对数收益率的经验分布：在拉伸指数与幂律之间？
Empirical Distributions of Log-Returns: between the Stretched Exponential and the Power Law?
摘要：
一个大的共识，现在似乎认为理所当然的是，金融时间序列的经验收益率的分布规律变化，尾部指数接近3。我们重新审视这一结果，并使用标准测试，以及开发一个新的非参数和参数测试（特别是与拉伸指数（SE）分布）的电池来表征金融时间序列的经验回报率的分布，与应用程序的100年的道琼斯工业平均指数的每日回报率和超过1年的5分钟回报率的纳斯达克综合指数。基于SE分布在一定范围内趋向于Pareto分布，使得通过适当调整SE分布的参数，可以在任意区间上以任意精度逼近Pareto（或幂律）分布的发现，我们证明了Wilks的嵌套假设检验仍然适用于SE分布和Pareto分布之间的非精确嵌套比较. SE分布在整个分位数范围内显着更好，但与帕累托定律相比，超过95%的分位数就变得不必要了。类似的结论适用于对数威布尔模型的帕累托分布。我们的主要结果是，尾部最终衰减慢于任何拉伸指数分布，但可能快于幂律与合理的指数。我们的研究结果的影响的“时刻条件失败”和风险估计和管理。
英文摘要：
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 close to 3. We revisit this results and use standard tests as well as develop a battery of new non-parametric and parametric tests (in particular with stretched exponential (SE) distributions) to characterize the distributions of empirical returns of financial time series, with application to the 100 years of daily return of the Dow Jones Industrial Average and over 1 years of 5-minutes returns of the Nasdaq Composite index. 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. Our main result is that the tails ultimately decay slower than any stretched exponential distribution but probably faster than power laws with reasonable exponents. Implications of our results on the ``moment condition failure'' and for risk estimation and management are presented.
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