Evolving Macroeconomic Perceptions and the Term Structure of Interest Rates

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Evolving Macroeconomic Perceptions and the Term Structure of Interest Rates Athanasios Orphanides and Min Wei 2010-01 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Evolving Macroeconomic Perceptions and the Term Structure of Interest Rates∗ Athanasios Orphanides† Min Wei‡ This Version: April 14, 2008 JEL Classification: E43, E44, E47, G12 Keywords: Macro Term Structure Model, Recursive VAR, Survey Forecasts, Anticipated Utility ∗WethankAndrewAng,ThomasLaubach,AndrewLevin,JonathanWright,TackYunandseminar participants at the 2008 AEA Meetings and the Federal Reserve Board for helpful discussions and comments. The opinions expressed are those of the authors and do not necessarily reflect views of the CentralBankofCyprusortheBoardofGovernorsoftheFederalReserveSystem. †Central Bank of Cyprus, 80 Kennedy Avenue, Nicosia, Cyprus; ph: +357-22714471; fax: +357- 22378151;email: Athanasios.Orphanides@centralbank.gov.cy;web: www.centralbank.gov.cy ‡Board of Governors of the Federal Reserve System, Division of Monetary Affairs, Washington, DC 20551, USA; ph: (202) 736-5619; fax: (202) 452-2301; email: min.wei@frb.gov; web: www.federalreserve.gov/research/staff/weiminx.htm

Abstract We explore the role of evolving beliefs regarding the structure of the macroeconomy in improving our understanding of the term structure of interest rates within the context of a simple macro-finance model. Using quarterly vintages of real-time data and survey forecasts for the United States over the past 40 years, we show that a recursively estimated VAR on real GDP growth, inflation and the nominal short-term interest generates predictions that are more consistent with surveyforecasts than a benchmark fixed-coefficientcounterpart. Wethen estimate a simple term structure model under the assumption that the investors’ risk attitude is driven by near-term expectations of the three state variables. When we allow for evolving beliefs about the macroeconomy, the resulting term structure model provides a better fit to the cross sectionofyieldsthanthebenchmarkmodel,especiallyatlongermaturities,andexhibitsbetter performanceinout-of-samplepredictionsoffutureyieldmovements.

1 Introduction Economictheorysuggeststhatthetermstructureofinterestratesatanymomentoughttoreflect agent’s perceptions regarding the current state of the macroeconomy as well as its dynamic structure. The endogenous response of monetary policy to inflation and economic conditions providesastronglinkbetweenthesefactorsandcurrentandexpectedfutureshort-terminterest rates. And to the extent investor appetite for risk varies with business conditions, premia on long-termyieldswouldalsoreflectcurrentandexpectedbusinesscycledevelopments. In this light, the recent emergence of no-arbitrage term structure models with macroeconomicfactorsinfittingjointlythetermstructureofinterestratesandmacroeconomicdynamics oftheU.S.economy,hasbeenawelcomedevelopmentinmacroeconomicsandfinance. These models typically posit that the macroeconomy is governed by a simple fixed-coefficient dynamic structure and that agents know this structure and form expectations consistent with the model. Whilesuchsimplefixed-coefficientsdynamicmodelshaveprovenuseful,manyresearchers also find that these models must be supplemented with additional latent factors and unobservable shocks to provide a satisfactory fit of yields across the spectrum of maturities. The key difficultyseemstobethatsuchafixed-coefficientmodelimpliestootightalinkbetweenmacro variablesandbondyieldsbyassumingthatspanthesameinformationsetandarelinkedtoeach otherviaatime-invariantfunctionalform,animplicationthathaslimitedempiricalsupport. In this paper, we relax the restriction of a time-invariant relationship between macro variablesandbondyieldsbyallowingevolvingperceptionsregardingthedynamicstructureofthe economy. In particular, we posit that agents engage in real-time re-estimation and updating of a vector autoregression (VAR) model assumed to govern the dynamics of the macroeconomy and, in each period, form expectations based on the estimation results with data available duringthatperiod. Inthismanner,weobtainananticipated-utilityversionofano-arbitragemodel of the term structure. We show that such a model generates forecasts about future path of the economy that are more consistent with the survey evidence and explore its role in improving theempiricalperformanceofthemacrofinancemodels. Weestimatethemodelusingreal-timevintagesofquarterlydataandcorrespondingsurvey forecasts for inflation, output growth and the short-term interest rate from the Federal Reserve Bank of Philadelphia’s survey of professional forecasters. To recover the evolution of percep- 1

tions about macroeconomic dynamics, in each quarter we estimate the VAR parameters that fit the historical data as well as the panel of survey forecasts in that quarter. We then use the recursiveVARestimatestofitourdynamictermstructuremodel. The main findings from this exercise can be summarized as follows. First, our results suggest significant deviations from the fixed-coefficient model-consistent benchmark model of expectations. Allowing for evolving perceptions regarding economic dynamics results in a significantly improved understanding of the evolution of expectations over time. Second, allowing for evolving macroeconomic expectations leads to large and economically significant improvementinthefitofthetermstructure,especiallyasthematuritylengthens. Contribution from an additional latent factor, albeit still large, become less important. Finally, survey forecasts provide useful information regarding the perceived future path of the economy and help improveboththein-samplefitandtheout-of-sampleforecastsofyieldsattheshorterend. Ourpaperisrelatedtothelargeliteratureonlearning. Comparedtomodelsimposingrationalexpectationsandafixedknownrulegoverninghowtheeconomyevolvesovertime,models inwhichagentshavetoinferinrealtimethestructureoftheeconomyappeartoprovideabetter description of the inflation dynamics 1 and the monetary policydecision making process2, and generate forecasts about the future path of the economy that are more consistent with the survey evidence3. Term structure implications of learning have been examined by Cogley (2005) based on a two-yield-factor model and Piazzesi and Schneider (2006) in a consumption-based asset pricing framework. However, using yield curve factors in the former study prevents an examination of the economic driving forces behind interest rate variations; the relative few numberofthefactorsinbothstudiesalsoleadstoalessthansatisfactoryfitofthecrosssection ofyields. Ourpaperbuildsontherapidlyexpandingmacrofinanceliteraturethatexaminesbondpricing implications of New Keyesian models by superimposing either an exogenous specified or anendogenouslyderivedpricingkernel.4 Morerecently,learningisincorporatedintothistype ofmodels,whereagentscontinuouslyupdatetheirbeliefsregardingthecentralbank’sinflation target(KozickiandTinsley(2001a,b),DewachterandLyrio(2006))orthedegreeofmonetary 1See,forexample,CogleyandSargent(2002). 2Eg. OrphanidesandWilliams(2005a,b). 3BranchandEvans(2006) 4Among others, see Ang and Piazzesi (2003), Rudebusch and Wu (2004), and Ho¨rdahl et al. (2006) for the former,andBekaertetal.(2004)andDewachterandLyrio(2006)forthelatter. 2

policy activism in general (Ang et al. (2007)). In comparison, the current paper makes no a prioriassumptionsaboutthepotentialsourceofstructuralinstabilities,butallowstheagentsto learnaboutallaspects—thedrift,theslopecoefficientsandtheconditionalvolatilities—ofthe economy. One paper that is most closely related to ours is Laubach et al. (2007), who approximate agents’ changing expectations about the economy using a constant-gain VAR similar to ours and examine the term structure implications. However, they do not employ real-time informationfromsurveydatatoestimateorevaluatethemodelaswedohere. Finally, a number of papers use survey information in term structure estimation. Kim and Orphanides (2006) show that incorporating additional information from survey forecasts of short-term interest rates help alleviate the small-sample problem when estimating a latentfactor term structure model. Pennacchi (1991) and D’Amico et al. (2007) use survey forecasts ofinflationtoidentifyexpectedinflationinarealtermstructuremodel,wheremostoftherisk factors remain unobserved. In contrast, Chun (2007) directly employs the one-period ahead survey forecasts of the nominal short rate, real GDP growth and inflation as state variables, andassumesthatinvestorexpectationsdependsolelyontheirownlagswithnofeedbackfrom subsequentrealizationsofthemacrovariables. Hisanalysisalsoignoresinformationcontained intheentiretermstructureofforecasts. The rest of the paper is structured as follows. Section 2 motivates the paper and describes the data used in this study. We summarize the various models in Section 3 and review the estimation methodology in Section 4. The main empirical results are presented in Section 5 whileSection6containssomefurtherdiscussions. Finally,Section7concludes. 2 Data and Motivation Figure1plotsthe3-monthnominalshortrate,thefinal-vintagedataonannualrealGDPgrowth andannualGDPdeflatorinflation,togetherwiththecorrespondingSPFforecastsfourquarters ago,forthefullsampleof1965Q4to2006Q2. AppendixAprovidesdetailsonthedata. The macro term structure literature typically motivates the link between the term structure and the real economy by referring to a forward-looking monetary policy rule, in which the central bank judiciously selects an optimal short-term policy rate based on the predicted path of future economy. An important question is whether the presumed law of motion used in the 3

empirical testing of the policy rule indeed generates forecasts consistent with investor expectations observed at the time of monetary policy decision. More specifically, under the further assumption that the economy evolves over time according to a fixed-coefficient VAR, these models have the future implication that the yield curve contain as much information about futuremacrovariablesasdocurrentmacrovariables,sinceyieldsandtheunderlyingmacrostate variablesareflipsidesofthesamecoininsuchaneconomy. Such a prediction can be easily tested. We estimate two predictive regressions for real GDPgrowthandinflation,wheretheexplanatoryvariablesareeithertheshortrate,laggedreal GDP growth and lagged inflation, or the 3-month, 1-year and 10-year nominal yields. When predictingrealGDPgrowth,thetworegressionsbecome g = α+β y1 +β g +β π , (1) t 1 t−1 2 t−1 3 t−1 and g = α+β y1 +β y4 +β y40 , (2) t 1 t−1 2 t−1 3 t−1 whereg istherealGDPgrowth,π istheGDPdeflatorinflation,andyn isthen-quarteryield. t t t The results from these regressions based on the final-vintage data are reported in the first two panels of Table 1. As can be seen from the first panel, when predicting next-quarter real GDP growth, the R2 goes down from 12% using lagged macro variables to 7% using yield curve variables alone. A more dramatic reduction in explanatory power is observed for quarterly inflation, as shown in the second panel, where the R2 falls from 73% using macro regressorsto 39%usingyieldcurveregressors. Atfirstsight,thisseemstosuggestthatmuchoftheyieldcurvevariationsarenotrelatedto macrovariables,bodingillforanyattempttoextractinformationaboutfutureeconomiccondition from the current term structure or to explain yield curve variations using macro variables. A different interpretation, however, is that the assumption of a known fixed-coefficient data generating process is the reason for the disparate results. Indeed, ample empirical evidence that monetary policy practice and the structure of the macro economy may have shifted over time would suggest that such an assumption is unlikely to hold. Even if the true underlying structure of the economy were fixed over time, but economic agents had to estimate and discover this structure, beliefs would evolve over time as additional data were incorporated into the agents’s discovery process. Depending on how real-time perceptions about the structure 4

of the economy evolved over time, real-time expectations, and the term-structure of interest rates mirroring these expectations, would correspondingly differ. For instance, for any given history of economic growth rates and inflation, differences in the perceived persistence or the perceived long-term asymptotes of these variables could have vastly different implications for longer-term interest rates. Obviously, in such circumstances, yields continue to be driven by expectations about future macro variables, which are in turn linked to current macro variables inatime-varyingfashion,suggestingthatfixed-coefficientregressionslike(1)and(2)aremisspecified. To examine the empirical validity of this conjecture, we re-run regressions (1) and (2) but replace the realized GDP growth and inflation by their SPF forecasts on the left hand side. As can be seen in Table 1, a different result emerges from this exercise. For example, the third panel of the table shows that using yield curve regressors alone, we can explain 32% of the variations in the median SPF forecasts of next-period real GDP growth, much higher than the proportion explained when predicting realized real GDP growth. This is not surprising given thatexpectationscontainlessnoises. Moreencouragingly,however,thisnumberisalsohigher than the proportion explained in a regression of future SPF forecasts of real GDP growth on lagged macro variables, which account for a slightly smaller 30% of the observed variations in median SPF forecasts. Similarly, the last panel shows that the predictable proportion of the movement in SPF inflation forecasts goes down from around 85% using macro variables to a loweryetstillrespectable 52%usingyieldcurvevariablesalone. Overall, Table 1 seems to suggest that yield curve contains important information about investorexpectationsaboutfuturemacrovariables;however,suchalinkisshroudedbyatimevarying relationship between realized macro variables and their expected future values. In the rest of this paper, we introduce time-varying coefficients to the perceived law of motion of the economy, taking seriously the restriction that any forecast generated by the model should be a reasonably good approximation to the true investor expectations, as measured by survey forecasts,andexaminethepricingimplicationsforlonger-termfixed-incomeassets. 5

3 Models 3.1 Time-Varying VAR At each quarter t, investors observe last quarter’s real GDP growth rate, g , last quarter’s t−1|t inflation,π ,thecurrent3-monthnominalshortrate,r ,wherethesubscriptt−1|tdenotes t−1|t t quarter-(t−1)valuesobservedatquartertandreflectsthetimelaginmacrodatareleases. At (cid:169) (cid:170) (cid:48) time t, investors fit a VAR(2) to the vector of macro variables, Z = g ,π ,r , based t t−1|t t−t|t t onarollingsampleof40quarters: Z = µ +Φ Z +Φ Z +Σ (cid:178) , s = t−39,...,t (3) s z,t 1,t s−1 2,t s−2 z,t s Investors update their VAR estimates in two steps. In the first step, they estimate the longrunmeanofeachvariableasadiscountedweightedaveragebasedontherollingsample (cid:195) (cid:33) (cid:195) (cid:33) −1 (cid:88)39 (cid:88)39 µ = vi viZ , z,t t−i i=0 i=0 where the gain parameter, v, controls how aggressively the past history is discounted when forming the mean estimates. In the rest of the analysis we fix the gain parameter at v = 0.98.5 In the second step, they estimate the rest of the VAR parameters by standard OLS without a drifttermusingdemeaneddatafromthesamerollingsample. Thistwo-stepschemeallowsthe investors to pick up the low-frequency variations in the trend components and helps avoid the problemthatnear-unit-rootslopeestimatesoccasionallyariseandleadtoimplausiblebehavior for long-horizon forecasts if all parameters are estimated in one step. Alternatively, we could usehistoricallong-termforecastsfrompublicorprivatesources,suchasthosepublishedbythe Congressional Budge Office or from Blue Chip Blue Economic Indicators, to emulate agent’s real-timedemeaningofthedata. Let X = {Z ,Z } be the extended state space and rewrite the VAR in the companion t t t−1 formintheusualwayas X = µ +Φ X +Σ (cid:178) , s = t−39,...,t (4) s t t s−1 t s where (cid:34) (cid:35) (cid:34) (cid:35) (cid:34) (cid:35) µ Φ Φ Σ z,t 1,t 2,t z,t µ = , Φ = , Σ = t t t 0 I 0 0 5Wealsoexperimentedwithothervaluesforthegainparameterandobtainsimilarresults. 6

We assume that investors form expectations of future realizations of the macro variables basedoncurrentparameterestimates: (cid:163) (cid:161) (cid:162) (cid:164) E∗[Z ] = FE∗[X ] = F (I −Φ )−1 I −Φk µ +ΦκX (5) t t+k t t+k t t t t t where F = [I , 0 ] selects current macro variables and E∗ is an expectation operator based 3 3×3 t on the assumption that current parameter estimates are the true parameters. In other words, although investors are fully aware that their parameter estimates might change when new data arrives in the future, at each point in time they act as if current estimates will persist in the future, ignoring parameter uncertainties. In agreement with Sargent (1999) and others, we viewthisassumptionof“anticipatedutility”asagoodapproximationtohowinvestorsactually behaveintherealworld. 3.2 Term Structure The nominal short rate is given by r = e(cid:48)X , where e is a selecting vector. Agents observe t 3 t 3 current level of the short rate; their expectations about future short rates, on the other hand, dependonthecurrentstatevariablesaswellascurrentparameterestimates. Thelognominalpricingkernelisspecifiedintheusualfashionas 1 m = −r − λ(cid:48)λ −λ(cid:48)(cid:178) . (6) t+1 t 2 t t t t+1 whereλ isthenominalpriceofriskandisafunctionofexpectednext-periodmacrovariables, t E∗[Z ]. Morespecifically,thenominalpriceofriskisgivenby t t+1 λ = λ +λ E∗[Z ] = λ +λ F (µ +Φ X ). (7) t 0 1 t t+1 0 1 t t t Two things are worth noting here. First, our price of risk loads on both lags of the macro variables albeit in a restricted fashion. This contrasts with the usual practice in the macro termstructureliteratureofrestrictingthepriceofrisktoloadoncurrent-periodvariablesonly.6 Second, we assume that term structure parameters, λ and λ , are fixed throughout the full 0 1 sample period, reflecting our prior that investor preferences are relatively more stable over timecomparedtothestructureoftheeconomy. 6Wedidestimateseparateversionsofthemodelsimposingtheusualrestrictionthatλ onlyloadsoncurrent t macrovariablesZ ,andfoundthatsuchmodelsalwaysgenerateaworsefitwithobservedyieldsatallmaturities. t 7

Given time-t VAR estimates, it is straightforward to show that the price of an n-period nominal bond is an exponential affine function of the state variables with time-varying coefficients: Pn = exp(A +B X ), (8) t n,t n,t t whereA andB followrecursiveequations n,t n,t 1 A = A +B [µ −Σ (λ +λ ϕ )]+ B Σ Σ(cid:48)B(cid:48) n,t n−1,t n−1,t t t 0 1 0,t 2 n−1,t t t n−1,t B = −e(cid:48) +B (Φ −Σ λ ϕ ) n,t 3 n−1,t t t 1 1,t withinitialconditionsA = 0andB = −e(cid:48). Bondyieldsarethereforealsoaffinefunctions 1,t 1,t 3 ofthestatevariables 1 yn (cid:44) − logPn = a +b(cid:48) X , (9) t n t n,t n,t t withcoefficientsa = −A /nandb = −B /n. n,t n,t n,t n,t We also estimate some alternative models where the term structure is driven by one additionallatentfactor,l ,whichisassumedtobeconditionallyuncorrelatedwiththemacrofactors t andfollowstheprocess l = ρ X +ρ l +(cid:178)l, t+1 m t l t t where (cid:178)l is distributed standard normal and uncorrelated with (cid:178) at all lags. In this case the t t pricesofriskareparameterizedas (cid:34) (cid:35) (cid:34) (cid:35) λm λm λ = 0 = 0 , 0 λl 0 0 (cid:34) (cid:35) (cid:34) (cid:35) λmm λml λmm λml λ = 1 1 = 1 1 . 1 λlm λll 0 0 1 1 The price of risk parameters associated with shocks to the latent factor, λl, λlm and λll, are 0 1 1 unidentified using nominal bond yields only, as the nominal short rate and hence the entire nominal yield curve are not exposed to risks associated with (cid:178)l. We therefore set those paramt eterstozeros. Incontrast,λml controlshowthelatentfactorl affectsnominalbondpricingby 1 t influencing risk loadings on macro factors and can be identified from excess bond returns. In otherwords,thelatentfactorinthiseconomyispurelyapriceofriskfactorandonlyinfluence bond yields through the term premium component. Bond pricing in this framework is similar tothatinthestandardmodelandisoutlinedintheappendix. 8

3.3 Summary of Models Our main analysis will focus on three models. We start from a benchmark model (Model FC) where all VAR parameters are assumed to be time-invariant and are estimated once over the fullsampleusingthefinal-vintagedata,ascommonlyseenintheliterature. Thesecondmodel is our preferred model as specified above, which we hereafter refer to as Model TVC. Finally, we re-estimate Model TVC using SPF forecasts of macro variables as additional data inputs, which we will call Model TVC-S. Neither SPF forecasts of yields nor Blue Chip forecasts are usedintheestimation. For illustration purposes, we also estimate three alternative models. The first alternative model is motivated by Kozicki and Tinsley (2001b), who show that allowing time variations in the perceived inflation target is crucial for explaining the movement in the long end of the yield curve. To evaluate the relative contribution of time variations in the drifts versus in the rest of the parameters, we estimate an alternative model (Model PFC) where the drift terms, but not the slope or the volatility coefficients, vary over time. More specifically, we first estimate the time-varying drifts as the discounted weighted averages based on the last ten years of data, the same way as in Model TVC, and then estimate the slope and volatility coefficients using the demean data over the full sample. The second alternative model (Model TVC-L)isavariantofModelTVC,whereweallowanadditionallatentfactortodrivetheterm structure, as outlined in Section 3.2. To compare the role of such an additional latent factor when perceptions about macroeconomic dynamics are allowed to evolve over time versus the alternativefixed-coefficientassumption,wecomparethismodelwithModelFC-L,whichadds alatentfactortoModelFC. Table2summarizesallmodelsestimatedinthispaper. 4 Estimation We use a two-step maximum likelihood procedure to estimate the model. The parameters Θ can be partitioned into the parameters µ, Φ and Σ that govern the VAR dynamics (3), and the parametersλ ,λ ,andρ = [ρ ,ρ ]inModelsFC-LandTVC-L,thatgovernthetermstructure 0 1 m l dynamics. In the first step, we estimate the VAR parameters µ, Φ and Σ based on either the final vintage (Model FC) or the current vintage (Model TVC) of data by standard OLS if no 9

SPF forecasts are used in the estimation. If SPF forecasts are used, we estimate the model by maximizingthefollowingloglikelihoodfunction (cid:88)T (cid:88)5 (cid:179) (cid:180) max logf (Z |Z ,...Z )+ logf Zspf |Z ,...Z , (10) t t−1 t−p T+j|T T T−p+1 {µt,Φt,Σt,σ t spf} t=p+1 j=1 where σspf represents the standard deviation of measurement errors on SPF forecasts. We t fix σspf at an admittedly arbitrary number of 75 basis points annual rate, similar to Kim and t Orphanides(2006). Inthesecondstep,weestimatethepriceofriskparameters, λ andλ ,andρifapplicable, 0 1 by maximizing the likelihood for observed yields, holding the history of VAR parameter estimates,µ ,Φ andΣ ,fixedfromthefirststep. Moreprecisely,ateachtimepointt,weobserve t t t yields on N zero coupon nominal bonds, Y = {yn}N . we compute model-implied yields t t n=1 yˆ(n) = a +b(cid:48) X based on the observed state variable and current parameter estimates, and t n,t n,t t findvaluesoftheparametersthatsolvetheproblem (cid:88)T max logf (Y |µ ,Φ ,Σ ) (11) t t t t {λ0,λ1,ρ} t=1 One of the observed factors, the short rate, makes direct use of the yields y(1), and is t considered to be measured without any observation error. When estimating models with an extra latent factor, we make the additional assumption that the 7-year yield is also observed without error. Other yields are functions of the state variables, X , according to the model t pricingequation(9),andaretreatedasbeingmeasuredwithsmallsamplingerrors. Weassume that the sampling errors have mean zero and estimate their standard deviation Ω in the second stage. 5 Empirical results 5.1 Parameter Estimates Table 3 reports the parameter estimates for Model FC, where all coefficients are held fixed throughout the sample. The unconditional moments of the VAR variables implied by this model are constant over time and are plotted as the blue line in Figure 2. The short rate is the 10

mostpersistentfactoramongthethreewithanautocorrelationcoefficientof0.93,whilethereal GDP growth is the least persistent with an autocorrelation coefficient of 0.3. Unconditionally, thepriceofriskisnegative(positive)forrealGDPgrowth(inflation)shocks,withthepuzzling implication that an asset with returns positively correlated with real GDP growth (inflation) shocks receives a negative (positive) risk premium on average. The price of real GDP growth (inflation) risks loads negatively (positively) on both the expected real GDP growth and the expected short rate, suggesting that investors are more sensitive to both types of shocks when the expected economic growth is relatively strong or when the expected short rate is high. In contrast,whentheexpectedinflationisrunningaboveaverage,investorsbecomemoresensitive to real GDP growth shocks but less sensitive to inflation risks. The fitting errors in yields are muchlargerthanintypicallatent-factormodels,especiallyatlongerhorizons. Table 4 reports the parameter estimates for Model TVC, where VAR parameters are estimated on a rolling sample of 40 quarters and without using additional information from SPF forecasts. The time-series average of the VAR parameter estimates reported here are quite similar to what we see in Model FC. The price of risk parameters, however, are quite different. Unconditionally, the price of risks is positive (negative) for real GDP growth (inflation) shocks, which seems more plausible given that an average investors would prefer to hold an asset that has a higher return when the economy is weaker or when inflation is running high. The first two diagonal terms in the λ matrix is positive and negative, respectively, suggesting 1 that investors become more sensitive to real GDP growth risks when the economy is expected toslowdownandmoresensitivetoinflationriskswheninflationisexpectedtopickup. Fitting errors in yields of maturities of longer than two years are uniformly smaller than what we see in 3. The biggest improvement is seen at the ten-year maturity with its fitting errors shrunk by morethan40%. Finally, we introduce additional information from SPF forecasts and report the parameter estimates for Model TVC-S in Table 5. The fitting errors are larger than in Model TVC (Table 4)beyondtheone-yearmaturitybutstillsmallerthanthosein ModelFC (Table3). Figure2plotstheunconditionalmean,shockvolatilityandpersistenceoftheVARvariables as implied by the three models. Results based on Model FC are plotted in the blue lines and areconstantovertime. ResultsbasedonModelsTVCandTVC-S(theredandgreenlines)are identicalpriorto1986andclosetoeachotherthereafter,andcanexhibitsizabletimevariations. The top panels plots the unconditional means of the macro variables. As can be seen from the 11

redlines,variationsintheunconditionalmeanaremorenotableforinflationandtheshortrate, whoseimpliedmeansrosefromaround4%inearly1970’stoabout7%and10%,respectively, in 1983, before declining to their current respective levels of about 1.5% and 4%. The middle panels plot the volatilities of shocks to these variables. The red and the green lines show that real GDP growth and short rate shocks exhibit more variations in their conditional volatilities, whichhavebeenonadownwardpathsincemid1980’s,roughlycoincidingwithwhatisusually referred as the “Great Moderation.” In comparison, inflation shock volatilities fluctuate within a relatively narrow band during the entire sample period. The bottom panels plot the firstorder autocorrelation coefficient of each variable.7 Generally speaking, real GDP growth and inflation are less persistent today than in the 70’s and 80’s, while the persistence of the short rateisrelativelyunchanged. 5.2 Expectations of Future VAR Variables Table 6 summarizes how model forecasts of future VAR variables differ from their survey counterparts. Forallthreemodels,wecomputeimpliedforecastsofrealGDPgrowth,inflation and the 3-month short rate 1-, 2- and 4-quarters from now, and report the root mean squared differences between those forecasts and the corresponding survey counterparts, relative to a random walk benchmark. We also look at two measures of long-term expectations, including the expected 1-quarter variables 40 quarter hence and the expected average values five to ten years ahead. Panel A of the table looks at the entire sample period of 1965Q4 to 2006Q2. Introducingtime-varyingVARcoefficientinModelTVCresultsinlargerdiscrepanciesbetween model forecasts and survey forecasts at shorter horizons, but seems to approximate survey forecastsmuchbetteratforecastinghorizonsbeyondoneyear. Notsurprisingly,directlyusing informationfromsurveyforecastsinModelTVS-Sfurtheralignthemodel-impliedandsurvey forecastsatallhorizonsandallsampleperiods. Thesamepatterncanbeseenfromthedifferent sub-samples,showninPanelsBtoD. 7Noteherewedonottakeintoaccounttheadditionaldegreeofpersistenceduetotheslow-varyingtrendcomponent. WhatisreportedherecanbethoughtofasthepersistenceofthetransitorycomponentsinapermanenttransitorycomponentdecompositionasininStockandWatson(2007). Undertheirspecification,thepersistence ofinflationcomesentirelyfromtherandom-walktrendcomponent,whereasthetransitorycomponentisawhite noise process. Similarly, we see here after purging the influence of the persistent trend component, inflation revertstoits(time-varying)long-termmeanmuchfasterthanwhatisimpliedbyModelFCwithafixedmean. 12

