Remark on the asymptotic distribution of the OLS estimator in a simple Gaussian unit-root autoregression

Abstract

This paper considers the asymptotic distribution of the OLS estimator in a simple, Gaussian unit-root AR(1) with fixed, non-zero startup. All asymptotic possibilities are considered. The approach is new, relatively simple, and relies on observing and determining the asymptotic/limiting behavior of the underlying finite sample distribution. It does not rely on inversion of joint moment generating or characteristic functions to derive limiting distributions. The paper introduces small-sigma/parameter-based asymptotic theory and re-examines large-sample asymptotic theory. In addition, combinations of these asymptotic approaches are considered explicitly. The analysis provides a set of very interesting and sometimes surprising results.AR(1) OLS estimator Unit-root process Parameter-based asymptotic theory Small-sigma asymptotic theory Large-sample asymptotic theory