2.3912) (2.9137) (3.4706) (4.0743) (4.5679) (5.0014) (5.3250) (5.4416) (5.5712)[2.1453] [2.6160] [3.1198] [3.6915] [4.1908] [4.6390] [4.9895] [5.1587] [5.3296]
R2 0.11 0.17 0.23 0.28 0.31 0.34 0.35 0.36 0.36
Panel D: r x(n)t+1 = θ(n)
0 +θ(n)RF RFt +u(n)
t+1.
θ(n)RF 0.1491 0.3549 0.5820 0.8148 1.0337 1.2476 1.4307 1.6114 1.7759
(3.3827) (4.2065) (5.1274) (6.0775) (6.7765) (7.4030) (7.7950) (8.1552) (8.3024)[3.1809] [3.9295] [4.7679] [5.6782] [6.4071] [7.0624] [7.5099] [7.9237] [8.1237]
R2 0.14 0.23 0.31 0.39 0.45 0.49 0.52 0.54 0.56
Panel E: r x(n)t+1 = θ(n)
0 +θ(n)RF?
RF?t +u(n)t+1.
θ(n)RF?
0.1569 0.3596 0.5865 0.8151 1.0317 1.2434 1.4287 1.6107 1.7675(3.4902) (4.3139) (5.2037) (6.0681) (6.7201) (7.2527) (7.6151) (7.9892) (8.1217)[3.2458] [4.0108] [4.8154] [5.6413] [6.3212] [6.8763] [7.2817] [7.7078] [7.9039]
R2 0.16 0.24 0.32 0.40 0.45 0.50 0.53 0.55 0.56
60 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
In Figure 2.4, I plot the p-values associated with the null hypothesis H0 : θ(n)F = 0.
Across all maturities, the null hypothesis cannot be rejected for any conventional
Figure 2.4: Additional Information in Prominent Excess Return Predictors
This figure plots the p-values associated with the null hypothesis H0 : θ(n)F = 0, where θ(n)
F is estimated
from the regressions r x(n)t+1 = θ(n)
0 +θ(n)RF RFt +θ(n)
F Ft +u(n)t+1, with Ft being PCt , C Pt , and C Ft in turn.
The leftmost graph is based on Newey and West (1987) corrected standard errors with 18 lags, and therightmost graph is based on Hansen and Hodrick (1980) corrected standard errors with 12 lags.
significance levels. These results are clearcut evidence that PCt , C Pt , and C Ft do not
contain additional predictive power, as RFt subsumes all information embedded in
these predictors.
2.3.6 The Unspanned Factor Phenomenon
The RFt factor builds on a four-factor structure. Principal component analysis of the
yield curve should capture this. However, allowing for a fourth principal component
in the PCt factor does not change the results much. In fact, even a fifth factor does
not add much. Essentially, a five-factor model is equivalent to the C Pt factor, which
is build from five forward rates.
Consider — for the sake of exposition — a two-factor affine model. That is, the
yield curve has the form
y (n)t =An +B1,nP1,t +B2,nP2,t , (2.14)
where An , B1,n , and B2,n are factors loadings. Suppose that B2,n = aB1,n . Then the
model in equation (2.14) does not allow recovery of P1,t and P2,t using yield data only.
2.3. BOND RETURN PREDICTABILITY 61
But the model instead suggests a one-factor structure, i.e.
y (n)t =An +B1,nP1,t +aB1,nP2,t =An +B1,nPt , (2.15)
where Pt = P1,t +aP2,t . Thus, the model in equation (2.15) is only invertible for Pt .
Assuming dynamics for Pt of AR(1) form, Pt+1 = φPt + vt+1, the one-factor model
implies yield forecasts of the form
Et
[y (n)
t+h |Pt
]=An +B1,nφ
hPt . (2.16)
These forecasts may be severely misspecified if the two pricing factors P1,t and P2,t
have distinct dynamics, e.g. Pi ,t+1 = φi Pi ,t + vi ,t+1 for i ∈ {1,2} and φ1 6= φ2. In this
case, the correct forecast would be
Et
[y (n)
t+h |P1,t ,P2,t
]=An +B1,nφ
h1 P1,t +B2,nφ
h2 P2,t . (2.17)
Hence, forecasting solely based on Pt induces the predictable forecast error
Et
[y (n)
t+h |P1,t ,P2,t
]−Et
[y (n)
t+h |Pt
]=B1,n
(φh
1 −φh)
P1,t +B2,n
(φh
2 −φh)
P2,t . (2.18)
If one of the factors where observed, say P1,t , then equation (2.14) could be inverted
for P2,t . Then controlling for P1,t in the forecasting regression would eliminate the
forecast error, although controlling for P1,t would leave the cross-sectional yield
curve fit unchanged. Following this line of logic, the macroeconomic factors might
appear as risk unspanned by the current yield curve even if it is perfectly priced into
the term structure. The invertibility issue naturally generalizes to more than two
factors and potentially explains why the RFt factor provides superior forecasts. I find
that both τπ,t and P(1)t are level factors, and both τc,t and P(2)
t are slope factors.
To provide some evidence on this, I run the regressions
PC1,t = ξ0,1 +ξ1,1τc,t +ξ2,1τπ,t +ξ3,1P(1)t +ϑ1,t
PC2,t = ξ0,2 +ξ1,2τc,t +ξ2,2τπ,t +ξ3,2P(2)t +ϑ2,t .
(2.19)
The R2 of the first regression is literally equal to 1, and the R2 from the second
regression is 0.998. This strongly suggests that the usual level and slope factors are
linear combinations of the term structure factors identified in this paper. Further,
restricted versions of the regressions in (2.19), imposing ξ1,1 = ξ1,2 = ξ2,1 = ξ2,2 = 0,
show that the latent factors P(1)t and P(2)
t have very different information from that
in the level PC1,t and slope PC2,t factors. These restricted regressions have R2’s
of 0.12 and 0.71, respectively. Finally, the level PC1,t and slope PC2,t factors have
autoregressive coefficients of 0.995 and 0.962, respectively. In comparison, the cycle
factors P(1)t and P(2)
t have autoregressive coefficients of 0.940 and 0.946, respectively.
Thus, the macroeconomic factors capture the low-frequency variation in the level
and slope factors, and the yield curve cycle factors capture the higher-frequency
variation.
62 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
The restrictions in Duffee (2011b) and Joslin et al. (2014) for non-invertibility of
the factor loading matrix is not exact in the data, and so in theory (linear combinations
of) the macroeconomic factors should be identifiable from yield data. However, the
studies by Duffee (2011b), Cieslak and Povala (2015), and Bauer and Rudebusch
(2017b) document that it is difficult to recover such factors when yields are measured
with small measurement errors. In particular, this is expected to be the case when the
factor loading matrix has close to linearly dependent columns, as I document is the
case here.
2.3.7 The Bauer and Hamilton (2018) Test
The predictive regressions for bond returns are known to be plagued by small-sample
distortions. I have applied both Newey and West (1987) and Hansen and Hodrick
(1980) corrections when computing standard errors to account for these problems.
However, Bauer and Hamilton (2018) argue that even these robust standard errors
are subject to serious small-sample distortions. To overcome the problem, they pro-
pose a bootstrap procedure. The bootstrap is under the null hypothesis that only
the first three principal components contain predictive information about future
bond returns. I follow their example and implement the bootstrap procedure. The
intuition is to re-sample from the first three principal components and construct
bootstrap yield curves. For each bootstrap sample, re-run the predictive regressions
with the additional proposed factors, which by construction should have no addi-
tional explanatory power. I compute the critical value for the Wald test as the 95th
percentile from the bootstrap distribution of χ2NW and χ2
H H for each maturity. That
is, the value for the Wald statistics that one would expect to see less than 5% of the
times if the three principal components were the only factors with predictive power
for excess returns.5 In Figure 2.5, I plot the bootstrapped critical values along with
the empirical values for the Wald test statistics. The bootstrap exercise indicates
severe small-sample distortions as argued by Bauer and Hamilton (2018). In fact,
the critical value at the 5% percent significance level is more than three times larger
than what asymptotic theory suggests. However, the empirical Wald test statistics
are far bigger than the critical values. Thus, the null that the first three principal
components of yields are the only factors with predictive power for excess returns is
strongly rejected. Accordingly, the improvements in performance by considering the
proposed forecasting factor is not an artifact of small-sample distortions.
2.3.8 Out-of-Sample Predictability: Excess Returns
Although the in-sample projections show significant predictive power of the return
forecasting factor, it is unclear if the predictive gains could have been realized over
5I provide details on the bootstrap procedure in the online appendix.
2.3. BOND RETURN PREDICTABILITY 63
Figure 2.5: The Bauer and Hamilton (2018) Bootstrap Test
This figure plots bootstrapped critical values for the Wald statistics χ2NW and χ2
H H for testing the hypoth-
esis that the factors τc,t , P(1)t , and P(2)
t have no predictive power beyond that in the first three principle
components of yields. Formally, the critical values are for the hypothesisH0 : θ(n)F = 0 in the regressions
r x(n)t+1 = θ(n)
0 +θ(n)PC
PCt +θ(n)F Ft +u(n)
t+1, where PCt =[PC1,t PC2,t PC3,t
]′ and Ft =[τc,t P(1)
t P(2)t
]′.
the sample period. In fact, it has proven remarkably difficult to outperform the no-
predictability benchmark, see e.g. Thornton and Valente (2012). For this reason, I
perform a pseudo out-of-sample exercise. The predictive ability is evaluated over the
latest 25 years of data, i.e. I start the test period in January 1990. Factor identification
and predictive coefficients are recursively updated on the expanding training sample.
For comparison, I implement univariate regressions for each of the excess returns
onto the forecasting factors PCt , C Pt , C Ft , and RFt . The RFt factor is based on
the constant-gain parameters νc = νπ = 0.016, and thus is subject to some look-
ahead bias. I also implement a RF?t factor, where the constant-gain parameters are
calibrated to the initial training sample (November 1971 - January 1990). I present the
results with root mean squared forecast errors as measure of forecast performance.
All numbers are relative to the root mean squared forecasting errors from using the
historical average for the same maturity as forecast, i.e. relative to a weak expectation
hypothesis benchmark that rules out predictability. This means that numbers less
than one imply that the specific model’s predictions outperform the expectation
hypothesis. I implement the encompassing test in Harvey et al. (1998), where the
null hypothesis is that the RFt factor encompasses the other prominent bond return
factors.
The first two columns iterate the finding that in-sample predictability is difficult to
64 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
Table 2.6: Out-of-Sample Prediction Results: Excess Returns
This table reports out-of-sample root mean squared forecasting errors for the predictive regressions
r x(n)t+1 = θ(n)
0 +θ(n)F Ft +u(n)
t+1 for Ft being PCt , C Pt , C Ft , RFt , and RF?t . Forecasting horizon is yearly, butobservations are sampled monthly. All numbers are relative to the root mean squared forecasting errorsfrom using the historical mean for the same maturity as forecast. Test statistics for the hypothesis that RFtencompasses a given variable is in parenthesis (), and that RF?t encompasses a given variable is in brackets[]. See Harvey, Leybourne, and Newbold (1998). Standard errors are Newey and West (1987) corrected with18 lags. The test sample is 1990:1 to 2014:12. Factor identification and predictive coefficients are recursivelyupdated as the training sample expands.
PCt C Pt C Ft RFt RF?t
r x(2)t+1 1.173 1.129 0.982 0.944 0.938
(0.621) (0.427) (-1.840)[-0.543] [-1.074] [-0.465]
r x(3)t+1 1.162 1.141 0.976 0.929 0.916
(0.782) (0.806) (-1.665)[-0.441] [-0.843] [-0.084]
r x(4)t+1 1.147 1.145 0.955 0.907 0.875
(0.840) (1.133) (-1.607)[-0.288] [-0.560] [0.242]
r x(5)t+1 1.139 1.138 0.950 0.891 0.852
(0.919) (1.351) (-1.426)[-0.206] [-0.501] [-0.546]
r x(6)t+1 1.138 1.147 0.957 0.878 0.850
(0.897) (1.487) (-1.281)[-0.433] [-0.673] [0.558]
r x(7)t+1 1.128 1.151 0.941 0.884 0.825
(0.865) (1.651) (-1.394)[-0.191] [-0.295] [0.689]
r x(8)t+1 1.137 1.159 0.958 0.880 0.834
(0.720) (1.539) (-1.277)[-0.510] [-0.663] [0.655]
r x(9)t+1 1.113 1.142 0.968 0.866 0.837
(0.584) (1.468) (-1.065)[-0.853] [-0.954] [0.696]
r x(10)t+1 1.110 1.160 0.974 0.905 0.833
(0.506) (1.605) (-1.167)[-0.539] [-0.391] [0.979]
realize over the sample period in real time. Both the first three principal components
and five forward rates do not outperform the no-predictability benchmark. In fact, the
two models have root mean squared forecasting errors that are in the order of 11−17%
larger than the forecasts from simple historical averages. The third column shows that
the C Ft factor from Cieslak and Povala (2015) does outperform the benchmark. Across
maturities, the gains are in the order 2−6%. Finally, in the two rightmost columns,
the return forecasting factor developed in this paper delivers root mean squared
forecasting errors that are in the order of 6−17% smaller than the no-predictability
benchmark. For all maturities considered, the RF?t factor delivers the lowest root
2.3. BOND RETURN PREDICTABILITY 65
mean squared error, and the factor does particularly well for the longer maturities.
This is especially true for maturities beyond five years, where the gain in root mean
squared error relative to the C Ft factor is greater than 10%. Across all maturities,
the null hypothesis that the RF?t factor (or the RFt factor) encompasses the other
prominent bond return factors is never rejected.
2.3.9 Further Robustness
In addition to the above results, I have conducted a range of robustness checks. The
results are reported in the online appendix accompanying the paper. Here, I briefly
summarize these results.
First, holding periods of three and six months are also often considered in the
literature, because they rely less on overlapping data compared to the benchmark
one-year holding period. It is comforting that the conclusions are not sensitive to
the length of the holding period. For both three and six months holding periods, the
relative ranking of the four considered predictors remains unchanged in terms of
predicted variation. The relative improvements of the RFt factor over the C Ft factor
remain comparable to those achieved over the one-year holding period. The relative
improvements are in the order of 5−26% and 5−22% for the three and six months
holding periods, respectively.
Second, I have also checked that the results are not only present when using
Fama and Bliss (1987) zero-coupon yields by implementing the predictive regressions
using the Gurkaynak, Sack, and Wright (2007) dataset. The two datasets are highly
correlated, and the results from the predictive regressions inherits the similarity in
the two data sets.
Third, I have documented that the out-of-sample results do not only hold for
returns. I have done a similar out-of-sample exercise as above for yield levels. Because
of the extreme persistence in yields, a great number of studies have documented
difficulties in improving forecasting performance over a simple random walk. The
factor structure from this paper outperforms the random walk benchmark, and
provides the best forecasts. The forecast from the learning from macroeconomic
experiences forecasts encompasses the forecasts from the other factor models.
Finally, I have checked the robustness against variations on the learning al-
gorithm for the macroeconomic expectations. In particular, I have implemented
an optimal-gain learning algorithm and a recursive least squares algorithm. The
optimal-gain learning algorithm sets the gain parameters using the Kalman filter
recursions, whereas the recursive least squares applies a decreasing gain parameter
νc,t = νπ,t = t−1. The two algorithms are different in an important way. Recursive
least squares implies equal weighting of all historical macroeconomic data at each
point in time, and thus does not capture down-weighting of macroeconomic data as
new generations emerge and old generations passes. This is unlike the constant-gain
learning algorithm (and potentially the optimal-gain learning algorithm). I construct
66 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
bond return forecasting factors using the same methodology as outlined for the
constant-gain learning algorithm. The bond return factor implied by the recursive
least squares algorithm only does marginally better than level, slope, and curvature.
On the contrary, the bond return factor implied by the optimal-gain learning algo-
rithm predicts a similar proportion of the excess return variation as the RFt factor
studied here.
2.3.10 Predicting Short Rate Changes
A successful decomposition of the yield curve should account for both expected
excess returns as well as expected short rates. To address whether the model-implied
factors also capture well variation in future short rates, I run the regressions
∆y (1)t ,t+h = Ztω+ηt+h , (2.20)
where ∆y (1)t ,t+h = y (1)
t+h − y (1)t and h = 1,2,3,4,5 in turn. Again, I consider four sets of
short rate factors. First, I consider the level PC1,t , slope PC2,t , and curvature PC3,t
factors. Second, I use the five forward rates. Third, I use inflation expectation and the
short rate cycle factor c(1)t . Finally, I use the four term structure factors identified in
this paper. Panel D in Table 2.7 indicates that my four term structure factors have
strong predictive power for future short rate changes 1 through 5 years ahead. The
four factors capture 34-65% of the variation in future short rate changes across the
different horizons. The two latent factors are eminently important, whereas the two
factors capturing macroeconomic expectation are not significant predictors for any
of the considered horizons. In fact, the more parsimonious specification that omits
the two macroeconomic factors does equally well. Although the two macroeconomic
factors do not add much in the predictive regressions, they are critical for the strong
predictive power. This is because the macroeconomic factors are key to identify the
latent factors P(1)t and P(2)
t . This is also evident from panel A, where the short rate
changes are regressed on the first three principal components of yields, i.e. without
considering the macroeconomic factors. The predicted variation is in the range of 4-
24%, and thus the predictive power is much weaker. PanelB reveals that the predictive
power in the five forward rates, by and large, resembles that of the first three principal
components. In contrast, panel C shows that conditioning on inflation expectations
increases the predictable variation across all horizons compared to the first three
principal components of yields. As expected inflation and the short rate cycle factor
capture 15-48% of the future short rate variation, the predictive power is still not
as strong as the predictive power when accounting for both expected consumption
growth and expected inflation.
2.4. TERM PREMIA 67
Table 2.7: Short Rate Changes: Predictive Regressions
This table reports regression coefficients for ∆y (1)t ,t+h
= Ztω+ηt+h . Results are for h = 1,2,3,4,5 years,
but observations are sampled monthly. Newey and West (1987) corrected t-statistics using 18 lags are inparentheses () and Hansen and Hodrick (1980) corrected t-statistics using 12 lags are in brackets [].
Horizon, h
1 2 3 4 5
Panel A: Zt =[1 PC1,t PC2,t PC3,t
].
R2 0.04 0.11 0.20 0.24 0.19
Panel B: Zt =[
1 f (1,1)t f (2,1)
t f (3,1)t f (4,1)
t f (5,1)t
].
R2 0.07 0.10 0.15 0.21 0.17
Panel C: Zt =[
1 τπ,t c(1)t
].
R2 0.15 0.34 0.46 0.48 0.35
Panel D: Zt =[
1 τc,t τπ,t P(1)t P(2)
t
].
