Document Type
Discussion Paper
Publication Date
10-1-2006
CFDP Number
1586
CFDP Pages
22
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 1-2 d ), 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.
Recommended Citation
Lieberman, Offer and Phillips, Peter C.B., "A Complete Asymptotic Series for the Autocovariance Function of a Long Memory Process" (2006). Cowles Foundation Discussion Papers. 1879.
https://elischolar.library.yale.edu/cowles-discussion-paper-series/1879