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A New Perspective on Gaussian Dynamic Term Structure Models

ANew Perspective on Gaussian DynamicTerm Structure ModelsScottJoslinMIT Sloan School of ManagementKenneth J. SingletonGraduate School of Business, Stanford University, and NBERH aoxiang ZhuGraduate School of Business, Stanford UniversityInany canonical Gaussian Dynamic term Structure model (GDTSM), the conditional fore-casts of the pricing factors are invariant to the imposition of no- arbitrage restrictions. Thisinvariance is maintained even in the presence of a variety of restrictions on the factorstructure of bond yields. To establish these results, we develop a novel canonicalGDTSMin which the pricing factors are observable portfolios of yields. For our normalization,standard maximum likelihood algorithms converge to the global optimum almost instanta-neously.

A New Perspective on Gaussian Dynamic Term Structure Models We show that, within any canonical GDTSM and for any sample of bond yields, imposing no-arbitrage does not affect the conditionalPexpectation of

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1 ANew Perspective on Gaussian DynamicTerm Structure ModelsScottJoslinMIT Sloan School of ManagementKenneth J. SingletonGraduate School of Business, Stanford University, and NBERH aoxiang ZhuGraduate School of Business, Stanford UniversityInany canonical Gaussian Dynamic term Structure model (GDTSM), the conditional fore-casts of the pricing factors are invariant to the imposition of no- arbitrage restrictions. Thisinvariance is maintained even in the presence of a variety of restrictions on the factorstructure of bond yields. To establish these results, we develop a novel canonicalGDTSMin which the pricing factors are observable portfolios of yields. For our normalization,standard maximum likelihood algorithms converge to the global optimum almost instanta-neously.

2 We present empirical estimates and out-of-sample forecasts for severalGDTSM susing data on Treasury bond yields. (JELE43, G12, C13)Dynamicmodels of the term Structure often posit a linear factor Structure for acollection of yields, with these yields related to underlying factorsPthrougha no- arbitrage relationship. Does the imposition of no- arbitrage in a Gaussiandynamic term Structure model (GDTSM) improve the out-of-sample forecastsof yields relative to those from the unconstrained factor model, or sharpenmodel-implied estimates of expected excess returns? In practice, the answersto these questions are obscured by the imposition of over-identifying restric-tions on the risk-neutral (Q) or historical (P) distributions of the risk factors,or on their market prices of risk, in addition to the cross-maturity restrictionsimplied by are grateful for helpful comments from Greg Duffee, James Hamilton, Monika Piazzesi, Pietro Veronesi(the Editor), an anonymous referee, and seminar participants at the AFA annual meeting, MIT, the New YorkFederal Reserve Bank, and Stanford.

3 Send correspondence to Scott Joslin, Assistant Professor of Finance,MIT Sloan School of Management E62-639, Cambridge, MA 02142-1347; telephone: (617) : that explore the forecasting performance ofGDTSMs includeDuffee(2002),Ang and Piazzesi(2003),Christensen, Diebold, and Rudebusch(2007),Chernov and Mueller(2008), andJardet, Monfort, andPegoraro(2009), among many The Author 2011. Published by Oxford University Press on behalf of The Society for Financial rights reserved. For Permissions, please e-mail: Access publication January 4, 2011 at Stanford Business Library on June 7, from ANew Perspective on Gaussian Dynamic Term Structure ModelsWe show that,within any canonical GDTSM and for any sample of bondyields, imposing no- arbitrage does not affect the conditionalPexpectation ofP, EP[Pt|Pt 1].

4 GDTSM-implied forecasts ofPare thus identical to thosefrom the unrestricted vector-autoregressive (VAR) model forP. To establishthese results, we develop an all-encompassing canonical model in which thepricing factorsPare linear combinations of the collection of yieldsy(suchas the firstNprincipal components (PCs))2andin which these yield fac-tors follow an unrestrictedVAR. Within our canonicalGDTSM, as long asPis measured without error, unconstrained ordinary least squares (OLS) givesthe maximum likelihood (ML) estimates ofEP[Pt|Pt 1].Therefore, enforcingno- arbitrage has no effect on out-of-sample forecasts ofP. This result holdsforanyother canonicalGDTSM, owing to observational equivalence (Dai andSingleton 2000) and, as such, is a generic feature , under the assumption that the yield factorsPare observedwithout error, these propositions follow from the factorization of the condi-tional density ofyinto the product of the conditionalPdensity ofPtimes theconditional density of measurement ofPis determinedby parameters controlling its conditional mean and its innovation covariancematrix.

5 The measurement error density is determined by the no- arbitrage cross-sectional relationship among the yields. We show thatGDTSMs can beparameterized so that the parameters governing thePforecasts ofPdo notappear in the measurement-error density. Given this separation, the only linkbetween the conditionalPdensity and the measurement density is the covari-ance of the innovations. However, a classic result ofZellner(1962) implies thattheMLestimates ofEP[Pt|Pt 1]are independent of this covariance. Conse-quently,OLSrecovers theMLestimates ofEP[Pt|Pt 1]and the no-arbitragerestriction is irrelevant for the conditionalPforecast to seeing this irrelevance is our choice of canonical anyN-factor model with portfolios of yieldsPas factors, bond prices depend on theN(N+1)parameters governing the risk-neutral conditional mean ofPand the(N+1)parameters linking the short rate toP, for a total of(N+1) all of these parameters are free, however, because internal consistencyrequires that the model-implied yields reproduce the yield-factorsP.

