On the empirical characteristic function process of the residuals in GARCH models and applications

The class of generalized autoregressive conditional heteroscedastic (GARCH) models has been proved to be particularly valuable in modeling financial data. This paper is devoted to study the empirical characteristic function process of the residuals. Specifically, it is shown that such process uniformly converges to the population characteristic function (CF) of the innovations in compact sets. The weak convergence of this empirical process, suitably normalized, is also studied. The limit depends on the population CF of the innovations, the equation defining the GARCH model and the parameter estimators employed to calculate the residuals. Applications of the obtained results for testing symmetry and goodness-of-fit to the law of the innovations are given. Copyright Sociedad de Estadística e Investigación Operativa 2014