Optimal predictions of powers of conditionally heteroskedastic processes

Abstract

The standard method for estimating powers of conditionally heteroskedastic processes is a two-step procedure in which the volatility is estimated by ga.us-sian quasi-maximum likelihood (QML) in a first step, and an empirical mean of the rescaled innovations is computed in a second step. This paper proposes an alternative one-step procedure, based on an appropriate non-gaussian QML estimation of the model, and establishes the asymptotic properties of the two ap¬proaches. Their performances are compared for finite-order GARCH models and for the ARCH(oo). For the standard GARCH(p, q) and the Asymmetric Power GARCH(p, g), it is shown that the asymptotic relative efficiency of the estimators only depends on the prediction problem and on some moments of the independent process. An application to indexes of major stock exchanges is proposed.