Regime-Shifts, Risk Premiums in the Term Structure, and the Business Cycle

Regime-Shifts, Risk Premiums in the Term Structure, and the Business Cycle∗ Ravi Bansal† George Tauchen‡ Hao Zhou§ First Draft: January 10, 2003 This Draft: April 15, 2003 Abstract WeexaminevariousdynamictermstructuremodelsformonthlyUSTreasuryyields from 1964 to 2001. Of particular interest is the predictability of bond excess returns. Recent evidence indicates that using multiple forward rates can sharply predict future excess returns on bonds; the R2 of this predictability regression can be as high as 30%. In addition, the projection coefficients in these predictability regressions exhibit a tent shaped pattern that relates to the maturity of the forward rate. This dimension of the data in conjunction with the transition dynamics of bond yields (i.e., conditional volatility and cross-correlation of bond yields) poses an serious challenge to term structure models. In this paper we present and estimate a regime-shifts term structure model—our findings show that this model can account for all aspects of the predictability regression and the transition dynamics of yields. Alternative models, such as, affine factor models cannot account for these features of the data. We find that the regimes in the model are related to the NBER business cycle indicator. Keywords: Regime Switching, Term Structure of Interest Rate, Expectation Hypothesis, Business Cycle, Efficient Method of Moments. JEL classification: E43, G12, C51, C52. ∗The views expressed in this paper reflect those of the authors and do not represent those of the Board of Governors of the Federal Reserve System or other members of its staff. †Fuqua School of Business, Duke University, Box 90120, Durham NC 27708, Email rb7@mail.duke.edu, Phone 919-660-7758, Fax 919-660-8038 ‡Corresponding author: George Tauchen, Department of Economics, Duke University, Box 90097, Durham NC 27708, Email get@econ.duke.edu, Phone 919-660-1812, Fax 919-684-8974. §Trading Risk Analysis Section, Division of Research and Statistics, Federal Reserve Board, Washington DC 20551 USA, Email hao.zhou@frb.gov, Phone 202-452-3360, Fax 202-728-5887. 1

1 Introduction Term structural models with regime-shifts as in Naik and Lee (1997), Evans (1998), and BansalandZhou(2002),capturetheimportantfeaturethattheaggregateeconomyissubject to discrete and persistent changes in the business cycle. The business cycle fluctuations together with the monetary policy response to them have significant impacts on not only the short interest rate, but also the entire term structure. Regime-switching term structure models represent a parsimonious way to introduce the nonlinear interactions between the business cycle effects and the term structure dynamics. Using the US treasury yield data from 1964 to 1995, Bansal and Zhou (2002) find that the model-implied regime changes usually lead or coincide with economic recessions. Therefore the term structure regimes seem to confirm and complement the real business cycles. One important benchmark for comparing term structure models is the characterization of risk premiums on bonds of various maturities. The most common strategy for understanding bond risk premiums is to study deviations from the the so-called Expectations Hypothesis. One form of the violation that regression of yield changes on yield spreads produce negative slope coefficient instead of unity (Campbell and Shiller, 1991), has been addressed is many recent papers (see, Roberds and Whiteman (1999), Dai and Singleton (2002), and Bansal and Zhou (2002)). Another form of the violation of Expectations Hypothesis is that the forward rate can predict the excess bond return (Fama and Bliss, 1987). More recently, Cochrane and Piazzesi (2002) document that using multiple forward rates to predict bond excess returns generates very high predictability of bond excess returns—the adjusted R2’s from the regression being around 30%. Further, they show that the coefficients of multiple forward-rate regressors form a tent shape pattern related to the maturity of the forward rate. The size of the predictability and the tent shape pattern of the projection coefficients is quite puzzling and constitutes a challenge to term structure models. The main contribution of this paper to provide an interpretation for the predictability evidence from the perspective of latent factor term structure models. When evaluating the plausibility of various term structure models it is important to not focus exclusively on the predictability issue; previous work (see Dai and Singleton (2000), Bansal and Zhou (2002), Ahn et al. (2002)) highlight the difficulties that many received models have in capturing the transition dynamics of yields (i.e. conditional volatility and conditional cross-correlation across yields). The predictability evidence, in conjunction with the transition dynamics 2

constitutes a sufficiently rich set of data-features to discriminate across alternative term structure models and evaluate their plausibility. The main empirical finding of this paper is to show that the regime-shifts term structure models can simultaneously justify the nature of bond return predictability and the transition dynamics of yields. More specifically, we find that models with regime-shifts model can reproduce the high predictability and the tentshaped regression coefficients documented in Cochrane and Piazzesi (2002). Additionally, the regime-shifts term structure model reproduces the dynamics of conditional volatility and cross-correlation across yields. On the other hand, commonly used multi-factor CIR and affine model cannot capture these dimensions of the data. Our overall evidence indicates that incorporating regime shifts are important for interpreting key aspects of treasury bonds markets data. We use US treasury yield data from 1964 through 2001. The period from 1996 to 2000 poses a tough challenge for standard asset pricing models, with unprecedented long economic growth and bull market run. At the same time this stretch of the data has several economic recessions and periods of economic boom. Using the whole sample we find that the conditional correlation between the long and short yields vary over a range from about 40% to 80%. The conditional volatilities of the long and short yields also reveal very large variation. Despite this, when confronting the U.S. treasury yields data from 1964 to 2001, our regime-shifts model still stands out as the best performing candidate. The regime indicator is related to business cycles in the data; for example, the model-based regime indicator predicts the 2001-2002 recession. To estimate various models under consideration we use the Efficient Method of Moments (EMM), developed in Bansal et al. (1995) and Gallant and Tauchen (1996). Tests of over-identifying restrictions based on the EMM method provide a way to compare different, potentially non-nested models. This estimation technique forces the model to confront several important aspects of the data, such as the conditional volatility and correlation across different yields. To generate diagnostic evidence to help discrimination across models, we rely on the reprojection methods developed by Gallant and Tauchen (1998). Our empirical evidence suggests that the benchmark CIR and affine model specifications with up to three factors are sharply rejected with p-values of zero. The only model specification that finds support in the data (with p-value of 1%) is our preferred two-factor regime switching model where the market price of risks depends on regime shifts. Our diagnostics of the various models show that the our preferred regime shifts model specification produces the smallest 3

