Abstract
Expected Shortfall (\(\mathrm {ES}\)) is one of the most heavily used measures of financial risk. It is defined as a scaled integral of the quantile of the profit-and-loss distribution up to a certainly confidence level. As such, quantile regression (QR) and the closely related expectile regression (ER) methods are natural techniques for estimating \(\mathrm {ES}\). In this paper, we survey QR and ER based estimators of ES and introduce several novel variants. We compare the performance of these methods through simulation and through a data analysis based on four major US market indices: the S&P 500 Index, the Russell 2000 Index, the Dow Jones Industrial Average, and the NASDAQ Composite Index. Our results suggest that QR and ER methods often work better than other, more standard, approaches.
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Notes
A dimension reduction subspace is the column space of any matrix \(\mathbf {\Gamma }\) such that the response \(r_k\) and the vector of predictors \({\mathbf {r}}_{k,p}\) are conditionally independent given \(\mathbf {\Gamma }^{\top }{\mathbf {r}}_{k,p}\). The central subspace is the dimension reduction subspace with the smallest dimension.
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The authors wish to thank the referees for comments that lead to improvements in the presentation of this paper. The second author’s work was funded, in part, by the Russian Science Foundation (Project No. 17-11-01098).
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Christou, E., Grabchak, M. Estimation of Expected Shortfall Using Quantile Regression: A Comparison Study. Comput Econ 60, 725–753 (2022). https://doi.org/10.1007/s10614-021-10164-z
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DOI: https://doi.org/10.1007/s10614-021-10164-z