Macroeconomic Implications of Changes in the Term Premium · FEDERAL RESERVE BANK OF ST.LOUIS R EVI W JULY /AUGUST 2 07 241 MacroeconomicImplicationsof ChangesintheTermPremium GlennD.Rudebusch,BrianP.Sack,andEricT.Swanson

FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2007 241

Macroeconomic Implications ofChanges in the Term PremiumGlenn D. Rudebusch, Brian P. Sack, and Eric T. Swanson

Linearized New Keynesian models and empirical no-arbitrage macro-finance models offer littleinsight regarding the implications of changes in bond term premiums for economic activity. Thispaper investigates these implications using both a structural model and a reduced-form framework.The authors show that there is no structural relationship running from the term premium to eco-nomic activity, but a reduced-form empirical analysis does suggest that a decline in the term pre-mium has typically been associated with stimulus to real economic activity, which contradictsearlier results in the literature. (JEL E43, E44, E52, G12)

Federal Reserve Bank of St. Louis Review, July/August 2007, 89(4), pp. 241-69.

The puzzlement over the recent low and rela-tively stable levels of long-term interest rates hasgenerated much interest in trying to understandboth the source of these low rates and their eco-nomic implications. In addressing these issues,it is useful to divide the yield on a long-term bondinto an expected-rate component that reflects theanticipated average future short rate for the matu-rity of the bond and a term-premium componentthat reflects the compensation that investorsrequire for bearing the interest rate risk fromholding long-term instead of short-term debt.Chairman Greenspan’s later July 2005 monetarypolicy testimony suggested that the conundrumlikely involved movements in the latter compo-nent, noting that “a significant portion of thesharp decline in the ten-year forward one-yearrate over the past year appears to have resultedfrom a fall in term premiums.” This interpreta-tion has been supported by estimates from vari-ous finance and macro-finance models thatindicate that the recent relatively stable 10-yearTreasury yield reflects that the upward revisions

F rom June 2004 through June 2006, theFederal Reserve gradually raised thefederal funds rate from 1 percent to 5¼percent. Despite this 425-basis-point

increase in the short-term rate, long-term interestrates remained at remarkably low levels, with the10-year Treasury yield averaging 4¼ percent inboth 2004 and 2005 and ending September 2006at just a little above 4½ percent. The apparentlack of sensitivity of long-term interest rates tothe large rise in short rates surprised manyobservers, as such behavior contrasted sharplywith interest rate movements during past policy-tightening cycles.1 Perhaps the most famousexpression of this surprise was provided bythen-Chairman of the Federal Reserve AlanGreenspan in monetary policy testimony beforeCongress in February 2005, in which he notedthat “the broadly unanticipated behavior of worldbond markets remains a conundrum.”

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

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

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

to expected future short rates that accompaniedthe monetary policy tightening were offset, onbalance, by a decline in the term premium (e.g.,Kim andWright, 2005, and Rudebusch, Swanson,and Wu, 2006).2

It is this recent experience of a declining termpremium in long-term rates that motivates ourpaper. We examine what is known—both in the-ory and from the data—about the macroeconomicimplications of changes in the term premium.This topic is especially timely and importantbecause of the practical implications of the recentlow term premium for the conduct of monetarypolicy. Specifically, as noted by Federal ReserveGovernor Donald Kohn (2005), “the decline interm premiums in the Treasury market of late mayhave contributed to keeping long-term interestrates relatively low and, consequently, may havesupported the housing sector and consumerspending more generally.” Furthermore, any suchmacroeconomic impetus would alter the appro-priate setting of the stance of monetary policy, asdescribed by Federal Reserve Chairman BenBernanke (2006):

To the extent that the decline in forward ratescan be traced to a decline in the term pre-mium…the effect is financially stimulativeand argues for greater monetary policyrestraint, all else being equal. Specifically, ifspending depends on long-term interest rates,special factors that lower the spread betweenshort-term and long-term rates will stimulateaggregate demand. Thus, when the term pre-mium declines, a higher short-term rate isrequired to obtain the long-term rate and theoverall mix of financial conditions consistentwith maximum sustainable employment andstable prices.

Under this “practitioner” view, which is alsoprevalent among market analysts and privatesector macroeconomic forecasters, the recent fallin the term premium provided a boost to realeconomic activity and, therefore, optimal mone-

tary policy should have followed a relativelymore restrictive path as a counterbalance.3

Unfortunately, this practitioner view of themacroeconomic andmonetary policy implicationsof a drop in the term premium is not supportedby the simple linearized New Keynesian modelof aggregate output that is currently so popularamong economic researchers. In that model, out-put is determined by a forward-looking IS curve:

(1)

where yt denotes aggregate output and it – Etπt+1is the one-period ex ante real interest rate. Solvingthis equation forward, output can be expressed asa function of short-term real interest rates alone:

(2)

According to this equation, it is the expectedpath of the short-term real interest rate that deter-mines the extent of intertemporal substitutionand hence current output. Long-term interest ratesmatter only because they embed expectations offuture short-term interest rates (as in McGough,Rudebusch, and Williams, 2005). Taken literally,this simple analytic framework does not allowshifts in the term premium to affect output; there-fore, according to this model, the recent declinein the term premium should be ignored whenconstructing optimal monetary policy, and theonly important consideration should be therestraining influence of the rising expected-ratecomponent.

Given these contradictory practitioner andNew Keynesian views about the macroeconomicimplications of changes in the term premium,this paper considers what economic theory moregenerally implies about this relationship as wellas what the data have to say. We start in the nextsection by examining a structural dynamic sto-chastic general equilibrium (DSGE) framework

y E i et tj

jt j t j t= − − +

=+ + +∑1

01γ

β π`

( ) .

y E y i E et t t t t t t= − − ++ +βγ

π1 11

( ) ,

Rudebusch, Sack, Swanson

242 JULY/AUGUST 2007 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW

2 Of course, as we discuss in detail below, such decompositions ofthe long rate into expected rates and a term premium are subjectto considerable uncertainty.

3 For example, in a January 2005 commentary, the private forecastingfirm Macroeconomic Advisers argued that the low term premiumwas keeping financial conditions accommodative and “wouldrequire the Fed to ‘do more’ with the federal funds rate to achievethe desired rate of growth.”

that can completely characterize the relation-ship between the term premium and the economy.In this framework, unlike its linearized NewKeynesian descendant, there are important con-nections between term premiums and the econ-omy. Unfortunately, given theoretical uncertaintiesand computational complexities, the model can-not be taken directly to the data, so it providesonly qualitative insights about themacroeconomicimplications of changes in term premiums, notquantitative empirical assessments.

To uncover such empirical assessments, thethird section surveys the recent empirical macro-finance literature, which links the behavior oflong-term interest rates to the economy, withvarying degrees of economic structure (e.g., Angand Piazzesi, 2003, and Rudebusch andWu, 2003,denoted RW). However, although this new litera-ture has made interesting advances in understand-ing howmacroeconomic conditions affect the termpremium, it has made surprisingly little progresstoward understanding the reverse relationship.Indeed, restrictions are typically imposed in thesemodels that either eliminate any effects of theterm premium on the economy or require theterm premium to affect the economy in the sameway as other sources of long-rate movements.Accordingly, as yet, this literature is not veryuseful for investigating whether there are impor-tant macroeconomic implications of movementsin the term premium.

In contrast, as reviewed in the fourth section,several papers have directly investigated the pre-dictive power of movements in the term premiumon subsequent gross domestic product (GDP)growth (e.g., Favero, Kaminska, and Söderström,2005, and Hamilton and Kim, 2002), but becausethese analyses rely on simple reduced-form regres-sions, their structural interpretation is unclear.Nevertheless, taken at face value, the bulk of theevidence suggests that decreases in the term pre-mium are followed by slower output growth—clearly contradicting the practitioner view (as wellas the simple New Keynesian view). However, wereconsider such regressions and provide somenew empirical evidence that supports the view

taken bymany central bankers andmarket analyststhat a decline in the term premium typically hasbeen associated with stimulus to the economy.

The final section concludes by describingsome practical lessons for monetary policymakerswhen confronted with a sizable movement in theterm premium.

