  Propensity score matching (PSM) and augmented inverse propensity weighting
(AIPW) are widely used in observational studies to estimate causal effects. The
two approaches present complementary features. The AIPW estimator is doubly
robust and locally efficient but can be unstable when the propensity scores are
close to zero or one due to weighting by the inverse of the propensity score.
On the other hand, PSM circumvents the instability of propensity score
weighting but it hinges on the correctness of the propensity score model and
cannot attain the semiparametric efficiency bound. Besides, the fixed number of
matches, K, renders PSM nonsmooth and thus invalidates standard nonparametric
bootstrap inference.
  This article presents novel augmented match weighted (AMW) estimators that
combine the advantages of matching and weighting estimators. AMW adheres to the
form of AIPW for its double robustness and local efficiency but it mitigates
the instability due to weighting. We replace inverse propensity weights with
matching weights resulting from PSM with unfixed K. Meanwhile, we propose a new
cross-validation procedure to select K that minimizes the mean squared error
anchored around an unbiased estimator of the causal estimand. Besides, we
derive the limiting distribution for the AMW estimators showing that they enjoy
the double robustness property and can achieve the semiparametric efficiency
bound if both nuisance models are correct. As a byproduct of unfixed K which
smooths the AMW estimators, nonparametric bootstrap can be adopted for variance
estimation and inference. Furthermore, simulation studies and real data
applications support that the AMW estimators are stable with extreme propensity
scores and their variances can be obtained by naive bootstrap.
