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Valid Edgeworth Expansions for the Whittle Maximum Likelihood Estimator for Stationary Long-memory Gaussian Time Series

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Author Info

  • Donald W.K. Andrews

    ()
    (Cowles Foundation)

  • Offer Lieberman

    (Technion-Israel Institute of Technology)

Abstract

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 theta in Theta subset R^{d_{theta}} . The error of the (s-2)-order expansion is shown to be o(n^{(s-2)/2}) -- the usual iid 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 spectral density and its derivatives in the neighborhood of the origin. As a by-product, we generalize a Theorem by Fox and Taqqu (1987) concerning the asymptotic behavior of Toeplitz matrices. Lieberman, Rousseau, and Zucker (2002) (LRZ) 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 LRZ in that the results provide an Edgeworth expansion for an asymptotically efficient estimator, as LRZ 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

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1361.

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Length: 25 pages
Date of creation: Apr 2002
Date of revision:
Publication status: Published in Econometric Theory (2005), 21(4): 710-734
Handle: RePEc:cwl:cwldpp:1361

Note: CFP 1162.
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Related research

Keywords: ARFIMA; Edgeworth expansion; Long Memory; Whittle estimator;

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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. Morten ´┐Żrregaard Nielsen & Per Houmann Frederiksen, 2005. "Finite Sample Comparison of Parametric, Semiparametric, and Wavelet Estimators of Fractional Integration," Econometric Reviews, Taylor & Francis Journals, vol. 24(4), pages 405-443.
  3. Andrews, Donald W.K. & Lieberman, Offer & Marmer, Vadim, 2006. "Higher-order improvements of the parametric bootstrap for long-memory Gaussian processes," Journal of Econometrics, Elsevier, vol. 133(2), pages 673-702, August.

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