A STRUCTURAL MODEL OF THETERM PREMIUM AND THEECONOMY

In this section, we use a standard structuralmacroeconomic DSGE framework to study therelationship between the term premium and theeconomy. In principle, such a framework cancompletely characterize this relationship; how-ever, in practice the DSGE asset-pricing frameworkhas a number of well-known computational andpractical limitations that keep it from being auseful empirical workhorse. Nevertheless, theframework can provide interesting qualitativeinsights, as we will now show.

An Asset-Pricing Representation ofthe Term Premium

As in essentially all asset pricing, the funda-mental equation that we assume prices assets inthe economy is the stochastic discountingrelationship:

(3)

where pt denotes the price of a given asset attime t andmt+1 denotes the stochastic discountfactor that is used to value the possible state-contingent payoffs of the asset in period t+1(where pt+1 implicitly includes any dividend orcoupon payouts).4 Specifically, the price of adefault-free n-period zero-coupon bond thatpays one dollar at maturity, pt

(n), satisfies

p E m pt t t t= + +[ ],1 1



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

(4)

where pt(0) = 1 (the price of one dollar delivered

at time t is one dollar).We can use this framework to formalize the

decomposition of bond yields described in theintroduction, with the term premium defined asthe difference between the yield on an n-periodbond and the expected average short-term yieldover the same n periods.5 Let it

(n) denote the con-tinuously compounded n-period bond yield(with it � it

(1) ); then the term premium can becomputed from the stochastic discount factor ina straightforward manner:

(5)

Of course, equation (5) does not have an easyinterpretation without imposing additional struc-ture on the stochastic discount factor, such asconditional log-normality. Nonetheless, even inthis general form, equation (5) highlights an impor-tant point: The term premium is not exogenous,as a change in the term premium can only bedue to changes in the stochastic discount factor.Thus, to investigate the relationship between theterm premium and the economy in a structuralmodel, we must first specify why the stochasticdiscount factor in the model is changing.

In general, the stochastic discount factor willrespond to all of the various shocks affecting theeconomy, including innovations to monetarypolicy, technology, and government purchases.Of course, these different types of shocks alsohave implications for the determination of outputand other economic variables. Thus, we wouldexpect the correlation between the term premiumand output to depend on which structural shock

in

E in

pn

E ptn

t t jj

n

tn

t t( ) ( )log log− = − ++

=

−

∑1 1 1

0

1

++=

−

+=

∑

∏= −

+

jj

n

t t jj

n

nE m

nE

( )

log

1

0

1

1

1 1tt t j t j

j

n

E mlog .+ − +=∑ 1

1

p E m ptn

t t tn( ) ( )[ ],= + +

−1 1

1 was driving the change in the term premium. Wenext elaborate on this point using a simple struc-tural model.

A Benchmark DSGE Structural Model

The expression for the term premiumdescribed by equation (5) is quite general butnot completely transparent, because it does notimpose any structure on the stochastic discountfactor. Thus, to illuminate the structural relation-ship between the term premium and the macro-economy, we introduce a simple benchmark NewKeynesian DSGE model.

The basic features of the model are as follows.Households are representative and have prefer-ences over consumption and labor streams givenby

(6)

where β denotes the household’s discount factor, ctdenotes consumption in period t, lt denotes labor,ht denotes a predetermined stock of consumptionhabits, and γ, χ, χ0, and b are parameters. We setht = Ct–1, the level of aggregate consumption inthe previous period, so that the habit stock isexternal to the household. There is no invest-ment in physical capital in the model, but thereis a one-period nominal risk-free bond and along-term default-free nominal consol that paysone dollar every period in perpetuity (under ourbaseline parameterization, the duration of theconsol is about 25 years). The economy also con-tains a continuum of monopolistically competi-tive firms with fixed, firm-specific capital stocksthat set prices according to Calvo contracts andhire labor competitively from households. Thefirms’ output is subject to an aggregate technol-ogy shock. Furthermore, we assume there is agovernment that levies stochastic, lump-sumtaxes on households and destroys the resourcesit collects. Finally, there is a monetary authoritythat sets the one-period nominal interest rateaccording to a Taylor-type policy rule:

(7)

i i i g y y gt i t i y t t t ti= + − + − + +−

∗−ρ ρ π επ1 11( ) ( ) ,,

max( )

Ec bh l

tt

t t t t

=

− +

∑ −−

−+

0

1

0

1

1 1

`

βγ

χχ

γ χ

,,

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



where i* denotes the steady-state nominal interestrate, yt denotes output, πt denotes the inflation rate(equal to Pt/Pt–1–1), εti denotes a stochastic mone-tary policy shock, and ρi, gy, and gπ are parame-ters.6 This basic structure is very common in themacroeconomics literature, so details of the speci-fication are presented in the appendix.

In equilibrium, the representative household’soptimal consumption choice satisfies the Eulerequation:

(8)

where Pt denotes the dollar price of one unit ofconsumption in period t. The stochastic discountfactor is given by

(9)

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

(10)

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

(11)

The implied term premium is then given by7

(12)

Having specified the benchmark model, wecan now solve the model and compute theresponses of the term premium and the other vari-ables of themodel to economic shocks. Parameters

log log( )

( )

( )

( )

pp

pp

t

t

trn

trn

`

`

`

`−

−−

1 1

.

p i E ptrn

t t trn( ) ( )exp( ) .` `= + − +1 1

p E m pt t t t( ) ( ) .` `= + + +1 1 1

mc bc

c bcP

Ptt t

t t

t

t+

+−

−−

+= −

−11

1 1

β γ

γ( )

( ).

( ) exp( ) ( ) ,c bc i E c bc P Pt t t t t t t t− = −−−

+−

+1 1 1γ γβ /

of the model are given in the appendix. We solvethe model by the standard procedure of approxi-mation around the nonstochastic steady state, butbecause the term premium is zero in a first-orderapproximation and constant in a second-orderapproximation, we compute a third-order approxi-mation to the solution of the model using thenth-order approximation package described inSwanson, Anderson, and Levin (2006), calledperturbation AIM.

In Figures 1, 2, and 3, we present the impulseresponse functions of the term premium andoutput to a 1-percentage-point monetary policyshock, a 1 percent aggregate technology shock,and a 1 percent government purchases shock,respectively. These impulse responses demon-strate that the relationship between the termpremium and output depends on the type ofstructural shock. For monetary policy and tech-nology shocks, a rise in the term premium isassociated with current and future weakness inoutput. By contrast, for a shock to governmentpurchases, a rise in the term premium is associatedwith current and future output strength. Thus,even the sign of the correlation between the termpremium and output depends on the nature ofthe underlying shock that is hitting the economy.

A second observation to draw from Figures 1,2, and 3 is that, in each case, the response of theterm premium is quite small, amounting to lessthan one-third of 1 basis point, even at the peakof the response! Indeed, the average level of theterm premium for the consol in this model is only15.7 basis points.8 This finding foreshadows



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

7 The continuously compounded yield to maturity of the consol isgiven by log[p/(p–1)]. To express the term premium in annualizedbasis points rather than in logs, equation (12) must be multipliedby 40,000. We obtained qualitatively similar results using alterna-tive term-premium measures in the model, such as the term pre-mium on a two-period zero-coupon bond.

8 From the point of view of a second- or third-order approximation,this result is not surprising, because only under extreme curvatureor large stochastic variances do second- or third-order terms mattermuch in a macroeconomic model. Some research has arguablyemployed such model modifications to account for the term pre-mium. For example, Hördahl, Tristani, and Vestin (2006b) assumethat the technology shock has a quarterly standard deviation of2.5 percent and a persistence of 0.986. Adopting these two param-eter values in our model causes the term premium to rise to 141basis points. Ravenna and Seppälä (2006) assume a shock to themarginal utility of consumption, with a persistence of 0.95 and aquarterly standard deviation of 8 percent. A similar shock in ourmodel boosts the term premium to 41 basis points. Wachter (2006)assumes a habit parameter (b) of 0.961, which in our model booststhe term premium to 22.3 basis points. Thus, we are largely able toreplicate some of these authors’ findings; nonetheless, we believethat our benchmark parameter values are the most standard onesin the macroeconomics literature (e.g., Christiano, Eichenbaum,and Evans, 2005).