ωc 0.3188 0.2468 0.0423 0.4379 1.5887(1.1265) (0.7048) (0.0878) (0.6686) (2.0484)[0.9920] [0.6219] [0.0765] [0.5848] [1.7975]
ωπ -0.0364 -0.0394 -0.0417 -0.0909 -0.1595(-0.2598) (-0.2610) (-0.2298) (-0.5030) (-0.9469)[-0.2273] [-0.2294] [-0.1994] [-0.4364] [-0.8242]
ω1 -0.3104 -0.5336 -0.5802 -0.5913 -0.5155(-5.9047) (-6.5370) (-5.5188) (-4.9728) (-3.9219)[-5.4721] [-5.7382] [-4.9181] [-4.4031] [-3.4568]
ω2 0.1261 -0.0844 -0.4643 -0.6893 -0.6138(0.8234) (-0.3104) (-1.8681) (-3.3185) (-2.3110)[0.7257] [-0.2711] [-1.6531] [-2.9547] [-2.0275]
R2 0.34 0.56 0.62 0.65 0.56
2.4 Term Premia
The results suggest that the four-factor model is well-suited for decompositions of
long-term yields into an expectation hypothesis component — average expected
short rates over the lifetime of the bond — and a term premia component. This
decomposition is summarized by
y (n)t = 1
nEt
[n−1∑i=0
y (1)t+i
]+ψ(n)
t , (2.21)
68 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
where ψ(n)t denotes the n-year term premia. Term premia is then the difference
between two strategies; buying the 10-year bond and rolling over in 1-year bonds as
they expire. To evaluate the roll-over strategy it is convenient to have a dynamically
complete model. The constant-gain learning algorithm provides a forecasting model
for the macroeconomic expectation factors. I use an annual VAR(1) sampled at the
monthly frequency for the cycle factors; that is, time t cycle factors are regressed on
their one-year lagged values, but the model is estimated using monthly observations.
Cochrane and Piazzesi (2005) argue that the dynamic properties of bond risk premia
are more evident at the annual horizon. The one-year horizon is also consistent with
the previous predictive regressions. The VAR(1) model is usual practice for affine term
structure models (Dai and Singleton, 2000; Duffee, 2002). For comparison, I have
implemented the three-factor affine term structure model using the same procedure.
The dynamically complete model for the decomposition in (2.21) does not en-
force no-arbitrage restrictions on factor loadings estimates in the short rate equa-
tion. Adrian et al. (2013) and Joslin, Le, and Singleton (2013) note that enforcing
no-arbitrage restrictions has little effect on the implied decomposition.6 In Figure
Figure 2.6: Model-Implied Term Premia
This figure plots the decomposition of model-implied yields with 10 years to maturity into term premiaand short rate expectations. The leftmost graph plots the decomposition for the proposed learning frommacroeconomic experiences model, and the rightmost graph plots the decomposition for the benchmarkthree-factor affine term structure model.
6I have implemented a version of the model enforcing no-arbitrage restrictions. The results are verysimilar. The results are also similar for a monthly VAR model for the cycle factors. For details on theserobustness results, see the online appendix.
2.5. THE DECLINE IN THE EQUILIBRIUM REAL RATE 69
2.6, I plot the 10-year yield along with the resulting decompositions into expected
short rates and term premia. The three-factor affine term structure model implies a
stationary VAR for the factors. This implies long-horizon short rate expectations that
display strong mean reversion. As term premia is extracted as the residual compo-
nent, the estimates of term premia will account for some of the trend component in
the 10-year yield. On the other hand, the constant-gain learning expectations implies
time-variation in the long-run mean of the nominal short rate. To see this, consider
y (1)∞ = lim
h→∞Et
[y (1)
t+h
]=α(1) +β(1)
τ τt +β(1)P
(I−ΦP
)−1µP, (2.22)
where(I−ΦP
)−1µP is the unconditional mean of the VAR(1) system for the cycle
factors; ΦP is the autoregressive loading matrix and µP the constant term. The long-
run mean of the macroeconomic factors is limh→∞Et[τt+h
] = τt , because of the
constant-gain learning structure. The time-variation in the long-run short rate expec-
tations now captures the trend component in the 10-year yield. In turn, term premia
estimates are more cyclical. This is because they are predominantly driven by the
cycle factors — as is also evident from the previous predictive regressions. Allowing
for the dynamic trend and cycle distinction of the affine term structure measures of
level and slope thus implies very different term premia dynamics. That expectations
of short rates capture the trend component in long-term yields is consistent with the
observed trends in survey responses (Kozicki and Tinsley, 2001; Kim and Orphanides,
2012). A cyclical term premium is consistent with the behaviour of risk premia in
other assets (Fama and French, 1989).
The mechanism generating the expected short rate and term premia decom-
position — that aligns with macro-finance priors — is the underlying equilibrium
consumption-based structure. In the long-run, the yield curve level and slope are
determined by the underlying economics behind the consumption-savings decision.
The level and slope deviates from this equilibrium yield curve over the short run, but
does so in cycles that mean-revert towards the long-run equilibrium. This mechanism
is different from other efforts that have been made to bring the affine term structure
model decompositions into alignment with the macro-finance priors. Cochrane and
Piazzesi (2008) and Bauer (2017) restrict market prices of risk, Bauer, Rudebusch,
and Wu (2012) bias correct the factor VAR, Kim and Orphanides (2012) incorporate
survey measures of yield curve expectations, and Duffee (2011b) and Christensen
and Rudebusch (2016) impose a unit root on the level factor. All are attempts to
increase the persistence of long-horizon short rate expectations, but there is no direct
underlying economic equilibrium interpretation.
2.5 The Decline in the Equilibrium Real Rate
Subtracting inflation expectations from the model of the nominal term structure of
yields implies that a model of the real term structure naturally follows. Measuring the
70 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
equilibrium real rate — the real short rate that would prevail over the long run — is of
great importance for both policy makers and investors that are assessing the current
stance of the economy. The equilibrium real rate r?t is defined to be
r?t = limh→∞
Et[rt+h
], (2.23)
where rt = y (1)t −Et
[πt+1
]denotes the 1-year real short rate. Using the model structure,
the equilibrium real rate is given by
r?t =α(1) +β(1)τ,cτc,t +
(β(1)τ,π−1
)τπ,t +β(1)
P
(I−ΦP
)−1µP, (2.24)
since long-run inflation expectations are given by limh→∞
Et[πt+h
]= τπ,t . Importantly,
the model implies time-variation in the equilibrium real rate. This is consistent with
a recent literature documenting a decline in the equilibrium real rate over the last
15-20 years (Hamilton et al., 2016; Holston et al., 2017; Bauer and Rudebusch, 2017a,
among others). Time-variation in the equilibrium real rate is a novel feature of the
proposed model, since equilibrium models of the term structure generally imply a
constant equilibrium real rate, see Wachter (2006), Piazzesi and Schneider (2007),
and Bansal and Shaliastovich (2013). The same is true for the workhorse three-factor
affine term structure model. Here, both long-run inflation and consumption growth
expectations affect the equilibrium real rate.
Figure 2.7 plots the estimated time series for the equilibrium real rate r?t . Consis-
tent with the recent evidence, the equilibrium real rate was fairly constant around
the 3-4% level until 1990. After 1990 the equilibrium real rate has been gradually
declining. Over the recent recession the equilibrium real rate even went into nega-
tive territory and has remained remarkably low since then. This extraordinarily low
equilibrium real rate is consistent with the view expressed in Yellen (2015).
The model-implied decline in the equilibrium real rate comes from two sources.
First, during the 1990’s, inflation expectations were still gradually coming down,
which affected the real rate as the gap between the central banks inflation target and
inflation expectations narrowed. Second, after 2000 consumption growth expecta-
tions have declined and caused a fall in the real rate. Of the estimated 241 basis point
decline in the equilibrium real rate over the period 2000-2015, the biggest contributor
was the drop in expected consumption growth — accounting for 196 basis points.
This result is consistent with the interpretation of a declining equilibrium real rate
because of a slowdown in productivity, see e.g. Holston et al. (2017).
The decomposition is also consistent with the recent interpretation of Bauer and
Rudebusch (2017a). Although nominal yields started to fall in the early- to mid-1980’s,
this decline was early on due to inflation expectations coming down as the equilib-
rium real rate remained fairly constant over this period. As inflation expectations
settled around 2% during the 1990’s, the equilibrium real rate started to fall, causing
nominal yields to continue their downward trajectory. The main difference compared
2.6. CONCLUSION 71
Figure 2.7: Equlibrium Real Rate
This figure plots the model-implied equilibrium real rate, r?t . The equilibrium real rate is defined to be
r?t = limh→∞
Et[rt+h
], where rt = y (1)
t −Et[πt+1
]denotes the real short rate. Using the model structure, the
equilibrium real rate is given by r?t =α(1)+β(1)τ,cτc,t +
(β(1)τ,π−1
)τπ,t +β(1)
P
(I−ΦP
)−1µP, since the long-run
consumption growth and inflation expectations are given by limh→∞
Et[∆ct+h
]= τc,t and limh→∞
Et[πt+h
]=τπ,t , respectively.
to Bauer and Rudebusch (2017a) is the structural underpinning of the decline in the
equilibrium real rate.
2.6 Conclusion
Usual measures of level and slope of the yield curve each have a separate trend and
cycle component. Level and slope trends have a clear equilibrium interpretation
— inflation expectations capture the level trend, consumption growth expectations
capture the slope trend. Accounting for the trend components identifies two yield
curve cycles; a level cycle and a slope cycle.
The yield curve cycles are important sources of variation in bond risk premia.
Decomposing the trend and cycle in the level and slope of the yield curve helps
predict excess bond returns with R2’s as high as 56%. The model-implied measure of
bond risk subsumes and add to the information in the most successful measures in
the literature. More importantly, the decomposition implies bond risk premia that
are predominantly cyclical; this is in contrast to the implications of the workhorse
affine term structure models.
Macroeconomic expectations capture the most persistent variations in the yield
72 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
curve, and implies time-variation in the equilibrium real rate. The macroeconomic
expectations provide an equilibrium underpinning for the recent decline in the long-
run real rate. Equilibrium real rate variation driven by macroeconomic expectations
predominantly affect the expectation hypothesis component in long-term yields. The
cycle variation — deviations from the consumption-based equilibrium yield curve —
captures the risk premia component. Affine term structure models using yield data
only understates the persistence of the expectations hypothesis component, because
there is no potential driver of variation in the equilibrium real rate.
Acknowledgements
I thank Martin M. Andreasen, Michael D. Bauer, Ambrogio Cesa-Bianchi, Anna Cies-
lak, Hans Dewachter, Tom Engsted, James D. Hamilton, Emanuel Möench, Stig V.
Møller for insightful comments, and Anh Le for sharing his zero-coupon bond data.
Remarks and suggestions from seminar participants at Aarhus University, Bank of
England, Deutsche Bundesbank, Fed Board, KU Leuven, Norwegian School of Eco-
nomics (NHH), University of Gothenburg and conference participants at the 11th
International Conference on Computational and Financial Econometrics are also
much appreciated. I acknowledge support from CREATES - Center for Research
in Econometric Analysis of Time Series (DNRF78), funded by the Danish National
Research Foundation.
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78 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
Appendix
B.1 Model Details
B.1.1 The Stochastic Discount Factor
This section derives the real stochastic discount factor. The representative household
has lifetime utility Vt given by
Vt = Et
[ ∞∑i=0
δiΓt+i logCt+i
]= Γt logCt +δEt
[Vt+1
], (B.1)
where Vt denotes the value function, Ct denotes the consumption good, Γt is an
exogenous demand shock. The subjective discount factor is denoted by δ. A complete
market for state contingent claims At (s) paying one unit of consumption in period
t if state s materializes are available to the household. Thus, the budget constraint
reads
Ct +Et[Mt ,t+1 At+1
]= At , (B.2)
where all terms are measured in real terms, i.e. consumption units. Mt ,t+1 denotes
the real stochastic discount factor. For notational convenience the state dependence
is suppressed. This implies that the maximization problem reads
maxCt ,At+1
Vt = Γt logCt +δEt[Vt+1
]s.t. Ct +Et
[Mt ,t+1 At+1
]= At ,(B.3)
and a standard no-Ponzi game condition. The Bellman equation reads
V(
At)= max
Ct ,At+1{Γt logCt +δEt
[Vt+1
]+Λt
(At −Ct −Et
[Mt ,t+1 At+1
])}. (B.4)
First order conditions are
Γt C−1t −Λt = 0
−Λt Mt ,t+1P (s)+δ ∂Vt+1
∂At+1P (s) = 0,
(B.5)
where P (s) denotes the probability of state s. By the envelope theorem, it follows that∂Vt∂At
=Λt . Thus,
Mt ,t+1 = δΛt+1
Λt= δ
(Ct+1
Ct
)−1 Γt+1
Γt, (B.6)
which hold for all states s. The log real stochastic discount factor then reads
mt+1 = logδ−∆ct+1 +∆γt+1, (B.7)
where γt+1 = logΓt+1 − logΓt . This implies a (log) nominal stochastic discount factor
given by
m$t+1 = logδ−∆ct+1 +∆γt+1 −πt+1, (B.8)
B.1. MODEL DETAILS 79
where πt+1 is net inflation. Finally, plugging in for the consumption growth and infla-
tion dynamics as perceived by the representative household, the nominal stochastic
discount factor is given by
m$t+1 = logδ−τc,t −εc,t+1 +∆γt+1 −τπ,t −επ,t+1, (B.9)
where τc,t and τπ,t are perceived conditional expectations of consumption growth
and inflation, respectively. εc,t and επ,t are forecast errors.
B.1.2 Exogenous Demand Shocks
The exogenous demand shock growth is specified as
∆γt+1 = γ0 +γττt +γPPt + 1
2λ′τ,tλτ,t +
(λ′τ,tΣ
−1τ − [1 1]
)εt+1
+ 1
2λ′P,tλP,t +λ′
P,tΣ−1P vt+1,
(B.10)
where τt = [τc,t τπ,t
]′ and εt = [εc,t επ,t
]′ ∼ N(0,ΣτΣ′
τ
). Pt is an nP × 1 vector of
latent factors. vt+1 ∼N(0,ΣPΣ
′P
)are one-step ahead forecast errors associated with
Pt . This specification implies a nominal stochastic discount factor given by
m$t+1 =−α−βττt−βPPt−1
2λ′τ,tλτ,t−λ′
τ,tΣ−1τ εt+1−1
2λ′P,tλP,t−λ′
P,tΣ−1P vt+1, (B.11)
where α= γ0 − logδ, βτ = γτ+ [1 1], and βP = γP. The market prices of risk (or risk
sensitivity functions) are given by
Στλτ,t =λ0,τ+λ1,ττt
ΣPλP,t =λ0,P+λ1,PPt ,(B.12)
where λ0,τ, λ1,τ, λ0,P, and λ1,P are parameters.
B.1.3 Perceived Law of Motions
The household recursively updates conditional expectations using a constant-gain
learning algorithm, i.e.
τt+1 = τt +νεt+1, (B.13)
where ν is the constant-gain matrix. This matrix is assumed diagonal. The latent
factors have law of motion
Pt+1 =µP+ΦPPt + vt+1. (B.14)
The innovations vt are unrelated to εt at all leads and lags.
80 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
B.1.4 The Nominal Short Rate
The pricing equation for the nominal short rate it is given by
Et
[em$
t+1
]= e−it . (B.15)
Since the stochastic discount factor is conditionally normal, and it hold for any
X ∼N(µ,σ2
)that E
[e X
]= eµ+
12σ
2, it follows that
it =−Et
[m$
t+1
]− 1
2Vt
[m$
t+1
]. (B.16)
The conditional expectation of the stochastic discount factor is
Et
[m$
t+1
]=−α−βττt −βPPt − 1
2λ′τ,tλτ,t − 1
2λ′P,tλP,t , (B.17)
and so, the innovation is given by
m$t+1 −Et
[m$
t+1
]=−λ′
τ,tΣ−1τ εt+1 −λ′
P,tΣ−1P vt+1. (B.18)
It follows that
Vt
[m$
t+1
]=λ′
τ,tλτ,t +λ′P,tλP,t , (B.19)
and so, the short nominal interest rate is given by
it =α+βττt +βPPt . (B.20)
B.1.5 Affine Yield Curve Representation
For all bonds with maturity longer than one period, the pricing equations are given
recursively as
Et
[em$
t+1+p(n−1)t+1
]= ep(n)
t . (B.21)
Inserting for the stochastic discount factor, the pricing equations read
Et
[e−α−βττt−βPPt− 1
2λ′τ,tλτ,t−λ′τ,tΣ
−1τ εt+1− 1
2λ′P,tλP,t−λ′P,tΣ
−1P
vt+1+p(n−1)t+1
]= ep(n)
t . (B.22)
Then, guessing and verifying, a solution for log bond prices that is linear in the state
variables, i.e. p(n)t = an +bτ,nτt +bP,nPt , implies that the left hand side exponent
reads
−α−βττt −βPPt − 1
2λ′τ,tλτ,t −λ′
τ,tΣ−1τ εt+1 − 1
2λ′P,tλP,t −λ′
P,tΣ−1P vt+1
+an−1 +bτ,n−1τt+1 +bP,n−1Pt+1.(B.23)
B.1. MODEL DETAILS 81
Inserting for law of motions of the state variables, it follows that
−α−βττt −βPPt − 1
2λ′τ,tλτ,t −λ′
τ,tΣ−1τ εt+1 − 1
2λ′P,tλP,t −λ′
P,tΣ−1P vt+1
+an−1 +bτ,n−1τt +bτ,n−1νεt+1 +bP,n−1µP+bP,n−1ΦPPt +bP,n−1vt+1,(B.24)
or
−α+an−1 +bP,n−1µP+ (bτ,n−1 −βτ
)τt +
(bP,n−1ΦP−βP
)Pt
−1
2λ′τ,tλτ,t − 1
2λ′P,tλP,t +
(bτ,n−1ν−λ′
τ,tΣ−1τ
)εt+1 +
(bP,n−1 −λ′
P,tΣ−1P
)vt+1.
(B.25)
Thus, applying the log-normal expectation formula, it follows that
Et
[em$
t+1+p(n−1)t+1
]= e
−α+an−1+bP,n−1µP+(bτ,n−1−βτ
)τt+
(bP,n−1ΦP−βP
)Pt
− 12λ
′τ,tλτ,t− 1
2λ′P,tλP,t+ 1
2
(bτ,n−1ν−λ′τ,tΣ
−1τ
)ΣτΣ
′τ
(bτ,n−1ν−λ′τ,tΣ
−1τ
)′+ 1
2
(bP,n−1−λ′P,tΣ
−1P
)ΣPΣ
′P
(bP,n−1−λ′P,tΣ
−1P
)′.
(B.26)
Lifting the two last parenthesis,
Et
[em$
t+1+p(n−1)t+1
]= e
−α+an−1+bP,n−1µP+(bτ,n−1−βτ
)τt+
(bP,n−1ΦP−βP
)Pt
− 12λ
′τ,tλτ,t+ 1
2 bτ,n−1νΣτΣ′τν
′b′τ,n−1+ 1
2λ′τ,tλτ,t−bτ,n−1νΣτλτ,t
− 12λ
′P,tλP,t+ 1
2 bP,n−1ΣPΣ′P
b′P,n−1+ 1
2λ′P,tλP,t−bP,n−1ΣPλP,t ,
(B.27)
and then rearranging,
Et
[em$
t+1+p(n−1)t+1
]= e−α+an−1+bP,n−1µP+ 1
2 bτ,n−1νΣτΣ′τν
′b′τ,n−1+ 1
2 bP,n−1ΣPΣ′P
b′P,n−1
+(bτ,n−1−βτ
)τt−bτ,n−1νΣτλτ,t
+(bP,n−1ΦP−βP
)Pt−bP,n−1ΣPλP,t .