cross-sectional pricing errors across all the specifications considered in the paper. Using reprojections we compute the conditional correlations and volatility under the null of the various models. Our results show that only the regime-shifts models can capture the large variations in conditional correlations and conditional volatility that are observe in the data. The remainder of this paper is organized in the following manner. Section 2 reviews the regime shifts term structure model developed in Bansal and Zhou (2002). Section 3 discusses the empirical estimation results, the specifications tests, and an array of diagnostics based on the conditional correlation and volatility. It also examines cross-sectional implications on pricing errors, violations of the expectation hypothesis of forward rate predictability and the link between regime classification and business cycles, especially the recent economic recession. Section 4 contains the concluding remarks. 2 Term Structure Model with Regime-Shifts In this section, we review the term structure model with regime shifts that is proposed by Bansal and Zhou (2002) and is related to earlier models developed by Naik and Lee (1997) and Evans (1998). The derivation focuses on a single factor (i.e., state variable) as the extensions to the multi-factor case follow naturally. To capture the idea that the aggregate economyis subject to regime shifts, wemodel the regime shifts process as a twostate Markov process as in Hamilton (1989). Suppose that the evolution of tomorrow’s regime s = 0,1 t+1 given today’s regime s = 0,1 is governed by the transitional probability matrix of a Markov t chain   π π  00 01  Π =  , (1) π π 10 11 (cid:80) where π = 1 and 0 < π < 1. In addition to the discrete regime shifts, the economy j=0,1 ij ij is also affected by a continuous state variable, √ x −x = κ (θ −x )+σ x u , (2) t+1 t st+1 st+1 t st+1 t t+1 where κ , θ , and σ , are the regime-dependent mean reversion, long run mean, and st+1 st+1 st+1 volatility parameters respectively. All these parameters are subject to discrete regime shifts. √ Specifically, x − x = κ (θ − x ) + σ x u if the regime s = 0, and x − x = t+1 t 0 0 t 0 t t+1 t+1 t+1 t √ κ (θ − x ) + σ x u if the regime s = 1. Note that the innovation in the process 1 1 t 1 t t+1 t+1 4

(2), u , is conditionally normal given x and s .1 It is also assumed that the agents in t+1 t t+1 the economy observe the regimes, though the econometrician may possibly not observe the regimes. The pricing kernel for this economy, is similar to that in standard models, except for incorporating regime shifts λ x λ √ M = exp{−r −( st+1)2 t − st+1 x u }. (3) t+1 f,t t t+1 σ 2 σ st+1 st+1 In the standard representative agent economy with log preferences, the equilibrium pricing kernel (or IMRS) depends on the consumption and inflation process—the above specification of the pricing kernel captures the intuition that these aggregate processes are latent and subject to regime shifts (as in Hamilton (1989)). Note that the λ parameter that affects the risk premia on bonds is also subject to regime shifts and hence depends on s . t+1 With regime shifts, we conjecture that the bond price with n periods to maturity, at date t depends on the regime s = i, i = 0,1, and x t t P (t,n) = exp{−A (n)−B (n)x }. i i i t The one period ahead bond price, analogously depends on s and x t+1 t+1 P (t+1,n−1) = exp{−A (n−1)−B (n−1)x }. st+1 st+1 st+1 t+1 In addition we impose the boundary condition A (0) = B (0) = 0 and the normalization i i A (1) = 0, B (1) = 1, for i = 0,1, that is, r = x . Setting r = x and using a log-linear i i f,t t f,t t approximation, the key asset pricing condition can be shown as (cid:88) σ2 π [−x +µ + n,st+1,t +B (n−1)λ x ] = 0 (4) ist+1 t n,st+1,t 2 st+1 st+1 t st+1=0,1 withi = 0,1. Theconditionalmeanandvolatilityofthebondreturninregimes isµ t+1 n,st+1,t and σ2 respectively. With discrete regime shifts, (4) is the asset pricing restriction on n,st+1,t the risk premium. Specifically note that (4) can equivalently be stated as σ2 E [µ + n,st+1,t −r |x ,s ] = −x E [B (n−1)λ |s ] (5) t n,st+1,t 2 f,t t t t t st+1 st+1 t 1For analytical tractability we assume that the process for regime shifts s is independent of x , t+1 t+1−l l=0···∞, this is similar to the assumptions made in Hamilton’s regime switching models. 5

