A New Regression-Based Tail Index Estimator: An Application to Exchange Rates

In this paper, a new regression-based approach for the estimation of the tail index of heavy-tailed distributions is introduced. Comparatively to many procedures currently available in the literature, our method does not involve order statistics and can be applied in more general contexts than just Pareto. The procedure is in line with approaches used in experimental data analysis with fixed explanatory variables, and has several important features which are worth highlighting. First, it provides a bias reduction when compared to available regression-based methods and a fortiori over standard least-squares based estimators of the tail index. Second, it is more resilient to the choice of the tail length used in the estimation of the index than the widely used Hill estimator. Third, when the effect of the slowly varying function at infinity of the Pareto distribution (the so called second order behaviour of the Taylor expansion) vanishes slowly our estimator continues to perform satisfactorily, whereas the Hill estimator rapidly deteriorates. Fourth, our estimator performs well under dependence of unknown form. For inference purposes, we also provide a way to compute the asymptotic variance of the proposed estimator under time dependence and conditional heteroscedasticity. An empirical application of the procedure to exchange rates is also provided.