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ON TAIL INDEX ESTIMATION FOR DEPENDENT, HETEROGENEOUS DATA

Published online by Cambridge University Press:  17 February 2010

Jonathan B. Hill
Jonathan B. Hill*
Affiliation:
University of North Carolina–Chapel Hill
*
*Address correspondence to Jonathan B. Hill, Dept. of Economics, University of North Carolina–Chapel Hill; e-mail: [email protected].
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Abstract

In this paper we analyze the asymptotic properties of the popular distribution tail index estimator by Hill (1975) for dependent, heterogeneous processes. We develop new extremal dependence measures that characterize a massive array of linear, nonlinear, and conditional volatility processes with long or short memory. We prove that the Hill estimator is weakly and uniformly weakly consistent for processes with extremes that form mixingale sequences and asymptotically normal for processes with extremes that are near epoch dependent (NED) on some arbitrary mixing functional. The extremal persistence assumptions in this paper are known to hold for mixing, Lp-NED, and some non-Lp-NED processes, including ARFIMA, FIGARCH, explosive GARCH, nonlinear ARMA-GARCH, and bilinear processes, and nonlinear distributed lags like random coefficient and regime-switching autoregressions.

Finally, we deliver a simple nonparametric estimator of the asymptotic variance of the Hill estimator and prove consistency for processes with NED extremes.

Type
ARTICLES
Information
Econometric Theory , Volume 26 , Issue 5 , October 2010 , pp. 1398 - 1436
Copyright
Copyright © Cambridge University Press 2010

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