# Asymptotic Properties of Residual Based Tests for Cointegration

https://doi.org/0012-9682(199001)58:1<165:APORBT>2.0.CO;2-V
p. 165-193

P. C. B. Phillips, S. Ouliaris

This paper develops an asymptotic theory for residual based tests for cointegration. These tests involve procedures that are designed to detect the presence of a unit root in the residuals of (cointegrating) regressions among the levels of economic time series. Attention is given to the augmented Dickey-Fuller (ADF) test that is recommended by Engle-Granger (1987) and the $Z_\alpha$ and $Z_t$ unit root tests recently proposed by Phillips (1987). Two new tests are also introduced, one of which is invariant to the normalization of the cointegrating regression. All of these tests are shown to be asymptotically similar and simple representations of their limiting distributions are given in terms of standard Brownian motion. The ADF and $Z_t$ tests are asymptotically equivalent. Power properties of the tests are also studied. The analysis shows that all the tests are consistent if suitably constructed but that the ADF and $Z_t$ tests have slower rates of divergence under cointegration than the other tests. This indicates that, at least in large samples, the $Z_\alpha$ test should have superior power properties. The paper concludes by addressing the larger issue of test formulation. Some major pitfalls are discovered in procedures that are designed to test a null of cointegration (rather than no cointegration). These defects provide strong arguments against the indiscriminate use of such test formulations and support the continuing use of residual based unit root tests. A full set of critical values for residual based tests is included. These allow for demeaned and detrended data and cointegrating regressions with up to five variables.