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 ESare 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.
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.
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(ES)
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The foundation for this joint regression is a recent result by Fissler and Ziegel (2016), who show that the quantile and the ESare jointly elicitable. This joint elicitability
This regression is based on a strictly consistent loss function for the pair quantile and ES, which
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Such a parameter estimation is not possible for an Expected Shortfall regression alone as Expected Shortfall is not elicitable.
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both,
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the
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loss function used for the M-estimation
underlying loss function
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choices affect
choice affects
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In an extensive simulation study, we
Extensive simulations
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under different choices for the
for different
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We find that the Z-estimator is numerically unstable and thus, we rely on M-estimation of the model parameters.
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Expected Shortfall
ES
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upcoming introduction of Expected Shortfall
recent introduction of ES