On the Independence of the Standardized One-Step-Ahead Prediction Errors in Arch Models

Abstract

In statistical modeling contexts, the use of one-step-ahead prediction errors for testing hypotheses on the forecasting ability of an assumed model has been widely considered (see, e.g., Xekalaki et al. (2003, in Stochastic Musings, J.Panaretos (ed.), Laurence Erlbaum), Degiannakis and Xekalaki (2005, Journal of Applied Stochastic Models in Business and Industry). Quite often, the testing procedure requires independence in a sequence of recursive standardized prediction errors, which cannot always be readily deduced particularly in the case of econometric modeling. In this paper, the results of a series of Monte Carlo simulations reveal that independence can be assumed to hold. They are also indicative of a chi-square distribution for the sum of squared standardized one-step-ahead prediction errors. Some theoretical justification of these findings can be traced in Degiannakis and Xekalaki's results (2005, Journal of Applied Stochastic Models in Business and Industry).

Keywords: ARCH models, Standardized Prediction Error Criterion, Model selection, Monte Carlo Simulation, Predictability, One-step-ahead Prediction Errors

JEL Classification: C14, C22, C53