Adaptiveness of the empirical distribution of residuals in semi-parametric conditional location scale models

This paper addresses the problem of deriving asymptotic distribution empirical function Fˆn residuals in a general class time series models, including conditional mean and heteroscedaticity, whose independent identically distributed errors have unknown F. We show that, for large models (including standard ARMA-GARCH with symmetric innovations), n{Fˆ n(·)?F(·)} is impacted by estimation but does not depend on model parameters. It thus neither asymptotically free, as case purely linear nor dependent, some nonlinear models. The stochastic equicontinuity also established. consider an application to Value-at-Risk.