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-domain 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-domain estimator is superior. We show in this paper, however, that the finite-sample efficiency of approximate frequency-domain ML relative to exact time-domain ML rises dramatically when the mean is unknown and so must be estimated. The intuition for our result is straightforward: the frequency-domain ML estimator is invariant to the true but unknown mean of the process, while the time-domain ML estimator is not. Feasible time-domain estimation must therefore be based upon de-meaned data, but the long memory associated with fractional integration makes precise estimation of the mean difficult. We conclude that the frequency-domain estimator is an attractive and efficient alternative for situations in which large sample sizes render time-domain estimation impractical.