Empirical distributions of stock returns: between the stretched exponential and the power law?
Empirical distributions of stock returns: between the stretched exponential and the power law?
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DOI：
10.1080/14697680500151343
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发表时间：
2005-08-01
影响因子：
1.3
通讯作者：
Sornette, D
中科院分区：
文献类型：
--
作者：
Malevergne, Y;Pisarenko, V;Sornette, D
现在，人们似乎理所当然地认为，金融时间序列的经验回报分布有规律地变化，尾部指数 h 接近 3。我们开发了一系列新的非参数和参数测试来表征中等规模金融时间序列的经验回报分布，并应用于道琼斯工业平均指数 100 年的日回报率、纳斯达克综合指数 1 年的 5 分钟回报率和 17 年的 1 分钟回报率。标准普尔 500 指数。我们提出了回报分布尾部的参数化表示，包括在参数的一个极限内的规则变化的分布以及在其他极限下的拉伸指数 (SE) 和对数威布尔或拉伸对数指数 (SLE) 分布类别的快速变化的分布。使用嵌套假设检验方法（威尔克斯定理），我们得出结论，SE 分布和 Pareto 分布都提供了数据的可靠描述，但对于足够高的阈值很难区分。基于发现SE分布在一定限度内趋于Pareto分布。我们证明，Wilks 的嵌套假设检验仍然适用于 SE 和 Pareto 分布之间的非精确嵌套比较。与帕累托定律相比，SE 分布在整个分位数范围内明显更好，但在 95% 分位数之外就变得不必要了。类似的结论也适用于帕累托分布的对数威布尔模型，但有关甚高频数据的明显例外。总结我们的测试提供的所有证据，似乎尾部最终的衰减速度比任何 SE 都慢，但可能比具有合理指数的幂律更快。因此，从实际角度来看，log-Weibull 模型提供了 SE 和 PD 之间的平滑插值，可以被视为样本分布的适当近似。
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 h close to 3. We develop a battery of new non-parametric and parametric tests to characterize the distributions of empirical returns of moderately large financial time series, with application to 100 years of daily returns of the Dow Jones Industrial Average, to I year of 5-min returns of the Nasdaq Composite index and to 17 years of 1-min returns of the Standard & Poor's 500. We propose a parametric representation of the tail of the distributions of returns 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 the log-Weibull or Stretched Log-Exponential (SLE) 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 but are hardly distinguishable for sufficiently high thresholds. Based on the discovery that the SE distribution tends to the Pareto distribution in a certain limit. 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 to be 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, with a noticeable exception concerning the very-high-frequency data. Summing up all the evidence provided by our 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 viewpoint, the log-Weibull model, which provides a smooth interpolation between SE and PD, can be considered as an appropriate approximation of the sample distributions.