Difference-in-difference with multiple periods and one more interaction

Question

I have a question with the non traditional diff in diff with multiple periods. I currently have a balanced panel data with simple diff in diff in multiple periods treatment which could be written as:

$Y_{ist}=α+γs(\text{Treatments})+λ(\text{year dummy}_t)+δD_{st}+ϵ_{ist}$

The above equation allows me to measure the policy after treatment time ($D_{st}$) effect on $Y$.

Now, I would like to measure the effect of another variable (say $Z$) at the post-treatment period from treated group on $Y$ controlling the nature difference of the effect from $Z$ on $Y$ on other dimensions (treat and control, pre and post).

I wonder if this is a triple difference, or a diff-in-diff but with one more interaction of the variable $Z$ on everything.

Answer 1

If theory tells you treatment effect should be larger for particular subset of observations, then you can interact additional variable (in your case Z) with other variables in your regression. In that sense, it measures sensivity to treatment. But you have to use ex-ante measures because your variable (e.g. Zt-1), Z itself might be affected from your treatment. It is called triple difference, but you can still call it diff-in-diff with interaction if you will.