Triple Difference vs Difference in Differences?

[Yevgeniy Feyman]

Hi,

I'm trying to figure out the right way to implement an estimator.

Some notes:

I'm using an individual-month level dataset, my treatment occurs at the state level, at different times for different states, and turns off at some later point. I'm using state and time fixed effects in my model as well. I hypothesize that my treatment effect will vary by gender.

"treat" is an indicator that in the current month-state, the treatment is either on (1) or off (0). I don't include a control for the control vs treated group because that appears to be subsumed by state FEs.

My base specification is as follows:

 reghdfe emp i.treat $controls [pw=panlwt] if child==1 & age>=18 & age<65, absorb(month state) vce(cluster state)

I have reason to think that the effect I expect to see will differ by gender. So I modify my specification as follows:

 reghdfe emp i.treat##i.sex $controls [pw=panlwt] if child==1 & age>=18 & age<65, absorb(month state) vce(cluster state)

My questions are as follows:

1) Would I just describe the second equation as a difference-in-differences with treatment effect heterogeneity?
2) As I understand, a triple difference estimator isn't intended for treatment effect heterogeneity, but rather to weaken the parallel trends assumption. Is that right?
3) If (2) is true, then a triple differences specification (where sex is interacted with everything) would be incorrect as a way of identifying treatment effect heterogeneity?

Thanks in advance.

[Clyde Schechter]

1) Yes.
2) Yes.
3) It depends, but it has nothing to do with 2). If you believe that the associations of the covariates ("control variables") with the emp outcome differ by sex, then you should interact them with sex as well. If you believe sex does not modify the those relationships, then sex need only be interacted with treat.

[Yevgeniy Feyman]

Thanks Clyde. Very helpful.