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.

  • $\begingroup$ Be careful when using the propensity score for common support; the reason you might see an apparent lack of common support in the propensity score could be that you estimated the propensity score poorly, and there is no way to verify if you estimated it correctly. Several forms of weighting do not even estimate propensity scores (e.g., entropy balancing). See Cole & Hernán (2008) for a discussion about common support (in the context of positivity). $\endgroup$ – Noah Jan 26 '18 at 3:43
  • $\begingroup$ Yes, i know that this may be a problem. However, I haven´t found many alternative approaches. King & Zeng (2005) use the convex hull of the data to determine common support, but this seems to be a bit conservative as it possibly disregards observations which are very similar. Porro & Iacus (2009) introduced a method using hyper-rectangles for which they claim it is less conservative. $\endgroup$ – Michael79 Jan 31 '18 at 9:10

The basic intuition of the support problem is that if you are going to estimate the counterfactual for a given person by someone matched to that person, then you need to observe someone similar to the person in the counterfactual state.

For the ATT parameter, you only need to find people that are similar to the treated, so you only discard the treated observations that don't have untreated counterparts with a similar propensity score. There's "free disposal" for untreated observations with low probability of treatment.

  • $\begingroup$ Thank you for your answer! But those controls with low probability of treatment, which also are dissimilar to the treated if there are no treated with those low values of the propensity score, are used in the calculation of the weighting estimator. Although they will receive low weights, I can´t see why they are relevant to the estimation of the counterfactuals, since they are dissimilar. $\endgroup$ – Michael79 Jan 24 '18 at 9:26
  • $\begingroup$ That's where the "free disposal" part comes in. If you have some untreated observations with low PSs, then the weighting will essentially downplay the or drop them altogether (depending on how you do the matching), unless there are also some treated observations with low PSs. They don't need to be used unless they are truly necessary to form the comparison group for the treated units. But you don't need to exclude them from the potential donor pool. $\endgroup$ – Dimitriy V. Masterov Jan 24 '18 at 17:14
  • $\begingroup$ In the context of weighting, control units with low probabilities of receiving treatment will be downweighted anyway. $\endgroup$ – Noah Jan 26 '18 at 3:41
  • $\begingroup$ @Michael79 Did this clarify things? $\endgroup$ – Dimitriy V. Masterov Jan 29 '18 at 15:53
  • $\begingroup$ Yes, this helps a lot! $\endgroup$ – Michael79 Jan 31 '18 at 8:59

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