The Annals of Statistics

Limit theory for the sample autocorrelations and extremes of a GARCH (1,1) process

Thomas Mikosch and C{\u{a}}t{\u{a}}lin St{\u{a}}ric{\u{a}}

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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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Ann. Statist., Volume 28, Number 5 (2000), 1427-1451.

First available in Project Euclid: 12 March 2002

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Primary: 62P20: Applications to economics [See also 91Bxx]
Secondary: 90A20 60G55: Point processes 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 62F10: Point estimation 62F12: Asymptotic properties of estimators

GARCH sample autocorrelations stochastic recurrence equation Pareto tail extremes extremal index point processes foreign exchange rates


Mikosch, Thomas; St{\u{a}}ric{\u{a}}, C{\u{a}}t{\u{a}}lin. Limit theory for the sample autocorrelations and extremes of a GARCH (1,1) process. Ann. Statist. 28 (2000), no. 5, 1427--1451. doi:10.1214/aos/1015957401.

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