  Caliper matching is used to estimate causal effects of a binary treatment
from observational data by comparing matched treated and control units. Units
are matched when their propensity scores, the conditional probability of
receiving treatment given pretreatment covariates, are within a certain
distance called caliper. So far, theoretical results on caliper matching are
lacking, leaving practitioners with ad-hoc caliper choices and inference
procedures. We bridge this gap by proposing a caliper that balances the quality
and the number of matches. We prove that the resulting estimator of the average
treatment effect, and average treatment effect on the treated, is
asymptotically unbiased and normal at parametric rate. We describe the
conditions under which semiparametric efficiency is obtainable, and show that
when the parametric propensity score is estimated, the variance is increased
for both estimands. Finally, we construct asymptotic confidence intervals for
the two estimands.
