COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 781R "Regression Theory for Near Integrated Time Series" Peter C.B. Phillips January 1986 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 a 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 Granger and Engle (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. |
