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.
Highlights
We introduce a novel semiparametric regression framework for the Expected Shortfall (ES) by jointly modeling both, regression equations for the conditional quantile and the conditional ES
This regression approach relies on the class of strictly consistent joint loss functions introduced by Fissler and Ziegel (2016), which permits the joint elicitation of the quantile and the ES
Given a set of standard regularity conditions, we show consistency and asymptotic normality for both estimators, which we verify numerically through extensive simulations
Summary
We introduce a novel semiparametric regression framework for the Expected Shortfall (ES) by jointly modeling both, regression equations for the conditional quantile and the conditional ES. The situation for the Z-estimator and the availability of underlying identification functions (moment conditions) is equivalent to the loss functions used for the M-estimator and only allows for joint Z-estimation of both regression equations Such a regression framework for the ES is essential for a variety of academic disciplines which consider measuring, forecasting and the evaluation of extreme risks. In a simultaneous and independent work, Patton et al (2019) introduce a similar M-estimator for such joint regression models for the quantile and the ES in the autoregressive context and show its asymptotic behavior This approach differs from our paper in the following ways.
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