# Johansen's $\Pi$ is full rank except variables are non-stationary

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)?

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If two variables are I(1) but are not co-integrated, it simply means that there is no long run-relationship between the two variables. However, there may be short run relationship. So you need to run the VAR in first difference. If two variables are I(0),we can always use OLS. There is no justification to use co-integration.
This doesn't answer my question. I asked about the scenario of $\Pi$ being full rank while at the same time both variables being $I(1)$ (hence apparent contradiction). –  Jase Jan 10 at 17:29
But I didn't ask how to specify a VECM. I asked what causes a full rank $\Pi$ when both variables are non-stationary. –  Jase Jan 11 at 2:57