# Difference in Difference - how to choose the control group (difference with no clear trend, parallel series or cointegrated?)

I'd like to use difference in difference methodology to test the impact of a treatment. Being the formula of the regression:

y = flag_test_period + flag_test_group + flag_test_group * flag_of_test_period

Checking the p value of the coefficient for the last term (the combination of both flags) is less than the desired threshold (i.e 0.05) would imply that there's an impact due to the treatment.

What I understand from this is that choosing the control group implies (considering the method) that at least there's no clear trend of the difference of y(control) and y(test) in the period previous to test. So I was running a regression in the previous period to test like:

y(control) - y(test) = gamma * t + alpha

and checking that the gamma has p_value > 0.9 (that would mean that there's no trend in the difference between both groups).

Questions:

1. Is it enough to require not having a clear trend in period previous to test in the difference between both groups? Or should I also need that the groups move in parallel?

2. Checking that the zones are cointegrated is necessary or even sufficient? I guess is none, because the series might be cointegrated and the difference might have a clear trend, right? What I understood from cointegration is that one series is a linear combination (+ error ~ N) of the other, so the series might not be parallel and want would be worse for the approach of DiD is that the difference between them has a clear trend. I guess, I could control by that (the trend in the diff) if I can identify the functional form. Would that be a good approach?

Thanks.