The first three panels in Figure 3 provide a visual comparison of the long-horizon inflation forecasts based on these models against the future realized value. A fixed-coefficient model like Model FC implies that state variables reverts to their time-invariant unconditional means fairly quickly and hence has trouble generating 10-year inflation expectations as variable as what we see from survey forecasts. In particular, the 10-year inflation forecasts during the early 1980s generated by this model only edged slightly higher and quickly came down to its average level, while survey forecasts from that period shot up and stay well above realized inflation for quite some time even as inflation moderated. Model TVC and TVC-S, neither of which uses survey information during this period, are able to match the substantial increase andthesubsequentgradualdeclineoflong-terminflationforecastsrelativelywell. 5.3 Expectations of Future Yields Aremainingquestioniswhetheramodelthatbetterdescribesagents’expectationsaboutfuture macro economy also generates forecasts of future yields that are more consistent with the survey evidence. To answer this question, Table 7 compares model-implied and survey forecasts of 2- and 10-year yields and reports the root mean squared differences relative to a random walk model. Survey forecasts of longer-term yields are available only recently. In particular, forecasts of average 2- and 10-year yields during the next five to ten years are from the Blue Chip survey and are available since 1986Q1, while the SPF forecasts of 10-year yield is available since 1992Q1. Note that these forecasts are not used in estimating any of the models. Evidence based on this short sample period seems to suggest that allowing time variations in the VAR estimates in Model TVC generates forecasts of future yields that are closer to survey forecasts at the 10-year maturity but not at the shorter 2-year maturity. Consistent with Table 6whichsuggeststhatsurveyinformationbringsthebiggestimproveattheshorterendofVAR dynamics, the forecasts based on Model TVC-S also shows a smaller departure from survey forecastsattheshortertwo-yearmaturity. Figure 4 looks at model predictions for long-term interest rates in more details. The top left panel shows that Model FC consistently under-predicts the 10-year yield one year hence for much of the 1980’s and almost completely misses the second spike in long yields around 1984. More recently, the model generates forecasts that lies consistently above future realized value throughout the late 1990’s and predicts that the 10-year yield will rise quickly above 13

7% since the last monetary tightening cycle started in 2004, a trend that is absent both in the realized data and in SPF forecasts. In contrast, the top right and the bottom left panels show that Models TVC and TVC-S generate forecasts that correspond generally better with future realizedvaluesinallthreecasesandalsowithSPFforecastsinthelastepisode. Movingtowardsevenlongerforecastinghorizons,Figure5showsthatModelFCgenerates almostnovariationsin5-to10-yearahead,10-yearyieldexpectations,whereastheBlueChip survey forecasts declined from around 9% in mid 1980’s to about 5.5% by mid 1990’s. Both ModelsTVCandTVC-Sareabletocapturethisdecline;ModelTVC-Salsogeneratesalonghorizon forecast of the 10-year yield that fluctuates about 5.5% since mid 1990’s, consistent withthesurveyevidence. 5.4 Out-of-Sample Forecasting Another way to test the model is too examine how well it performs in out-of-sample forecasting. It’s conceivable that a model with more free parameters, such as the type of models with time-varying coefficients estimated in this paper, could fare better in sample but less well out of sample. To see whether this is the case, Table 8 reports the RMSEs in out-of-sample forecasting of VAR variables and long-term yields based on all three models, where both the VAR coefficients and the term structure parameter estimates are updated recursively based on the current sample, together with the corresponding SPF forecasts. Panel A of Table 8 shows that thetwotime-varyingcoefficientmodels(TVCandTVC-S)indeedperformslightlyworsethan the fixed-coefficient model (FC) in forecasting VAR variables out of sample, most notably for forecastinginflation. However,theyarestillcomparabletotheSPFforecasts.8 Turning to forecasting longer-term yields out of sample, Panel B shows that Model TVC outperformsmodelFCformaturitiesoffiveyearsandbeyondandforhorizonsaboveoneyear, withtheRMSE65%lowerwhenpredicting10-yearyieldtwoyearshence. Introducingsurvey information on macro variables post 1986 in Model TVC-S mostly improves on the model’s ability to predict the shorter end of the yield curve at the expense of a slightly worse perfor- 8Notethatinthisexercise,theVARcoefficientsinModelFCarenolongerfixedovertimebutarerecursively estimated using all data up to the forecasting period. Therefore, the difference between VAR variable forecasts based on Models FC and TVC is essentially the difference between a VAR estimated using a recursive sample versusthatestimatedusingarollingsample. 14

mance when forecasting the long end of the yield curve, although it continues to outperform ModelFCforthe10-yearbondmaturityandatthetwo-yearforecastinghorizon. 5.5 Term Structure Implications AllowingtimevariationsinVARparametersmightleadtodifferenttermstructureimplications of the model, to which we now turn. Figures 6 and 7 plot the realized and model-implied 2and 10-year nominal yields, together with the model-implied hypothetical yields when the ExpectationsHypothesis(EH)holds. ThecorrespondingtermpremiumsaregraphedinFigure 8. Due to the persistent nature of interest rates, a stationary term structure model most likely willgeneratealong-terminterestrateforecastinthenearfuturethatisclosetoitscurrentvalue. Therefore,thetopleftpanelofFigure7exhibitsroughlythesamepatternasseeninthetopleft panel of Figure 4: the ten-year yield as implied by Model FC lies below (above) its realized level in late 1980’s (1990’s) and is predicted to rise quickly since 2004 rather than fluctuating around the same level as seen in the data. The model-implied 10-year yield also bears too much similarity to the short rate. Comparing the red solid line and the red dashed line shows that the lower level of model-implied yields in the late 1980’s largely reflects the expectation that the short rate will trend down and revert back to its lower long-term mean during the next ten years. On the other hand, Models TVC (top right panel) and TVC-S (bottom left panel) imply that the long-term mean of the short rate has shifted higher during this period, which pushesuptheEHcomponentandthetotallevelofthelong-termyield. In comparison, the high level of model-implied 10-year yields in the 1990’s as implied by ModelFCisprimarilyduetoanincreaseinthetermpremiumratherthanintheEHcomponent, whichinturnresultsfromapositivecorrelationbetweentheleveloftheshortrateandtheterm premium (see Figure 8), as this model predicts that investors become more sensitive to both realGDPgrowthandinflationrisksanddemandahighertermpremiumastheshortraterises. Incontrast,ModelsTVCandTVC-Simplythatahighershortrateprimarilyactstoreducerisk premiums associated with all three shocks, as can be seen from the signs, leading to a lower term premium estimate in the 1990’s, as shown in Figure 8, and a better fit with the realized data,asshowninFigure7. Figure9plotstheimpulseresponsesof1-quarter,1-yearand10-yearyieldstoonestandard 15

deviation shocks to real GDP growth, inflation and the nominal short rate on three dates— 1978Q1 and 1983Q1, two dates representing the periods immediately before and after the Volckerdisinflation,and2006Q2,thelastdatapointinoursample—allbasedonModelTVC- S.9 Except for the negative yet imprecisely estimated contemporaneous response of the short rate,yieldsofallmaturitiesrespondtoinflationshocksmorestronglyattheendof1983thanin early1978orearly2006,consistentwiththeempiricalevidencethattheFedcombatsinflation more vigorously post 1982.10 Shocks to the short rate also have the biggest effect on yields in late 1983, mainly reflecting a larger volatility of short rate shocks and a resulting larger term premiumassociatedwithshortrateshocksduringthatperiod. Incomparison,monetarypolicy during the most recent period is characterized by a response to real GDP growth shocks more aggressivethaninpreviousperiods. Table 9 reports the results from an in-sample variance decomposition of yields of various maturities into components due to time-varying parameters, each state factor, and a remainder term.11 ModelFCprecludesvariationsintheparametersandattributesnearlyallthevariations in yields to movement in the short rate. In contrast, Models TVC and TVC-S attribute a considerableproportionofvariationsinyieldstotimevariationsinparameterestimates,especially atlongermaturities. Shortratevariationscontinuetoexplainmostoftheremainingmovement, but changes in inflation now plays a slightly more important role in driving shorter-maturity yields. 6 Discussions 6.1 Shifts in Mean versus Shifts in All Parameters Kozicki and Tinsley (2001b) show that allowing time-varying endpoints is important for explainingthevariationsinlong-terminterestrates. Toaddressthequestionwhethertheimprovement in performance of Models TVC and TVC-S comes mainly from allowing a time-varying mean, we estimate an alternative model, Model PVC, where we allow time variations in the unconditional mean of the state variables but not in their persistence or volatilities. In particu- 9ResultsbasedonModelTVCarenearlyidentical. 10SeeClaridaetal.(2000),forexample. 11SeeAppendixDfordetailsofthisdecomposition. 16

lar,wemodeltheshiftingendpointsusingadiscountedweightedaveragewitharollingsample of 40 quarters and a quarterly gain of 98%, the same way as in Models TVC and TVC-S, but estimate the remaining VAR parameters once using the de-meaned final-vintage data over the full sample. The parameter estimates for this model are reported in Table 10, with the implied unconditional moments of VAR variables plotted as the black dashed lines in Figure 2. The unconditionalmeansareclosetothoseimpliedbyModelsTVCandTVC-S,12 whiletheshock volatilitiesandthepersistenceofthevariablesareclosetothoseimpliedbyModelFC. The main implications of this model are shown in the bottom left panels of Figures 3 to 8. Here both inflation and the short rate slowly reverts to a time-varying mean that rises over time until around mid 1980s and then declines since then. As can be seen from Figures 3 and 4, the presence of these persistent yet time-varying asymptotes enables this model to capture most of the variations in long-horizon inflation expectations and long yield expectations since the corresponding SPF forecasts became available around 1980 and 1985, respectively. This model also fits two-year yield relatively well while attributing almost all the variations to the EH component, as can be seen from Figure 6. In addition, Figure 7 shows that it is able to capture the downward trend in the 10-year yields since early 1990s. Nevertheless, there are several episodes when this model provides a poor fit with the data. For example, it fails to match the magnitude of the two spikes in long yields during early and mid 1980s, as it by construction rules out the channel through which the rising volatilities of shocks to the real GDPgrowthandthenominalshortrateleadtheinvestorstodemandatermpremium. Similarly,inthelate1980’s,thismodelgeneratesimpliedyieldsthataretoohighcompared to the realized data, and overstates the subsequent decline in 10-year yield forecasts 5- to 10year ahead when compared to the Blue Chip survey forecasts (Figure 5). During this period, the3-monthshortrateisexpectedtoreverttoitsunconditionalmeanofabout8.5%fromalevel of around 5%, which pushes up the EH component of 10-year yield, while the model-implied term premium is largely unchanged around that time. In comparison, Model TVC-S implies thattheshortrateisexpectedtomean-revertataslowerpaceinearly1987thanintheprevious period, leading to a slightly lower EH component. More importantly, both Model TVC and Model TVC-S imply that volatilities of real GDP growth shocks are revised down during this period,leadingtoalargereductioninthemodel-impliedtermpremiuminlate1980’s. 12The small differences are due to the fact that this model is estimated using the final-vintage data, while ModelsTVCandTVC-Sareestimatedusingreal-timedata. 17

