Limit Theory for Forecasts of Extreme Distortion Risk Measures and Expectiles

Abstract

Abstract We develop central limit theory for tail risk forecasts in general location–scale models. We do so for a wide range of risk measures, viz. distortion risk measures (DRMs) and expectiles. Two popular members of the class of DRMs are the Value-at-Risk and the Expected Shortfall. The forecasts we consider are motivated by a Pareto-type tail assumption for the innovations and allow for extrapolation beyond the range of available observations. Simulations reveal adequate coverage of the forecast intervals derived from the limit theory. An empirical application demonstrates that our estimators outperform nonparametric alternatives when forecasting extreme risk in sufficiently large samples.