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 a strictly consistent loss function for the pair quantile and ES, which allows for M- and Z-estimation of the joint regression parameters. We show consistency and asymptotic normality for both, the M- and the Z-estimator under standard regularity conditions. The underlying loss function depends on two specification functions, whose choice affects the properties of the resulting estimators. Extensive simulations verify the asymptotic properties and analyze the small sample behavior of the M-estimator for different specification functions. We find that the Z-estimator is numerically unstable and thus, we rely on M-estimation of the model parameters. 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 ES in the Basel Accords.