Staggered two way fixed effects with time heterogenic effects and binary outcome variable TL;DR: Does Callaway & Sant'Anna (2021) work for binary outcomes? If not, what should I do to estimate a time heterogenic policy effect on a binary variable?
Full question: I am trying to estimate the effect of a policy on a binary outcome. Given my data, the effect is likely to be heterogenous across time, i.e., the effect will increase over time, so the effect size is larger for units that have been treated for 3 periods than for units that have been treated for 1 period. The effect is not likely to be heterogenous across groups (that get treated at different periods), i.e., it is length of exposure that matters, and not the time period of the treatment. For example, looking at a group in period 4, when the group was treated in period 3, the effect size should be the same as when looking at another group in periode 7 that was treated in period 6.
Chaisemartin & D'Haultfæuille (2022) https://www.nber.org/papers/w29691 reviews a litterature that documents the problems of using standard TWFE estimators in staggered applications when treatment effects are heterogenous over time. Callaway & Sant'Anna (2021) 10.1016/j.jeconom.2020.12.001 provides a solution to this. My question is if the estimator they provide can also be used in applications where the outcome is binary? If not, what can I do to estimate the effect of a binary policy on a binary outcome in an application with 1) panel data 2) staggered policy adoption 3) time heterogenous effects 4) a justfiable conditional parallel trends assumption.

I'm only familiar with the general outline of Callaway and Sant'Anna's approach so I can't say anything definite about this. However, the other main approach that's arisen in response to this problem is to simply run two regressions, the first utilizing untreated observations to estimate (impute) group and time fixed effects, and the second to estimate the average treatment effect after the group and time effects previously estimated are subtracted from observed outcomes.
This approach is briefly discussed in the paper you reference. It's been independently proposed in a few different papers (all referenced by Chaisemartin and D'Haultfæuille). Gardner (2022) offers a very intuitive outline of the idea. I'd recommend you start there.
