The Annals of Statistics

Asymptotic theory of least squares estimators for nearly unstable processes under strong dependence

Boris Buchmann and Ngai Hang Chan

Full-text: Open access


This paper considers the effect of least squares procedures for nearly unstable linear time series with strongly dependent innovations. Under a general framework and appropriate scaling, it is shown that ordinary least squares procedures converge to functionals of fractional Ornstein–Uhlenbeck processes. We use fractional integrated noise as an example to illustrate the important ideas. In this case, the functionals bear only formal analogy to those in the classical framework with uncorrelated innovations, with Wiener processes being replaced by fractional Brownian motions. It is also shown that limit theorems for the functionals involve nonstandard scaling and nonstandard limiting distributions. Results of this paper shed light on the asymptotic behavior of nearly unstable long-memory processes.

Article information

Ann. Statist., Volume 35, Number 5 (2007), 2001-2017.

First available in Project Euclid: 7 November 2007

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] 62E20: Asymptotic distribution theory
Secondary: 60F17: Functional limit theorems; invariance principles

Autoregressive process least squares fractional noise fractional integrated noise fractional Brownian motion fractional Ornstein–Uhlenbeck process long-range dependence nearly nonstationary processes stochastic integrals unit-root problem


Buchmann, Boris; Chan, Ngai Hang. Asymptotic theory of least squares estimators for nearly unstable processes under strong dependence. Ann. Statist. 35 (2007), no. 5, 2001--2017. doi:10.1214/009053607000000136.

Export citation


  • Anderson, T. W. (1959). On asymptotic distributions of estimates of parameters of stochastic difference equations. Ann. Math. Statist. 30 676--687.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Brockwell, P. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • Chan, N. H. and Wei, C. Z. (1987). Asymptotic inference for nearly nonstationary AR$(1)$ processes. Ann. Statist. 15 1050--1063.
  • Dahlhaus, R. (1985). Data tapers in time series analysis. Habilitation thesis, Universität-GHS, Essen.
  • Davidson, J. (2002). Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes. J. Econometrics 106 243--269.
  • Davidson, J. and de Jong, R. M. (2000). The functional central limit theorem and weak convergence to stochastic integrals. II. Fractionally integrated processes. Econometric Theory 16 643--666.
  • Davydov, Yu. A. (1970). The invariance principle for stationary processes. Theory Probab. Appl. 15 487--498.
  • de Jong, R. M. and Davidson, J. (2000). The functional central limit theorem and weak convergence to stochastic integrals. I. Weakly dependent processes. Econometric Theory 16 621--642.
  • Dobrushin, R. L. and Major, P. (1979). Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 27--52.
  • Doukhan, P., Oppenheim, G. and Taqqu, M. S., eds. (2003). Theory and Applications of Long-Range Dependence. Birkhäuser, Boston.
  • Duncan, T. E., Hu, Y. and Pasik-Duncan, B. (2000). Stochastic calculus for fractional Brownian motion. I. Theory. SIAM J. Control Optim. 38 582--612.
  • Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
  • Jeganathan, P. (1991). On the asymptotic behavior of least-squares estimators in AR time series with roots near the unit circle. Econometric Theory 7 269--306.
  • Jeganathan, P. (1999). On asymptotic inference in cointegrated time series with fractionally integrated errors. Econometric Theory 15 583--621.
  • Larsson, R. (1998). Bartlett corrections for unit root test statistics. J. Time Ser. Anal. 19 425--438.
  • Le Breton, A. and Pham, D. T. (1989). On the bias of the least squares estimator for the first order autoregressive process. Ann. Inst. Statist. Math. 41 555--563.
  • Mann, H. B. and Wald, A. (1943). On the statistical treatment of linear stochastic difference equations. Econometrica 11 173--220.
  • Marinucci, D. and Robinson, P. M. (1999). Alternative forms of fractional Brownian motion. J. Statist. Plann. Inference 80 111--122.
  • Phillips, P. C. B. (1987). Towards a unified asymptotic theory for autoregression. Biometrika 74 535--547.
  • Rao, M. M. (1978). Asymptotic distribution of an estimator of the boundary parameter of an unstable process. Ann. Statist. 6 185--190.
  • Robinson, P. M., ed. (2003). Time Series with Long Memory. Oxford Univ. Press.
  • Robinson, P. M. (2005). Efficiency improvements in inference on stationary and nonstationary fractional time series. Ann. Statist. 33 1800--1842.
  • Robinson, P. M. (2005). The distance between rival nonstationary fractional processes. J. Econometrics 128 283--300.
  • Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall, New York.
  • Sowell, F. (1990). The fractional unit root distribution. Econometrica 58 495--505.
  • Tanaka, K. (1996). Time Series Analysis: Nonstationary and Noninvertible Distribution Theory. Wiley, New York.
  • Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287--302.
  • Velasco, C. and Robinson, P. M. (2000). Whittle pseudo-maximum likelihood estimation for nonstationary time series. J. Amer. Statist. Assoc. 95 1229--1243.
  • White, J. S. (1958). The limiting distribution of the serial correlation coefficient in the explosive case. Ann. Math. Statist. 29 1188--1197.
  • Wu, W. B. (2006). Unit root testing for functionals of linear processes. Econometric Theory 22 1--14.
  • Wu, W. B. and Min, W. (2005). On linear processes with dependent innovations. Stochastic Process. Appl. 115 939--958.
  • Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67 251--282.
  • Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields 111 333--374.