Residuals Based Tests for the Null of No Cointegration: An Analytical Comparison

This paper computes the asymptotic distribution of five residuals-based tests for the null of no cointegration under a local alternative when the tests are computed using both OLS and GLS detrended variables. The local asymptotic power of the tests is shown to be a function of Brownian Motion and Ornstein-Uhlenbeck processes, depending on a single nuisance parameter, which is determined by the correlation at frequency zero of the errors of the cointegration regression with the shocks to the right-hand variables. The tests are compared in terms of power in large and small samples. It is shown that, while no significant improvement can be achieved by using different unit root tests than the OLS detrended t-test originally proposed by Engle and Granger (1987), the power of GLS residuals tests can be higher than the power of system tests for some values of the nuisance parameter.