Abstract: We prove Hill’s (1975) tail index estimator is asymptotically normal
when the employed data are generated by a stationary parametric time
series {xt(θ0) : t ∈ ℤ} and θ0 is an unknown k × 1 vector. We assume
xt(θ0) is unobservable but θ0 is estimable with estimator
n and sample
size n ≥ 1, and that the filtered series xt(
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 xt(θ0) to be i.i.d., nor generated by a
linear process with geometric dependence. We assume xt(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.