IDEAS home Printed from https://ideas.repec.org/a/eee/dyncon/v177y2025ics0165188925000946.html

Extreme conditional tail risk inference in ARMA–GARCH models

Author

Listed:
  • Ma, Yaolan
  • Wei, Bo

Abstract

In this study, we investigate the estimation of extreme conditional Value-at-Risk (CVaR) and conditional Expected Shortfall (CES) within the framework of ARMA-GARCH models, where innovations are assumed to follow a Pareto-type tail distribution and have no finite fourth moments. Building on the two-stage self-weighted estimation procedure proposed by He et al. (2022), we develop a robust methodology for forecasting extreme CVaR and CES. Using extreme value theory, we derive a unified asymptotic theory for the extreme CVaR and CES estimators. Through comprehensive simulation studies, we evaluate the performance of our approach and compare it with several recently proposed estimators in the literature. Additionally, we apply our methodology to forecast extreme CVaR and CES for daily negative log-returns (i.e., losses) of four financial assets, demonstrating its practical applicability in financial risk management.

Suggested Citation

  • Ma, Yaolan & Wei, Bo, 2025. "Extreme conditional tail risk inference in ARMA–GARCH models," Journal of Economic Dynamics and Control, Elsevier, vol. 177(C).
    • Handle: RePEc:eee:dyncon:v:177:y:2025:i:c:s0165188925000946
    DOI: 10.1016/j.jedc.2025.105128
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0165188925000946
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jedc.2025.105128?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Natalia Nolde & Johanna F. Ziegel, 2016. "Elicitability and backtesting: Perspectives for banking regulation," Papers 1608.05498, arXiv.org, revised Feb 2017.
    2. Keith Kuester & Stefan Mittnik & Marc S. Paolella, 2006. "Value-at-Risk Prediction: A Comparison of Alternative Strategies," Journal of Financial Econometrics, Oxford University Press, vol. 4(1), pages 53-89.
    3. Gao, Feng & Song, Fengming, 2008. "ESTIMATION RISK IN GARCH VaR AND ES ESTIMATES," Econometric Theory, Cambridge University Press, vol. 24(5), pages 1404-1424, October.
    4. Yi He & Liang Peng & Dabao Zhang & Zifeng Zhao, 2022. "Risk Analysis via Generalized Pareto Distributions," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 40(2), pages 852-867, April.
    5. Peter R. Hansen & Asger Lunde & James M. Nason, 2011. "The Model Confidence Set," Econometrica, Econometric Society, vol. 79(2), pages 453-497, March.
    6. Yannick Hoga, 2022. "Limit Theory for Forecasts of Extreme Distortion Risk Measures and Expectiles [Coherent Measures of Risk]," Journal of Financial Econometrics, Oxford University Press, vol. 20(1), pages 18-44.
    7. McNeil, Alexander J. & Frey, Rudiger, 2000. "Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach," Journal of Empirical Finance, Elsevier, vol. 7(3-4), pages 271-300, November.
    8. 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.
    9. Li, Shuo & Peng, Liuhua & Song, Xiaojun, 2023. "Simultaneous Confidence Bands For Conditional Value-At-Risk And Expected Shortfall," Econometric Theory, Cambridge University Press, vol. 39(5), pages 1009-1043, October.
    10. Danielsson, Jon & Ergun, Lerby M. & Haan, Laurens de & Vries, Casper G. de, 2016. "Tail index estimation: quantile driven threshold selection," LSE Research Online Documents on Economics 66193, London School of Economics and Political Science, LSE Library.
    11. Pan, Xiaoqing & Leng, Xuan & Hu, Taizhong, 2013. "The second-order version of Karamata’s theorem with applications," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1397-1403.
    12. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    13. Debbie J. Dupuis & Nicolas Papageorgiou & Bruno Rémillard, 2015. "Robust Conditional Variance and Value-at-Risk Estimation," Journal of Financial Econometrics, Oxford University Press, vol. 13(4), pages 896-921.
    14. Hang Chan, Ngai & Deng, Shi-Jie & Peng, Liang & Xia, Zhendong, 2007. "Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations," Journal of Econometrics, Elsevier, vol. 137(2), pages 556-576, April.
    15. Hu, Jie & Chen, Yu & Tan, Keqi, 2021. "Estimation Of High Conditional Tail Risk Based On Expectile Regression," ASTIN Bulletin, Cambridge University Press, vol. 51(2), pages 539-570, May.
    16. Yannick Hoga, 2019. "Confidence Intervals for Conditional Tail Risk Measures in ARMA–GARCH Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(4), pages 613-624, October.
    17. Martins-Filho, Carlos & Yao, Feng & Torero, Maximo, 2018. "Nonparametric Estimation Of Conditional Value-At-Risk And Expected Shortfall Based On Extreme Value Theory," Econometric Theory, Cambridge University Press, vol. 34(1), pages 23-67, February.
    18. Hill, Jonathan B., 2010. "On Tail Index Estimation For Dependent, Heterogeneous Data," Econometric Theory, Cambridge University Press, vol. 26(5), pages 1398-1436, October.
