A Complete Asymptotic Series for the Autocovariance Function of a Long Memory Process

Offer Lieberman and Peter C. B. Phillips ()
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Offer Lieberman: Technion-Israel Institute of Technology

No 1586, Cowles Foundation Discussion Papers from Cowles Foundation, Yale University

Abstract: An infinite-order asymptotic expansion is given for the autocovariance function of a general stationary long-memory process with memory parameter d in (-1/2,1/2). The class of spectral densities considered includes as a special case the stationary and invertible ARFIMA(p,d,q) model. The leading term of the expansion is of the order O(1/k^{1-2d}), where k is the autocovariance order, consistent with the well known power law decay for such processes, and is shown to be accurate to an error of O(1/k^{3-2d}). The derivation uses Erdélyi's (1956) expansion for Fourier-type integrals when there are critical points at the boundaries of the range of integration - here the frequencies {0,2}. Numerical evaluations show that the expansion is accurate even for small k in cases where the autocovariance sequence decays monotonically, and in other cases for moderate to large k. The approximations are easy to compute across a variety of parameter values and models.

Keywords: Autocovariance; Asymptotic expansion; Critical point; Fourier integral; Long memory (search for similar items in EconPapers)
JEL-codes: C13 C22 (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-ecm and nep-ets
Date: Written
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Published in Journal of Econometrics (2008), 147: 99-103

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Journal Article: A complete asymptotic series for the autocovariance function of a long memory process (2008)
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