Note that all risk-premiums and bond prices at date t depend only on s and x . To further t t √ get some intuition regarding the risk premium result, note that −σ B (n−1) x is the st+1 st+1 t exposure of the bond return to the standardized (i.e., N(0,1)) shock u in regime s . t+1 t+1 √ Further, [ λst+1 x ] is the exposure of the pricing kernel (more precisely, −M /E [M ]) σst+1 t t+1 t t+1 to u in regime s . The covariance between these exposures determine the compensation t+1 t+1 for risk in regime s . Hence, the risk compensation for regime s is the product t+1 t+1 √ λ √ −σ B (n−1) x ×[ st+1 x ] = −B (n−1)λ x st+1 st+1 t σ t st+1 st+1 t st+1 Given information regarding s , x , and the regime transition probabilities; agents integrate t t out the future regime, s , which leads to the risk premium result stated in (5). In the t+1 absence of regime shifts, the risk premium in (5), would simply be −x B(n − 1)λ. Hence t incorporating regime shifts makes the “beta” of the asset, that is the coefficient on x , be t time-varying and dependent on the current regime. The market price of risk, as discussed in Cochrane (2001), can be computed as the risk premiumonanassetwithaunitexposuretothesourceof(standardized)risk. Consequently, using the above analysis, an asset return with an exposure of −1 to the standardized shock √ u would have a risk premium of −E [ λst+1|s ] x ; this is the risk premium per unit of t+1 t σst+1 t t standard deviation, and hence the market price of risk. Using exactly the same logic, it also √ follows that the market price of risk in the 1-Factor[CIR] model is, −λ x . From (4) it σ t is evident that the risk premium is also governed by the regime-dependent risk parameters λ and λ . If the risk parameters across two regimes are restricted to be the same (i.e., 0 1 λ = λ ), we arrive at a specification related to that in Naik and Lee (1997). Allowing the 0 1 risk parameters to depend on the regimes opens an additional channel for regime shifts to affect bond prices. This feature, as discussed below, is important for understanding many of the quantitative features of term structure data. Given (4), the solution for the bond prices can be derived, by solving for the unknown coefficients A and B, in particular we show;      B (n) π π (1−κ −λ )B (n−1)− 1σ2B2(n−1)+1   0   =   00 01     0 0 0 2 0 0   (6) B (n) π π (1−κ −λ )B (n−1)− 1σ2B2(n−1)+1 1 10 11 1 1 1 2 1 1 and      A (n) π π A (n−1)+κ θ B (n−1)  0   00 01  0 0 0 0    =    (7) A (n) π π A (n−1)+κ θ B (n−1) 1 10 11 1 1 1 1 6

with initial conditions A (0) = A (0) = B (0) = B (0) = 0. Note that bond price coefficients 0 1 0 1 aremutuallydependentonboththeregimes—currentbondpricesreflectagent’sexpectations regarding regime shifts in the future. Finally, the bond yield of a K factor regime-switching model can be derived in an analogous manner, lnP (t,n) A (n) (cid:88)K B (n)x s s ks kt Y (t,n) = − = + . (8) s n n n k=1 3 Empirical Estimation and Model Evaluation 3.1 Estimation Methodology To utilize a consistent approach for evaluation and estimation across the different models we rely on the simulation-based EMM (efficient method of moments) estimator, developed in Bansal et al. (1995) and Gallant and Tauchen (1996). The EMM estimator consists of three steps. The first, called the projection step, entails estimating a reduced form model, termed the auxiliary model, that provides a good statistical description of the data,. Multivariate bonds yields are difficult data to model as they exhibit extreme persistence in location and scale, time varying correlations, and non-Gaussian innovations. Since we do not have good a priori information on the specification of a model to capture all of these features, we utilize a semi-nonparametric (SNP) series expansion. The SNP expansion has a VAR-ARCH Gaussian density as its leading term, and the departures from the leading term are captured by a Hermite polynomial expansion. We elected to use a simpler, ARCH-like leading term, instead of a GARCH-type leading term because of some of the problems with multivariate GARCH-type models of bond yields noted by Ahn et al. (2002). In the second step, termed the estimation step, the score function from the log-likelihood estimation of the auxiliary model is used to generate moments for a GMM-type criterion function. The score function provides a set of moment conditions that are true by construction and are to be confronted by all term structure models under consideration. In the computations, the score function is averaged over the simulation output from a given term structure model and the criterion function is minimized with respect to the parameters on the term structure model under consideration. By using the scores from the non-parametric SNP density as the moment conditions, each model is forced to match the conditional distribution of the observed 6-month and 5-year yields. Being a GMM-type estimator, EMM provides a chi-squared measure of fit. In particular, the normalized objective function acts as 7

an omnibus specification test, which is distributed as a chi-square (as in GMM) with degrees of freedom equal to the number of scores (moment conditions) less the number of parameters in the particular term structure model. The distance matrix (the weight matrix in GMM) used in constructing the specification test is identical across different model specifications (the null hypotheses). Consequently, the p-values based on this specification test can be directly compared across different structural models to identify the best model specification.2 It is well recognized in the literature, that tests for the absence of regime-shifts against a regime shifting alternative require non-standard approaches (see Hansen (1992) and Garcia (1992)). Our approach of comparing all the considered models to a common non-parametric density (the SNP density), allows us to rank order all the considered models according to the p-values implied by the EMM criterion function. The advantage of using the non-parametric SNP, as discussed in Gallant and Tauchen (1999), is that it can asymptotically converge to virtually any smooth distributions, including mixture distributions (as is the case with a model of regime shifts). The third step is reprojection, or post-estimation analysis of model simulations. Since EMM is a simulation-based estimator, there are available for analysis long simulated realizations from each estimated model. These simulations can be used to compute statistics of interest that can be compared to analogous values computed from the observed data. The reprojected statistics should be thought of as population quantities implied by the model at the estimated parameter values. Among other things, we compute the reprojected Cochrane- Piazzia forward rate regressions for models with and without regime switching. 3.2 Data Description The data set is monthly, June 1964 to December 2001, bond yield data obtained from the Center for Research in Security Prices (CRSP). There are total 451 monthly observations, with eight maturities 1, 3, 6, month and 1, 2, 3, 4, 5 year. It is important to recognize that the data period 1964–2001 contains six major recessions and six major expansions, which as stated earlier provides potential economic motivation for incorporating regime shifts. The summary statistics of these monthly yields are displayed in Table 1. On average, the yield curve is upward sloping. The standard deviation, positive skewness, and kurtosis are systematically higher for short maturities than for long ones. To incorporate important 2Foradiscussionoftheimportanceofhavingthesamedistancematrix,foraconsistentcomparisonacross models, see Hansen and Jagannathan (1997). 8

