Empirical Distributions of Log-Returns: Between the Stretched Exponential and the Power Law?

71 Pages Posted: 25 Jun 2003

See all articles by Yannick Malevergne

Yannick Malevergne

Université Paris I Panthéon-Sorbonne - Laboratoire PRISM

Vladilen Pisarenko

Russian Academy of Sciences (RAS) - International Institute of Earthquake Prediction Theory

Didier Sornette

Risks-X, Southern University of Science and Technology (SUSTech); Swiss Finance Institute; ETH Zürich - Department of Management, Technology, and Economics (D-MTEC); Tokyo Institute of Technology

Abstract

A large consensus now seems to take for granted that the distributions of empirical returns of financial time series are regularly varying, with a tail exponent b close to 3. First, we show by synthetic tests performed on time series with time dependence in the volatility with both Pareto and Stretched-Exponential distributions that for sample of moderate size, the standard generalized extreme value (GEV) estimator is quite inefficient due to the possibly slow convergence toward the asymptotic theoretical distribution and the existence of biases in presence of dependence between data. Thus it cannot distinguish reliably between rapidly and regularly varying classes of distributions. The Generalized Pareto distribution (GPD)estimator works better, but still lacks power in the presence of strong dependence. Then, we use a parametric representation of the tail of the distributions of returns of 100 years of daily return of the Dow Jones Industrial Average and over 1 years of 5-minutes returns of the Nasdaq Composite index, encompassing both a regularly varying distribution in one limit of the parameters and rapidly varying distributions of the class of the Stretched-Exponential (SE) and Log-Weibull distributions in other limits. Using the method of nested hypothesis testing (Wilks' theorem), we conclude that both the SE distributions and Pareto distributions provide reliable descriptions of the data and cannot be distinguished for sufficiently high thresholds. However, the exponent b of the Pareto increases with the quantiles and its growth does not seem exhausted for the highest quantiles of three out of the four tail distributions investigated here. Correlatively, the exponent c of the SE model decreases and seems to tend to zero. Based on the discovery that the SE distribution tends to the Pareto distribution in a certain limit such that the Pareto (or power law) distribution can be approximated with any desired accuracy on an arbitrary interval by a suitable adjustment of the parameters of the SE distribution,we demonstrate that Wilks' test of nested hypothesis still works for the non-exactly nested comparison between the SE and Pareto distributions. The SE distribution is found significantly better over the whole quantile range but becomes unnecessary beyond the 95% quantiles compared with the Pareto law. Similar conclusions hold for the log-Weibull model with respect to the Pareto distribution. Summing up all the evidence provided by our battery of tests, it seems that the tails ultimately decay slower than any SE but probably faster than power laws with reasonable exponents. Thus, from a practical view point, the log-Weibull model, which provides a smooth interpolation between SE and PD, can be considered as an appropriate approximation of the sample distributions. We finally discuss the implications of our results on the "moment condition failure" and for risk estimation and management.

Keywords: Extreme-Value Estimators, Non-Nested Hypothesis Testing, Pareto distribution, Weibull distribution

Suggested Citation

Malevergne, Yannick and Pisarenko, Vladilen and Sornette, Didier, Empirical Distributions of Log-Returns: Between the Stretched Exponential and the Power Law?. Available at SSRN: https://ssrn.com/abstract=410261 or http://dx.doi.org/10.2139/ssrn.410261

Yannick Malevergne

Université Paris I Panthéon-Sorbonne - Laboratoire PRISM ( email )

17 rue de la Sorbonne
Paris, 75005
France

HOME PAGE: http://perso.univ-paris1.fr/ymalevergn

Vladilen Pisarenko

Russian Academy of Sciences (RAS) - International Institute of Earthquake Prediction Theory ( email )

79, kor. 2
Moscow 113556
Russia

Didier Sornette (Contact Author)

Risks-X, Southern University of Science and Technology (SUSTech) ( email )

1088 Xueyuan Avenue
Shenzhen, Guangdong 518055
China

Swiss Finance Institute ( email )

c/o University of Geneva
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CH-1211 Geneva 4
Switzerland

ETH Zürich - Department of Management, Technology, and Economics (D-MTEC) ( email )

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Zurich, ZURICH CH-8092
Switzerland
41446328917 (Phone)
41446321914 (Fax)

HOME PAGE: http://www.er.ethz.ch/

Tokyo Institute of Technology ( email )

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Tokyo 152-8550, 52-8552
Japan