The Annals of Statistics

Time series regression with long-range dependence

F. J. Hidalgo and P. M. Robinson

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A central limit theorem is established for time series regression estimates which include generalized least squares, in the presence of long-range dependence in both errors and stochastic regressors. The setting and results differ significantly from earlier work on regression withlong-range-dependent errors. Spectral singularities are permitted at any frequency. When sufficiently strong spectral singularities in the error and a regressor coincide at the same frequency, least squares need no longer be $n^{1/2}$-consistent, where n is the sample size. However, we show that our class of estimates is $n^{1/2}$-consistent and asymptotically normal. In the generalized least squares case, we show that efficient estimation is still possible when the error autocorrelation is known only up to finitely many parameters. We include a Monte Carlo study of finite-sample performance and provide an extension to nonlinear least squares.

Article information

Ann. Statist., Volume 25, Number 1 (1997), 77-104.

First available in Project Euclid: 10 October 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 60G18: Self-similar processes
Secondary: 62F12: Asymptotic properties of estimators 62J05: Linear regression 62J02: General nonlinear regression

Long-range dependence linear regression generalized least squares nonlinear regression


Robinson, P. M.; Hidalgo, F. J. Time series regression with long-range dependence. Ann. Statist. 25 (1997), no. 1, 77--104. doi:10.1214/aos/1034276622.

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