In this paper, we consider autoregressive models with conditional autoregressive variance, including the case of homoscedastic AR models and the case of ARCH models. Our aim is to test the hypothesis of normality for the innovations in a completely non-parametric way, that is, without imposing parametric assumptions on the conditional mean and volatility functions. To this end, the Cramér–von Mises test based on the empirical distribution function of non-parametrically estimated residuals is shown to be asymptotically distribution-free. We demonstrate its good performance for finite sample sizes in a small simulation study.
AMS 2010 Classification: Primary 62 M10, Secondary 62 G10