Tail Index Estimation for Parametric Families
Using Log Moments
Tucker McElroy
KEY WORDS: Extreme Value Theory, Heavy Tails, Stable Distributions
ABSTRACT
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 in¯nite 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.
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