This article is a critical survey of models designed for pricing fixed-income securities and their associated term structures of market yields. Our primary focus is on the interplay between the theoretical specification of dynamic term structure models and their empirical fit to historical changes in the shapes of yield curves. We begin by overviewing the dynamic term structure models that have been fit to treasury or swap yield curves and in which the risk factors follow diffusions, jump-diffusion, or have "switching regimes." Then the goodness-of-fit of these models is assessed relative to their abilities to (i) match linear projections of changes in yields onto the slope of the yield curve; (ii) match the persistence of conditional volatilities, and the shapes of term structures of unconditional volatilities, of yields; and (iii) to reliably price caps, swaptions, and other fixed-income derivatives. For the case of defaultable securities we explore the relative fits to historical yield spreads.

[This article develops a rich class of discrete-time, nonlinear dynamic term structure models (DTSMs). Under the risk-neutral measure, the distribution of the state vector X t resides within a family of discrete-time affine processes that nests the exact discrete-time counterparts of the entire class of continuous-time models in Duffie and Kan (1996) and Dai and Singleton (2000). Under the historical distribution, our approach accommodates nonlinear (nonaffine) processes while leading to closed-form expressions for the conditional likelihood functions for zero-coupon bond yields. As motivation for our framework, we show that it encompasses many of the equilibrium models with habit-based preferences or recursive preferences and long-run risks. We illustrate our methods by constructing maximum likelihood estimates of a nonlinear discrete-time DTSM with habit-based preferences in which bond prices are known in closed form. We conclude that habit-based models, as typically parameterized in the literature, do not match key features of the conditional distribution of bond yields.]

[This article develops and empirically implements an arbitrage-free, dynamic term structure model with "priced" factor and regime-shift risks. The risk factors are assumed to follow a discrete-time Gaussian process, and regime shifts are governed by a discrete-time Markov process with state-dependent transition probabilities. This model gives closed-form solutions for zero-coupon bond prices, an analytic representation of the likelihood function for bond yields, and a natural decomposition of expected excess returns to components corresponding to regime-shift and factor risks. Using monthly data on U.S. Treasury zero-coupon bond yields, we show a critical role of priced, state-dependent regime-shift risks in capturing the time variations in expected excess returns, and document notable differences in the behaviors of the factor risk component of the expected returns across high and low volatility regimes. Additionally, the state dependence of the regime-switching probabilities is shown to capture an interesting asymmetry in the cyclical behavior of interest rates. The shapes of the term structure of volatility of bond yield changes are also very different across regimes, with the well-known hump being largely a low-volatility regime phenomenon.]

This article is a critical survey of models designed for pricing fixed-income securities and their associated term structures of market yields. Our primary focus is on the interplay between the theoretical specification of dynamic term structure models and their empirical fit to historical changes in the shapes of yield curves. We begin by overviewing the dynamic term structure models that have been fit to treasury or swap yield curves and in which the risk factors follow diffusions, jump-diffusion, or have “switching regimes.” Then the goodness-of-fit of these models is assessed relative to their abilities to (i) match linear projections of changes in yields onto the slope of the yield curve; (ii) match the persistence of conditional volatilities, and the shapes of term structures of unconditional volatilities, of yields; and (iii) to reliably price caps, swaptions, and other fixed-income derivatives. For the case of defaultable securities we explore the relative fits to historical yield spreads.

Linear projections of returns on the slope of the yield curve have contradicted the implications of the traditional “expectations theory”. This paper shows that these findings are not puzzling relative to a large class of richer dynamic term structure models. Specifically, we match all the key empirical findings reported by Fama and Bliss ((1987) American Economic Review 77 (4), 680–692) and Campbell and Shiller ((1991) Review of Economic Studies 58, 495–514), among others, within large subclasses of affine and quadratic-Gaussian term structure models. Additionally, we show that certain “risk-premium adjusted” projections of changes in yields on the slope of the yield curve recover the coefficients of unity predicted by the models. Key to this matching are parameterizations of the market prices of risk that let the risk factors affect the market prices of risk directly, and not only through factor volatilities. The risk premiums have a simple form consistent with Fama's findings on the predictability of forward rates, and are also shown to be consistent with interest-rate feedback rules used by a monetary authority in setting monetary policy.

This paper explores the structural differences and relative goodness‐of‐fits of affine term structure models (ATSMs). Within the family of ATSMs there is a trade‐off between flexibility in modeling the conditional correlations and volatilities of the risk factors. This trade‐off is formalized by our classification of N ‐factor affine family into non‐nested subfamilies of models. Specializing to three‐factor ATSMs, our analysis suggests, based on theoretical considerations and empirical evidence, that some subfamilies of ATSMs are better suited than others to explaining historical interest rate behavior.

This article develops a rich class of discrete-time, nonlinear dynamic term structure models (DTSMs). Under the risk-neutral measure, the distribution of the state vector Xt resides within a family of discrete-time affine processes that nests the exact discrete-time counterparts of the entire class of continuous-time models in Duffie and Kan (1996) and Dai and Singleton (2000). Under the historical distribution, our approach accommodates nonlinear (nonaffine) processes while leading to closed-form expressions for the conditional likelihood functions for zero-coupon bond yields. As motivation for our framework, we show that it encompasses many of the equilibrium models with habit-based preferences or recursive preferences and long-run risks. We illustrate our methods by constructing maximum likelihood estimates of a nonlinear discrete-time DTSM with habit-based preferences in which bond prices are known in closed form. We conclude that habit-based models, as typically parameterized in the literature, do not match key features of the conditional distribution of bond yields.

This article develops and empirically implements an arbitrage-free, dynamic term structure model with “priced” factor and regime-shift risks. The risk factors are assumed to follow a discrete-time Gaussian process, and regime shifts are governed by a discrete-time Markov process with state-dependent transition probabilities. This model gives closed-form solutions for zero-coupon bond prices, an analytic representation of the likelihood function for bond yields, and a natural decomposition of expected excess returns to components corresponding to regime-shift and factor risks. Using monthly data on U.S. Treasury zero-coupon bond yields, we show a critical role of priced, state-dependent regime-shift risks in capturing the time variations in expected excess returns, and document notable differences in the behaviors of the factor risk component of the expected returns across high and low volatility regimes. Additionally, the state dependence of the regime-switching probabilities is shown to capture an interesting asymmetry in the cyclical behavior of interest rates. The shapes of the term structure of volatility of bond yield changes are also very different across regimes, with the well-known hump being largely a low-volatility regime phenomenon.