0 2 4 6 8 10 12 14 16 18 20–0.20

–0.15

–0.10

–0.05

0.00

Percent

Term Premium

0 2 4 6 8 10 12 14 16 18 200.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Output

Basis Points

Quarters

Quarters

Figure 1

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



0 2 4 6 8 10 12 14 16 18 200.00

0.05

0.10

0.15

0.20

Percent

Term Premium

0 2 4 6 8 10 12 14 16 18 20–0.35

–0.30

–0.25

–0.20

–0.15

–0.10

–0.05

0.00

Output

Basis Points

Quarters

Quarters

Figure 2

Impulse Responses to a 1 Percent Technology Shock



0 2 4 6 8 10 12 14 16 18 200.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Percent

Term Premium

0 2 4 6 8 10 12 14 16 18 200.00

0.05

0.10

0.15

0.20

0.25

Output

Basis Points

Quarters

Quarters

Figure 3

Impulse Responses to a 1 Percent Government Purchases Shock

one of the primary limitations of the structuralapproach to modeling term premiums, which wewill discuss in more detail below.

Finally, we note that, although this structuralmodel is very simple, in principle there is no rea-son why the same analysis cannot be performedusing larger and more realistic DSGE models,such as Smets and Wouters (2003), Christiano,Eichenbaum, and Evans (2005), or the extensionsof these in use at a number of central banks andinternational policy institutions.9 Even with theselarger models, we can describe the term-premiumresponse to any given structural shock and thebroader implications of the shock for the economyand optimal monetary policy.

Limitations of the DSGE Model of theTerm Premium

Using a structural DSGE model to investigatethe relationship between the term premium andthe economy has advantages in terms of concep-tual clarity, but there are also a number of limi-tations that prevent the structural-modelingapproach from being useful at present as anempirical workhorse for studying the term pre-mium. This remains true despite the increasinguse of structural macroeconomic models at policy-making institutions for the study of other macro-economic variables, such as output and inflation.These limitations generally fall into two categories:theoretical uncertainties and computationalintractabilities.

Regarding the former, even though someDSGE models—sometimes crucially augmentedwith highly persistent structural shocks—appearto match the empirical impulse responses ofmacroeconomic variables, such as output andinflation, researchers do not agree on how tospecify these models to match asset prices. Forexample, a variety of proposals to explain theequity-premium puzzle include habit formation

in consumption (Campbell and Cochrane, 1999),time-inseparable preferences (Epstein and Zin,1989), and heterogeneous agents (Constantinidesand Duffie, 1996, and Alvarez and Jermann, 2001).This lack of consensus implies that there is muchuncertainty about the appropriate DSGE specifi-cation for analyzing the term premium.

The possibility that a heterogeneous-agentmodel is necessary to understand risk premiumsposes perhaps the most daunting challenge forstructural modelers of the term premium. In thecase of heterogeneous agents with limited partici-pation in financial markets, different households’valuations of state-contingent claims are not equal-ized, so determining equilibrium asset prices canbecome much more complicated than in the rep-resentative-household case. Although a stochasticdiscount factor still exists under weak assump-tions even in the heterogeneous-household case,it need not conform to the typical utility func-tions that are in use in current structural macro-economic models.10

The structural approach to asset pricing alsofaces substantial computational challenges, par-ticularly for the larger-scale models that arebecoming popular for the analysis of macroeco-nomic variables. Closed-form solutions do notexist in general, and full numerical solutions arecomputationally intractable except for the simplestpossible models.11 The standard approach oflog-linearization around a steady state that hasproved so useful in macroeconomics is unfortu-nately not applicable to asset pricing, because byconstruction it eliminates all risk premiums inthe model. Some extensions of this procedure toa hybrid log-linear log-normal approximation(Wu, 2006, and Bekaert, Cho, and Moreno, 2005)



10 One might even question the assumptions required for a stochasticdiscount factor to exist. For example, if there are large traders andsome financial markets are thin, then it is no longer the case thatall investors can purchase any amount of a security at a constantprice, contrary to the standard assumptions.

11 See Backus, Gregory, and Zin (1989), Donaldson, Johnsen, andMehra (1990), Den Haan (1995), and Chapman (1997) for examplesof numerical solutions for bond prices in very simple real businesscycle models. Gallmeyer, Hollifield, and Zin (2005) provide aclosed-form solution for bond prices in a simple New Keynesianmodel, under the assumption of a very special reaction functionfor monetary policy.

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

and to a full second-order approximation aroundthe steady state (Hördahl, Tristani, and Vestin,2006b) are only moderately more successful,because they imply that all risk premiums in themodel are constant (in other words, these authorsall assume the weak form of the expectationshypothesis). Obtaining a local approximationthat actually produces time-varying risk or termpremiums requires a full third-order approxima-tion, as in our analysis above and in Ravenna andSeppälä (2006). Even then, the implied time vari-ation in the term premium is very small, due tothe inherently small size of third-order terms,unless one is willing to assume very large valuesfor the curvature of agents’ utility functions, verylarge stochastic shock variances, and/or very highdegrees of habit persistence (which goes back tothe theoretical limitations discussed above). Thus,the challenges in computing the asset-pricingimplications of DSGE models, while becomingless daunting over time, remain quite substantial.

MACRO-FINANCE MODELS OFTHE TERM PREMIUM

Because of the significant limitations inapplying the structural model discussed above,researchers interested in modeling the term pre-mium in a way that can be taken to the data havehad no choice but to pursue a less-structuralapproach. Although one can model “yields withyields” using a completely reduced-form, latent-factor, no-arbitrage asset-pricing model, as inDuffie and Kan (1996) and Dai and Singleton(2000), recent research has focused increasinglyon hybrid macro-finance models of the termstructure, in which some connections betweenmacroeconomic variables and risk premiums aredrawn, albeit not within the framework of a fullystructural DSGEmodel (see Diebold, Piazzesi, andRudebusch, 2005). The approaches employed inthis macro-finance literature have generally falleninto two categories: vector autoregression (VAR)macro-finance models and New Keynesian macro-finance models. We consider each in turn.

VAR-Based Macro-Finance Models

The first paper in the no-arbitrage macro-finance literature was Ang and Piazzesi (2003).12

They assume that the economy follows a VAR:

(13)

where the state vector, Xt, contains output, infla-tion, the one-period nominal interest rate, and twolatent factors (discussed below). The stochasticshock, εt, is i.i.d. over time. In this model, the one-period nominal interest rate, it, is determined bya Taylor-type monetary policy rule based on Xt,so that the model-implied expected path of theshort-term interest rate is known at any point intime.

The VAR, however, does not contain anyinformation about the stochastic discount factor.Ang and Piazzesi simply assume that the sto-chastic discount factor falls into the essentiallyaffine class, as in standard latent-factor financemodels, so it has the functional form

(14)

where εt is assumed to be conditionally log-normally distributed and the prices of risk, λt,are assumed to be affine in the state vector, Xt:

(15)

Estimation of this model is complicated bythe inclusion of two unobserved, latent factorsin the state vector, Xt, which are typical of no-arbitrage models in the finance literature. To makeestimation tractable, Ang and Piazzesi imposethe restriction that the unobserved factors do notinteract at all with the observed macroeconomicvariables (output and inflation) in the VAR.Because of this very strong restriction, the macro-

X Xt t t= + +−µ εΦ Σ1 ,

m it t t t t t+ += − − ′ − ′

1 1

12

exp ,λ λ λ ε

λ λ λt tX= +0 1 .

12 A number of papers before Ang and Piazzesi (2003) investigatedthe dynamic interactions between yields and macroeconomicvariables in the context of unrestricted VARs, including Evansand Marshall (2001) and Kozicki and Tinsley (2001). Diebold,Rudebusch, and Aruoba (2006) and Kozicki and Tinsley (2005)provide follow-up analysis. As with the no-arbitrage papers dis-cussed below, however, none of these papers has exploredwhether the term premium implied by their models feeds back tothe macroeconomy, the question of interest in the present paper.



economic variables in the model are determinedby a VAR that essentially excludes all interestrates (both short-term and long-term rates). Thus,while the Ang-Piazzesi model can effectivelycapture the extent to which changes in macro-economic conditions affect the term premium, itcannot capture any aspects of that relationshiprunning in the reverse direction.13 In this regard,their model falls short of addressing the topic ofinterest in the present paper.14

Bernanke, Reinhart, and Sack (2004, denotedBRS), employ a similar model but assume that thestate vector, Xt, consists entirely of observablemacroeconomic variables, which determine bothshort-rate expectations (through the VAR) andthe prices of risk (15). By eliminating the use oflatent variables, the empirical implementation ofthe model is simplified tremendously. Of course,as in Ang and Piazzesi, the BRS framework willcapture effects of movements in the term premiumdriven by observable factors included in the VAR,but it does not empirically separate the role ofthe term premium from that of lagged macroeco-nomic variables. Note that the BRS specification,as in the Ang and Piazzesi model, does not includelonger-term interest rates in the VAR (but in thiscase does include the short-term interest rate),implying that movements in the term premiumnot captured by the included variables areassumed to have no effect on the dynamics ofthe economy.

Ang, Piazzesi, and Wei (2006, denoted APW),also estimate a no-arbitrage macro-finance modelbased on a VAR of observed state variables. How-ever, in contrast to BRS, APW explicitly includethe five-year Treasury yield as an element of thestate vector. Thus, to a very limited extent, theirmodel begins to address the types of effects thatare the focus of the present paper. However,their VAR does not distinguish between the risk-neutral and term-premium components of thefive-year yield, so it is only able to capture dis-tinct effects from these two components if theyare correlated (in different ways) with the othervariables in the VAR (which are, specifically, theshort-term interest rate and GDP growth). Eventhen, it would not be possible in their model todisentangle the direct effects of the short-terminterest rate and GDP growth on future outputfrom the indirect effects that changes in thosevariables have on the term premium; it is in thisrespect that the APWmodel cannot help answerthe question we are interested in, even though itallows a separate role for longer-term yields inthe VAR.15

Finally, Dewachter and Lyrio (2006a,b) con-sider a model that is very similar in spirit to APWand BRS, only they work in continuous time andallow for a time-varying long-run inflation objec-tive of the central bank, as argued for by Kozickiand Tinsley (2001) and Gürkaynak, Sack, andSwanson (2005). However, just as with the otherpapers discussed above, Dewachter and Lyrio donot allow changes in the term premium to feedback to themacroeconomic variables of themodel.

New Keynesian Macro-Finance Models

A separate strand of the macro-finance litera-ture has attempted to bridge the gulf betweenDSGE models and VAR-based macro-financemodels by incorporating more economic structureinto the latter. Specifically, these papers replacethe reduced-form VAR in the macro-financemodels with a structural New Keynesian macro-



13 Even when movements in the term premium are driven by theobserved macroeconomic variables (output and inflation) ratherthan the latent factors, the Ang-Piazzesi model fails to identifyeffects of the term premium on the macroeconomy. For example,suppose higher inflation is estimated to raise the term premiumand lead to slower growth in the future. We cannot ascribe theslower growth to the term premium, because the higher inflationmay also predict tighter monetary policy or other factors that wouldbe expected to slow the economy. Note that the VAR does at leastpartially address the issue that not all movements in the term pre-mium are created equal, because the predictive power of a changein the term premium will depend on the specific combination ofeconomic factors driving it.

14 Cochrane and Piazzesi (2006) also focus on the interaction betweenmacroeconomic conditions and the term premium. They use thepredictable component of the ex post returns from holding longer-term securities as a measure of the term premium. Their findingssupport the case that the term premium varies importantly overtime, and they link those movements to macroeconomic conditions.However, they do not address whether the term premium itselfaffects economic activity.

15 APW also present some related reduced-form results on the fore-casting power of the term premium for future GDP growth, whichwe discuss in more detail in the next section.

economic model that governs the dynamics ofthe macroeconomic variables.

An early and representative paper in this lit-erature was written by Hördahl, Tristani, andVestin (2006a, denoted HTV). They begin with abasic New Keynesian structural model in whichoutput, inflation, and the short-term nominalinterest rate are governed by the equations

(16)

(17)

(18)

Equation (16) describes a New Keynesian curvethat allows for some degree of habit formation onthe part of households through the lagged outputterm; equation (17) describes a New KeynesianPhillips curve that allows for some rule-of-thumbprice setters through the lagged inflation term;and equation (18) describes the monetary author-ity’s Taylor-type short-term interest rate reactionfunction. Equations (16) and (17) are structuralin the sense that they can be derived from a log-linearization of household and firm optimalityconditions in a simple structural New KeynesianDSGE model along the lines of our benchmarkmodel (although HTV modify this structure byallowing the long-run inflation objective, π t

*, tovary over time).

In contrast to a DSGE asset-pricing model,however, HTV model the term premium usingan ad hoc affine structure for the stochastic dis-count factor, as in the VAR-based models above.Although this approach is not completely struc-tural, it makes the model computationally tract-able and provides a good fit to the data whileallowing the term premium to vary over time ina manner determined by macroeconomic condi-tions that are determined structurally (to firstorder). The true appeal of this type of model isthat it is parsimonious and simple while allowingfor expectations to influence macroeconomic

i i g E g yt i t i t t t y t ti= + − − + +− +

∗ρ ρ π π επ1 11( ) ( ) ..

y E y y i Et y t t y t i t t t ty= + − − − ++ − +µ µ ζ π ε1 1 11( ) ( ) ,

π µ π µ π δ επ ππ

t t t t y t tE y= + − + −+ −1 11( ) ,

dynamics and for the term premium to vary non-trivially to macroeconomic developments.

However, as was the case in the VAR-basedmodels, the HTV model does not allow the termpremium to feed back tomacroeconomic variables.As discussed in the introduction, the structureof the IS curve in the HTV model assumes thateconomic activity depends only on expectationsof the short-term real interest rate and not on theterm premium. Thus, this approach is also unableto address the issue considered in the currentpaper.

RW develop a New Keynesian macro-financemodel that comes a step closer to addressing thetopic of this paper by allowing for feedback fromthe term structure to the macroeconomic variablesof the model. In particular, RW incorporate twolatent term-structure factors into the model andgive those latent factors macroeconomic inter-pretations, with a level factor that is tied to thelong-run inflation objective of the central bankand a slope factor that is tied to the cyclical stanceof monetary policy. Thus, the latent factors inthe RWmodel can affect economic activity, andthe term structure does provide information aboutthe current values of those latent factors. How-ever, RWmake no effort to decompose the effectsof long-term interest rates on the economy intoan expectations component and a term-premiumcomponent, so there is no sense in which theterm premium itself affects macroeconomicvariables.

Wu (2006) and Bekaert, Cho, and Moreno(2005) come closer to a true structural NewKeynesian macro-finance model by deriving thestochastic discount factor directly from the utilityfunction of the representative household in theunderlying structural model. Thus, like a DSGEmodel, their papers impose the cross-equationrestrictions between the macroeconomy and thestochastic pricing kernel that are ignored whenthe kernel is specified in an ad hoc affine manner.However, these analyses also suffer from the com-putational limitations of working within theDSGE framework (discussed above), because bothpapers are unable to solve the model as specified.



Instead, those authors use a log-linear, log-normalapproximation, which implies that the term pre-mium in the model is time-invariant.16 Thus,their papers do not address the question we haveposed in this paper.17

REDUCED-FORM EVIDENCEON THE EFFECTS OF THETERM PREMIUM

Because of the limitations discussed above,the models in the previous two sections do notprovide us with much insight into the empiricaleconomic implications of changes in the termpremium. The benchmark structural model islargely unable to reproduce the magnitude andvariation of the term premium that is observedin the bond market, and, although the macro-finance models are more successful at capturingthe observed behavior of term premiums, theytypically impose very restrictive assumptions thateliminate any macroeconomic implications ofchanges in term premiums. A separate literaturethat has provided a direct examination of theseimplications is based on reduced-form empiricalevidence. Specifically, in the large literature thatuses the slope of the yield curve to forecast sub-sequent GDP growth, several recent papers havetried to estimate separately the predictive powerof the term premium. In this section, we summa-rize these papers and contribute some new evi-dence on this issue.

