Electronic Journal of Statistics
- Electron. J. Statist.
- Volume 13, Number 1 (2019), 1823-1871.
A joint quantile and expected shortfall regression framework
Timo Dimitriadis and Sebastian Bayer
Full-text: Open access
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 strictly consistent loss functions for the pair consisting of the quantile and the ES, which allow 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 functions depend on two specification functions, whose choices affect 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 the ES into the Basel Accords. We illustrate this through two exemplary empirical applications in forecasting and forecast combination of the ES.
Article information
Source
Electron. J. Statist., Volume 13, Number 1 (2019), 1823-1871.
Dates
Received: May 2018
First available in Project Euclid: 7 June 2019
Permanent link to this document
https://projecteuclid.org/euclid.ejs/1559872834
Digital Object Identifier
doi:10.1214/19-EJS1560
Mathematical Reviews number (MathSciNet)
MR3959874
Zentralblatt MATH identifier
07080063
Keywords
Expected shortfall joint elicitability joint regression M-estimation quantile regression
Rights
Creative Commons Attribution 4.0 International License.
Citation
Dimitriadis, Timo; Bayer, Sebastian. A joint quantile and expected shortfall regression framework. Electron. J. Statist. 13 (2019), no. 1, 1823--1871. doi:10.1214/19-EJS1560. https://projecteuclid.org/euclid.ejs/1559872834
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