6 We showthat, given theNyield factors, the entire time-tyield curve can be constructedby specifying (a)rQ ,the long-run mean of the short rate underQ; (b) Q,thespeeds of mean reversion of the yield-factors underQ; and (c) P,the2 Althoughstandard formulations of affine term Structure Models use latent (unobservable) risk factors ( ,Daiand Singleton 2000, Duffee 2002), byDuffie and Kan(1996) we are free to normalize a model so that the factorsare portfolios of yields on bonds and we ,for example,Chen and Scott(1993) andPearson and Sun(1994).4To emphasize, our canonical form is key toseeingthe result; due to observational equivalence, the result holdsforanycanonical at Stanford Business Library on June 7, from TheReview of Financial Studies / v 24 n 3 2011conditionalcovariance matrix of yields factors from theVAR.

7 That is, given P,the entire cross-section of bond yields in anN-factorGDTSMis fully de-termined by only theN+1 parametersrQ and ,(rQ , Q, P)canbe efficiently estimated independently of thePconditional mean ofPt,renderingno- arbitrage irrelevant for these results in place, we proceed to show that the conditional fore-castEP[Pt|Pt 1]from a no-arbitrageGDTSM remains identical to its coun-terpart from an unrestrictedVAReven in the presence of a large class ofover-identifying restrictions on the factor Structure ofy. In particular,regard-less of the constraints imposed on the risk-neutral distribution of the yield-factorsP, the GDTSM- and VAR-implied forecasts of these factors differently,OLSrecovers the conditional forecasts of the yieldfactors even in the presence of further cross-sectional restrictions on the shapeof the yield curve beyond does the Structure of aGDTSM improve out-of-sample forecasts ofP?

8 We show that if constraints are imposed directly on thePdistribution ofPwithin a no-arbitrageGDTSM, then theMLestimate ofEP[Pt|Pt 1]is moreefficient than itsOLScounterpart from aVAR. Thus, our theoretical results,as well as subsequent empirical illustrations, show that gains from forecast-ing using aGDTSM, if any, must come from auxiliary constraints on thePdistribution ofP, and not from the no- arbitrage restrictionper example of such auxiliary constraints is the number of riskfactors that determine risk premiums. Motivated by the descriptive analysis ofCochrane and Piazzesi(2005,2008) andDuffee(2008), we develop methodsfor restricting expected excess returns to lie in a space of dimensionL(<N),without restricting a priori which of the N factorsPtrepresent priced <N, then there are necessarily restrictions linking the historical andrisk-neutral drifts this case, the forecasts of future yields implied byaGDTSMare in principle different than those from an unrestrictedVAR, andwe investigate the empirical relevance of these constraints within three-factor(N=3) , we show that our canonical form allows for the computa-tionally efficient estimation ofGDTSMs.

9 The conditional density of observedyields is fully characterized byrQ and Q,as well as the parameters con-trolling any measurement errors in yields. Importantly,(rQ , Q)constitutesalow-dimensional, rotation-invariant (and thus economically meaningful) pa-rameter space. Using standard search algorithms, we obtain near-instantaneousconvergence to the global optimum of the likelihood function. Convergence is5 Thoughone might conclude from reading the recent literature that enforcing no- arbitrage improves out-of-sample forecasts of bond yields, our theorems show that this is not the case. What underlies any documentedforecast gains in these studies from usingGDTSMs is the combined Structure of no-arbitrageandthe auxiliaryrestrictions they impose on thePdistribution at Stanford Business Library on June 7, from ANew Perspective on Gaussian Dynamic Term Structure Modelsfast regardless of the number of risk factors or bond yields used in estimation,or whether the pricing factorsPare measured with convergence to global optima using our canonicalGDTSM makes itfeasible to explore rolling out-of-sample forecasts.

10 For a variety ofGDTSMs with and without measurement error in yield factors, and with and withoutconstraints on the dimensionalityLof risk premia we compare the out-of-sample forecasting performance relative to a benchmark unconstrainedVAR,and confirm our theoretical predictions in the A CanonicalGDTSM withObservableRisk FactorsIn this section, we develop our JSZ canonical representation this end, we start with a generic representation of aGDTSM, in whichthe discrete-time evolution of the risk factors (state vector)Xt2 RNisgov-erned by the following equations:71Xt=KP0X+KP1 XXt 1+ X Pt,(1)1Xt=KQ0X+KQ1 XXt 1+ X Qt,(2)rt= 0X+ 1X Xt,(3)wherertisthe one-period spot interest rate, X X0isthe conditional covari-ance matrix ofXt,and Pt, Qt N(0,IN).


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