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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
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