The Annals of Statistics

Semiparametric Analysis of Long-Memory Time Series

P. M. Robinson

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We study problems of semiparametric statistical inference connected with long-memory covariance stationary time series, having spectrum which varies regularly at the origin: There is an unknown self-similarity parameter, but elsewhere the spectrum satisfies no parametric or smoothness conditions, it need not be in $L_p$, for any $p > 1$, and in some circumstances the slowly varying factor can be of unknown form. The basic statistic of interest is the discretely averaged periodogram, based on a degenerating band of frequencies around the origin. We establish some consistency properties under mild conditions. These are applied to show consistency of new estimates of the self-similarity parameter and scale factor. We also indicate applications of our results to standard errors of least squares estimates of polynomial regression with long-memory errors, to generalized least squares estimates of this model and to estimates of a "cointegrating" relationship between long-memory time series.

Article information

Ann. Statist., Volume 22, Number 1 (1994), 515-539.

First available in Project Euclid: 11 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62M15: Spectral analysis
Secondary: 62G05: Estimation 60G18: Self-similar processes

Long-memory time series semiparametric inference regular variation autocorrelation-consistent standard errors cointegration


Robinson, P. M. Semiparametric Analysis of Long-Memory Time Series. Ann. Statist. 22 (1994), no. 1, 515--539. doi:10.1214/aos/1176325382.

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