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)?