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October2000 Limit theory for the sample autocorrelations and extremes of a GARCH (1,1) process
Thomas Mikosch, C{\u{a}}t{\u{a}}lin St{\u{a}}ric{\u{a}}
Ann. Statist. 28(5): 1427-1451 (October2000). DOI: 10.1214/aos/1015957401

Abstract

The asymptotic theory for the sample autocorrelations and extremes of a GARCH (1, 1) process is provided. Special attention is given to the case when the sum of the ARCH and GARCH parameters is close to 1, that is, when one is close to an infinite variance marginal distribution. This situation has been observed for various financial log-return series and led to the introduction of the IGARCH model. In such a situation, the sample autocorrelations are unreliable estimators of their deterministic counterparts for the time series and its absolute values, and the sample autocorrelations of the squared time series have nondegenerate limit distributions. We discuss the consequences for a foreign exchange rate series.

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Thomas Mikosch. C{\u{a}}t{\u{a}}lin St{\u{a}}ric{\u{a}}. "Limit theory for the sample autocorrelations and extremes of a GARCH (1,1) process." Ann. Statist. 28 (5) 1427 - 1451, October2000. https://doi.org/10.1214/aos/1015957401

Information

Published: October2000
First available in Project Euclid: 12 March 2002

zbMATH: 1105.62374
MathSciNet: MR1805791
Digital Object Identifier: 10.1214/aos/1015957401

Subjects:
Primary: 62P20
Secondary: 60G55 , 60J10 , 62F10 , 62F12 , 90A20

Keywords: extremal index , Extremes , foreign exchange rates , GARCH , Pareto tail , Point processes , sample autocorrelations , stochastic recurrence equation

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.28 • No. 5 • October2000
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