Diagnosing the distribution of GARCH innovations

The Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model, designed to model volatility clustering, exhibits heavy-tailedness regardless of the distribution of its innovation term. When applying the model to financial time series, the distribution of innovations plays an important role for risk measurement and option pricing. We investigate methods on diagnosing the distribution of GARCH innovations. For GARCH processes that are close to integrated-GARCH (IGARCH), we show that the method based on estimated innovations is not reliable, whereas an alternative approach based on analyzing the tail index of a GARCH series performs better. The alternative method leads to a formal test on the distribution of GARCH innovations.