Abstract: In this paper, we analyze the returns of stocks comprising the German stock index DAX with respect to the -stable distribution. We apply nonparametric estimation methods such as the Hill estimator as well as parametric estimation methods conditional on the -stable distribution. We find for both the nonparametric and parametric estimation methods that the -stable hypothesis cannot be rejected for the return distribution. We then employ the GARCH model; the fit of innovations modeled with an underlying -stable distribution is compared to the fit obtained from modelling the innovations with the skew- distribution. The -stable distribution is found to outperform the skew- distribution.

2000 AMS Mathematics Subject Classification: 91B28, 62M10, 62M15.

Key words and phrases: Stable distributions, heavy-tails, tail estimation, ARMA-GARCH, DAX 30.