(B.28)
Inserting for the market prices of risk, λτ,t and λP,t , implies
Et
[em$
t+1+p(n−1)t+1
]= e−α+an−1+bP,n−1µP+ 1
2 bτ,n−1νΣτΣ′τν
′b′τ,n−1+ 1
2 bP,n−1ΣPΣ′P
b′P,n−1
+(bτ,n−1−βτ
)τt−bτ,n−1νλ0,τ−bτ,n−1νλ1,ττt
+(bP,n−1ΦP−βP
)Pt−bP,n−1λ0,P−bP,n−1λ1,PPt ,
(B.29)
or
Et
[em$
t+1+p(n−1)t+1
]= e
−α+an−1+bP,n−1
(µP−λ0,P
)−bτ,n−1νλ0,τ
+ 12 bτ,n−1νΣτΣ
′τν
′b′τ,n−1+ 1
2 bP,n−1ΣPΣ′P
b′P,n−1
+[
bτ,n−1(I−νλ1,τ
)−βτ]τt
+[
bP,n−1
(ΦP−λ1,P
)−βP
]Pt
.
(B.30)
82 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
Thus, for
ean+bτ,nτt+bP,nPt = e−α+an−1+bP,n−1
(µP−λ0,P
)−bτ,n−1νλ0,τ
+ 12 bτ,n−1νΣτΣ
′τν
′b′τ,n−1+ 1
2 bP,n−1ΣPΣ′P
b′P,n−1
+[
bτ,n−1(I−νλ1,τ
)−βτ]τt
+[
bP,n−1
(ΦP−λ1,P
)−βP
]Pt
.
(B.31)
to hold for all states, it must be that
an = −α+an−1 +bP,n−1(µP−λ0,P
)−bτ,n−1νλ0,τ
+ 1
2bτ,n−1νΣτΣ
′τν
′b′τ,n−1 +
1
2bP,n−1ΣPΣ
′Pb′
P,n−1
bτ,n = bτ,n−1(I−νλ1,τ
)−βτbP,n = bP,n−1
(ΦP−λ1,P
)−βP(B.32)
Then yields are affine in the state variables, i.e.
y (n)t =An +Bτ,nτt +BP,nPt , (B.33)
where An =− 1n an , Bτ,n =− 1
n bτ,n , and BP,n =− 1n bP,n .
B.2 Data Construction
I use data that is standard in the literature. Monthly real per capita consumption
growth is constructed from the NIPA tables at the U.S. BEA. I use data on non-durables
and services. Each of the series are divided by their price indexes before they are added
together. Finally, the sum is divided by the population size, which is downloaded
from the Federal Reserve Bank of St. Louis (FRED) database. Inflation is constructed
from core U.S. CPI, which also is downloaded from the FRED database. Consumption
growth and inflation are both calculated as year-over-year log differences of the
consumption and CPI levels. Real per capita consumption is available from January
1959, whereas core U.S. CPI is available from January 1957.
Yield curve data are end-of-month unsmoothed Fama and Bliss (1987) zero-
coupon yields constructed in Le and Singleton (2013) for maturities of 12, 24, 36,
48, 60, 72, 84, 96, 108, and 120 months. I start the sample in November 1971 as in
Cieslak and Povala (2015). Prior to this period, data on bond prices with maturities
longer than 5 years is sparse (see Fama and Bliss (1987) and Le and Singleton (2013))
and thus excluded here. Data is available through December 2014, giving a total of
T = 518 observations.
B.3. ROBUSTNESS RESULTS 83
B.3 Robustness Results
B.3.1 Sample split - 1985:M1 to 2014:M12
Table B.1: Predictive Regressions
The table reports regression coefficient for r x(n)t+1 = Ztθ
(n) +u(n)t+1. Newey and West (1987) corrected t-
statistics using 18 lags are in parentheses () and Hansen and Hodrick (1980) corrected t-statistics using 12
lags are in brackets [].
Maturities2 3 4 5 6 7 8 9 10
Panel A: r x(n)t+1 = θ(n)
0 +θ(n)PC
PCt +u(n)t+1.
θ(n)PC
0.1850 0.3561 0.5844 0.7843 1.0243 1.2119 1.4241 1.6581 1.7718(1.8970) (1.9504) (2.2649) (2.5405) (2.6756) (2.8665) (3.0617) (3.1168) (3.1577)
[1.6590] [1.7056] [1.9843] [2.2296] [2.3581] [2.5346] [2.7165] [2.7738] [2.8085]
R2 0.11 0.11 0.14 0.16 0.18 0.19 0.21 0.22 0.21
Panel B: r x(n)t+1 = θ(n)
0 +θ(n)C P C Pt +u(n)
t+1.
θ(n)C P 0.2036 0.3865 0.6370 0.8142 1.0452 1.2124 1.3986 1.6246 1.6779
(3.0063) (3.0499) (3.6125) (3.8840) (3.8411) (4.0733) (4.1501) (4.0617) (3.9262)
[2.6529] [2.7022] [3.2098] [3.4772] [3.4522] [3.6854] [3.7730] [3.6881] [3.5794]
R2 0.15 0.15 0.20 0.20 0.22 0.23 0.24 0.25 0.22
Panel C: r x(n)t+1 = θ(n)
0 +θ(n)C F C Ft +u(n)
t+1.
θ(n)C F 0.1774 0.3813 0.6043 0.8036 1.0188 1.2346 1.4155 1.6012 1.7633
(3.9934) (4.6492) (5.3454) (6.0371) (6.6336) (7.0493) (7.3807) (7.1803) (7.3176)
[3.7148] [4.4221] [5.1594] [6.0092] [6.8913] [7.3253] [7.9218] [7.8297] [8.0063]
R2 0.17 0.22 0.27 0.30 0.32 0.36 0.36 0.36 0.37
Panel D: r x(n)t+1 = θ(n)
0 +θ(n)RF RFt +u(n)
t+1.
θ(n)RF 0.1678 0.3668 0.5876 0.7864 1.0099 1.2306 1.4187 1.6252 1.8070
(4.5082) (5.5549) (6.2337) (6.6635) (7.3047) (7.8272) (8.3678) (8.5503) (9.7131)
[4.3015] [5.4834] [6.1619] [6.6618] [7.5589] [8.0382] [8.9703] [9.4654] [11.5637]
R2 0.21 0.27 0.34 0.38 0.42 0.47 0.49 0.50 0.51
84 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
B.3.2 Holding Period: 3 months
Table B.2: Predictive Regressions
The table reports regression coefficient for r x(n)t+3/12
= Ztθ(n) +u(n)
t+3/12. Newey and West (1987) corrected
t-statistics using 5 lags are in parentheses () and Hansen and Hodrick (1980) corrected t-statistics using 3
lags are in brackets [].
Maturities2 3 4 5 6 7 8 9 10
Panel A: r x(n)t+3/12
= θ(n)0 +θ(n)
PCPCt +u(n)
t+3/12.
θ(n)PC
0.0941 0.3425 0.5141 0.7765 0.8228 1.1322 1.3727 1.6261 1.6256(0.9361) (1.6912) (1.8841) (2.2702) (1.9810) (2.3908) (2.6350) (2.9719) (2.5602)
[0.9280] [1.6512] [1.8314] [2.1904] [1.8838] [2.2800] [2.4988] [2.8433] [2.4292]
R2 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.04
Panel B: r x(n)t+3/12
= θ(n)0 +θ(n)
C P C Pt +u(n)t+3/12
.
θ(n)C P 0.2071 0.4389 0.5372 0.8639 0.9149 1.0930 1.3189 1.4520 1.5026
(2.2254) (2.4662) (2.1792) (2.9784) (2.7577) (2.8590) (3.0874) (3.1406) (2.8884)
[2.1504] [2.4024] [2.1543] [2.9875] [2.7823] [2.8884] [3.2231] [3.1706] [2.9137]
R2 0.06 0.06 0.05 0.07 0.06 0.06 0.07 0.07 0.06
Panel C: r x(n)t+3/12
= θ(n)0 +θ(n)
C F C Ft +u(n)t+3/12
.
θ(n)C F 0.1595 0.3861 0.5847 0.7982 0.9052 1.1224 1.3137 1.4918 1.5460
(4.3066) (5.0386) (5.4537) (5.7831) (5.4811) (5.8550) (6.1013) (6.3384) (5.8329)
[4.1907] [4.8226] [5.2015] [5.5012] [5.1700] [5.5158] [5.6985] [5.9749] [5.4895]
R2 0.07 0.10 0.11 0.12 0.11 0.13 0.14 0.14 0.13
Panel D: r x(n)t+3/12
= θ(n)0 +θ(n)
RF RFt +u(n)t+3/12
.
θ(n)RF 0.1684 0.3728 0.5865 0.7791 0.8902 1.1121 1.2960 1.4979 1.5528
(4.5246) (5.0169) (5.6531) (5.8983) (5.6622) (6.1930) (6.4921) (6.7693) (6.3247)
[4.2596] [4.6984] [5.3015] [5.5413] [5.3029] [5.7750] [6.0340] [6.3137] [5.8956]
R2 0.09 0.10 0.13 0.13 0.12 0.14 0.15 0.16 0.15
B.3. ROBUSTNESS RESULTS 85
B.3.3 Holding Period: 6 months
Table B.3: Predictive Regressions
The table reports regression coefficient for r x(n)t+6/12
= Ztθ(n) +u(n)
t+6/12. Newey and West (1987) corrected
t-statistics using 9 lags are in parentheses () and Hansen and Hodrick (1980) corrected t-statistics using 6
lags are in brackets [].
Maturities2 3 4 5 6 7 8 9 10
Panel A: r x(n)t+6/12
= θ(n)0 +θ(n)
PCPCt +u(n)
t+6/12.
θ(n)PC
0.0678 0.3144 0.5166 0.7565 0.9342 1.1630 1.3214 1.5722 1.6580(1.4846) (2.6257) (2.8141) (3.1122) (3.2951) (3.3767) (3.3504) (3.7370) (3.3813)
[1.3671] [2.4352] [2.6083] [2.8760] [3.0525] [3.1090] [3.0792] [3.4430] [3.0954]
R2 0.02 0.05 0.06 0.08 0.08 0.09 0.09 0.10 0.09
Panel B: r x(n)t+6/12
= θ(n)0 +θ(n)
C P C Pt +u(n)t+6/12
.
θ(n)C P 0.1173 0.3557 0.5495 0.8029 0.9654 1.1313 1.3009 1.4644 1.5969
(3.1620) (3.6965) (4.0030) (4.6613) (4.8839) (4.7665) (4.9991) (5.2116) (4.9227)
[3.0978] [3.6211] [3.8732] [4.5280] [4.7505] [4.6037] [4.8465] [5.0401] [4.7328]
R2 0.08 0.11 0.11 0.14 0.14 0.14 0.14 0.14 0.14
Panel C: r x(n)t+6/12
= θ(n)0 +θ(n)
C F C Ft +u(n)t+6/12
.
θ(n)C F 0.1064 0.3472 0.5646 0.7780 0.9265 1.1348 1.3162 1.4923 1.6082
(4.7192) (5.9206) (6.4044) (6.7024) (6.9512) (7.0751) (7.2896) (7.4352) (7.0344)
[4.5394] [5.7175] [6.1081] [6.3045] [6.6013] [6.6643] [6.8042] [6.9119] [6.5324]
R2 0.13 0.19 0.22 0.24 0.24 0.25 0.26 0.27 0.26
Panel D: r x(n)t+6/12
= θ(n)0 +θ(n)
RF RFt +u(n)t+6/12
.
θ(n)RF 0.1119 0.3399 0.5574 0.7559 0.9070 1.1250 1.3097 1.5019 1.6082
(4.8136) (5.9278) (6.5530) (6.8471) (7.1438) (7.4625) (7.8117) (8.0493) (7.6499)
[4.5512] [5.6603] [6.2077] [6.3965] [6.7329] [6.9948] [7.2702] [7.4720] [7.1026]
R2 0.15 0.21 0.24 0.25 0.25 0.28 0.29 0.30 0.29
86 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
B.3.4 Gurkaynak et al. (2007) data
Table B.4: Predictive Regressions
The table reports regression coefficient for r x(n)t+1 = Ztθ
(n) +u(n)t+1. Newey and West (1987) corrected t-
statistics using 18 lags are in parentheses () and Hansen and Hodrick (1980) corrected t-statistics using 12
lags are in brackets [].
Maturities2 3 4 5 6 7 8 9 10
Panel A: r x(n)t+1 = θ(n)
0 +θ(n)PC
PCt +u(n)t+1.
θ(n)PC
0.1956 0.3995 0.6059 0.8105 1.0119 1.2093 1.4023 1.5907 1.7744(2.5902) (2.8533) (3.0878) (3.2960) (3.4813) (3.6428) (3.7783) (3.8864) (3.9675)
[2.2848] [2.5055] [2.7071] [2.8894] [3.0541] [3.1993] [3.3226] [3.4227] [3.4995]
R2 0.09 0.12 0.14 0.16 0.17 0.18 0.19 0.20 0.20
Panel B: r x(n)t+1 = θ(n)
0 +θ(n)C P C Pt +u(n)
t+1.
θ(n)C P 0.1947 0.3955 0.5969 0.7982 1.0001 1.2024 1.4043 1.6048 1.8032
(2.7828) (3.1165) (3.4108) (3.6765) (3.9193) (4.1372) (4.3253) (4.4800) (4.5994)
[2.4583] [2.7406] [2.9945] [3.2262] [3.4397] [3.6321] [3.7993] [3.9379] [4.0464]
R2 0.10 0.13 0.15 0.17 0.19 0.21 0.22 0.23 0.24
Panel C: r x(n)t+1 = θ(n)
0 +θ(n)C F C Ft +u(n)
t+1.
θ(n)C F 0.2304 0.4436 0.6438 0.8352 1.0201 1.1996 1.3742 1.5440 1.7092
(5.6404) (6.1164) (6.5804) (7.0062) (7.3516) (7.5938) (7.7312) (7.7761) (7.7470)
[5.0562] [5.4695] [5.8861] [6.2794] [6.6071] [6.8433] [6.9822] [7.0331] [7.0124]
R2 0.32 0.36 0.39 0.42 0.44 0.45 0.47 0.47 0.48
Panel D: r x(n)t+1 = θ(n)
0 +θ(n)RF RFt +u(n)
t+1.
θ(n)RF 0.2299 0.4412 0.6403 0.8317 1.0177 1.1992 1.3762 1.5483 1.7155
(6.3949) (7.5433) (8.4357) (9.1154) (9.5904) (9.8788) (10.0109) (10.0203) (9.9376)
[5.7936] [6.8626] [7.6983] [8.3480] [8.8150] [9.1069] [9.2459] [9.2614] [9.1832]
R2 0.36 0.40 0.44 0.47 0.49 0.51 0.53 0.53 0.54
B.3.5 Out-of-Sample Predictability: Yield Levels
Because of the extreme persistence in yields, a great number of studies have docu-
mented difficulties in improving forecasting performance over a simple random walk,
i.e. y (n)t+h = y (n)
t for all h. Here, I show that accounting for the macroeconomic origin
of such persistence does in fact improve forecasting performance. Using the same
methodology as before, I construct forecasts from performing the regressions
y (n)t+h = Zt%
(n) +ζ(n)t+h , (B.34)
B.3. ROBUSTNESS RESULTS 87
on an expanding sample. As before, I start the test period in January 1990. The set of
predictor variables Zt is in turn i) the first three principle components ii) the first five
forward rates, iii) inflation expectations and two interest rate cycles, and finally iv)
the proposed four factors, i.e. expected consumption growth, expected inflation, and
two latent factors.
Table B.5: Out-of-Sample Prediction Results: Yield Level
This table reports out-of-sample root mean squared forecasting errors for the predictive regressions
y (n)t+h
= Zt%(n) +ζ(n)
t+hfor Zt being i) the first three principal components, ii) five forward rates, iii) inflation
expectations and two interest rate cycles, iv) and consumption growth expectations, inflation expectations,and two latent factors, (v) optimal and consumption growth expectations, inflation expectations, and twolatent factors. All numbers are relative to the root mean squared forecasting errors from using a simplerandom walk for the same maturity as forecast. Test statistics for the hypothesis that (iv) encompasses agiven variable are in parenthesis (), and that (v) encompasses a given variable are in brackets []. See Harveyet al. (1998). Standard errors are Newey and West (1987) corrected with ceil(h ×1.5) lags. The test sample is1990:1 to 2014:12. Factor identification and predictive coefficients are recursively updated as the trainingsample expands.
(i) (ii) (iii) (iv) (v)
Panel A: 5-year yieldh = 3/12 1.076 1.0742 1.020 0.988 0.984
(-0.068) (-0.200) (0.488)[0.075] [-0.062] [0.858]
h = 6/12 1.112 1.116 0.978 0.933 0.927(0.350) (0.333) (0.498)[0.458] [0.455] [0.949]
h = 1 1.249 1.233 1.008 0.926 0.924(0.064) (-0.169) (0.537)[-0.039] [-0.253] [0.7253]
Panel B: 7-year yieldh = 3/12 1.052 1.056 0.966 0.964 0.962
(0.057) (-0.124) (-0.608)[0.1101] [-0.056] [-0.294]
h = 6/12 1.085 1.081 0.929 0.917 0.912(0.381) (0.071) (-0.254)[0.349] [0.063] [0.077]
h = 1 1.228 1.215 0.957 0.895 0.896(-0.011) (-0.280) (0.277)[-0.259] [-0.527] [0.477]
Panel C: 10-year yieldh = 3/12 1.053 1.167 1.014 1.011 1.006
(-2.304) (0.269) (-0.648)[-2.010] [0.334] [-0.330]
h = 6/12 1.091 1.107 0.928 0.942 0.937(-0.513) (-0.597) (-1.283)[-0.471] [-0.563] [-1.066]
h = 1 1.227 1.206 0.931 0.926 0.929(-0.522) (-1.042) (-0.825)[-0.733] [-1.226] [-0.724]
I use root mean squared forecasting errors as measure of forecast performance.
The numbers are relative to the root mean squared forecasting errors from the simple
random walk for the same maturity. I implement the encompassing test in Harvey
et al. (1998), where the null hypothesis is that the expected consumption growth,
expected inflation, and two latent factors encompasses the other prominent bond
return factors.
