Optimal predictions of powers of conditionally heteroscedastic processes

In conditionally heteroscedastic models, the optimal prediction of powers, or logarithms, of the absolute value has a simple expression in terms of the volatility and an expectation involving the independent process. A natural procedure for estimating this prediction is to estimate the volatility in the first step, for instance by Gaussian quasi-maximum-likelihood or by least absolute deviations, and to use empirical means based on rescaled innovations to estimate the expectation in the second step. The paper proposes an alternative one-step procedure, based on an appropriate non-Gaussian quasi-maximum-likelihood estimator, and establishes the asymptotic properties of the two approaches. Asymptotic comparisons and numerical experiments show that the differences in accuracy can be important, depending on the prediction problem and the innovations distribution. An application to indices of major stock exchanges is given.