Plug‐in Selection of the Number of Frequencies in Regression Estimates of the Memory Parameter of a Long‐memory Time Series

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 long‐memory 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(opt)=Cn4/5, 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 log‐periodogram regression and derive its consistency. We also derive an asymptotically valid confidence interval for d when the number of frequencies used in the regression is deterministic and proportional to n4/5. 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(opt) above. We also study the performance of a bias‐corrected version of the plug‐in estimator of d. Comparisons with the choice m=n1/2 frequencies, as originally suggested by Geweke and Porter‐Hudak (The estimation and application of long memory time series models. Journal of Time Ser. Anal. 4 (1983), 221–37), are provided.