zbMATH — the first resource for mathematics

The mean squared error of Geweke and Porter-Hudak’s estimator of the memory parameter of a long-memory time series. (English) Zbl 0920.62108
The aim of the paper is to obtain the asymptotic mean squared error of the GPH estimator [J. Geweke and S. Porter-Hudak, J. Time Ser. Anal. 4, 221-238 (1983; Zbl 0534.62062)] for the memory parameter of a long-memory time-series, assuming that the time-series is Gaussian. Expressions for the estimator’s asymptotic bias, variance, and mean squared error are obtained as functions of the number of periodogram ordinates, $$m$$, used in the regression. The optimal $$m$$ is shown to be of order $$O(n^{4/5})$$, where $$n$$ is the sample size. The obtained formula for the mean squared error implies that the GPH estimator is consistent under the additional condition that $$(m\log m)/n\to 0$$, as long as $$m\to\infty$$ and $$n\to\infty$$. Finally, the asymptotic normality of the GPH estimator is established.
The paper presents the results of a simulation study that asses the accuracy of the proposed asymptotic theory on the mean squared error for finite sample sizes. Another finding is that the choice $$m=n^{1/2}$$, originally suggested by Geweke and Porter-Hudak, can lead to performance which is markedly inferior to that of the optimal choice, even in reasonably small samples.
Reviewer: N.Curteanu (Iaşi)

MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62H12 Estimation in multivariate analysis 93E10 Estimation and detection in stochastic control theory 93E24 Least squares and related methods for stochastic control systems
Full Text: