Tips from TIPS: the informational content of Treasury Inflation-Protected Security prices

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Tips from TIPS: the informational content of Treasury Inflation-Protected Security prices Stefania D’Amico, Don H. Kim, and Min Wei 2008-30 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.

Tips from TIPS: the informational content of Treasury Inflation-Protected Security prices∗ Stefania D’Amico†, Don H. Kim‡, and Min Wei§ This version: February 27, 2008 JEL Classification: G12, E31, E43 Keywords: Term structure model; Inflation expectation; Inflation risk premium; SPF; Treasury Inflation-Protected Securities (TIPS) ∗Part of the work on this paper was done while one of us (Kim) was at the Bank for International Settlements. We thank Andrew Ang, Claudio Borio, Mike Chernov, Jim Clouse, Greg Duffee, Peter Ho¨rdahl, Athanasios Orphanides, Frank Packer, George Pennacchi, Jennifer Roush, Brian Sack, JonathanWright, andseminarparticipantsatthe2008AFAAnnualMeetings, DallasFedConference onPriceMeasurementforMonetaryPolicy,ECBWorkshoponInflationRiskPremium,the11thInternational Conference on Computing in Economics and Finance, and the San Francisco Fed for helpful comments or discussions. We also thank Michelle Steinberg for providing information from the NY Fed’ssurveyofTIPSdealers. Wealoneareresponsibleforanyerrors. Theopinionsexpressedinthis paper do not necessarily reflect those of the Federal Reserve Board or the Federal Reserve System or theBankforInternationalSettlements. †Division of Monetary Affairs, Federal Reserve Board, stefania.d’amico@frb.gov, +1-202-452- 2567. ‡DivisionofMonetaryAffairs,FederalReserveBoard,don.h.kim@frb.gov,+1-202-452-5223. §DivisionofMonetaryAffairs,FederalReserveBoard,Min.Wei@frb.gov,+1-202-736-5619.

Abstract WeexaminetheinformationalcontentofTIPSyieldsfromtheviewpointofageneral3-factor no-arbitragetermstructuremodelofinflationandinterestrates. Ourempiricalresultsindicate that TIPS yields contained a “liquidity premium” that was until recently quite large (∼ 1%). Key features of this premium are difficult to account for in a rational pricing framework, suggestingthat TIPSmay nothavebeen pricedefficientlyinits earlyyears. Besides theliquidity premium, a time-varying inflation risk premium complicates the interpretation of the TIPS breakeven inflation rate (the difference between the nominal and TIPS yields). Nonetheless, high-frequency variation in the TIPS breakeven rates is similar to the variation in inflation expectations implied by the model, lending support to the view that TIPS breakeven inflation ratesareausefulproxyforinflationexpectations.

1 Introduction Since its inception in 1997, the market for Treasury Inflation-Protected Securities (TIPS) has grown substantially and now comprises about 10% of the outstanding Treasury debt market. Almost a decade’s TIPS data thus accumulated is a rich source of information to academic researchers and market participants alike. Because TIPS are securities whose coupon and principal payments are indexed to the price level, information about yields on these “real bonds” has direct implications for asset pricing models, many of which are written in terms ofrealconsumption. Meanwhile,real-timeTIPSdatahaveattractedmuchattentionfrompolicy makers and market participants as a source of information about the state of economy. In particular, the differential between yields on nominal Treasury securities and on TIPS of comparablematurities,oftencalledthe“breakeveninflation(BEI)rate”or“inflationcompensation”,hasbeenoftenusedinpolicycirclesandthefinancialpressasaproxyforthemarket’s inflationexpectations. However, certain complications arise in interpreting the information from TIPS yields. First, TIPS might not have been “efficiently” priced, due to their lower liquidity, the relative newness of the TIPS market, and other factors. Second, besides such institutional idiosyncrasies, the interpretation of the TIPS breakeven rate as an inflation expectation depends on the validity of the “Fisher hypothesis”, which states that the nominal yield is the real yield plus expected inflation. This relation is only approximate, however, as it ignores the potential correlationbetweeninflationandtherealeconomy. Moreformally,wecandefinethenominal yield as the sum of the real yield, expected inflation, and the inflation risk premium. The last component may be substantial; indeed, the presence of an inflation risk premium in nominal bonds,translatingtoanadditionalfinancingcostforTreasurytoissuenominalbonds,wasone oftheargumentforissuingTIPSinthefirstplace. These issues naturally lead to the following questions: (1) Can we take TIPS yields at face value? Are they suitable for use as an input for other studies that involve ex ante real interest rates? (2) Can we take the TIPS breakeven rate as a reasonable measure of inflation expectations? Do movements in the breakeven rates reflect “fundamentals” or extraneous factors? Thepurposeofourpaperistoprovidequantitativeevidencethatbearsonthesequestions. Specifically,wemodelthedynamicsofnominalyields,inflation,andTIPSyieldsinageneral 1

no-arbitrage term structure model setting, and examine the extent to which these data are consistent with each other. Furthermore, we seek to establish some basic facts about the real term structure and the inflation risk premia implicit in nominal bond yields and to obtain an estimateofthe“liquiditypremium”inTIPSyields. Although there have been other studies that use a no-arbitrage framework to model the U.S. real term structure, they have not explicitly explored the efficiency of TIPS pricing or estimated a liquidity premium in TIPS yields. Also, these studies do not employ TIPS data in estimation, with the exception of Chen, Liu and Cheng (2005).1 Studies such as Ang, Bekaert and Wei (2007a), Chernov and Mueller (2007) and Buraschi and Jiltsov (2005) produce “shadow real yields” which could be compared with the TIPS yields. However, the estimatesofrealyields(aswellasotherquantities,liketheinflationriskpremium)fromthese studies differ a lot from each other, and in many cases are too much at odds with the priors of practitioners,possiblyindicatingproblemsinthespecificationorestimationofthemodel. Inviewofthefactthethepricingmechanismsbehindnominalandrealbondsarenotwell understood, in this paper we take a statistical perspective, and use flexibly specified affine- Gaussian latent-factor models, which may face less misspecification concerns than some of the models in the existing literature. Also, in the estimation stage, we strive to address the small sample and overfitting problems that can lead to poor results, by utilizing additional informationinsurveyforecastsandexperimentingwithdifferentauxiliaryconditionstosearch forrobustconclusions. Our main results can be summarized as follows. In all the cases that we have examined, estimatingthemodeltakingTIPSyieldsattheirfacevaluefailstoproduceplausibleestimates of inflation expectations or inflation risk premia. The difference between the observed TIPS yields and the model-implied real yields estimated without TIPS data indicates that the “liquiditypremium”wasquitelargeintheearlyyearsofTIPS’sexistence,buthasbecomesmaller recently. Thisliquiditypremiumturnsouttobedifficulttoaccountforwithinasimplerational pricing framework, suggesting that TIPS may not have been priced efficiently in their early years. Nonetheless,timevariationinTIPS-basedandmodel-impliedbreakevenratesarequite similar, suggesting that changes in the TIPS breakeven rates largely reflects changes in infla- 1Mostoftheexistingstudies(includingRisa(2001)andEvans(2003))withinflation-indexedbonddatahave focusedontheUKtermstructures. ArecentpaperbyHo¨rdahlandTristani(2007)explorestheeuro-areaterm structureusingtheFrenchindexedbonddata. 2

tion expectations or in the investors’ attitude toward inflation risks, rather than being random movements. The rest of this paper is organized as follows. In Section 2, we discuss the informational content of TIPS yields in simple terms as well as the related literature. Section 3 specifies the no-arbitrage approach to modeling inflation, nominal yields, and real yields jointly. Section 4 describes the empirical strategies for estimating the model and presents the empirical results. Section 5 provides additional discussion of the model estimates and the interpretation of the TIPS information, and Section 6 concludes. Throughout the main text, we strive to keep the discussionrelativelynon-technical,relegatingmostofthetechnicaldetailstotheAppendices. 2 TIPS: Preliminary considerations 2.1 TIPS breakevens as measures of expected inflation Despite potential complications associated with the inflation risk premium and the liquidity premium,TIPSbreakevenrateshavebeenfrequentlyusedasaproxyforinflationexpectations by policy makers and market practitioners. The minutes of FOMC meetings often take note ofchangesinTIPSyieldssincethepreviousmeeting,2 anditisnotuncommontoseeexplicit references to TIPS breakeven rates in Fed officials’ speeches.3 Similarly, TIPS breakeven ratesarefrequentlycitedinthefinancialpresswhendiscussinginflationexpectations. Such usages indicate that many practitioners find TIPS breakeven rates to be a plausible measure of market inflation expectations. Indeed, empirical evidence indicates that TIPS breakeven rates respond to news arrivals or important economic events in the “right” direc- 2For example, the minutes of the June 2006 FOMC meeting includes the following sentence: “Yields on inflation-indexedTreasurysecuritiesincreasedbymorethanthoseonnominalsecurities,andtheresultingdecline ininflationcompensationretracedasubstantialshareoftherisethathadoccurredovertheprecedingintermeeting period.” 3Fed Vice Chairman Kohn (2006)’s speech on April, 2006, for example, includes the following remark: “[L]onger-term inflation expectations remain well contained. For example, the median expected inflation rate during the next five to ten years, as reported in the University of Michigan’s survey of consumers, has barely edgedupinrecentyears... Meanwhile,inflationcompensationforinvestorsimpliedbythespreadsbetweenthe rates on nominal and CPI-indexed Treasury notes at both five- and ten-year maturities also has not shown any tendencytomovehigheronbalance.” 3

tion.4 It is, however, difficult to tell from such event studies whether the magnitude of the reaction is right, which is also of considerable interest. This question about the magnitude of TIPS breakeven rates’ reaction to data announcements is a part of the larger discussion about whetherthevariabilityandthelevelofTIPSbreakevenratesarereasonable,towhichwenow turn. One way to examinethe reasonableness of the leveland the variabilityof TIPS breakeven rates is to compare them with another measure based on survey forecasts of inflation. There arelargelytwokindsofsurveysavailable. OneistheMichigansurvey,whichpollshouseholds (consumers),andtheother,suchastheBlueChipsurveyortheSPFsurvey,polls“professionals” (business forecasters). Figure 1(a) shows the the Michigan survey of long-term inflation forecast and the 10-year SPF inflation forecast, together with the 10-year TIPS breakeven inflation rate.5 It can be seen that until recently the TIPS breakeven rate has been lower than these survey forecasts. In the case of the Michigan survey, the TIPS breakeven rate has remainedsubstantiallylowerthroughoutthe1999-2007period. [insertFigure1abouthere] Note that comparing the level of TIPS breakeven rates with the Michigan survey may not befair. Aninflationmeasurecandependsignificantlyonthedefinitionoftheconsumptionbasket,howqualitychangesaretakenintoaccount,andotherissues.6 WhiletheTIPSbreakeven raterelatestoCPIinflation,theMichigansurveyaskstheparticipantsfortheirviewsaboutthe change in the “prices of things you buy” rather than inflation based on a specific price index. However, even the 10-year SPF inflation forecast, which is specifically about CPI inflation, tendedtobehigherthanthe10-yearTIPSbreakevenrateuntilabout2004. Thisissurprising: one would have expected the TIPS breakevens to be higher, since it includes an inflation risk premium,whichisnormallybelievedtobe positive. One might argue that the fault lies with the survey forecast rather than with TIPS. Survey 4Forexample,intheworkingpaperversionofGu¨rkaynak,Sack,andSwanson(2005),theyfindthatahigherthan-expectedcoreCPIdatareleasetypicallyleadstoariseinthebreakevenrates,suggestinganupwardrevision ininflationexpectations. 5Forprofessional’ssurveyforecastofinflation,weshallusetheSPFforecastsonly,astheBlueChipsurveys (BlueChipEconomicIndicatorsandBlueChipFinancialForecasts)arequitesimilar. 6Note, for instance, that (annual) CPI inflation has been persistently higher than PCE inflation by about 40 basispoints. 4

forecasts cannot be expected to be a completely reliable measure of inflation expectations. That said, a quick comparison of the 1-year ahead CPI inflation forecasting performance of the 5-year TIPS breakeven rates versus the SPF survey forecast7 for the 1999:Q1-2007:Q3 periodproducesthefollowingroot-mean-squareerrors(RMSE): RMSE(TIPS5Y) = 1.07, RMSE(SPF1Y) = 0.92. NotethatthesurveymeasurehasdonebetterthantheTIPSbreakevenrate,makingitdifficult to dismiss offhand. A recent study by Ang, Bekaert, and Wei (2007b) also provides extensive evidence that surveys perform better in forecasting inflation than various model-based measuresthattheyhaveexamined. TIPS breakeven rates differ from survey forecasts not only in levels but also in variability. Figure 1(a) shows that TIPS breakeven rates exhibit greater time variation than the survey forecasts. The Michigan survey may contain a substantial amount of sampling error and other noise.8 However, once we look beyond the monthly noise and focus on the systematic movement,thelong-horizonMichigansurveyforecastseemstobelessvariablethantheTIPS breakeven rate. An even greater contrast with TIPS is offered by the 10-year SPF inflation forecast, which has been pretty much immobile in the 1999-2006 period. This may appear suspicious: eveniflong-terminflationexpectationsare“wellanchored”,itisdifficulttoimagine that they have become virtually immovable as suggested by the SPF survey. Nonetheless, the qualitatively similar messages from the two surveys raise the possibility that the greater variabilityofTIPSbreakevensmaybearobustfact. One possible explanation is that part of the TIPS breakeven rate variation is due to variationsintheinflationriskpremiumratherthanchangesininflationexpectations. Thedifficulty with this argument is that the level consideration seems to leave little room for inflation risk premium,atleastonethatispositive. 7A reliable TIPS breakeven rate for near-term maturities are not available (especially in the early years of TIPS),soweusethe5-yearTIPSbreakevenrateasaTIPS-basedforecastofthe1-yearinflation. 8ThegroupofMichigansurveyparticipantschangesfrommonthtomonth. Also,thedistributionofforecasts isextremelywide,containingmanyresponsesthatareveryhighorverylow(negative). Forexample,intheJan 2007Michigansurvey,15%ofrespondentspredicted0%ornegativeinflationinthenext12months,whileabout 10%ofrespondentspredictedinflationof10%orhigher. 5