Finally, repeating the out-of-sample forecasting exercise in Section 5.4 based on the PVC model produces RMSEs for 10-year yields that are uniformly larger than the TVC models discussedabove. (Detailedresultsnotshownhereforbrevity). Theseresultssuggestthatallowingforvariationintheperceivedmeansofmacroeconomic variables is not sufficient to capture the role of evolving beliefs about the structure of the economyonthetermstructureofinterestrates. Rather,theevolvingbeliefsaboutthenatureof short-termmacroeconomicdynamics,asreflectedinslopeparameters,mustalsobeaccounted fortoimproveourunderstandingofthetermstructure. 6.2 Contribution of Additional Latent Factor So far we’ve shown that allowing time dependence in the perceived dynamics of underlying state variables helps improve the model’s fit with observed longer-term yields; nonetheless, the yield fitting errors are still large compared to those from a typical latent-factor term structure model. In this section we examine whether we can further improve on Model TVC by introducing an additional unobserved factor, as outlined in Section 3.2. Recall that under this specification, the latent factor has no effect on how the macro variables, including the short rate, are perceived to evolve over time, but can affect longer-term yields by influencing the termpremium. The parameter estimates of the resulting model, Model TVC-L, are reported in Table 11. Theyieldfittingerrorsaremuchsmallercomparedtothecorrespondingmodelwithoutalatent factor,ModelTVC,withtheRMSE40%loweratthe10-yearmaturity. Thisimprovementcan also be seen from the top two panels of Figure 10, which shows much smaller discrepancies between model-implied and realized yields at both the 2-year and the 10-year maturities. This better fit has to come through the term premium channel, as the perceived short rate process in this model is identical to that in Model TVC at each point in time. The bottom panel of Figure 10 plots the 2-year and 10-year term premiums. A comparison of these and the corresponding series implied by Model TVC, shown in the top right panel of Figure 8, shows that term premiums exhibit more high-frequency variations in this model, while their rise in the early 1980’s and the subsequent decline assume a smaller magnitude. The 10-year term premiumappearstobelowerafter1985,whenitfluctuatesaround50basispoints,thanbefore 1980, when it fluctuates around 150 basis points. The price of risk parameters, λ , loads 1 18

positively and significantly on the latent factor for inflation and nominal short rate shocks, as shown in Table 11, implying that a more positive latent factor leads to a more negative price on inflation and nominal short rate risks and reduce the term premium. On the other hand, the loading of the price of real GDP growth risks on the latent factor is not significantly different fromzero. Comparing Panels E and B in Table 9 shows that the latent factor absorbs some of the variations previously attributed to time-varying coefficients, especially at longer maturities, and explains about one quarter of the variations in the 10-year yield. In comparison, when we re-estimate the benchmark model, Model FC, with an additional latent factor, (the FC-L model)thelatentfactoraccountsforabout2/3ofthe10-yearyieldmovement. Attheshortend oftheyieldcurve,shortratevariationscontinuetoplayadominantrole,whilethelatentfactor explainsabout15%ofthe1-yearyieldmovement. These results suggest that although allowing for evolving beliefs regarding the dynamics ofthemacroeconomycannotfullyaccountfortheexplanatorypoweroflatentfactorsinfixedcoefficientmodels,itdoesgoalongwaytowardssuchanaccounting. 7 Conclusion In this paper we build a simple model that can accommodate the presence of evolving beliefs regardingmacroeconomicdynamics,andexaminetheirroleinexplainingthetermstructureof interest rates. In each period, agents re-estimate a VAR on real GDP growth, inflation and the nominal short-term interest rate, and use this recursively estimated VAR to form expectations. Usingquarterly-vintagesofreal-timedataandsurveyforecastsfortheUnitedStates,weshow that allowing for evolving macroeconomic perceptions in this manner generates predictions about the future path of the economy that are more consistent both with survey forecasts and with future realized values, relative to those from a benchmark model that imposes rational expectationsandafixed-coefficientVAR. Wethenexploretheroleofthetime-variationinbeliefsregardingthestructureoftheeconomy for understanding the term-structure of interest rates. To that end, we price zero-coupon bonds of different maturities under the assumption that the investors’ risk attitude is driven by expectations about the three macro variables in the following period. We find that when 19

we allow for evolving beliefs about the macroeconomy, the resulting term structure model provides a better fit to the cross section of yields than the benchmark model—especially at longermaturities—andexhibitsbetterperformanceinout-of-samplepredictionofyieldmovements. Supplementing the data with information from survey forecasts during the first-step VARestimationfurtherreducesthediscrepanciesbetweenmodel-impliedforecastsandsurvey expectationsnotonlyformacrovariablesbutalsoforbondyieldsatshortermaturities. These findings demonstrate the usefulness of imposing additional discipline on the estimation of term structure models using information from survey forecasts. Existing work in a latent-factor setting has shown that such information can materially improve estimation of the expectedfutureshortrateandtheexpectedexcessreturnsonlong-termbonds. Inamacroterm structure framework, it also helps to ensure that the underlying macroeconomic model correctlyapproximatestheevolvingnatureoftheprocessgoverningtheformationofexpectations about the outlook of the economy by bond market participants at the time when bond yields areobserved. Our main result is that allowing for time variation in the perceived mean, slope and conditional volatilities of macroeconomic variables can greatly facilitate our understanding of the linkages between the macroeconomy and the term structure. In addition, when we introduce an additional latent factor that is uncorrelated with the macro variables, we find that the latent factor accounts for a smaller portion of yield curve variations in our preferred time-varying modelthaninthebenchmarkfixed-coefficientmodel. Accounting for evolving macroeconomic perceptions, as reflected by parameter variations in the perceived dynamic process governing the economy, can help reconcile the seemingly conflictingevidencethatontheonehand,interestratesrespondstronglytonewsaboutthekey macroeconomic variables (as demonstrated by even studies), while on the other hand, yields appeartohavelowexplanatorypowerforsubsequentrealizationsofthemacrovariables. In summary, we conclude that accounting for evolving macroeconomic perceptions is a criticalsteptowardsa betterunderstandingofthe termstructureofinterest ratesinthe context ofmacro-financemodels. 20

Appendix A Data Real-timedataonseasonallyadjustedrealandnominalGDPisobtainedfromFederalReserveBankofPhiladelphia’swebsiteforthesampleperiodof1954Q1to2006Q2.13 WeconstructtheimpliedGDPdeflatorfromthese twoseriesandmeasureinflationasthelogarithmofquarterlychangesintheimpliedGDPdeflator. Median SPF forecasts of 3-month T-Bill rate, nominal and real GDP level, GDP deflator and 10-year T- Bond yields are also obtained from Federal Reserve Bank of Philadelphia. SPF forecasts are available starting from 1968Q4 for nominal GDP and GDP deflator, from 1981Q3 for real GDP and 3-month T-Bill rates, and from 1992Q1 for 10-year T-Bond yields. We fill in real GDP forecasts for the period of 1968Q4 to 1981Q2 usingforecastsofnominalGDPandGDPdeflator. Surveyparticipantsforecastthelevelofeachvariableforthe preceding,thecurrentandthenextfourquarters,whichallowustoconstructquarterlygrowthrateforecastsfor realGDPandGDPdeflatorforthecurrentandthefollowingfourquarters. Wealsousefive-toten-yearahead forecastsofrealGDPgrowth,GDPdeflatorinflation,andyieldsofmaturities3months,2and10yearsfromBlue ChipEconomicIndicators,availabletwiceayearinFebruaryandSeptemberbetween1986Q1and2006Q2.14 Nominal yields for the maturities of 3-month and 1 to 5 years from 1965Q4 to 2006Q2 are obtained from CRSP.Forlongermaturities,weuse7-and10-yearyieldsbasedonazerocouponnominalyieldcurvefittedat theFederalReserveBoardusingtheSvensson(1995)method,availablesince1961Q3forthe7-yearmaturityand since1971Q4forthe10-yearmaturity.15 Weselectyieldsattheendofthefirstmonthwithineachquartertobest approximatethereleasedatesofreal-timemacrodataaswellastheSPFandBlueChipforecasts. B Nominal Bond pricing B.1 Time-varying µ, Φ and Σ Assumingthatthepriceofann-periodbondattimetisanexponentialaffinefunctionofthestatevariables (cid:179) (cid:180) Pn =exp A +B X(cid:101) , t n,t n,t t 13The real GDP series measures real, fixed-weight GNP before 1992Q2, real, fixed-weight GDP between 1992Q2 and 1995Q4, and real, chain-weight GDP thereafter. The nominal GDP series measures nominal GNP priorto1992Q1andnominalGDPthereafter. 14ForecastedvariablesarerealGNPgrowthandGNPdeflatorinflationuptoSeptember1991, 3-monthyield throughout,and3-and30-yearyieldsuptoSeptember1987. 15SeeGurkaynaketal.(2006)fordetails. 21

