Measuring Tail Thickness to Estimate the Stable Index α: A Critique

Abstract

A generalized Pareto or simple Pareto tail-index estimate above 2 has frequently been cited as evidence against infinite-variance stable distributions. It is demonstrated that this inference is invalid; tail index estimates greater than 2 are to be expected for stable distributions with α as low as 1.65. The nonregular distribution of the likelihood ratio statistic for a null of normality and an alternative of symmetric stability is tabulated by Monte Carlo methods and appropriately adjusted for sampling error in repeated tests. Real stock returns yield a stable α of 1.845 and reject iid normality at the .996 level.

KEY WORDS:

• Bond returns
• Foreign-exchange returns
• Generalized Pareto distribution
• Hill estimator
• Infinite variance
• Monte Carlo distribution of stable likelihood ratio statistic
• Pareto distribution
• Stable distributions
• Stock returns
• Tail index