Thread by @jmwooldridge

There are lots of DID experts on here. Dare I go there?

For the T = 2 panel data before/after, we know that linear two-way FE is the same as the DID regression

y on 1, treat, post, treat*post

If y is binary, we should not try "probit fixed effects" -- incidental parameters.

But to estimate ATT, we can also use

gen w = treat*post
probit y treat post i.w, vce(cluster id)
margins, dydx(w)

And we know it's important to use margins in this way. Do not use i.treat#.lpost and compute margins WRT treat.

The above probit is what the Chamberlain correlated RE probit delivers.

As is well known due to Athey & Imbens, the parallel trend assumption is different if we use a linear model versus probit or logit or cloglog or ....

Using several binary response functional forms in place of linear and probit is a crude way of essentially finding bounds on the ATT.

All fit the data equally well. It's saturated, and so all fitted probabilities are the same. But the estimated ATTs can be very different.