Reducing the number of interaction terms in a diff-in-diff model? Is there a way to reduce the number of interaction terms in a diff-in-diff model to make the results easier to interpret/present?
Background: I'm trying to run the following diff-in-diff regression:
Y = POST + TREAT + A + B + C + POST*TREAT + POST*A + POST*B + POST*C + TREAT*A + TREAT*B + TREAT*C + A*B + B*C + POST*TREAT*A + POST*TREAT*B + POST*TREAT*C + e
POST, TREAT = dummies A,B,C = continuous variables
I'm actually interested in POST*TREAT*C, but I also want to allow my controls A and B to change through time/sample/time and sample.
This is terrible long and somewhat difficult to present. So my idea was to run:
Y = POST + A + B + C + POST*A + POST*B + POST*C for TREAT = 0
Y = POST + A + B + C + POST*A + POST*B + POST*C for TREAT = 1
and simply compare POST*C (TREAT=0) = POST*C (TREAT=1) using an F-test. However, p-values are higher using this approach, so I'm wondering if there is something else I can do?
PS: I'm using Stata.
 A: The only reason for interacting your controls with the treatment, time and difference in differences dummies would be if you suspect that those controls are also affected by the treatment and hence they vary by group and period. If this is true then you better not include these controls in your regression because in that case they are bad controls. Those are variables which are outcomes of the treatment themselves, hence putting them on the right-hand side of the regression equation will lead to a bias in your estimation.
The reasoning behind difference in differences is that you have two groups that are similar before the treatment but then one group changes because of the treatment in the post-treatment period. And the treatment must be the only(!) reason that has led to this change. Otherwise your difference in differences regression cannot have a causal interpretation but this is probably something you want because that's what difference in differences is for. So either your controls are bad as described above or they also "jump" at the treatment date (or have a structural change themselves) in which case your regression cannot distinguish whether the change in the outcome of the treatment group is because of the treatment or because of one or more of your controls. For a nice discussion of the purpose of covariates in difference in differences see slides 7 and 8 in the notes by Pischke (2010).
