In the context of I(1) time series, we provide some asymptotic results for the Davidson-MacKinnon J-type test. We examine both the case where our regressor sets x1t and x2t are not cointegrated, and the case where they are. In the former case, the OLS estimator of the weighting coefficient from the artificial compound model converges at rate T to a mixed normal distribution, and the associated t-statistic has an asymptotic standard normal distribution. In the latter case, we find that the J-test also has power against violation of weak exogeneity (with respect to the short-run coefficients of the null model), which is caused by correlation between the disturbance of the null model and that of the cointegrating equation linking x1t and x2t. Moreover, unlike the previous case, the OLS estimator of the weighting coefficient from the artificial compound model converges at \sqrt{T} to an asymptotic normal distribution when the null model is specified correctly. In an empirical illustration, we use the tests to examine an industrial production data set for six countries.