The Annals of Statistics

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

Abstract

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

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

Dates
First available in Project Euclid: 4 April 2002

https://projecteuclid.org/euclid.aos/1017938932

Digital Object Identifier
doi:10.1214/aos/1017938932

Mathematical Reviews number (MathSciNet)
MR1740105

Zentralblatt MATH identifier
0962.62085

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

Citation

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. https://projecteuclid.org/euclid.aos/1017938932

References

• ARCONES, M. 1994. Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 2243 2274. Z.
• BERAN, J. 1993. Fitting long memory models by generalized regression. Biometrika 80 817 822. Z.
• BERAN, J. 1994. Statistics for long memory processes. Chapman and Hall, New York. Z.
• DAHLHAUS, R. 1989. Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749 1766. Z.
• FOX, R. and TAQQU, M. S. 1986. Large samples properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14 517 532. Z.
• GEWEKE, J. and PORTER-HUDAK, S. 1983. The estimation and application of long memory time series models. J. Time Anal. 4 221 238. Z.
• GIRAITIS, L., ROBINSON, P. and SAMAROV, A. 1997. Rate optimal semiparametric estimation of the memory parameter of the Gaussian time series with long range dependence. J. Time Series Anal. 18 49 61. Z.
• GIRAITIS, L. and SURGAILIS, D. 1990. A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittles's estimate. Probab. Theory Related Fields 86 87 104. Z.
• GRANGER, C. W. J. and JOYEUX, R. 1980. An introduction to long memory time series and fractional differencing. J. Time Series Anal. 1 15 30. Z.
• HART, J. D. 1997. Nonparametric smoothing and lack-of-fit tests. Springer Series in Statistics. New York, NY: Springer. Z.
• HURVICH, C. and BELTRAO, K. 1993. Asymptotics of the low-frequency ordinates of the periodogram of a long-memory time series. J. Time Series Anal. 14 455 472.
• HURVICH, C., DEO, R. and BRODSKY, J. 1998. The mean squared error of Geweke and PorterHudak's estimator of a long-memory time-series. J. Time Ser. Anal. 19 19 46. Z.
• JOHNSON, N. L. and KOTZ, S. 1970. Continuous Univariate Distributions 1 Wiley, New York. Z.
• KUNSCH, H. R. 1986. Discrimination between monotonic trends and long-range dependence. ¨ J. Appl. Probab. 23 1025 1030. Z.
• MOULINES, E. and SOULIER, P. 1998. Data-driven order selection for long range dependent time series. J. Time Ser. Anal. To appear. Z.
• ROBINSON, P. M. 1994. Time series with strong dependence. In Advances in Econometrics. Proceedings of the Sixth World Congress 1 47 95. Cambridge Univ. Press. Z.
• ROBINSON, P. M. 1995a. Log-periodogram regression of time series with long range dependence. Annal. Statist. 23 1048 1072. Z.
• ROBINSON, P. M. 1995b. Gaussian semi-parametric estimation of long range dependence. Annal. Statist. 23 1630 1661. Z.
• SOULIER, P. 1998. Some new bounds and a central limit theorem for nonstationary sequences of Gaussian vectors. Universite d'Evry-Val d'Essonne. Submitted to Statistics and Prob´ ability Letters. Z.
• TAQQU, M. S. 1977. Law of the iterated logarithm for sums of nonlinear functions of Gaussian variables that exhibit long range dependence. Z. Wahrsch. Verw. Gebiete. 40 203 238. Z.
• VELASCO, C. 1999. Gaussian semiparametric estimation of non-stationary time series. J. Time Ser. Anal. 20 87 127.