Difference-in-differences estimator for time-invariant subgroups

[Ruben Da Costa]

I have a conceptual question. I have a standard difference in differences (DiD) model

 Y_{i,t} = \beta_0 + \beta_1 treat_i + \beta_2 time_t + \beta_3 (treat_i \cdot time_t) + \varepsilon_{i,t}

where $treat_i$ is a dummy variable for the group membership (treatment vs. control group), and $time_t$ is a dummy variable for the period (before vs. after). Individuals (in my case: firms) and time are indexed by $i$ and $t$, respectively.

The parameter of interest is the DiD estimator $\beta_3$.

How do I estimate $\beta_3$ for subgroups of firms determined by time-invariant characteristics? Example: I want to determine $\beta_3$ for both small and large firms separately, and see if (and by how much) $\beta_3$ differs across those subgroups. Notably, both small and large firms are affected by the treatment $treat_i$, which is why a triple DDD seems not suitable.

How can I do this without subsetting the data and estimating the DiD model for small and large firms, separately?

How can I extend this to subgroup characteristics that are not binary (small vs. large firms), but multinomial (e.g. regions of firms)?

[Clyde Schechter]

Create a dichotomous 0/1 variable for firm size and then add the three way interaction of that variable with treat and time to the model. If you use factor variable notation, you can do the same thing with multinomial category variables using the same code.

regression_command outcome_variable i.categorical_variable##i.treat##i.time ...

The coefficient of the three-way interaction term(s) estimates the difference in the treatment effect across categories of the categorical variable. Note that if you are using a fixed-effects regression, the terms for the categorical variable itself will be omitted due to colinearity with the fixed effects, but this is not a problem. Don't worry about that.

[Ruben Da Costa]

Thank you for your reply, Clyde.

The three-way interaction produces a triple DDD, right? I thought this was not suitable as it estimates the effect of the treatment on - say - large firms, assuming that large firms where treated and small firms were not. Yet, there are small and large firms in both, the treatment and the control groups.

[Clyde Schechter]

I thought this was not suitable as it estimates the effect of the treatment on - say - large firms, assuming that large firms where treated and small firms were not.

No, that's not true. The triple interaction specifically distinguishes the intervention effect among the large and the intervention effect among the small, but it does not assume that one group is treated and the other is not.

(I don't use the terminology "triple DDD" because different people use it to mean different things, and that can cause confusion.)

[Ruben Da Costa]

Thank you so much, Clyde. I sat down and did the math, and see it now.
Going to leave this here, maybe it'll help someone else, too.