Narrow-band analysis of nonstationary processes
D. Marinucci and P. M. Robinson
Source: Ann. Statist. Volume 29, Number 4
(2001), 947-986.
Abstract
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.
First Page:
Show
Hide
Keywords: Nonstationary processes; long range dependence; least squares estimation; narrow-band estimation; cointegration analysis
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1013699988
Mathematical Reviews number (MathSciNet): MR1869235
Digital Object Identifier: doi:10.1214/aos/1013699988
Zentralblatt MATH identifier: 1012.62100
References
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.
Mathematical Reviews (MathSciNet): MR38:1718
Box, G. E. P. and Jenkins, G. M. (1971). Time Series Analysis. Forecasting and Control. HoldenDay, San Francisco.
Mathematical Reviews (MathSciNet): MR436499
Brillinger, D. R. (1975). Time Series, Data Analysis and Theory. Holt, Rinehart and Winston, New York.
Mathematical Reviews (MathSciNet): MR443257
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.
Zentralblatt MATH: 0843.62084
Mathematical Reviews (MathSciNet): MR1370302
Digital Object Identifier: doi:10.1214/aos/1176324318
Project Euclid: euclid.aos/1176324318
Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1-37.
Mathematical Reviews (MathSciNet): MR98b:62168
Zentralblatt MATH: 0871.62080
Digital Object Identifier: doi:10.1214/aos/1034276620
Project Euclid: euclid.aos/1034276620
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.
Mathematical Reviews (MathSciNet): MR80i:62072
Zentralblatt MATH: 0413.62075
Digital Object Identifier: doi:10.2307/2286348
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.
Mathematical Reviews (MathSciNet): MR86i:60066
Zentralblatt MATH: 0569.60016
Digital Object Identifier: doi:10.1214/aop/1176993001
Project Euclid: euclid.aop/1176993001
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.
Mathematical Reviews (MathSciNet): MR87k:62149
Zentralblatt MATH: 0606.62096
Digital Object Identifier: doi:10.1214/aos/1176349936
Project Euclid: euclid.aos/1176349936
Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long memory time series models. J. Time Ser. Anal. 4 221-238.
Mathematical Reviews (MathSciNet): MR85c:62237
Zentralblatt MATH: 0534.62062
Digital Object Identifier: doi:10.1111/j.1467-9892.1983.tb00371.x
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.
Mathematical Reviews (MathSciNet): MR53:1884
Zentralblatt MATH: 0328.62059
Digital Object Identifier: doi:10.1214/aos/1176343415
Project Euclid: euclid.aos/1176343415
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.
Mathematical Reviews (MathSciNet): MR95e:62101a
Zentralblatt MATH: 0782.62086
Digital Object Identifier: doi:10.1111/j.1467-9892.1993.tb00157.x
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.
Mathematical Reviews (MathSciNet): MR95m:62199
Zentralblatt MATH: 0813.62081
Digital Object Identifier: doi:10.1111/j.1467-9892.1995.tb00221.x
Jeganathan, P. (1999). On asymptotic inference in cointegrated time series with fractionally integrated errors. Econometric Theory 15 583-621.
Mathematical Reviews (MathSciNet): MR2000h:62100
Zentralblatt MATH: 0985.62070
Digital Object Identifier: doi:10.1017/S0266466699154057
JSTOR: links.jstor.org
Jeganathan, P. (2001). On asymptotic inference in cointegrated time series with fractionally integrated errors. Preprint, Dept. Statistics, Univ. Michigan.
Mathematical Reviews (MathSciNet): MR1717968
Zentralblatt MATH: 0985.62070
Digital Object Identifier: doi:10.1017/S0266466699154057
JSTOR: links.jstor.org
Kokoszka, P. S. and Taqqu, M. S. (1996). Parameter estimation for infinite variance fractional ARIMA. Ann. Statist. 24 1880-1913.
Mathematical Reviews (MathSciNet): MR98f:62257
Zentralblatt MATH: 0896.62092
Digital Object Identifier: doi:10.1214/aos/1069362302
Project Euclid: euclid.aos/1069362302
K ¨unsch, H. R. (1986). Discrimination between monotonic trends and long-range dependence. J. Appl. Probab. 23 1025-1030.
Mathematical Reviews (MathSciNet): MR88d:62161
Zentralblatt MATH: 0623.62085
Digital Object Identifier: doi:10.2307/3214476
JSTOR: links.jstor.org
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.
Mathematical Reviews (MathSciNet): MR1092336
Lobato, I. G. (1997). Consistency of the averaged cross-periodogram in long memory series. J. Time Ser. Anal. 18 137-156.
Mathematical Reviews (MathSciNet): MR98d:62163
Zentralblatt MATH: 0938.62103
Digital Object Identifier: doi:10.1111/1467-9892.00043
Marinucci, D. (2000). Spectral regression for cointegrated time series with long-memory innovations. J. Time Ser. Anal. 21 685-705.
Mathematical Reviews (MathSciNet): MR2001i:62095
Zentralblatt MATH: 0972.62089
Digital Object Identifier: doi:10.1111/1467-9892.00204
Marinucci, D. and Robinson, P. M. (1999). Alternative forms of fractional Brownian motion. J. Statist. Plann. Inference 80 111-122.
Mathematical Reviews (MathSciNet): MR2000e:60135
Zentralblatt MATH: 0934.60071
Digital Object Identifier: doi:10.1016/S0378-3758(98)00245-6
Marinucci, D. and Robinson, P. M. (2000). Weak convergence of multivariate fractional processes. Stochastic Process. Appl. 86 103-120.
Mathematical Reviews (MathSciNet): MR2001a:60040
Zentralblatt MATH: 1028.60030
Digital Object Identifier: doi:10.1016/S0304-4149(99)00088-5
Parzen, E. (1963). On spectral analysis with missing observations and amplitude modulation. Sankhya A 25 383-392.
Mathematical Reviews (MathSciNet): MR30:2654
Zentralblatt MATH: 0136.40701
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.
Mathematical Reviews (MathSciNet): MR1174982
Zentralblatt MATH: 0754.62077
Robinson, P. M. and Marinucci, D. (1997). Semiparametric frequency-domain analysis of fractional cointegration. Preprint.
Mathematical Reviews (MathSciNet): MR2077521
Robinson, P. M. and Marinucci, D. (2000). The averaged periodogram for nonstationary vector time series. Statist. Inference Stochastic Process. 3 149-160.
Mathematical Reviews (MathSciNet): MR1819292
Digital Object Identifier: doi:10.1023/A:1009925202524
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.
Mathematical Reviews (MathSciNet): MR91g:62072
Zentralblatt MATH: 0727.62025
Digital Object Identifier: doi:10.2307/2938213
JSTOR: links.jstor.org
Stock, J. H. (1987). Asymptotic properties of least squares estimators of cointegrating vectors. Econometrica 55 1035-1056.
Mathematical Reviews (MathSciNet): MR88m:62138
Zentralblatt MATH: 0651.62105
Digital Object Identifier: doi:10.2307/1911260
JSTOR: links.jstor.org
Whittle, P. (1951). Hypothesis testing in time series analysis. Thesis, Uppsala Univ.
Mathematical Reviews (MathSciNet): MR12,726c
Zentralblatt MATH: 0045.41301
Yong, C. H. (1974). Asymptotic Behaviour of Trigonometric Series. Chinese Univ., Hong Kong.
Zygmund, A. (1977). Trigonometric Series. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR58:29731
Zentralblatt MATH: 0367.42001