wehave (cid:163) (cid:164) Pn =E exp(m )Pn−1 t t (cid:183) (cid:181) t+1 t+1 (cid:182)(cid:184) 1 =E exp −r − λ(cid:48)λ −λ(cid:48)(cid:178) +A +B X(cid:101) t t 2 t t t t+1 n−1,t n−1,t t+1 (cid:183) (cid:181) (cid:179) (cid:180)(cid:182)(cid:184) 1 =E exp −e(cid:48)X(cid:101) − λ(cid:48)λ −λ(cid:48)(cid:178) +A +B µ(cid:101) +Φ(cid:101) X(cid:101) +Σ(cid:101) (cid:178) t 1 t 2 t t t t+1 n−1,t n−1,t t t t t t (cid:183) (cid:179) (cid:180) (cid:184) 1 =exp −e(cid:48)X(cid:101) +A +B µ(cid:101) +Φ(cid:101) X(cid:101) + B Σ(cid:101) Σ(cid:101)(cid:48)B(cid:48) −B Σ(cid:101) λ 1 t n−1,t n−1,t t t t 2 n−1,t t t n−1,t n−1,t t t (cid:183) (cid:179) (cid:180) 1 =exp −e(cid:48)X(cid:101) +A +B µ(cid:101) +Φ(cid:101) X(cid:101) + B Σ(cid:101) Σ(cid:101)(cid:48)B(cid:48) 1 t n−1,t n−1,t t t t 2 n−1,t t t n−1,t (cid:179) (cid:180)(cid:105) −B Σ(cid:101) (λ +λ ϕ )+λ ϕ X(cid:101) . n−1,t 0 1 0,t 1 1,t t ThereforeA andB followrecursiveequations: n n (cid:104) (cid:105) 1 A =A +B µ(cid:101) −Σ(cid:101) (λ +λ ϕ ) + B Σ(cid:101) Σ(cid:101)(cid:48)B(cid:48) , n,t n−1,t n−1,t t t 0 1 0,t 2 n−1,t t t n−1,t (cid:179) (cid:180) B =−e(cid:48) +B Φ(cid:101) −Σ(cid:101) λ ϕ , n,t 1 n−1,t t t 1 1,t withinitialconditionsA =0andB =−e(cid:48). 1 1 1 B.2 Time-varying µ, Fixed Φ and Σ Assumingthatthepriceofann-periodbondattimetisanexponentialaffinefunctionofthestatevariables (cid:179) (cid:180) Pn =exp A +B X(cid:101) +C µ(cid:101) , t n n t n t wehave (cid:163) (cid:164) Pn =E exp(m )Pn−1 t t (cid:183) (cid:181) t+1 t+1 (cid:182)(cid:184) 1 =E exp −r − λ(cid:48)λ −λ(cid:48)(cid:178) +A +B X(cid:101) +C µ(cid:101) t t 2 t t t t+1 n−1 n−1 t+1 n−1 t (cid:183) (cid:181) (cid:179) (cid:180) (cid:182)(cid:184) 1 =E exp −e(cid:48)Y(cid:101) − λ(cid:48)λ −λ(cid:48)(cid:178) +A +B µ(cid:101) +Φ(cid:101)X(cid:101) +Σ(cid:101)(cid:178) +C µ(cid:101) t 1 t 2 t t t t+1 n−1 n−1 t t t n−1 t (cid:183) (cid:179) (cid:180) (cid:184) 1 =exp −e(cid:48)X(cid:101) +A +B µ(cid:101) +Φ(cid:101)X(cid:101) +C µ(cid:101) + B Σ(cid:101)Σ(cid:101)(cid:48)B(cid:48) −B Σ(cid:101)λ 1 t n−1 n−1 t t n−1 t 2 n−1 n−1 n−1 t (cid:183) (cid:179) (cid:180) 1 =exp −e(cid:48)X(cid:101) +A +B µ(cid:101) +Φ(cid:101)X(cid:101) +C µ(cid:101) + B Σ(cid:101)Σ(cid:101)(cid:48)B(cid:48) 1 t n−1 n−1 t t n−1 t 2 n−1 n−1 (cid:179) (cid:180)(cid:105) −B Σ(cid:101) (λ +λ ϕ )+λ ϕ X(cid:101) . n−1 0 1 0 1 1 t whereϕ =0 inthefirstcaseandϕ =Fµ(cid:101) inthesecondcase.ThereforeA ,B andC followrecursive 0,t 3×1 0,t t n n n equations: 1 A =A −B Σ(cid:101)λ + B Σ(cid:101)Σ(cid:101)(cid:48)B(cid:48) , n n−1 n−1 0 2 n−1 n−1 (cid:179) (cid:180) B =−e(cid:48) +B Φ(cid:101) −Σ(cid:101)λ ϕ , n 1 n−1 1 1 and C =C +B n n−1 n−1 inthefirstcaseand (cid:179) (cid:180) C =C +B I−Σ(cid:101)λ F n n−1 n−1 1 inthesecondcase,withinitialconditionsA =0,B =−e(cid:48),C =0. 1 1 1 1 22

C Real Bond Pricing mR =m +π t+1 t+1 t+1 1 m =−r − λ(cid:48)λ −λ(cid:48)ε(cid:48) t+1 t 2 t t t t+1 1 mR =−r +π − λ(cid:48)λ −λ(cid:48)ε(cid:48) t+1 t t+1 2 t t t t+1 1 =−(δ +δ(cid:48)X )+e(cid:48) (µ+ΦX +Σε )− λ(cid:48)λ −λ(cid:48)ε 0 1 t 2 t t+1 2 t t t t+1 (cid:181) (cid:182) 1 =− δ −e(cid:48)µ+ λ(cid:48)λ −(δ(cid:48) −e(cid:48)Φ)X +(e(cid:48)Σ−λ(cid:48))ε 0 2 2 t t 1 2 t 2 t t+1 (cid:161) (cid:162) rR =−logE exp mR t t t+1 (cid:181) (cid:182) 1 1 = δ −e(cid:48)µ+ λ(cid:48)λ +(δ(cid:48) −e(cid:48)Φ)X − (e(cid:48)Σ−λ(cid:48))(e(cid:48)Σ−λ(cid:48)) (cid:48) 0 2 2 t t 1 2 t 2 2 t 2 t (cid:181) (cid:182) 1 = δ −e(cid:48)µ− e(cid:48)ΣΣ(cid:48)e +e(cid:48)Σλ +(δ(cid:48) −e(cid:48)Φ)X 0 2 2 2 2 2 t 1 2 t (cid:181) (cid:182) 1 = δ −e(cid:48)µ− e(cid:48)ΣΣ(cid:48)e +e(cid:48)Σ(λ +λ (ϕ +ϕ X )) +(δ(cid:48) −e(cid:48)Φ)X 0 2 2 2 2 2 0 1 0 1 t 1 2 t (cid:181) (cid:182) 1 = δ − e(cid:48)Σ(cid:101) Σ(cid:101)(cid:48)e −e(cid:48) [µ−Σ(λ +λ ϕ )] +[δ(cid:48) −e(cid:48) (Φ−Σλ ϕ )]X 0 2 2 t t 2 2 0 1 0 1 2 1 1 t (cid:179) (cid:180) 1 δR =δ − e(cid:48)Σ(cid:101) Σ(cid:101)(cid:48)e −e(cid:48) µ(cid:101) −Σ(cid:101) (λ +λ ϕ ) 0 0 2 2 t t 2 2 t t 0 1 0,t (cid:179) (cid:180) δR(cid:48) =δN(cid:48)−e(cid:48) Φ(cid:101) −Σ(cid:101) λ ϕ 1 1 2 t t 1 1,t λR =λ +Σ(cid:101)(cid:48)e 0 0 t 2 λR =λ 1 1 D Variance Decomposition Then-quarternominalyieldisafunctionofunderlyingstatevariables yn =a +b X =a +bm Xm+bL XL, t n,t n,t t n,t n,t t n,t t whereXmandXLdenotemacroandlatentfactors,respectively. t t Itsvariancecanbecomputedas var[yn]=var[a ]+2cov[a ,b E(X )]+var[b E(X )] t n,t n,t n,t t n,t t +var[E(b )X ]+trace{var(b )var(X )}, n,t t n,t t whereweusetheequality (cid:163) (cid:164) var[a(cid:48)b]=var[a(cid:48)E[b]]+var E(a)(cid:48)b +trace[var(a)var(b)] foruncorrelatedcolumnvectorsaandb. 23

Thethirdtermontherighthandsidecanbefurtherdecomposedintotwocomponentsdrivenbymacroand latentfactors: (cid:163) (cid:161) (cid:162) (cid:164) (cid:163) (cid:161) (cid:162) (cid:164) var[E(b )X ]=cov E(b )X ,E bm Xm +cov E(b )X ,E bl Xl . n,t t n,t t n,t t n,t t n,t t We therefore decompose the variance of n-quarter nominal yield into a component due to time variations in parameterestimates, var[a ]+2cov[a ,b E(X )]+var[b E(X )] ρ = n,t n,t n,t t n,t t , tvp var[yn] t acomponentduetovariationsinmacrofactors, (cid:163) (cid:161) (cid:162) (cid:164) cov E(b )X ,E bm Xm ρ = n,t t n,t t , macro var[yn] t acomponentduetovariationsinthelatentfactor, (cid:163) (cid:161) (cid:162) (cid:164) cov E(b )X ,E bl Xl ρ = n,t t n,t t , latent var[yn] t andaremainderterm ρ =1−ρ −ρ −ρ . other tvp macro latent thatisduetotrace{var(b )var(X )}aswellasin-samplecorrelationsbetweenparameterandstatevariable n,t t estimates. 24

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Table1: Predictiveregressions Dependent Coefficients R2 variable constant y1 g π y4 y40 t−1 t−1 t−1 t−1 t−1 g 0.008 -0.145 0.291 -0.070 0.12 t (5.885) (-1.321) (4.385) (-0.513) 0.008 -1.322 1.075 0.105 0.07 (3.011) (-1.988) (1.323) (0.317) π 0.001 0.108 0.013 0.766 0.73 t (1.060) (2.728) (0.562) (15.672) 0.007 0.644 0.220 -0.526 0.39 (3.777) (1.533) (0.428) (-2.513) gspf 0.009 -0.318 0.228 -0.026 0.30 t+1|t (8.406) (-4.478) (4.472) (-0.301) 0.008 -1.995 1.612 0.127 0.32 (4.523) (-4.922) (3.257) (0.628) πspf 0.000 0.205 0.017 0.638 0.82 t+1|t (0.365) (6.105) (0.683) (15.860) 0.005 0.551 0.214 -0.340 0.52 (3.300) (1.664) (0.531) (-2.063) Note:ThistablereportsslopecoefficientsandR2fromregressionsofrealizedquarterlyrealGDPgrowth (g ),realizedquarterlyinflation(π )andtheirSPFforecasts(gspf andπspf )onvariouslaggedexplanatory t t t+1|t t+1|t variables. y1, y4 and y40 denote 1-quarter, 1- and 10-year yields, respectively. T-statistics are reported in t t t parentheses. 27