2.2 Evidence on inflation risk premium The above discussions highlight the potential importance of the inflation risk premium. Let us now review here what is known, or has been said, about the inflation risk premium in the existingliterature: Oneargumentforthepresenceoftheinflationriskpremiumintheearlyliteratureisbased on the need to make sense of the high nominal interest rates in the 1980s. Many studies from that period took note of the “extremely high level” of real interest rates in the 1980s.9 As can be seen from Figure 1(b), inflation in the early 1980s fell below 4%. Although inflation expectationsbasedonsurveyforecastsremainedfairlyhigh,nominalinterestratesweremuch higher, e.g., about 13% for the 10-year yield in 1984. The Fisher hypothesis then implies a very high ex ante real rate, e.g., 6% or even higher. Unlike inflation expectations which are known to be quite persistent, it is difficult to think of mechanisms that would make the exanterealratessimilarlypersistentorgeneratesuchhighvaluesofrealinterestrates,10 thus leavingroomforapositiveinflationriskpremium. Thepossibilityofapositiveandsubstantial inflationriskpremiuminthe1980sseemstobealsoinlinewithobservations(e.g.,Goodfriend (1993)) that during these years the Federal Reserve had less credibility than in later periods, bond markets thus counting in an extra premium in nominal yields beyond those attributable toinflationexpectations. Theoretically,theinflationriskpremiumarisesfromapotentialcorrelationbetweeninflation and the real aggregate marginal utility. One simple way to assess the magnitude of the inflation risk premium is, therefore, modeling the inflation covariance risks in a (C)CAPM framework,eitherbycomparingtheexpectedexcessreturnsonthenominalandrealbonds,11 or by estimating the risk premium due to covariances with the expected or the unexpected inflation shocks.12 The estimated inflation risk premium ranges from 10 to 100 basis points. The interpretation of such results, however, hinges on the validity of the proxy used for the marginal utility of wealth. The general failure of the current generation of (C)CAPM models in accounting for the time-series and cross-sectional properties of equity returns casts some doubts on this framework. This methodology also ignores information from the entire yield 9See,amongothers,ClaridaandFriedman(1984),BlanchardandSummers(1984),andPoole(1988). 10However, thefindingsinthelaterpartsofourpaper(especiallySec. 5)suggestthatrealtermpremiamay helpexplainpartofthepuzzle. 11See,forexample,CampbellandShiller(1996) 12See,forexample,Chen,RollandRoss(1986),FersonandHarvey(1991)andChan(1994). 6

curve,and,givenitstraditionalregression-basedimplementation,cannotbeusedtoaccurately pindowntheinflationriskpremiumatlongerhorizonsorgenerateaconsistentestimateofthe termstructureoftheinflationriskpremia. Lastly, the inflation risk premium can be estimated using a no-arbitrage term structure model, which avoids the aforementioned criticisms of (C)CAPM-type models by adopting a no-arbitrage approach and by modeling the entire yield curve simultaneously. These can be grouped into two categories: those that use indexed-bond data and those that do not.13 Due to the relatively short history of the indexed debt market in the U.S., there has not been a lot of “time-consistent” no-arbitrage model-based studies with TIPS data, Chen, Liu and Cheng (2005) being the only study we are aware of.14 Many of these studies find an inflation risk premiuminnominalyieldsthatisquitesubstantial,hasasignificantcross-sectional(maturity) dependence, and exhibits substantial variation over time.15 However, if we look at specific qualitative and quantitative aspects of the inflation risk premia, there is much disagreement between various studies. This disagreement is also reflected in real yields: for example, Ang, BekaertandWei(2007a),BuraschiandJiltsov(2005),andChernovandMueller(2007)obtain model-implied real yields that are quite different from each other, underscoring how little is establishedaboutrealyieldsandinflationriskpremiaintheextantliterature. Some of these studies may face misspecification concerns. For example, the model considered by Chen, Liu and Cheng (2005), which is similar to that of Richard (1978), assumes that the nominal short rate rN is the sum of two factors that both follow square-root (CIR) t processes: theinstantaneousinflationexpectation,π ,andtherealshortrate,rR. Althoughthe t t shocks to π and rR in Chen, Liu, and Cheng (2005)’s model are correlated (unlike Richard t t (1978)’s),thistypeofmodeldoesnotpermitaflexiblefeedbackstructurebetweenthefactors. In their model π and rR are thus restricted to be univariate; hence the term structure of inflat t tion expectations and the real term structure are each described by only one factor. Another potential problem with this type of model is that the market price of risk specification is too 13See,e.g.,Chen,Liu,andCheng(2005),Risa(2001),andEvans(2003),Ho¨rdahlandTristani(2007)forthe former,andAng,BekaertandWei(2007a,b),BuraschiandJiltsov(2005),CampbellandViceira(2001)forthe latter. 14JarrowandYildirim(2003)alsomodelnominalyields,TIPSyields,andinflationjointlyinanHJMframework, taking nominal and TIPS term structures as inputs. Their focus is on hedging and inflation derivative pricingratherthanonrealyieldandinflationriskpremiummodeling. 15See,e.g.,Ho¨rdahlandTristani(2007)andRisa(2001)forthenon-UScase,andChen,LiuandCheng(2005), BuraschiandJiltsov(2005),Ang,Bekaert,andWei(2007a),ChernovandMueller(2007)fortheUScase. 7

restrictive (see, e.g., Duffee (2002)). A recent study by Ang, Bekaert, and Wei (2007b) finds that the 3-factor no-arbitrage models that they consider perform worse than not only survey forecastsbutalsomanysimpler(regression)models,suggestingtheremightbeproblemswith someofthericherno-arbitragemodelsintheliteratureaswell. Existing studies may also face technical difficulties with estimation. In particular, the small-sample problems can have serious consequences on the behavior of the estimated inflation risk premia. The conventional estimation of stationary models has a tendency to understate the true persistence of the time series, making the process appear to be converging to their long-run means sooner than they actually do. This can lead to an artificially stable long-horizon inflation expectations; as a result, the inflation risk premium might pick up part ofthevariabilityofinflationexpectationsthatislostintheestimation. Intheremainderofthepaper,weexploreajointnominalandrealno-arbitragetermstructure model and try to address these specification and estimation issues. Further, differently from Chen, Liu, and Cheng (2005) who take TIPS yields at the face value, we are mindful of thepotentialproblemswiththisassumption. 3 Joint model of inflation and interest rates 3.1 Fisher hypothesis and beyond ItisusefultostartwiththeFisherhypothesis, yN = yR +I , (1) t,τ t,τ t,τ where yN is the nominal yield at time t with a time-to-maturity τ, yR is the real yield (“ex t,τ t,τ anterealrate”),and I isinflationexpectationbetweentime0and τ,i.e., t,τ (cid:183) (cid:181) (cid:182)(cid:184) 1 Q t+τ I = E log , (2) t,τ t τ Q t where Q is the price level. Given the cross-sectional data on real and nominal yields at time t t,onecanintroducetheinterpolationschemes yN = f (τ), yR = g (τ) (3) t,τ t t,τ t 8

and approximate inflation expectations as I = f (τ)−g (τ). Thus, the Fisher relation (1), t,τ t t togetherwithdataonrealandnominalbondyields,definesasimplewaytocomputeinflation expectations for any maturity τ;16 note that we do not need to model the dynamics of the yieldsinthissetting. As mentioned earlier, however, the Fisher hypothesis ignores potential correlation effects. To go beyond Fisher hypothesis, one needs to model the dynamics of real rates and inflation together. For this purpose, it is convenient to utilize a no-arbitrage relation between the socalled“pricingkernels”or“stochasticdiscountfactors”. Areal(nominal)pricingkernel,MR t (MN), has the property that it gives today’s market value of a future payoff in real (nominal) t terms.17 Inparticular,realandnominalbondprices, PR andPN,aregivenby PR = E (MR )/MR, PN = E (MN )/MN. (4) t,τ t t+τ t t,τ t t+τ t Notethatinnominalterms,arealbondisanassetwhosepayoffisproportionaltotheprice level. Thismeansthattherealandthenominalpricingkernelsarelinkedbythe“no-arbitrage relation”18 MN = MRQ−1. (5) t t t Usingthisrelation,wehave PN = E (MR Q−1 )/(MRQ−1). (6) t,τ t t+τ t+τ t t Thus,byspecifyingthejointdynamicsofMRandQ,onecanobtaintheno-arbitrage-consistent nominal yields; equivalently, one could also specify MN and Q to obtain the no-arbitrageconsistentrealyields.19 Fromeq(6),itisstraightforwardtoshowthat yN = yR +I +℘I , (7) t,τ t,τ t,τ t,τ 16Tohaveamanageablescope,inthispaperwefocusoninflationcompensationbasedonyields. Butwenote thatitisstraightforwardtocomputetheinflationcompensationbasedonforwardrates(called“forwardinflation compensation”). 17Apricingkernel(stochasticdiscountfactor)M inthecontinuous-timeformulationhasthepropertythatit t pricesanassetthatpaysX afterτ periodasP =E (X M )/M . Indiscretetime,apricingkernelforthe t t t+τ t+τ t nextperiodM givesthepriceofanassetthatpaysX attimet+1asP =E (X M ). t,t+1 t t t+1 t,t+1 18See,Campbell,Lo,MacKinlay(1996,p443)forthederivationofthediscrete-timeversionofthisresult. 19“No-arbitrage”inthispaperisintwosenses: theconsistencybetweenrealandnominalbondsasembodied bytherelation(5),andtheconsistencybetweenbondyieldsofvariousmaturities. 9

wheretheinflationriskpremiumterm ℘I capturesdeparturesfromtheFisherhypothesis.20 t,τ It may be useful to discuss briefly the advantages and disadvantages of the no-arbitrage setup. The no-arbitrage principle is more general than the Fisher hypothesis, but in order to operationalize the no-arbitrage idea, one has to make assumptions about the dynamics of pricing kernels and the price level, and can expect to incur some amount of specification error in the process. For example, some of the studies in the literature adopt a structural specification of the real pricing kernel, MR.21 While this approach has the potential benefit of making a definite connection between macroeconomic fundamentals and bond yields, it is not clear how well the current generation of structural models can capture the mechanisms that underlie asset price variation and inflation.22 Reduced-form models are not exempt from misspecification concerns, either. One example is Richard (1978)’s model, whose potential specification problems were discussed earlier in Sec. 2.2. Note also that if a model has too small a number of factors (e.g., a 2-factor model), it may have difficulty capturing the crosssectionofthenominalyieldcurvewell,resultinginsizablefittingerrorsandmakingitdifficult todiscussactualmarketdevelopmentsbasedonthemodel. AlthoughtheFisherrelationineq. (1)ignorestheinflationriskpremium,itleavessmaller room for cross-sectional fitting errors. Often it is unclear whether the specification errors in no-arbitrage models are any smaller than the error of ignoring the inflation risk premium. Indeed,wearenotawareofanyno-arbitragemodel-basedproceduresforcomputinginflation expectations from bond yields that is as widely used by practitioners as the method based on the cross-sections of nominal and indexed bond yields (Fisher hypothesis). Even so, there are issueswhichtheFisherhypothesissimplycannotaddress. First,onecannotlearnthedynamics of inflation expectations or the real and nominal term structures just from the cross-sectional 20The ℘I term is given by ℘I = J˜ +c˜ , where the covariance effect term is c˜ ≡ −(1/τ)log[1+ t,τ t,τ t,τ t,τ cov (MR /MR,Q /Q )/(E (MR /MR)E (Q /Q ))]andtheJensen’sinequalityeffecttermisJ˜ ≡ t t+τ t t t+τ t t+τ t t t t+τ t,τ −(1/τ)[log(E (Q /Q ))−E (log(Q /Q ))]. Therefore, the Fisher relation obtains if we ignore the cot t t+τ t t t+τ varianceeffectandtheJensen’sinequalityeffect. 21Forexample,BuraschiandJiltsov(2005)andBoudoukh(1993)usesimpleconsumption-basedmodelsfor MR, of the form MR = e−ρtu(cid:48)(C ), where u(C ) is often a log utility or CRRA utility function. Buraschi t t t t and Jiltsov (2005) also have a specific mechanism for inflation dynamics, while studies like Boudoukh (1993) introduceanexogenousprocessforinflationdynamicsandanexogenouscorrelationbetweeninflationandconsumption/outputgrowth. 22Thestill-ongoingdebateabouttheequitypremiumpuzzleisjustonereminderofpotentialmisspecification riskinstructuralmodels. 10

analysis of TIPS yields and nominal yields. Second, there are periods in which the indexed bondmarketdidnotexist,suchaspre-1997intheU.S.Third,eveniftheindexedbondmarket exists, one might wish to examine whether indexed bonds are priced efficiently. To address theseissues,oneneedsadynamicmodelliketheno-arbitragemodel. 3.2 No-arbitrage model Inthissectionweprovidea“bigpicture”descriptionofourmodel,relegatingtechnicaldetails ofthemodeltoAppendixA. Basically,weviewourmodelingproblemasastatisticalone: findaffinefunctions yN = aN +bN(cid:48)x (8) t,τ τ τ t yR = aR +bR(cid:48)x (9) t,τ τ τ t I = aI +bI(cid:48)x (10) t,τ τ τ t of an n-dimensional Gaussian state vector x = [x ,...,x ](cid:48) such that they are as general as t 1t nt possible, and at the same time consistent with no-arbitrage.23 This can be done by specifying a general form for nominal pricing kernel MN and the price level Q , and imposing suitable t t normalization conditions so that the model is econometrically identified (“maximally flexible”). We use the affine-Gaussian model for the nominal yields, and specify inflation as the sum of the instantaneous expectation, which is an affine function of the state vector, and the unexpectedinflation. Therealpricingkernel(hencetherealtermstructure)isthendetermined bytheno-arbitragerelationshipineq. (5). Here,thestatevariables x arestatisticalvariables it whosemeaningisonlyimplicitlydefinedbythedataonnominalyields,inflation,andTIPSif available. The affine-Gaussian model of the type used here is attractive as a model for capturing the variation in term premia (departures from the expectations hypothesis); it is reasonable to expect that some of the variation in the term premium in nominal bonds reflect variations in the inflation risk premia. Furthermore, the affine-Gaussian models allow for a flexible factor correlation structure, which is important since the departure from Fisher hypothesis involves acorrelationeffect. 23Notethattheinflationriskpremium,implicitlydefinedbyeq. (7),isthusalsoaffineinthestatevariables. 11

