  This paper reexamines Abadie and Imbens (2016)'s work on propensity score
matching for average treatment effect estimation. We explore the asymptotic
behavior of these estimators when the number of nearest neighbors, $M$, grows
with the sample size. It is shown, hardly surprising but technically
nontrivial, that the modified estimators can improve upon the original
fixed-$M$ estimators in terms of efficiency. Additionally, we demonstrate the
potential to attain the semiparametric efficiency lower bound when the
propensity score achieves "sufficient" dimension reduction, echoing Hahn
(1998)'s insight about the role of dimension reduction in propensity
score-based causal inference.
