Valid Asymptotic Expansions for the Maximum Likelihood Estimator of the Parameter of a Stationary, Gaussian, Strongly Dependent Process

Offer Lieberman, Judith Rousseau and David M. Zucker
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Offer Lieberman: Technion - Israel Institute of Technology
Judith Rousseau: Universite de Paris
David M. Zucker: Hebrew University

No 1351, Cowles Foundation Discussion Papers from Cowles Foundation for Research in Economics, Yale University

Abstract: We establish the validity of an Edgeworth expansion to the distribution of the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process. The result covers ARFIMA type models, including fractional Gaussian noise. The method of proof consists of three main ingredients: (i) verification of a suitably modified version of Durbin's (1980) general conditions for the validity of the Edgeworth expansion to the joint density of the log-likelihood derivatives; (ii) appeal to a simple result of Skovgaard (1986) to obtain from this an Edgeworth expansion for the joint distribution of the log-likelihood derivatives; (iii) appeal to and extension of arguments of Bhattacharya and Ghosh (1978) to accomplish the passage from the result on the log-likelihood derivatives to the result for the maximum likelihood estimators. We develop and make extensive use of a uniform version of Dahlhaus's (1989) Theorem 5.1 on products of Toeplitz matrices; the extension of Dahlhaus's result is of interest in its own right. A small numerical study of the efficacy of the Edgeworth expansion is presented for the case of fractional Gaussian noise.

Keywords: Edgeworth expansions; long memory processes; ARFIMA models (search for similar items in EconPapers)
JEL-codes: C10 C13 (search for similar items in EconPapers)
Pages: 37 pages
Date: 2002-01
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Citations: View citations in EconPapers (1)

Published in The Annals of Statistics (2003), 31(2): 586-612

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