Allriskfactors(statevariables)inourmodelarelatentfactors. Thismaysoundunappealing as latent factors are not easy to interpret economically. While it would be desirable to be able to interpret the yield curve movements macroeconomically, in view of the fact that so little is empirically established about quantities like real yields and the inflation risk premia embedded in nominal yields (as discussed in Section 2.2), in this paper we set a more modest goalof“measuring”them. Thelatentfactorapproachseemsattractiveforthispurpose,asitis moregeneralthanno-arbitragemodelsthatuseobservedmacrovariablesasriskfactors. Kim (2007a) has argued that a substantial part of short-run inflation is unrelated to bond yields; hence the use of realized inflation as a risk factor (e.g., Ang, Bekaert and Wei (2007a) and ChernovandMueller(2007))involvesstrongassumptions. In this paper, we focus on the three-factor (n = 3) case. Note that numerous variables could influence nominal yields, including the instantaneous inflation expectation, π , the real t short rate, rR, variables that affect their future movements (such as a time-varying perceived t inflation target), variables that underlie time variation in the real term premium, the inflation risk premium, and inflation and interest rate uncertainties, among others. However, all these variables are not independent, and we can envision a relatively low-dimensional vector of latent factors summarizing the information in these variables. We do not mean to suggest that three factors are sufficient. Nonetheless, in view of the fact that many of the nominalyields-onlymodelsintheliteraturehavebeenestimatedwith3factors(e.g.,DaiandSingleton (2000), Duffee (2002), Kim and Orphanides (2005)) and that fitting errors for nominal yields are fairly small for n = 3, the three-factor case can be viewed as an important benchmark to be explored. With fewer factors, the cross-section of yields would be fit less well, so that it might become harder to describe actual market movements. With large number of factors, on theotherhand,empiricaldifficulties(e.g.,overfittingconcerns)mayincrease. Traditionally, the real term structure is often modeled as of a lower dimension than the nominal term structure.24 In this paper we let the real term structure have as many factors as the nominal term structure; if the real term structure is truly lower-dimensional than the nominal term structure, we let the data decide on that. A related point is that in a reducedform setup like ours, one cannot make a distinction between the real and the nominal factors, asthecorrelationeffectinageneralmodelmakesuchadistinctionmeaningless. 24Forexample,inthemodelsofChen,Liu,andCheng(2005)andCampbellandViceira(2001),thenominal termstructureisdescribedbya2-factormodel,whiletherealtermstructureisdescribedbya1-factormodel. 12

Whiletheaffine-Gaussianmodeloffersarichandflexibleframeworkfordescribinginflation and the interest rates, it does have a shortcoming, namely that it implies time-invariant interest rate volatilities and inflation uncertainties. Intuitively we would expect that the term premia in nominal and real bonds, as well as the inflation risk premia, woulddepend not only on the price of risk but also on the quantity of risk, but affine-Gaussian models assume the latterisconstant. Nonetheless,affine-Gaussianmodelsmaystillprovideareasonableestimate ofvariousquantitiesofeconomicinterestasdocumentedbyDuffee(2002)andothers,despite thecounterfactualassumptionofconstantyieldvolatility. Itwouldcertainlybedesirabletomodeltime-varyinguncertaintyexplicitly,buttheremay be a greater risk of misspecification as well as implementation difficulties with stochastic volatilitymodels. Furthermore,intheaffinemodelsetting,thereisatrade-offbetweenaflexiblefactorcorrelationstructureandflexiblestochasticvolatilitymodeling,asdiscussedbyDai and Singleton (2000). Modeling of stochastic volatility is also complicated by the debates as to whether and how to incorporate unspanned stochastic volatility (USV) effects.25 Furthermore,asdiscussedinKim(2007a),anexplicitmodelingoftime-varyinginflationuncertainty could be especially challenging, as the short-term and long-term inflation uncertainties may behaveinaqualitativedifferentmanner. Inviewoftheseopenissueswithstochasticvolatility models, we focus on the affine-Gaussian models in this paper, with the presumption that they wouldbeusefulbenchmarkresults,beforethese“moreadvanced”issuesaretackled. 4 Empirical results 4.1 Overall empirical strategy Our model (sketched in Sec. 3.2 and spelled out in Appendix A) can be estimated with or without the TIPS data. If the model is estimated with the nominal yields and inflation but without the indexed-bond data, the resulting model-implied real yields can be viewed as the shadow real yield. This “hypothetical” yield can be useful in many contexts. For example, this is the relevant yield when thinking about the ex ante real interest rates implied by asset pricing models, the majority of which are written in real terms. In the case of pre-1997 U.S., 25Collin-DufresneandGoldstein(2002),LiandZhao(2006),amongothers,arguefortheneedtoincorporate theUSVeffectsinthemodel,butJoslin(2007),Kim(2007b),andothers,argueotherwise. 13

indexedbondsdidnotexist;thereforethe“hypothetical”realyieldsareallonecantalkabout. If indexed bonds are available, as in the U.S. post-1997, one can include them in the estimation, equating the model-implied real yields with the traded indexed-bond yield up to anerrorterm. Thiscanpotentiallyimprovetheefficiencyoftheestimation. Ontheotherhand, shouldthemodel-impliedrealyieldandthetradedindexedbondyielddifferforsomereason, or if there is a failure of relative pricing between the indexed bonds and the nominal bonds, theinclusionofTIPSintheestimationcanleadtopoorerresults. The conditions that the model be (1) Gaussian, (2) linear in some basis, and (3) maximally flexible (econometrically identified), lead to an almost unique specification that we can analyze.26 However, the implementation of the model faces many challenging issues, and the estimates can depend materially on the implementation. In particular, the conventional (Kalman-filter-based) ML estimation of the model, using just nominal yields and inflation (with or without TIPS) data, leads to poor estimates which most practitioners would dismiss immediately. For example, as mentioned briefly in Sec. 2.2, one often obtains long-horizon expectations of inflation and interest rates that are too stable and fixed near the sample means of these variables. Kim and Orphanides (2005), for example, provide Monte Carlo evidence that conventional estimations tend to understate the variability of long-horizon short rate expectationsandoverstatethevariabilityoftermpremia. The problem is that due to the persistence of interest rates and inflation, a typical sample used in the literature (e.g., 15 years’ data) is not long enough, no matter how frequently it is sampled. Basically, we have a “small-sample problem” which manifests itself in two ways: (1) a biased estimate that leads to artificially stable long-horizon expectations, and (2) a very imprecise estimate, arising from the fact that parts of the model that are important for the description of physical (real-world) dynamics of interest rates and inflation are difficult to estimatereliably. Furthermore,fora3-factormodel,thenumberofparameterstobeestimated isalreadyquitelarge,raisingconcernsaboutoverfitting.27 Another difficulty is with the evaluation of the estimated model. Though in-sample and 26We add the term “almost”, because in addition to the popular normalization that has the mean reversion matrixK withrealeigenvalues, onecouldhaveother(inequivalent)normalizationsinwhichonehasthemean reversionmatrixwithcomplexeigenvalues,asdiscussedinAppendixA. 27Ontheotherhand, restrictingthemodelinadhocwaysorusingsimplermodelsrisksstrongassumptions materiallyaffectingthemodeloutput,asdiscussedinSec. 3.2. 14

out-of-sample forecast root-mean-square errors (RMSE) can help detect problematic models, selecting a model (estimate) based on the “smallest RMSE” criterion would be inadvisable: in-sample RMSEs may have been artificially pushed down due to the use of the future information and the especially flexible nature of latent factor models, and out-of-sample RMSEs may have low statistical power especially in view of the considerable volatility of short-run inflation in the recent several years. Testing for a bias in the forecast is also ambiguous: an unbiasedforecasthasbeentraditionallyregardedasdesirableintheacademicliterature,buta mild bias may be a more realistic description of the market expectations in the sample period consideredhere.28 In light of these considerations, rather than relying on a single implementation, we shall explore several different implementations of the model (different options for the data and the auxiliaryconditionsforaddressingthesmall-sampleproblems)andseektoestablishrelatively robust empirical conclusions, focusing on the basic question of whether disparate pieces of inputdatacanbemadeconsistentwithreasonablepriors. 4.2 Data and estimations Weuse3-and6-month,1-,2-,4-,7-,and10-yearnominalyieldsandCPI-UdatafromJanuary 1990 to March 2007. When TIPS yields are used, they cover a shorter period from either January 1999 or January 2005 to March 2007, and are treated as missing observations in the rest of the sample. Both the nominal and the TIPS yields are based on zero-coupon yield curves fitted at the Federal Reserve Board29 and are sampled at the weekly frequency, while CPI-UinflationisavailablemonthlyandassumedtobeobservedonthelastWednesdayofthe current month.30 Due to the complications associated with the shorter-maturity TIPS yields whicharediscussedindetailinAppendixB,onlythe10-year(zero-coupon)TIPSyieldisused in estimations with TIPS data. This focus on the 10-year TIPS yield also reflects our special interest in long-term inflation expectations; as discussed in Kim (2007a), a key information embedded in bond yields is about the “trend component” of inflation, which can be better proxied by long-term, rather than short-term, inflation expectations. Because the model we 28ModelevaluationdifficultiesarefurtherdiscussedinKim(2007a). 29SeeGu¨rkaynak,Sack,andWright(2007a,2007b)fordetails. 30Hereweabstractfromthereal-timedataissuebyassumingthatinvestorscorrectlyinferthecurrentinflation rateinatimelyfashion. 15

estimate does not accommodate seasonality, we use the seasonally-adjusted CPI inflation in the estimate. TIPS are indexed to non-seasonally-adjusted CPI, but our use of seasonally adjustedCPIisnotexpectedtomattermuchforarelativelylongmaturitylike10years. The sample period 1990-2007 was chosen as a compromise between having more data so as to improve the efficiency of estimation, and having a more homogeneous sample so as to avoid possible structural breaks in the relation between term structure variables and inflation (e.g., the 1979-83 episode of Fed’s experiment with reserve targeting). This sample period roughlyspansGreenspan’stenureandalittlebitofBernanke’saswell. When TIPS data are used in the estimation, the zero-coupon TIPS yield, denoted yT , is t,τ takentobe yT = yR +∆R +(cid:178)R , (cid:178)R ∼ N(0,δ2 ). (11) t,τ t,τ τ t,τ t,τ R,τ whereyR isthemodelimpliedrealyield,∆Risanallowanceforaconstantliquiditypremium, t,τ τ and(cid:178)R representsthemeasurement errorsor themodel fittingerrors. Themeasurement error t,τ standard deviation for TIPS, δ , and the constant liquidity premium term, ∆R, are deter- R,τ τ minedinsidetheestimation,asarethemeasurementerrorstandarddeviationδ fornominal N,τ bondyields.31 Toaddresstheaforementionedsmall-sampleandoverfittingproblems,wesupplementthe nominal yields, TIPS yields, and CPI data with survey data on the forecasts of future shortterm (3-month) nominal rates in all estimations reported here.32 We experiment with additionally including the survey forecasts of inflation in the estimation.33 These survey-based forecasts are quite straightforward to incorporate within the Kalman-filter-based maximumlikelihood framework: they are taken as the model’s forecast plus a measurement error, i.e., weassume Esvy(yN ) = E (yN )+(cid:178)F , (cid:178)F ∼ N(0,δ2 ) (12) t t+u,3m t t+u,3m t,u t,u F,u Isvy = I +(cid:178)I , (cid:178)I ∼ N(0,δ2 ), (13) t,τ t,τ t,τ t,τ I,τ wherethesuperscript svy denotessurveyforecasts. 31TheobservednominalyieldsaremodeledasyN +(cid:178)N ,(cid:178)N ∼N(0,δ2 ). t,τ t,τ t,τ N,τ 32Some of the output from the conventional estimation of our model (without the use of any auxiliary data) areavailableuponrequest. KimandOrphanides(2005)presentsomeresultsfromaconventionalestimationof thenominal-yields-onlymodelanddiscusstheirproblematicaspects. 33In the term structure estimation literature, Pennacchi (1991) is the first paper to use survey forecasts of inflation. 16

As in Kim and Orphanides (2005), we use the 6-month- and 12-month-ahead forecasts of the 3-month T-bill yield constructed from Blue Chip Financial Forecast that are available monthly, letting the estimation decide the size of the measurement errors δ ,δ . F,u=6m F,u=12m The long-horizon forecast of the 3-month T-bill yield (available twice a year) is also used, withthemeasurementerrorδ fixedatafairlylargevalueof0.75%atanannualrate. F,long−term For estimations that also incorporate the survey inflation forecast information, we use the survey forecasts of business economists instead of the consumer survey forecasts, for reasons discussed earlier. Specifically, we use the 1- and 10-year inflation forecasts from the SPF. We could let the size of the measurement errors be free variables to be estimated. However, given that our interest is in uncovering information about inflation expectations contained in nominalandTIPSyields,wefixthemeasurementerrorvariabilityatalarge,butnotirrelevant, valueof0.75%atanannualratetoavoidmakingthesurveyinflationforecastinformationtoo influential. This can be viewed in the Bayesian spirit as providing a quasi-informative prior. Insum,weperformtwoversionsofestimationinthispaper: I : δ = ∞, δ = ∞ (14) I,1y I,10y II : δ = 0.75%, δ = 0.75%. (15) I,1y I,10y The measurement error size at ∞ (or at a very large value) for version-I corresponds to not usingthesurveyinflationforecastdata.34 4.3 Estimation results In the rest of the paper we examine the results from five estimation methods with different auxiliary conditions: the version I estimation without TIPS data (denoted NT-I), the version II estimation without TIPS data (denoted NT-II), the version-I estimation with 10-year TIPS yield from 1999 (denoted T99-I), and the version-II estimation with 10-year TIPS yield from 1999(denotedT99-II).ForreasonsthatwillbecomeclearattheendofthissectionandinSec. 5, we also examine an estimation using TIPS data starting from 2005 (denoted T05-II). Table 1providesasummary.35 34Thechoiceof75basispointsforδ ,δ inversion-II,aswellasδ =0.75%,areadmittedly I,1y I,10y F,long−term somewhatarbitrary,butwehaveexperimentedwithotherchoicesandobtainedsimilarresults. 35Inallestimationsinthispaper,inordertofacilitatetheestimationandalsotomaketheresultseasilyreplicablebyothers,weperforma“pre”-estimationwithonlythenominalyieldsandthesurveyforecastsof3-month 17

