The uncertainty in extreme risk forecasts from covariate-augmented volatility models

For a GARCH-type volatility model with covariates, we derive asymptotically valid forecast intervals for risk measures, such as the Value-at-Risk or Expected Shortfall. To forecast these, we use estimators from extreme value theory. In the volatility model, we allow for leverage effects and the inclusion of exogenous variables, e.g., volatility indices or high-frequency volatility measures. In simulations, we find coverage of the forecast intervals to be adequate for sufficiently extreme risk levels and sufficiently large samples, which is consistent with theory. Finally, we investigate if covariate information from volatility indices or high-frequency data improves risk forecasts for major US stock indices. While—in our framework—volatility indices appear to be helpful in this regard, intra-day data are not.