We introduce a novel regression framework which simultaneously models the quantile and the Expected Shortfall of a response variable given a set of covariates. The foundation for this joint regression is a recent result by Fissler and Ziegel (2016), who show that the quantile and the ES are jointly elicitable. This joint elicitability allows for M- and Z-estimation of the joint regression parameters. Such a parameter estimation is not possible for an Expected Shortfall regression alone as Expected Shortfall is not elicitable. We show consistency and asymptotic normality for the M- and Z-estimator under standard regularity conditions. The loss function used for the M-estimation depends on two specification functions, whose choices affect the properties of the resulting estimators. In an extensive simulation study, we verify the asymptotic properties and analyze the small sample behavior of the M-estimator under different choices for the specification functions. This joint regression framework allows for various applications including estimating, forecasting and backtesting Expected Shortfall, which is particularly relevant in light of the upcoming introduction of Expected Shortfall in the Basel Accords.
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