Abstract: We prove Hill’s (1975) tail index estimator is asymptotically normal when the employed data are generated by a stationary parametric time series {x_t(θ⁰) : t ∈ ℤ} and θ⁰ is an unknown k × 1 vector. We assume x_t(θ⁰) is unobservable but θ⁰ is estimable with estimator _n and sample size n ≥ 1, and that the filtered series x_t( _n) is observed and used to estimate the tail index. Natural applications include regression residuals, GARCH filters, and weighted sums based on an optimization problem like optimal portfolio selection. Our main result substantially extends Resnick and Stărică (1997)’s theory for estimated AR i.i.d. errors and Ling and Peng (2004)’s theory for estimated ARMA i.i.d. errors to a wide range of filtered time series since we do not require x_t(θ⁰) to be i.i.d., nor generated by a linear process with geometric dependence. We assume x_t(θ⁰) is β-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 Resnick and Stărică (1997) and Ling and Peng (2004) we do not require _n to be super--convergent when x _t(θ⁰) has an infinite variance. We allow a far greater variety of plug-ins, including those that are slower than , such as QML-type estimators for GARCH models.

Key words and phrases: GARCH filter, regression residuals, tail index estimation, weak dependence.