A note on nonparametric testing for Gaussian innovations in AR-ARCH models

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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 nonparametric way, i. e. 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 nonparametrically estimated residuals is shown to be asymptotically distribution-free. We demonstrate its good performance for finite sample sizes in a simulation study.