A joint quantile and expected shortfall regression framework

We introduce a novel regression framework which simultaneously models the quantile and the Expected Shortfall (ES) of a response variable given a set of covariates. This regression is based on strictly consistent loss functions for the pair consisting of the quantile and the ES, which allow for M- and Z-estimation of the joint regression parameters. We show consistency and asymptotic normality for both estimators under weak regularity conditions. The underlying loss functions depend on two specification functions, whose choices affect the properties of the resulting estimators. We find that the Z-estimator is numerically unstable and thus, we rely on M-estimation of the model parameters. Extensive simulations verify the asymptotic properties and analyze the small sample behavior of the M-estimator for different specification functions. This joint regression framework allows for various applications including estimating, forecasting and backtesting ES, which is particularly relevant in light of the recent introduction of the ES into the Basel Accords. We illustrate this through two exemplary empirical applications in forecasting and forecast combination of the ES.