I have a dataset in the following form:
<id> <city> <treated> <time> <after>
where id identifies the individuals in my panel, city is the location where the individual live (non-time varying), treated is a dummy indicating those individual that are eventually treated (0: non-treated, 1: treated), time is a year-month variable, and after is a dummy (0: before, 1: after) indicating the period in which the treated unit are under treatment.
Withi this data I am running a DD specification:
y = i.treated##i.after + i.time
Now, as is common practice I want to add treatment specific time trends (this should relax the parallel trend assumption), so I run:
y = i.treated##i.after + i.time + c.treated#c.time
To make the model even more flexible I want to introduce city-treatment specific linear trends:
y = i.treated##i.after + i.time + i.city#c.treated#c.time
My question is, does it make sense to include city-treatment trends? Results are quite different using treatment specific or city-treatment specific trends.
Moreover, can someone explain to me what's the statistical difference between adding trends using:
My understanding is that the second approach creates two trends for every city , one for treated and one for untreated units (which I think it is what I want), while the first approach creates a trend for every city-treated group.
If someone can help me understand the statistical difference between the two approaches, or suggest some papers to read, it would be really useful.
Thanks a lot!
EDIT: sorry but I was wrong, I didn't figure out the statistical difference between adding i.city#c.treated#c.time or i.city#i.treated#c.time, so if anyone knows what is the correct way to add city-treatment specific linear trends, this would be very helpful -- thanks