88 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
The first two columns confirm the finding that the random walk is difficult to
beat with the usual term structure factors. Across maturities, the performance is
always worse than the random walk forecast by 5-25%. The performance worsen
as the forecast horizon increases, which is consistent with the term structure fac-
tors displaying excess mean reversion compared to what is seen in yields. Column
(iii) shows that accounting for persistent inflation expectations does improve the
forecasting performance relative to the usual term structure factors. Relative to the
random walk forecast, the forecasting performance is also improved for six and twelve
month horizons by by 4-7% for the 7-year and 10-year yields. For the 5-year yield,
the forecasting performance is comparable to the random walk. Finally, column
(iv) shows that controlling for expected consumption growth further improves the
forecasting performance. Accounting for the macroeconomic expectation factors
improves forecasting performance relative to the random walk benchmark across
maturities and forecasting horizons except for the 10-year yield with a three month
horizon. The forecasting performance relative to the random walk benchmark is
improved with the forecasting horizon, which is consistent with the interpretation
that macroeconomic expectations capture the most persistent source of variation
in yields. The null that expected consumption growth, expected inflation, and two
latent factors encompasses the other prominent bond return factors is not rejected
(with the exception of the level, slope, and curvature factors for the 10-year yield at
the three month horizon).
B.4 The Bauer and Hamilton (2018) Bootstrap
1. Estimate a VAR(1) for the level, slope, and curvature factors, i.e.
PCt+1 = µPC+ ΦPCPCt + vPC,t+1, (B.35)
and a VAR(1) for the proposed new factors. In this case, estimate a VAR(1) for
Ft =[τc,t P(1)
t P(2)t
]′, i.e.
Ft+1 = µF + ΦF Ft + vF,t+1. (B.36)
Finally, estimate the measurement equation for yields under the null, i.e.
y (n)t = α(n)
PC+ β(n)
PCPCt + κ(n)
t (B.37)
2. Draw 10,000 bootstrap samples, each with length T = 518 as in the original
sample, by iterating on
PC?s+1 = µPC+ ΦPCPC?s + v?PC,s+1
F?s+1 = µF + ΦF F?
s + v?F,s+1,(B.38)
B.5. OTHER LEARNING ALGORITHMS 89
where[
v?PC,s v?F,s
]′is drawn with replacement from the joint empirical dis-
tribution of[vPC,t vF,t
]′. Finally, the bootstrapped yield curve is computed
from
y?,(n)s = α(n)
PC+ β(n)
PCPC?s +κ?,(n)
s , (B.39)
where κ?,(n)s
i .i .d .∼ N(0,σ2
κ,n
). The measurement error standard deviation, σκ,n ,
is set to the standard error of the empirical measurement errors, κ(n)t .
3. For each bootstrap sample, run the regressions7
r x?,(n)s+1 = θ(n)
0 +θ(n)PC
PC?s +θ(n)F F?
s +u(n)s+1, (B.40)
and compute the Wald statistics χ2?NW and χ2?
H H associated with the null hypoth-
esisH0 : θ(n)F = 0.
B.5 Other Learning Algorithms
I consider two other learning algorithms outlined in Branch and Evans (2006). Both
learning algorithms allow for time-variation in the gain parameter, but in two impor-
tantly different ways.
B.5.1 Recursive Least Squares
The recursive least squares algorithm is a simple formulation of the recursively up-
dated mean. Here,
τt = τt−1 +νt(Mt −τt−1
), (B.41)
with νt = t−1. This structure implies,
τt = 1
t
t−1∑i=0
Mt−i . (B.42)
This implies that all of macroeconomic history receives equal weight at any point in
time. In this way, macroeconomic experiences are not weighted more heavily.
B.5.2 Kalman Filter Learning Rule
The Kalman Filter learning rule sets the gain parameter, νt , in an optimal way given
the observed data. Here, the recursions are given by
τt = τt−1 +νt(Mt −τt−1
)νt = Pt−1
σ2 +Pt
Pt = Pt−1 −P 2
t−1
σ2 +Pt−1+χ2σ2.
(B.43)
7The bootstrap excess returns are computed as r x?,(n)s+1 = ny?,(n)
s − (n −1) y?,(n−1)s+1 − y?,(1)
s .
90 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
Pt is an estimate of the variance of τt , and νt here is the optimal Kalman gain (Hamil-
ton, 1994). I follow Branch and Evans (2006) and estimate σ2 as the variance of the
residuals from an AR(1) model for Mt . I set χc = 0.009 and χπ = 0.016. This is in line
with evidence that inflation experienced greater structural change over the sample
period compared to real activity (Branch and Evans, 2006).
B.5.3 Results
I construct two bond return factor, RF RLSt and RF K F
t , following the same procedure
as in the main paper. The results from the maturity-specific predictive regressions
are reported below.
Table B.6: Predictive Regressions
The table reports regression coefficient for r x(n)t+1 = Ztθ
(n) +u(n)t+1. Newey and West (1987) corrected t-
statistics using 18 lags are in parentheses () and Hansen and Hodrick (1980) corrected t-statistics using 12
lags are in brackets [].
Maturities2 3 4 5 6 7 8 9 10
Panel A: r x(n)t+1 = θ(n)
0 +θ(n)RLS RF RLS
t +u(n)t+1.
θ(n)RLS 0.2197 0.4289 0.6261 0.8355 1.0337 1.1810 1.4288 1.5610 1.6852
(3.3135) (3.7794) (4.1957) (4.6530) (4.7691) (4.6853) (5.0825) (4.9936) (4.7582)
[2.9172] [3.3283] [3.7036] [4.1120] [4.2206] [4.1352] [4.4942] [4.4051] [4.2038]
R2 0.16 0.18 0.20 0.23 0.25 0.24 0.27 0.27 0.25
Panel B: r x(n)t+1 = θ(n)
0 +θ(n)K F RF K F
t +u(n)t+1.
θθ(n)K F 0.2154 0.4319 0.6379 0.8285 1.0215 1.2036 1.3974 1.5400 1.7239
(5.5073) (6.6604) (7.6016) (8.3371) (9.1651) (9.9005) (10.0281) (10.2180) (10.7496)
[4.9678] [6.0326] [6.8962] [7.5999] [8.4485] [9.1668] [9.3866] [9.5759] [10.1642]
R2 0.33 0.39 0.44 0.48 0.51 0.53 0.55 0.56 0.55
B.6 Information in Cycle Factors
The natural question to ask is, what kind of variation does the latent factors cap-
ture? In order to address this question, I augment the Mincer and Zarnowitz (1969)
regressions for consumption and inflation with the two latent factors.
Table B.7 shows that the latent slope factor predicts future consumption growth
one month, one quarter, and half a year ahead with a negative regression slope.
The significance of the added predictive power increases with the horizon, and is
statistically significant at the 10 percent significance level for h = 3/12 and at the 5
percent significance level for h = 6/12. Both latent factors are strong predictors of
future inflation at a 1 percent significance levels for all horizons. Rudebusch and
B.6. INFORMATION IN CYCLE FACTORS 91
Table B.7: Augmented Predictive Macroeconomic Regressions
Results are for the regressions ∆ct+h = ρ(h)0,c +ρ(h)
1,c τc,t +ρ(h)2,c P
(1)t +ρ(h)
3,c P(2)t + εc,t+h and πt+h = ρ
(h)0,π +
ρ(h)1,πτπ,t +ρ(h)
2,πP(1)t +ρ(h)
3,πP(2)t + επ,t+h . Newey and West (1987) corrected t-statistics in absolute values
using 5 lags for the null hypothesisH0 : ρ(h)2,i = 0 andH0 : ρ(h)
3,i = 0 for i ∈ {c,π} are in parenthesis.
Panel A : ∆ct+h
h = 1/12 h = 3/12 h = 6/12
ρ(h)2,c −0.0097
(0.2234)−0.0069
(0.3401)0.0050(0.1387)
ρ(h)3,c −0.1441
(1.2701)−0.2043
(1.8521)−0.2765
(2.5723)
R2 0.21 0.17 0.13
Panel B : πt+h
h = 1/12 h = 3/12 h = 6/12
ρ(h)2,π −0.1220
(2.8195)−0.1361
(3.0415)−0.1717
(3.4322)
ρ(h)3,π 0.8233
(5.6388)0.8781(5.4770)
0.9483(5.3548)
R2 0.68 0.67 0.64
Williams (2009) document that the yield curve has information useful for forecasting
recessions above and beyond that of professional forecasters. This is in line with the
interpretation of the latent factors governing variation in marginal utilities that is not
reflected in macroeconomic expectations. Because the latent factors are reflected in
marginal utilities of consumption, they are priced into the bond market, although
the factors are not processed when the representative household forms conditional
expectations of macroeconomic variables. In this sense, the representative household
has irrational conditional expectations, although expectations are not systematically
wrong and thus not irrational in the sense of Mincer and Zarnowitz (1969).
92 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
B.7 Term Premia Estimates
B.7.1 Unrestricted Model: Monthly VAR
Table B.8: Unrestricted Model: Learning from Macro Experiences
This table reports the estimated model parameters. Estimation is by OLS.
Short Rate Parameters:
α(12) β(12)c β(12)
π β(12)1 β(12)
2−0.0255(0.0007)
1.4319(0.0365)
1.5902(0.0132)
0.2815(0.0058)
0.7154(0.0211)
Time Series Parameters:
µP(1,1
)µP
(2,1
)ΦP
(1,1
)ΦP
(1,2
)ΦP
(2,1
)ΦP
(2,2
)4.26×10−5
(0.0005)2.35×10−5
(0.0002)0.9318(0.0178)
0.0472(0.0504)
−0.0002(0.0062)
0.9467(0.0188)
Table B.9: Unrestricted Model: Three-factor DTSM
This table reports the estimated model parameters. Estimation is by OLS.
Short Rate Parameters:
α(12) β(12)1 β(12)
2 β(12)3
0.0001(0.0002)
0.3103(0.0007)
0.6461(0.0046)
0.5451(0.0183)
Time Series Parameters:
µPC
(1,1
)ΦPC
(1,1
)ΦPC
(1,2
)ΦPC
(1,3
)0.0019(0.0020)
0.9918(0.0084)
0.0435(0.0450)
−0.0079(0.1989)
µPC
(2,1
)ΦPC
(2,1
)ΦPC
(2,2
)ΦPC
(2,3
)−0.0005
(0.0007)0.0006(0.0025)
0.9580(0.0155)
−0.0543(0.0794)
µPC
(3,1
)ΦPC
(3,1
)ΦPC
(3,2
)ΦPC
(3,3
)0.0012(0.0003)
−0.0037(0.0010)
0.0112(0.0076)
0.8188(0.0320)
B.7. TERM PREMIA ESTIMATES 93
Figure B.1: Model Implied Term Premia
This figure reports the decomposition of model implied yields with 10 years to maturity into term premiaand short rate expectations. The leftmost graph plots the decomposition for the proposed learning frommacroeconomic experiences model, and the rightmost graph plots the decomposition for the benchmarkthree-factor DTSM.
94 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
Figure B.2: Equlibrium Real Rate
This figure reports the model implied equilibrium real rate, r?t . The equilibrium real rate is defined to be
r?t = limh→∞
Et[rt+h
], where rt = y (12)
t −Et[πt+12
]denotes the real short rate. Using the model structure,
the equilibrium real rate is given by r?t = α(12) +β(12)τ,c τc,t +
(β(12)τ,π −1
)τπ,t +β(12)
P
(I−ΦP
)−1µP, since
the long-run consumption growth and inflation expectations are given by limh→∞
Et[∆ct+h
] = τc,t and
limh→∞
Et[πt+h
]= τπ,t , respectively.
B.7.2 Unrestricted Model: Annual VAR
Table B.10: Unrestricted Model: Learning from Macro Experiences
This table reports the estimated model parameters. Estimation is by OLS.
Short Rate Parameters:
α(1) β(1)c β(1)
π β(1)1 β(12)
2−0.0255(0.0007)
1.4319(0.0365)
1.5902(0.0132)
0.2815(0.0058)
0.7154(0.0211)
Time Series Parameters:
µP(1,1
)µP
(2,1
)ΦP
(1,1
)ΦP
(1,2
)ΦP
(2,1
)ΦP
(2,2
)0.0007(0.0043)
0.0002(0.0014)
0.2515(0.1200)
0.7253(0.3332)
−0.0512(0.0398)
0.5884(0.1073)
B.7. TERM PREMIA ESTIMATES 95
Table B.11: Unrestricted Model: Three-factor DTSM
This table reports the estimated model parameters. Estimation is by OLS.
Short Rate Parameters:
α(1) β(1)1 β(1)
2 β(1)3
0.0001(0.0002)
0.3103(0.0007)
0.6461(0.0046)
0.5451(0.0183)
Time Series Parameters:
µPC
(1,1
)ΦPC
(1,1
)ΦPC
(1,2
)ΦPC
(1,3
)0.0234(0.0234)
0.8999(0.0964)
0.6319(0.4474)
0.4884(1.7878)
µPC
(2,1
)ΦPC
(2,1
)ΦPC
(2,2
)ΦPC
(2,3
)−0.0041
(0.0058)0.0004(0.0202)
0.5971(0.1097)
−0.5058(0.4039)
µPC
(3,1
)ΦPC
(3,1
)ΦPC
(3,2
)ΦPC
(3,3
)0.0035(0.0011)
−0.0121(0.0035)
−0.0063(0.0280)
0.3793(0.0717)
B.7.3 No-Arbitrage: Annual VAR
B.7.3.1 Estimation
I estimate the model by maximum likelihood treating the four factors as observed.
All yields included in the estimation are taken to be measured with a small amount of
noise, i.e.
yot − yt = εt , (B.44)
where yot is the vector of observed yields, yt is the vector of model-implied yields.
The measurement errors εt ∼ N(0,σ2
εI)
are uncorrelated with other innovations
at all leads and lags. I implement the model using an annual VAR(1) for the two
latent pricing factors, but sampled at the monthly frequency. That is, the latent
pricing factors are regressed on a constant and their one-year lagged values. These
regressions are estimated using the monthly observations. As suggested by the results
in Cochrane and Piazzesi (2005) and Cochrane and Piazzesi (2008), the dynamics
of risk premia are more clear at an annual horizon. Further, the annual horizon is
consistent with the one-year holding period considered throughout this text. This
implies the likelihood function
logL(Θ
)= T∑t=1
log fy
(y0
t |Pt ,τt ,Θ)+ log fP
(Pt |Pt−12,Θ
), (B.45)
where fy and fP are normal pdfs. The structure of the model admits a separation of
the parametersΘ into time series parameters governing the dynamics of the factors
and cross-sectional parameters determining the factor loadings. This separation is
such that Θ1 = {α,βτ,βP,λ0,τ,λ1,τ,λ0,P,λ1,P,ΣP} and Θ2 = {µP,ΦP}, and the likeli-
hood reads
logL(Θ
)= T∑t=1
log fy
(y0
t |Pt ,τt ,Θ1
)+ log fP
(Pt |Pt−12,Θ1,Θ2
). (B.46)
96 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
The only content of Θ1 which affects fP is the covariance matrix ΣP, and from the
well-known Zellner (1962) result, this covariance matrix does not affect the maximum
likelihood estimates of the parameters inΘ2. Thus, as suggested in Joslin, Singleton,
and Zhu (2011), I adopt a two-step procedure where, firstly, I estimateΘ2 by simple
OLS, and then secondly, maximize overΘ1 conditional on the estimates ofΘ2. Here,
the first step provides good starting values for ΣP in the second step.
B.7.3.2 Three-factor ATSM: Estimation Results
Table B.12: No-Arbitrage Restrictions: Three-factor ATSM
This table reports the estimated model parameters when enforcing the no-arbitrage restrictions. Asymp-totic standard errors computed from the outer product of the gradient of the likelihood function areprovided in parenthesis.
Short-rate Parameters:
α βPC
(1,1
)βPC
(1,2
)βPC
(1,3
)0.0001(0.0521)
0.3106(0.1213)
0.6461(0.1532)
0.5250(0.1659)
Market Price of Risk Parameters:
λ0,PC
(1,1
)λ1,PC
(1,1
)λ1,PC
(1,2
)λ1,PC
(1,3
)0.0225(0.0136)
−0.1067(0.0437)
1.4613(0.2466)
1.0281(0.6762)
λ0,PC
(2,1
)λ1,PC
(2,1
)λ1,PC
(2,2
)λ1,PC
(2,3
)−0.0035
(0.0037)−0.0080
(0.0122)−0.0590
(0.0685)0.7876(0.2655)
λ0,PC
(3,1
)λ1,PC
(3,1
)λ1,PC
(3,2
)λ1,PC
(3,3
)0.0040(0.0007)
−0.0138(0.0024)
0.0011(0.0144)
−0.3700(0.0582)
Variance-Covariance of Innovations:
ΣPC
(1,1
)ΣPC
(2,1
)ΣPC
(2,2
)0.0398(0.0014)
0.0055(0.0005)
0.0088(0.0004)
ΣPC
(3,1
)ΣPC
(3,2
)ΣPC
(3,3
)−0.0001
(0.0001)0.0006(0.0001)
0.0024(0.0001)
Time-Series Parameters:
µPC
(1,1
)ΦPC
(1,1
)ΦPC
(1,2
)ΦPC
(1,3
)0.0234(0.0081)
0.8999(0.0267)
0.6319(0.1984)
0.4884(0.6260)
µPC
(1,1
)ΦPC
(1,1
)ΦPC
(1,2
)ΦPC
(1,3
)−0.0041
(0.0021)0.0004(0.0081)
0.5971(0.0571)
−0.5058(0.1911)
µPC
(1,1
)ΦPC
(1,1
)ΦPC
(1,2
)ΦPC
(1,3
)0.0035(0.0006)
−0.0121(0.0022)
−0.0063(0.0121)
0.3793(0.0468)
B.7. TERM PREMIA ESTIMATES 97
B.7.3.3 Learning from Macroeconomic Experiences: Estimation Results
Table B.13: No-Arbitrage Restrictions: Model Parameters
This table reports the estimated model parameters when enforcing the no-arbitrage restrictions. Asymp-totic standard errors computed from the outer product of the score function are provided in parenthesis.
Short-rate Parameters:
α βτ(1,1
)βτ
(1,2
)βP
(1,1
)βP
(1,2
)−0.0263
(0.0108)1.4536(0.2738)
1.6019(0.1529)
0.2833(0.1113)
0.7113(0.2513)
Market Price of Risk Parameters:
λ0,τ(1,1
)λ0,τ
(2,1
)λ1,τ
(1,1
)λ1,τ
(1,2
)λ1,τ
(2,1
)λ1,τ
(2,2
)−0.0033
(0.0080)−0.0088
(0.0162)0.0635(0.1499)
−0.0634(0.0603)
0.1035(0.0212)
0.0570(0.0336)
λ0,P(1,1
)λ0,P
(2,1
)λ1,P
(1,1
)λ1,P
(1,2
)λ1,P
(2,1
)λ1,P
(2,2
)0.0287(0.0935)
0.0029(0.0107)
−0.7827(0.1039)
1.6033(0.4191)
−0.1059(0.0316)
0.0834(0.0815)
Variance-Covariance of Innovations:
ΣP(1,1
)ΣP
(2,1
)ΣP
(2,2
)0.0304(0.0010)
0.0045(0.0005)
0.0085(0.0004)
Time Series Parameters:
µP(1,1
)µP
(2,1
)ΦP
(1,1
)ΦP
(1,2
)ΦP
(2,1
)ΦP
(2,2
)0.0007(0.0019)
0.0002(0.0005)
0.2515(0.0569)
0.7253(0.1686)
−0.0512(0.0168)
0.5884(0.0513)
98 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
B.7.3.4 Model-implied Term Premia and Equilibrium Real Rate
Figure B.3: Model Implied Term Premia
This figure reports the decomposition of model implied yields with 10 years to maturity into term premiaand short rate expectations. The leftmost graph plots the decomposition for the proposed learning frommacroeconomic experiences model, and the rightmost graph plots the decomposition for the benchmarkthree-factor DTSM.
