No-Arbitrage Taylor Rules∗ Andrew Ang† Columbia University and NBER Sen Dong Lehman Brothers Monika Piazzesi‡ University of Chicago, FRB Minneapolis, NBER and CEPR September 2007 ∗WethankRuslanBikbov,SebastienBlais,DaveChapman,MikeChernov,JohnCochrane,Charlie Evans, Michael Johannes, Andy Levin, David Marshall, Thomas Philippon, Tom Sargent, Martin Schneider,GeorgeTauchen,andJohnTaylorforhelpfuldiscussions. WeespeciallythankBobHodrick for providing detailed comments. We also thank seminar participants at the American Economics Association,AmericanFinanceAssociation,aCEPRFinancialEconomicsmeeting,theCEPRSummer Institute, the European Central Bank Conference on Macro-Finance, the Federal Reserve Bank of San Francisco Conference on Fiscal and Monetary Policy, an NBER Monetary Economics meeting, the Society of Economic Dynamics, the Western Finance Association, the World Congress of the Econometric Society, Bank of Canada, Carnegie Mellon University, Columbia University, European CentralBank,FederalReserveBoardofGovernors,LehmanBrothers,MorganStanley,PIMCO,andthe UniversityofSouthernCaliforniaforcomments. AndrewAngandMonikaPiazzesibothacknowledge financial support from the National Science Foundation. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. †ColumbiaBusinessSchool,3022Broadway805Uris,NewYork,NY10027;ph: (212)854-9154; fax: (212)662-8474;email: [email protected];WWW:http://www.columbia.edu/∼aa610 ‡University of Chicago, Graduate School of Business, 5807 S. Woodlawn, Chicago, IL 60637; ph: (773) 834-3199; email: [email protected]; WWW: http://faculty.chicagogsb.edu/monika. piazzesi/research/ No-Arbitrage Taylor Rules Abstract We estimate Taylor (1993) rules and identify monetary policy shocks using no-arbitrage pricing techniques. Long-term interest rates are risk-adjusted expected values of future short rates and thus provide strong over-identifying restrictions about the policy rule used by the Federal Reserve. The no-arbitrage framework also accommodates backward-looking and forward-looking Taylor rules. We find that inflation and output gap account for over half of the variation of time-varying excess bond returns and most of the movements in the term spread. Taylor rules estimated with no-arbitrage restrictions differ significantly from Taylor rules estimated by OLS, and monetary policy shocks identified with no-arbitrage techniques arelessvolatilethantheirOLScounterparts. 1 Introduction Mostcentralbanks,includingtheU.S.FederalReserve(Fed),conductmonetarypolicytoonly influence the short end of the yield curve. However, the entire yield curve responds to the actions of the Fed because long interest rates are conditional expected values of future short rates,afteradjustingforriskpremia. Theserisk-adjustedexpectationsoflongyieldsareformed basedonaviewofhowtheFedconductsmonetarypolicy. Thus,thewholeyieldcurvereflects the monetary actions of the Fed, so the entire term structure of interest rates can be used to estimatemonetarypolicyrulesandextractestimatesofmonetarypolicyshocks. According to the Taylor (1993) rule, the Fed sets the short-term interest rate by reacting to CPIinflationandtheoutputgap. Toexploitthecross-equationrestrictionsonyieldmovements impliedbytheassumptionofnoarbitrage,weplacetheTaylorruleintoatermstructuremodel. The no-arbitrage assumption is reasonable in a world of large investment banks and active hedge funds, who takepositions eliminating arbitrage opportunities arising in bond prices that are inconsistent with each other in either the cross-section or their expected movements over time. Moreover, the absence of arbitrage is a necessary condition for an equilibrium in most macroeconomic models. Imposing no arbitrage, therefore, can be viewed as a useful first step towardsafullyspecifiedgeneralequilibriummodel. We describe expectations of future short rates by versions of the Taylor rule and a Vector Autoregression (VAR) for macroeconomic variables. Following the approach taken in many papersinmacro(see,forexample,FuhrerandMoore,1995),wecouldinferthevaluesoflong