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.