Title: A Joint Quantile and Expected Shortfall Regression Framework,
Abstract: 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
estimators under weak regularity conditions. The underlying loss function
depends on two specification functions, whose choice affects 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 ES into the Basel Accords.