ON MAXIMUM-LIKELIHOOD-ESTIMATION OF THE DIFFERENCING PARAMETER OF FRACTIONALLY-INTEGRATED NOISE WITH UNKNOWN MEAN

There are two approaches to maximum likelihood (ML) estimation of the parameter of fractionally-integrated noise: approximate frequency-doma in ML [Fox and Taqqu (1986)] and exact time-domain ML [Sowell (1992b)] . If the mean of the process is known, then a clear finite-sample mean -squared error ranking of the estimators emerges: the exact time-domai n estimator is superior. We show in this paper, however, that the fini te-sample efficiency of approximate frequency-domain ML relative to ex act time-domain ML rises dramatically when the mean is unknown and so must be estimated. The intuition for our result is straightforward: th e frequency-domain ML estimator is invariant to the true but unknown m ean of the process, while the time-domain ML estimator is not. Feasibl e time-domain estimation must therefore be based upon de-meaned data, but the long memory associated with fractional integration makes preci se estimation of the mean difficult. We conclude that the frequency-do main estimator is an attractive and efficient alternative for situatio ns in which large sample sizes render time-domain estimation impractic al.