I have two variables. They're both $I(1)$ even when I fit constant and trend terms into the ADF test. The $p$-values for the stationarity tests are around 0.5 so it's not a marginal case.
However, when I execute the Johansen procedure with a constant in the error correction term the $\Pi$ matrix is full rank by the trace and maximum eigenvalue tests ($p < 0.01$). I've read a textbook and a paper on the consequences of $\Pi$ being full rank and they both state that this implies that both variables in the system are $I(0)$. It's important to note that if I exclude a constant from the ECT or if I add a deterministic time trend, then the rank of $\Pi$ is estimated as 1 (instead of full rank).
Is this most likely a type 1 error or could there be more going on here (such as partial integration)?
Best Answer
You have answered your own question. If you get cointegration when adding trend, or removing constant, this means that these terms are important, especially when without these terms you get conflicting results. To test the intuition I suggest doing Monte-Carlo simulation with simple unit roots. Also if you have two variables you can also test cointegration using Engle-Granger approach.