Thread by @Jason__Cook

I just taught @CdeChaisemartin & D'Haultfoeuille (2018) Fuzzy DID and thought I would share some insights I have about the nature of the problem that Fuzzy DID introduces.

*These insights are in their paper, this is just my take on an explanation

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Consider two groups g, and time periods t.

You can write the expected value of the outcome for a group-year combo as a function of the avg. outcome in the state of the world that units are not treated + the ATT * the share that are treated.

Makes sense.

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Let's consider the standard DID estimator, we can write is as function of these four expectations. Why do we do this? Because we are trying to remove the unobserved determinants of the outcomes (in levels) that are potential sources of bias. I.e., we want to kill Y(0) term.

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Plug in the first equation into the DID we get the following. As desired, all the Y(0) terms cancel out from the Common Trends Assumption. So we can celebrate!

Or can we?

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Our efforts to remove unobserved confounders left us with an odd equation. Our DID gives us a weighted sum of the ATTs specific to each group and time period. This would be fine if we added all the ATTs together, but instead two terms are subtracted!

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As a result, even if the ATTs are positive in each period and group, we can end up with a null or even negative DID estimate! Had different ATTs been negative, it would makes sense for them to cancel out, but it is odd to cancel out based on negative weights alone.

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In sharp case, where only treated group is treated in post period. The equation simplifies and DID estimates the ATT for the treatment group in the post period alone. This is sensible. When units in other groups and periods are treated, then we have this odd weighted sum.

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One solution is to assume that ATT is homogenous for each group and period, but this is likely false in most contexts.

What to do?

They provide Stata code to allow you to estimate these weights to see how prevalent negative weighting is in your context.

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Further, they show less restrictive assumptions that let you still estimate a sensible treatment effect. Finally, they propose "Time-Corrected" estimator that relaxes required assumptions further and isn't plagued by these negative weights. (I can tweet more on this later)

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I love this paper, as well as the TWFE follow-up piece. It is accessible, extremely important (this applies to nearly every panel data design!), and has practical applications in applied work. I recommend a thorough read!

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