This paper derives the asymptotic distribution for a number of rank-based and classical residual specification tests in AR–GARCH type models. We consider tests for the null hypotheses of no linear and quadratic serial residual autocorrelation, residual symmetry, and no structural breaks. We also apply our method to backtesting Value-at-Risk. For these tests we show that, generally, no size correction is needed in the asymptotic test distribution when applied to AR–GARCH residuals obtained through Gaussian quasi maximum likelihood estimation. To be precise, we give exact expressions for the limiting null distribution of the test statistics applied to (standardized) residuals, and find that standard critical values often, though not always, lead to conservative tests. For this result, we give simple necessary and sufficient conditions. Simulations show that our asymptotic approximations work well for a large number of AR–GARCH models and parameter values. We also show that the rank-based tests often, though not always, have superior power properties over the classical tests, even if they are conservative. An empirical application illustrates the relevance of these tests to the AR–GARCH models for weekly stock market return indices of some major and emerging countries.

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