Tail Index Estimation for a Filtered Dependent Time Series

Abstract

We prove Hill's (1975) tail index estimator is asymptotically normal where the employed data are generated by a stationary parametric process {x(t)}. We assume x(t) is an unobservable function of a parameter q that is estimable. Natural applications include regression residuals and GARCH filters. Our main result extends Resnick and Stărică's (1997) theory for estimated AR i.i.d. errors and Ling and Peng's (2004) theory for estimated ARMA i.i.d. errors to a wide range of filtered time series since we do not require x(t) to be i.i.d., nor generated by a linear process with geometric dependence. We assume x(t) is b-mixing with possibly hyperbolic dependence, covering ARMA-GARCH filters, ARMA filters with heteroscedastic errors of unknown form, nonlinear filters like threshold autoregressions, and filters based on mis-specified models, as well as i.i.d. errors in an ARMA model. Finally, as opposed to existing results we do not require the plug-in for q to be super-n1/2-convergent when x(t) has an infinite variance allowing a far greater variety of plug-ins including those that are slower than n1/2 , like QML-type estimators for GARCH models.

Keywords: tail index estimation, regression residuals, GARCH filter, weak dependence