B.7. TERM PREMIA ESTIMATES 99
Figure B.4: Equlibrium Real Rate
This figure reports the model implied equilibrium real rate, r?t . The equilibrium real rate is defined to be
r?t = limh→∞
Et[rt+h
], where rt = y (1)
t −Et[πt+1
]denotes the real short rate. Using the model structure, the
equilibrium real rate is given by r?t =α+βτ,cτc,t +(βτ,π−1
)τπ,t +βP
(I−ΦP
)−1µP, since the long-run
consumption growth and inflation expectations are given by limh→∞
Et[∆ct+h
]= τc,t and limh→∞
Et[πt+h
]=τπ,t , respectively.
B.7.3.5 Model-Implied Term Premia: Specification Tests
Since the contribution of Dai and Singleton (2002), a popular way of assesing if model-
implied term premia are well-specified is by means of the ordinary Campbell and
Shiller (1991) regressions, i.e.
y(n−h)t+h − y (n)
t = θ(n)0 +θ(n)
1
h
n −h
(y (n)
t − y(h)t
)+ε(n)
t+h , (B.47)
and the adequately risk-adjusted Campbell and Shiller (1991) regressions, i.e.
y(n−h)t+h − y (n)
t −∆ψ(n−h)t ,t+h + h
n −hζ
(n−h,h)t = δ(n)
0 +δ(n)1
h
n −h
(y (n)
t − y(h)t
)+e(n)
t+h , (B.48)
where ∆ψ(n−h)t ,t+h =ψ
(n−h)t+h −ψ(n−h)
t and ζ(n,h)t = f (n,h)
t −Et[rt+n
]is the forward pre-
mium. The intuition behind these specification tests is straightforward. If model-
implied term premia are correctly specified they should account for the empirically
observed deviations from the expectation hypothesis, e.g. the evidence against the
expectation hypothesis presented in Campbell and Shiller (1991). That is, model-
implied yields should display the same pattern of negative and decreasing regression
100 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
slopes θ(n)1 as function of maturity as seen in the data. Further, adequately risk-
adjusting yields, i.e. accounting for ∆ψ(n−h)t ,t+h and ζ(n,h)
t as in (B.48), should recover
δ(n)1 = 1 for all n as implied by the expectation hypothesis.
I simulate 100,000 observations from my four-factor model and the three-factor
Gaussian DTSM. To simulate from the macroeconomic variables in my model, I fit an
annual VAR(1) sampled at a monthly frequency. I use the estimated annual VAR(1)
to sample from consumption growth and inflation, and I then compute implied
conditional expectations as perceived by the representative household using the
constant-gain algorithm. In the leftmost column of figure B.5, I report results for
my four-factor model. The model-implied ordinary Campbell and Shiller (1991)
regression slopes are negative and decreasing with maturity as seen in the data.
Although negative and decreasing, the model-implied regression slopes are slightly
above those observed in the data. However, the model-implied regression slopes are
not outside the 95% confidence intervals, with the exception of the 10-year maturity
which is a borderline case. Thus it is likely that the model could have generated the
pattern observed in the data. In the same way, the risk-adjusted regressions show
that the expectations hypothesis implied unity regression slopes are within the 95%
confidence intervals for all maturities except the 10-year bond, which is again a
borderline case. And so, the proposed model has well-specified model-implied risk
corrections judged from the metric proposed in Dai and Singleton (2002).
B.7. TERM PREMIA ESTIMATES 101
Figure B.5: Ordinary and Risk-Adjusted Campbell-Shiller Regressions
This figure plots ordinary and risk-adjusted Campbell and Shiller (1991) regressions. Learning fromMacroeconomic Experiences is in the leftmost column, and the three-factor DTSM is in the rightmostcolumn. Model-implied ordinary Campbell and Shiller (1991) regression coefficients are obtained fromsimulated samples of 100,000 observations. Confidence bounds are computed from Hansen and Hodrick(1980) corrected standard errors using 12 lags.
B.7.3.6 Out-of-Sample Forecasting Performance
Another way to study the specification of conditional moments is by evaluating how
well the models predict future yields out of sample. Although I have documented
out-of-sample forecasting gains by using the four yield curve factors identified in this
paper, it is interesting how well the models perform when enforcing the no-arbitrage
restrictions. I use the data from November 1971 through January 1990 to estimate
the model parameters and then use the model structure to forecast yields 12 months
ahead. I then add one month of data, re-estimate the model parameters, and use the
model structure to forecast yields 12 months ahead. I iterate on this procedure until
December 2013 as the last year of data is reserved for evaluating the forecasts.
I evaluate the performance over two test samples. The first is the period 1990-
2008, which excludes the period with yields near their zero lower bound (ZLB) for the
shortest maturities. Second, I consider the full period 1990-2015 which includes the
ZLB period. Over both test samples, my model shows sizeable improvements over the
standard three-factor Gaussian DTSM. Over the pre-ZLB period the model further
systematically outperforms the random walk benchmark. Here, the improvements
are in the order of 11-23%, whereas the three-factor Gaussian DTSM has root mean
squared forecasting errors that are in the order of 16-25% greater than the random
102 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
walk. Over the full test sample, the relative ranking of the two models favours my
model over the standard three-factor Gaussian DTSM for all maturities. However,
the random walk does better when including the ZLB period. This is to be expected,
since short rates are constrained by the ZLB and thus are extraordinarily persistent
in this part of the sample. The rest of the maturity spectrum inherits much of this
persistence, and this implies that a random walk forecasts extraordinarily well. Over
the full test sample, my model and the random walk display very similar forecasting
performance. Over the sample where yields were unconstrained by their ZLB my
model shows sizeable gains over both the standard three-factor Gaussian DTSM and
the random walk. I cannot reject the null that the learning from macroeconomic
experience model forecasts encompass the standard three factor model forecasts for
any of the considered maturities.
Table B.14: No-Arbitrage Restrictions: Out-of-Sample Forecasts
This table reports one year ahead out-of-sample forecast for (i) the three-factor DTSM and (ii) the learningfrom macroeconomic experiences four-factor model when enforcing the no-arbitrage restrictions. Allnumbers are relative to the root mean squared forecasting errors from using a simple random walk forthe same maturity as forecast. Test statistics for the hypothesis that the learning from macroeconomicexperiences forecast encompasses the three factor model forecasts are in parenthesis (). Parameters arere-estimated as the training sample expands.
Maturity, n 1990-2008 Sample 1990-2015 Sample
(i) (ii) (i) (ii)
1 1.246(0.214)
0.891 1.297(0.125)
0.998
2 1.212(0.158)
0.875 1.297(0.470)
1.048
3 1.181(−0.177)
0.845 1.279(0.593)
1.057
4 1.162(−0.471)
0.816 1.254(0.489)
1.048
5 1.166(−0.493)
0.796 1.247(0.586)
1.035
6 1.170(−0.567)
0.787 1.236(0.459)
1.024
7 1.171(−0.674)
0.776 1.219(0.402)
1.010
8 1.173(−0.782)
0.767 1.212(0.357)
1.005
9 1.190(−0.789)
0.773 1.215(0.403)
1.006
10 1.204(−0.779)
0.771 1.207(0.504)
0.983
B.7. TERM PREMIA ESTIMATES 103
B.7.3.7 Simulation Evidence
I argued that term structure data is insufficient to identify the true risk factors, i.e. a
principal component analysis would account for all relevant variation in the cross-
section of bond yields, but would fail to identify the true underlying factor structure.
Here, I provide some simulation evidence to support this statement further. I simulate
1,000 test data sets from the proposed four-factor model and the standard three-factor
Gaussian DTSM with T = 518 observations as in the original sample. I simulate using
the model parameters as estimated from the data and the fitted annual VAR(1) for
consumption growth and inflation. The simulated macroeconomic variables are then
used to form conditional expectations using the constant-gain learning algorithm.
Then, I perform a singular value decomposition for each test data set to detect the
model-implied factor structure. If the factor loading matrices do not suffer from
invertibility issues the singular value decomposition should identify (i) the three-
factor structure in the Gaussian DTSM, and (ii) the four-factor structure in my model.
In Table B.15, I report the average variation explained by each principal compo-
nent across the test data sets. The singular value decompositions correctly identify
three risk factors in the test data sets simulated from the Gaussian DTSM. Although
the variation explained by the third principal component is low, the significant drop
in explanatory power is going from three to four factors. However, the singular value
decomposition of the test data sets simulated from my model displays a two-factor
structure. This is evidence that yield data is insufficient to recover the true under-
lying factor structure. Further, in column (iii) I find that when conditioning on the
macroeconomic factors, the two-factor structure remains. This is consistent with
the identification of the latent factors from the variation that is orthogonal to the
macroeconomic factors. That is, only when conditioning on the macroeconomic
factors is it possible to identify the true four-factor structure in the data generating
process.
I then investigate the properties of the implied model factor structure, if the
econometrician misspecifies the learning parameters. The investigation is based on
1,000 test economies with T = 518 observations each as in the original sample. Table
B.16 reports the explained variation by each principal component of the residual vari-
ation in yields after conditioning on the (misspecified) conditional macroeconomic
expectations.
In conclusion, misspecification of the learning parameter does not wash away
the factor structure in yields conditional on macroeconomic expectation factors.
Next, I investigate if misspecification of the learning parameter induces bias in
the VAR transition matrix of the latent factor dynamics. Table B.17 reports the bias
estimates obtained from the simulated test data sets from above.
In conclusion, small misspecifications does not induce severe bias in the transi-
tion matrix of the VAR dynamics for the latent variables. However, extreme misspeci-
fication of the learning parameters can induce a bias.
104 CHAPTER 2. MACROECONOMIC EXPERIENCES AND THE YIELD CURVE
Table B.15: Model-Implied Factor Structure: Simulation Evidence
This table reports the average variation explained by the principal components of 1,000 test data setsobtained by simulating from (i) the three-factor DTSM, and (ii) the learning from macroeconomic experi-ences four-factor model. The third column (iii) reports the average variation explained by the principalcomponents of the 1,000 test data sets obtained by simulating from the learning from macroeconomicexperiences four-factor model and conditioning on the macroeconomic expectation factors. Each testdata set consists of T = 518 observations as in the original sample, and the simulations are conductedusing the estimated parameter values.
Variation explained by i ’th PC
Three-factor model Four-factor model Four-factor model
unconditional conditional on macro factors
i (i) (ii) (iii)
1 98.913% 96.081% 92.877%
2 0.983% 3.507% 6.233%
3 0.065% 0.075% 0.158%
4 0.007% 0.063% 0.143%
5 0.006% 0.057% 0.130%
6 0.006% 0.053% 0.118%
7 0.006% 0.048% 0.106%
8 0.005% 0.044% 0.093%
9 0.005% 0.039% 0.079%
10 0.004% 0.033% 0.062%
Table B.16: Simulation Evidence: Factor Structure
This table reports the average variation explained by the principal component of 1,000 test data sets
obtained by simulating from the learning from macroeconomic experiences model. Principal components
are of the residual variation in model-implied yields after conditioning on the (misspecified) conditional
macroeconomic expectations. The true data generating process has ν= 0.016.
Variation explained by i ’th PC
i ν= 0.016 ν= 0.018 ν= 0.020 ν= 0.050
1 92.872% 92.893% 92.946% 94.202%2 6.237% 6.221% 6.176% 5.125%3 0.159% 0.158% 0.156% 0.119%4 0.143% 0.142% 0.141% 0.106%5 0.130% 0.129% 0.128% 0.097%6 0.118% 0.117% 0.116% 0.088%7 0.106% 0.106% 0.105% 0.080%8 0.094% 0.093% 0.092% 0.072%9 0.078% 0.079% 0.078% 0.062%
10 0.062% 0.061% 0.061% 0.051%
B.7. TERM PREMIA ESTIMATES 105
Table B.17: Simulation Evidence: Bias in VAR Dynamics
This table reports the estimated bias induced in the VAR transition matrix of the latent factor dynamics bymisspecification of the learning parameter ν. The true data generating process has ν= 0.021.
Bias, ν= 0.016 Bias, ν= 0.018
-0.011 -0.004 -0.007 -0.0080.000 -0.019 0.001 -0.019
Bias, ν= 0.020 Bias, ν= 0.050
0.002 -0.018 0.199 -0.2210.002 -0.020 0.024 -0.034
C H A P T E R 3BOND RISK PREMIA AT
THE ZERO LOWER BOUND
Martin M. AndreasenAarhus University, CREATES, and the Danish Finance Institute
Kasper JørgensenAarhus University and CREATES
Andrew C. MeldrumBoard of Governors of the Federal Reserve System
Abstract
We study bond risk premia at the zero lower bound. In predictive regressions of
excess bond returns onto yield spreads, we document a structural break in regression
coefficients over the recent low interest rate regime. The standard three-factor shadow
rate model fails to account for this empirical pattern. Instead, we propose a shadow
rate model with market prices of risk that switch across non-binding and binding zero
lower bound regimes. Our shadow rate model with regime-dependent market prices
of risk is consistent with the provided regression evidence. The regime-switching
shadow rate model suggests that markets expected monetary policy lift-off to occur
later than otherwise thought.
107
108 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
3.1 Introduction
The classical studies by Fama and Bliss (1987) and Campbell and Shiller (1991) relate
the slope of the current yield curve to risk premia in bond markets. This evidence is in
essence based on predictive linear regressions of excess bond returns onto measures
of yield curve slope. However, the recent episodes with prolonged periods of short-
term interest rates being restricted by their zero lower bound (ZLB) across several
countries poses a serious challenge to this linear relation. As the short end of the
yield curve becomes constrained from below, this in turn affects the slope of the yield
curve and generates a "slope compression effect". That is, the slope of the yield curve
is flatter than it would otherwise have been in the absence of a ZLB, meaning that a
given slope of the yield curve carries a stronger signal at the ZLB. Furthermore, the
recent low interest rate environment has called for unconventional monetary policies
such as forward guidance and quantitative easing. This is likely to affect the required
compensations for risk by bond investors, meaning that we also may have a "price of
risk effect".
We study modified regressions of 1-year excess bond holding period returns onto
the yield spread that allow for separate intercepts and regression slope coefficients
over non-binding and binding ZLB periods. Our modified excess bond return regres-
sions reveal a structural break in the predictive regression coefficients. In "normal"
times — when the federal funds rate is deemed unrestricted by its lower bound — we
find the typical pattern. The regressions slope coefficients are positive and increasing
in the maturity of the considered bond. This pattern is amplified when the federal
funds rate is restricted by its ZLB; the regression slope coefficients are larger and
increase faster in the maturity of the considered bond. The estimated differences in
predictive regression coefficients are statistically significant for bonds with maturities
in the 3- through 10-year range. This new empirical fact is robust to: (i) measuring
the slope of the yield curve by a forward spread or the second principal component
of yields instead of the yield spread, (ii) including yield curve level and curvature
factors as control variables, and (iii) including macroeconomic information as control
variables.1
Dai and Singleton (2002) use the classical Campbell-Shiller regression evidence
as a specification test for the popular affine term structure models developed in
Duffie and Kan (1996) and Duffee (2002). In a similar spirit, we use our modified
excess return regressions to conduct a specification test of the popular shadow rate
model (SRM) developed in Black (1995) that enforces the ZLB.2 The idea is simple; a
1We focus on real activity measured by the Chicago Fed National Activity Index (Joslin et al., 2014) andtrend inflation (Cieslak and Povala, 2015). Other macroeconomic variables that have been found to havepredictive information about future bond returns include the output gap (Cooper and Priestley, 2008),factors extracted from a large macroeconomic data set (Ludvigson and Ng, 2009), and Treasury bondsupply (Greenwood and Vayanos, 2014).
2Kim and Singleton (2012), Christensen and Rudebusch (2015), Bauer and Rudebusch (2016), and Wuand Xia (2016) study multi-factor versions of the SRM. The ZLB may also be enforced by affine models with
3.1. INTRODUCTION 109
well-specified model of bond risk premia at the ZLB should capture the model-free
empirical patterns from our modified excess return regressions. The SRM seems like
a natural candidate, since it maintains the properties of the affine term structure
model away from the ZLB.
Our main results from estimating a standard three-factor SRM on monthly U.S.
data from January 1990 through December 2017 are as follows. First, the standard
three-factor SRM does not pass our new specification test. The SRM does achieve
differences in regression coefficients when regressing excess bond returns onto the
yield spread across non-binding and binding ZLB periods. These differences, however,
are quantitatively far too small to explain the patterns observed empirically. That is,
the slope compression effect of the standard SRM is not strong enough to explain the
structural break in bond risk premia dynamics at the ZLB. Second, we also show that
the inability of the standard SRM to match this break is that it implies spells of binding
ZLB that are too short-lived. This is contradictory to one of the arguments that have
popularized the SRM; its capability to generate potentially long spells of zero short-
term interest rates. The implication is that the ZLB regime is characterized by strong
mean-reversion forces pulling the short term interest rate towards its unconditional
mean — far above zero in our sample. These mean-reversion forces contaminate the
ZLB dynamics and explain why the standard three-factor SRM fail our specification
test.
To evaluate the importance of the price of risk effect, we propose a three-factor
SRM with regime-dependent market prices of risk. Here, the marginal bond investor
requires different risk compensations at and away from the ZLB. The ZLB has called
for unconventional policies such as forward-guidance and large scale asset purchases.
Contrary to conventional policies, these policies aim at affecting the slope of the
yield curve through long-maturity bond prices. As these unconventional policies
change the nature of risks in bond markets, we expect bond investors to update the
compensations they require to take on this risk.
We document the following results from estimating our three-factor SRM with
regime-dependent market prices of risk. First, the regime-switch in market prices of
risk across non-binding and binding ZLB periods is statistically significant. Second,
our extension of the SRM passes the proposed specification test. Our extension of
the SRM matches the empirical patterns in regressions slope coefficients both at and
away from the ZLB. One particular reason is that the extended model implies dura-
tions of ZLB spells that are potentially much longer than implied by the standard SRM.
This is because the extended model implies mean-reversion towards a much lower
long-run short rate within the ZLB regime. That is, the SRM with regime-dependent
market prices of risk does not have very strong mean-reversion forces that pulls the
square-root processes, as in Cox, Ingersoll, and Ross (1985) and Dai and Singleton (2000), or by quadraticterm structure models as in Ahn, Dittmar, and Gallant (2002) and Leippold and Wu (2002). More recently, anumber of alternative ways to enforce the ZLB has been proposed by Feunou, Fontaine, Le, and Lundblad(2015), Filipovic, Larsson, and Trolle (2017), and Monfort, Pegoraro, Renne, and Roussellet (2017).
