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 on function spaces. This theory generalizes the limit theory that was derived in Phillips and Durlauf (1985) for integrated processes. It is also shown to imply a general central limit theory for standardized sums of stationary processes. The theory is applied to the study of vector autoregressions and cointegrating regressions of the type recently advanced by Granger and Engle (1985). A noncentral limiting distribution theory is derived for the unit root tests that have been proposed by Dickey and Fuller (1979, 1981), by Evans and Savin (1981, 1984) and by the author (1985a). This noncentral distribution theory yields some interesting insights into the asymptotic power properties of the various tests.
Phillips, Peter C.B., "Regression Theory for Near-Integrated Time Series" (1986). Cowles Foundation Discussion Papers. 1022.