Analysis of Multifactor Affine Yield Curve Models

In finance and economics much work has been done on the theoretical modeling and statistical estimation of the yield curve, defined as the relationship between -1/tau logp(t)(tau) and tau, where p(t)(tau) is the time t price of a zero-coupon bond with payoff I at maturity date t + tau. Of considerable current interest are models of the yield curve in which a collection of observed and latent factors determine the market price of factor risks, the stochastic discount factor, and the arbitrage-free bond prices. The model is particularly interesting from a statistical perspective, because the yields are complicated nonlinear functions of the underlying parameters (e.g., those appearing in the evolution dynamics of the factors and those appearing in the model of the factor risks). This nonlinearity tends to produce a likelihood function that is multimodal. In this article we revisit the question of how such models should be fit from the Bayesian viewpoint. Key aspects of the inferential framework include (a) a prior on the parameters of the model that is motivated by economic considerations, in particular, those involving the slope of the implied yield curve; (b) posterior simulation of the parameters in ways to improve the efficiency of the MCMC output, for example, through sampling of the parameters marginalized over the factors and tailoring of the proposal densities in the Metropolis-Hastings steps using information about the mode and curvature of the current target based on the output of a simulating annealing algorithm; and (c) measures to mitigate numerical instabilities in the fitting through reparameterizations and square root filtering recursions. We apply the techniques to explain the monthly yields on nine U.S. Treasury Bills (with maturities ranging from 1 month to 120 months) over the period January 1986-December 2005. The model contains three factors, one latent and two observed. We also consider the problem of predicting the nine yields for each month of 2006. We show that the (multi-step-ahead) prediction regions properly bracket the actual yields in those months, thus highlighting the practical value of the fitted model.