Table2: SummaryofModels VAR Latent Model Mean Slope Usesurvey DataVintage factor FC fixed fixed N finaldata N TVC discountedrollingaverage rolling N real-timedata N TVC-S discountedrollingaverage rolling Y real-timedata N PFC discountedrollingaverage fixed N finaldata N TVC-L discountedrollingaverage rolling N real-timedata Y FC-L fixed fixed N finaldata Y Note: Thistablesummarizesvariousmodelsestimatedinthispaper. “FC”standsforfixedcoefficients, “TVC”standsfortime-varyingcoefficients,and“PFC”standsforpartiallyfixedcoefficients.Modelsdenoted with the suffix “-S” are the same as corresponding models without the suffix but are estimated with SPF forecastsasadditionalinputs. Modelsdenotedwiththesuffix“-L”includesoneadditionallatentfactorthat drivesthetermstructure. 28

Table3: ParameterEstimates: ModelFC µ Φ GDPgrowth 3.581 0.221 0.074 0.939 0.045 -0.239 -1.049 (0.218) (0.017) (0.005) (0.043) (0.004) (0.011) (0.033) inflation 0.132 0.006 0.516 0.097 0.017 0.347 -0.043 (0.149) (0.000) (0.019) (0.014) (0.002) (0.012) (0.004) shortrate 0.086 0.047 0.184 0.846 0.006 -0.065 0.031 (0.138) (0.004) (0.021) (0.022) (0.000) (0.005) (0.001) Σ λ λ 0 1 GDPgrowth 3.243 -0.980 -0.995 -0.248 -0.989 (0.227) (0.072) (0.019) (0.015) (0.071) inflation -0.192 1.142 2.882 0.072 -0.104 0.574 (0.112) (0.099) (0.558) (0.003) (0.016) (0.031) shortrate 0.116 0.096 0.925 -1.519 0.340 0.265 -0.079 (0.090) (0.088) (0.237) (0.071) (0.013) (0.016) (0.009) StandardDeviationofMeasurementErrorsofYields 1-yr 2-yr 3-yr 4-yr 5-yr 7-yr 10-yr 0.453 0.669 0.791 0.894 0.959 1.037 1.093 (0.050) (0.108) (0.028) (0.253) (0.056) (0.039) (0.177) Note: Parametersµ,Σandstandarddeviationsofmeasurementerrorsaremultipliedby400. λ 1 isdividedby400. Standarderrors(inparentheses)arecomputedfollowingthemethodinAppendix A. 29

Table4: ParameterEstimates: ModelTVC µ Φ GDPgrowth 6.844 0.105 -0.389 1.200 0.037 -0.300 -1.514 (4.098) (0.179) (0.393) (0.719) (0.126) (0.533) (0.652) inflation 0.423 0.020 0.375 0.201 -0.042 0.028 0.128 (1.468) (0.075) (0.156) (0.359) (0.046) (0.192) (0.349) shortrate 0.719 0.026 0.218 1.003 0.024 -0.063 -0.242 (0.672) (0.055) (0.198) (0.264) (0.048) (0.135) (0.180) Σ λ λ 0 1 GDPgrowth 2.509 -1.256 0.399 0.300 -0.100 (0.715) (0.152) (0.043) (0.049) (0.026) inflation -0.117 0.954 -0.756 0.019 -0.023 0.102 (0.180) (0.375) (0.082) (0.020) (0.024) (0.015) shortrate 0.120 0.041 0.711 0.288 -0.225 -0.223 0.114 (0.097) (0.159) (0.455) (0.060) (0.012) (0.014) (0.008) StandardDeviationofMeasurementErrorsofYields 1-yr 2-yr 3-yr 4-yr 5-yr 7-yr 10-yr 0.582 0.678 0.683 0.677 0.679 0.673 0.635 (0.058) (0.196) (0.358) (0.314) (0.376) (0.181) (0.084) Note:Numbersinboldaresamplemeansandsamplestandarddeviations(inparentheses)ofparameterestimates.Therestareparameterestimatesandstandarderrors(inparentheses).Parameters µ,Σandstandarddeviationsofmeasurementerrorsaremultipliedby400. λ isdividedby400. 1 30

Table5: ParameterEstimates: ModelTVC-S µ Φ GDPgrowth 6.482 0.111 -0.370 1.217 0.038 -0.254 -1.513 (4.325) (0.181) (0.400) (0.732) (0.121) (0.521) (0.654) inflation 0.417 0.020 0.373 0.200 -0.042 0.029 0.129 (1.467) (0.074) (0.156) (0.357) (0.047) (0.195) (0.348) shortrate 0.635 0.026 0.220 1.008 0.022 -0.060 -0.237 (0.622) (0.056) (0.198) (0.260) (0.047) (0.137) (0.186) Σ λ λ 0 1 GDPgrowth 2.509 -1.240 0.234 0.237 -0.061 (0.780) (0.120) (0.033) (0.037) (0.021) inflation -0.115 0.924 -0.521 -0.004 -0.138 0.130 (0.184) (0.447) (0.054) (0.015) (0.017) (0.011) shortrate 0.126 0.037 0.354 0.723 -0.183 -0.181 0.049 (0.094) (0.161) (0.770) (0.035) (0.008) (0.009) (0.005) StandardDeviationofMeasurementErrorsofYields 1-yr 2-yr 3-yr 4-yr 5-yr 7-yr 10-yr 0.573 0.670 0.700 0.729 0.745 0.775 0.717 (0.051) (0.166) (0.325) (0.338) (0.327) (0.154) (0.083) Note:Numbersinboldaresamplemeansandsamplestandarddeviations(inparentheses)ofparameterestimates.Therestareparameterestimatesandstandarderrors(inparentheses).Parameters µ,Σandstandarddeviationsofmeasurementerrorsaremultipliedby400. λ isdividedby400. 1 31

Table6: DifferencebetweenModelandSurveyForecastsofVARVariables(RMSE;RW=1) PanelA:FullSample1965:Q4–2006:Q2 ModelFC ModelTVC ModelTVC-S Horizon GDPG inflation 3myld GDPG inflation 3myld GDPG inflation 3myld 1 0.671 0.769 1.266 0.752 0.809 1.799 0.685 0.788 1.675 2 0.686 0.806 1.373 0.767 0.939 1.443 0.665 0.917 1.163 4 0.586 0.787 1.311 0.759 1.015 1.163 0.670 1.002 0.904 40 n/a 0.980 n/a n/a 0.806 n/a n/a 0.811 n/a 20-40 0.722 0.599 0.309 0.280 0.326 0.531 0.286 0.265 0.410 PanelB:Sub-sample1965:Q4–1981:Q4 ModelFC ModelTVC ModelTVC-S Horizon GDPG inflation 3myld GDPG inflation 3myld GDPG inflation 3myld 1 0.645 0.802 0.501 0.726 0.733 0.718 0.726 0.733 0.718 2 0.691 0.801 0.695 0.780 0.870 0.504 0.780 0.870 0.504 4 0.571 0.768 1.011 0.823 1.027 0.717 0.823 1.027 0.717 40 n/a 2.620 n/a n/a 1.220 n/a n/a 1.220 n/a 20-40 n/a n/a n/a n/a n/a n/a n/a n/a n/a PanelC:Sub-sample1982:Q1–1995:Q4 ModelFC ModelTVC ModelTVC-S Horizon GDPG inflation 3myld GDPG inflation 3myld GDPG inflation 3myld 1 0.757 0.686 1.391 0.877 1.129 2.100 0.656 1.071 1.978 2 0.597 0.765 1.652 0.707 1.312 1.701 0.408 1.242 1.406 4 0.519 0.724 1.905 0.655 1.103 1.601 0.385 1.056 1.282 40 n/a 0.605 n/a n/a 0.701 n/a n/a 0.710 n/a 20-40 0.619 0.182 0.217 0.247 0.290 0.525 0.255 0.209 0.388 PanelD:Sub-sample1996:Q1–2006:Q2 ModelFC ModelTVC ModelTVC-S Horizon GDPG inflation 3myld GDPG inflation 3myld GDPG inflation 3myld 1 0.691 0.729 1.266 0.698 0.466 1.374 0.369 0.415 1.167 2 0.901 0.891 1.091 0.865 0.495 1.319 0.404 0.434 0.938 4 0.828 1.074 0.675 0.691 0.564 0.911 0.346 0.497 0.524 40 n/a 1.480 n/a n/a 1.104 n/a n/a 1.099 n/a 20-40 1.085 1.868 0.762 0.403 0.567 0.584 0.402 0.569 0.586 Note: This table summarizes differences between model and survey forecasts for the real GDP growth (“GDPG”),inflationandthe3-monthyield(“3myld”)atvariousforecastinghorizons.Thestatisticsreportedare theratiosofRMSEsoverthoseofcorrespondingRandomWalkmodels. 32

