Abstract

As a coherent risk measure, Expected Shortfall (ES) has garnered in

creasing attention due to its merits in quantitative risk management, particularly

its ability to capture tail risks. Consequently, the Expected Shortfall regression

model has recently been proposed in conjunction with quantile regression to investigate the conditional effect of predictors on a response variable of interest.

However, existing approaches have encountered challenges in effectively estimating the conditional expected shortfall regression at extreme levels, primarily due

to the scarcity of observations in the tails.

To address this issue, this paper

first fits a joint regression model of conditional quantile and conditional ES at

an intermediate level using a two-step procedure. Subsequently, three extrapolative approaches are proposed to study the extreme conditional ES estimation.

We also develop the asymptotic properties of all proposed estimators within a

conditional heteroscedastic extreme framework.

Furthermore, simulations are

conducted to examine the finite sample performance of our methods. Finally, a

real-world example underscores the practical advantages of extreme conditional

ES regression.

Information

Preprint No.SS-2025-0187
Manuscript IDSS-2025-0187
Complete AuthorsQingzhao Zhong, Jingyu Ji, Liujun Chen, Yanxi Hou, Deyuan Li
Corresponding AuthorsJingyu Ji
Emailsjingyuji@cueb.edu.cn

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Acknowledgments

Jingyu Ji’s research was partially supported by the National Natural Science

Foundation of China grant 72403172. Liujun Chen’s research was partially

supported by the National Natural Science Foundation of China grants

12301387 and 12471279. Yanxi Hou’s research was partially supported by

the National Natural Science Foundation of China grant 72171055. Deyuan

Li’s research was partially supported by the National Natural Science Foundation of China grant 12471279.

Supplementary Materials

The online Supplementary Material contains some simulation results, auxiliary results and all technical proofs.

Supplementary materials are available for download.