<p>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 &quot;scattered rollout,&quot; 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:</p>
<p><strong>Does the following estimating equation correctly identify the effect of T1, T2 and the interaction between T1 and T2?</strong></p>
<p><span class="math-container">$$
y_{it} = \alpha_0 + \alpha_1T1_{it} + \alpha_1T2_{it} + \alpha_3(T1_{it} \times T2_{it}) + \alpha_4 timeFE + \alpha_5 unitFE + \alpha_6 (T1_{it} \times unitFE) + \alpha_7 (T2_{it} \times unitFE)  + \alpha_8 (T1_{it} \times timeFE) + \alpha_9 (T2_{it} \times timeFE) + \epsilon_{it}
$$</span></p>
<p>Where <span class="math-container">$i$</span> is the unit of analysis and the level at which the treatments vary, and <span class="math-container">$t$</span> denotes time. My understanding is that we must include interactions of each treatment with the time and unit FEs so that the <span class="math-container">$\alpha_1, \alpha_2, \alpha_3$</span> identify the treatment effect of each treatment and the interaction effect, respectively.</p>

CC BY-SA 4.0
2023-01-16T14:36:36.300Z
https://www.nber.org/papers/w30108
Papayapap
CC BY-SA 4.0
2023-01-16T16:58:21.260Z
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
CC BY-SA 4.0
2023-01-16T19:18:00.687Z
I don't know, but I admire the question ;)
Papayapap
<applied-econometrics><difference-in-difference><treatment-effect>
Two-way fixed effects with two treatments and interaction between both treatments
aeiz
# Two-way fixed effects with two treatments and interaction between both treatments

(Asked by: username_0 on 2023-01-16)

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.

**Tags:** applied-econometrics, difference-in-difference, treatment-effect
> **Comments:**
> **username_1**: https://www.nber.org/papers/w30108  
> **username_0**: 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.  
> **username_1**: I don't know, but I admire the question ;)  

---

## 0 answers

