The mean squared error of Geweke and Porter‐Hudak's estimator of the memory parameter of a long‐memory time series

We establish some asymptotic properties of a log‐periodogram regression estimator for the memory parameter of a long‐memory time series. We consider the estimator originally proposed by Geweke and Porter‐Hudak (The estimation and application of long memory time series models. Journal of Time Ser. Anal. 4 (1983), 221–37). In particular, we do not omit any of the low frequency periodogram ordinates from the regression. We derive expressions for the estimator's asymptotic bias, variance and mean squared error as functions of the number of periodogram ordinates, m, used in the regression. Consistency of the estimator is obtained as long as m←∞ and n←∞ with (m log m)/n← 0, where n is the sample size. Under these and the additional conditions assumed in this paper, the optimal m, minimizing the mean squared error, is of order O(n4/5). We also establish the asymptotic normality of the estimator. In a simulation study, we assess the accuracy of our asymptotic theory on mean squared error for finite sample sizes. One finding is that the choice m = n1/2, originally suggested by Geweke and Porter‐Hudak (1983), can lead to performance which is markedly inferior to that of the optimal choice, even in reasonably small samples.