Two-way fixed effects with two treatments and interaction between both treatments

I am interested in estimating the effect of two different treatments (T1 and T2) and the interaction of the two (T1 x T2) in a two-way fixed effects event-study design. Both T1 and T2 have a "scattered rollout," in the sense that different units become treated at different times. I have been unable to find any papers discussing this sort of set up, where one is interested in two treatments and their interaction. Hence the following question:

Does the following estimating equation correctly identify the effect of T1, T2 and the interaction between T1 and T2?

$$y_{it} = \alpha_0 + \alpha_1T1_{it} + \alpha_1T2_{it} + \alpha_3(T1_{it} \times T2_{it}) + \alpha_4 timeFE + \alpha_5 unitFE + \alpha_6 (T1_{it} \times unitFE) + \alpha_7 (T2_{it} \times unitFE) + \alpha_8 (T1_{it} \times timeFE) + \alpha_9 (T2_{it} \times timeFE) + \epsilon_{it}$$

Where $$i$$ is the unit of analysis and the level at which the treatments vary, and $$t$$ denotes time. My understanding is that we must include interactions of each treatment with the time and unit FEs so that the $$\alpha_1, \alpha_2, \alpha_3$$ identify the treatment effect of each treatment and the interaction effect, respectively.

• nber.org/papers/w30108 Commented Jan 16, 2023 at 14:36
• Thank you, this is a helpful and relevant paper. However, while it sheds light on estimating the separate treatment effects (T1 and T2) , it does not say much (if anything) about the interaction of multiple treatments (unless I misunderstood something). Any thoughts on this would be much appreciated. Thank you.
– aeiz
Commented Jan 16, 2023 at 16:58
• I don't know, but I admire the question ;) Commented Jan 16, 2023 at 19:18