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I have panel data for a number of counties for two years only (t = 1,2). The treatment, let's assume it's a policy implementation programme, happens in between year 1 and 2 and it is a continuous treatment: certain counties receive it already in year 1, while other countries receive it in year 2, and with different intensities. (this means that there is no specific control group).

My question is:

Should I use a cut-off point (above and below the median) instead of a continuous treatment?

Thanks!

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The following paper shoud be relevant:

Abstract.Differences-in-differences (DID) is a method to evaluate the effect ofa treatment. In its basic version, a “control group” is untreated at two dates,whereas a “treatment group” becomes fully treated at the second date. However,in many applications of this method, the treatment rate only increases more inthe treatment group. In such fuzzy designs, de Chaisemartin and D’Haultfœuille(2018b) propose various estimands that identify local average and quantile treat-ment effects under different assumptions. They also propose estimands that canbe used in applications with a non-binary treatment, multiple periods and groupsand covariates. This paper presents the Stata commandfuzzydid, which com-putes the various corresponding estimators. We illustrate the use of the commandby revisiting Gentzkow et al. (2011).

de Chaisemartin, C., D’Haultfœuille, X., & Guyonvarch, Y. (2019). Fuzzy differences-in-differences with Stata. The Stata Journal, 19(2), 435-458.

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