110 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
model away from the ZLB regime. Third, the extended SRM implies expected excess
bond returns that were up to 43% more volatile over the recent ZLB episode compared
to its SRM implied counterparts. The additional volatility in bond risk premia does
not show up in the pre-ZLB regime. The model-implied conditional expectations of
excess bond returns from the SRM with regime-dependent market prices of risk does
better than the SRM in Mincer and Zarnowitz (1969)-type specification tests over
the ZLB period. Over the non-binding ZLB period, the two models show comparable
performance. Finally, the SRM with regime-dependent market prices of risk imply
monetary policy lift-off probabilities that were substantially lower than their SRM
counterparts all the way up until shortly before the fact. This suggests that markets
expected monetary policy lift-off to occur later than otherwise thought.
The paper proceeds as follows. Section 3.2 presents our new empirical evidence
on bond risk premia at the ZLB. We conduct our specification test of the baseline
SRM in Section 3.3. In Section 3.4, we present our extension of the SRM with regime-
dependent market prices of risk and show that it passes our proposed specification
test. Section 3.5 discusses the economic implication of our SRM specification with
regime-dependent market prices of risk, while Section 3.6 concludes.
3.2 Bond Return Predictability at the ZLB
Bond risk premia measure the risk compensation required by bond investors to hold
bonds with long maturities. By now, there is substantial evidence for sizeable and
time-varying bond risk premia. The strongest and most robust predictor appears to
be the slope of the yield curve (Fama and Bliss, 1987; Campbell and Shiller, 1991).
More recently, a number of other yield curve and macroeconomic variables have
been found to be significant predictors as well (e.g. Cochrane and Piazzesi, 2005;
Ludvigson and Ng, 2009; Joslin et al., 2014; Cieslak and Povala, 2015). The common
practice to asses the time-variation and dynamics of bond risk premia is by means of
predictive regressions, where the typical specifications admit the representation
r x(n)t+1 =β(n)
0 +β(n)1 St +γ(n)′Zt +ε(n)
t+1. (3.1)
Here, r x(n)t+1 = p(n−1)
t+1 − p(n)t + p(1)
t denotes excess holding returns — borrow in the
1-year bond, buy a n-year bond and sell it one year later — and p(n)t is the log price
of a bond with n-years to maturity at time t . The variable St denotes the slope
of the current yield curve, and Zt denotes a vector of control variables including
additional yield curve factors or macroeconomic variables. The forecast errors ε(n)t+1
are orthogonal to the included regressors. The expectation hypothesis predicts β(n)1 =
γ(n) = 0 for all n, whereas consistent deviations from this null hypothesis is evidence
of predictable time-variation in bond risk premia.
The specification in (3.1) imposes a linear relation between the current slope of
the yield curve and bond risk premia. Prolonged ZLB episodes — as experienced
3.2. BOND RETURN PREDICTABILITY AT THE ZLB 111
recently — are an unavoidable yield curve non-linearity. Thus, the relation between
the slope of the yield curve and bond risk premia is likely to inherent this non-linearity.
This poses a serious challenge to the linear relation in (3.1).
One interpretation is in terms of a shadow rate model. As the short end of the
yield curve becomes constrained from below, this would in turn constrain the slope
of the yield curve — a slope compression effect. Following this line of reasoning, the
information in the observable slope of the yield curve is quite different when the ZLB
is binding. In particular, the signal from a given slope of the current yield curve about
future bond returns is expected to be stronger when the lower bound is binding. This
is because the current yield curve is flatter than it would have been, had there been no
lower bound on the short rate. That is, the so-called shadow rate is negative, whereas
the short rate is zero.
An alternative interpretation is in terms of re-pricing of yield curve risks. The ZLB
period has called for unconventional monetary policies such as forward-guidance
and large scale asset purchases. Contrary to conventional policy, the unconventional
policies aim at affecting longer-term yields. These policy differences could trigger
changes in required risk compensations by the marginal bond investor. In particular,
the nature of duration — or yield curve slope — risk is potentially very different with
such unconventional policies being active components of monetary policy. Changes
in market prices of risk alter the transmission of risk factors to required risk premia
and ultimately yield curve dynamics.
3.2.1 Modified Excess Bond Return Regressions
In our baseline specification, we initially strip down the specification in (3.1) to focus
on the information in the yield curve slope variable. That is, we impose γ(n) = 0 and
study the modification of the regression in (3.1),
r x(n)t+1 =β(n)
0 +β(n)1
(y (n)
t − y (1)t
)+δ(n)
0 I{rt<c} +δ(n)1 I{rt<c}
(y (n)
t − y (1)t
)+ε(n)
t+1. (3.2)
Here, I{·} is the indicator function, which takes on a value of one if the short rate
rt falls below a certain treshold c. The treshold value is set to c = 0.01 to let the
indicator function identify periods where the short rate is taken to be constrained
by the ZLB. We take the yield spread, y (n)t − y (1)
t , as a measure of yield curve slope.
This specification immediately allows for an assessment of the difference in bond risk
premia dynamics when the short rate is deemed to be constrained from below. Here,
the main null hypothesis of interest is H0 : δ(n)1 = 0. We implement the regressions
in (3.2) for maturities of n = 2,3, . . . ,10 years using end-of-month U.S. nominal zero-
coupon Treasury yields from Gurkaynak et al. (2007). We take the effective federal
funds rate to be the short rate. The sample is January 1990 through December 2017.
The start date is chosen to avoid the structural break in U.S. yield dynamics during
the 1980’s (Rudebusch and Wu, 2007).
112 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
Figure 3.1: Modified Bond Return Regressions: Yield Spread
This figure reports modified 1-year excess bond return regression coefficients estimated over the January
1990 through December 2017 sample. The estimated specification is r x(n)t+1 = β(n)
0 +β(n)1
(y (n)
t − y (1)t
)+
δ(n)0 I{rt<c} + δ(n)
1 I{rt<c}
(y (n)
t − y (1)t
)+ ε(n)
t+1. Confidence bands are constructed using the 2.5 and 97.5
percentiles from 5,000 block bootstrap repetitions with a block length of 24 months. Each bootstrap
sample is required to have a minimum of 50 ZLB observations.
Figure 3.1 plots the estimated regression coefficients along with 95% confidence
bands for the difference between non-ZLB and ZLB periods. The confidence bands
are computed using a block bootstrap with 5,000 repetitions and a block window
of two years of data.3 Prior to the ZLB period, we find the usual empirical pattern;
the slope regression coefficients are greater than zero and increasing as a function
of maturity. Both the intercept and slope regression coefficients are significantly
different when the ZLB is binding. In particular, we find that the usual empirical
pattern is amplified at the ZLB. The slope regression coefficients are larger and
increase faster as a function of maturity when the ZLB is binding. This evidence
suggests that the slope of the current yield curve carries a stronger signal about bond
risk premia when the shortest maturities are constrained from below. Additionally,
the regression intercepts are consistently smaller over the ZLB period. The differences
3We use a stationary block bootstrap, where the data is resampled in blocks of consecutive observa-tions of both the left- and right-hand side variables of the regressions. This way we account for possibletime series dependencies and preserve cross-sectional dependencies in the data. We require each boot-strap sample to have a minimum of 50 ZLB observations, since this achieves a conservative lower boundon the fraction of binding ZLB periods of 15% in simulated samples. In the data, binding ZLB periodsaccount for 31% of the observations.
3.2. BOND RETURN PREDICTABILITY AT THE ZLB 113
in intercepts and regression slope coefficients are statistically significant at a 5% level
for all maturities beyond three years.
3.2.2 Robustness
For robustness, we implement the regressions in (3.2) with alternative measures of
yield curve slope; (i) the forward spread f (n)t − y (1)
t , where f (n)t = p(n−1)
t −p(n)t denotes
the forward rate at time t for a loan between time t +n −1 and t +n, and (ii) the
second principal component of the yield curve. We further implement the regressions
controlling for typical macroeconomic variables; (i) inflation trend (Cieslak and
Povala, 2015) and (ii) real activity measured by the Chicago Fed National Activity
Index (Joslin et al., 2014).
Figure 3.2: Modified Bond Return Regressions: Other Slope Measures
This figure reports modified 1-year excess returns regression coefficients estimated over the January 1990
through December 2017 sample using two alternative measures of yield curve slope; (i) forward spread
and (ii) the second principal component of yields. Confidence bands are constructed using the 2.5 and
97.5 percentiles from 5,000 block bootstrap repetitions with a block length of 30 months. Each bootstrap
sample is required to have a minimum of 50 ZLB observations.
Across the different measures of yield curve slope and controlling for the macroe-
conomic variables, the results remain largely consistent with the findings from Figure
3.1. The slope of the yield curve predicts higher bond risk premia and the regression
slopes are increasing in time to maturity. At the ZLB, this effect is amplified — a given
slope of the yield curve predicts higher bond risk premia when the ZLB binds, and
114 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
Figure 3.3: Modified Bond Return Regressions: Macroeconomic Controls
This figure reports modified 1-year excess returns regression coefficients estimated over the January 1990
through December 2017 sample using yield spreads as measure of yield curve slope. Two macroeconomic
variables are further included as control variables; inflation trend and the Chicago Fed National Activity
Index. Confidence bands are constructed using the 2.5 and 97.5 percentiles from 5,000 block bootstrap
repetitions with a block length of 30 months. Each bootstrap sample is required to have a minimum of 50
ZLB observations.
the effect increases faster with maturity. The differences in slope regression coeffi-
cients are statistically significant across the specifications considered. For the forward
spread regressors, the differences are significant at a 5% level for maturities beyond
three years. For the second principal component, the difference is significant at a 5%
level for maturities beyond two years. Controlling for macroeconomic variables, the
differences remain statistically significant at a 5% level for maturities beyond two
years.
The baseline result is further robust to (i) extending the data sample back to
November 1971, (ii) holding periods of 3 and 6 months, (iii) thresholds of 25 and 50
bps for identifying when the short rate was constrained by its lower bound. For the
regression with the second principal component as measure of slope, the results are
also robust to including level and curvature as controls; that is, controlling for the first
and third principal components. These results are relegated to the online appendix.
3.3. A SHADOW RATE MODEL 115
3.3 A Shadow Rate Model
Following Dai and Singleton (2002), the empirical patterns observed above serve as a
natural specification test for the popular SRM; if the SRM constitutes a well-specified
model of the yield curve and bond risk premia dynamics, the model should account
for the apparent differences in slope regression coefficients observed in the actual
data.
We consider the SRM outlined in e.g. Kim and Singleton (2012), Christensen and
Rudebusch (2015), Bauer and Rudebusch (2016), and Wu and Xia (2016). Following
Black (1995), the short rate rt is modelled as,
rt = max{st ,0}, st =α+β′X t , (3.3)
where st denotes the shadow rate that is affine in NX pricing factors X t . We assume
that no arbitrage opportunities exist, and thus that there exist a risk-neutral mea-
sure Q. The pricing factors have risk-neutral first-order vector auto-regression (VAR)
dynamics given by
X t+1 =Φµ+(I−Φ)
X t +ΣεQt+1. (3.4)
where εt+1 is an i.i.d. standard normally distributed vector. I denotes the identity
matrix, and Σ is a lower triangular matrix with dimension NX ×NX identifying the
covariance of the factor innovations. Further, the pricing factors have physical proba-
bility measure P dynamics that is given by
X t+1 = h0 +hX X t +ΣεPt+1, (3.5)
where εPt+1 is an i.i.d. standard normally distributed vector. This assumption implies
an essentially affine stochastic discount factor (Duffee, 2002). The price of a n-year
bond is given recursively as P (n)t = EQt
[exp
(−rt)
P (n−1)t+1
]with yield to maturity y (n)
t =−n−1 logP (n)
t . Yields do not have closed-form expressions, and we therefore use the
second-order approximation of Priebsch (2013). For identification, we impose the
standard restrictions: (i) β′ = 1, (ii) µ = 0, (iii) Σ is lower triangular, and (iv) Φ is in
Jordan form with increasing diagonal elements. The physical measure transition
parameters, h0 and hx , are free parameters.
3.3.1 Estimation
We estimate the model using the sequential regression (SR) approach of Andreasen
and Christensen (2015). The SR approach has known asymptotic properties and
is computationally easy to implement even for non-linear dynamic term structure
models. This is in contrast to the alternative non-linear filtering techniques and
quasi-maximum likelihood estimators.
Our data set is as above. We consider a three factor specification, that is NX = 3.
The SR approach is constructed for large cross-sections. For this reason, we include
116 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
more yields than typically considered in the estimation. That is, we include Ny = 25
yields at each date; yields in the 3-month through 2.5-year range at three month
intervals, and yields in the 3-year through 10-year range at six month intervals. All
maturities are taken to be observed with a small measurement error v (n)t , i.e. y (n)
t =gn
(X t
)+ v (n)t . Here, gn
(X t
)is the function relating the factors to the cross-section
of yields. This function is governed by the risk-neutral VAR dynamics in (3.4). We
assume the measurement errors are mean zero and have finite, positive-definite
variance-covariance matrix.
The SR approach has three steps. In step 1, the risk-neutral parameters and latent
factors are estimated jointly from cross-sectional regressions. For given values of risk-
neutral parameters θ1 =[α di ag
(Φ
)vech
(Σ
)′ ], we estimate the factors at each
point in time by cross-sectional non-linear regressions. Estimates of the risk-neutral
parameters θ1 are obtained from minimizing the pooled squared residuals from the
cross-sectional regressions.
In step 2, the P dynamics are estimated using the factor estimates from step 1.
The physical measure factor dynamics h0, hX , and Σ can be estimated by a linear
regression that is adequately modified for estimation uncertainty in the factors.
Both step 1 and step 2 provide consistent estimates of Σ. In step 3, we condition
on Σstep2, since unreported results show that Σstep2 tends to be the most efficient
estimate.4 The remaining risk-neutral parameters and factor estimates are then
updated by re-running step 1 conditional on Σstep2. Finally, the estimates of h0 and
hX are updated by re-running step 2 using the new factor estimates. See Andreasen
and Christensen (2015) for further details on the estimation procedure.
3.3.2 Cross-Sectional Fit
The model provides a good fit to the cross-section of yields at each point in time.
Short maturities are heavily affected by the ZLB restriction, whereas long maturities
are less restricted and move more freely.
Table 3.1 and 3.2 report the estimated model parameters. The estimates are in
line with the typical findings; the largest eigenvalue of the risk-neutral transition
matrix I−Φ is close to unity, and the two remaining eigenvalues are smaller and
nearly identical (Christensen et al., 2011). The physical measure dynamics also show
the typical high persistence for the factors. Model-implied yields track closely the
observed data both at short maturities and the long maturities. In the short end of
the maturity-spectrum, the model occasionally produces 3-month yields that are
too low. This happens in particularly when the shadow rate is negative and the short
rate is truncated at zero, whereas over our sample the observed 3-month yield stays
slightly positive. These deviations are, however, only in the order of magnitude of a
few basis points. The fit is even closer at the long end of the maturity-spectrum.
4This is consistent with the results in Joslin et al. (2011) for affine term structure models and Andreasenand Meldrum (2018) for SRMs.
3.3. A SHADOW RATE MODEL 117
Table 3.1: Risk-Netural Parameters
Asymptotic standard errors are provided in parentheses and are robust to yield measurement errorsdisplaying heteroscedasticity in the time series dimension, cross-sectional correlation, and autocorrelation.We use ωD = 5 and ωT = 10 in the estimator provided in Andreasen and Christensen (2015).
SRM R-SRM
α 0.0168 (0.0070) 0.0169 (0.0072)Φ11 0.0012 (0.0007) 0.0012 (0.0007)Φ22 0.0461 (0.0020) 0.0467 (0.0020)Φ33 0.0555 (0.0017) 0.0553 (0.0018)
Table 3.2: Time Series Parameters
This table presents the estimated model parameters. The estimation procedures are outlined in Andreasenand Christensen (2015) and Andreasen, Engsted, Møller, and Sander (2016). Asymptotic standard errorsare provided in parentheses and are computed as described in Andreasen et al. (2016).
Panel A: SRM
h0 hX hX(·,1
)hX
(·,2)
hX(·,3
)Σ Σ
(·,1)
Σ(·,2
)Σ
(·,3)
−1.57×10−4(1.53×10−4
) hX(1, ·) 0.9875
(0.0116)0.0188(0.0104)
0.0179(0.0123)
Σ(1, ·) 3.16×10−4(
2.29×10−5) · ·
4.99×10−4(0.0016)
hX(2, ·) 0.0556
(0.1143)1.0362(0.1605)
0.0965(0.1769)
Σ(2, ·) −0.0012(
3.94×10−4) 0.0038(
2.34×10−4) ·
−9.42×10−4(0.0016)
hX(3, ·) −0.0865
(0.1128)−0.1037
(0.1607)0.8267(0.1761)
Σ(3, ·) 9.42×10−4(
3.86×10−4) −0.0039(
2.50×10−4) 2.27×10−4(
2.79×10−5)
Panel B: R-SRM
h(1)0 h(1)
X h(1)X
(·,1)
h(1)X
(·,2)
h(1)X
(·,3)
Σ Σ(·,1
)Σ
(·,2)
Σ(·,3
)−2.45×10−4(
2.32×10−4) h(1)
X
(1, ·) 0.9814
(0.0176)0.0033(0.0164)
8.17×10−4(0.0201)
Σ(1, ·) 3.12×10−4(
2.25×10−5) · ·
−0.0010(0.0027)
h(1)X
(2, ·) −0.0654
(0.2081)0.9466(0.2612)
0.0182(0.2906)
Σ(2, ·) −0.0013(
3.95×10−4) 0.0041(
2.37×10−4) ·
4.15×10−4(0.0027)
h(1)X
(3, ·) 0.0229
(0.2039)−0.0164(0.2580)
0.8997(0.2850)
Σ(3, ·) 0.0010(
3.86×10−4) −0.0042(
2.53×10−4) 2.19×10−4(
2.85×10−5)
h(2)0 h(2)
X h(2)X
(·,1)
h(2)X
(·,2)
h(2)X
(·,3)
−4.24×10−4(8.59×10−4
) h(2)X
(1, ·) 0.9707
(0.0503)0.0282(0.0428)
0.0294(0.0405)
−0.0074(0.0114)
h(2)X
(2, ·) −0.4066
(0.6831)0.7539(0.5785)
−0.1712(0.5634)
0.0068(0.0116)
h(2)X
(3, ·) 0.3698
(0.6940)0.14388(0.5959)
1.0631(0.5814)
Figure 3.4 shows the root mean squared measurement errors in basis points for
all the included maturities. Over the entire sample, the measurement errors are only
a few basis points for all maturities. This pattern largely remains over the binding and
non-binding ZLB periods.
118 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
Figure 3.4: In-Sample Fit: SRM
This figure reports root mean squared measurement errors in basis points for all maturities considered in
the estimation.
3.3.3 The Modified Linear Projection Test
We expose the SRM to our modified linear projection test. If the SRM is a well-
specified model, we would expect the non-linearity imposed by the truncation of
the short rate to account for the apparent shift in the regression coefficient that we
observe in the data. Specifically, we ask if the population regression coefficients δ(n)1
implied by the SRM match the pattern from Figure 3.1.