yieldsfromtheseexpectationsbyimposingtheExpectationsHypothesis(EH).However,there is strong empirical evidence against the EH (see, for example, Campbell and Shiller, 1991; Cochrane and Piazzesi, 2005, among many others). Term structure models can account for deviationsfromtheEHbyexplicitlyincorporatingtime-varyingriskpremia(see,forexample, DaiandSingleton,2002). We present a setup that embeds Taylor rules in an affine term structure model with time- varying risk premia. The structure accommodates standard Taylor rules, backward-looking Taylor rules that allow multiple lags of inflation and output gap to influence the actions of the Fed (for example, Eichenbaum and Evans, 1995; Christiano, Eichenbaum and Evans, 1996), and forward-looking Taylor rules where the Fed responds to anticipated inflation and output gap (Clarida, Gal´ı and Gertler, 2000). The framework also accommodates monetary policy shocks that are serially correlated but uncorrelated with macro factors. The model specifies standard VAR dynamics for the macro indicators, inflation and output gap, together with an 1 additionallatentfactorthatdrivesinterestrates. Thislatentfactorcapturesothermovementsin yields that may be correlated with inflation and output gap, including monetary policy shocks. Ourframeworkalsoallowsriskpremiatodependonthestateofthemacroeconomy. Bycombining no-arbitrage pricing with the Fed’spolicyrule, we extractinformation from the entire term structure about monetary policy, and vice versa, use our knowledge about monetarypolicytomodelthetermstructureofinterestrates. Themodelallowsustoefficiently measure how different yields respond to systematic changes in monetary policy, and how they respond to unsystematic policy shocks. Interestingly, the model implies that a large amount of interest rate volatility is explained by systematic changes in policy that can be traced back to movements in macro variables. For example, 74% of the variance of the 1-quarter yield and 66% of the variance of the 5-year yield can be attributed to movements in inflation and the output gap. Over 78% of the variance of the 5-year term spread is due to time-varying output gap and output gap risk. The estimated model also captures the counter-cyclical properties of time-varyingexpectedexcessreturnsonbonds. WeestimateTaylorrulesfollowingthelargemacroliteraturethatuseslowfrequencies(we usequarterlydata)atwhichtheoutputgapandinflationarereported. Underthecross-equation restrictions for yields implied by the no-arbitrage model, we estimate a flexible specification for the macro and latent factors. This setup offers a natural solution to the usual identification probleminVARdynamicsthatcontainfinancialdata,suchasbondyields(forexample,Evans andMarshall1998,2001;Piazzesi2005). TheFed’sendogenouspolicyreactionsaredescribed by the Taylor rule as movements in the short rate which can be traced to movements in the macro variables that enter the rule: inflation and output. While the Fed may take current yield dataintoaccount,itdoessoonlybecausecurrentyieldscontaininformationaboutfuturevalues ofthesemacrovariables. Our paper is related to a growing literature on linking the dynamics of the term structure withmacrofactors. However,theotherpapersinthisliteraturearelessinterestedinestimating various Taylor rules, rather than embedding a particular form of a Taylor rule, sometimes pre-estimated, in a macroeconomic model. For example, Bekaert, Cho, and Moreno (2003), Gallmeyer, Hollifield, and Zin (2005), Rudebusch and Wu (2005), and Ho¨rdahl, Tristani, and Vestin(2006)estimatestructuralmodelswithinterestratesand macrovariables. Incontrastto these studies, we do not impose any structural restrictions, but only assume no arbitrage. This makes our approach more closely related to the identified VAR literature in macroeconomics (for a survey, see Christiano, Eichenbaum and Evans, 1999) and this provides us additional flexibility in matching the dynamics of the term structure. Other non-structural