The Annals of Statistics

Time series regression with long-range dependence

F. J. Hidalgo and P. M. Robinson

Full-text: Open access

Abstract

A central limit theorem is established for time series regression estimates which include generalized least squares, in the presence of long-range dependence in both errors and stochastic regressors. The setting and results differ significantly from earlier work on regression withlong-range-dependent errors. Spectral singularities are permitted at any frequency. When sufficiently strong spectral singularities in the error and a regressor coincide at the same frequency, least squares need no longer be $n^{1/2}$-consistent, where n is the sample size. However, we show that our class of estimates is $n^{1/2}$-consistent and asymptotically normal. In the generalized least squares case, we show that efficient estimation is still possible when the error autocorrelation is known only up to finitely many parameters. We include a Monte Carlo study of finite-sample performance and provide an extension to nonlinear least squares.

Article information

Source
Ann. Statist., Volume 25, Number 1 (1997), 77-104.

Dates
First available in Project Euclid: 10 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1034276622

Digital Object Identifier
doi:10.1214/aos/1034276622

Mathematical Reviews number (MathSciNet)
MR1429918

Zentralblatt MATH identifier
0870.62072

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 60G18: Self-similar processes
Secondary: 62F12: Asymptotic properties of estimators 62J05: Linear regression 62J02: General nonlinear regression

Keywords
Long-range dependence linear regression generalized least squares nonlinear regression

Citation

Robinson, P. M.; Hidalgo, F. J. Time series regression with long-range dependence. Ann. Statist. 25 (1997), no. 1, 77--104. doi:10.1214/aos/1034276622. https://projecteuclid.org/euclid.aos/1034276622


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References

  • ADENSTEDT, R. K. 1974. On large-sample estimation of the mean of a stationary random sequence. Ann. Statist. 2 1095 1107. Z.
  • ANDERSON, T. W. 1971. The Statistical Analy sis of Time Series. Wiley, New York. Z.
  • BINGHAM, N. H., GOLDIE, C. M. and TEUGELS, J. L. 1987. Regular Variation. Cambridge Univ. Press. Z.
  • DAHLHAUS, R. 1989. Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749 1766. Z.
  • DAHLHAUS, R. 1995. Efficient location and regression estimation for long range dependent regression models. Ann. Statist. 23 1029 1047. Z.
  • DAVIES, R. B. and HARTE, D. S. 1987. Tests for Hurst effect. Biometrika 74 95 101. Z.
  • DEGROOT, M. H. 1970. Optimal Statistical Decisions. McGraw-Hill, New York. Z.
  • EICKER, F. 1967. Limit theorems for regressions with unequal and dependent errors. Proc. Fifth Berkeley Sy mp. Math. Statist. Probab. 1 59 82. Univ. California Press, Berkeley. Z.
  • 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. Z.
  • GALLANT, A. R. and GOEBEL, J. J. 1976. Nonlinear regression with autocorrelated errors. J. Amer. Statist. Assoc. 71 961 967.
  • GIRAITIS, L., KOUL, H. L. and SURGAILIS, D. 1994. Asy mptotic normality of regression estimators with long memory errors. Statist. Probab. Lett. To appear. Z.
  • GIRAITIS, L. and SURGAILIS, D. 1991. A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asy mptotic normality of Whittle's estimate. Probab. Theory Related Fields 86 87 104. Z.
  • GRAY, H. L., ZHANG, N.-F. and WOODWARD, W. A. 1989. On generalized fractional processes. J. Time Ser. Anal. 10 233 257. Z.
  • GRAY, H. L., ZHANG, N.-F. and WOODWARD, W. A. 1994. On generalized fractional processes a correction. J. Time Ser. Anal. 15 561 562. Z.
  • HAMON, B. V. and HANNAN, E. J. 1963. Estimating relations between time series. J. Geophy s. Res. 68 1033 1041. Z.
  • HANNAN, E. J. 1967. The estimation of a lagged regression relation. Biometrika 54 409 418. Z.
  • HANNAN, E. J. 1971. Non-linear time series regression. J. Appl. Probab. 8 767 780. Z.
  • HANNAN, E. J. 1979. The central limit theorem for time series regression. Stochastic Process. Appl. 9 281 289. Z.
  • HOSOy A, Y. 1993. A limit theory in stationary processes with long-range dependence and related statistical models. Unpublished manuscript. Z.
  • IBRAGIMOV, I. A. and LINNIK, Y. V. 1971. Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen. Z.
  • JENNRICH, R. I. 1969. Non-linear least squares estimators. Ann. Math. Statist. 40 633 643. Z.
  • KASHy AP, R. L. and EOM, K.-B. 1988. Estimation in long-memory time series model. J. Time Ser. Anal. 9 35 41. Z.
  • KOUL, H. L. 1992. M-estimators in linear models with long range dependent errors. Statist. Probab. Lett. 14 153 164. Z.
  • KOUL, H. L. and MUKHERJEE, K. 1993. Asy mptotics of R-, MDand LAD estimators in linear regression models with long range dependent errors. Probab. Theory Related Fields 95 538 553. Z.
  • KUNSCH, H., BERAN, J. and HAMPEL, R. 1993. Contrasts under long-range correlations. Ann. ¨ Statist. 21 943 964. Z.
  • MALINVAUD, E. 1970. The consistency of nonlinear regression. Ann. Math. Statist. 41 959 969. Z.
  • ROBINSON, P. M. 1972. Non-linear regression for multiple time series. J. Appl. Probab. 9
  • ROBINSON, P. M. 1994a. Time series with strong dependence. In Advances in Econometrics: Z. Sixth World Congress C. A. Sims, ed. 1 47 95. Cambridge Univ. Press. Z.
  • ROBINSON, P. M. 1994b. Efficient tests of nonstationary hy potheses. J. Amer. Statist. Assoc. 89 1420 1437. Z.
  • SCOTT, D. J. 1973. Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach. Adv. in Appl. Probab. 5 119 137. Z.
  • STOUT, W. F. 1974. Almost Sure Convergence. Academic Press, New York. Z.
  • TAQQU, M. S. 1975. Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287 302. Z.
  • YAJIMA, Y. 1988. On estimation of a regression model with long-memory stationary errors. Ann. Statist. 16 791 807. Z.
  • YAJIMA, Y. 1991. Asy mptotic properties of the LSE in a regression model with long-memory stationary errors. Ann. Statist. 19 158 177. Z.
  • YONG, C. H. 1974. Asy mptotic Behaviour of Trigonometric Series. Chinese Univ. Hong Kong. Z.
  • Zy GMUND, A. 1977. Trigonometric Series. Cambridge Univ. Press.