Regression Theory for Near-Integrated Time Series

The concept of a near-integrated vector random process is introduced. Such processes
help us to work towards a general asymptotic theory of regression for multiple time series
in which some series may be integrated processes of the ARIMA type, others may be stable
ARMA processes with near unit roots, and yet others may be mildly explosive. A limit
theory for the sample moments of such time series is developed using weak convergence
and is shown to involve simple functionals of a vector diffusion. The results suggest finite
sample approximations which in the stationary case correspond to conventional central
limit theory. The theory is applied to the study of vector autoregressions and cointegrating
regressions of the type recently advanced by Engle and Granger (1987). A noncentral
limiting distribution theory is derived for some recently proposed multivariate unit root
tests. This yields some interesting insights into the asymptotic power properties of the
various tests. Models with drift and near-integration are also studied. The asymptotic
theory in this case helps to bridge the gap between the nonnormal asymptotics obtained by
Phillips and Durlauf (1986) for regressions with integrated regressors and the normal
asymptotics that usually apply in regressions with deterministic regressors.
KEYWORDS:Brownian motion, cointegration, diffusion, near-integration, unit root tests.
1. INTRODUCTION
MANYOBSERVED TIME SERIES in economics seem to be modeled rather well by
integrated processes. The simplest model generating an integrated process is, of
course, a random walk; and this is a model that has been widely used in financial
and commodity market studies, in theories of rational expectations, and in recent
work with aggregate economic time series. More general models of the ARIMA
type have also been used frequently in econometric work and have been found to
represent very adequately the movements in many different economic series.
Moreover, in a recent study Nelson and Plosser (1982) provide substantial
empirical evidence that a wide selection of macroeconomic time series for the
U.S. are modeled better in terms of integrated processes than as stationary
processes about a deterministic trend. In fact, their findings support autoregressive
representations with unit roots for all but one of the historical time series in
their study.