term structure 2 models with macro factors, like Ang and Piazzesi (2003), and Dewachter and Lyrio (2006), amongmanyothers,alsodonotinvestigatehowno-arbitragerestrictionscanidentifydifferent monetarypolicyrules. We do not claim that no-arbitrage techniques are superior to estimating monetary policy rulesusingstructuralmodels. Rather,webelievethatestimatingpolicyrulesusingno-arbitrage restrictions are a useful addition to existing methods. Our framework enables the entire cross-section and time-series of yields to be modeled and provides a unifying framework to jointly estimate standard, backward-, and forward-looking Taylor rules in a single, consistent framework. Indeed, we show that many formulations of policy rules imply term structure dynamicsthatareobservationallyequivalent. Naturally,ourmethodologycanbeusedinmore structuralapproachesthateffectivelyconstrainthefactordynamicsandriskpremia. The rest of the paper is organized as follows. Section 2 outlines the model and develops themethodologyshowinghowTaylorrulescanbeidentifiedwithno-arbitrageconditions. We briefly discuss the estimation strategy in Section 3. In Section 4, we lay out the empirical results. After describing the parameter estimates, we attribute the time-variation of yields and expectedexcessholdingperiodreturnsoflong-termbondstoeconomicsources. Wedescribein detailtheimpliedTaylorruleestimatesfromthemodelandcontrastthemwithOLSestimates. Section5concludes. 2 The Model We describe the setup of the model in Section 2.1. Section 2.2 derives closed-form solutions for bond prices (yields) and expected returns. In Sections 2.3 to 2.8, we explain how various Taylorrulescanbeidentifiedintheno-arbitragemodel. 2.1 General Set-up Our state variables are the output gap at quarter t, g ; the continuously compounded year-on- t year inflation rate from quarter t−4 to t, π ; and a latent term structure state variable, fu. We t t measureyear-on-yearinflationusingtheGDPdeflator. Oursystemusesfourlagsoftheoutput gapandyear-on-yearinflationvariablesbutparsimoniouslycapturesthedynamicsofthelatent factorwithonlyonelag. Thisspecificationisflexibleenoughtomatchtheautocorrelogramof year-on-year inflation and the output gap at a quarterly frequency. We assume that in the full 3 state vector, X , potentially up to four lags of the output gap and inflation Granger-cause g t−1 t andπ ,butonlythefirstlagofthevariables,g ,π ,fu ,Granger-causethelatentfactorfu. t t−1 t−1 t−1 t Below we show that this assumption is not restrictive (for example, in the sense of matching impulseresponses.) Thus,wecanwritethedynamicsofthestatevariablesas:   (cid:195) (cid:33) fo (cid:179) (cid:180) fo (cid:179) (cid:180) t−2 fo = µ + Φ Φ t−1 + Φ Φ Φ fo +vo t 1 11 12 fu 13 14 15  t−3 t t−1 fo t−4 (cid:195) (cid:33) (cid:179) (cid:180) fo fu = µ + Φ Φ t−1 +vu, (1) t 2 21 22 fu t t−1 for fo = [g π ](cid:62) the vector of observable macro variables, the output gap and inflation, fu the t t t t latentfactor,and (cid:195) (cid:33) vo v = t ∼ IIDN(0,Σ Σ(cid:62)). t vu v v t Foreaseofnotation,wecollectthefourlagsofallthestatevariablesinavectorofK = 12 elements: (cid:163) (cid:164) X = g π fu ... g π fu (cid:62), t t t t t−3 t−3 t−3 andwritetheVARinequation(1)incompanionformas: X = µ+ΦX +Σε , (2) t t−1 t where (cid:195) (cid:33) (cid:195) (cid:33) v Σ 0 t v 3×9 ε = Σ = t 0 0 0 9×1 9×3 9×9 andµandΦcollecttheappropriateconditionalmeansandautocorrelationmatricesoftheVAR inequation(1),respectively. We use only one latent state variable because this is the most parsimonious set-up with Taylor rule residuals (as the next section makes clear). This latent factor, fu, is a standard t latent factor in the tradition of the term-structure literature. Our focus is to show how this factor is related to monetary policy and how the no-arbitrage restrictions can identify various policyrules. Wespecifytheshortrateequationtobe: r = δ +δ(cid:62)X , (3) t 0 1 t 4 forδ ascalarandδ aK×1vector. Tokeepthemodeltractable,ourbaselinesystemhasonly 0 1 contemporaneous values