Estimating Discrete-Time Gaussian Term Structure Models in Canonical Companion Form

Abstract

This article formally introduces a convenient parametrization for the popular class of discrete-time essentially-affine term structure models in the spirit of Duffee (2002), and Ang and Piazzesi (2003). First, I show that if the term structure is spanned by N_f latent state variables, all pricing information must also be contained in N_f shortest-maturity forward rates. Every no-arbitrage model of the type studied is therefore observationally equivalent to a unique canonical model in which these forward rates act as factors. Second, the risk-neutral transition matrix of the canonical model is conveniently parametrized by N_f unrestricted real numbers, and the risk neutral drift is a function of factor covariance matrix, plus one extra parameter. Third, although it may appear restrictive to specify the shortest-maturity forward rates as factors, the model can be estimated using all information in observed bond prices, either by Kalman filter, or assuming perfect observability of certain combinations of yields. Monte-Carlo evidence suggests that both approaches lead to similar out-of-sample forecasting performance in artificial data sets. Finally, by using unique insights offered by the canonical companion form, I discuss some difficulties in fitting term structure models of the essentially-affine class to the standard set of Fama-Bliss discount bonds. The problems stem from the existence of factors seemingly inconsistent with the assumption of no arbitrage. This discussion may have implications for interpreting the evidence of bond return predictability, and for the question of whether imposing no arbitrage can improve yield forecasts.