[insertTable1abouthere] The parameter estimates and the corresponding standard errors from all five estimations are given in Appendix C, while Appendix D provides some discussions on the unconditional momentsofvarioustermstructurevariablesasimpliedbythemodel. 4.3.1 EstimationswithoutTIPS Let us first examine the estimations without the TIPS data (NT-I and NT-II). As can be seen fromFigures2and3,theresultsfromthesetwoestimationsarebroadlysimilar. Forexample, comparing the top left panels in both figures shows that the 10-year inflation expectations in bothestimationstrenddownfromabout4%inearly1990toabout2%bymid2005,pickedup again since then and now average around 2.5%, largely in line with the SPF long-horizon inflationforecast. ThismightnotbesurprisingforNT-II,assurveyinflationforecastswereused in estimation, albeit with an allowance for arguably large deviations from the true forecasts. It is encouraging to see that NT-I also generates a similar downward trend rather than a flat long-term forecast; apparently the survey interest rate forecast information is helping to pin down those parameters that are relevant for describing long-horizon inflation expectations. In bothNT-IandNT-II,the“termspread”betweenthe1-yearand10-yearinflationexpectations isfairlynarrow,afeaturethatisalsosharedwiththesurvey-basedinflationforecasts. [insertFigure2abouthere] TheNT-I-impliedinflationexpectationsaresomewhatlowerthanboththemodelforecasts from NT-II and SPF forecasts, which in turn implies a somewhat better in-sample inflation forecastingperformancebyNT-IrelativetothatofNT-II,sincerealizedinflationtendedtobe low in much of the 1990s. This does not necessarily put the NT-I results in a better light, as sucharesultislikelyduetoalook-aheadbias.36 T-billyieldstoobtainapreliminaryestimateoftheparametersunderlyingthenominalyieldcurvemodel. From theseestimatedparametersandnominalyieldsdata,wecanobtainapreliminaryestimateofx .Fromtheregrest sionofmonthlyinflationontox ,wecanobtainapreliminaryestimateoftheparametersrelatedtotheinflation t dynamics. These estimates are then used as initial parameter guesses in the full (one-step) estimation of all parameters. 36When we compared the out-of-sample inflation forecast performance of NT-I and NT-II in the 1999-2007 period(estimatedwithsamplesfrom1990),theNT-Iestimationresultswerelessstableovertimethanthoseof 18

The model-implied 10-year breakevenrates, defined as the differencebetween the modelimpliednominalandrealyields,areshowninthetoprightpanelsofFigures2and3,together withthe10-yearTIPSbreakevenrate. BasedonbothNT-IandNT-II,thereisaleveldifference between the model-implied and TIPS breakeven rates up to about 2004, with the latter being substantially lower than the former. Nonetheless, the time variation of the two are broadly similar. Forexample,boththemodel-impliedandtheTIPSbreakevenratespeaklocallyatthe beginningof2000,inthemiddleof2001and2002,andsoon,andthescalesoftheirvariation are also similar. Since late 2004, the two series largely move together within 30 basis points fromeachother. [insertFigure3abouthere] The bottom left panels show the inflation risk premia at the 1- and 10-year maturities. In both NT-I and NT-II, the 10-year inflation risk premium is positive and fluctuates in a range of about 50 basis points in the 1990-2007 period. It is also encouraging that for maturities of 1-yearandbelow,theinflationriskpremiumisquitesmall. Themodel-impliedrealyieldsareplottedinthebottomrightpanelsforthe1-and10-year maturities. A comparison across maturities reveals that the model-implied real yield is more variable at the shorter maturity, with changes in the real yield accounting for about 75 (60) percentofthevariationinthenominalyieldatthe1-year(10-year)maturity,whichlendssome support to the usual wisdom that expected inflation affects the longer-term nominal yields to a larger extent (see, for example, Fama (1975) and Mishkin (1981)). The model-implied real yieldisalsohighlycorrelatedwiththe10-yearinflationexpectation. 4.3.2 EstimationswithTIPS Next we look at how the model implications change when TIPS yields are used in the estimation at their face value, up to a constant liquidity premium. Figures 4 and 5 display the corresponding results from the estimation with 10-year TIPS yield data from 1999-2007, eitherwithorwithoutthesurveyinflationforecastdata(T99-IIandT99-I).37 Themodel-implied NT-II,andtheNT-Iforecastingerrorswerelarger. 37The earlier part of the sample for which TIPS data are unavailable (i.e. 1990-98) is viewed as a case of missingdata. 19

inflation expectations from the T99-I estimation, shown in the top left panel of Figure 4, are not very plausible. Few would take seriously the notion that the 10-year inflation expectation has not changed much in the past 15 years. The near constancy of model-implied long-term inflation expectation observed here arises from the tension between the upward trend in the TIPS breakeven rates, shown in the top right panel, and the downward trend in the long-term inflation expectation implied by nominal yields and interest rate forecasts, demonstrated by the NT-I estimation and shown in the top left panel of Figure 2. In addition, the variation in the1-yearinflationexpectationistoolargetobecredible,andthe levelofthe1-yearinflation expectation in early 1990s is too low. Comparison with the survey forecasts makes the last twopointsclear. [insertFigure4abouthere] ThetoprightpanelofFigure4pointstofurtherproblemswiththeT99-Iestimation. While theT99-I-basedbreakevenratematchesthe“level”oftheTIPSbreakevenratebetterthanthe NT-IandNT-IIestimations,itmissestheshort-runtimevariationsintheTIPSbreakevenrates; infactitisalmostconstantovertime.38 Inaddition,themodel-impliedinflationriskpremium is negative and increases over the sample period. This is at odds with the general perception that the inflation risk premium has been historically positive. As discussed in Sec. 2, the inflationrisk premium likelywaspositiveand substantial in the early 1980sand probably has come down since then, whereas we observe almost the opposite behavior in the bottom right panel of Figure 4. The 95% confidence interval for the 10-year inflation risk premium in this estimation is shown in Figure A3 in Appendix C. Even allowing for sampling uncertainties, theresultsstillappearimplausible. The results from the T99-II estimation, which uses survey inflation forecast data, are shown in Figure 5. The model-implied 1-year and 10-year inflation expectations, shown in the top left panel, are now in better accordance with survey forecasts. However, the modelimplied breakeven rate, shown in the top right panel, again misses most of the short-run variability of the TIPS breakeven rate, and the inflation risk premia, shown in the bottom right 38Inviewoftheflexiblenatureoflatent-factormodelusedinthispaper,theremaybeanotherlocalmaximum ofthelikelihoodfunctioninwhichtheTIPSyieldsarefittedbetter,producingaclosermatchbetweenthemodelimplied breakeven rate and the “measured” breakeven rate. However, such a fit would have to come at the expenseofotherfeatures. 20

panel, are even more unreasonable, implying a 10-year inflation risk premium of about -2% in 1990. Even the “short-term” (1-year) inflation risk premium departs a lot from zero, being quitenegativein1990. [insertFigure5abouthere] We have experimented with other auxiliary estimation conditions (e.g., using the 5-year and 7-year TIPS yields in addition to the 10-year TIPS yield, using different measurement error structure for survey forecasts, etc.), but obtained similarly implausible results for inflation expectations and/or inflation risk premia. Basically, when TIPS data from 1999 to 2007 are taken at the face value up to a constant liquidity premium, it seems impossible to obtain sensibleandconsistentresultswithina3-factorno-arbitragemodelsetting.39 As we can see from NT-I and NT-II estimations without TIPS data, the model-implied and TIPS breakeven rates line up reasonably well in the last several years. Thus, we also perform an estimation using the 10-year TIPS yield only from 2005 to the present (T05-II). The results from this estimation are shown in Figure 6. This shorter-sample-TIPS estimation produces more reasonable inflation risk premium estimates and agrees with the NT-I and NT- II estimation results better: The T05-II-implied 10-year breakeven rate (upper right panel of Figure6)nowshowssimilarvariationsastheTIPSbreakevenrate,andcontrastssharplywith thoseofT99-IandT99-IIestimations. The10-yearinflationriskpremiumbasedontheT05-II estimation (lower left panel of Figure 6) is somewhat lower than those from NT-I and NT-II estimations,butitsoverallvariationissimilar. [insertFigure6abouthere] 39Insomesense, thesefindingsmirrorthoseChen, Liu, andCheng(2005), whotakeTIPSatfacevalueand obtain results that are also quite implausible. For example, their estimate of instantaneous inflation rate drops tonear-zerolevelin2001andstaysthereuntilmid-2003. (BesidespotentialproblemswithtakingTIPSatface value,specificationproblemsandotherissuesmayhavealsocontributedtothisresult.) 21

5 Further discussions 5.1 Real yield and inflation risk premium Onecommonfeatureamongthealternativeestimationresultsreportedintheprevioussection isthatthemodel-implied10-yearrealyield(long-termrealyield)movesaroundsubstantially, and more specifically, it mirrors much of the movement in the 10-year nominal yield. This is the case with the 10-year TIPS yield as well. Figure 7 illustrates this point by comparing the 10-year nominal yield with the model-implied 10-year real yields from the NT-II and T05-II estimations(panel(a))andtheTIPS10-yearyield(panel(b)). Thestrongpositivecovariation between the nominal yield and the model-implied real yield (or TIPS yield) works toward reducing the variability of the breakeven inflation (the differential between the nominal yield andtherealyield);evenso,thebreakeveninflationrateismorevariablethanthesurveylongterminflationexpectation,aswehavediscussedinSec. 2.1. [insertFigure7abouthere] Althoughthefeaturethatrealyieldstendtotracknominalyieldsmayappearunremarkable topractitionerswholargelyequatetheTIPSyieldwiththerealyield,thatisnotthecaseinthe extant academic literature. For example, it is harder to see a similarity between the long-term realyieldandthenominalyieldinAng,BekaertandWei(2007a,Figures1and2). Evenmore striking is the result of Chernov and Mueller (2007, Figure 7), in which their preferred model (Model AO) generates a 10-year real yield that stays almost constant during the entire post- 1970 sample period. Note that the real yields from these studies would imply an even more variablebreakeveninflationrate,whichisevenhardertoreconcilewiththesurveyevidence. The term structure model allows us to decompose the real yield into a “real expectations” componentanda“realtermpremium”component,i.e., yR = yR,EH +(yR −yR,EH), (16) t,τ t,τ t,τ t,τ whereyR,EH istheexpectedaveragefutureshortratesoverthelifeofthebond, t,τ (cid:90) t+τ yR,EH = (1/τ) E (rR )ds, (17) t,τ t t+s t 22

while the terms inside the parentheses in eq. (16) represent deviations from the expectations hypotheses. The variance decomposition results, reported in the first two columns of Table 2, show that changes in the expected future short rates account for most of the variation in realyieldsatshortermaturities,whilemuchofthevariationinthe10-yearrealyieldisdueto movementintherealtermpremium. [insertTable2abouthere] Figure 8 plots the decomposition of the 10-year real yield, based on the NT-II and T05- II estimations. In both cases, the real expectations component displays a fairly “stationary” behavior, the variation being mainly the ups and downs associated with business cycles. The real term premium, on the other hand, displays a visible downward trend. This may be due in part to the gradual reduction in risk, manifested as declining uncertainty associated with key macroeconomic variables since the Volcker disinflation, a phenomenon often dubbed the “great moderation.”40 The real term premium component also contains substantial short-run variation. In some sense, this is not surprising: a visual inspection of the 10-year TIPS yield andthe10-yearnominalyieldinthe1999-2007periodshowsalotofsimilarvariationinthem; thus,ifweacceptthatnominalyieldscontainsasubstantialamountoftermpremiumvariation (as indicated by various expectations hypothesis tests), we can expect a similar effect for real yieldsaswell. [insertFigure8abouthere] Let us now turn to the inflation risk premium. It is worth noting that, although our estimates of the inflation risk premium (NT-I, NT-II, T05-II) display interesting variation over time,ourinflationriskpremiumisalessprominentdriverofthe10-yearnominalyieldthanin someotherstudies. Forinstance,ChernovandMueller(2007)’sresultsimplythatmostofthe variation in the 10-year nominal yield is due to the variations in the inflation risk premium, because of the near-constancy of their long-term real yield estimates. Our inflation risk premium estimates also differs substantially from those in Ang, Bekaert and Wei (2007a, Figure 5). In particular, their estimates sometimes show sharp jumps even though no such changes areobservedinthecorrespondingnominalyields(e.g.,early1987,early1990andlate2000). 40SeeBernanke(2004)foradiscussionofthisphenomenonfromapolicymaker’sperspective. 23