of g , π and fu and no lags of these factors determining r , so only t t t t thefirstthreeelementsof δ arenon-zero. 1 Tocompletethemodel,wespecifythepricingkerneltotakethestandardform: (cid:181) (cid:182) 1 m = exp −r − λ(cid:62)λ −λ(cid:62)ε , (4) t+1 t 2 t t t t+1 withthepricesofrisk: λ = λ +λ X , (5) t 0 1 t fortheK×1vectorλ andtheK×K matrixλ . Tokeepthenumberofparametersdown,we 0 1 onlyallowtherowsof λ thatcorrespondtocurrentvariablestodifferfromzero. Wespecify t (cid:195) (cid:33) ¯ λ 0 λ = , 0 0 9×1 ¯ whereλ isa3×1vector. Likewise,wespecifythatthetime-varyingcomponentsoftheprices 0 ofriskλ ,dependsoncurrentandpastvaluesofmacrovariables,butonlythecontemporaneous 1 valueofthelatentfactor: [g π fu g π g π g π ](cid:62).Thatis,wecanwrite: t t t t−1 t−1 t−2 t−2 t−3 t−3 (cid:195) (cid:33) ¯ λ 1 λ = , 1 0 9×12 whereλ¯ isa3×12matrixwithzerocolumnscorrespondingto fu ,fu andfu . 1 t−1 t−2 t−3 The pricing kernel determines the prices of zero-coupon bonds in the economy from the recursiverelation: P(n) = E [m P(n−1)], (6) t t t+1 t+1 where P(n) is the price of a zero-coupon bond of maturity n quarters at time t. Equivalently, t wecansolvethepriceofazero-couponbondas: (cid:34) (cid:195) (cid:33)(cid:35) (cid:88)n−1 P(n) = EQ exp − r , t t t+i i=0 where EQ denotes the expectation under the risk-neutral probability measure, under which t the dynamics of the state vector X are characterized by the risk-neutral constant and t autocorrelationmatrix: µQ = µ−Σλ 0 ΦQ = Φ−Σλ . 1 Ifinvestorsarerisk-neutral, λ = 0andλ = 0,andnoriskadjustmentisnecessary. 0 1 5 2.2 Bond Prices and Expected Returns The model (2)-(5) belongs to the Duffie and Kan (1996) affine class of term structure models, but incorporates both latent and observable macro factors. The model implies that bond yields taketheform: y(n) = a +b(cid:62)X , (7) t n n t wherey(n) istheyieldonann-periodzerocouponbondattimetthatisimpliedbythemodel, t whichsatisfies P(n) = exp(−ny(n)). t t Thescalara andtheK ×1vectorb aregivenbya = −A /nandb = −B /n,where n n n n n n A andB satisfytherecursiverelations: n n 1 A = A +B(cid:62)(µ−Σλ )+ B(cid:62)ΣΣ(cid:62)B −δ n+1 n n 0 2 n n 0 B(cid:62) = B(cid:62)(Φ−Σλ )−δ(cid:62), (8) n+1 n 1 1 where A = −δ and B = −δ . The recursions (8) are derived by Ang and Piazzesi (2003). 1 0 1 1 Intermsofnotation,theone-periodyield y(1) isthesameastheshortrate r inequation(3). t t Since yields take an affine form and the conditional mean of the state vector is affine, expected holding period returns on bonds are also affine in X . We define the one-period t excessholdingperiodreturnas: (cid:195) (cid:33) P(n−1) rx(n) = log t+1 −r t+1 P(n) t t = ny(n) −(n−1)y(n−1) −r . (9) t t+1 t Theconditionalexpectedexcessholdingperiodreturncanbecomputedusing: 1 E [rx(n)] = − B(cid:62) ΣΣ(cid:62)B +B(cid:62) Σλ +B(cid:62) Σλ X t t+1 2 n−1 n−1 n−1 0 n−1 1 t ≡ Ax +Bx(cid:62)X , (10) n n t which again takes an affine form for the scalar Ax = −1B(cid:62) ΣΣ(cid:62)B + B(cid:62) Σλ and n 2 n−1 n−1 n−1 0 the K ×1 vector Bx = λ Σ(cid:62)B . From equation (10), we can see directly that the expected n 1 n−1 excessreturncomprisesthreeterms: (i)aJensen’sinequalityterm,(ii)aconstantriskpremium, and (iii) a time-varying risk premium. The time variation is governed by the parameters in the matrix λ . Since both bond yields and the expected holding period returns of bonds are 1 affine functions of X , we can easily compute variance decompositions following standard t VARmethods. 