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I am trying to estimate a non-standard DID model.

There are two time-periods, pre and post treatment. All units are treated in period 2 yet some more intensely than others. Suppose this variance in treatment intensity in the 2nd period is exogenous. Finally, the treatment dosage is not 0 in the pre-treatment in period 1.

So for instance, let's say I wanted to estimate the effect of minimum wages on employment. Suppose all counties have some minimum-wage laws in place in period 1. In period 2, all counties raise the minimum wage but some more so than others. Suppose this variance in treatment intensity in the 2nd period is plausibly exogenous. Suppose I want to test the hypothesis that employment decreased more strongly in those counties that raised the minimum wage higher (i.e. that had higher treatment intensity).

How do I estimate this model in a regression framework?

I have seen the following specification in the literature and on this board for multiple time-period DIDs but am not sure whether this also applies to my special case described above:

$Y_{i, t} = \lambda_i + post_{i,t} + treatment_{i,t} + post_{i,t} * treatment_{i,t} + \epsilon_{i,t}$

where $Y_{i, t}$ is the outcome DV, $post$ is a dummy variable for the post-treatment period, and $treatment$ is a continuous variable for the treatment intensity in unit $i$ in both period 1 and 2. $\lambda_i$ are county-fixed-effects.

PS: How would the interpretation of the model change if I do / don't estimate county-fixed-effects?

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