The Annals of Statistics

Broadband log-periodogram regression of time series with long-range dependence

Eric Moulines and Philippe Soulier

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This paper discusses the properties of an estimator of the memory parameter of a stationary long-memory time-series originally proposed by Robinson. As opposed to ‘‘narrow-band’’ estimators of the memory parameter (such as the Geweke and Porter-Hudak or the Gaussian semiparametric estimators) which use only the periodogram ordinates belonging to an interval which degenerates to zero as the sample size $n$ increases, this estimator builds a model of the spectral density of the process over all the frequency range, hence the name, “broadband.” This is achieved by estimating the ‘‘short-memory’’ component of the spectral density, $f*(x) = |1 - e^{ix}|^{2d}f(x)$, where $d \epsilon (-1/2, 1/2)$ is the memory parameter and $f(x)$ is the spectral density, by means of a truncated Fourier series estimator of log $f*$. Assuming Gaussianity and additional conditions on the regularity of $f*$ which seem mild, we obtain expressions for the asymptotic bias and variance of the long-memory parameter estimator as a function of the truncation order. Under additional assumptions, we show that this estimator is consistent and asymptotically normal. If the true spectral density is sufficiently smooth outside the origin, this broadband estimator outperforms existing semiparametric estimators, attaining an asymptotic mean-square error $O(\log(n)/n)$ .

Article information

Ann. Statist., Volume 27, Number 4 (1999), 1415-1439.

First available in Project Euclid: 4 April 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 60F05: Central limit and other weak theorems

Long-range dependence log-periodogram regression central limit theorems for dependent variables


Moulines, Eric; Soulier, Philippe. Broadband log-periodogram regression of time series with long-range dependence. Ann. Statist. 27 (1999), no. 4, 1415--1439. doi:10.1214/aos/1017938932.

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