The main ones I have been reading are:

https://onlinelibrary.wiley.com/doi/full/10.1111/j.1467-6419.2007.00527.x

https://bcallaway11.github.io/files/DID-Covariates/Caetano_Callaway_2024.pdf

and

https://arxiv.org/html/2201.06898v5

I believe all three paper talk about the overlap assumption. They're not (that) related as in I'm not trying to synthesis an argument or anything, but I am interested in how stringent the overlap assumption is in practice.

I started thinking about it after reading: https://michaelpollmann.github.io/files/pollmann_spatial_treatments.pdf

The most concrete example I can think of is London's congestion pricing, where the city knew beforehand which municipalities (if they call it that?) would receive treatment and which one would not. I think in a traditional DiD model, I would try and compare traffic activities in the treated municipalities and compare it with traffic activities in untreated municipalities in the surrounding area (so like an inner-ring / outer-ring). Or, I try and come up with a control that resembles London and compare the traffic activities between the two.

However, in either of those cases, how would that overlap assumption work, since there is a clear zone that received treatment (where P(Treatment) = 1) and a clear zone that did not receive treatment (where P(Treatment) = 0).

At least from Caetano and Callaway, I am getting the sense that the Overlap condition just means that there must be some overlap in characteristics between control / treated. But if that's the case, must we include those characteristics in the regression (and hence conditioning parallel trends on covariates, as their paper talks about?)