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Beyond the Sample: Extreme Quantile and Probability Estimation

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  • Jon Danielsson
  • Casper G. de Vries

Abstract

Economic problems such as large claims analysis in insurance and value-at-risk in finance, require assessment of the probability P of extreme realizations Q. This paper provides a semi-parametric method for estimation of extreme (P,Q) combinations for data with heavy tails. We solve the long standing problem of estimating the sample threshold of where the tail of the distribution starts. This is accomplished by the combination of a control variate type device and a subsample bootstrap technique. The subsample bootstrap attains convergence in probability, whereas the full sample bootstrap would only provide convergence in distribution. This permits a complete and comprehensive treatment of extreme (P,Q) estimation.

Suggested Citation

  • Jon Danielsson & Casper G. de Vries, 1998. "Beyond the Sample: Extreme Quantile and Probability Estimation," FMG Discussion Papers dp298, Financial Markets Group.
    • Handle: RePEc:fmg:fmgdps:dp298
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    1. Jansen, Dennis W & de Vries, Casper G, 1991. "On the Frequency of Large Stock Returns: Putting Booms and Busts into Perspective," The Review of Economics and Statistics, MIT Press, vol. 73(1), pages 18-24, February.
    2. Hendry, David F., 1984. "Monte carlo experimentation in econometrics," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 2, chapter 16, pages 937-976, Elsevier.
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    Cited by:

    1. Odening, Martin & Hinrichs, Jan, 2003. "Die Quantifizierung von Marktrisiken in der Tierproduktion mittels Value-at-Risk und Extreme-Value-Theory," German Journal of Agricultural Economics, Humboldt-Universitaet zu Berlin, Department for Agricultural Economics, vol. 52(02), pages 1-11.
    2. Cotter, John, 2001. "Margin exceedences for European stock index futures using extreme value theory," Journal of Banking & Finance, Elsevier, vol. 25(8), pages 1475-1502, August.
    3. Engle, Robert F. & Manganelli, Simone, 2001. "Value at risk models in finance," Working Paper Series 0075, European Central Bank.
    4. 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.
    5. Marcia M. A. Schafgans, 2000. "On Intercept Estimation in the Sample Selection Model," Econometric Society World Congress 2000 Contributed Papers 0730, Econometric Society.
    6. Tsourti, Zoi & Panaretos, John, 2003. "Extreme Value Index Estimators and Smoothing Alternatives: A Critical Review," MPRA Paper 6390, University Library of Munich, Germany.
    7. Odening, Martin & Hinrichs, Jan, 2002. "Assessment Of Market Risk In Hog Production Using Value-At-Risk And Extreme Value Theory," 2002 Annual meeting, July 28-31, Long Beach, CA 19907, American Agricultural Economics Association (New Name 2008: Agricultural and Applied Economics Association).
    8. Manfred Gilli & Evis këllezi, 2006. "An Application of Extreme Value Theory for Measuring Financial Risk," Computational Economics, Springer;Society for Computational Economics, vol. 27(2), pages 207-228, May.
    9. John Cotter, 2004. "Downside risk for European equity markets," Applied Financial Economics, Taylor & Francis Journals, vol. 14(10), pages 707-716.
    10. Carol Alexander & Emese Lazar & Silvia Stanescu, 2011. "Analytic Approximations to GARCH Aggregated Returns Distributions with Applications to VaR and ETL," ICMA Centre Discussion Papers in Finance icma-dp2011-08, Henley Business School, University of Reading.
    11. Francis X. Diebold & Til Schuermann & John D. Stroughair, 2000. "Pitfalls and Opportunities in the Use of Extreme Value Theory in Risk Management," Journal of Risk Finance, Emerald Group Publishing Limited, vol. 1(2), pages 30-35, January.
    12. Marius Galabe Sampid & Haslifah M Hasim & Hongsheng Dai, 2018. "Refining value-at-risk estimates using a Bayesian Markov-switching GJR-GARCH copula-EVT model," PLOS ONE, Public Library of Science, vol. 13(6), pages 1-33, June.
    13. Lucas, André & Straetmans, Stefan & Klaassen, Pieter, 1999. "Tail behavior of credit loss distributions," Serie Research Memoranda 0060, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    14. Robert F. Engle & Simone Manganelli, 1999. "CAViaR: Conditional Value at Risk by Quantile Regression," NBER Working Papers 7341, National Bureau of Economic Research, Inc.
    15. Niklas Wagner & Terry Marsh, 2004. "Tail index estimation in small smaples Simulation results for independent and ARCH-type financial return models," Statistical Papers, Springer, vol. 45(4), pages 545-561, October.
    16. Guang Bi & David E. Giles, 2007. "An Application of Extreme Value Theory to U.S. Movie Box Office Returns," Econometrics Working Papers 0705, Department of Economics, University of Victoria.
    17. Tsourti, Zoi & Panaretos, John, 2001. "Extreme Value Index Estimators and Smoothing Alternatives: Review and Simulation Comparison," MPRA Paper 6384, University Library of Munich, Germany.
    18. Tsourti, Zoi & Panaretos, John, 2004. "Extreme-value analysis of teletraffic data," Computational Statistics & Data Analysis, Elsevier, vol. 45(1), pages 85-103, February.
    19. Mendes, Beatriz Vaz de Melo & Lopes, Hedibert Freitas, 2004. "Data driven estimates for mixtures," Computational Statistics & Data Analysis, Elsevier, vol. 47(3), pages 583-598, October.
    20. Sebastian Schich, 2004. "European stock market dependencies when price changes are unusually large," Applied Financial Economics, Taylor & Francis Journals, vol. 14(3), pages 165-177.
    21. Engle, Robert F. & Manganelli, Simone, 2001. "Value at risk models in finance," Working Paper Series 75, European Central Bank.
    22. Carmela E. Quintos & Zhenhong Fan & Peter C.B. Phillips, 2000. "Structural Change in Tail Behavior and the Asian Financial Crisis," Cowles Foundation Discussion Papers 1283, Cowles Foundation for Research in Economics, Yale University.
    23. Brito, Margarida & Freitas, Ana Cristina Moreira, 2010. "Consistent estimation of the tail index for dependent data," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1835-1843, December.
    24. Ho, Lan-Chih & Burridge, Peter & Cadle, John & Theobald, Michael, 2000. "Value-at-risk: Applying the extreme value approach to Asian markets in the recent financial turmoil," Pacific-Basin Finance Journal, Elsevier, vol. 8(2), pages 249-275, May.
    25. Kilic, Ekrem, 2006. "Violation duration as a better way of VaR model evaluation : evidence from Turkish market portfolio," MPRA Paper 5610, University Library of Munich, Germany.

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    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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