time-series and cross-sectional aspects of term structure data we focus on a short term and a long term yield—the yield on the six month bill and the five year note. Time series plots of the basis yields are in Figure 1. We very briefly summarize the first step estimation results for the non-parametric SNP specification which was guided by the BIC information criterion; details are available upon request. The leading term of the bivariate SNP density has 1 lag in the VAR based conditional mean (L = 1) and 5 lags in ARCH specification (L = 5). The preferred specification µ r accommodates departures from conditional normality via a Hermite polynomial of degree 4 (K = 4). This “semiparametric ARCH” specification is similar to that proposed by Engle z and Gonz´alez-Rivera (1991). This specification allows for skewness and kurtosis in the error distribution. The total number of parameters for the specification is l = 28; hence, there θ are a total of 28 data-determined moments conditions that each model must confront. The conditional moments of the estimated SNP density for the observed interest rates are available analytically. It is fairly instructive to focus on some specific aspects of the estimated non-parametric SNP bivariate density. The top panel in Figure 5 gives the estimated conditional volatilities and cross correlations of the 6-month and 5-year yields, which seem to be very persistent and fairly volatile. The short interest rate has a wide range for the conditional volatility which peaks around 1980, while the range for the five year yield volatility is narrow. The range for the conditional correlation is from about 40% to 80%—a wide range indeed. The most volatile period for bond yields, the early 80’s sees, is associated with a considerable drop in the conditional correlation. The behavior of the conditional variance and the cross-correlation, as documented above, poses a serious challenge to the various term structure models under consideration. 3.3 Model Estimation Results Table 2 gives the main EMM estimation results for four different models: one-factor regimeswitching (1-Factor[RS]), two-factor square-root (2-Factor[CIR]), two-factor regime switching (2-Factor[RS]), and three-factor affine (3-Factor[AF]).3 The results reported here are for simulation size of 50,000. The one factor model with regime-shifts (1-Factor[RS]) is rejected with a p-value less than 0.0000. The two factor square-root model (2-Factor[CIR]) 3Additionalthreemodels,one-factorsquare-root,two-factorNaikandLee(1997),andthree-factorsquareroot, are also estimated with similar results as reported in Bansal and Zhou (2002), and none of those can replicate the Expectation Hypothesis puzzle. 9

improves things but this specification is still sharply rejected—the model specification test drops to 56.066 with p-value smaller than 0.0003. The best model amongst all specifications is the two-factor regime switching specification with p-value reaching 1%. The estimated regime switching probabilities are both just under 0.95. All the parameters of the model are estimated rather accurately. The transition probabilities reported for the 2-Factor[RS] specification are comparable to those found in other papers (see Gray (1996), Hamilton (1988), and Cai (1994)). The 2-Factor[RS] model can be viewed as a three factor model with the regime switching factor being a multiplicative or nonlinear third factor. For a fair comparison of this two factorregimeswitchingmodel,wealsoestimateathree-factoraffinetermstructuremodel,(3- Factor[AF]) preferred by Dai and Singleton (2000)—they find considerable empirical support for this specification using the post 1987 swap yield data. The discrete time counterpart to this affine specification is; √ x −x = κ (θ −x )+σ x u 1t+1 1t 1 1 1t 1 1t 1t+1 √ x −x = κ (θ −x )+σ u +σ x u (9) 2t+1 2t 2 2 2t 2 2t+1 23 1t 3t+1 √ √ x −x = κ (x −x )+ x u +σ σ x u +σ σ u 3t+1 3t 3 2t 3t 1t 3t+1 31 1 1t 1t+1 32 2 2t+1 Associated with this 3-Factor[AF] specification are three market price of risk parameters, which as before we label as λ , k = 1,2,3. In all there are 13 parameters to estimate. As k reported in Table 2 the 3-Factor[AF] specification is sharply rejected with a X2(15) = 42.803 and a p-value of 0.0017. Table 3 reports the t-ratio diagnostics for the 28 moment conditions implied by each of the four specifications. These 28 scores (moment conditions) should, for a correctly specified model, be close to zero. If the structural model under consideration matches the particular moment under consideration, then at conventional 5% level of significance the t-ratio should be smaller than 1.96. The reported t-ratios are not adjusted for parameter estimation so these t’s are therefore asymptotically slightly downward biased relative to 2.0. They thus must be interpreted with cautious intuition guided by the overall chi-square diagnostics, which are free of such asymptotic bias. For the 1-Factor[RS] model, 17 out of 28 moment tests are rejected, with fitting of conditional volatility especially bad. The 2-Factor[CIR] model only has 9 t-ratios higher than 1.96, and adding one more linear factor dramatically improves the fitting of conditional volatility and conditional mean. It is remarkable that our favored 2-Factor[RS] matches well all the mean, volatility, and polynomial scores, except for 10

the single ARCH(1) score of the six month yield that is just over 2.0. The 3-Factor[AF] is certainly a improvement than the one or two factor models, but it still has 4 out of 13 ARCH scores and 2 out of 9 Hermite scores are not well matched. Overall, our preferred 2-Factor[RS] specification seems to have the greatest advantage in matching the conditional volatility and covariance, i.e., the ARCH scores and the non-Gaussian polynomials (i.e., the Hermite polynomial parameters), relative other multifactor CIR or affine specifications. 3.4 Risk Premium Analysis An important diagnostic is to evaluate if the different model specifications can justify the observed patterns of violations of the Expectations Hypothesis, in particular, as documented in Fama and Bliss (1987), the predictability of forward rates on excess returns.4 The simple existence of the predictability from forward rate to excess return—R-square significantly higherthanzero—iseasilyexplainedbyanydynamictermstructuremodelwithtime-varying risk premium. However, the greater challenge, as recently popularized by Cochrane and Piazzesi (2002), is to explain the robust tent-shaped pattern of the slope coefficients when multiple forward rates are used as regressors. Following the same conventions in Cochrane and Piazzesi (2002), we work with log bond prices, i.e., pk is the log of the price at t of a k year bond, and geometric (log) yields and t returns, so y1 = −p1 is the geometric yield on the 1-year bond. Cochrane and Piazzesi (2002) t t consider the regression of excess returns of bonds on the yields and the forward rates: (cid:88)5 exk = β +β y1 + β fi +(cid:178)k , k = 2,..,5 (10) t+12 k0 k1 t ki t t+12 i=2 where exk = pk−1 −pk −y1 is the excess return on the k year bond and fk = pk−1 −pk t+12 t+12 t t t t t is the forward rate. Note that exk is effectively the return on holding a k year bond for t+12 one year in excess of the one year yield. This excess return data is collected on a monthly frequency which leads to the usual overlap in return data. We first check the robustness of the Cochrane and Piazzesi (2002) findings. As shown in the top panel of Table 4, the regression R-square with five forward rates reaches 36%, which confirms their findings. An important note is that the difference between using three, four, 4Another form of the EH violation is the negative slope in stead of unity when regressing yield changes onyieldspreads(CampbellandShiller,1991). BansalandZhou(2002)provideevidencethatthetwo-factor regime-shifts model is the only one can replicate this type of EH violation at the shorter maturities, while all multi-factor models fair well at the longer maturities. 11

