The stable Paretian hypothesis and the frequency of large returns: an examination of major German stocks

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Abstract
A statistical analysis is provided of daily returns for 30 German stocks forming the DAX share index as well as the DAX itself during the period 1988–1994. Estimating the parameters of the stable laws and performing standard tests of fit, some evidence in favour of the stable Paretian hypothesis is found. However, application of a more recently developed semiparametric technique for analysis of the limiting behaviour in the tails of a distribution (Hill's tail index estimator) suggests that the empirical tail regions are thinner than expected under a stable distribution. Since the reliability of tail index estimation rests on the appropriateness of the chosen tail regions, it is also examined whether the tails indeed follow approximately the expected limiting distributions of extremes. It turns out that convergence to the limiting extreme value distributions cannot be rejected in the vast majority of cases for tails covering the most extreme 15% of observations or less. Furthermore, strong similarity in the extremal behaviour of the 30 series is found and the hypothesis of identical limit laws governing their extreme value distributions is not rejected.