PLUG-IN SELECTION OF THE NUMBER OF FREQUENCIES IN REGRESSION ESTIMATES OF THE MEMORY PARAMETER OF A LONG- MEMORY TIME SERIES By Clifford M. Hurvich and Rohit

We consider the problem of selecting the number of frequencies, m, in a log-periodogram regression estimator of the memory parameter d of a Gaussian longmemory time series. It is known that under certain conditions the optimal m, minimizing the mean squared error of the corresponding estimator of d, is given by m ˆ Cn, where n is the sample size and C is a constant. In practice, C would be unknown since it depends on the properties of the spectral density near zero frequency. In this paper, we propose an estimator of C based again on a logperiodogram regression and derive its consistency. We also derive an asymptotically valid con®dence interval for d when the number of frequencies used in the regression is deterministic and proportional to n. In this case, squared bias cannot be neglected since it is of the same order as the variance. In a Monte Carlo study, we examine the performance of the plug-in estimator of d, in which m is obtained by using the estimator of C in the formula for m above. We also study the performance of a bias-corrected version of the plug-in estimator of d. Comparisons with the choice m ˆ n frequencies, as originally suggested by Geweke and Porter-Hudak (The estimation and application of long memory time series models. J. Time Ser. Anal. 4 (1983), 221±37), are provided.