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I have a question regarding some potentially existing empirical tests in the difference-in-difference context.

Assume that I have the usual setting of observing some hypothetical outcome for a treatment and a non-treatment group both before and after the treatment. As commonly known, the difference between:

$Outcome_{treated, after} - Outcome_{treated, before}$

and

$Outcome_{non-treated,after} - Outcome_{non-treated, before}$

is known as the difference-in-difference estimator, which I call $\delta_1$.

Now assume that for some reasons, I can consider a part of the non-treated group as also being treated. Hence I simply assign some of the non-treated entities the treatment status. All entities that already had a treatment status remain in the treatment group. Compared to the previous setting, the treatment group hence increases and the non-treatment group decreases. We can moreover consider the 'old' treatment group as a subset of the 'new' treatment group. I estimate the model again and obtain a new difference-in-difference estimator $\delta_2$.

Now my question: I'm very much interested in knowing whether the treatment effect is now significantly stronger, i.e. whether $\delta_1$ - $\delta_2 \neq 0$ at some common significance level.

Does anyone of you knows how I could test for this? I know that I can put everything in a regression setting with two binary variables (for the treatment vs. non treatment and the after vs. before) plus an interaction term. The regression coefficient of this term is then $\delta$. But how can I compare these two coefficients in the two models with each other?

I'd appreciate any feedback. Many thanks in advance

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