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New statistic for financial return distributions: Power-law or exponential?

Author

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  • Pisarenko, V.
  • Sornette, D.

Abstract

We introduce a new statistical tool (the TP-statistic and TE-statistic) designed specifically to compare the behavior of the sample tail of distributions with power-law and exponential tails as a function of the lower threshold u. One important property of these statistics is that they converge to zero for power-laws or for exponentials correspondingly, regardless of the value of the exponent or of the form parameter. This is particularly useful for testing the structure of a distribution (power-law or not, exponential or not) independently of the possibility of quantifying the values of the parameters. We apply these statistics to the distribution of returns of one century of daily data for the Dow Jones Industrial Average and over 1 year of 5-min data of the Nasdaq Composite index. Our analysis confirms previous works showing the tendency for the tails to resemble more and more a power-law for the highest quantiles but we can detect clear deviations that suggest that the structure of the tails of the distributions of returns is more complex than usually assumed; it is clearly more complex that just a power-law. Our new TP- and TE-statistic should also be useful for other applications in the natural sciences as a powerful non-parametric test for power-laws and exponentials.

Suggested Citation

  • Pisarenko, V. & Sornette, D., 2006. "New statistic for financial return distributions: Power-law or exponential?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 366(C), pages 387-400.
    • Handle: RePEc:eee:phsmap:v:366:y:2006:i:c:p:387-400
    DOI: 10.1016/j.physa.2005.10.015
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    References listed on IDEAS

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    1. Danielsson, J. & de Haan, L. & Peng, L. & de Vries, C. G., 2001. "Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation," Journal of Multivariate Analysis, Elsevier, vol. 76(2), pages 226-248, February.
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    2. Pan, Raj Kumar & Sinha, Sitabhra, 2008. "Inverse-cubic law of index fluctuation distribution in Indian markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(8), pages 2055-2065.
    3. Negrea, Bogdan, 2014. "A statistical measure of financial crises magnitude," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 397(C), pages 54-75.
    4. Marcin Wk{a}torek & Jaros{l}aw Kwapie'n & Stanis{l}aw Dro.zd.z, 2021. "Financial Return Distributions: Past, Present, and COVID-19," Papers 2107.06659, arXiv.org.
    5. Coronel-Brizio, H.F. & Hernández-Montoya, A.R., 2010. "The Anderson–Darling test of fit for the power-law distribution from left-censored samples," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(17), pages 3508-3515.
    6. Tao, Chen & Zhong, Guang-Yan & Li, Jiang-Cheng, 2023. "Dynamic correlation and risk resonance among industries of Chinese stock market: New evidence from time–frequency domain and complex network perspectives," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 614(C).
    7. Rakhee Dinubhai Patel & Frederic Paik Schoenberg, 2011. "A graphical test for local self-similarity in univariate data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(11), pages 2547-2562, January.

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