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Valid Edgeworth Expansions For The Whittle Maximum Likelihood Estimator For Stationary Long-Memory Gaussian Time Series


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  • Andrews, Donald W.K.
  • Lieberman, Offer


In this paper, we prove the validity of an Edgeworth expansion to the distribution of the Whittle maximum likelihood estimator for stationary long-memory Gaussian models with unknown parameter . The error of the (s 2)-order expansion is shown to be o(n (s 2) 2) the usual independent and identically distributed rate for a wide range of models, including the popular ARFIMA(p,d,q) models. The expansion is valid under mild assumptions on the behavior of the spectral density and its derivatives in the neighborhood of the origin. As a by-product, we generalize a theorem by Fox and Taqqu (1987, Probability Theory and Related Fields 74, 213 240) concerning the asymptotic behavior of Toeplitz matrices.Lieberman, Rousseau, and Zucker (2003, Annals of Statistics 31, 586 612) establish a valid Edgeworth expansion for the maximum likelihood estimator for stationary long-memory Gaussian models. For a significant class of models, their expansion is shown to have an error of o(n 1). The results given here improve upon those of Lieberman et al. in that the results provide an Edgeworth expansion for an asymptotically efficient estimator, as Lieberman et al. do, but the error of the expansion is shown to be o(n (s 2) 2), not o(n 1), for a broad range of models.

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Bibliographic Info

Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 21 (2005)
Issue (Month): 04 (August)
Pages: 710-734

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Handle: RePEc:cup:etheor:v:21:y:2005:i:04:p:710-734_05

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Cited by:
  1. D.S. Poskitt & Simone D. Grose & Gael M. Martin, 2012. "Higher Order Improvements of the Sieve Bootstrap for Fractionally Integrated Processes," Monash Econometrics and Business Statistics Working Papers 9/12, Monash University, Department of Econometrics and Business Statistics.
  2. Donald W.K. Andrews & Offer Lieberman, 2002. "Higher-order Improvements of the Parametric Bootstrap for Long-memory Gaussian Processes," Cowles Foundation Discussion Papers 1378, Cowles Foundation for Research in Economics, Yale University.
  3. Morten ├śrregaard Nielsen & Per Frederiksen, 2005. "Finite Sample Comparison of Parametric, Semiparametric, and Wavelet Estimators of Fractional Integration," Working Papers 1189, Queen's University, Department of Economics.


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