Time Series Regression with a Unit Root

TLDR

The paper investigates random walks in a general time‑series framework that allows weak dependence and heterogeneous innovations. The study develops new tests of the random‑walk hypothesis that accommodate a wide class of dependent and heterogeneous innovation sequences. The authors construct a new limiting‑distribution theory using functional central‑limit theory and continuous data recording, and develop tests that handle dependent and heterogeneous innovations. Least‑squares regression consistently estimates a unit root even with autocorrelated errors, and the derived limiting distribution and asymptotic expansion explain experimental results reported by Evans and Savin.

Abstract

This paper studies the random walk, in a general time series setting that allows for weakly dependent and heterogeneously distributed innovations. It is shown that simple least squares regression consistently estimates a unit root under very general conditions in spite of the presence of autocorrelated errors. The limiting distribution of the standardized estimator and the associated regression t statistic are found using functional central limit theory. New tests of the random walk hypothesis are developed which permit a wide class of dependent and heterogeneous innovation sequences. A new limiting distribution theory is constructed based on the concept of continuous data recording. This theory, together with an asymptotic expansion that is developed in the paper for the unit root case, explain many of the interesting experimental results recently reported in Evans and Savin (1981, 1984).