Objects in propensity-matched pairs are farther than they appear

To match closely on a propensity, prognostic or other index score is
harder than it would appear. Nearest-neighbor matching on an estimate
of the score may pair subjects quite closely, particularly if the
procedure imposes a radius or caliper restriction, such that subjects
with scores falling outside the region of overlap are left out.  Even
after matching within the strictest of such calipers, however, paired
differences on the actual score can be surprisingly large.  A first
contribution of this research is to marshall evidence that univariate
propensity score matching typically does leave large discrepancies on
the underlying true score. Even such luxuries as covariates that are
(a) moderate in number (d=o(√n)) and (b) outlier-free (sub-gaussian),
together with (c) strict overlap between the groups (all propensities
within [a,b] ⊆ (0,1)), do not in themselves address the problem.

There's good news as well.  When (a) and (b) are satisfied, estimation
errors of the scores tend uniformly to zero, having sizes that are
themselves estimable and can be summarized with a single number. That
number in turn suggests a new caliper criterion with the attractions
of being likely to reject few or no pairings that are close on the
underlying score, while excluding sufficiently many matches of
genuinely poor quality that among the remaining eligible pairs, true
differences on the underlying score tend uniformly to zero.  None of
(a)-(c) is required for this, although all will be true of the matched
sample if the matching procedure has been governed by a caliper of the
proposed type.

This caliper may be wider or narrower than others in broad use, but often it's wider, that is more permissive, consistent with matched counterparts often being farther on the true score than they would appear.  Nonetheless, it suffices to confer favorable large-sample properties on matched estimators. The argument for these conclusions combines theory and a case study.  The presentation focuses on propensity scores, but similar considerations apply to prognostic scores, principal scores and predictive mean matching.

https://lsa.umich.edu/stats/people/faculty/bbh.html