Intuitively, the inflation risk premium can be expected to be positively linked to inflation uncertainty. WethereforecorrelateourinflationriskpremiumestimatesfromtheNT-IIandthe T05-II estimations with two measures that could be viewed as proxies for such uncertainties. Our first measure of inflation uncertainty is the dispersion of long-term inflation forecasts, constructedasthedistancebetweentheaverageofthetoptenforecastsandthatofthebottom ten forecasts of CPI inflation 5 to 10 years hence from the Blue Chip Economic Indicators survey, available twice a year since 1987. As can be seen from Figure 9(b), this measure was trendinglowerduringmuchofthe1990sandhasbeenrelativelystablesince2000,suggesting diminished inflation uncertainties in recent years. Inflation risk premium estimates based on the NT-II estimation are highly correlated with this measure with a correlation coefficient of 0.55. The corresponding correlation coefficient based on the T05-II estimates, however, is negative at -0.15 over the full post-1990 sample. A closer inspection reveals that the negative correlation comes primarily from the upward trend in the T05-II inflation risk premium estimates during the early sample period. Using data after 1996, the correlation coefficient becomeshighlypositiveat0.45. Ournextmeasureistheabsoluteimpliedvolatilityofthe10-yearswaptionswithanunderlyingswaptenorofoneyear(Figure9(b)),whichhasbeenavailablesince2000. Thisvariable measures long-run uncertainty associated with future nominal interest rates, and can also be viewed as a rough measure of inflation uncertainty if we are willing to assume that investors are relatively certain about future real yields. Our inflation risk premium estimates are also positivelycorrelatedwiththismeasurewithcorrelationcoefficientsof0.63and0.58,basedon theNT-IIandT05-IIestimations,respectively. [insertFigure9abouthere] Finally, we could assess the relative contribution of each of the three components of the nominalyieldineq. (7)bydecomposingthevarianceofthenominalyieldintoitscovariances withtherealyield,inflationexpectation,andtheinflationriskpremium,respectively: var(yN ) = cov(yN ,yR )+cov(yN ,I )+cov(yN ,℘I ). t,10y t,10y t,10y t,10y t,10y t,10y t,10y The last four columns of Table 2 report the results from this variance decomposition for yield maturitiesof3months,1,2,5and10yearsbasedonin-samplemomentsandtheNT-IImodel estimates.41 Realyieldmovementaccountsforabout3/4ofthevariationattheshorterendof 41ResultsbasedontheT05-IIestimationaresimilar. 24

the yield curve. At longer maturities, real yield variation continues to play a dominant role, although changes in the inflation risk premia become more important. The contribution of expected inflation stays relatively stable at about 30 percent at all maturities.42 Note that the contribution of the inflation risk premium is modest (e.g., 6 percent for 10-year yield), consistent with our earlier remark that the inflation risk premium in our paper is a less prominent driverofnominalyieldsthaninsomeotherstudies. 5.2 TIPS “liquidity premium” The problems we have encountered with the T99 estimations in Sec. 4.3.2 indicate that it is difficulttoequateTIPSyieldswith“hypothetical”realyieldsthatareimplicitinnominalbond pricesandthatthereexistsacomponentinTIPSyieldsthatisnotcapturedbyourmodel. This conclusion is further supported by simple regression evidence.43 A regression of the 10-year breakevenrate(1999-2007)on3-month,2-year,and10-yearnominalyields,i.e., yN −yT = c +c yN +c yN +c yN +e (18) t,10y t,10y o 1 t,3m 2 t,2y 3 t,10y t gives an R2 of only 33%; i.e., a significant part of breakeven rate variation is unexplained by nominalyieldcurvefactors. FromthemorereasonableNTandT05estimationswecanobtainanestimateofthe“TIPSspecificcomponent”. Specifically,wetakethethedifferentialbetweentheTIPSyieldandthe model-impliedrealyield,i.e. L = yT −yR , (19) t,τ t,τ t,τ and explore the interpretation of this variable as a measure of the TIPS liquidity premium, though it could be also reflecting other unaccounted-for effects. We view an examination of this measure as a useful first step before attempting to model TIPS-specific factors explicitly withintheno-arbitrageframework. Figure 10 shows this object for maturities of five and ten years based on the NT-II and T05-IIestimations. Theseliquiditypremiumestimatesarequalitativelysimilaracrossthetwo estimationsandexhibitseveralinterestingfeatures. Boththe5-yearand10-yearliquiditypremia were large until about 2002 and then came down to a level close to zero. The dashed 42However,ifunconditionalvariancesareusedinsteadinthedecomposition,theinflationexpectationcomponenthasahigherweight,becauseofthehighlypersistentnatureoftheinflationexpectationdynamics. 43WethankGregDuffeeforthispoint. 25

lines in the figure mark the “old level” and the “new level”. The 5-year liquidity premium appears more jagged, possibly due in part to the fact that the 5-year TIPS yield is more “contaminated” by the indexation lag effect. The 10-year liquidity premium also contains some relatively high-frequency variations, perhaps reflecting the limitation of our 3-factor model. Still, the large magnitude of L in the early years and the clearly visible trend component t,τ appeartobefairlyrobustfindings.44 [insertFigure10abouthere] To examine the validity of our interpretation of this variable as primarily a measure of TIPSliquiditypremium,weregressL (fromNT-IIandT05-II)ontoasetofvariablesthat t,10y may be related to the liquidity conditions in the TIPS market: the 3-month moving average of the weekly turnover in TIPS (defined as the ratio of TIPS transaction volume over the total amount of TIPS debt outstanding), the implied volatilities derived from options on ten-year Treasury note futures and on the S&P 500 stock index, and the spread between the overnight LIBOR and the effective federal funds rate. The results from these regressions, reported in Table3,showthatL loadssignificantlyonallvariableswithintuitivesigns: thecoefficient t,10y on the turnover is negative (a higher trading turnover implies a more liquid TIPS market and hence a lower liquidity premium), the coefficients on Treasury and S&P implied volatilities are positive (a higher liquidity premium during periods of heightened market uncertainties), and the coefficient on the LIBOR spread is also positive (a higher liquidity premium during periodsofstrainsintheLIBORmarket). Togetherthefourvariablesaccountformorethan80 percentofthevariationsinourliquiditypremiumestimates. [insertTable3abouthere] The weekly turnover measure exhibits the highest explanatory power, with a one percent increase in this variable leading to a 25 and 44 basis point decrease in the liquidity premia. As can be seen in Figure 11, the trading turnover in TIPS remained low up to 2002 and then rose substantially in 2003, the latter coinciding roughly with the decline in the TIPS liquidity premium(Figure10),suggestinganimprovementintheliquidityintheTIPSmarketinrecent 44Meyer and Sack (2005) also find a liquidity premium with similar declining trend (their Chart 5). Their liquiditypremiumissmallerthanoursbecausetheirsetupdoesnotincludeaninflationriskpremium. 26

years.45 We caution, however, that the turnover as a liquidity proxy may be quite imperfect andlikelyaffectedbyotherfactorsaswell. WedonothaveameasureofTIPSbid-askspread that is available regularly enough to be used in our regression. However, a recent survey by the NY Fed finds that the TIPS bid-ask spread has narrowed modestly since its last survey in 2003,consistentwiththedeclineinourmeasureof L .46 t,τ [insertFigure11abouthere] While the qualitative behavior of the TIPS liquidity premium thus seems plausible, it is questionable whether our estimates of L are consistent with rational pricing. One issue is t the magnitude of the TIPS liquidity premium in early years. A comparison with corporate bonds may be useful here. Corporate bonds, including those with the highest credit rating (AAA/AA), tend to trade infrequently, e.g., once a day. TIPS have traded more often than AAA/AA corporate bonds even during the early years when liquidity was poorer; the bidask spread in TIPS has also been substantially smaller than those of corporate bonds. Thus the TIPS liquidity premium should be smaller than the liquidity premium on an AAA/AA corporate bond. A typical magnitude of the AAA/AA spread (over the swap yield or the Treaury yield) is 50 ∼ 100 basis points; the liquidity premium on a AAA/AA bond would be some fraction of that.47 Alternatively, one can also estimate the liquidity premium in the corporate bond by taking the difference between the CDS premium and the bond spread;48 this number also tends to be 50 basis points or less for AAA/AA bonds. Therefore, a TIPS liquiditypremiumexceeding1%intheearlyyears(inFigure10)maybedifficulttoreconcile withtheusualconceptofliquiditypremium(ala AmihudandMendelson(1986)). One possibility is that the TIPS liquidity premium reflects some amount of “mispricing” which took time to get corrected. Such mispricing is not unheard of in financial markets; 45Thedeclineintheliquiditypremiuminthe2003-2004periodmayhavebeenalsohelpedbytheincreased marketattentiontoinflationriskamidaboomingeconomyandrisingoilprices. 46AninformalsurveyofsevenprimarydealersbyNYFedin2007foundthattheTIPSbid-askspreadisabout 1/2-1tickatthetwo-yearmaturity,1tickatthefive-yearmaturity,1-2ticksatthe10-yearmaturity,4-6ticksat the twenty-year maturity, and 6-10 ticks at the thirty-year maturity. A similar survey conducted by NY Fed in 2003 and quoted in Sack and Elsasser (2004) found a TIPS bid-ask spread of 1/2-1 ticks for maturities of five yearsorless,2ticksformaturitiesoffivetotenyears,and4-16ticksformaturitiesbeyondtenyears. Onetick is1/32sofapoint,whereapointroughlyequals1percentofthesecurity’sfacevalue. 47TheAAAyieldmayalsocontainsomeamountofdefaultpremiumandtaxpremium. 48See,e.g.,Longstaffetal(2005). 27

a notable example of securities being mispriced in a relatively new market is the convexity premium mispricing in the swap/eurodollar-futures markets prior to 1995 or so.49 A lesson there is that when a relatively complex security is first introduced, it could be mispriced to some extent and the mispricing might last for some time. The case of the TIPS market bears some similarity to the early swap-eurodollar markets in the sense that TIPS was a new security, and is a fairly complex one involving elaborate calculations of indexed coupon and principal payments with the reference CPIs. Furthermore, a popular belief that TIPS are taxdisadvantageousfortaxableinvestors50 mayhavefurtherdepressedthedemandforTIPS. The results in this section imply a considerable challenge for modeling the TIPS-specific factors along the lines of liquidity premium modeling in the reduced-form defaultable bond pricing literature (e.g., Driessen (2005) and Longstaff, Mithal and Neis (2005)). Note from Figure 10 that in early years not only is L large but also L is. In order to explain 5y,t 10y,t this in terms of the (physical or risk neutral) expectation of a liquidity factor l (e.g., L = t t (cid:82) (1/τ)EQ( t+τ l ds)) it has to follow a unit-root process (or something close to it). However, t t s theunit-root-likedescriptionoftheliquidityprocessmaybeunpalatableonintuitivegrounds, especially if a substantial part of the downward movement seen in Figure 10 reflects a onetimeadjustmentassociatedwiththeinceptionoftheTIPSmarket. AppendixEdiscussesthese issuesfurther. The existing TIPS data is still rather short; more data in the future would certainly help shedmorelightonthesourcesoftheTIPS-specificvariation. 5.3 Interpreting the TIPS breakeven rates We now revisit the question of whether TIPS breakeven rates are too variable and whether they are informative about inflation expectations of the bond market participants. Although the level of TIPS breakeven rates may have been too low in its early years due to nontrivial liquiditypremium,thechangesinTIPSbreakevenratesatweeklyormonthlyfrequenciesmay stillbeinformative,astheadjustmentoftheTIPSyieldstoamorenormallevelmaybeaslow process. Indeed,asdiscussedinSection4,avisualinspectionofthe10-yearmodel-impliedbreakeven 49SeeBurghardtandHoskins(1995)andGuptaandSubrahmanyam(2000)fordetails. 50See,forexample,thediscussioninHeinandMercer(2003). 28

ratesfromtheNTandtheT05estimationsrevealthat,apartfromthedifferentlong-termtrend, theirtimevariationsarequitesimilartothoseofthe10-yearTIPSbreakevenrate;inthissense, TIPSbreakevenratesmightnotbenecessarilybeexcessivelyvolatile. Partofthisvolatilityin TIPS breakeven rates is due to the variation of the inflation risk premium (as seen in the bottom left panels of Figures 3 and 6), which in turn may be linked to time variation in inflation uncertainty(thechangingperceptionofthecredibilityofmonetarypolicy,thenormalcyclical variationinbusinessuncertainties,etc.) andinthepriceofinflationrisk. Figure 12(a) shows in blue circles a scatter plot of the weekly changes in the TIPS-based andtheNT-IImodel-implied10-yearbreakevenrates. Thetwobreakevenrateslineupclosely against the 45-degree line, with a correlation coefficient of 0.76. A regression of the weekly changes in the TIPS breakeven rate onto the weekly changes in the model-implied breakeven rate produces a beta coefficient of 1.06, representing a roughly one-for-one relationship. On the other hand, the weekly changes in the breakeven rate implied by the T05-II model and those in the TIPS breakeven rate, shown in red pluses in Figure 12(a), are somewhat further awayfromtheone-for-onerelationship,withthebetacoefficientinthesameregressionbeing 1.32. Itisalsointerestingtocomparetheweeklychangesinthemodel-implied10-yearinflation expectation with the weekly changes in the 10-year TIPS breakeven rates, plotted in Figure 12(b). BoththeNT-IIandtheT05-IIestimationsrevealafairlyclearpositiverelationbetween theweeklychangesintheTIPSbreakevenrateandthoseinthemodel-impliedinflationexpectation, with the correlation in both cases being about 0.7. A regression of the weekly changes intheTIPSbreakevenrateontotheweeklychangesinthemodel-impliedinflationexpectation gives a beta coefficient of 1.4 and an R2 of 46% for NT-II, and a beta coefficient of 1.3 and an R2 of 50% for T05-II. Thus, the TIPS breakeven rates are informative about the direction of the change in inflation expectations (but somewhat overstating the magnitude), though a substantialpartoftheweeklychangesinTIPSbreakevenratesremainstobeaccountedfor. [insertFigure12abouthere] 29

6 Concluding remarks In this paper we analyze the inflation-related information in nominal and TIPS yields from a “measurement perspective”, i.e., using a flexibly-specified reduced-form model which has little structure beyond the no-arbitrage assumption. Our framework allows for nontrivial term premia and inflation risk premia, as opposed to the use of the expectations hypothesis and the Fisher hypothesis. We were particularly motivated by two questions: whether TIPS“breakevenrates”areinformativeaboutmarketinflationexpectations,andwhetherTIPS yieldsareconsistentwiththeeconomicfundamentals(orwithno-arbitragepricing). We find that, at least from the viewpoint of a flexibly-specified 3-factor term structure model,wecannotreconcileTIPSdatawithreasonablepriorsaboutinflationexpectationsand inflationriskpremia. TIPSyieldsseemtohavebeentoohighintheearlyyears,possiblydueto the newness of the security, the poor liquidity, and some degrees of mispricing. This implies that it may be problematic to use early years’ TIPS data as “no-arbitrage consistent” real yields in other applications. On a more encouraging note, we find that the liquidity premium componentofTIPSyieldshasbecomemuchsmallerinrecentyears,suggestingthattheTIPS informationcanbetakenatitsfacevaluecurrentlyandinthefuture(assumingnodeterioration oftheTIPSmarketliquidityconditionsgoingforward)morethaninthepast. TheanswertothequestionofwhethertheTIPSbreakevenratecanbetakenasinflationexpectationismorecomplicated. Wefindthattheweeklychangesinthemodel-implied10-year inflation expectation tend to line up with the weekly changes in the 10-year TIPS breakeven rate. However, we also find that time variationin the inflation risk premium and the TIPS liquidity premium, the latter of which may also include other unaccounted-for effects, are often significant enough to drive a wedge between the qualitative behavior of the breakeven rates andinflationexpectations. Our findings in this paper provide support for the use of TIPS breakeven rate information as a proxy for inflation expectations, but also provide a justification for caution. Indeed, in speeches that touch on inflation, policy makers often refer to the TIPS breakeven rate, but they also recognize that the interpretation of this measure is complicated by inflation risk premiaandliquidityissuesandthencontinuetomonitoralargenumberofvariablestogauge inflationexpectationsandunderlyinginflationpressures.51 MoredataandmoreworkonTIPS 51SeeBernanke(2007),forexample. 30

modelinginthefuturewillundoubtedlyshedmorelightontheinformationalcontentofTIPS prices. 31

