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A Joint Quantile and Expected Shortfall Regression Framework

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  • Timo Dimitriadis
  • Sebastian Bayer

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 a strictly consistent loss function for the pair quantile and ES, which allows 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 function depends on two specification functions, whose choice affects 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 ES into the Basel Accords.

Suggested Citation

  • Timo Dimitriadis & Sebastian Bayer, 2017. "A Joint Quantile and Expected Shortfall Regression Framework," Papers 1704.02213, arXiv.org, revised Aug 2017.
    • Handle: RePEc:arx:papers:1704.02213
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    References listed on IDEAS

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    5. Song Xi Chen, 2008. "Nonparametric Estimation of Expected Shortfall," Journal of Financial Econometrics, Oxford University Press, vol. 6(1), pages 87-107, Winter.
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    Cited by:

    1. Fortin, Alain-Philippe & Simonato, Jean-Guy & Dionne, Georges, 2023. "Forecasting expected shortfall: Should we use a multivariate model for stock market factors?," International Journal of Forecasting, Elsevier, vol. 39(1), pages 314-331.
    2. Patton, Andrew J. & Ziegel, Johanna F. & Chen, Rui, 2019. "Dynamic semiparametric models for expected shortfall (and Value-at-Risk)," Journal of Econometrics, Elsevier, vol. 211(2), pages 388-413.
    3. Taylor, James W., 2020. "Forecast combinations for value at risk and expected shortfall," International Journal of Forecasting, Elsevier, vol. 36(2), pages 428-441.
    4. Joanna Bruzda, 2020. "Multistep quantile forecasts for supply chain and logistics operations: bootstrapping, the GARCH model and quantile regression based approaches," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 28(1), pages 309-336, March.
    5. Caporale, Guglielmo Maria & Zekokh, Timur, 2019. "Modelling volatility of cryptocurrencies using Markov-Switching GARCH models," Research in International Business and Finance, Elsevier, vol. 48(C), pages 143-155.
    6. Sebastian Bayer & Timo Dimitriadis, 2018. "Regression Based Expected Shortfall Backtesting," Papers 1801.04112, arXiv.org, revised Sep 2019.
    7. Ziegel, Johanna F. & Krueger, Fabian & Jordan, Alexander & Fasciati, Fernando, 2017. "Murphy Diagrams: Forecast Evaluation of Expected Shortfall," Working Papers 0632, University of Heidelberg, Department of Economics.

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