This note provides some explanations and extensions for the interesting results in Ghose and Kroner (1995). Specifically, we address the following points: (1) It is shown that the stable distribution and the stationary ARCH distributions are partially nested with respect to their tail shapes; (2) A novel interpretation of the McCulloch estimator is developed from the vantage point of extreme value theory; (3) This interpretation not only explains the apparent bias in some of the reported estimates, but it also helps in remedying the problem. Taken together, all three points reinforce the main conclusion of Ghose and Kroner.

GARCH, stable distributions, tail index estimators
Semiparametric and Nonparametric Methods (jel C14), Time-Series Models; Dynamic Quantile Regressions (jel C22),
Journal of Empirical Finance
Erasmus School of Economics

Groenendijk, P.A, Lucas, A, & de Vries, C.G. (1995). A note on the relationship between GARCH and symmetric stable processes. Journal of Empirical Finance, 2(3), 253–264. doi:10.1016/0927-5398(95)00005-F