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Interval estimation of the tail index of a GARCH(1,1) model

Author

Listed:
  • Ngai Chan
  • Liang Peng
  • Rongmao Zhang

Abstract

It is known that the tail index of a GARCH model is determined by a moment equation, which involves the underlying unknown parameters of the model. A tail index estimator can therefore be constructed by solving the sample moment equation with the unknown parameters being replaced by its quasi-maximum likelihood estimates (QMLE). To construct a confidence interval for the tail index, one needs to estimate the non-trivial asymptotic variance of the QMLE. In this paper, an empirical likelihood method is proposed for interval estimation of the tail index. One advantage of the proposed method is that interval estimation can still be achieved without having to estimate the complicated asymptotic variance. A simulation study confirms the advantage of the proposed method. Copyright Sociedad de Estadística e Investigación Operativa 2012

Suggested Citation

  • Ngai Chan & Liang Peng & Rongmao Zhang, 2012. "Interval estimation of the tail index of a GARCH(1,1) model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(3), pages 546-565, September.
    • Handle: RePEc:spr:testjl:v:21:y:2012:i:3:p:546-565
    DOI: 10.1007/s11749-011-0264-0
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    References listed on IDEAS

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    1. Song Chen & Ingrid Van Keilegom, 2009. "A review on empirical likelihood methods for regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(3), pages 415-447, November.
    2. Ling, Shiqing, 2007. "Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH/IGARCH models," Journal of Econometrics, Elsevier, vol. 140(2), pages 849-873, October.
    3. Carmela Quintos & Zhenhong Fan & Peter C. B. Phillips, 2001. "Structural Change Tests in Tail Behaviour and the Asian Crisis," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 68(3), pages 633-663.
    4. Basrak, Bojan & Davis, Richard A. & Mikosch, Thomas, 2002. "Regular variation of GARCH processes," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 95-115, May.
    5. Iglesias, Emma M. & Linton, Oliver, 2009. "Estimation of tail thickness parameters from GJR-GARCH models," UC3M Working papers. Economics we094726, Universidad Carlos III de Madrid. Departamento de Economía.
    6. Song Chen & Ingrid Van Keilegom, 2009. "Rejoinder on: A review on empirical likelihood methods for regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(3), pages 468-474, November.
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    Cited by:

    1. Sun, Haoze & Jiang, Yuexiang, 2014. "Empirical likelihood based confidence intervals for the tail index when γ<−1/2," Statistics & Probability Letters, Elsevier, vol. 84(C), pages 149-157.
    2. Nieto, Maria Rosa & Ruiz, Esther, 2016. "Frontiers in VaR forecasting and backtesting," International Journal of Forecasting, Elsevier, vol. 32(2), pages 475-501.
    3. León, Ángel & Ñíguez, Trino-Manuel, 2021. "The transformed Gram Charlier distribution: Parametric properties and financial risk applications," Journal of Empirical Finance, Elsevier, vol. 63(C), pages 323-349.
    4. Ma, Yaolan & Jiang, Yuexiang & Huang, Wei, 2018. "Empirical likelihood based inference for conditional Pareto-type tail index," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 114-121.

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    More about this item

    Keywords

    Empirical likelihood; GARCH model; Tail index; 62M10; 62E20; 60F17;
    All these keywords.

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