This question concerns the overlap / common support issue for estimating the Average Treatment Effect on the Treated (ATT) in a two group setting via Propensity Score (PS) Weighting.
One possibility to check for overlap in the covariate distributions is to plot the density of the estimated Propensity Score seperatly in the treatment and in the control group. Two situations are of interest:
1) There are PS values in the treatment group which are not present in the control group, and
2) There are PS values in the control group which are not present in the treatment group.
The goal in estimating the ATT is to weight the distribution of the covariates of the control group so that it becomes equal to that of the treated group.
So in case 1), this is clearly not possible because among the controls, there exist no observations in certain covariate regions which are occupied in treatment group.
In case 2), if the controls not on the common support receive positive weights, this will also prevent an equalization of the distributions.
I hope thus far, my understanding is correct.
However, among the trimming rules presented on page 28 and 29 here, all discard only treated observations, no controls not on the common support.
Why is this the case? Is it really not necessary to drop control not on the common support for estimating the ATT? Remember that I´m not asking about matching, where dissimilar controls unit are dropped by the matching algorithm, but about weighting.