6 2.3 The Benchmark Taylor Rule The Taylor (1993) rule describes the Fed as adjusting short-term interest rates in response to movements in inflation and real activity. The rule is consistent with a monetary authority that minimizes a quadratic loss function that tries to stabilize inflation and output around a long- run inflation target and the natural output rate (see, for example, Svensson 1997). Following Taylor’soriginalspecification,wedefinethebenchmarkTaylorruletobe: r = γ +γ g +γ π +εMP,T, (11) t 0 1,g t 1,π t t where the short rate is set by the Federal Reserve in response to current output and inflation. The basic Taylor rule (11) can be interpreted as the short rate equation (3) in a standard affine term structure model, where the unobserved monetary policy shock εMP,T corresponds to a t latent term structure factor, εMP,T = γ fu. This corresponds to the short rate equation (3) in t 1,u t thetermstructuremodelwith     δ γ 1,g 1,g     δ  γ  δ ≡  1,π =  1,π, 1     δ  γ  1,u 1,u 0 0 9×1 9×1 whichhaszerosforallcoefficientsonlagged g andπ. The Taylor rule (11) can be estimated consistently using OLS under the assumption that εMP,T, or fu, is contemporaneously uncorrelated with the output gap and inflation. This t t assumption is satisfied if the output gap and inflation only react slowly to policy shocks. However, there are several advantages to estimating the policy coefficients, γ and γ , and 1,g 1,π extractingthemonetarypolicyshock,εMP,T,usingno-arbitragerestrictionsratherthansimply t runningOLSonequation(11). First,no-arbitragerestrictionscanfreeupthecontemporaneous correlation between the macro and latent factors. Second, even if the macro and latent factors are conditionally uncorrelated, OLS is consistent but not efficient. By imposing no arbitrage, we use cross-equation restrictions that produce more efficient estimates by exploiting information contained in the whole term structure in the estimation of the Taylor rule coefficients, while OLS only uses data on the short rate. Third, the term structure model provides estimates of the effect of a policy or macro shock on any segment of the yield curve, which an OLS estimation of equation (11) cannot provide. Finally, our term structure model allows us to trace the predictability of risk premia in bond yields to macroeconomic or monetarypolicysources. 7 The Taylor rule in equation (11) does not depend on the past level of the short rate. Therefore, OLS regressions typically find that the implied series of monetary policy shocks from the benchmark Taylor rule, εMP,T, is highly persistent (see, for example, Rudebusch t and Svensson, 1999). The statistical reason for this finding is that the short rate is highly autocorrelated, and its movements are not well explained by the right-hand side variables in equation (11). This causes the implied shock to inherit the dynamics of the level of the persistent short rate. In affine term structure models, this finding is reflected by the properties of the implied latent variables. In particular, εMP,T corresponds to δ fu, which is the scaled t 1,u t latent term structure variable. For example, Ang and Piazzesi (2003) show that the first latent factorimpliedbyanaffinemodelwithbothlatentfactorsandobservablemacrofactorsclosely correspondstothetraditionalfirst,highlypersistent,latentfactorintermstructuremodelswith only unobservable factors. This latent variable also corresponds closely to the first principal componentofyields,ortheaverageleveloftheyieldcurve,whichishighlyautocorrelated. 2.4 Backward-Looking Taylor Rules Eichenbaum and Evans (1995), Christiano, Eichenbaum and Evans (1996), Clarida, Gal´ı and Gertler (1998), among others, consider modified Taylor rules that include current as well as laggedvaluesofmacrovariablesandthepreviousshortrate: r = γ +γ g +γ π +γ g +γ π +γ r +εMP,B, (12) t 0 1,g t 1,π t 2,g t−1 2,π t−1 2,r t−1 t whereεMP,B istheimpliedmonetarypolicyshockfromthebackward-lookingTaylorrule. This t formulationhasthestatisticaladvantagethatwecomputemonetarypolicyshocksrecognizing that the short rate is a highly persistent process. The economic mechanism behind such a backward-looking rule may be that the objective of the central bank is to smooth interest rates (seeGoodfriend,1991). In the setting of our model, we can modify the short rate equation (3) to take the same formasequation(12). Usingthenotationfo andfu torefertotheobservablemacroandlatent t t factors,respectively,wecanrewritetheshortratedynamics(3)as: r = δ +δ(cid:62) fo +δ fu, (13) t 0 1,o t 1,u t where   δ  1,o  δ ≡ δ . 1  1,u 0 9×1 8