Appendix A Joint no-arbitrage model of inflation and interest rates Then-dimensionalvectorofstatevariablesx = [x ,...,x ](cid:48)followsamultivariateGaussianprocess, t 1t nt dx = K(µ−x )dt+ΣdB , (A-1) t t t where B is an n-dimensional vector of standard Brownian motion, µ is an n-dimensional constant t vector, and K,Σ are n×n constant matrix. The nominal pricing kernel and the price level processes arespecifiedas dMN/MN = −rN(x )dt−λ(x )(cid:48)dB , (A-2) t t t t t dlogQ = π(x )dt+σ(cid:48)dB +σ⊥dB⊥. (A-3) t t q t q t Eq. (A-2) is a standard way of specifying the nominal pricing kernel, which describes nominal interestratesthat followdiffusionprocesses. The nominalterm structurein thispaper isdescribed by the “essentially affine” A (3) specification of Duffee (2002), i.e., the nominal short rate rN(x ) and 0 t themarketpriceofriskλ(x )arespecifiedtobeaffinefunctionsofthestatevariables: t rN(x ) = ρN +ρN(cid:48) x (A-4) t 0 1 t λ(x ) = λN +ΛNx , (A-5) t 0 t where ρN is a constant, ρN and λN are both n-dimensional constant vectors, and ΛN is an n × n 0 1 0 constant matrices. The specification (A-3) is also a standard specification of the diffusion model for inflation,consistingofthe(instantaneous)expectedinflation,π(x ),andthe“unexpectedinflation”(or t theinflationshock),σ(cid:48)dB +σ⊥dB⊥. Theinstantaneousexpectedinflationπ(x )isalsospecifiedas q t q t t anaffinefunctionofthestatevariables,i.e., π(x ) = ρπ +ρπ(cid:48) x . (A-6) t 0 1 t The “unexpected inflation” is allowed to depend on shocks that move the nominal interest rates (or expectedinflation),dB ,andalsoonanorthogonalshock,dB⊥ (i.e.,dB dB⊥ = 0).52 t t t t In some simple cases, for example, as in Campbell and Viceira (2001), the state vector x can be t represented intuitively: x = [π ,rR](cid:48), ρπ = [1,0](cid:48), etc. In general, however, the state variables x ’s t t t 1 it 52The dB⊥ part is included to accommodate short-run inflation shocks that are not spanned by yield curve t movements,discussedinKim(2007a) 32

arestatisticalvariableswhosemeaningofdeterminedonlyimplicitlybythedataonyieldsandinflation. This is similar to “nominal yields only” model of Duffee (2002) and Dai and Singleton (2000); as we shall discuss below, some normalization restrictions have to be imposed to obtain an econometrically identified model. By having a joint model of nominal yields and inflation (eqs. (A-2) and (A-3)), we aregivingmoremeaningtothestatevariables(thanthenominal-yields-onlymodel), thoughtheystill remain“latentvariables.” ApplyingIto’slemmatoeq. (5),wecanderivetherealpricingkernelprocessas dMR/MR = dMN/MN +dQ /Q +(dMN/MN)·(dQ /Q ) (A-7) t t t t t t t t t t = −rR(x )dt−λR(x )(cid:48)dB −(·)dB⊥ (A-8) t t t t where rR(x ) = ρR+ρR(cid:48) x (A-9) t 0 1 t λR(x ) = λR+ΛRx (A-10) t 0 t with 1 ρR = ρN −ρπ − (σ(cid:48)σ +σ⊥2)+λN(cid:48) σ (A-11) 0 0 0 2 q q q 0 q ρR = ρN −ρπ +ΛN(cid:48) σ (A-12) 1 1 1 q λR = λN −σ (A-13) 0 0 q ΛR = ΛN. (A-14) It is straightforward to show that in this formulation, the nominal yields, the real yields and inflation expectations all take the affine form – eq. (8) to (10), where factor loadings aN,bN,aR,bR,aI,bI depend on the basic parameters of the model.53 Specifically, the time-t τ-period nominal and real bondyields,yN andyR ,aregivenbyeqs. (8)and(9)with t,τ t,τ ai = −Ai/τ, bi = −Bi/τ, τ τ τ τ where dAi (cid:161) (cid:162) 1 τ = −ρi +Bi(cid:48) Kµ−Σλi + Bi(cid:48)ΣΣ(cid:48)Bi dτ 0 τ 0 2 τ τ dBi (cid:161) (cid:162) τ = −ρi − K+ΣΛi (cid:48) Bi dτ 1 τ 53SeeDaiandSingleton(2000),forexample. 33

with the initial conditions Ai = 0 and Bi = 0 , for i = N,R. The factor loadings aI and bI for 0 0 n×1 inflationexpectation,I ,ineq. (10)aregivenby t,τ (cid:90) τ aI = ρπ +(1/τ)ρπ(cid:48) ds(I −e−Ks)µ τ 0 1 (cid:90) 0 τ bI = (1/τ) dse−K(cid:48)sρπ, τ 1 0 where eM (M being a square matrix) denotes the matrix exponential. These expressions for factor loadingsai,bi (i = N,R,I)canbefurtherworkedouttoyieldanalyticalexpressions. Tomakethemodeloperational,weneedtoimposesomeidentificationrestrictions(normalizations) toruleout“equivalent”models. Inthispaper,weimposethenormalizationrestriction     0.01 0 0 K 0 0 11     µ = 0 , Σ =  Σ 0.01 0 , K =  0 K 0 , (A-15) 3×1  21   22  Σ Σ 0.01 0 0 K 31 32 33 (otherparametersρN,ρN,λN,ΛN,ρπ,ρπ,σq,σ⊥ remainunrestricted). Itcanbeshownthatanyspec- 0 1 0 0 1 q ificationoftheaffineGaussianmodelthathasaKmatrixwithall-realeigenvaluescanbetransformed totheform(A-15).54 Notethatthisdoesnotexhausttheempiricalpossibilitiesforallaffine-Gaussian models. Forexample,theK matrixcouldcontaincomplexeigenvalues,   K −K 0 a b   K =  K K 0 . (A-16)  b a  0 0 K c However, the normalization (A-15) does cover a large set of possibilities, and is equivalent to those usedinotherstudiessuchasDuffee(2002);thereforewefocusonthiscaseinthispaper.55 B More on the TIPS data ThisappendixisdevotedtoamoredetaileddiscussionontheTIPSdata. FigureA1showsthesmoothed TIPS par yield curves on June 9, 2005 in the top panel and on June 9, 1999 in the bottom panel, togetherwiththeactualtradedTIPSyieldsthatwereusedtocreatethesmoothedTIPSparyieldandzero 54Withnormalization(A-15),thespecificationweestimateinthispapercanbeshowntobeequivalenttothat ofSangvinatsosandWachter(2005). Themaindifferencefromtheirpaperisempirical: theyuseamuchlonger sample(assumingthestabilityofinflation-yieldsrelationship)anddonotusesurveyforecastinformation. 55WehavealsoexaminedtheempiricalcontentsofthespecificationwithKgivenbyeq. (A-16),andobtained similarresults. 34

coupon yield curves. The smoothing is done by assuming that the zero-coupon TIPS yield curve followsthe4-parameterNelson-Siegel(1987)functionalformuptotheendof2003andthe6-parameter Svensson (1994) functional form thereafter,56 and minimizing the fitting error for the actual traded TIPSsecurities. Thesubstantialincreaseinthenumberofpointsinthetoppanelreflectsthegrowthof the TIPS market. Note that in 1999 there is essentially one data point on the curve between the 0 and 5yearsmaturity(correspondingtothe5-yearTIPSissuedin1997),thustheTIPStermstructureinthe short-maturityregionof0-5yearscannotbedeterminedreliably. Withmorepointsacrossthematurity spectrumin2005,shortermaturityTIPSyieldscanbedeterminedmorereliablythanin1999. [insertFigureA1abouthere] Still,theanalysisoftheshort-maturityTIPSarecomplicatedbytheindexationlagandtheseasonality issues. Note that the TIPS payments are indexed to the CPI 2.5 months earlier, thus the TIPS yields containsomeamountofrealizedinflation,oftenreferredtoasthe“carryeffect”. Theyieldthatismore relevanttopolicymakersistheonethattakesoutthisrealizedinflation—theso-calledcarry-adjusted yields—whichcanbeheuristicallyrepresentedas yT,CA = yT +(δ/τ)π , (B-17) t,τ t,τ t−δ,t where π = log(Q /Q )/δ denotes the inflation realized between time t − δ and t, with δ = t−δ,t t t−δ 2.5months.57 Becausetherealizedinflationπ canbequitevolatile,thecarry-unadjustedyieldyT t−δ,t andthecarry-adjustedyieldyT,CAcandiffersignificantly,thoughthedifferencebecomessmallerwith an increasing maturity, due to the δ/τ factor in eq. (B-17). Figure A2 shows the carry-adjusted and theunadjustedTIPSyieldsforthe5-yearand10-yearmaturities. Itcanbeseenthatindeedthe5-year carry-adjustedandunadjustedTIPSyieldsshowgreaterdiscrepanciesthanthe10-yearones. Thishas been particularly the case in 2005, during which large fluctuations in oil prices caused sharp shorttermfluctuationsininflation. Theexpression(B-17)forthecarryadjustmentisonlyaschematicone. The actual carry-adjustment is further complicated by the fact that TIPS is indexed to the seasonallyunadjustedCPI,ratherthantheseasonally-adjustedCPI.Whileonecouldinprincipleexplicitlymodel seasonality(andcarryeffects)withinthedynamicmodelofinflationandtermstructure,suchaproceduremaybemorepronetospecificationerrorsthanthecaseinwhichtheseeffectsarecorrectedatthe inputstage.58 56Incomparison,thezero-couponnominalyieldcurveisassumedtofollowthe6-parameterSvensson(1994) functionalformduringtheentiresampleperiod. InthecaseofTIPS,however,therewerenotenoughsecurities intheearlyyearstopindownasmanyparameters. SeeGu¨rkaynak,Sack,andWright(2007a,2007b)fordetails. 57Notethateq. (B-17)takesouttherealizedinflationintheprevious2.5monthsbutmakesnoadjustmentfor thelackofinflationprotectionduringthelast2.5monthspriortothematuritydate. 58SeeGhysels(1993)foradiscussionoftheSims-Sargentdebatethatbearsonthisissue. 35

[insertFigureA2abouthere] Asnotedinthemaintext,inviewofthereliabilityproblemsandtheindexationlag(carryadjustment) problems for shorter maturity TIPS, in this paper we focus mainly on the 10-year maturity (“longmaturity”) TIPS yield for which the effects of the indexation lag and seasonality are less important. While the analysis of the shorter-maturity TIPS yields is an important problem in itself, it deserves a fuller treatment elsewhere.59 The 10-year (carry-unadjusted) TIPS yield used in the estimation in this paper is viewed as the carry-corrected TIPS yield plus a measurement error (as suggested by eq. (B-17)). C More on the Estimation and the Estimates Here we show how to rewrite the model in a state-space form, which consists of a state equation that describesthedynamicofthestatevariables,andanobservationequationthatdescribestherelationship betweentheobservablevariablesandtheunobservablestatevariables. Whenthetimeintervalhissmall,wecandiscretizethecontinuous-timeequation(A-1)as x = κµh+(I −κh)x +Ση = K +HX +Ση , (C-18) t t−h t t−h t whereη ∼ N(0,hI ). Similarly,thediscretizedprocessforthepricelevelis t 3×3 (cid:161) (cid:162) logQ = logQ + ρπ +ρπ(cid:48)X h+σ(cid:48)η +σ⊥η⊥. (C-19) t t−h 0 1 t−h q t q t Definethestatevectors = [logQ ,x(cid:48)](cid:48). WecanwritetheKalman-filterstateequationsinamatrix t t t form s = G +Γ s +ηs. (C-20) t h h t−h t where       ρπh 1 ρπ(cid:48)h σ(cid:48)η +σ⊥η⊥ G =  0 , Γ =  1  andηs =  q t q t  h h t K 0 H Ση t Ateachtimepoint,weobserveN nominalyieldsandN TIPSyields,bothwithmeasurementer- 1 2 rors. Wealsoobservethepricelevelaswellassurveyforecastsoffuturenominalshortrateorinflation, allatalowerfrequencythanthatoftheyields. WedenotethevectorofnominalyieldsasyN,thevector t ofTIPSyieldsasyT,thevectorofsurveyforecastsofthe3-monthyield(thevectorofEsvy(yN ) t t+u,3m 59Taking a proper account of the seasonality and carry effects is important to TIPS traders, but here in this paperweareconcernedwithmorebasicquestions. 36

for various u’s) as ysvy,N, and the vector of survey inflation forecast as Isvy, respectively. Define the t t observationvectoras z = [logQ ,yN(cid:48) ,yT(cid:48) ,ysvy,N(cid:48) ,Isvy(cid:48) ](cid:48). (C-21) t t t t t t Whenallvariablesareavailable(attimet),wehaveanobservationequationthattakestheform z = a+Fs +ν (C-22) t t t where       0 1 0 0              aN  0 bN   (cid:178)N  t       a =   aR   , F =  0 bR   andν t =   (cid:178)R t         asvy,N 0 bsvy,N (cid:178)svy,N      t  asvy,I 0 bsvy,I (cid:178)svy,I t where ai and bi stack the ai and bi terms for nominal and real yields, for i = N,R, asvy,i and bsvy,i τ τ u collectthefactorloadingsofsurveyforecastsoffuturenominalshortratesandinflation,fori = N,I, and (cid:178)’s are measurement errors. When only a subset of z is available, either due to the less frequent t observations of logQ , ysvy,N or Isvy, or due to the TIPS yields not available or not used in the t t t estimation,theKalmanfilterisrunwiththeavailablesubsetofthedataonly. WeestimatethemodelusingthemaximumlikelihoodmethodwiththeKalmanfilter.60 TableA1 reportstheparameterestimatesandthecorrespondingstandarderrorsfortheNT-I,NT-II,T99-I,T99-II andT05-IIestimations. Theconfidence intervals(standarderrors)forquantities ofeconomicinterest, suchastheinflationriskpremium,canbecomputedusingthedeltamethod. Asanexample,inFigure A3weplotthe95%confidenceintervalsforthe10-yearinflationriskpremiumbasedtheNT-Iandthe T99-Iestimates. Because the state variables in the model are statistical variables that are only defined up to an invariant transformation, individual parameters of the model are not easy to interpret. However, we note that in all estimations in Table A1, there is a diagonal element (eigenvalue) of the K matrix whichisquitesmall(e.g.,min(diag(K))=0.0419forNT-II);thisisanecessaryconditionforthemodel to generate long-horizon expectations that have substantial variation over time. Note also that the size of the nominal yield measurement errors (δ ’s) is quite small (e.g., the 10-year nominal yield N,τ measurement error is about 5 basis points in all five estimations). In other words, the model fits the nominalyieldcurvefairlywell. Lastly,notethatthesizeofthe“orthogonalshock”toinflation(σ⊥)is q quitelarge;thiscouldleadtoamaterialdifferencebetweenthelatent-factormodelofthepresentpaper andmodelsintheliteraturethatdonotaccommodatesuchashock. 60Thex partofthestatevectors = [logQ ,x(cid:48)](cid:48) isstartedfromtheunconditionaldistributionofx ,while t t t t t thelogQ isstartedfromadiffusepriorasitisnonstationary. t 37

