What is the positivity assumption required for matching and ATT estimand? Does ATT estimand require a less stringent positivity assumption in matching?
For example, if a small treated group is matched to a large control group, most of the control subjects will be discarded after matching. Do we only require positivity assumption to hold in the remaining sample?
 A: The positivty assumption required for identifying the ATT is indeed less strict than the positivity assumption required for the ATE. Stated in "overlap" terms, positivity for the ATT requires that the distribution of covariates for the treated group is contained within the distribution of covariates for the control group. That is, there is no place in the covariate distribution where there are only treated units but no control units. However, there can be areas of the covariate distribution in the controls that are not occupied by the treated. This is not true of the positivity requirement for the ATE. Intuitively, you can see that such control units would simply be dropped in a matching analysis or down-weighted in a weighting analysis; such units would have a propensity score of zero.
More formally, the positivity requirement for the ATT stated in the usual "probability of treatment" terms is
$$
P(A=1|X=x) < 1
$$
for all $x$. That is, there are no areas of the covariate distribution where only the treatment is possible; in all areas of the covariate distribution, it is possible to receive control. This is general a much easier criterion to satisfy than the positivity requirement for the ATE.
