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I am working on a standard DID research design where all the assumptions are met - focused on a policy reform affecting a number of sectors in a country.

I would like to improve the design by exploring which sectors (if any in particular) are driving the overall effect, and possibly doing it with one regression.

Can I simply adding to my equation sector-dummies, proven parallel trend assumption holds for each sector?

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Simply adding dummies will shift your intercept, but if you believe some sectors follow a much stronger relationship than others, you can add the interaction of the sector dummies with the variable of concern. If your $i$ is at a much lower unit than sector, you could also estimate the model by sector (I am assuming there aren't a lot of sectors) and do a simple test of the equality of the coefficients from the different regressions.

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