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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 $treat_i$.

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)?

Ruben
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  • So all firms (big/small) receive the intervention at precisely the same time? Or, do you have early/adopter firms? – Thomas Bilach Sep 11 '20 at 01:42
  • @ThomasBilach: Small and big firms receive the intervention at the same time. Treatment is only determined by the `treat' dummy, and firm size is simply a characteristic of the firms. – Ruben Sep 11 '20 at 06:40
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    You could estimate this with one big equation by including separate treatment indicators for the two groups: one dummy for the group of large firms and one dummy for the group of small firms. Interacting each of those with $time_t$ is similar to working on subsets of your data. However, what you seek is something analogous to a triple difference (i.e., heterogeneous treatment effects for different subgroups). I think the answers [here](https://stats.stackexchange.com/questions/183302/difference-in-difference-with-interaction) should help. Follow-up if anything is still unclear! – Thomas Bilach Sep 11 '20 at 23:03

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