Exact predictive densities for linear models with arch disturbances

Abstract

It is shown how exact predictive densities may be formed in the ARCH linear model by means of Monte Carlo integration with importance sampling. Several improvements in computational efficiency over earlier implementations of this procedure are developed, including use of the exact likelihood function rather than an asymptotic approximation to construct the importance sampling distribution, and antithetic acceleration of convergence. A numerical approach to the formulation of posterior odds ratios and the combination of non-nested models is also introduced. These methods are applied to daily quotations of closing stock prices. Forecasts are formulated using linear models, ARCH linear models and an integrated model constructed from the posterior probabilities of the respective models. The use of the exact predictive density in a decision-theoretic context is illustrated by deriving the optimal day-to-day portfolio adjustments of a trader with constant relative risk aversion.