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Plug‐in Selection of the Number of Frequencies in Regression Estimates of the Memory Parameter of a Long‐memory Time Series
Author(s) -
Hurvich Clifford M.,
Deo Rohit S.
Publication year - 1999
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/1467-9892.00140
Subject(s) - mathematics , estimator , statistics , series (stratigraphy) , consistency (knowledge bases) , mean squared error , monte carlo method , bias of an estimator , consistent estimator , minimum variance unbiased estimator , discrete mathematics , paleontology , biology
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) = Cn 4/5 , 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 n 4/5 . 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 = n 1/2 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.