An important caveat worth repeating is thatthere is only a reduced-form relationship—not astructural one—between the term premium andfuture output growth, so even the sign of their pair-

wise correlation over a given sample will dependon which types of shocks are most influential.Nevertheless, it may be of interest to considerthe average correlation between future outputgrowth and changes in the term premium oversome recent history. If the mixture of shocks isexpected to remain relatively stable, then theaverage estimated reduced-form relationshipbetween the term premium and future economicgrowth could be useful for forecasting. For thisreason, the historical relationship may provideuseful information to a policymaker who has todecide whether and how to respond to a givenchange in the term premium.

Evidence in the Literature

Recent research relating the term premium tosubsequent GDP growth has been part of a muchlarger literature on the predictive power of theslope of the yield curve. A common approach inthis literature is to investigate whether the spreadbetween short-term and long-term interest rateshas significant predictive power for future GDPgrowth by estimating a regression of the form

(19)

where yt is the log of real GDP at time t and it(n) is

the n-quarter interest rate (usually a longer-termrate such as the 10-year Treasury yield).18 Thestandard finding is that the estimated coefficientβ2 is significant and positive, indicating that theyield-curve slope helps predict growth.

Note that equation (19) is a reduced-formspecification that has no economic structure.However, it can be motivated by thinking of thelong-term interest rate as a proxy for the neutrallevel of the nominal funds rate, so that the yield-curve slope captures the current stance of mone-tary policy relative to its long-run level. Forexample, a steep yield-curve slope (with shortrates unusually low relative to long rates) wouldindicate that policy is accommodative and would

y y y y i it t t t tn

t t+ −− = + − + − +4 0 1 4 2β β β ε( ) ( )( ),



18 This equation assumes that the dependent variable is future GDPgrowth (a continuous variable). Other papers in this literature usea dummy variable for recessions (a discrete variable). In eithercase, the motivation for the approach is the same and the resultsare qualitatively similar.

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

17 A related paper by Gallmeyer, Hollifield, and Zin (2005) providesa full nonlinear solution to a very similar model. However, they areable to solve the model only under the assumption of an extremelyspecial reaction function for monetary policy; thus, their methodhas no generality and is invalid in cases in which that policyreaction function is not precisely followed.

be associated with faster subsequent growth, thusaccounting for the positive coefficient.

In this respect, the use of the long-term interestrate in the regression (19) is motivated entirelyby the component related to the expected long-run level of the short rate. But the long-term ratealso includes a term premium; hence, any varia-tion in this premium will affect the performanceof the equation. Indeed, it is useful to decomposethe yield-curve slope into these two components,as follows:

(20)

The first term captures the expectations compo-nent, or the proximity of the short rate to itsexpected long-run level. The second componentis the term premium, or the amount by which thelong rate exceeds the expected return from invest-ing in a series of short-term instruments. Fornotational simplicity, we will denote the firstcomponent in (20) as exspt, the expected-ratecomponent of the yield spread, and the secondcomponent as tpt, the term premium.

With this decomposition, the predictionequation (19) can be generalized as follows:

(21)

The standard equation (19) imposes the coeffi-cient restriction β2 = β3. Loosening that restric-tion allows the term premium to have a differentimplication for subsequent growth than theexpected-rate component.19 Several recent papershave considered this issue, as we will brieflysummarize.

The first paper to examine the importance ofthe above decomposition for forecasting was

i in

E i i in

Etn

t t t j tj

n

tn( ) ( )− = −

+ −+

=

−

∑1 1

0

1

tt t jj

ni +

=

−

∑

0

1.

y y y y exsp tpt t t t t t t+ −− = + − + + +4 0 1 4 2 3β β β β ε( ) .

Hamilton and Kim (2002), which forecasts futureGDP growth using a spread between the 10-yearand 3-month Treasury yields in equation (19). Theinnovation of their paper is that it then separatesthe yield spread into the expectations and term-premium components considered in equation (21).The authors achieve this separation by consider-ing the ex post realizations of short rates, usinginstruments known ex ante to isolate the expec-tations component. They find that the coefficientsβ2 and β3 are indeed statistically significantlydifferent from one another, although both coeffi-cients are estimated to be positive. Note that apositive value for β3 implies that a decline in theterm premium is associated with slower futuregrowth.

A second paper that decomposes the predic-tive power of the yield spread into its expectationsand term-premium components is Favero,Kaminska, and Söderström (2005). These authorsdiffer from Hamilton and Kim (2002) by using areal-time VAR to compute short-rate expectationsrather than a regression of ex post realizations ofshort rates on ex ante instruments. As in Hamiltonand Kim (2002), they find a positive sign for thecoefficient β3, so that a lower term premium againpredicts slower GDP growth.

A third relevant paper is by Wright (2006),who touches on this issue in the context of aprobit model for forecasting recessions. Wrightconsiders the predictive power of the yield slope,and then he investigates whether the return fore-casting factor from Cochrane and Piazzesi (2005)also enters those regressions significantly. Sincethis factor is correlated with the term premium,he is implicitly controlling for the term premium,as in equation (21). He finds that this factor isinsignificant for predicting recessions over hori-zons of two or four quarters but has a significantnegative coefficient for predicting recessions overa six-quarter horizon; that is, a lower term pre-mium raises the odds of a recession, consistentwith the findings of the other papers that it wouldpredict slower growth.

A final reference is APW. As noted above,they use a VAR that includes long rates, GDPgrowth, and a short rate, but they cannot separateout the effects of the term premium from other

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



movements in long-term interest rates. However,the authors perform an additional exercise inwhich they calculate the expected-rate and term-premium components of the long rate as impliedby the VAR and then estimate the forecastingequation (21), allowing for different effects fromthese two components. In contrast to the previ-ously discussed papers, APW find that the termpremium has no predictive power for future GDPgrowth; that is, the coefficient β3 is zero.

Overall, the handful of papers that havedirectly tackled the predictive power of the termpremium have produced results that starkly con-trast with the intuition that Chairman Bernankeexpressed in his March 2006 speech (see the intro-duction). The empirical studies to date suggestthat, if anything, the relationship has the oppositesign from the practitioner view. According to theseresults, policymakers had no basis for worryingthat the decline in the term premium might bestimulating the economy and instead should haveworried that it was a precursor to lower GDPgrowth.

Empirical Estimates of the TermPremium

Estimation of equation (21) requires a measureof the term premium, and there are a variety ofpossibilities in the literature. We begin our empiri-cal analysis by collecting a number of the promi-nent term-premium measures and examiningsome of the similarities and differences amongthem.

Specifically, we consider five measures ofthe term premium on a zero-coupon nominal 10-year Treasury security20:

1.VAR measure: The first of these measures,whichwe label the “VAR”measure, is basedon a straightforward projection of the shortrate from a simple but standard three-

variable macroeconomic VAR comprisingfour lags each of the unemployment rate,quarterly inflation in the consumer priceindex, and the 3-month Treasury bill rate.At each date the VAR can be used to fore-cast the short rate over a given horizon, andthe average expected future short rate canbe used as an estimate of the risk-neutrallong-term rate of that maturity.21 The differ-ence between the observed long-term rateand the risk-neutral long-term rate thenprovides a simple estimate of the termpremium. This approach has been usedby Evans and Marshall (2001), Favero,Kaminska, and Söderström (2005), Diebold,Rudebusch, and Aruoba (2006), andCochrane and Piazzesi (2006).

2. Bernanke-Reinhart-Sack measure:A poten-tial shortcoming of using a VAR to estimatethe term premium is that it does not imposeany consistency between the yield curveat a given point in time and the VAR’s pro-jected evolution of those yields. Such pric-ing consistency can be imposed by using ano-arbitrage model of the term structure.As discussed in the previous section, a no-arbitrage structure can be laid on top of aVAR to estimate the behavior of the termpremium, as in BRS. Here, we considerthe term-premium estimate from that paper,as updated by Rudebusch, Swanson, andWu (2006).

