Using a Nonlinear Filter to Estimate a Multifactor Term Structure Model with Gaussian Mixture Innovations

Based on an idea in Backus, Foresi, and Telmer (1998) we extend the class of discrete-time affine multifactor Gaussian models by allowing factor innovations to be distributed as Gaussian mixtures. This is motivated by the observation that bond yield changes for some maturities are distinctly nonnormal. We derive an analytical formula for bond yields as a function of factors. For the model with Gaussian mixture innovations these functions are still affine. The model allows the resulting distribution of yields and yield changes to assume a wide variety of shapes. In particular, it can account for non-vanishing skewness and excess kurtosis that varies with maturity. For estimation, the model is cast into state space form. If the model were purely Gaussian, the corresponding state space model would be linear and Gaussian and could be estimated by maximum likelihood based on the Kalman filter. For the class of term structure models considered in this paper, however, the corresponding state space model has a transition equation for which the innovation is distributed as a Gaussian mixture. The exact filter for such a state space model is nonlinear in observations. Moreover, the exact filtering density at time t is a Gaussian mixture for which the number of components is exponentially growing with time, rendering a practical application of the exact filter impossible. To deal with this problem we propose a new approximate filter that preserves the nonlinearity of the exact solution but that restricts the number of components in the mixture distributions involved. As an application, a two-factor model with Gaussian mixture innovations is estimated with US data using the approximate nonlinear filter. The model turns out to be superior to its purely Gaussian counterpart, as it captures nonnormality in bond yield changes