Time Series Regression with a Unit Root

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).
KEYWORDS:Unit root, time series, functional limit theory, Wiener process, weak
dependence, continuous record, asymptotic expansion.
1. INTRODUCTION
AUTOREGRESSITVIEM E SERIES with a unit root have been the subject of much
recent attention in the econometrics literature. In part, this is because the unit
root hypothesis is of considerable interest in applications, not only with data
from financial and commodity markets where it has a long history but also with
aggregate time series. The study by Hall (1978) has been particularly influential
with regard to the latter, advancing theortical support for the random walk
hypothesis for consumption expenditure and providing further empirical
evidence. Moreover, the research program on vector autoregressive (VAR) modeling
of aggregate time series (see Doan et al. (1984) and the references therein)
has actually responded to this work by incorporating the random walk hypothesis
as a Bayesian prior in the VARspecification. This approach has helped to attenuate
the dimensionality problem of VAR modeling and seems to lead to decided
improvements in forecasting performance (Litterman (1984)).