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Monte Carlo-based tail exponent estimator

  • Barunik, Jozef
  • Vacha, Lukas

In this paper we propose a new approach to estimation of the tail exponent in financial stock markets. We begin the study with the finite sample behavior of the Hill estimator under α-stable distributions. Using large Monte Carlo simulations, we show that the Hill estimator overestimates the true tail exponent and can hardly be used on samples with small length. Utilizing our results, we introduce a Monte Carlo-based method of estimation for the tail exponent. Our proposed method is not sensitive to the choice of tail size and works well also on small data samples. The new estimator also gives unbiased results with symmetrical confidence intervals. Finally, we demonstrate the power of our estimator on the international world stock market indices. On the two separate periods of 2002–2005 and 2006–2009, we estimate the tail exponent.

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Article provided by Elsevier in its journal Physica A: Statistical Mechanics and its Applications.

Volume (Year): 389 (2010)
Issue (Month): 21 ()
Pages: 4863-4874

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Handle: RePEc:eee:phsmap:v:389:y:2010:i:21:p:4863-4874
Contact details of provider: Web page: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/

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  1. Plerou, V & Gopikrishnan, P & Rosenow, B & Amaral, L.A.N & Stanley, H.E, 2000. "A random matrix theory approach to financial cross-correlations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 374-382.
  2. Hall, Peter, 1990. "Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 177-203, February.
  3. Rafal Weron, 2003. "Levy-stable distributions revisited: tail index > 2 does not exclude the Levy-stable regime," Econometrics 0305003, EconWPA.
  4. Xavier Gabaix & Rustam Ibragimov, 2011. "Rank - 1 / 2: A Simple Way to Improve the OLS Estimation of Tail Exponents," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(1), pages 24-39, January.
  5. Stanley, H.Eugene, 2003. "Statistical physics and economic fluctuations: do outliers exist?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 318(1), pages 279-292.
  6. repec:ner:tilbur:urn:nbn:nl:ui:12-125712 is not listed on IDEAS
  7. Stanley, H.E & Amaral, L.A.N & Gopikrishnan, P & Plerou, V, 2000. "Scale invariance and universality of economic fluctuations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 283(1), pages 31-41.
  8. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
  9. Szymon Borak & Wolfgang Härdle & Rafal Weron, 2005. "Stable Distributions," SFB 649 Discussion Papers SFB649DP2005-008, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  10. 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.
  11. Einmahl, J. H.J. & Dekkers, A. L.M. & de Haan, L., 1989. "A moment estimator for the index of an extreme-value distribution," Other publications TiSEM 81970cb3-5b7a-4cad-9bf6-2, Tilburg University, School of Economics and Management.
  12. Mantegna, Rosario N & Palágyi, Zoltán & Stanley, H.Eugene, 1999. "Applications of statistical mechanics to finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 274(1), pages 216-221.
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