3. Rudebusch-Wu measure: No-arbitragerestrictions can also be imposed on top ofa New Keynesian macroeconomic model.Here we take the term premium estimatedfrom one such model, Rudebusch and Wu(2003 and 2007), discussed earlier. As withthe Bernanke-Reinhart-Sack measure, thisterm-premium measure was extended to a



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

21 Of course, there are several reasons for not taking these VAR pro-jections too seriously as good measures of the actual interest rateexpectations of bond traders at the time. Rudebusch (1998) describesthree important limitations of such VAR representations: (i) theuse of a time-invariant, linear structure, (ii) the use of final reviseddata and full-sample estimates, and (iii) the limited number ofinformation variables. We examined several rolling-sample esti-mated VARs as well and obtained similar results.

longer sample by Rudebusch, Swanson,and Wu (2006), and we use this extendedversion below.

4. Kim-Wright measure:One can also estimatethe term premium using a standard no-arbitrage dynamic latent-factor model fromfinance (with no macroeconomic structureunderlying the factors). In these models,risk-neutral yields and the term premiumare determined by latent factors that arethemselves linear functions of the observedbond-yield data. We use the term-premiummeasure from a three-factor model dis-cussed by Kim and Wright (2005), whichwe extend back to 1961.22

5. Cochrane-Piazzesi measure: Cochrane andPiazzesi (2005) analyze excess returns fora range of securities over a one-year hold-ing period. Their primary finding is that asingle factor—a particular combination ofcurrent forward rates—predicts a consid-erable portion of the excess returns froma one-year holding period for Treasurysecurities. For our purposes, however, weare interested in the term premium on a10-year security, or the (annualized) excessreturn expected over the 10-year period.Sack (2006a) provides a straightforwardapproach for converting the Cochrane-Piazzesi one-year holding-period resultsinto a measure of the term premium. Speci-fically, the expected one-period excessreturns implied by the Cochrane-Piazzesiestimates, together with the one-year risk-free rate, imply an expected set of zero-coupon yields one year ahead (because theonly way to generate expected returns onzero-coupon securities is through changes

in yield). Those expected future yields canthen be used to compute the expectedCochrane-Piazzesi factor one year aheadand, hence, the expected excess returnsover the one-year period beginning oneyear ahead. By iterating forward, one cancompute the expected excess return foreach of the next 10 years, thereby yieldinga measure of the term premium on the 10-year security.

As is clear from the above descriptions, theapproaches used to derive the five term-premiummeasures differ considerably in the variablesincluded and the theoretical restrictions incor-porated. Nevertheless, the measures show manysimilar movements over time, as can be seen inFigure 4, which plots the five measures of theterm premium for the 10-year zero-couponTreasury yield back to 1984.

Three of themeasures, in particular—the VAR,Bernanke-Reinhart-Sack, and Kim-Wright—areremarkably highly correlated over this period.23

As shown in Table 1, the correlation coefficientsamong these measures range from 0.94 to 0.98.The other two measures—Rudebusch-Wu andCochrane-Piazzesi—are less correlated with theothers. For example, the correlation coefficientswith the VAR measure are 0.68 for Rudebusch-Wu and 0.88 for Cochrane-Piazzesi. These lowercorrelations largely reflect that the Rudebusch-Wu measure is more stable than the others andthat the Cochrane-Piazzesi measure is morevolatile.

The greater stability of the Rudebusch-Wumeasure can be easily understood. Their under-lying model attributes much of the variation inthe 10-year Treasury yield to changes in theexpected future path of short rates, reflecting, intheir framework, variation in the perceived infla-tion target of the central bank. That assumptionis supported by other research. For example,Gürkaynak, Sack, and Swanson (2005) foundsignificant systematic variation in far-ahead for-



22 We extend the Kim-Wright measure back to 1961 by regressingthe three Kim-Wright latent factors on the first three principalcomponents of the yield curve and using these coefficients to esti-mate the Kim-Wright factors in prior years. Because the term pre-mium in the model is a linear function of observed yields, andbecause the Kim-Wright model fits the yield-curve data very well,this exercise should come very close to deriving the same factorsthat would be implied if we extended their model back to 1961.Over the period where our proxy and the actual Kim-Wright termpremium overlap, the correlation between the two measures is0.998 and the average absolute difference between them is lessthan 4 basis points.

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

ward nominal interest rates in response to macro-economic news in a way that suggested changesin inflation expectations rather than changes interm premiums. Similarly, Kozicki and Tinsley(2001) found that statistical models that allowfor a “moving endpoint” are able to fit interestrate and inflation time series much better thanstandard stationary or difference-stationary VARs.By attributing more of the movement in long ratesto short-rate expectations, the Rudebusch-Wu

analysis does not need as much variation in theterm premium to explain the observed variationin yields.24

The behavior of the measure based onCochrane and Piazzesi (2005) is harder to under-stand. This measure is well below the other meas-



1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004–3

–2

–1

0

1

2

3

4

5

6

7

8

Bernanke-Reinhart-SackCochrane-PiazzesiKim-WrightRudebusch-WuVAR

Percent

Figure 4

Five Measures of the 10-Year Term Premium

Table 1Correlations Among the Five Measures of the Term Premium

BRS RW KW CP VAR

BRS 1.00

RW 0.76 1.00

KW 0.98 0.81 1.00

CP 0.92 0.87 0.96 1.00

VAR 0.96 0.68 0.94 0.88 1.00

24 One could argue that a weakness of the other term-premium esti-mates is that they are based on models that assume that the long-run features of the economy, such as the steady-state real interestrate and rate of inflation, are completely anchored.

ures and is much more volatile. To a large extent,this behavior simply mimics the one-periodexpected excess returns computed by Cochraneand Piazzesi. Indeed, Sack (2006b) and Wright(2006) have pointed out that the implied one-period expected excess returns are surprisinglyvolatile and are currently very negative. Thisbehavior partly shows through to the impliedterm-premium measure.

Overall, Figure 4 provides us with a menu ofchoices for the analysis that follows.25 Even withthe differences noted above, the five measuresshow considerable similarities in their variationover this sample. Indeed, the first principal com-ponent captures 95 percent of the variation inthese five term-premium estimates. In the analysisin the next section, we focus our attention on the

Kim-Wright measure. This measure appears to berepresentative of the other measures considered.In fact, it is very highly correlated (0.99) with thefirst principal component of all five measures.Moreover, it has the advantage that it can beextended back to the early 1960s, allowing us toconduct our analysis over a longer sample.

The 10-year zero-coupon yield is shown inFigure 5 along with the two components based onthe Kim-Wright term-premium estimate.26 As canbe seen, both short-rate expectations and the termpremium contributed to the run-up in yieldsthrough the early 1980s and, since then, to thedecline in yields. As noted by Kim and Wright(2005), the term premium recently has fallen tovery low levels, a pattern consistent with the



0

2

4

6

8

10

12

14

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

10-Year Zero-Coupon Yield

Risk-Neutral10-Year Zero-Coupon Yield

10-Year Term Premium

Percent

Figure 5

Kim-Wright Decomposition of the 10-Year Zero-Coupon Yield

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

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

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

conundrum discussed by former Chairman AlanGreenspan.

Figure 6 plots this term-premium measurealong with the Congressional Budget Office (CBO)output gap and provides the first hint of a negativerelationship between the two. It is this relation-ship that we now explore in more detail.

New Evidence on the Implications ofthe Term Premium

We begin by estimating the standard relation-ship between the slope of the yield curve andsubsequent GDP growth, using the specificationin equation (19). The long rate is a 10-year zero-coupon Treasury yield, taken from the Gürkaynak,Sack, and Wright (2006) database. The short rateis the 3-month Treasury bill rate from the FederalReserve’s H.15 data release. All data are quarterlyaverages, and the sample ranges from 1961:Q3 to2005:Q4. We examine both this full sample anda shorter subsample beginning in 1984, which

arguably has a more consistent monetary policyregime (e.g., Rudebusch and Wu, 2007).

Results are presented in the first column ofTable 2. Over the full sample, we find that thecoefficient for the yield-curve slope is highly sta-tistically significant and has a positive sign. Thisestimate implies that a flatter yield curve predictsslower GDP growth, the standard finding in theacademic literature. Over the shorter sample, theestimated coefficient loses its significance, reflect-ing another fact that is well-appreciated amongresearchers—that the predictive power of theyield-curve slope for growth appears to havediminished in recent decades.

