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

TL;DR

This paper investigates the asymptotic behavior of least squares estimators for nearly unstable long-memory processes with strong dependence, revealing convergence to fractional Ornstein-Uhlenbeck functionals under nonstandard scaling.

Contribution

It introduces a general framework showing that least squares estimators for nearly unstable processes converge to fractional Ornstein-Uhlenbeck functionals, extending classical results to strongly dependent innovations.

Findings

Convergence of least squares estimators to fractional Ornstein-Uhlenbeck functionals.

Use of fractional integrated noise to illustrate key ideas.

Identification of nonstandard scaling and limiting distributions.

Abstract

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.