Chapter 6 - Testing for a finite variance in stock return distributions

Abstract

This chapter proposes a test statistic to discriminate between finite variance distributions and infinite variance distributions for stock returns. The test statistic is the ratio of the sample inter-quartile range and the sample standard deviation. The distributions proposed include both unconditional and conditional distributions. All unconditional distributions or time-independent models can be divided between two families. One family has finite variance. Examples include the normal distribution, lognormal distribution, student distribution, mixture of normals (MN), compound log-normal and normal distribution, mixed diffusion-jump (MDJ) process and more recent distributions, such as the generalized beta (GB) distribution, Weibull distribution, Variance Gamma (VG) distribution, hyperbolic distribution, and generalized lambda distribution. The other family has infinite variance. A widely used infinite variance model is the stable distribution. The stable distribution has been appreciated as a distribution to model stock returns for both statistical and economic reasons. Statistically speaking, the stable distribution has domains of attraction and belongs to its own domain of attraction. It seems natural to use sample variance or sample standard deviation to discriminate between finite variance and infinite variance distributions. Unfortunately, the power based on the sample variance or sample standard deviation may not be good because a finite variance distribution can generate a larger sample variance than an infinite variance distribution, even when the sample size is large.