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Suppose that we are estimating a treatment effect using propensity score matching. We assume selection-on-observable. What is an additional assumption such that ATE (average treatment effect) equals ATE1 (average treatment effect on the treated).

I was thinking that the additional assumption would be that the propensity score is neither zero or one. Is that correct?

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  • $\begingroup$ Selection-observables is the same as the ignorability of treatment assumption $\endgroup$
    – OGC
    Dec 8, 2019 at 23:48

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See my answer here for a discussion of the differences between the ATE and ATT. The only difference between them is the population to which the effect generalizes. There are two situations in which the ATE and ATT are equal: when the distribution of covariates among the treated is the same as the distribution of covariates in the population (which implies no confounding, i.e., the result of a randomized experiment), and when there is no effect modification by the covariates that are associated with selection into treatment.

If you're using matching, you've probably ruled out a randomized experiment, so the only way an estimated ATT from matching will be unbiased for the ATE is if the effect of treatment doesn't differ across levels of the variables that cause selection into treatment. This is a strong assumption to make, so it's important to think carefully about to which population you want to generalize your treatment effect. Note that the matching of King, Stuart, Imai, and others in that camp (which only targets the ATT) is distinct from the matching of Abadie and Imbens (which can target the ATT or ATE).

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