قیمت گذاری ساختار اصطلاح با رگرسیون خطی

We show how to price the time series and cross section of the term structure of interest rates using a three-step linear regression approach. Our method allows computationally fast estimation of term structure models with a large number of pricing factors. We present specification tests favoring a model using five principal components of yields as factors. We demonstrate that this model outperforms the Cochrane and Piazzesi (2008) four-factor specification in out-of-sample exercises but generates similar in-sample term premium dynamics. Our regression approach can also incorporate unspanned factors and allows estimation of term structure models without observing a zero-coupon yield curve.

Affine models of the term structure of interest rates are a popular tool for the analysis of bond pricing. The models typically start with three assumptions: (1) the pricing kernel is exponentially affine in the shocks that drive the economy, (2) prices of risk are affine in the state variables, and (3) innovations to state variables and log yield observation errors are conditionally Gaussian (for examples, see Chen and Scott, 1993, Dai and Singleton, 2000, Collin-Dufresne and Goldstein, 2002, Duffee, 2002 and Kim and Wright, 2005). These assumptions give rise to yields that are affine in the state variables and whose coefficients on the state variables are subject to constraints across maturities (for overviews, see Duffie and Kan, 1996, Piazzesi, 2003 and Singleton, 2006). Empirically, the affine term structure literature has primarily used maximum likelihood (ML) methods to estimate coefficients and pricing factors, thus exploiting the distributional assumptions as well as the no-arbitrage constraints. In this paper, we propose an alternative, regression-based approach to the pricing of interest rates. We start with observable pricing factors and develop a three-step ordinary least squares (OLS) estimator. In the first step, we decompose pricing factors into predictable components and factor innovations by regressing factors on their lagged levels. In the second step, we estimate exposures of Treasury returns with respect to lagged levels of pricing factors and contemporaneous pricing factor innovations. In the third step, we obtain the market price of risk parameters from a cross-sectional regression of the exposures of returns to the lagged pricing factors onto exposures to contemporaneous pricing factor innovations. We provide analytical standard errors that adjust for the generated regressor uncertainty. We also discuss the advantages of our method with respect to the recently suggested approaches by Joslin, Singleton, and Zhu (2011) and Hamilton and Wu (2012). In particular, we point out that the assumption of serially uncorrelated yield pricing errors underlying these likelihood-based methods imply excess return predictability not captured by the pricing factors. In contrast, because our approach is based on return regressions, we do not need to make assumptions about serial correlation in yield pricing errors. We report a specification with five principal components of zero coupon yields as pricing factors. We show that models with fewer factors are rejected in specification tests. The pricing errors in the five-factor specification are remarkably small and return pricing errors do not exhibit autocorrelation. We further find that level risk is priced and that the time variation of level risk is best captured by the second (the slope factor) and fifth principal components. The five-factor specification exhibits substantial variation in risk premiums and at the same time gives reasonable maximal Sharpe ratios. We next present a four-factor specification following Cochrane and Piazzesi (2008, CP) which includes the first three principal components of Treasury yields and a linear combination of forward rates designed to predict Treasury returns (the CP factor) as pricing factors. Unlike Cochrane and Piazzesi (2008), we allow for unconstrained prices of risk and find that the CP factor significantly prices all factors except slope. The magnitude and time pattern of the price of risk specification of the four-factor CP model is akin to that of the five-factor model, indicating that the two models capture term premium dynamics in a similar way. However, in-sample yield pricing errors are somewhat larger in the four-factor model. To compare the four- and five-factor models, we perform two out-of-sample exercises. In the first, we use the model-implied term premiums to infer the future path of average short-term interest rates. In the second, we estimate the models using returns on bonds maturing in less than or equal to ten years and then impute the model-implied yields of bonds with longer maturities. In both of these exercises, the five-factor model outperforms the four-factor specification. Hence, we choose the five-factor model to be our preferred specification. Our procedure can potentially be applied to any set of fixed income securities. In this paper, we use our approach to estimate an affine term structure model from returns on maturity-sorted portfolios of coupon-bearing Treasury securities. The availability of a zero coupon term structure is, therefore, not necessary to estimate the model. Yet, estimation from the returns on maturity sorted bond portfolios with pricing factors extracted from coupon bearing yields generates a zero coupon curve that is very similar to the Fama and Bliss discount bond yields. We present a number of extensions. First, we show how to estimate the model in the presence of unspanned factors. Such factors do not improve the cross-sectional fit of yields but do affect the time variation of prices of risk through their predictive power for the yield curve factors. In contrast to the four- and five-factor specifications, incorporating an unspanned real activity factor produces a significant price of slope risk. Second, we show how to impose linear restrictions on risk exposures and market prices of risk in the estimation of the model. Third, we show that the maximal Sharpe ratios implied by the four- and five-factor specifications are very reasonable, with an average level below one and peaks below 2.5. Fourth, we explain how the model can be used to fit the yield curve at the daily frequency. Finally, we show that the implied principal component loadings from the term structure model are statistically indistinguishable from the actual principal component loadings. Our paper is organized as follows. In Section 2, we present the model and our three-step estimator, and we show how to obtain model-implied yields from the estimated parameters. We further discuss the relation between our approach and other estimation methods in the literature. In Section 3, we present our main empirical findings. Section 4 discusses extensions and robustness checks. Section 5 concludes.

We outline an empirical approach to the estimation of dynamic term structure models. Our approach is computationally fast, gives rise to small pricing errors, and provides asymptotic standard errors for the model parameters of interest. Our method can be used for applications with observable factors and allows for unspanned factors. Our empirical analysis uncovers a number of new results and revisits certain controversies. First of all, we show in specification tests that the first three principal components of Treasury yields are not sufficient to span the cross section of Treasury returns. We, therefore, study a baseline specification that uses the first five principal components as pricing factors. Second, we show that the five-factor model gives rise to similar risk premiums and pricing kernel dynamics as a specification with three principal components and the Cochrane and Piazzesi (2008) forecasting factor. In both specifications, we find that practically all of the time variation in risk premiums is associated with level risk. However, the dynamics of the risk premium are mainly explained by the second and fifth principal component in the five-factor model and by the CP factor in the four-factor model. Nevertheless, we reject the restriction that the CP factor influences only the price of risk of the level shock and find that it significantly prices other sources of risk as well. For both the five-factor and the four-factor specifications, we find that slope risk is not priced. When comparing the two models based on out-of-sample predictions, we find that the five-factor model outperforms the four-factor model. We, therefore, designate it as our preferred specification. We also allow for certain factors to be unspanned and provide asymptotic standard errors in this case. Once we add unspanned macroeconomic factors to a model with the first three principal components as pricing factors, we find that slope risk is significantly time varying as a function of a real activity indicator. This result suggests that there is time variation in the pricing kernel that is not spanned by the yield curve. Our estimation method can be easily adapted and extended, as it solely relies on linear regressions. We present several examples of such extensions in the paper. First, we demonstrate that affine term structure models can be estimated even if zero coupon yields are not available. We estimate such a model by using five principal components of coupon bearing yields to price the cross section of maturity sorted returns. The resulting zero coupon yield curve is very similar to the Fama and Bliss discount curve, even though the estimation methods are vastly different. We furthermore present estimation results at the daily frequency, which are readily computed due to the ease of our estimation method. We leave it to future work to adapt the model to further applications, such as the estimation of inflation risk premiums, or to credit risk models.