Prediction Problems for Square-Transformed Stationary Processes

Abstract

This paper discusses the prediction problems for square-transformed process, Yt = Xt2, where Xt is a stationary process with spectral density g(λ). The square-transformation is important in prediction of the volatility of ARCH models. First, we evaluate the mean square prediction error for square-transformed process when the predictor is constructed from the true spectral density g(λ). However, it is often that the true structure g(λ) is not completely specified. Hence, we consider the problem of misspecified prediction when a conjectured spectral density fθ(λ), θ ∈ Θ, is fitted to g(λ). Then, constructing the best linear predictor based on fθ(λ), we can evaluate the prediction error for square-transformed process. Also, we consider a bias adjusted prediction problem for the above two cases. Furthermore, we may suppose that Xt is a non-Gaussian process. Then, we evaluate the mean square prediction errors when the best linear predictor is constructed by the true spectral density g(λ) and the conjectured spectral density fθ (λ), respectively. Since θ is usually unknown we estimate it by a quasi-MLE \(\hat \theta _Q\). The second-order asymptotic approximations of the mean square errors of the predictors based on g(λ) and f\(\hat \theta _Q\)(λ) are given. Finally, we provide some numerical examples, which show some unexpected features.