Specification Analysis of Affine Term Structure Models

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 tradeoff 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 N + 1 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. IN SPECIFYING A DYNAMIC TERM STRUCTURE MODEL-one that describes the comovement over time of shortand long-term bond yields-researchers are inevitably confronted with trade-offs between the richness of econometric representations of the state variables and the computational burdens of pricing and estimation. It is perhaps not surprising then that virtually all of the empirical implementations of multifactor term structure models that use time series data on longand short-term bond yields simultaneously have focused on special cases of "affine" term structure models (ATSMs). An ATSM accommodates time-varying means and volatilities of the state variables through affine specifications of the risk-neutral drift and volatility coefficients. At the same time, ATSMs yield essentially closed-form expressions for zero-coupon-bond prices (Duffie and Kan (1996)), which greatly facilitates pricing and econometric implementation. The focus on ATSMs extends back at least to the pathbreaking studies by Vasicek (1977) and Cox, Ingersoll, and Ross (1985), who presumed that the instantaneous short rate r (t) was an affine function of an N-dimensional state vector Y(t), r(t) = 50 + 8YY(t), and that Y(t) followed Gaussian and square-root diffusions, respectively. More recently, researchers have explored formulations of ATSMs that extend the one-factor Markov represen* Dai is from New York University. Singleton is from Stanford University. We thank Darrell Duffie for extensive discussions, Ron Gallant and Jun Liu for helpful comments, the editor and the referees for valuable inputs, and the Financial Research Initiative of the Graduate School of Business at Stanford University for financial support. We are also grateful for comments from seminar participants at the NBER Asset Pricing Group meeting, the Western Finance Association Annual Meetings, Duke University, London Business School, New York University, Northwestern University, University of Chicago, University of Michigan, University of Washington at St. Louis, Columbia University, Carnegie Mellon University, and CIRANO.