Asymptotic Theory for Maximum Likelihood Estimation of the Memory Parameter in Stationary Gaussian Processes

Consistency, asymptotic normality and e ciency of the maximum likelihood estimator for stationary Gaussian time series, were shown to hold in the short memory case by Hannan (1973) and in the long memory case by Dahlhaus (1989). In this paper, we extend these results to the entire stationarity region, including the case of intermediate memory and noninvertibility. In the process of proving the main results, we provide a useful theorem on the limiting behavior of a product of Toeplitz matrices under strictly weaker conditions than those employed by Dahlhaus (1989).