Triple DiD, 3 different subgroups rather than 2

[Noemi Seng]

Dear community,

I know triple DiD, so a DiD for which the treatment group has two subgroups, e.g. female and male.

With panel data, I would estimate the following

X_it = a₀ + b₁ x Treated_it (= 1 if year is after the treatment onset and if unit belongs to treatment group) + b₂ x Treated_it x Female_i + d_i + d_t + d_it + e_it

where b₂ accounts for the heterogenous treatment effects for women vs. men and d stands for the fixed effects.

My question is now: What do I do I have 3 rather than only 2 subgroups, so e.g. if I have female, male and non-binary as different subgroups (just as an example, in my work it's about primary, secondary and tertiary economic activity)?

Would I do sth like:

X_it = a₀ + b₁ x Treated_it + b₂ x Treated_it x Female_i + b₃ x Treated_it x Male_i d_i + d_t + d_it + e_it

such that the reference group are non-binary units?

Or would I do 2 regressions of:

X_it = a₀ + b₁ x Treated_it + b₂ x Treated_it x Female_i + d_i + d_t + d_it + e_it
and
X_it = a₀ + b₁ x Treated_it + b₂ x Treated_it x Male_i + d_i + d_t + d_it + e_it?

Or is this not possible at all?

I appreciate any help.

Best
Noemi

[Andreas Backhaus]

Dear Noemi,

the Triple DiD can be understood as the difference of two DiD - maybe it would be helpful for your approach to clarify first between which groups you could estimate these two DiD?
https://academic.oup.com/ectj/article/25/3/531/6545797

[Noemi Seng]

Dear Andreas Backhaus

thank you for your answer. So the first DiD would be for men and the second one for women. The first DiD estimates the difference in the difference in the outcome after treatment relative to before treatment between treated men and untreated men. The second DiD estimates difference in the difference in the outcome after treatment relative to before treatment between treated women and untreated women. So these would be the groups between which I would estimate the two DiD. And now, I have a third group, non-binary, for which I could a third DiD: the estimate the difference in the difference in the outcome after treatment relative to before treatment between treated non-binary people and untreated non-binary people. (again, this is just an example). But it is still not clear to me how I can integrate this group into my regression equation. Would it be a quadruple DiD then?

Best
Noemi

[George Ford]

X_it = a₀ + b₁ x Treated_it + b₂ x Treated_it x Female_i + b₃ x Treated_it x Male_i d_i + d_t + d_it + e_it

Say you have 3 groups: Male, Female, Binary.

If Female, you have b1 + b2.
If Male, you have b1 + b3
If Binary (neither male or female), you have b1.

So, you've already accounted for 3 groups.