or five forward rates is negligible, while reducing to two or one forward rates dramatically decreases the R-square. This seems to suggest the existence of three latent factors, and the use of five regressors creates near perfect co-linearity problem, up to cross-sectional measurement errors that can mask the singularity. We concentrate on the regressions with three forward rates. The estimated coefficients are plotted in the top-left panel of Figure 2 and the tent shape finding of Cochrane and Piazzesi (2002) is quite apparent. Next, we examine if any of the dynamic term structure models under consideration can meet the challenge of replicating this unique tent-shape phenomenon. Using the estimated parameters of the four models, we simulate 50,000 monthly data and run the same regressions of excess bond returns on forward rates. As seen in the lower panel of Table 4, the 2-Factor[RS] not only achieves the highest predicting R-squares (20-36%) and it clearly mimics closely the tent-shape regression coefficients. On the other hand, the 2-Factor[CIR] model produces a skewed and inverted tent shape, and the 3-Factor[AF] model produces a inverted tent shape. Both models achieve R-squares around 10-20%. Interestingly, even the 1-Factor[RS] model can replicate to some degree the tent shape, even though its R-square is only about 1%. These patterns are quite apparent in Figure 2. These results suggest that the prediction capability of forward rates for excess returns may be explained but two or three linear factors while the tent pattern of regression coefficients appears to be due to the regime-switching nature of the yield curve. 3.5 Pricing Error and Business Cycle Implication We now explore the cross sectional implications of the term structure models over the maturities that are not used in the model estimation. We also look at the association between the bond market implied regimes and the real business cycle. For the 2-Factor[CIR] and 3-Factor[AF] models, a standard method is used to calculate the pricing errors—since the yield curve solution is linear in the factors, we first invert from two or three basis yields to get the latent factors and then using the linear pricing solution to calculate the non-basis yields. For the 1-Factor[RS] and 2-Factor[RS] models, the yield curve solution is non-linear when current regime is unobserved by the econometrician. However, the modelling assumption that the agent in the economy know the current regime implies a strategy to recover the regime from one of the two yield curves that produces the smallest pricing error. Under the null of correct specification, the pricing error should be zero given correct regime choice. For more details see Bansal and Zhou (2002). 12

(cid:80) Table 5 report the time-series average of pricing errors 1/T T PE or other statistics t=1 t (cid:80) from the cross-sectional average PE = 1/N N |Y ˆ (n) − Y(n) |, where Y ˆ (n) is the calt n=1 t t t culated yield and Y(n) is the observed yield for maturity n at time t. It is clear from the t sample statistics that the 2-Factor[RS] model has the smallest average pricing error and also the smallest standard deviation in the pricing error. The maximal pricing error associated with the 2-Factor[RS] classification is also the smallest. Further, on average the pricing error is only about 27 basis points for the annualized percentage yields. The 3-Factor[AF] specifications have average pricing errors of 31 basis points, which in an absolute sense is also quite small. The 1-Factor[RS] and 2-Factor[CIR] models achieve similar pricing result as 44-45 basis points. Figure 3 displays results from regime classification. Most of the time, it seems that the economy is in regime 1. The total number of regime switches recovered from the sample period is 44, which is large enough for identifying the regime probabilities. The regime classification by and large coincides with the NBER business cycles. It seems that regime 0 obtains during or before recessions in the economy. The relatively frequent regime shifts in the 1960s and 1990s, as shown in Figure 3, reflect the fact that during the long economic expansions, there exists some temporary economic slowdowns, international financial crises, andthe“softlandings”ofmonetarypolicy. ThecorrelationbetweentheNBERbusinesscycle (recession is regime 0 and boom is regime 1) indicator and the recovered regime indicator is 0.1117. Correlation between NBER business cycle indicator and the yield spread (5 year yield minus 6 month yield) is 0.1501. Correlation between the model based regime indicator and the yield spread is 0.2337–that is, regime 1 has high yield spread and regime 0 has low yield spread. The most recent prolonged stay in regime 0 since the October of 2000, seems to predict a protracted economic recession or a very slow economic recovery. The out-of-sample experience of 2002-2003 largely confirms that. 3.6 The Reprojected Conditional Density, Volatility, and Correlation As a final diagnostic we assess the various models’ capabilities to match the shape and track the conditional mean and variance characteristics of the data. Following Gallant and Tauchen(1998)wecomputethereprojectedconditionaldensityofthetwobasisyields. Given the estimated null model and the simulated output for yields, the reprojected conditional density is obtained by re-estimating the parameters of the SNP density. Moments of interest, 13

