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

Abstract

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