Annals of Statistics

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

Boris Buchmann and Ngai Hang Chan

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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.

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

First available in Project Euclid: 7 November 2007

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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.

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