such as the conditional variances and correlations implied by the model specification can then be computed. These conditional moments are simply functions of the conditioning information used to estimate the reprojected density. Given the conditioning information, theimplicationsofagivennullmodelforanyconditionalmomentofinterestcanbeevaluated on the observed data and compared to the conditional moment implied by the unrestricted SNP density. Figure 4 plots the reprojected conditional density (evaluated at the sample mean), for the different models under consideration. The unrestricted 6 month yield SNP density has high peak and narrow shoulders and the unrestricted density for the 5 year yield is skewed to the left and moderately peaked. The reprojected densities for the 3-Factor[AF] model do capture the peakedness of the 5 year yield but miss the peak of the 6 month yield and the skew of 5 year yield. On the other hand, the reprojected densities for the 1-Factor[RS] and 2-Factor[CIR] models capture somewhat the skewness of the 5 year yield but largely miss the peak of both yields. The 2-factor regime shift models have greater success in capturing the left skew of the five year yield and the peak of both yields. Figure 5 displays the conditional volatility and cross correlation for the various model specifications as implied by the reprojected densities. Note that, in the data, the dynamics of the conditional variance of the 6-month yield is quite different from that of the 5-year yield. The range for the conditional volatility for the 6-month yield rate is much larger than for the 5-year yield—the high end being almost three times the lowest for the 6-month, and two times for the 5-year yield. The short yield volatility is more persistent, while the long yield volatility seems more choppy. The 1-Factor[RS] model does not reflect any timevariations of short and long rate volatilities, although the levels of volatility are matched. The 2-Factor[CIR] has difficulty in matching the short rate volatility and does somewhat better in matching the volatility of the 5-year yield. The 2-Factor[RS] model is capable of duplicating the projected volatility of the short rate extremely well, and that of the long yield volatility almost completely. The 3-Factor[AF] model seem to capture the volatility of the short rate much better than the 2-Factor[CIR] model, however, its capability to mimic the long rate volatility is diminished relative to the 2-Factor[CIR] model. TherightmostsubplotsofFigure5providesevidenceregardingtheconditionalcorrelation between the 6-month and 5-year yields. The 2-Factor[RS] model succeeds in capturing the widerangeofthecorrelationobservedacrosstheseyields. Thecorrelationvariesfrom40%to 90%. Note that although the conditional volatility increases during the volatile period of the 14

early 80’s, the conditional correlation decreases—suggesting that the volatilities of the two yields rise more rapidly relative to the conditional covariance. The 1-Factor[RS] model, with only one linear factor, not surprisingly presents a nearly constant correlation very close to unity. The 2-Factor[CIR] and the 3-Factor[AF] specifications have difficulty in capturing the conditional covariance. However, the 3-Factor[AF] specification seems doing a considerably better job of capturing the conditional covariance relative to the 2-Factor[CIR] specification. The 2-Factor[RS] model comes quite close to capturing virtually all the observed dynamics of the conditional correlation between these yields. 4 Concluding Remarks Businesscyclemovementsbetweeneconomicexpansionsandrecessionsaffectmacroeconomic variables, financial markets, and in particular, the term structure of interest rates. In this paper, we incorporate the well documented feature of regime-shifts as in Hamilton (1988) into the standard term structure model such as Cox et al. (1985). We uncover additional important new evidence on the empirical success of regime switching models beyond that reported in Bansal and Zhou (2002). The empirical work is conducted on nominal U.S. treasury bill and bond yields from 1964–2001. For estimation and specification tests of the various models, we use the Efficient Method of Moments estimation technique developed in Bansal et al. (1995) and Gallant and Tauchen (1996). A two factor regime shifting model is the only specification that fits the data according to the usual chi squared test of the restrictions; other models, including the multi-factor CIR and affine, are rejected. Furthermore, the preferred two-factor regime switching model matches the semiparametric moments with acceptable t-ratio diagnostics. In terms of cross-sectional implications, the preferred model achieves the smallest pricing error among all the specifications considered. More important, the evidence also indicates that only models with regime switching can account for the tent-shape structure of regression coefficients in the generalized Expectations Hypothesisregressionsofexcessbondreturnsonforwardrates(CochraneandPiazzesi,2002). Regime switching and the risk premium for holding bonds appear closely connected. Also, the evidence suggests that there is an intimate link between business cycles and regimes extracted from our term structure model, and a recent change of regime accurately predicts the 2001-2002 economic recession and slow recovery. 15

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Table 1: Summary Statistics Thereare451monthlyobservationsoftheyieldswitheightmaturities. Thedataareobtained from CRSP (Center for Research in Security Prices) Treasury bill and bond files, ranging from June 1964 to December 2001. 6 Month 1 Year 2 Year 3 Year 4 Year 5 Year Mean 6.5971 6.8106 7.0156 7.1711 7.2909 7.3545 Stdv 2.6038 2.5239 2.4559 2.3814 2.3491 2.3240 Skew 1.3041 1.1737 1.1288 1.1283 1.1003 1.0565 Kurt 4.8778 4.4157 4.1226 4.0313 3.9196 3.7344 19

Table 2: Model Estimation by Efficient Method of Moments The four term structure models are laid out in Section II. The 1-Factor[RS] or 2-Factor[RS] model refers to the regime shifts specification. The 2-Factor[CIR] model is the Cox-Ingersoll-Ross model with two factors. The 3-Factor[AF] model is the affine specification mentioned in the main text. The simulation size of EMM (efficient method of moments) is chosen to be 50,000 for all the seven models. 1-Factor[RS] 2-Factor[CIR] 2-Factor[RS] 3-Factor[AF] Factor 1 Regime 0 θ .00566 (.00021) .00548(.00051) .00501(.00069) .14e-6(.01e-6) 10 κ .01678 (.00201) .03515(.00304) .01109(.00285) .03530(.00247) 10 σ .00652 (.00034) .00508(.00032) .00504(.00039) .00006(.00000) 10 λ -.00721 (.00165) .02624(.00178) .01877(.00273) -.04136(.00223) 10 Factor 1 Regime 1 θ .00218(.00031) .00629(.00060) 11 κ .01498(.00243) .04655(.00971) 11 σ .00194(.00018) .00075(.00021) 11 λ -.00324(.00276) -.00673(.00310) 11 Factor 2 Regime 0 θ .00091(.00008) .00039(.00310) .00340(.00024) 20 κ .02666(.00305) .01817(.00004) .02487(.00660) 20 σ .00545(.00011) .00305(.00502) -.00005(.00001) 20 λ -.04212(.00389) -.04938(.00024) .00097(.00012) 20 σ -.27376(.05107) 23 Factor 2 Regime 1 θ .00031(.00003) 21 κ .02982(.00603) 21 σ .00476(.00020) 21 λ -.05977(.00576) 21 Factor 3 κ .01925(.00074) 3 σ -344.37(43.686) 31 σ -.45467(.00257) 32 λ 336.76(2.9700) 3 Transitional Probability Pr{s |s } t+1 t π .97564(.00565) .94007(.00008) 00 π .94489(.00001) .93005(.00005) 11 Specification Test X2 94.523 56.066 23.211 42.803 p-Value .00000 .00003 0.0100 .00017 d.o.f. 18 20 10 15 20