    19. James, Robert & Leung, Henry & Leung, Jessica Wai Yin & Prokhorov, Artem, 2023. "Forecasting tail risk measures for financial time series: An extreme value approach with covariates," Journal of Empirical Finance, Elsevier, vol. 71(C), pages 29-50.
    20. Daniele Massacci, 2017. "Tail Risk Dynamics in Stock Returns: Links to the Macroeconomy and Global Markets Connectedness," Management Science, INFORMS, vol. 63(9), pages 3072-3089, September.
    21. John H. J. Einmahl & Laurens Haan & Chen Zhou, 2016. "Statistics of heteroscedastic extremes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 31-51, January.
    22. 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.
    23. Ma, Yaolan & Zhou, Mo & Peng, Liang & Zhang, Rongmao, 2022. "Test For Zero Median Of Errors In An Arma–Garch Model," Econometric Theory, Cambridge University Press, vol. 38(3), pages 536-561, June.
    24. Moosup Kim & Sangyeol Lee, 2016. "On the tail index inference for heavy-tailed GARCH-type innovations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(2), pages 237-267, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hoga, Yannick, 2021. "The uncertainty in extreme risk forecasts from covariate-augmented volatility models," International Journal of Forecasting, Elsevier, vol. 37(2), pages 675-686.
    2. Yannick Hoga, 2023. "The Estimation Risk in Extreme Systemic Risk Forecasts," Papers 2304.10349, arXiv.org.
    3. Nieto, Maria Rosa & Ruiz, Esther, 2016. "Frontiers in VaR forecasting and backtesting," International Journal of Forecasting, Elsevier, vol. 32(2), pages 475-501.
    4. Timo Dimitriadis & Yannick Hoga, 2022. "Dynamic CoVaR Modeling and Estimation," Papers 2206.14275, arXiv.org, revised Jan 2025.
    5. Santos, Douglas G. & Candido, Osvaldo & Tófoli, Paula V., 2022. "Forecasting risk measures using intraday and overnight information," The North American Journal of Economics and Finance, Elsevier, vol. 60(C).
    6. Julia S. Mehlitz & Benjamin R. Auer, 2021. "Time‐varying dynamics of expected shortfall in commodity futures markets," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(6), pages 895-925, June.
    7. Ning Zhang & Yujing Gong & Xiaohan Xue, 2023. "Less disagreement, better forecasts: Adjusted risk measures in the energy futures market," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 43(10), pages 1332-1372, October.
    8. Candia, Claudio & Herrera, Rodrigo, 2024. "An empirical review of dynamic extreme value models for forecasting value at risk, expected shortfall and expectile," Journal of Empirical Finance, Elsevier, vol. 77(C).
    9. 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.
    10. Chao Wang & Richard Gerlach, 2019. "Semi-parametric Realized Nonlinear Conditional Autoregressive Expectile and Expected Shortfall," Papers 1906.09961, arXiv.org.
    11. Daouia, Abdelaati & Hachem, Joseph & Stupfler, Gilles, 2026. "Extreme value inference for heterogeneous heavy-tailed data: A derandomization theory," TSE Working Papers 26-1727, Toulouse School of Economics (TSE).
    12. David Happersberger & Harald Lohre & Ingmar Nolte, 2020. "Estimating portfolio risk for tail risk protection strategies," European Financial Management, European Financial Management Association, vol. 26(4), pages 1107-1146, September.
    13. Simon Fritzsch & Maike Timphus & Gregor Weiss, 2021. "Marginals Versus Copulas: Which Account For More Model Risk In Multivariate Risk Forecasting?," Papers 2109.10946, arXiv.org.
    14. Taylor, James W., 2020. "Forecast combinations for value at risk and expected shortfall," International Journal of Forecasting, Elsevier, vol. 36(2), pages 428-441.
    15. Fiszeder, Piotr & Małecka, Marta & Molnár, Peter, 2024. "Robust estimation of the range-based GARCH model: Forecasting volatility, value at risk and expected shortfall of cryptocurrencies," Economic Modelling, Elsevier, vol. 141(C).
    16. 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.
    17. Fritzsch, Simon & Timphus, Maike & Weiß, Gregor, 2024. "Marginals versus copulas: Which account for more model risk in multivariate risk forecasting?," Journal of Banking & Finance, Elsevier, vol. 158(C).
    18. Lucas, André & Schwaab, Bernd & Zhang, Xin & D’Innocenzo, Enzo, 2026. "Joint extreme value-at-risk and expected shortfall dynamics with a single integrated tail shape parameter," Working Paper Series 3166, European Central Bank.
    19. Taylor, James W., 2022. "Forecasting Value at Risk and expected shortfall using a model with a dynamic omega ratio," Journal of Banking & Finance, Elsevier, vol. 140(C).
    20. Gery Geenens & Richard Dunn, 2017. "A nonparametric copula approach to conditional Value-at-Risk," Papers 1712.05527, arXiv.org, revised Oct 2019.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:dyncon:v:177:y:2025:i:c:s0165188925000946. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jedc .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.