The Annals of Statistics

Stable limits of martingale transforms with application to the estimation of GARCH parameters

Thomas Mikosch and Daniel Straumann

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In this paper we study the asymptotic behavior of the Gaussian quasi maximum likelihood estimator of a stationary GARCH process with heavy-tailed innovations. This means that the innovations are regularly varying with index α∈(2,4). Then, in particular, the marginal distribution of the GARCH process has infinite fourth moment and standard asymptotic theory with normal limits and $\sqrt{n}$-rates breaks down. This was recently observed by Hall and Yao [Econometrica 71 (2003) 285–317]. It is the aim of this paper to indicate that the limit theory for the parameter estimators in the heavy-tailed case nevertheless very much parallels the normal asymptotic theory. In the light-tailed case, the limit theory is based on the CLT for stationary ergodic finite variance martingale difference sequences. In the heavy-tailed case such a general result does not exist, but an analogous result with infinite variance stable limits can be shown to hold under certain mixing conditions which are satisfied for GARCH processes. It is the aim of the paper to give a general structural result for infinite variance limits which can also be applied in situations more general than GARCH.

Article information

Ann. Statist., Volume 34, Number 1 (2006), 493-522.

First available in Project Euclid: 2 May 2006

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Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators
Secondary: 62G32: Statistics of extreme values; tail inference 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems 60G42: Martingales with discrete parameter 60G70: Extreme value theory; extremal processes

GARCH process Gaussian quasi-maximum likelihood regular variation infinite variance stable distribution stochastic recurrence equation mixing


Mikosch, Thomas; Straumann, Daniel. Stable limits of martingale transforms with application to the estimation of GARCH parameters. Ann. Statist. 34 (2006), no. 1, 493--522. doi:10.1214/009053605000000840.

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