Table 3: Diagnostic t-Ratios The SNP score generator is explained in Section 3.2. The t-ratios are testing whether the fitted sample moments are equal to zero, as predicted by population moments of the SNP density. Parameter Description 1-Factor[RS] 2-Factor[CIR] 2-Factor[RS] 3-Factor[AF] Hermite A(1) 00 00 A(2) 01 00 0.30 -1.038 -0.752 0.528 A(3) 10 00 2.13 0.240 -0.646 0.898 A(4) 02 00 1.47 1.874 1.809 2.215 A(5) 11 00 -3.13 -2.258 1.251 -1.402 A(6) 20 00 2.36 -2.752 1.921 -1.538 A(7) 03 00 0.08 -0.072 -0.152 1.431 A(8) 30 00 0.40 -1.093 -0.442 -0.582 A(9) 04 00 1.05 2.018 1.634 2.384 A(10) 40 00 2.20 -1.230 1.423 -0.389 Mean ψ(1) u( 1) 2.61 0.263 -1.022 1.100 ψ(2) u( 2) -0.69 -0.716 -0.299 -0.487 ψ(3) u( 1), y( 1), lag 1 -1.75 0.859 0.963 0.568 ψ(4) u( 2), y( 1), lag 1 -0.11 -0.407 -0.342 -0.213 ψ(5) u( 1), y( 2), lag 1 -2.31 0.534 1.312 0.017 ψ(6) u( 2), y( 2), lag 1 0.29 -0.047 -0.219 0.085 ARCH τ(1) R( 1) 1.85 -3.402 1.264 -2.140 τ(2) R( 2) -4.27 -2.924 0.155 -2.692 τ(3) R( 3) 3.98 3.579 1.369 2.962 τ(4) R( 1), z( 1), lag 5 2.56 -1.606 1.576 -0.640 τ(9) R( 3), z( 2), lag 5 2.76 2.063 0.104 1.641 τ(10) R( 1), z( 1), lag 4 2.57 -1.307 1.858 -0.467 τ(15) R( 3), z( 2), lag 4 2.80 1.916 0.933 1.891 τ(16) R( 1), z( 1), lag 3 1.68 -2.097 1.008 -1.621 τ(21) R( 3), z( 2), lag 3 4.41 3.474 1.963 3.198 τ(22) R( 1), z( 1), lag 2 2.99 -0.212 1.644 -0.003 τ(27) R( 3), z( 2), lag 2 2.25 1.846 0.879 1.597 τ(28) R( 1), z( 1), lag 1 3.46 -0.529 2.061 0.325 τ(33) R( 3), z( 2), lag 1 2.62 1.893 1.294 1.811 21

Table 4: Predictability of Bond Excess Returns Using Multiple Forward Rates The dependent variable in all the regressions below is the one year return from holding a bond with n years tomaturitylesstheyieldonabondwithoneyeartomaturity. Thisannualexcessreturnistrackedmonthly. All R2s are adjusted for degrees of freedom. The sample size in the data is 451 observations. In the top panel the predictability regression is run using 1, 2, 3, 4, and 5 year forward rates as regressors. As the R2 using 1yr, 3yr, 5yr forward rates is almost the same as using additional forward rates (see 1yr, 3-5yr, and 1-5yr) we focus on the 1yr, 3yr, 5yr projection. Newey-West robust standard errors are reported in parenthesis in the panel “Regression Coefficients and R2 in Data” for this projection. The results reported in1-Factor[RS],2-Factor[CIR],2-Factor[RS],2-Factor[AF]panelsarebasedonsimulating50,000observations from the estimated term structure model and running the same regression as reported in the “Regression Coefficients and R2 in Data” panel R-Square 4yr 1yr, 3yr 1yr, 3yr, 5yr 1yr, 3-5yr 1-5yr R2’s in the Data 2 Year Bond 0.1744 0.2619 0.3088 0.3187 0.3280 3 Year Bond 0.1322 0.2538 0.3326 0.3357 0.3373 4 Year Bond 0.1368 0.2634 0.3406 0.3617 0.3639 5 Year Bond 0.1297 0.2640 0.3163 0.3308 0.3336 Coefficient Intercept 1yr 3yr 5yr R-Square Regression Coefficients and R2 in Data 2 Year Bond -2.2222 -0.6753 1.7041 -0.7245 0.3088 (0.5747) (0.1743) (0.2527) (0.2109) 3 Year Bond -3.5737 -1.4040 3.5688 -1.6963 0.3326 (1.0078) (0.3207) (0.4704) (0.3657) 4 Year Bond -4.9032 -2.0580 5.0008 -2.3245 0.3406 (1.4403) (0.4597) (0.6552) (0.4864) 5 Year Bond -6.2848 -2.5018 5.6134 -2.3573 0.3163 (1.7667) (0.5674) (0.8329) (0.6004) Coefficient Intercept 1yr 3yr 5yr R-Square 1-Factor[RS] 2 Year Bond 8.3712 -0.4714 4.4622 -4.9444 0.0164 3 Year Bond 15.0127 -0.8423 7.9971 -8.8619 0.0149 4 Year Bond 20.1520 -1.1259 10.7298 -11.8906 0.0138 5 Year Bond 24.0829 -1.3394 12.8183 -14.2055 0.0129 Coefficient Intercept 1yr 3yr 5yr R-Square 2-Factor[CIR] 2 Year Bond -1.8475 -0.2066 -0.0302 0.3613 0.1741 3 Year Bond -3.6219 -0.3211 -0.0105 0.6765 0.2209 4 Year Bond -5.5087 -0.3954 0.0380 0.9938 0.2538 5 Year Bond -7.6055 -0.4542 0.1060 1.3377 0.2718 Coefficient Intercept 1yr 3yr 5yr R-Square 2-Factor[RS] 2 Year Bond -3.3175 -0.8523 1.9875 -0.6116 0.1914 3 Year Bond -6.1451 -1.4279 3.2531 -0.8669 0.2308 4 Year Bond -8.9064 -1.8229 4.0214 -0.8262 0.2936 5 Year Bond -11.9532 -2.1004 4.4245 -0.5051 0.3621 Coefficient Intercept 1yr 3yr 5yr R-Square 3-Factor[AF] 2 Year Bond 9.3180 0.6074 -1.8067 1.3361 0.1256 3 Year Bond 16.4960 1.2536 -3.7574 2.8143 0.1745 4 Year Bond 22.6622 1.9470 -5.8732 4.4451 0.2206 5 Year Bond 28.6284 2.6990 -8.1863 6.2503 0.2579 22

