Second Order Expansions for the Distribution of the Maximum Likelihood Estimator
of the Fractional Difference Parameter

Offer Lieberman and Peter C.B. Phillips

July 2001

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/pi² ). This paper develops a second order asymptotic expansion to the distribution of this statistic. 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 make significant improvements in accuracy.

Key Words: ARFIMA, Edgeworth expansion, fractional differencing, pivotal statistic.