As discussed above, this approach is purelya reduced-form exercise that is not explicitly tiedto a theoretical structure. However, a commonmotivation for using the yield-curve slope as apredictor is that it serves as a proxy for the stanceof monetary policy relative to its neutral level.Given this motivation, one would prefer to meas-ure the yield-curve slope based strictly on the



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

–8

–6

–4

–2

0

2

4

6

8

10-Year Term Premium

Output Gap

Percent

Figure 6

Kim-Wright Term Premium and the CBO Output Gap

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

portion of the long-term interest rate associatedwith expectations of the short-term rate. In thatcontext, we can also ask how the other componentof the long rate—the term premium—affectsgrowth. This consideration leads to specification(21) above, in which the two components of theyield-curve slope are allowed to have differentpredictive effects for subsequent GDP growth.

We can implement this approach using theterm-premium measure described above.27 Theresults are shown in column 2. For both samples,the expectations-based component of the yieldslope has slightly stronger predictive power thanthe pure yield-curve slope (that is, the coefficient

on this component is slightly larger and moresignificant than the coefficient on the overallslope reported in column 1), and the coefficienton the term premium, β3, is not significantlydifferent from zero. However, we are unable toreject the hypothesis that β2 = β3 at even the 10percent level over either the post-1962 or post-1985 sample.

Our findings are similar in spirit to the exist-ing empirical evidence that the term premiumhas a different effect on subsequent growth thanthe expectations-related component of the yieldcurve. Note that the only purpose of having a term-premium measure, according to these results, isto determine the expectations component of theyield slope more accurately. The term premiumitself has no predictive power for future growth.

27 In our analysis, we ignore any potential issues associated withgenerated regressors.



Table 2Prediction Equations for GDP GrowthDependent Variable: yt+4 – yt

(1) (2) (3) (4)

1962-2005 Sample

yt – yt–4 0.15 (1.57) 0.12 (1.18) 0.32 (3.04) 0.38 (4.22)

it(n) – it 0.64 (3.64)

exspt 0.68 (4.03) 1.03 (5.64)

exspt–4 –0.79 (–3.49)

tpt 0.30 (0.92) –0.61 (–1.34)

tpt–4 0.54 (1.24)

exspt – exspt–4 0.96 (5.62)

tpt – tpt–4 –0.77 (–1.95)

1985-2005 Sample

yt – yt–4 0.26 (2.54) 0.32 (2.31) 0.36 (2.30) 0.36 (2.68)

it(n) – it 0.28 (1.29)

exspt 0.35 (1.59) 0.46 (1.92)

exspt–4 –0.07 (–0.32)

tpt 0.07 (0.25) –0.46 (–1.15)

tpt–4 0.61 (2.18)

exspt – exspt–4 0.30 (1.37)

tpt – tpt–4 –0.59 (–1.93)

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

However, the specification of these regres-sion equations seems somewhat at odds with themodels we presented earlier. For example, theNew Keynesian IS curve (2) could be used tomotivate the use of the yield-curve slope, as itassumes that output is determined by the devia-tion of the real short-term interest rate from itsequilibrium level. The expectations componentof the yield-curve slope might capture this vari-able, but it should then be related to the level ofthe output gap. In contrast, the reduced-formspecifications (19) and (21) relate the slope ofthe yield curve to the growth rate of output. Thus,this specification seems to differ from the morestructural models by a derivative. Moreover, theterm premium in Figures 5 and 6 appears to benonstationary or nearly nonstationary, while GDPgrowth is much closer to being stationary. Thus,from a statistical point of view, specifications(19) and (21) are also highly suspect.

If we difference equation (2) to arrive at aspecification in growth rates, it would suggestthat it is changes in the stance of monetary policythat predict future GDP growth.28 This suggestsinvestigating whether GDP growth is tied tochanges in the stance of policy and changes inthe term premium, as opposed to the levels ofthose variables.

As an exploratory step in this direction, were-estimate equation (21) with an additional one-year lag of the right-hand-side variables includedin the regression. The results, shown in column3 of Table 2, strongly hint that there is greaterpredictive power associated with the changes inthese variables than with their levels. Indeed, onecan reject the hypothesis that the coefficients onthe lagged variables are zero (at the 1 percent sig-nificance level). Moreover, one cannot reject thatthe right-hand-side variables enter the regressiononly as changes. That is, the hypothesis that thecoefficients on the lag of these components equalthe negative of the coefficients on their currentlevels cannot be rejected even at the 10 percent

significance level. A similar (though less striking)pattern is found in the shorter sample.

Because both the theory and the hypothesistests in the preceding paragraph suggest that onlydifferences should matter, column (4) of the tablepresents results from estimating the baseline fore-casting regression equation in differences, namely,

(22)

The full-sample results indicate that both com-ponents of the yield-curve slope matter for futuregrowth. The coefficient on the risk-neutral expec-tations component of the yield-curve slope is nowlarger and more statistically significant than inany of the earlier specifications. We can also over-whelmingly reject the hypothesis that β2 = β3(with p-values less than 10–4). This finding indi-cates that GDP growth is expected to be highernot when the short-term interest rate is merelylow relative to its long-run level, but when it hasfallen relative to that level.

More importantly for this paper, we find thatthe estimated coefficient on the term premium isnow negative and (marginally) statistically sig-nificant. According to these results, a decline inthe term premium tends to be followed by fasterGDP growth—the opposite sign of the relationshipuncovered by previous empirical studies. (In theshorter sample, all of the coefficients are again lesssignificant. However, we still reject the hypothe-sis that β2 = β3 [with a p-value of 0.0395] in col-umn 4, and the coefficient on the change in theterm premium is again negative and borderlinestatistically significant.29)

Our findings line up with the intuitionexpressed by Chairman Bernanke when he sug-gested that the declining term premium signaledadditional stimulus to the economy. Our results

y y y y exsp exspt t t t t t+ − −− = + − + −4 0 1 4 2 4β β ββ

( ) ( )

+ 3(( )tp tpt t t− +−4 ε .



28 Some might argue that the dependent variable here should be thegrowth of the output gap rather than GDP. As discussed below, weobtained similar results using the change in the CBO output gapas the dependent variable.

29 Furthermore, using the year-on-year change in the CBO outputgap as the predicted variable rather than the year-on-year changein output itself gave similar results. Specifically, the coefficienton the term premium remains negative, with a p-value just lessthan 0.05. These results suggest that a decline in the term premiumpredicts a higher future value of the output gap and that policy-makers might want to take that prediction into account when for-mulating the optimal policy response.

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

CONCLUSIONSOur results can be usefully summarized from

the perspective of advising monetary policy-makers. Specifically, policymakers may wonderhow they should respond when confronted witha substantial change in the term premium, suchas the recent decline that appears to have takenplace during 2004 and 2005.

The first, and perhaps most important, con-clusion from our analysis is that policymakersshould always try to determine the source of thechange in the term premium. If that source canbe identified, then policymakers are advised toconsider the repercussions of that underlyingdriving force more broadly rather than focusingexclusively on the change in the term premium.In this way, policymakers can take into accountthe macroeconomic implications of the structuralshifts or disturbances that are affecting the termpremium.

Of course, policymakers often may be uncer-tain about the reasons for changes in the termpremium. Indeed, during the past few years, avariety of only tentative explanations have beenoffered for the seemingly low term premium. Insuch a situation, policymakers may find ourreduced-form analysis of the implications of theterm premium for future economic activity to bea useful baseline. Our results suggest that a declinein the term premium has typically been associatedwith higher future GDP growth, which appearsconsistent with the practitioner view. Indeed,according to our reduced-form analysis, the atten-tion that Federal Reserve officials paid to theseemingly large decline in the term premium in2004 and 2005 may have been justified.

Finally, our finding that changes in the termpremium have a significant correlationwith futureGDP growth is not captured by many macroeco-

nomic models. Understanding and incorporatingthis correlation within the framework of a modelwould appear to be a useful addition to theresearch agenda. In this regard, we only speculatethat our empirical findings may reflect a hetero-geneous population in which a decline in the termpremiummakes financial market conditions moreaccommodative for certain classes of borrowers.