[insertTableA1abouthere] [insertFigureA3abouthere] D Term structure of real and nominal yields Table A2 reports the model-implied unconditional moments for the nominal yield, the nominal term premium, the real yield, the real term premium, expected inflation and the inflation risk premium at maturities of 1 quarter, 1-, 2-, 5- and 10 years, based on the NT-II estimation.61 As can be seen from the third column, this model implies a slightly upward sloping nominal term structure, with the mean levels of yields gently rising from 4.31% at the 1-quarter maturity to 4.74% at the 10-year maturity. Themodel-impliedrealtermstructureisessentiallyflat,similartowhatAng,BekaertandWei(2007a) find, implying a real term premium that is close to zero at all maturities. The nominal term premium canthereforebeattributedalmostentirelytoanupwardslopingtermstructureofinflationriskpremia, as shown in the last column, with the point estimates of the inflation risk premia ranging from 10 to 50basis points for maturities up to ten years. The steady-state CPI inflation is estimated to be around 2.4%. Afteradjustingforatypicaldifferenceofaround50basispointsbetweentheCPIandcoreCPI inflationmeasures,thisestimatefallsjustwithinthe1%to2%rangeforcoreCPIinflationcommonly referredtoastheFed’s“comfortzone.” [InsertTableA2abouthere] We do note, however, that these results about unconditional moments should be interpreted with muchcaution: thereisageneralconsensusamongpractitionersandpolicymakersthattheexpectations of inflation and interest rates, even at long horizons, have moved around substantially in the past few decades, which means that the data are close to the unit-root boundary (nonstationarity) where the unconditionalmomentsareill-defined. E Liquidity premium modeling In the corporate bond pricing literature (Duffie and Singleton (1999), Longstaffet al (2005), Driessen (2005)), the liquidity premium component is often modeled via a “liquidity factor” tacked on to the 61TheresultsfromtheNT-IandT05-IIestimationsarequalitativelysimilar. 38

shortrateinthediscountfunction. WecananalogouslytrytoexpresstheTIPSyieldsas (cid:82) y t T ,τ = −(1/τ)logE t Q(e− t t+τ(r s R+ls)ds), (E-23) wherel istheliquidityfactor. IfweassumeasinDriessen(2005)andLongstaffetal(2005)thatl is t t independentoffactorsdrivingtherealshortraterR,wehave t yTIPS = yR +℘L , (E-24) t,τ t,τ t,τ (cid:82) ℘L t,τ = −(1/τ)logE t Q(e− t t+τlsds), (E-25) where℘L istheliquiditypremiumforTIPSofτ maturity. Driessen(2005)andLongstaffetal(2005) t,τ modeltheliquidityfactorasaunivariateprocessundertheQmeasure:62 dl = α(l )dt+β(l )dWQ. (E-26) t t t t ItcanbeseenfromFigure10thatstationaryspecificationsofeq. (E-26)arenotlikelytobeapromising model of the liquidity premium in the historical TIPS yields: the liquidity premia seem to contain a quitelargesecular(nonstationary)component. Anotherpossibilityistoadoptadeterministicprocessforl ,i.e.,l = F(t),suchthattheliquidity t t premium term structure at a certain point in time is matched exactly, analogous to the practice of introducing deterministic components in term structure modeling to fit the current yield curve exactly (see, e.g., HullandWhite(1990)). Forexample, motivatedbytheideathattheliquiditydiscountsthat werelargeattheinceptionoftheTIPSmarketwoulddisappearovertime,onecouldwrite l = c e−c2(t−t0), t > t , c ,c > 0 (E-27) t 1 0 1 2 or l = c (1−tanh(c (t−t −c ))), t > t , c ,c ,c > 0, (E-28) t 1 2 0 3 0 1 2 3 where t is denotes the start date of the TIPS market. In both cases, l would monotonically decay 0 t to 0. However, TIPS liquidity premium implied by these forms are not consistent with observed time variation and term structure of ℘L . For example, assuming a liquidity factor of the form (E-27), the t,τ liquiditypremiumonaτ-yearTIPScanbederivedas (cid:90) t+τ ℘L = (1/τ) dsc e−c2(s−t0), (E-29) t,τ 1 t implyinglittlevariationin℘L forlargeτ,oramoresignificanttermstructure(τ-dependence)of℘L t,τ t,τ thanisseeninFigure10. √ 62Some examples are the CIR model dl = k(θ −l )dt+σ l dWQ and the Vasicek model dl = k(θ − t t t t l )dt+σdWQ. t 39

Anotherwaytointroducedeterministicdynamicsofl istofitthetermstructureof℘ atapoint t t,τ intime,say,att = t∗. Thatcanbedonebysetting ∂ l = (τ ·℘L ), (E-30) t ∂τ t∗,t+τ where℘L isthetermstructureofliquiditypremiumattimet∗. However,thiswouldbeaproblematic t∗,τ descriptionof℘L atanothertime,sayt∗∗,aphenomenonknownasthe“timeinconsistencyproblem.” 40

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Table1: SummaryofEstimations Estimation UsingTIPSyields? Usingsurveyforecastsofinflation? NT-I no no NT-II no yes T99-I since1999 no T99-II since1999 yes T05-II since2005 yes 45

Table2: In-SampleVarianceDecompositionofNominalandRealYields(NT-II) realyield nominalyield maturity realEH realterm real expected inflation (qtr) component premium yield inflation riskpremium 1 1.00 0.00 0.73 0.26 0.01 (0.01) (0.01) (0.00) (0.02) (0.02) 4 0.94 0.06 0.72 0.26 0.02 (0.03) (0.03) (0.00) (0.03) (0.03) 8 0.82 0.18 0.69 0.27 0.03 (0.05) (0.05) (0.00) (0.04) (0.04) 20 0.49 0.51 0.62 0.33 0.04 (0.08) (0.08) (0.00) (0.06) (0.06) 40 0.25 0.75 0.58 0.36 0.06 (0.11) (0.11) (0.00) (0.11) (0.11) Note: This table reports in-sample variancedecompositions of real yields into the realexpectationscomponentandtherealtermpremiumcomponent,andin-samplevariance decompositions of nominal yields into the real yield, expected inflation and the inflation risk premium, all based on NT-II model estimates. Variance decompositions oftherealyieldsarecalculatedaccordingto (cid:179) (cid:180) (cid:179) (cid:180) cov yR ,yR,EH cov yR ,yR −yR,EH t,τ t,τ t,τ t,τ t,τ 1= (cid:161) (cid:162) + (cid:161) (cid:162) , var yR var yR t,τ t,τ whilevariancedecompositionsofthenominalyieldsarecalculatedaccordingto (cid:161) (cid:162) (cid:161) (cid:162) (cid:161) (cid:162) cov yN ,yR cov yN ,I cov yN ,℘I 1= t(cid:161),τ t(cid:162),τ + t(cid:161),τ t (cid:162) ,τ + t(cid:161),τ t(cid:162),τ . var yN var yN var yN t,τ t,τ t,τ Standarderrorscalculatedusingthedeltamethodarereportedinparentheses. 46

Table3: Model-ImpliedTIPSLiquidityPremium L NT-II T05-II t,10y Constant 0.74 -0.16 [13.10] [-2.81] DailyTurnover -0.45 -0.26 [-28.96] [-16.38] Ten-yearimpliedvolatility 0.07 0.09 [9.03] [11.75] S&P500impliedvolatility 0.02 0.03 [11.77] [12.10] Libor-FFR 0.05 0.07 [1.42] [1.72] R2 0.87 0.81 Note: This table regresses model-implied TIPS liquidity premia, L ,fromNT-IIandT05-IIestimationsonmeasuresofliquiditycont,10y ditionsintheTIPSmarket. OLSt-statisticsarereportedinbrackets. 47

TableA1: ParameterEstimates NT-I NT-II T99-I T99-II T05-II StateVariables K 0.8597(0.5198) 0.8169(0.4666) 0.8630(0.1368) 0.8433(0.1555) 0.8720(0.4394) 11 K 0.0481(0.0677) 0.0419(0.0657) 0.0521(0.0810) 0.0488(0.0693) 0.0595(0.0620) 22 K 1.5080(0.8180) 1.4558(0.7827) 1.5175(0.1711) 1.5542(0.3420) 1.6065(0.7212) 33 100·Σ -0.3216(0.3141) -0.4376(0.3257) -0.3385(0.2728) -0.5437(0.2805) -0.2812(0.2602) 21 100·Σ -4.6373(9.5310) -4.7168(9.1773) -4.6357(1.0617) -4.6394(2.7987) -4.2647(6.6484) 31 100·Σ -0.5415(0.2378) -0.5558(0.2465) -0.5649(0.2574) -0.6195(0.2409) -0.5114(0.2162) 32 NominalPricingKernel ρN 0.0414(0.0137) 0.0429(0.0132) 0.0418(0.0138) 0.0438(0.0115) 0.0496(0.0057) 0 ρN 2.9036(5.5599) 3.0242(5.4197) 2.8956(0.5941) 2.9736(1.6467) 2.6107(3.8015) 1,1 ρN 0.4756(0.1130) 0.4747(0.1160) 0.4881(0.1246) 0.5206(0.1144) 0.4771(0.1024) 1,2 ρN 0.6183(0.0396) 0.6216(0.0338) 0.6135(0.0288) 0.6095(0.0258) 0.6090(0.0353) 1,3 λN 0.4459(0.2510) 0.3845(0.2091) 0.4283(0.1844) 0.3593(0.1862) 0.3330(0.3409) 0,1 λN -0.1673(0.8515) -0.2632(0.7712) -0.1785(0.8719) -0.2939(0.8008) -0.7861(0.4900) 0,2 λN 0.0656(3.4014) -0.3082(3.2911) -0.0031(3.4588) -0.4506(2.9862) -1.9781(1.3698) 0,3 [ΣΛN] -0.5324(1.7803) -0.4950(1.6063) -0.5637(0.3480) -0.7460(0.6339) -0.5476(1.4149) 11 [ΣΛN] 1.7443(5.0050) 1.5625(4.3385) 1.7923(1.0253) 2.0343(1.4944) 1.8403(4.0156) 21 [ΣΛN] 3.9083(16.7852) 3.8860(15.8683) 3.9766(2.1698) 4.6583(5.8307) 3.3807(1.5160) 31 [ΣΛN] -0.0368(0.2438) -0.0396(0.2299) -0.0193(0.1014) 0.0772(0.1054) -0.0247(0.2011) 12 [ΣΛN] -0.2734(0.1321) -0.2499(0.1296) -0.2886(0.1053) -0.3761(0.0782) -0.3128(0.1013) 22 [ΣΛN] -0.6572(0.8385) -0.6737(0.8239) -0.7407(0.4363) -1.2661(0.6819) -0.6546(0.7059) 32 [ΣΛN] -0.0684(0.3218) -0.0662(0.2910) -0.0777(0.0935) -0.1527(0.0950) -0.0858(0.2643) 13 [ΣΛN] 0.6062(0.2421) 0.5632(0.2340) 0.6181(0.2161) 0.7250(0.1882) 0.7139(0.2149) 23 [ΣΛN] 0.6822(2.1513) 0.7232(2.0002) 0.6877(0.3571) 0.9391(0.8475) 0.6115(1.7395) 33 Inflation ρπ 0.0232(0.0096) 0.0239(0.0108) 0.0268(0.0048) 0.0250(0.0078) 0.0281(0.0059) 0 ρπ 0.0764(0.6658) 0.0993(0.5811) -1.0700(0.5303) -1.2517(1.1124) -1.3801(2.7343) 1,1 ρπ 0.2786(0.1283) 0.3557(0.0923) 0.1051(0.0925) 0.2942(0.0709) 0.4084(0.0657) 1,2 ρπ -0.0174(0.2035) -0.0081(0.1702) -0.5109(0.1582) -0.5595(0.1642) -0.5348(0.2232) 1,3 σ -0.1136(0.0755) -0.1043(0.0761) -0.1774(0.0675) -0.1515(0.0722) -0.1821(0.0710) q,1 σ 0.0814(0.0767) 0.0896(0.0761) 0.0524(0.0812) 0.0451(0.0810) 0.1410(0.0764) q,2 σ 0.0295(0.0619) 0.0320(0.0589) 0.0038(0.0549) -0.1021(0.0503) 0.0820(0.0591) q,3 σ⊥ 0.7144(0.0241) 0.7168(0.0237) 0.7545(0.0271) 0.7661(0.0285) 0.7368(0.0264) q MeasurementErrorsandLiquidityPremium 100·δ 0.1011(0.0026) 0.1011(0.0026) 0.1009(0.0026) 0.1009(0.0026) 0.1010(0.0026) N,3m 100·δ 0.0220(0.0017) 0.0220(0.0017) 0.0224(0.0017) 0.0223(0.0017) 0.0223(0.0017) N,6m 100·δ 0.0530(0.0016) 0.0530(0.0016) 0.0531(0.0017) 0.0532(0.0017) 0.0530(0.0017) N,1y 100·δ 0 0 0 0 0 N,2y 100·δ 0.0293(0.0012) 0.0293(0.0012) 0.0292(0.0013) 0.0292(0.0013) 0.0293(0.0012) N,4y 100·δ 0 0 0 0 0 N,7y 100·δ 0.0491(0.0019) 0.0491(0.0019) 0.0491(0.0019) 0.0491(0.0019) 0.0491(0.0019) N,10y 100·δ 0.1757(0.0134) 0.1756(0.0133) 0.1757(0.0134) 0.1761(0.0135) 0.1763(0.0136) F,6m 100·δ 0.2259(0.0195) 0.2260(0.0195) 0.2259(0.0196) 0.2255(0.0195) 0.2266(0.0197) F,12m 100·δ 0.3028(0.0132) 0.3036(0.0132) 0.0868(0.0088) R,10y 100·∆R 0.0519(0.2251) -0.1385(0.1846) -0.1292(0.1810) 10y Note: StandarderrorsbasedontheBHHHformulaaregiveninparentheses. 48