Taking the estimated model parameters at face value, we simulate a sample path
of length T = 1,000,000 and run the regressions in (3.2) to identify the population
regression coefficients δ(n)1 . Figure 3.5 plots the results. The SRM fails to account for
the differences in regression coefficients. In particular, the differences in regression
intercepts are negative but far too small to line up with the data. Regression slope
differences are mostly positive — at least for the longer maturities — but again far too
small to be in line with what we observe in the data. Thus, the model predicts changes
in regression coefficient that are in the right direction, but the changes are too small
to account for the empirical patterns. This means that the slope compression effect is
insufficient to generate the observed structural break in bond risk premia dynamics
at the ZLB.
3.3. A SHADOW RATE MODEL 119
Figure 3.5: Modified Linear Projection Test: SRM
This figure reports the results from the modified linear projection test. The population regression co-
efficients for the shadow rate model are computed from a simulated sample of length T = 1,000,000.
Simulations are for the estimated parameter values. The regression specification is r x(n)t+1 = β(n)
0 +β(n)
1
(y (n)
t − y (1)t
)+δ(n)
0 I{rt<c} +δ(n)1 I{rt<c}
(y (n)
t − y (1)t
)+ε(n)
t+1. Confidence bands are constructed using
the 2.5 and 97.5 percentiles from 5,000 block bootstrap repetitions with a block length of 24 months. Each
bootstrap sample is required to have a minimum of 50 ZLB observations.
3.3.4 Diagnosing the Failure
The model-implied regression coefficients do not account for the apparent shift
in regression coefficients. However, the SRM does drive a change in the regression
coefficients in the correct direction. This is consistent with the intuition of truncation
of the yield curve slope from below. To address whether it is in fact the non-linearity
at the ZLB that drives the shift in regression coefficients, consider the same model
without imposing a ZLB. In this case, the model has the well-known closed form
solutions for yields that are affine in the underlying risk factors, y (n)t =An +Bn X t ,
where the factor loadings are given by the usual recursions. We redo the modified
linear projection test on simulated data from the affine yield curve specification using
the same model parameters and simulated state vectors as for the SRM above.
Figure 3.6 shows that the non-linearity imposed by the shadow rate specification
only result in minor changes in the regression coefficients. One particular reason for
this is highlighted in Table 3.3. The estimated SRM implies durations for ZLB spells
that are very short-lived. Because short rates at zero are far from the unconditional
mean of the risk factors, the VAR(1) dynamics of the factors imply strong mean
120 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
Figure 3.6: SRM vs. ATSM
The population regression coefficients for the shadow rate and affine models are computed from a
simulated sample of length T = 1,000,000. Simulations are for the estimated parameter values. The
regression specification is r x(n)t+1 =β(n)
0 +β(n)1
(y (n)
t − y (1)t
)+δ(n)
0 I{rt<c} +δ(n)1 I{rt<c}
(y (n)
t − y (1)t
)+ε(n)
t+1.
Table 3.3: Duration of ZLB Spells
The model-implied distributions of durations of ZLB spells are obtained from simulating a sample pathof length T = 1,000,000 from each model at the estimated parameter values. Durations are measured asconsecutive months with the short rate falling below the threshold c = 0.01.
Mean 50-prctile 75-prctile 95-prctile 99-prctile
SRM 13.10 4 14 58 98
R-SRM 18.65 6 26 75 106
reversion and hence short ZLB spells. In fact, the model-implied median duration of
a ZLB spell is only 4 months and the 95th percentile duration is 58 months. As short
rates are expected to lift-off from zero quickly, such episodes do not affect longer
maturities much. This result suggests that the non-linearity is unlikely to capture the
shift in dynamics of bond risk premia at the ZLB. Instead, it suggests that underlying
bond risk premia dynamics of the SRM (and of the affine model for that matter) is
misspecified.
3.4. REGIME-DEPENDENT MARKET PRICES OF RISK 121
3.4 Regime-Dependent Market Prices of Risk
To guide intuition, consider the affine model that does not enforce the ZLB. In this
case, the coefficient from regressing excess holding returns onto the yield spread is
given in closed form by
β(n)1 =
(n −1)Bn−1λXV[
X t](
1nBn +β′
)′(
1nBn +β′
)V
[X t
](1nBn +β′
)′ , (3.6)
where λX = hX − (I−Φ)
denotes the market price of risk loading on the factors X t .
Thus, a re-pricing of risk has the potential to capture the observed shift in the regres-
sion coefficients. Re-pricing of risk is motivated by the restrictions that the ZLB puts
on conventional short rate policies. As policy makers turn toward unconventional
policies aimed at affecting long-maturity bond prices, the marginal bond investor
realizes that the risk he faces have changed and updates his required risk compen-
sations. In particular, the nature of duration risk is likely to have been somewhat
different over the recent U.S. ZLB episode, where the Federal Reserve engaged in
unconventional policies such as forward-guidance and large scale asset purchases.
To the extend that this led bond investor’s to adjust their required duration risk
compensation, such policies could cause a shift in bond risk premia dynamics.
We reconcile the two-regime risk factor dynamics in Andreasen et al. (2016) to
fit our setting. In particular, we introduce regime-dependent market prices of risk,
which in turn implies regime-dependent factor dynamics. Except for the Pmeasure
dynamics, the model is defined as for the standard SRM. That is, we consider a regime-
switching shadow rate model — the R-SRM hereafter — where the factor dynamics
are given by
X t+1 = I{rt≥c}h(1)0 +I{rt≥c}h
(1)X X t +I{rt<c}h
(2)0 +I{rt<c}h
(2)X X t +ΣεPt+1. (3.7)
This specification is consistent with regime-dependent market prices of risk, while
leaving the risk-neutral distribution — and hence largely the cross-sectional yield
curve fit — unaffected. At the same time, the regime-dependent market price of risk
specification allows for potentially longer durations of ZLB spells. The ZLB-regime
can exhibit extraordinarily factor autocorrelation, which can prolong the spells at zero
short term interest. Further, the parameters in the vector h(2)0 capture the point that
the risk factors mean-reverts towards in the ZLB-regime. This point can potentially
be low, and thus further prolonging the duration of ZLB spells.
3.4.1 Estimation
Since the set of parameters governing the risk-neutral dynamics are unaffected by the
extension only the second step of the SR approach has to be adjusted. We adjust the
time-series regression in step 2 to account for the regime-dependence of the market
122 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
prices of risk, while continuing to adequately adjust for estimation uncertainty in
the risk factors. Importantly, the P dynamics remains given in closed form. Thus,
the estimation of the regime-dependent dynamic parameters h(1)0 , h(2)
0 , h(1)X , and h(2)
Xcomes at no additional complexity or computational cost. See Andreasen et al. (2016)
for further details.
3.4.2 Cross-Sectional Fit and Significance
Table 3.1 shows that the risk-neutral parameter estimates are largely unaffected by
the shift in market prices of risk. The extension maintains the good cross-sectional fit
of yields at each point in time. Model-implied yields track closely the observed data
and are hardly distinguishable from the baseline SRM. This result is not surprising,
since only the P dynamics are affected by the regime-switch at the ZLB. The shift
in P dynamics affect the risk-neutral parameters through their common variance-
covariance matrix of factor innovations Σ, although this effect is small. Measurement
errors remain in the order of a few basis points, and the maturity-specific pattern
mirrors that of the baseline model. The similarity also remain over the period with
binding and non-binding ZLB. This evidence is in Figure 3.7.
Figure 3.7: In-Sample Fit: R-SRM
This figure reports root mean squared measurement errors in basis points for all maturities considered in
the estimation.
The extension is nevertheless important for understanding the dynamics of the
yield curve when the short end is constrained from below. The two regimes are
different in potentially important ways. Our dynamic distinction — at and away
3.4. REGIME-DEPENDENT MARKET PRICES OF RISK 123
from the ZLB — is statistically significant with a 0.03 p-value based on a Wald test.
That is, we reject the null hypothesis that h(1)0 = h(2)
0 and h(1)X = h(2)
X at a five percent
significance level. In particular, the regime distinction implies very different long-run
shadow rate means within each regime. The shadow rate mean-reverts towards a
long-run mean of 3.4% in the "normal" regime, whereas the shadow rate reverts
towards a long-run mean of 0.4% in the ZLB regime. In the baseline SRM the shadow
rate reverts towards a long-run mean of 2.2%. This difference in dynamics near
the ZLB implies that we should expect to see longer durations of ZLB spells in the
regime-switching SRM. Table 3.3 shows that this is in fact the case. The distribution
of durations of spells at the ZLB is substantially shifted compared to the baseline
SRM. In particular, the short rate is expected to remain at the ZLB for a prolonged
period. The durations are longer by as much as 50% compared to the baseline SRM.
For example, the 75th and 95th percentiles increase from 14 and 58 months to 26
and 75 months, respectively. Further, the dynamic differences have the potential to
explain why we find that a given yield spread predicts higher bond returns at the ZLB
compared to "normal" times.
3.4.3 The Modified Linear Projection Test
As for the baseline SRM, we expose our extension to the modified linear projection
test. We do so by simulating a sample path of length T = 1,000,000 taking our esti-
mated model parameters as given. The resulting regression coefficients are plotted in
Figure 3.8.
The extended model with regime-dependent market prices of risk does much
better in terms of matching the empirical regression coefficients. Across all maturities,
the model-implied population coefficients are always within the 95% confidence
intervals. This is the case both for the shift in regression intercept and regression
slope on the yield spread.
3.4.4 Interpretation
To obtain insights into which parameters are driving the success of the R-SRM in
passing the modified linear projection test, we conduct a range of experiments. First,
consider the counterfactual exercise, where we omit regime-switching by imposing
the same factor dynamics over the ZLB regime as estimated over the non-binding
ZLB regime. Under this assumption, we re-compute the model-implied regression
coefficients over the ZLB regime by simulating a sample path of length T = 1,000,000.
We then in turn allow for subsets of the parameters governing the factor dynamics
to switch to their estimated ZLB regime values. This is to isolate the effect on the
model-implied regression coefficients.
The first two rows of Table 3.4 iterate the finding that the single-regime model
does not capture the regression evidence in the data. In particular, the single-regime
124 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
Figure 3.8: Modified Linear Projection Test: R-SRM
This figure reports the results from the modified linear projection test. The population regression co-
efficients for the shadow rate model are computed from a simulated sample of length T = 1,000,000.
Simulations are for the estimated parameter values. The regression specification is r x(n)t+1 = β(n)
0 +β(n)
1
(y (n)
t − y (1)t
)+δ(n)
0 I{rt<c} +δ(n)1 I{rt<c}
(y (n)
t − y (1)t
)+ε(n)
t+1. Confidence bands are constructed using
the 2.5 and 97.5 percentiles from 5,000 block bootstrap repetitions with a block length of 24 months. Each
bootstrap sample is required to have a minimum of 50 ZLB observations.
model struggles to generate the large regression coefficients for the 5- to 10-year ma-
turities (e.g. 3.88 vs. 7.34 in the data for the 10-year bond). Row (3) in Table 3.4 allows
for the regime-switch in the intercepts of the factor dynamics only. This specification
substantially increases the regression coefficients across the maturity-spectrum. In
fact, the model-implied regression coefficients overshoot the data estimates for the
short- to medium-term maturities. For the 2-year bond, the model-implied regression
coefficient increases to 3.12 compared to the empirical value of 1.76. The intuition
behind this result is that the regime-shift in the constants affects the unconditional
factor means within the ZLB regime. As the shift in constants drives down the point
that the short rate mean-reverts towards, this in turn implies longer durations at
the ZLB, since there is no strong mean-reversion forces pulling the short rate away
from the ZLB regime. Row (4) in Table 3.4 instead allow only the transition coefficient
of the yield curve slope factor to shift. This specification again increase the regres-
sion coefficients, although the effect is less than for the intercepts. The regression
coefficient for the 2-year bond excess return largely matches the data counterpart
(1.87 vs. 1.76 in the data), whereas the longer-term bond excess return regression
coefficients fall short of the data counterparts (4.44 vs. 7.34 in the data). This pattern
3.5. ECONOMIC IMPLICATIONS 125
Table 3.4: Model-Implied Linear Projection Experiments
This table presents the effects of the regime-switching parameters on the model-implied excess returnregression coefficients. The regression coefficients are obtained by simulating a sample path with lengthT = 1,000,000 from the R-SRM evaluated at the specified parameters. The regression specification is
r x(n)t+1 = β(n)
0 +β(n)1
(y (n)
t − y (1)t
)+δ(n)
0 I{rt<c} +δ(n)1 I{rt<c}
(y (n)
t − y (1)t
)+ε(n)
t+1. The baseline specification
has factor dynamics that are equal across the two regimes and estimated based on the non-binding ZLBperiod. Column 1 outlines which parameters are allowed to switch to the values estimated over the bindingZLB period.
β(2)1 +δ(2)
1 β(5)1 +δ(5)
1 β(10)1 +δ(10)
1
(1) Data 1.76 4.66 7.34
(2) No regime-switch 1.45 2.46 3.88
(3) h(1)0 → h(2)
0 3.12 5.61 7.58
(4) h(1)X
(2, ·)→ h(2)
X
(2, ·) 1.87 2.89 4.44
(5) h(1)X → h(2)
X 2.13 3.34 4.63
(6) h(1)0 ,h(1)
X → h(2)0 ,h(2)
X 2.08 4.26 6.44
remains in row (5), where the entire factor transition matrix is allowed to switch.
Finally, the specification with both intercept and transition parameters switching
strikes a good balance between matching the short-, medium-, and long-term excess
return regression coefficients within the ZLB regime. The 2-, 5- and 10-year model-
implied regression coefficients are 2.08, 4.26, and 6.44 compared to the empirical
counterparts of 1.76, 4.66, and 7.34. The experiments highlight the importance of
prolonging the durations of the ZLB spells. Here, this is largely achieved by changing
the point towards which the short rate mean-reverts within the ZLB regime.
3.5 Economic Implications
The differences in market prices of risk are economically important. Because the
market prices of risk affect the factor dynamics, and thus ultimately yield dynamics
and persistence, the extended model predicts important differences in expected bond
returns compared to the standard SRM. These differences have important implica-
tions for measuring bond risk premia and doing inference on market expectations
for the monetary policy lift-off horizon.
3.5.1 Bond Risk Premia
The regime-dependent market prices of risk imply differences in yield curve dynamics
and model-implied bond risk premia.
Figure 3.9 shows the model-implied expected bond excess returns for 3-, 5-, 7-,
and 10-year bonds. The two model-implied estimates of expected bond returns are of
course highly correlated. For the reported maturities, the correlation is in the range
126 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
Figure 3.9: Model-Implied Expected Excess Returns
This figure plots the time series of model-implied expected excess returns for different maturities. The
model-implied expectations are computed from 2,000 Monte Carlo draws from the R-SRM and SRM
models, respectively.
0.91-0.93. However, the two models do display important differences. In particular,
over the ZLB period, the R-SRM implies expected excess returns that display greater
variability than the SRM counterpart. The variability in the R-SRM implied expected
excess returns are 19-43% higher than the SRM counterparts depending on the
considered maturity. For comparison, the variability of the R-SRM implied expected
excess returns over the full sample is only 6-8% greater than the SRM counterparts;
this difference is primarily driven by the ZLB period.
There is of course no information in Figure 3.9 useful for judging which of the two
models that provide a better measure of bond risk premia. Our proposed specifica-
tion test suggest that the R-SRM have well-specified bond risk premia, whereas the
standard SRM struggle to capture the empirical properties of excess return dynam-
ics. Another popular specification test for conditional expectations are the Mincer
and Zarnowitz (1969) regressions. For this particular application, the test is based
on a regression of realized excess returns onto a constant and the model-implied
conditional expected excess return. That is, we run the regressions
r x(n)t+1 =φ(n)
0 +φ(n)1 Et
[r x(n)
t+1
]+ε(n)
t+1. (3.8)
A well-specified measure of conditional expectations should return a constant and
regression slope of 0 and 1, respectively. Unreported results show that there is little
3.5. ECONOMIC IMPLICATIONS 127
evidence to distinguish between the SRM and R-SRM over the full sample as both
models return constants slightly above 0 and regressions slopes slightly below 1. For
both models, the model-implied expected excess returns cannot be rejected to be well-
specified when accounting for estimation uncertainty. However, we are particularly
interested in the period with a binding ZLB. Therefore, we perform a modified Mincer
and Zarnowitz (1969) test. We redo the regressions over two subsamples; (i) the
periods with a non-binding ZLB, and (ii) over the period with a binding ZLB. Formally,
the modified Mincer and Zarnowitz (1969) test reads
r x(n)t+1 = I{rt≥c}ν
(n)0 +I{rt≥c}ν
(n)1 Et
[r x(n)
t+1
]+
I{rt<c}ρ(n)0 +I{rt<c}ρ
(n)1 Et
[r x(n)
t+1
]+ε(n)
t+1.(3.9)
As before, a given month is deemed to be constrained by the ZLB if the short rate is
below the threshold level c = 0.01.
Figure 3.10: Mincer-Zarnowitz Regressions
This figure plots the results from the Mincer and Zarnowitz (1969) regressions, r x(n)t+1 = φ(n)
0 +φ(n)
1 Et
[r x(n)
t+1
]+ε(n)
t+1, where Et
[r x(n)
t+1
]are model-implied expectations from the R-SRM and SRM. The
regressions are done over two subsamples: (i) non-binding ZLB periods and (ii) binding ZLB periods.The model-implied expectations are computed from 2,000 Monte Carlo draws from the R-SRM and SRMmodels, respectively.
For both models, the constants ν(n)0 and ρ(n)
0 are all very close to zero. Over the
subsample with non-binding ZLB, the regression coefficients ν(n)1 are similar for the
two models. As for the full sample, the regression coefficients are below 1 for both
models. The binding ZLB period displays larger differences between the two models.
The R-SRM has regression coefficients ρ(n)1 that are somewhat closer to unity than the
128 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
SRM, although estimation uncertainty unavoidably makes the differences statistically
insignificant due to the short sample period. Thus, the R-SRM does somewhat better
than the SRM over the ZLB period without compromising the performance over the
non-binding ZLB period.
3.5.2 Lift-Off Probability
A key feature that enables the R-SRM to match the dynamics of bond risk premia at
the ZLB is the prolonged durations of ZLB episodes. One popular application of the
baseline SRM is to extract market expectations about the timing of policy rate lift-off
from the ZLB (e.g. Bauer and Rudebusch, 2016). Obviously, it is important to specify
the dynamics of the short rate correctly to obtain the most accurate predictions of
the timing of anticipated lift-off.
Figure 3.11: Lift-Off Probabilities
This figure plots the time series of R-SRM and SRM implied lift-off probabilities over the one-year horizon.Probabilities are computed from 100,000 Monte Carlo draws from the model-implied distribution of theshort rate. Lift-off is defined as the short rate exceeding the threshold level c = 0.01.
We compute model-implied lift-off probabilities at each point in time by simula-
tion. Starting from the estimated state vector, we simulate forward 100,000 sample
paths and calculate the proportion of sample paths where the short rate exceeds the
threshold value c = 0.01 one year from the considered date.
