Maximum likelihood estimation for α-stable autoregressive processes

Maximum likelihood estimation for α-stable autoregressive processes

Beth Andrews, Matthew Calder, and Richard A. Davis

Source: Ann. Statist. Volume 37, Number 4 (2009), 1946-1982.

Abstract

We consider maximum likelihood estimation for both causal and noncausal autoregressive time series processes with non-Gaussian α-stable noise. A nondegenerate limiting distribution is given for maximum likelihood estimators of the parameters of the autoregressive model equation and the parameters of the stable noise distribution. The estimators for the autoregressive parameters are n^1/α-consistent and converge in distribution to the maximizer of a random function. The form of this limiting distribution is intractable, but the shape of the distribution for these estimators can be examined using the bootstrap procedure. The bootstrap is asymptotically valid under general conditions. The estimators for the parameters of the stable noise distribution have the traditional n^1/2 rate of convergence and are asymptotically normal. The behavior of the estimators for finite samples is studied via simulation, and we use maximum likelihood estimation to fit a noncausal autoregressive model to the natural logarithms of volumes of Wal-Mart stock traded daily on the New York Stock Exchange.

Primary Subjects: 62M10

Secondary Subjects: 62E20, 62F10

Keywords: Autoregressive models; maximum likelihood estimation; noncausal; non-Gaussian; stable distributions

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aos/1245332837
Digital Object Identifier: doi:10.1214/08-AOS632
Zentralblatt MATH identifier: 05582015
Mathematical Reviews number (MathSciNet): MR2533476

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