The Annals of Statistics

Narrow-band analysis of nonstationary processes

D. Marinucci and P. M. Robinson

Full-text: Open access


The behavior of averaged periodograms and cross-periodograms of a broad class of nonstationary processes is studied. The processes include nonstationary ones that are fractional of any order, as well as asymptotically stationary fractional ones. The cross-periodogram can involve two nonstationary processes of possibly different orders, or a nonstationary and an asymptotically stationary one. The averaging takes place either over the whole frequency band, or over one that degenerates slowly to zero frequency as sample size increases. In some cases it is found to make no asymptotic difference, and in particular we indicate how the behavior of the mean and variance changes across the two-dimensional space of integration orders. The results employ only local-to-zero assumptions on the spectra of the underlying weakly stationary sequences. It is shown how the results can be applied in fractional cointegration with unknown integration orders.

Article information

Ann. Statist., Volume 29, Number 4 (2001), 947-986.

First available in Project Euclid: 14 February 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 60G18: Self-similar processes 62M15: Spectral analysis

Nonstationary processes long range dependence least squares estimation narrow-band estimation cointegration analysis


Robinson, P. M.; Marinucci, D. Narrow-band analysis of nonstationary processes. Ann. Statist. 29 (2001), no. 4, 947--986. doi:10.1214/aos/1013699988.

Export citation


  • Akonom, J. and Gourieroux, C. (1987). A functional central limit theorem for fractional processes. Preprint.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Box, G. E. P. and Jenkins, G. M. (1971). Time Series Analysis. Forecasting and Control. HoldenDay, San Francisco.
  • Brillinger, D. R. (1975). Time Series, Data Analysis and Theory. Holt, Rinehart and Winston, New York.
  • Chan, N. H. and Terrin, N. (1995). Inference for unstable long-memory processes with applications to fractional unit root autoregressions. Ann. Statist. 23 1662-1683.
  • Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1-37.
  • Dickey, D. A. and Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. J. Amer. Statist. Assoc. 74 427-431.
  • Dolado, J. and Marmol, F. (1998). Efficient estimation of cointegrating relationships among higher order and fractionally integrated processes. Banco de Espana-Servicio de Estudios, Documento de Trabajo 9617.
  • Fox, R. and Taqqu, M. S. (1985). Non-central limit theorems for quadratic forms in random variables having long-range dependence. Ann. Probab. 13 428-476.
  • Fox, R. and Taqqu, M. S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14 517-532.
  • Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long memory time series models. J. Time Ser. Anal. 4 221-238.
  • Hannan, E. J. (1963). Regression for time series with errors of measurement. Biometrika 50 293-302.
  • Hannan, E. J. (1976). The asymptotic distribution of serial covariances. Ann. Statist. 4 396-399.
  • Hurvich, C. M. and Beltrao, K. (1993). Asymptotics for the low-frequency ordinates of the periodogram of a long-memory time series. J. Time Ser. Anal. 14 455-472.
  • Hurvich, C. M. and Ray, B. K. (1995). Estimation of the memory parameter for nonstationary or noninvertible fractionally integrated processes. J. Time Ser. Anal. 16 17-41.
  • Jeganathan, P. (1999). On asymptotic inference in cointegrated time series with fractionally integrated errors. Econometric Theory 15 583-621.
  • Jeganathan, P. (2001). On asymptotic inference in cointegrated time series with fractionally integrated errors. Preprint, Dept. Statistics, Univ. Michigan.
  • Kokoszka, P. S. and Taqqu, M. S. (1996). Parameter estimation for infinite variance fractional ARIMA. Ann. Statist. 24 1880-1913.
  • K ¨unsch, H. R. (1986). Discrimination between monotonic trends and long-range dependence. J. Appl. Probab. 23 1025-1030.
  • K ¨unsch, H. R. (1987). Statistical aspects of self-similar processes. Proceedings First World Congress of the Bernoulli Society 1 67-74. VNU Science Press, Utrech.
  • Lobato, I. G. (1997). Consistency of the averaged cross-periodogram in long memory series. J. Time Ser. Anal. 18 137-156.
  • Marinucci, D. (2000). Spectral regression for cointegrated time series with long-memory innovations. J. Time Ser. Anal. 21 685-705.
  • Marinucci, D. and Robinson, P. M. (1999). Alternative forms of fractional Brownian motion. J. Statist. Plann. Inference 80 111-122.
  • Marinucci, D. and Robinson, P. M. (2000). Weak convergence of multivariate fractional processes. Stochastic Process. Appl. 86 103-120.
  • Parzen, E. (1963). On spectral analysis with missing observations and amplitude modulation. Sankhya A 25 383-392.
  • Phillips, P. C. B. (1991). Spectral regression for cointegrated time series. In Nonparametric and Semiparametric Methods in Econometrics and Statistics (W. A. Barnett, J. Powell and G. Tauchen, eds.) 413-435. Cambridge Univ. Press. Robinson, P. M. (1994a). Semiparametric analysis of long-memory time series. Ann. Statist. 22 515-539. Robinson, P. M. (1994b). Rates of convergence and optimal spectral bandwidth for long-range dependence. Probab. Theory Related Fields 99 443-473. Robinson, P. M. (1995a). Log-periodogram regression of time series with long-range dependence. Ann. Statist. 23 1048-1072. Robinson, P. M. (1995b). Gaussian semiparametric estimation of long-range dependence. Ann. Statist. 23 1630-1661.
  • Robinson, P. M. and Marinucci, D. (1997). Semiparametric frequency-domain analysis of fractional cointegration. Preprint.
  • Robinson, P. M. and Marinucci, D. (2000). The averaged periodogram for nonstationary vector time series. Statist. Inference Stochastic Process. 3 149-160.
  • Silveira, G. (1991). Contributions to strong approximations in time series with applications in nonparametric statistics and functional central limit theorems. Ph.D. thesis, Univ. London.
  • Sowell, F. B. (1990). The fractional unit root distribution. Econometrica 58 495-505.
  • Stock, J. H. (1987). Asymptotic properties of least squares estimators of cointegrating vectors. Econometrica 55 1035-1056.
  • Whittle, P. (1951). Hypothesis testing in time series analysis. Thesis, Uppsala Univ.
  • Yong, C. H. (1974). Asymptotic Behaviour of Trigonometric Series. Chinese Univ., Hong Kong.
  • Zygmund, A. (1977). Trigonometric Series. Cambridge Univ. Press.