As expected, the R-SRM generally implies lower lift-off probabilities than the SRM.
This result comes from the lower degree of mean-reversion within the ZLB regime
for the R-SRM. However, over the early part of the period with binding ZLB, the two
models generally agree that lift-off was an unlikely event. The two models disagree
3.6. CONCLUSION 129
with a wider margin in early to mid-2014 and afterwards. The SRM assigns substantial
probability (around 60%) to the short rate exceeding 1 percent as early as December
2015. On the other hand, the R-SRM assigns much smaller probability to this event;
around 20%. The Federal Reserve first raised its target range to 25-50 bps in December
2015, and proclaimed further increases in the federal funds rate target range at a
steady pace onwards. Over the course of 2015 and 2016, the R-SRM assigns increasing
probability of the short rate exceeding 1 percent. The same pattern is evident for the
SRM. The exact probabilities, however, still display substantial differences across the
two models. The SRM model assigns probabilities around 80% by the end of 2016,
whereas the R-SRM assigns a probability of around 50% to the short rate exceeding
1 percent by the end of 2017. By early 2017, the R-SRM catches up with the SRM as
both models assign a high probability of lift-off of approximately 80%. The actual fed
funds target range increased to 1-1.25 percent in June 2017.
The R-SRM model suggests that simply extrapolating the short rate dynamics
from the pre-ZLB period for forming short rate projections at the ZLB may lead
to lift-off probabilities that exceed those of market participants. The baseline SRM
model assigns substantial probabilities of the short rate exceeding 1 percent as early
as two-three years before they did. In comparison, the R-SRM — which estimates
the lift-off probabilities from the yield curve dynamics estimated over the actual
ZLB period — suggests that the SRM lift-off probabilities may be overstated. Further,
the R-SRM suggests that the anticipated timing of lift-off may have been later than
otherwise thought.
3.6 Conclusion
We document a structural break in the regression coefficients from linear projections
of excess bond returns onto yield spreads. This empirical observations serve as a
natural specification test of the standard SRMs. As the short end of the yield curve be-
comes constrained from below, the slope of the yield curve will be flatter than it would
have been in the absence of the lower bound. We find that the slope compression
effect of the three-factor SRM is not strong enough to explain the regression-based ev-
idence. One particular reason for this is that the SRM implies durations of ZLB spells
that are very short-lived. This implies that strong mean-reversion forces dominate
the dynamics of yields and bond risk premia near the ZLB.
Instead, we propose a SRM with regime-dependent market prices of risk. The
regime-dependence of risk prices is motivated by the changing nature of bond market
risks over the recent ZLB episode. Unconventional monetary policies have, unlike
conventional short rate policies, aimed at affecting longer-term bond prices. This
changing nature of risks in bond markets may have induce investors to update their
required risk compensations.
We find a significant switch in market prices of risk at the ZLB. Further, the SRM
130 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
with regime-dependent market prices of risk matches the regression-based empirical
evidence. Compared to the standard SRM, our extensions imply longer durations
of ZLB episodes. This is because the short rate mean-reverts towards a much lower
long-run mean within the ZLB regime. Contrary to the standard SRM, the extension
does not have strong mean-reversion forces pulling the short rate away from the ZLB
regime.
The extension has some interesting economic implications. First, model-implied
bond risk premia were up to 43% more volatile over the recent ZLB episode compared
to its SRM counterpart. Second, the regime-switching model assigns substantially
lower probability to monetary policy lift-off all the way up until shortly before the
fact.
Acknowledgements
Andreasen acknowledges financial support from the Danish e-Infrastructure Cooper-
ation (DeIC). Andreasen and Jørgensen acknowledges financial support to CREATES
(Center for Research in Econometric Analysis of Time Series; DNRF78) from the Dan-
ish National Research Foundation. The analysis and conclusions are those of the
authors and do not indicate concurrence by the Board of Governors of the Federal
Reserve System or other members of the research staff of the Board.
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134 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
Appendix
C.1 Robustness Results
C.1.1 Extended Sample: November 1971 - December 2017
Figure C.1: Modified Bond Return Regressions: Yield Spread
This figure reports modified 1-year excess bond return regression coefficients estimated over the November
1971 through December 2017 sample. The estimated specification is r x(n)t+1 = β(n)
0 +β(n)1
(y (n)
t − y (1)t
)+
δ(n)0 I{rt<c} + δ(n)
1 I{rt<c}
(y (n)
t − y (1)t
)+ ε(n)
t+1. Confidence bands are constructed using the 2.5 and 97.5
percentiles from 5,000 block bootstrap repetitions with a block length of 24 months. Each bootstrap
sample is required to have a minimum of 50 ZLB observations.
C.1. ROBUSTNESS RESULTS 135
C.1.2 Holding Period: 3 Months
Figure C.2: Modified Bond Return Regressions: Yield Spread
This figure reports modified 1-year excess bond return regression coefficients estimated over the January
1990 through December 2017 sample. The estimated specification is r x(n)t+1 = β(n)
0 +β(n)1
(y (n)
t − y (1)t
)+
δ(n)0 I{rt<c} + δ(n)
1 I{rt<c}
(y (n)
t − y (1)t
)+ ε(n)
t+1. Confidence bands are constructed using the 2.5 and 97.5
percentiles from 5,000 block bootstrap repetitions with a block length of 24 months. Each bootstrap
sample is required to have a minimum of 50 ZLB observations.
136 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
C.1.3 Holding Period: 6 Months
Figure C.3: Modified Bond Return Regressions: Yield Spread
This figure reports modified 1-year excess bond return regression coefficients estimated over the January
1990 through December 2017 sample. The estimated specification is r x(n)t+1 = β(n)
0 +β(n)1
(y (n)
t − y (1)t
)+
δ(n)0 I{rt<c} + δ(n)
1 I{rt<c}
(y (n)
t − y (1)t
)+ ε(n)
t+1. Confidence bands are constructed using the 2.5 and 97.5
percentiles from 5,000 block bootstrap repetitions with a block length of 24 months. Each bootstrap
sample is required to have a minimum of 50 ZLB observations.
C.1. ROBUSTNESS RESULTS 137
C.1.4 Threshold: 25 bps
Figure C.4: Modified Bond Return Regressions: Yield Spread
This figure reports modified 1-year excess bond return regression coefficients estimated over the January
1990 through December 2017 sample. The estimated specification is r x(n)t+1 = β(n)
0 +β(n)1
(y (n)
t − y (1)t
)+
δ(n)0 I{rt<c} + δ(n)
1 I{rt<c}
(y (n)
t − y (1)t
)+ ε(n)
t+1. Confidence bands are constructed using the 2.5 and 97.5
percentiles from 5,000 block bootstrap repetitions with a block length of 24 months. Each bootstrap
sample is required to have a minimum of 50 ZLB observations.
138 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
C.1.5 Threshold: 50 bps
Figure C.5: Modified Bond Return Regressions: Yield Spread
This figure reports modified 1-year excess bond return regression coefficients estimated over the January
1990 through December 2017 sample. The estimated specification is r x(n)t+1 = β(n)
0 +β(n)1
(y (n)
t − y (1)t
)+
δ(n)0 I{rt<c} + δ(n)
1 I{rt<c}
(y (n)
t − y (1)t
)+ ε(n)
t+1. Confidence bands are constructed using the 2.5 and 97.5
percentiles from 5,000 block bootstrap repetitions with a block length of 24 months. Each bootstrap
sample is required to have a minimum of 50 ZLB observations.
C.2. SHADOW RATE MODEL 139
C.1.6 Controlling for Level and Curvature
Figure C.6: Modified Bond Return Regressions: Second PC
This figure reports modified 1-year excess bond return regression coefficients estimated over the January
1990 through December 2017 sample. The estimated specification is r x(n)t+1 =β(n)
0 +β(n)1 PC1,t +β(n)
2 PC2,t +β(n)
3 PC3,t +δ(n)1 I{rt<c}PC2,t +ε(n)
t+1. Confidence bands are constructed using the 2.5 and 97.5 percentiles
from 5,000 block bootstrap repetitions with a block length of 24 months. Each bootstrap sample is required
to have a minimum of 50 ZLB observations.
C.2 Shadow Rate Model
Define the shadow rate st to be a linear function of the NX ×1 vector of pricing factors
X t , i.e.
st =α+β′X t . (C.1)
The short rate is given by
rt = max{st ,0}, (C.2)
where 0 is assumed to be the lower bound on the short rate. The risk factors have a
VAR(1) law of motion under the risk-neutralQmeasure
X t+1 =Φµ+(I−Φ)
X t +ΣεQt+1, (C.3)
where εQt+1 ∼NID(0, I
). Here, Φ is a NX ×NX matrix, µ is a NX ×1 vector, and Σ is a
NX ×NX matrix. Under the assumption of an essentially affine stochastic discount
140 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
factor, the physical measure P dynamics are
X t+1 = h0 +hX X t +ΣεPt+1, (C.4)
where εPt+1 ∼NID(0, I
). Here, hX is a NX ×NX matrix and h0 is a NX ×1 vector.
C.2.1 Risk-Neutral Pricing
No-arbitrage implies that the price P (n)t of an n-period zero-coupon bond at time t is
given by
P (n)t = EQt
[exp{−rt }P (n−1)
t+1
], (C.5)
or by recursive substitution
P (n)t = EQt
[exp{−
n−1∑i=0
rt+i }
]. (C.6)
Since the yield to maturity is defined as y (n)t =− 1
n logP (n)t , we have
y (n)t =− 1
nlogEQt
exp
(−
n−1∑i=0
rt+i
) . (C.7)
C.2.2 Cumulant Approximation
The quantity logEQt
[exp
(−∑n−1
i=0 rt+i
)]appearing in (C.7) is the conditional cumulant-
generating function underQ, evaluated at -1, of the random variable RNt ≡∑n−1
i=0 rt+i .
It can be expressed in terms of its series representation, i.e.
logEQt
[exp
(−RN
t
)]=
∞∑j=1
(−1) jκQ
j
j !, (C.8)
where κQj is the j ’th cumulant of RNt underQ. An approximation to (C.7) can therefore
be computed by truncating the sum in (C.8) after a finite number of terms.
C.2.2.1 First-Order Approximation
The first-order approximation of (C.7) reads
y (n)t = 1
nEQt
[n−1∑i=0
rt+i
]. (C.9)
In order to compute the expectation in (C.9), notice that if Z ∼N(µ,σ2
)then we have
E[max{Z ,0}
]=µΦ(µ
σ
)+σφ
(µ
σ
). (C.10)
C.2. SHADOW RATE MODEL 141
Note that since X t is normal, then st is normal as well. Thus (C.10) is applicable for
evaluating the expectation in (C.9). We simply need to evaluate the conditional mean
and standard deviation of st . Firstly, the conditional mean can be computed as
EQt
[st+i
]=α+β′EQt[
X t+i]
, (C.11)
where EQt[
X t+i]
can be computed by the recursive substitutions
X t+2 =Φµ+(I−Φ)
X t+1 +ΣεQt+2
=Φµ+ (I−Φ)(
Φµ+ (I−Φ)
X t +ΣεQt+1
)+ΣεQt+2
=Φµ+ (I−Φ)
Φµ+ (I−Φ)2 X t +
(I−Φ)
ΣεQt+1 +ΣεQt+2
X t+3 =Φµ+(I−Φ)
X t+2 +ΣεQt+3
=Φµ+ (I−Φ)(
Φµ+ (I−Φ)
Φµ+ (I−Φ)2 X t +
(I−Φ)
ΣεQt+1 +ΣεQt+2
)+ΣεQt+3
=Φµ+ (I−Φ)
Φµ+ (I−Φ)2
Φµ+ (I−Φ)3 X t +
(I−Φ)2
ΣεQt+1
+ (I−Φ)
ΣεQt+2 +ΣεQt+3
...
X t+i =i−1∑j=0
(I−Φ) j
Φµ+ (I−Φ)i X t +
i−1∑j=0
(I−Φ) j
ΣεQ
t+i− j .
(C.12)
This implies
EQt
[X t+i
]= i−1∑j=0
(I−Φ) j
Φµ+ (I−Φ)i X t . (C.13)
Hence,
EQt
[st+i
]=α+β′ i−1∑j=0
(I−Φ) j
Φµ+β′ (I−Φ)i X t . (C.14)
The conditional variance can be computed as
VQt
[st+i
]=β′VQt[
X t+i]β, (C.15)
where VQt[
X t+i]
can be computed from (C.12) as
VQt
[X t+i
]= EQt [(X t+i −EQt
[X t+i
])(X t+i −EQt
[X t+i
])′]
= EQt
i−1∑
j=0
(I−Φ) j
ΣεQ
t+i− j
i−1∑j=0
(I−Φ) j
ΣεQ
t+i− j
′ .(C.16)
142 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
Since the innovations are independent across time it follows that
VQt
[X t+i
]= EQti−1∑
j=0
(I−Φ) j
ΣεQ
t+i− j
(εQ
t+i− j
)′Σ′
((I−Φ) j
)′=
i−1∑j=0
(I−Φ) j
ΣΣ′((I−Φ) j
)′.
(C.17)
Equation (C.17) implies that the variances can be computed recursively for i = 1, . . . , N
by
VQt
[X t+i
]= i−1∑j=0
(I−Φ) j
ΣΣ′((I−Φ) j
)′=
i−2∑j=0
(I−Φ) j
ΣΣ′((I−Φ) j
)′+ (I−Φ)i−1
ΣΣ′((I−Φ)i−1
)′=VQt
[X t+i−1
]+ (I−Φ)i−1
ΣΣ′((I−Φ)i−1
)′,
(C.18)
with the initial condition VQt[
X t]= 0. Applying (C.10) it follows that
EQt
[rt+i
]=EQt [max{st+i ,0}
]=µt ,t+iΦ
(µt ,t+i
σt ,t+i
)+σt ,t+iφ
(µt ,t+i
σt ,t+i
),
(C.19)
where µt ,t+i =α+β′EQt[
X t+i]
and σ2t ,t+i =β′VQt
[X t+i
]β. It is then straight-forward
to compute
EQt
[n−1∑i=0
rt+i
]=
n−1∑i=0
EQt
[rt+i
], (C.20)
using the result from (C.19). This completes the computation of the first-order ap-
proximation.
C.2.2.2 Second-Order Approximation
The second-order approximation of (C.7) reads
y (n)t = 1
n
EQt[
n−1∑i=0
rt+i
]− 1
2VQt
[n−1∑i=0
rt+i
] . (C.21)
C.2. SHADOW RATE MODEL 143
The second term can be computed by noting that
VQt
[n−1∑i=0
rt+i
]=EQt
(n−1∑i=0
rt+i
)2−EQt
[n−1∑i=0
rt+i
]2
=EQt
n−1∑i=0
n−1∑j=0
rt+i rt+ j
−EQt[
n−1∑i=0
rt+i
]2
=n−1∑i=0
n−1∑j=0
EQt
[rt+i rt+ j
]−
(n−1∑i=0
EQt
[rt+i
])2
.
(C.22)
Equation (C.22) implies that only EQt
[rt+i rt+ j
]needs to be computed, since EQt
[rt+i
]is already known from the first-order approximation. Inserting for the short-rate it
follows that
EQt
[rt+i rt+ j
]=EQt
[max{st+i ,0}max{st+ j ,0}
]. (C.23)
Note that if [Z1
Z2
]∼N
[µ1
µ2
],
[σ2
1 σ12
σ12 σ22
] , (C.24)
then
E[max{Z1,0}max{Z2,0}
]=(µ1µ2 +σ12
)Φd
2
(−ζ1,−ζ2;χ)
+σ2µ1φ(ζ2
)Φ
(ζ1 −χζ2√
1−χ2
)
+σ1µ2φ(ζ1
)Φ
(ζ2 −χζ1√
1−χ2
)
+σ1σ2
√1−χ2
2πφ
√ζ2
1 −2χζ1ζ2 +ζ22
1−χ2
,
(C.25)
where ζ j = µ j
σ j, χ = σ12
σ1σ2and Φd
2
(ζ1,ζ2;χ
) = 1−Φ(ζ1
)−Φ(ζ2
)+Φ2(ζ1,ζ2;χ
). Thus
to evaluate EQt
[rt+i rt+ j
]it is necessary to compute CovQt
[st+i , st+ j
]. From (C.1) it
follows that
CovQt
[st+i , st+ j
]=β′CovQt
[X t+i , X t+ j
]β, (C.26)
and from (C.12) that
CovQt
[X t+i , X t+ j
]=CovQt
[i−1∑k=0
(I−Φ)k
ΣεQ
t+i−k ,i−1∑h=0
(I−Φ)h
ΣεQ
t+i−h
]
=i−1∑k=0
i−1∑h=0
CovQt
[(I−Φ)k
ΣεQ
t+i−k ,(I−Φ)h
ΣεQ
t+i−h
].
(C.27)
144 CHAPTER 3. BOND RISK PREMIA AT THE ZERO LOWER BOUND
Since the innovations are independent across time, only terms where i −k = j −h or
k = i − j +h are non-zero. Hence,
CovQt
[X t+i , X t+ j
]=
i−1∑h=0
CovQt
[(I−Φ)i− j+h
ΣεQ
t+ j−h ,(I−Φ)h
ΣεQ
t+i−h
]=
i−1∑h=0
(I−Φ)i− j+h
ΣCovQt
[εQ
t+ j−h ,εQt+i−h
]Σ′
((I−Φ)h
)′=
i−1∑h=0
(I−Φ)i− j+h
ΣΣ′((I−Φ)h
)′.
(C.28)
Covariances can then be computed recursively as
CovQt
[X t+i+1, X t+ j
]= (I−Φ)
CovQt
[X t+i , X t+ j
], (C.29)
with the initial condition CovQt
[X t+ j , X t+ j
]= V
Qt
[X t+ j
]. Using these result, it is
straight-forward to compute
EQt
[max{st+i ,0}max{st+ j ,0}
]=[
µt ,t+iµt ,t+ j +σt ,t+i ,t+ j
]Φd
2
(−ζt ,t+i ,−ζt ,t+ j ;χt ,t+i ,t+ j
)+σt ,t+ jµt ,t+iφ
(ζt ,t+ j
)Φ
ζt ,t+i −χt ,t+i ,t+ j ζt ,t+ j√1−χ2
t ,t+i ,t+ j
+σt ,t+iµt ,t+ jφ
(ζt ,t+i
)Φ
ζt ,t+ j −χt ,t+i ,t+ j ζt ,t+i√1−χ2
t ,t+i ,t+ j
+σt ,t+iσt ,t+ j
√1−χ2
t ,t+i ,t+ j
2πφ
√√√√ζ2
t ,t+i −2χt ,t+i ,t+ j ζt ,t+iζt ,t+ j +ζ2t ,t+ j
1−χ2t ,t+i ,t+ j
,
(C.30)
where σt ,t+i ,t+ j = β′CovQt
[X t+i , X t+ j
]β, ζt ,t+i = µt ,t+i
σt ,t+i, and χt ,t+i ,t+ j = σt ,t+i ,t+ j
σt ,t+iσt ,t+ j.
This completes the computation of the second-order approximation.