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February 2006 Stable limits of martingale transforms with application to the estimation of GARCH parameters
Thomas Mikosch, Daniel Straumann
Ann. Statist. 34(1): 493-522 (February 2006). DOI: 10.1214/009053605000000840


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.


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Thomas Mikosch. Daniel Straumann. "Stable limits of martingale transforms with application to the estimation of GARCH parameters." Ann. Statist. 34 (1) 493 - 522, February 2006.


Published: February 2006
First available in Project Euclid: 2 May 2006

zbMATH: 1091.62082
MathSciNet: MR2275251
Digital Object Identifier: 10.1214/009053605000000840

Primary: 62F12
Secondary: 60E07 , 60F05 , 60G42 , 60G70 , 62G32

Keywords: GARCH process , Gaussian quasi-maximum likelihood , infinite variance , Mixing , regular variation , stable distribution , stochastic recurrence equation

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 1 • February 2006
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