Expected Shortfall (ES) has become a central risk measure in modern financial risk management because it captures the magnitude of extreme losses beyond a given quantile level. Estimating conditional ES in the presence of covariates is therefore an important statistical problem.

This thesis studies the problem of linear conditional ES regression and investigates the behavior of several representative estimators under challenging data-generating environments. In particular, we focus on three estimators that embody distinct methodological principles: an iterative Rockafellar-type estimator (iRock), the classical Two-Step procedure based on quantile regression and tail averaging, and the Averaged Quantile (AQ) approach that approximates ES through aggregation of lower-tail quantile regressions.

 

To systematically evaluate these estimators, we conduct an extensive simulation study under a range of structural scenarios. Estimation performance is assessed using out-of-sample prediction error for the conditional ES function, supplemented by parameter recovery analysis in scenarios where the true linear coefficients are well defined.

 

The results illustrate how different estimators respond to structural challenges such as tail heaviness, variance heterogeneity, and model misspecification. The proposed iterative estimator demonstrates improved stability in several difficult regimes, while the comparison highlights characteristic strengths and limitations of existing approaches. These findings contribute to a clearer understanding of practical estimation strategies for conditional ES regression in complex environments.