Including dummy variables in VECM using Oxmetrics I've encountered a problem when doing rank test for an Vector Erro Correction Model using CATS. I'm looking at the relationship between the variables in the Purchasing Price Parity theory, especially that the following is stationary:
$$ p_t - \beta_1 \cdot p_t^* - \beta_2 \cdot e_t$$
When making a VAR(3)-model with dummies for year (month) 2005(2), 2005(9), 2007(2), 2008(10), 2015(12), 2016(1), 2020(4) and 2021(4), I get the following misspecification test using Oxmetrics:

The model seems to be well specified - only suffering from heteroskedasticity. Now I do I test for rank using the Trace test, and I find:

Apparently, the model seems to be stationary. However, this doesn't line up with the vector plots I get, which are the three first figures here:

If I instead specify the VAR(3)-model without dummies, I get:

The model is not well specified. However, if we proceed making a VECM-model using this, we get the following rank test:

Now the first rank is only significant. The vector graphs can be seen as the last three graph.
What I also notice is that the $\beta_2$ vector is almost identical in the second and fifth graph. They can't both be stationary and non-stationary. Graph three and six are clearly trending and should not be stationary.
Why does my model using dummies end up showing that the variables are cointegrated with rank 3 - eg. that they are stationary non-unit root?

I think I have an idea why this is the case.
The third figure should be a random walk process, however Oxmetrics puts it as a stationary process. This could be because the large amount of dummies included restructures the model in such a way that the random walk process is removed. The random walks are being given as a structural break which is negated by dummies. This can basically turn any non-stationary process into a stationary one if the amount of included dummies are large enough.
Can anyone confirm if this is somewhat true or not?
I found this by J. James Reade:

