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Quasi Maximum Likelihood Estimation of Value at Risk and Expected Shortfall

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  • Catania, Leopoldo
  • Luati, Alessandra

Abstract

Quasi maximum likelihood estimation of Value at Risk (VaR) and Expected Shortfall (ES) is discussed. The reference likelihood is that of a location-scale asymmetric Laplace distribution, related to a family of loss functions that lead to strictly consistent scoring functions for joint estimation of VaR and ES. The case of zero mean processes is considered, where quasi maximum likelihood estimators (QMLE) are consistent and asymptotically normal, as well as the case of non-zero mean processes, where quasi maximum likelihood estimators lead to inconsistent estimates due to lack of identification. In the latter situation, the asymptotic properties of two stage quasi maximum likelihood estimators (2SQMLE) are derived. QMLE and 2SQMLE are related with sample and M-estimators and compared in terms of asymptotic efficiency. A simulation study investigates the finite sample properties of QMLE, 2SQMLE, sample and M-estimators of expected shortfall.

Suggested Citation

  • Catania, Leopoldo & Luati, Alessandra, 2025. "Quasi Maximum Likelihood Estimation of Value at Risk and Expected Shortfall," Econometrics and Statistics, Elsevier, vol. 33(C), pages 23-34.
    • Handle: RePEc:eee:ecosta:v:33:y:2025:i:c:p:23-34
    DOI: 10.1016/j.ecosta.2021.08.003
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    as
    1. Komunjer, Ivana, 2005. "Quasi-maximum likelihood estimation for conditional quantiles," Journal of Econometrics, Elsevier, vol. 128(1), pages 137-164, September.
    2. Franco Peracchi & Andrei V. Tanase, 2008. "On estimating the conditional expected shortfall," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 24(5), pages 471-493, September.
    3. Natalia Nolde & Johanna F. Ziegel, 2016. "Elicitability and backtesting: Perspectives for banking regulation," Papers 1608.05498, arXiv.org, revised Feb 2017.
    4. Gourieroux, C. & Laurent, J. P. & Scaillet, O., 2000. "Sensitivity analysis of Values at Risk," Journal of Empirical Finance, Elsevier, vol. 7(3-4), pages 225-245, November.
    5. Gilbert W. Bassett, 2004. "Pessimistic Portfolio Allocation and Choquet Expected Utility," Journal of Financial Econometrics, Oxford University Press, vol. 2(4), pages 477-492.
    6. Gilbert W. Bassett Jr Bassett & Roger Koenker & Gregory Kordas, 2004. "Pessimistic portfolio allocation and Choquet expected utility," CeMMAP working papers 09/04, Institute for Fiscal Studies.
    7. Gerlach, Richard & Wang, Chao, 2020. "Semi-parametric dynamic asymmetric Laplace models for tail risk forecasting, incorporating realized measures," International Journal of Forecasting, Elsevier, vol. 36(2), pages 489-506.
    8. Gneiting, Tilmann, 2011. "Making and Evaluating Point Forecasts," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 746-762.
    9. O. Scaillet, 2004. "Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 115-129, January.
    10. Cai, Zongwu & Wang, Xian, 2008. "Nonparametric estimation of conditional VaR and expected shortfall," Journal of Econometrics, Elsevier, vol. 147(1), pages 120-130, November.
    11. 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.
    12. Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731, Enero.
    13. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    14. Poiraud-Casanova, Sandrine & Thomas-Agnan, Christine, 2000. "About monotone regression quantiles," Statistics & Probability Letters, Elsevier, vol. 48(1), pages 101-104, May.
    15. Cai, Zongwu, 2002. "Regression Quantiles For Time Series," Econometric Theory, Cambridge University Press, vol. 18(1), pages 169-192, February.
    16. Weiss, Andrew A., 1991. "Estimating Nonlinear Dynamic Models Using Least Absolute Error Estimation," Econometric Theory, Cambridge University Press, vol. 7(1), pages 46-68, March.
    17. Gabrielsen, Arne, 1978. "Consistency and identifiability," Journal of Econometrics, Elsevier, vol. 8(2), pages 261-263, October.
    18. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    19. Robert F. Engle & Simone Manganelli, 2004. "CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles," Journal of Business & Economic Statistics, American Statistical Association, vol. 22, pages 367-381, October.
    20. Kengo Kato, 2012. "Weighted Nadaraya--Watson Estimation of Conditional Expected Shortfall," Journal of Financial Econometrics, Oxford University Press, vol. 10(2), pages 265-291, 2012 15.
    21. Powell, James L., 1984. "Least absolute deviations estimation for the censored regression model," Journal of Econometrics, Elsevier, vol. 25(3), pages 303-325, July.
    22. Song Xi Chen, 2008. "Nonparametric Estimation of Expected Shortfall," Journal of Financial Econometrics, Oxford University Press, vol. 6(1), pages 87-107, Winter.
    23. White,Halbert, 1996. "Estimation, Inference and Specification Analysis," Cambridge Books, Cambridge University Press, number 9780521574464, Enero.
    24. McNeil, Alexander J., 1997. "Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory," ASTIN Bulletin, Cambridge University Press, vol. 27(1), pages 117-137, May.
    25. James W. Taylor, 2019. "Forecasting Value at Risk and Expected Shortfall Using a Semiparametric Approach Based on the Asymmetric Laplace Distribution," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(1), pages 121-133, January.
    26. Yamai, Yasuhiro & Yoshiba, Toshinao, 2002. "On the Validity of Value-at-Risk: Comparative Analyses with Expected Shortfall," Monetary and Economic Studies, Institute for Monetary and Economic Studies, Bank of Japan, vol. 20(1), pages 57-85, January.
    27. James W. Taylor, 2008. "Estimating Value at Risk and Expected Shortfall Using Expectiles," Journal of Financial Econometrics, Oxford University Press, vol. 6(2), pages 231-252, Spring.
    28. Gneiting, Tilmann & Raftery, Adrian E., 2007. "Strictly Proper Scoring Rules, Prediction, and Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 359-378, March.
    29. Bayer, Sebastian, 2018. "Combining Value-at-Risk forecasts using penalized quantile regressions," Econometrics and Statistics, Elsevier, vol. 8(C), pages 56-77.
    30. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    31. Jianqing Fan & Juan Gu, 2003. "Semiparametric estimation of Value at Risk," Econometrics Journal, Royal Economic Society, vol. 6(2), pages 261-290, December.
    32. Linton, Oliver & Xiao, Zhijie, 2013. "Estimation Of And Inference About The Expected Shortfall For Time Series With Infinite Variance," Econometric Theory, Cambridge University Press, vol. 29(4), pages 771-807, August.
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