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Exact inference and optimal invariant estimation for the stability parameter of symmetric [alpha]-stable distributions

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  • Dufour, Jean-Marie
  • Kurz-Kim, Jeong-Ryeol

Abstract

Hill estimation (Hill, 1975), the most widespread method for estimating tail thickness of heavy-tailed financial data, suffers from two drawbacks. One is that the optimal number of tail observations to use in the estimation is a function of the unknown tail index being estimated, which diminishes the empirical relevance of the Hill estimation. The other is that the hypothesis test of the underlying data lying in the domain of attraction of an [alpha]-stable law ([alpha]Â =Â 2) for finite samples, is performed on the basis of the asymptotic distribution, which can be different from those for finite samples. In this paper, using the Monte Carlo technique, we propose an exact test method for the stability parameter of [alpha]-stable distributions which is based on the Hill estimator, yet is able to provide exact confidence intervals for finite samples. Our exact test method automatically includes an estimation procedure which does not need the assumption of a known number of observations on the distributional tail. Empirical applications demonstrate the advantages of our new method in comparison with the Hill estimation.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Empirical Finance.

Volume (Year): 17 (2010)
Issue (Month): 2 (March)
Pages: 180-194

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Handle: RePEc:eee:empfin:v:17:y:2010:i:2:p:180-194

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Web page: http://www.elsevier.com/locate/jempfin

Related research

Keywords: [alpha]-Stable distribution Hill estimator Monte Carlo test Hodges-Lehmann estimator Exact confidence interval Finite sample;

References

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  1. Andrews, Donald W K, 1993. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Econometrica, Econometric Society, vol. 61(4), pages 821-56, July.
  2. Huisman, Ronald, et al, 2001. "Tail-Index Estimates in Small Samples," Journal of Business & Economic Statistics, American Statistical Association, vol. 19(2), pages 208-16, April.
  3. Einmahl, J. & Dekkers, A. & de Haan, L., 1989. "A moment estimator for the index of an extreme-value distribution," Open Access publications from Tilburg University urn:nbn:nl:ui:12-125712, Tilburg University.
  4. Kim Jeong-Ryeol & Mittnik Stefan & Rachev Svetlozar T., 1996. "Detecting Asymmetries in Observed Linear Time Series and Unobserved Disturbances," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 1(3), pages 1-15, October.
  5. Fama, Eugene F & French, Kenneth R, 1992. " The Cross-Section of Expected Stock Returns," Journal of Finance, American Finance Association, vol. 47(2), pages 427-65, June.
  6. Loretan, Michael Stanislaus & Kurz-Kim, Jeong-Ryeol, 2007. "A note on the coefficient of determination in regression models with infinite-variance variables," Discussion Paper Series 1: Economic Studies 2007,10, Deutsche Bundesbank, Research Centre.
  7. Jean-Marie Dufour, 2005. "Monte Carlo tests with nuisance parameters: a general approach to finite-sample inference and non-standard asymptotics," CIRANO Working Papers 2005s-02, CIRANO.
  8. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
  9. Quintos, Carmela & Fan, Zhenhong & Phillips, Peter C B, 2001. "Structural Change Tests in Tail Behaviour and the Asian Crisis," Review of Economic Studies, Wiley Blackwell, vol. 68(3), pages 633-63, July.
  10. Johan Segers, 2002. "Abelian and Tauberian Theorems on the Bias of the Hill Estimator," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 29(3), pages 461-483.
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Cited by:
  1. Beaulieu, Marie-Claude & Dufour, Jean-Marie & Khalaf, Lynda, 2014. "Exact confidence sets and goodness-of-fit methods for stable distributions," Journal of Econometrics, Elsevier, vol. 181(1), pages 3-14.
  2. John Goddard & Enrico Onali, 2014. "Self-affinity in financial asset returns," Papers 1401.7170, arXiv.org.

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