Risk management requires accurate estimates of the downside risk of financial investment. For financial return series, specific features such as heavy-tailedness and volatility clustering impose extra difficulties in downside risk evaluation. The Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model captures these features regardless of the distributional assumptions on the innovation process. Nevertheless, distribution of the innovations plays an important role when analyzing conditional and unconditional downside risk. We show that the diagnosis method on the heavy-tailedness of GARCH innovations in McNeil and Frey (2000) is not reliable for GARCH processes that are close to non-stationarity. With comparing different tail index estimates, we provide an alternative approach which leads to a formal test on the distribution of GARCH innovations. Empirical analysis on real data confirms similar finding as in McNeil and Frey (2000) that when modeling financial returns with the GARCH model, the downside distribution of innovations possesses heavier tail than the usual normality assumption.