Testing Hypotheses on the Innovations Distribution in Semi-Parametric Conditional Volatility Models

Testing symmetry or quantile assumptions on the innovations distribution can be of invaluable help to improve or simplify the statistical procedures designed for GARCH-type models. In particular, evaluation of the conditional value-at-risk (VaR) or construction of confidence intervals for predictions requires estimating quantiles of the innovations distribution. We propose tests of different hypotheses: adequacy of a set of parametric quantiles, mean–median equality, symmetry of extreme quantiles, and zero-median in presence of a conditional mean. The tests rely on the asymptotic distribution of the empirical distribution function of the residuals. They are generally model-free (though not estimation-free) and thus are simple to implement. Efficiency comparisons are made using the Bahadur approach. Numerical studies based on simulated and real data are provided to illustrate the usefulness of the proposed tests for risk management or statistical purposes.