EXPANSIONS FOR THE DISTRIBUTION OF THE MAXIMUM LIKELIHOOD ESTIMATOR OF THE FRACTIONAL DIFFERENCE PARAMETER

Offer Lieberman and Peter Phillips

Econometric Theory, 2004, vol. 20, issue 3, 464-484

Abstract: The maximum likelihood estimator (MLE) of the fractional difference parameter in the Gaussian ARFIMA(0,d,0) model is well known to be asymptotically N(0,6/π2). This paper develops asymptotic expansions to the distribution of this statistic under the assumption of a known unit variance. The correction term for the density is shown to be independent of d, so that the MLE is second-order pivotal for d. This feature of the MLE is unusual, at least in time series contexts. Simulations show that the normal approximation is poor and that the expansions can make a significant improvement in accuracy provided the correction terms are computed without further asymptotic approximation.This paper was commenced and revised while Lieberman was visiting the Cowles Foundation during 2000–2002. Lieberman thanks the Cowles Foundation for support and hospitality during this visit. Phillips thanks the NSF for support under grants SBR 97-30295 and SES 0092509.

Date: 2004
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