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Looking for efficient qml estimation of conditional value-at-risk at multiple risk levels

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  • Francq, Christian
  • Zakoian, Jean-Michel

Abstract

We consider joint estimation of conditional Value-at-Risk (VaR) at several levels, in the framework of general GARCH-type models. The conditional VaR at level $\alpha$ is expressed as the product of the volatility and the opposite of the $\alpha$-quantile of the innovation. A standard method is to estimate the volatility parameter by Gaussian Quasi-Maximum Likelihood (QML) in a first step, and to use the residuals for estimating the innovations quantiles in a second step. We argue that the Gaussian QML may be inefficient with respect to more general QML and can even be in failure for heavy tailed conditional distributions. We therefore study, for a vector of risk levels, a two-step procedure based on a generalized QML. For a portfolio of VaR's at different levels, confidence intervals accounting for both market and estimation risks are deduced. An empirical study based on stock indices illustrates the theoretical results.

Suggested Citation

  • Francq, Christian & Zakoian, Jean-Michel, 2015. "Looking for efficient qml estimation of conditional value-at-risk at multiple risk levels," MPRA Paper 67195, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:67195
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    File URL: https://mpra.ub.uni-muenchen.de/67195/8/MPRA_paper_67195.pdf
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    References listed on IDEAS

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    1. Christophe Hurlin & Sébastien Laurent & Rogier Quaedvlieg & Stephan Smeekes, 2017. "Risk Measure Inference," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 35(4), pages 499-512, October.
    2. Christian Francq & Jean-Michel Zakoian, 2014. "Multi-level Conditional VaR Estimation in Dynamic Models," Working Papers 2014-01, Center for Research in Economics and Statistics.
    3. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. "On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
    4. Francq, Christian & Lepage, Guillaume & Zakoïan, Jean-Michel, 2011. "Two-stage non Gaussian QML estimation of GARCH models and testing the efficiency of the Gaussian QMLE," Journal of Econometrics, Elsevier, vol. 165(2), pages 246-257.
    5. 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.
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    11. Peter Hall & Qiwei Yao, 2003. "Inference in Arch and Garch Models with Heavy--Tailed Errors," Econometrica, Econometric Society, vol. 71(1), pages 285-317, January.
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    More about this item

    Keywords

    Asymmetric Power GARCH; Distortion Risk Measures; Estimation risk; Non-Gaussian Quasi-Maximum Likelihood; Value-at-Risk;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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