## The Annals of Statistics

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

#### Abstract

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

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

Dates
First available in Project Euclid: 2 May 2006

https://projecteuclid.org/euclid.aos/1146576272

Digital Object Identifier
doi:10.1214/009053605000000840

Mathematical Reviews number (MathSciNet)
MR2275251

Zentralblatt MATH identifier
1091.62082

#### Citation

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. https://projecteuclid.org/euclid.aos/1146576272

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