Tail Index Estimation for Parametric Families Using Log Moments

For heavy-tailed econometric data it is of interest to estimate the tail index, a parameter that measures the thickness of the tails of the marginal distribution. Common models for such distributions include Pareto and t distributions, and in other applications (such as hydrology) stable distributions are popular as well. This paper constructs square root n consistent estimators of the tail index that are independent of the scale of the data, which are based on an assumed knowledge of the parametric family for the marginal distribution. Given the popularity of parametric modeling for economic time series, this method gives an appealing alternative to nonparametric tail index estimators – such as the Hill and Pickands estimators – that are appropriate when the modeler believes that the data belongs to a certain known parametric family of distributions. The method works fairly well for stationary time series with intermediate memory and infinite variance, and since it is parametric does not depend upon blocking or tuning parameters. Small sample results and full asymptotics are provided in this paper, and simulation studies on various heavy-tailed time series models are given as well.