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How do I interpret the ATE coefficient (i.e., the post-treatment indicator interacted with the continuous variable)? Does it make sense?

Should I break it down into subgroups and just run a fixed effects model instead (interact an indicator for each subgroup with post-treatment indicator)?

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Yes, it makes sense and in this case the coefficient for the interaction of the post-treatment indicator and the treatment variable gives you the effect on the outcome that results from an increase in the treatment intensity. An example of this is the paper by Acemoglu, Autor and Lyle (2004), where they estimate the effect of World War II on female labor supply in the US. In their model

$$y_{ist} = \delta_s + \gamma d_{1950} + X'_{ist}\beta + \varphi \left(d_{1950}\cdot m_s\right) + \epsilon_{ist}$$

$y$ are weeks worked by female $i$, in state $s$, in year $t$. They have two periods, 1940 and 1950 where $d_{1950}$ is a dummy for the latter year, $X$ is a vector of individual characteristics, $\delta_s$ are state dummies, and $m_s$ is the mobilization rate in each state. Their interaction estimates whether states with higher mobilization rates during WWII saw a stronger rise in females' weeks worked from 1940 to 1950. This is given by the coefficient $\varphi$.

This is also a difference in differences (DiD) setting with variable treatment intensity since mobilization rates $m_s$ are continuous and differ across states. They get a point estimate of 11.2 for $\varphi$, i.e. a 10 percentage points increase in the mobilization rate increased female labor supply by 1.1 weeks (note that their mobilization rate is between 0 and 100). States with higher treatment intensity therefore saw a bigger increase in female labor market participation as a result of the "treatment".

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    $\begingroup$ Is it possible to use a continuous treatment in a D-in-D if we have more than two periods? In Acemoglu, Autor and Lyle (2004) they seem to only consider 1940 and 1950. $\endgroup$
    – Elias
    Commented Sep 22, 2017 at 11:34
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    $\begingroup$ @Anna yes. In this case you would just interact it with a post-treatment dummy. If Acemoglu et al. had data till 1960, this would be a dummy that turns on for 1950 and 1960. $\endgroup$
    – Andy
    Commented Sep 24, 2017 at 18:35
  • $\begingroup$ Thanks for your great answer. Does an additional term $\alpha m_s$ need to be your equation? Or is your equation correct? $$y_{ist} = \delta_s + \gamma d_{1950} + X'_{ist}\beta + \varphi \left(d_{1950}\cdot m_s\right) +\alpha m_s + \epsilon_{ist}$$ $\endgroup$
    – acubens555
    Commented Nov 24, 2019 at 4:30
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    $\begingroup$ Adding $m_s$ would be absorbed by the state dummies $\delta_s$. $\endgroup$
    – Andy
    Commented Nov 25, 2019 at 19:01
  • $\begingroup$ @Andy Is it safe to say $m_{s}$ should be included as a main effect but would ultimately be dropped by software due to the inclusion of state fixed effects? Or, is the mobilization rate first calculated by 'multiplying out' the dummy with the mobilization rate, and then including that into the model? I imagine both ways would yield the same estimate of $\varphi$. $\endgroup$
    – Thomas Bilach
    Commented Jan 24, 2020 at 12:57

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