TableA2: UnconditionalMomentsofRealandNominalTermStructure(NT-II) maturity nominal nominal real realterm expected inflation (qtr) moment yield termpremium yield premium inflation riskpremium 1 mean 4.31 0.03 1.84 0.02 2.39 0.08 (1.48) (0.23) (0.29) (0.22) (1.07) (0.17) std. dev. 2.03 0.29 0.89 0.27 1.32 0.24 (1.23) (0.20) (0.14) (0.20) (1.00) (0.20) 4 mean 4.33 0.04 1.83 0.01 2.39 0.11 (1.92) (0.66) (0.69) (0.64) (1.07) (0.18) std. dev. 2.52 0.84 1.11 0.78 1.30 0.28 (1.67) (0.60) (0.48) (0.60) (1.02) (0.20) 8 mean 4.32 0.03 1.78 -0.04 2.39 0.15 (2.22) (0.97) (0.98) (0.94) (1.07) (0.18) std. dev. 2.86 1.23 1.36 1.12 1.28 0.31 (1.98) (0.88) (0.80) (0.90) (1.03) (0.19) 20 mean 4.38 0.09 1.72 -0.10 2.39 0.27 (2.53) (1.28) (1.32) (1.29) (1.07) (0.15) std. dev. 3.20 1.65 1.69 1.51 1.21 0.34 (2.32) (1.17) (1.18) (1.27) (1.08) (0.14) 40 mean 4.74 0.45 1.88 0.06 2.39 0.46 (2.51) (1.27) (1.40) (1.37) (1.07) (0.06) std. dev. 3.16 1.74 1.76 1.59 1.09 0.32 (2.32) (1.10) (1.28) (1.35) (1.12) (0.14) Note:Thistablereportstheunconditionalmeanandstandarddeviationsofnominalyields,nominaltermpremium,realyields,realtermpremia,expectedinflationandinflationriskpremiabased onNT-IImodelestimates. Standarderrorscalculatedusingthedeltamethodarereportedinparentheses. 49

(a) Survey−based inflation forecasts and TIPS breakeven rates 3.5 3 2.5 2 Michigan survey 1.5 SPF survey TIPS breakeven 1 1999 2000 2001 2002 2003 2004 2005 2006 2007 (b) Nominal yields and realized inflation 15 1−year yield 10−year yield annual inflation SPF forecast 10 5 0 1980 1985 1990 1995 2000 2005 Note: The top panel plots the 10-year TIPS breakeven rate (red line), long-horizon Michigan inflation forecast(blueline),and10-yearSPFinflationforecast(blackpluses).Thebottompanelplotsthe1-year(thin blueline)and10-year(thickblueline)nominalyields,togetherwiththerealizedannualinflation(redline) andthecorrespondingSPFforecast(blackpluses). Figure1: NominalandTIPSYieldsandInflation 50

(a) Inflation expectations (b) Ten−year breakeven rates 5 5 Model 1−year TIPS 4.5 Model 10−year 4.5 Model SPF 1−year 4 4 SPF 10−year 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 1990 1993 1996 1999 2002 2005 1990 1993 1996 1999 2002 2005 (c) Inflation risk premiums (d) Real yields 1.5 8 1−year 1 10−year 6 0.5 4 0 −0.5 2 −1 0 −1.5 1−year 10−year −2 −2 1990 1993 1996 1999 2002 2005 1990 1993 1996 1999 2002 2005 Note: ResultsfromtheNT-Iestimation: (a)Model-implied10-year(thickblueline)and1-year(thinred line)inflationexpectation.The10-yearand1-yearSPFsurveyinflationforecastsareshowninbluecirclesand redplussigns,respectively.(b)Model-implied10-yearbreakevenrate(thinredline)andTIPSbreakevenrate (thickblueline). (c)Model-implied1-year(thinredline)and10-year(thickblueline)inflationriskpremia. (d)Model-implied1-year(thinredline)and10-year(thickblueline)realyields. Figure2: ResultsfromtheNT-IEstimation 51

(a) Inflation expectations (b) Ten−year breakeven rates 5 5 Model 1−year TIPS 4.5 Model 10−year 4.5 Model SPF 1−year 4 4 SPF 10−year 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 1990 1993 1996 1999 2002 2005 1990 1993 1996 1999 2002 2005 (c) Inflation risk premiums (d) Real yields 1.5 8 1−year 1 10−year 6 0.5 4 0 −0.5 2 −1 0 −1.5 1−year 10−year −2 −2 1990 1993 1996 1999 2002 2005 1990 1993 1996 1999 2002 2005 Note: ResultsfromtheNT-IIestimation: (a)Model-implied10-year(thickblueline)and1-year(thinred line)inflationexpectation.The10-yearand1-yearSPFsurveyinflationforecastsareshowninbluecirclesand redplussigns,respectively.(b)Model-implied10-yearbreakevenrate(thinredline)andTIPSbreakevenrate (thickblueline). (c)Model-implied1-year(thinredline)and10-year(thickblueline)inflationriskpremia. (d)Model-implied1-year(thinredline)and10-year(thickblueline)realyields. Figure3: ResultsfromtheNT-IIEstimation 52

(a) Inflation expectations (b) Ten−year breakeven rates 5 5 Model 1−year TIPS 4.5 Model 10−year 4.5 Model SPF 1−year 4 4 SPF 10−year 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 1990 1993 1996 1999 2002 2005 1990 1993 1996 1999 2002 2005 (c) Inflation risk premiums (d) Real yields 1.5 8 1−year 1 10−year 6 0.5 4 0 −0.5 2 −1 0 −1.5 1−year 10−year −2 −2 1990 1993 1996 1999 2002 2005 1990 1993 1996 1999 2002 2005 Note: ResultsfromtheT99-Iestimation: (a)Model-implied10-year(thickblueline)and1-year(thinred line)inflationexpectation.The10-yearand1-yearSPFsurveyinflationforecastsareshowninbluecirclesand redplussigns,respectively.(b)Model-implied10-yearbreakevenrate(thinredline)andTIPSbreakevenrate (thickblueline). (c)Model-implied1-year(thinredline)and10-year(thickblueline)inflationriskpremia. (d)Model-implied1-year(thinredline)and10-year(thickblueline)realyields. Figure4: ResultsfromtheT99-IEstimation 53

(a) Inflation expectations (b) Ten−year breakeven rates 5 5 Model 1−year TIPS 4.5 Model 10−year 4.5 Model SPF 1−year 4 4 SPF 10−year 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 1990 1993 1996 1999 2002 2005 1990 1993 1996 1999 2002 2005 (c) Inflation risk premiums (d) Real yields 1.5 8 1−year 1 10−year 6 0.5 4 0 −0.5 2 −1 0 −1.5 1−year 10−year −2 −2 1990 1993 1996 1999 2002 2005 1990 1993 1996 1999 2002 2005 Note: Results from the T99-II estimation: (a) Model-implied 10-year (thick blue line) and 1-year (thin red line) inflation expectation. The 10-year and 1-year SPF survey inflation forecasts are shown in blue circles and red plus signs, respectively. (b) Model-implied 10-year breakeven rate (thin red line) and TIPS breakevenrate(thickblueline).(c)Model-implied1-year(thinredline)and10-year(thickblueline)inflation riskpremia. (d)Model-implied1-year(thinredline)and10-year(thickblueline)realyields. Figure5: ResultsfromtheT99-IIEstimation 54

(a) Inflation expectations (b) Ten−year breakeven rates 5 5 Model 1−year TIPS 4.5 Model 10−year 4.5 Model SPF 1−year 4 4 SPF 10−year 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 1990 1993 1996 1999 2002 2005 1990 1993 1996 1999 2002 2005 (c) Inflation risk premiums (d) Real yields 1.5 8 1−year 1 10−year 6 0.5 4 0 −0.5 2 −1 0 −1.5 1−year 10−year −2 −2 1990 1993 1996 1999 2002 2005 1990 1993 1996 1999 2002 2005 Note: Results from the T05-II estimation: (a) Model-implied 10-year (thick blue line) and 1-year (thin red line) inflation expectation. The 10-year and 1-year SPF survey inflation forecasts are shown in blue circles and red plus signs, respectively. (b) Model-implied 10-year breakeven rate (thin red line) and TIPS breakevenrate(thickblueline).(c)Model-implied1-year(thinredline)and10-year(thickblueline)inflation riskpremia. (d)Model-implied1-year(thinredline)and10-year(thickblueline)realyields. Figure6: ResultsfromtheT05-IIEstimation 55

(a) Ten−year nominal and model−implied real yields (b) Ten−year nominal and TIPS yields 10 10 Nominal Nominal Real (NT−II) TIPS 8 Real (T05−II) 8 6 6 4 4 2 2 0 0 1999 2002 2005 1999 2002 2005 Note: (a) The red line plots the actual 10-year nominal yield, and the blue solid (dashed) line plots the implied 10-year real yield based on Model NT-II (T05-II). (b) The red and the blue lines plot the 10-year nominalandTIPSyields,respectively. Figure7: Ten-YearNominalandRealYields 56

4 EH component (NT−II) TP component (NT−II) 3.5 EH component (T05−II) TP component (T05−II) 3 2.5 2 1.5 1 0.5 0 −0.5 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Note: Decompositionofthe10-yearrealyieldintotherealexpectationscomponent(solidlines)andthe realtermpremiumcomponent(dashedlines). TheredlinesarebasedontheNT-IIestimation,andtheblue linesarebasedontheT05-IIestimation. Figure8: DecompositionofTen-yearRealYield 57

(a) Model−implied 10−year inflation risk premiums 1.5 NT−II T05−II 1 0.5 0 −0.5 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 (b) Measures of inflation uncertainty 2.5 1.4 Blue Chip inflation uncertainy (left axis) Swaption implied volatility (right axis) 2 1.2 1.5 1 1 0.5 0.8 0 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Note: (a) Model-implied 10-year inflation risk premia based on NT-II (blue line) and T05-II (red line) estimations. (b)Theblueasterisksplotthedistancebetweentheaverageofthetoptenforecastsandthatof thebottomtenforecastsofCPIinflation5to10yearsaheadfromBlueChipEconomicIndicatorsurvey. The redlineplotsthebasis-pointimpliedvolatility(absoluteimpliedvolatility)from10-yearswaptionswithan underlyingswaplengthof1year. Figure9: InflationRiskPremiumandMeasuresofInflationUncertainty 58

2.5 10−year (a) 2 5−year 1.5 1 0.5 0 −0.5 −1 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2.5 10−year (a) 2 5−year 1.5 1 0.5 0 −0.5 −1 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 Note: (a) NT-II, (b) T05-II. The black (blue) line plots the model-implied 5-year (10-year) liquidity premium. Figure10: Model-ImpliedLiquidityPremium 59

30 25 20 15 10 5 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 Note: inmillionsofdollarsdividedbyTIPSoutstandinginthecorrespondingmonth. Figure11: WeeklyturnoverofTIPS 60

0.2 0.1 0 −0.1 −0.2 −0.2 −0.1 0 0.1 0.2 nevekaerb SPIT (a) 0.2 0.1 0 −0.1 −0.2 −0.2 −0.1 0 0.1 0.2 model breakeven nevekaerb SPIT (b) model inflation expectation Note: (a)ScatterplotsofTIPS10-yearbreakevenrateagainsttheNT-II(circles)andtheT-05-II(pluses) model-implied 10-year breakeven rates. (b) Scatter plots of TIPS 10-year breakeven rate against the NT-II (circles)andtheT05-II(pluses)model-implied10-yearinflationexpectations. Figure12: TIPSandModel-ImpliedBreakevenRates 61

Note: Thetop(bottom)panelplotsthefittedTIPSparyieldcurvetogetherwithindividualTIPSyields onJune9,2005(June9,1999). FigureA1: TIPSYieldCurves 62

3 2.5 2 1.5 10−year C−UA 1 10−year C−A 5−year C−UA 5−year C−A 0.5 2004 2004.5 2005 2005.5 2006 2006.5 2007 2007.5 Note: This figure plots 10-year carry-unadjusted (carry-adjusted) TIPS yields in red solid(blackdashed)lineand5-yearcarry-unadjusted(carry-adjusted)TIPSyieldsinblue solid(graydashed)line. FigureA2: TIPSYieldswithandwithoutCarryAdjustment 63

(a) NT−I 3 2 1 0 −1 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 (b) T−I99 1 0 −1 −2 −3 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Note:Thetop(bottom)panelplotsthe10-yearinflationriskpremiumtogetherwiththe 95%confidencebandsasimpliedbytheNT-I(T99-I)estimation. FigureA3: Ten-YearInflationRiskPremia 64