Table7: DifferencebetweenModelandSurveyForecastsofYields(RMSE;RW=1) PanelA:FullSample1986:Q1–2006:Q2 ModelFC ModelTVC ModelTVC-S Horizon 2yyld 10yyld 2yyld 10yyld 2yyld 10yyld 1 n/a 2.472 n/a 1.325 n/a 1.399 2 n/a 2.297 n/a 1.372 n/a 1.332 4 n/a 1.975 n/a 1.300 n/a 1.111 20-40 0.363 0.461 0.484 0.348 0.300 0.267 PanelB:Sub-sample1986:Q1–1995:Q4 ModelFC ModelTVC ModelTVC-S Horizon 2yyld 10yyld 2yyld 10yyld 2yyld 10yyld 1 n/a 1.434 n/a 1.553 n/a 1.510 2 n/a 1.389 n/a 1.565 n/a 1.490 4 n/a 1.453 n/a 1.608 n/a 1.553 20-40 0.228 0.209 0.484 0.309 0.282 0.271 PanelC:Sub-sample1996:Q1–2006:Q2 ModelFC ModelTVC ModelTVC-S Horizon 2yyld 10yyld 2yyld 10yyld 2yyld 10yyld 1 n/a 2.863 n/a 1.190 n/a 1.338 2 n/a 2.607 n/a 1.275 n/a 1.254 4 n/a 2.101 n/a 1.198 n/a 0.947 20-40 0.842 0.884 0.489 0.456 0.406 0.254 Note:Thistablesummarizesdifferencesbetweenmodelandsurveyforecastsforthe2-yearyield (“2yyld”)andthe10-yearyield(“10yyld”)atvariousforecastinghorizons. Thestatisticsreported aretheratiosofRMSEsoverthoseofcorrespondingRandomWalkmodels. 33

)1=WR;ESMR(stsaceroFelpmaS-fo-tuO :8elbaT selbairavRAV:AlenaP FPS S-CVTledoM CVTledoM CFledoM dlym3 noitaflni GPDG dlym3 noitaflni GPDG dlym3 noitaflni GPDG dlym3 noitaflni GPDG noziroH 859.0 578.0 257.0 278.0 220.1 767.0 979.0 850.1 228.0 499.0 719.0 977.0 1 249.0 549.0 960.1 368.0 351.1 089.0 120.1 402.1 950.1 069.0 089.0 279.0 2 030.1 083.1 750.1 629.0 187.1 939.0 960.1 538.1 120.1 649.0 533.1 649.0 4 a/n a/n a/n 779.0 735.1 897.0 320.1 695.1 308.0 329.0 552.1 477.0 8 sdleiymret-regnoL:BlenaP FPS S-CVTledoM CVTledoM CFledoM ry-01 ry-01 ry-5 ry-2 ry-1 ry-01 ry-5 ry-2 ry-1 ry-01 ry-5 ry-2 ry-1 noziroH 752.1 859.1 361.2 519.1 773.1 114.1 534.1 459.1 896.1 737.2 978.1 205.1 784.1 1 790.1 906.1 796.1 435.1 602.1 170.1 161.1 265.1 944.1 430.2 783.1 612.1 232.1 2 879.0 504.1 934.1 292.1 511.1 108.0 310.1 862.1 142.1 417.1 342.1 011.1 790.1 4 a/n 763.1 812.1 030.1 879.0 976.0 709.0 699.0 900.1 039.1 513.1 940.1 400.1 8 m3“( dleiy htnom-3 eht rof syevrus dna sledom tnereffid fo ecnamrofrep gnitsacerof elpmas-fo-tuo sezirammus elbat sihT :etoN –4Q:5991 si doirep elpmas-fo-tuo ehT .snoziroh gnitsacerofsuoirav ta noitaflni rotafled PDG dna )”GPDG“(htworg PDG laer ,)”dly .sledomklaWmodnaRgnidnopserrocfoesohtrevosESMRfosoitarehteradetroperscitsitatsehT .2Q:6002 34

Table9: VarianceDecompositionofNominalYields PanelA:ModelFC Maturity TVC GDPgrowth Inflation Shortrate Latentfactor Residue 4 0.00 0.00 -0.01 1.01 0.00 0.00 8 0.00 0.00 -0.02 1.02 0.00 0.00 20 0.00 0.00 -0.06 1.05 0.00 0.00 40 0.00 0.01 -0.09 1.08 0.00 0.00 PanelB:ModelTVC Maturity TVC GDPgrowth Inflation Shortrate Latentfactor Residue 4 0.16 -0.00 0.04 0.48 0.00 0.33 8 0.43 -0.00 0.02 0.22 0.00 0.33 20 0.71 0.00 0.01 0.06 0.00 0.22 40 0.85 0.00 0.00 0.02 0.00 0.13 PanelC:ModelTVC-S Maturity TVC GDPgrowth Inflation Shortrate Latentfactor Residue 4 0.10 -0.00 0.05 0.55 0.00 0.30 8 0.32 -0.00 0.04 0.28 0.00 0.36 20 0.64 -0.00 0.02 0.09 0.00 0.26 40 0.82 0.00 0.01 0.03 0.00 0.15 PanelD:ModelPFC Maturity TVC GDPgrowth Inflation Shortrate Latentfactor Residue 4 0.03 -0.00 0.11 0.81 0.00 0.06 8 0.08 -0.00 0.16 0.62 0.00 0.14 20 0.27 -0.00 0.16 0.31 0.00 0.25 40 0.49 -0.00 0.11 0.14 0.00 0.26 PanelE:ModelTVC-L Maturity TVC GDPgrowth Inflation Shortrate Latentfactor Residue 4 0.06 -0.01 0.05 0.54 0.13 0.23 8 0.14 -0.00 0.03 0.29 0.22 0.32 20 0.28 -0.00 0.03 0.10 0.27 0.33 40 0.43 -0.00 0.02 0.03 0.22 0.30 Note:Thistablereportsvariancedecompositionofnominalyieldsofvariousmaturities intocomponentsduetotime-varyingcoefficients(TVC),eachofthestatevariables,anda residueterm. 35

Table10: ParameterEstimates: ModelPFC µ Φ GDPgrowth 4.746 0.148 -0.076 0.745 0.091 -0.264 -0.895 (0.754) (0.155) (0.048) (0.352) (0.020) (0.210) (0.677) inflation 0.411 -0.014 0.596 0.125 0.008 0.211 -0.066 (0.278) (0.007) (0.343) (0.026) (0.004) (0.143) (0.029) shortrate 0.113 0.065 0.203 0.811 0.014 -0.072 0.043 (0.199) (0.009) (0.013) (0.034) (0.004) (0.005) (0.004) Σ λ λ 0 1 GDPgrowth 2.925 -0.251 -0.220 0.153 -0.098 (0.990) (0.048) (0.064) (0.015) (0.002) inflation -0.177 1.078 -1.097 0.137 0.043 0.021 (0.238) (0.212) (0.114) (0.030) (0.004) (0.002) shortrate 0.105 0.100 0.969 0.583 0.015 -0.099 0.007 (0.162) (0.209) (0.230) (0.183) (0.003) (0.018) (0.001) StandardDeviationofMeasurementErrorsofYields 1-yr 2-yr 3-yr 4-yr 5-yr 7-yr 10-yr 0.502 0.661 0.738 0.754 0.759 0.757 0.745 (0.053) (0.062) (0.087) (0.104) (0.076) (0.120) (0.067) Note:Numbersinboldaresamplemeansandsamplestandarddeviations(inparentheses)ofparameterestimates.Therestareparameterestimatesandstandarderrors(inparentheses).Parameters µ,Σandstandarddeviationsofmeasurementerrorsaremultipliedby400. λ isdividedby400. 1 36

Table11: ParameterEstimates: ModelTVC-L µ Φ realGDPgrowth 6.844 0.105 -0.389 1.200 0.037 -0.300 -1.514 0.000 (4.098) (0.179) (0.393) (0.719) (0.126) (0.533) (0.652) inflation 0.423 0.020 0.375 0.201 -0.042 0.028 0.128 0.000 (1.468) (0.075) (0.156) (0.359) (0.046) (0.192) (0.349) shortrate 0.719 0.026 0.218 1.003 0.024 -0.063 -0.242 0.000 (0.672) (0.055) (0.198) (0.264) (0.048) (0.135) (0.180) latentfactor 0.000 -0.025 -0.035 0.024 0.000 0.000 0.000 0.973 (0.002) (0.003) (0.002) (0.000) (0.000) (0.000) (0.003) Σ λ λ 0 1 realGDPgrowth 2.509 -0.063 0.187 0.308 -0.267 0.017 (0.715) (0.109) (0.018) (0.018) (0.015) (0.036) inflation -0.117 0.954 0.215 -0.008 0.002 -0.019 0.052 (0.180) (0.375) (0.089) (0.010) (0.014) (0.017) (0.017) shortrate 0.120 0.041 0.711 -1.569 -0.059 -0.122 0.242 0.445 (0.097) (0.159) (0.455) (0.238) (0.011) (0.020) (0.013) (0.024) latentfactor 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 StandardDeviationofMeasurementErrorsofYields 1-yr 2-yr 3-yr 4-yr 5-yr 10-yr 0.316 0.251 0.202 0.156 0.120 0.108 (0.024) (0.036) (0.037) (0.018) (0.012) (0.009) Note: Numbers in bold are sample means and sample standard deviations (in parentheses) of parameter estimates. The rest are parameter estimates and standard errors (in parentheses). Parameters µ, Σ and standard deviationsofmeasurementerrorsaremultipliedby400. λ isdividedby400. 1 37

Thisfigureplots3-monthT-billyields,4-quarterrealGDPgrowth,4-quarterGDPdeflatorinflationandthe correspondingSPFforecastslagged4quartersbehind. ShadedareasrepresentNBERrecessions. Figure1: Shortrate,realGDPgrowth,inflationandlaggedSPFforecasts 38

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This figure plots the impulse responses of 1-quarter, 1-year and 10-year yields to 1 standard deviation real GDPgrowth,inflationandshortrateshockson1968:Q2(blueline),1981:Q1(redline)and2006:Q2(green line),basedonModelTVC-S. Figure9: ImpulseResponsesofYields(ModelTVC-S) 46

Thetoptwopanelsplotactualandmodel-impliedtwo-andten-yearyields,respectively,andthecorrespondingEHcomponents. Thebottompanelplotsthemodel-impliedtwo-andten-yeartermpremiums. Shaded areasrepresentNBERrecessions. AllresultsarebasedonModelTVC-L. Figure10: TermStructureImplicationsofModelTVC-L 47