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My question relates to the following post:

How do I interpret a "difference-in-differences" model with continuous treatment?

I am reproducing an equation from Acemoglu, Autor and Lyle (2004), where they estimate the effect of World War II on female labor supply in the US. Their model takes the following form:

$$ y_{ist} = \delta_{s} + \gamma d_{1950} + X'_{ist}\beta_{t} + \varphi(d_{1950} \cdot m_{s}) + \epsilon_{ist}, $$

where $y$ 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, $X'_{ist}$ 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 𝜑.

I have seen this referenced in other posts as a difference-in-differences (DD) design. In my estimation, it is similar in form to the generalized DD approach, where you regress some outcome on a treatment indicator (or a continuous measure) and a set of unit (i.e., $\delta_s$) and time (i.e., $\gamma_{t}$) fixed effects.

Based upon my reading of the paper, the treatment in this case is the mobilization rate. Each state had a mobilization rate, which was expressed as the percentage of eligible males ages 18-44 mobilized. In my understanding, the DD approach requires some "group" of states (i.e., control group) in 1950 where the percentage of eligible males mobilized was zero (i.e., zero intensity). Without this condition (i.e., states with an absence of intensity), I believe we cannot adequately refer to this as a DD design. Yet, it is frequently referred to as such.

I would assume that low-mobilization states would still be considered "treated" in theory. I always assumed that under a DD approach, zero intensity is a necessary condition to disambiguate between treated and untreated units.

Questions:

(1) If this is, in fact, a DD design, what is the counterfactual? The interaction is representative of the wage differential for female workers due to an increase in the percentage of males mobilized. We only observe wage differences between low- and high-mobilization states.

(2) Since $m_{s}$ is time-invariant, it would be absorbed by the fixed effects if also included as a main effect. If this is so, aren't the authors just examining cross-state patterns in mobilization rates (see pp. 519-522)?

I think there are important distinctions to be made with respect to how we code continuous treatments in a DD design.

If there is anyone with working knowledge of how to do this using the classical or generalized DD approach, it would be great to see an example.

Update:

Lecture notes produced by Pischke 2005 address continuous treatments (see pp. 5-7). He outlines a study by Card (1992) examining the effect of the federal increase in minimum wage in 1990 using all the US states. His measure is the fraction of workers paid less than $3.80, something he calls the "fraction of affected workers." I am not sure but I assume the fraction of affected workers in high minimum wage states would be zero. This goes back to my original concern about how a treatment measure (continuous) is coded. See equation (2) on page 6:

$$ y_{st} = \gamma_{s} + \lambda_{t} + \beta M_{st} + \epsilon_{st}, $$

where $M_{st}$ is a measure of the "bite" of the minimum wage in state $s$ at time $t$. Pischke notes that this formulation retains the features of the DD model. I am most concerned with how the $M_{st}$ variable looks over the full panel series. This would help me, and maybe others, obtain a more intuitive understanding of how continuous treatments work in these settings.

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