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We introduce a novel regression framework which simultaneously models thequantile and the Expected Shortfall (ES) of a response variable given a set ofcovariates. This regression is based on a strictly consistent loss function forthe pair quantile and ES, which allows for M- and Z-estimation of the jointregression parameters. We show consistency and asymptotic normality for bothestimators under weak regularity conditions. The underlying loss functiondepends on two specification functions, whose choice affects the properties ofthe resulting estimators. We find that the Z-estimator is numerically unstableand thus, we rely on M-estimation of the model parameters. Extensivesimulations verify the asymptotic properties and analyze the small samplebehavior of the M-estimator for different specification functions. This jointregression framework allows for various applications including estimating,forecasting, and backtesting ES, which is particularly relevant in light of therecent introduction of ES into the Basel Accords.

A Joint Quantile and Expected Shortfall Regression Framework