COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1586 A Complete Asymptotic Series for the Autocovariance Function Offer Lieberman and Peter C.B. Phillips October 2006 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. |