Table 5: Average Absolute Pricing Error (Basis Points) There are eight maturities (1, 3, 6 month; 1, 2, 3, 4, 5 year) for each of 451 dates. The absolute pricing error of 1-Factor[RS] model is over 7 points; 2-Factor[CIR] over 6 points; 2-Factor[RS] over 6 points; and 3-Factor[AF] over 5 points. The summary statistics of the absolute pricing errors are calculated over the 451 dates for each of the four models. 1-Factor[RS] 2-Factor[CIR] 2-Factor[RS] 3-Factor[AF] Mean 45 44 27 31 Median 34 40 19 23 Std. 33 24 22 28 Min. 5 5 3 1 Max. 223 156 154 188 23

Six−Month Zero−Coupon Yield 18 16 14 12 10 8 6 4 2 1965 1970 1975 1980 1985 1990 1995 2000 Five−Year Zero−Coupon Yield 18 16 14 12 10 8 6 4 2 1965 1970 1975 1980 1985 1990 1995 2000 Figure 1: Observed Short Rate and Long Rate 24

Observed Data 6 4 2 0 −2 −4 1 2 3 4 5 1, 3, 5 Year Forward Rates stneiciffeoC noissergeR 2 Year Bond 1 Year Excess Return 3 Year Bond 1 Year Excess Return 4 Year Bond 1 Year Excess Return 5 Year Bond 1 Year Excess Return 1−Factor[RS] Model 15 10 5 0 −5 −10 −15 1 2 3 4 5 1, 3, 5 Year Forward Rates stneiciffeoC noissergeR 2−Factor[RS] Model 5 4 3 2 1 0 −1 −2 −3 1 2 3 4 5 1, 3, 5 Year Forward Rates stneiciffeoC noissergeR 2−Factor[CIR] Model 1.5 1 0.5 0 −0.5 1 2 3 4 5 1, 3, 5 Year Forward Rates stneiciffeoC noissergeR 3−Factor[AF] Model 10 5 0 −5 −10 1 2 3 4 5 1, 3, 5 Year Forward Rates stneiciffeoC noissergeR Figure 2: Predictability Regression Coefficients 25

Treasury Market Regime and NBER Business Cycle 1.5 0 = NBER Recession and 1 = NBER Expansion 1 0.5 0 Treasury Bond Market Regime NBER Business Cycle Indicator −0.5 1965 1970 1975 1980 1985 1990 1995 2000 Figure 3: Recovered Regimes and Business Cycle 26

6mn Yield: Observed 5yr Yield: Observed 3 3 2 2 1 1 0 0 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 6mn Yield: 1−Factor[RS] 5yr Yield: 1−Factor[RS] 3 3 2 2 1 1 0 0 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 6mn Yield: 2−Factor[CIR] 5yr Yield: 2−Factor[CIR] 3 3 2 2 1 1 0 0 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 6mn Yield: 2−Factor[RS] 5yr Yield: 2−Factor[RS] 3 3 2 2 1 1 0 0 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 6mn Yield: 3−Factor[AF] 5yr Yield: 3−Factor[AF] 3 3 2 2 1 1 0 0 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 Figure 4: Reprojected Densities 27

6mn Vol. Observed 5yr Vol. Observed 6mn−5yr Cor. Observed 2 1 1 1.5 1 0.5 0.5 0.5 0 0 0 1970 1980 1990 2000 1970 1980 1990 2000 1970 1980 1990 2000 6mn Vol. 1−Factor[RS] 5yr Vol. 1−Factor[RS] 6mn−5yr Cor. 1−Factor[RS] 2 1 1 1.5 1 0.5 0.5 0.5 0 0 0 1970 1980 1990 2000 1970 1980 1990 2000 1970 1980 1990 2000 6mn Vol. 2−Factor[CIR] 5yr Vol. 2−Factor[CIR] 6mn−5yr Cor. 2−Factor[CIR] 2 1 1 1.5 1 0.5 0.5 0.5 0 0 0 1970 1980 1990 2000 1970 1980 1990 2000 1970 1980 1990 2000 6mn Vol. 2−Factor[RS] 5yr Vol. 2−Factor[RS] 6mn−5yr Cor. 2−Factor[RS] 2 1 1 1.5 1 0.5 0.5 0.5 0 0 0 1970 1980 1990 2000 1970 1980 1990 2000 1970 1980 1990 2000 6mn Vol. 3−Factor[AF] 5yr Vol. 3−Factor[AF] 6mn−5yr Cor. 3−Factor[AF] 2 1 1 1.5 1 0.5 0.5 0.5 0 0 0 1970 1980 1990 2000 1970 1980 1990 2000 1970 1980 1990 2000 Figure 5: Reprojected Volatilities and Correlations 28