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In most propensity score literature on observational studies, the ATT is usually reported and sought after. However, the ATE is usually not reported and in some cases I've read papers that claim you cannot obtain it. What is the succinct reason for the fact the ATT is reported? Is the ATE not theoretically possible?

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In many policy contexts, it is the ATT that is of interest. In deciding whether a policy is beneficial, it does not matter whether on average the program is beneficial for a randomly plucked person from the population (ATE), but whether it is beneficial for those who are assigned or typically assign themselves to the treatment (ATT).

For example, consider a policy maker who is interested in the effect of Catholic schooling. Would that person care about the effect averaged over all the children who could potentially attend a Catholic school or in the effect for the children that typically would go to such schools? Most likely, it will be the latter. The ATE would not be the most relevant policy parameter. To generalize almost to the point of uselessness, science might require the ATE, while policy needs the ATT. However, there are many exceptions to this rule.

PSM can give you ATE, ATT, and ATU (average treatment effect on the untreated).

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In addition to the reasons why ATT is sometimes (oftentimes) more interesting than ATE, that Dimitriy V. Masterov pointed out, it is also the case that estimation of ATE through matching would require that we not only match treated to untreated, but also the untreated to the treated. If we have a vast number of untreated individuals but a relatively speaking small number of treated, then finding good matches for all untreated individuals might prove impossible. Estimation of ATE would then prove very difficult through matching, whereas the ATT is much easier to estimate.

Consider this small and fictional example. You have one variable you match on (e.g. the estimated propensity score), and the following observations for the treated:

$$(0.6, 0.7, 0.8),$$ while you have untreated with observed values of the variable as follows: $$(0.1, 0.1, 0.15, 0.2, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8).$$

Matching all the treated to an untreated individual is straight-forward. For each of your treated you can find a perfect match in this example. However, what will you do with the remaining untreated indivuals? Those with values in the range of $0.1$ through $0.5$? Sure, you could match them all to the closest observation in the treated group (i.e. $0.6$), but that's likely a rather bad match for an individual with a propensity score of $0.1$, which would introduce bias in your estimates.

In such a scenario, the ATT is easy to estimate, while the ATE is not.

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