Expected Shortfall Regression via Optimization

To provide a comprehensive summary of the tail distribution, the expected shortfall is defined as the average over the tail above (or below) a certain quantile of the distribution. The expected shortfall regression captures the heterogeneous covariate-response relationship and describes the covariate effects on the tail of the response distribution. Based on a critical observation that the superquantile regression from the operations research literature does not coincide with the expected shortfall regression, we propose and validate a novel optimization-based approach for the linear expected shortfall regression, without additional assumptions on the conditional quantile models. While the proposed loss function is implicitly defined, we provide a prototype implementation of the proposed approach with some initial expected shortfall estimators based on binning techniques. With practically feasible initial estimators, we establish the consistency and the asymptotic normality of the proposed estimator. The proposed approach achieves heterogeneity-adaptive weights and therefore often offers efficiency gain over existing linear expected shortfall regression approaches in the literature, as demonstrated through simulation studies.