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

TL;DR

This paper introduces a nonparametric Cramér-von Mises test for normality of innovations in AR-ARCH models, effective without parametric assumptions on mean and volatility, with demonstrated good finite-sample performance.

Contribution

It develops an asymptotically distribution-free nonparametric test for Gaussian innovations in AR-ARCH models, applicable to both homoscedastic and heteroscedastic cases.

Findings

Test is asymptotically distribution-free.

Simulation shows good finite-sample performance.

Applicable to various AR-ARCH model types.

Abstract

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 Cram\'er-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.