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APPENDIX

Benchmark New Keynesian Model

To better understand the structural relationship between the term premium and the macroeconomy,we define a simple New Keynesian DSGE model to use as a benchmark. This appendix provides adetailed description of the model, the benchmark parameter values we used in computing the impulseresponses in Figures 1 to 3, and our solution algorithm.

The economy contains a continuum of households with a total mass of unity. Households are rep-resentative and seek to maximize utility over consumption and labor streams given by

(23)

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

The economy also consists of a continuum of monopolistically competitive intermediate goodsfirms indexed by f∈[0,1]. Firms have Cobb-Douglas production functions:

(24)

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

(25)

where εtA denotes an i.i.d. aggregate technology shock with mean zero and variance σA2. Intermediategoods are purchased by a perfectly competitive final goods sector that produces the final good with aconstant elasticity of substitution production technology:

(26)

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

(27) y fp f

PYt

t

tt( )

( ),

( )

=

− +1 θ θ/

Y y f dft t=

++

∫ ( ) .( )1 1

0

1 1/ θ

θ

log log� ,A At A t tA= +−ρ ε1

y f A k l ft t t( ) ( ) ,= −α α1

mc bC

c bCP

Pt t jj t j t j

t t

t

t j,

( )

( )++ + −

−

−−

+

−−

;βγ

γ1

1

..

max( )

Ec bh l

tt

t

t t tβγ

χχ

γ χ

=

− +

∑ −−

−+

0

1

0

1

1 1

`

,,

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



where

(28)

is the constant elasticity of substitution aggregate price of a unit of the final good.Each firm sets its price, pt(f ), according to a Calvo contract that expires with probability 1 – ξ each

period. There is no indexation, so the price, pt(f ), is fixed over the life of the contract. When a contractexpires, the firm is free to reset its price as it chooses. In each period t, firms must supply whateveroutput is demanded at the posted price, pt(f ). Firms hire labor, lt(f ), from households in a competitivelabor market, paying the nominal market wage, wt. Marginal cost for firm f at time t is thus given by

(29)

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

(30)

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

(31)

To aggregate up from firm-level variables to aggregate variables, it is useful to define the cross-sectional price dispersion, ∆t:

(32)

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

(33)

where K– = k– and

(34)

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

(35) wP

lc bC

t

t

t

t t

=− −

−χ χ

γ0

1( ),

P p f dft t; ( )−−

∫

1

0

1 /θθ

L l f dft t;0

1

∫ ( )

Y A K Lt t t t= − −∆ 1 1α α ,

∆tj

jt jp f1 1

0

1 11/ /( ) ( ) ( (( ) ( )−

=−

∗ − + −− ∑α θ θ αξ ξ;`

))),

p fE m mc f y f

Et

tj

j t t j t j t j∗ = + + +=

+ ∑( )

( ) ( ) ( ),1 0θ ξ`

ttj

j t t j t jm y fξ= + +∑ 0`

, ( ).

E m p f y f w l ftj

jt t j t t j t j t jξ

=+ + + +∑ −

0

`

, ( ) ( ) ( ) ,,

mc fw l f

y ftt t

t

( )( )

( ) ( ).=

−1 α



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

and the intertemporal Euler equation,

(36)

where it denotes the continuously compounded interest rate on the riskless one-period nominal bond.There is no investment in physical capital in the model.

There is a government in the economy, which levies lump-sum taxes, Gt, on households anddestroys the resources it collects. The aggregate resource constraint implies that

(37)

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

(38)

where εtG denotes an i.i.d. government consumption shock with mean zero and variance σG2.Finally, there is a monetary authority in the economy that sets the one-period nominal interest rate

according to a Taylor-type policy rule:

(39)

where i* denotes the steady-state nominal interest rate, πt denotes the inflation rate (equal to Pt/Pt–1 – 1),εti denotes an i.i.d. stochastic monetary policy shock with mean zero and variance σi2, and ρi, gy, andgπ are parameters. Of course, the steady-state inflation rate, π*, in this economy must satisfy1 + π* = βexp(i*).

As noted above, households have access to a long-term default-free nominal consol that pays onedollar every period in perpetuity. The nominal consol’s price, pt

(�), thus satisfies

(40)

wheremt+1 � mt,t+1 is the representative household’s stochastic discount factor. We define the risk-neutral consol price, pt

(�)rn, to be

(41)

and the implied term premium is then given by

(42)

Note that under our baseline parameterization, the consol in our model has a duration of about 25 years.This completes the specification of the benchmark model referred to in the text. In computing

impulse response functions, we use the parameter values as specified in Table A1. A technical issuethat arises in solving the model above is the relatively large number of state variables, eight in all: Ct–1,At–1, Gt–1, it–1, ∆t–1, plus the three shocks εtA, εtG, εti.32 Because of dauntingly high dimensionality, value-function iteration-based methods, such as projection methods (or, even worse, discretization methods),are computationally intractable. We instead solve the model above using a standard macroeconomic

( ) exp( ) ( ) ,c bC i E c bC P Pt t t t t t t t− = −−−

+−

+1 1 1γ γβ /

log log( )

( )

( )

( )

pp

pp

t

t

trn

trn

`

`

`

`−

−−

1 1

.

p i E ptrn

t t trn( ) ( ) ( )exp( ) ,` `= + − +1 1

1

p E m pt t t t( ) ( ) ,` `= + + +1 1 1

i i i g Y Y gt i t i y t t t ti= + − + − + +−

∗−ρ ρ π επ1 11( ) ( ) ,,

log log ,G Gt G t tG= +−ρ ε1

Y C Gt t t= + ,



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

technique that approximates the model’s solu-tion around the nonstochastic steady state—a so-called perturbation method.

As discussed in the text, a first-order approx-imation (i.e., a linearization or log-linearization)of the model around the steady state eliminatesthe term premium from the model entirely,because equations (40) and (41) are identical inthe first-order approximation. A second-orderapproximation produces a nonzero but constantterm premium (a sum of the variances σA2, σG2,and σi2). Because our interest in this paper is notjust in the level of the term premium but also inits variation over time, we must compute a third-order approximation to the solution of the modelaround the nonstochastic steady state. We do sousing the nth-order perturbation AIM algorithm ofSwanson, Anderson, and Levin (2006). This algorithm requires that the equations of the model be putinto a recursive form, which for the model above is fairly standard. The most difficult equation is (31),which can be written in recursive form as

(43)

(44)

(45)

The computational time required to solve our model to the third order is minimal—no more than about10 seconds on a standard laptop computer.

Computing impulse responses for this model is actually simpler than the use of a third-orderapproximation might suggest. We are interested in the responses of output and the term premium toan exogenous shock to εtA, εtG, or εti. For output, we plot the standard first-order (i.e., log-linear) responsesof output to each shock. For small shocks, such as those of the size we are considering here (1 percent),these responses are highly accurate. For the term premium, of course, the first- and second-orderresponses of that variable to each shock would be identically zero, so we plot the third-order responsesof that variable. These third-order terms are all of the form σZ2X, where Z∈{A,G,i } and X is one of thestate variables of the model,33 so if we plug in the values of σA2, σG2, and σi2 given in Table A1, theseterms are linear as well, which makes them easy to plot.

z L En t t t t t,( )( )= +

−++ − −

+

+

11

0 1 1 11

1

θ χα

βξ πχ αθ

θ∆ /11

11

−+

α zn t, .

z Y C bC E zd t t t t t t d t, ,( )= − +−−

+ +1 11

1γ θβξ π /

p fP

zz

t

t

n t

d t

∗ +−

+

=( ) ,

,

11

1αα

θθ



Table A1Benchmark Model Parameter Values

α 0.3 ρA 0.9

β 0.99 ρG 0.9

θ 0.2 ρi 0.7

ξ 0.75 σA2 0.012

γ 2 σG2 0.0042

χ0 (1 – b)–γ σi2 0.0042

χ 1.5 K–

1

b 0.66 π * 0

gπ 2

gy 0.5

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


Documents

Macroeconomic Implications of Changes in the Term Premium · FEDERAL RESERVE BANK OF ST.LOUIS R EVI W JULY /AUGUST 2 07 241 MacroeconomicImplicationsof ChangesintheTermPremium GlennD.Rudebusch,BrianP.Sack,andEricT.Swanson