  We study semiparametric efficiency bounds and efficient estimation of
parameters defined through general moment restrictions with missing data.
Identification relies on auxiliary data containing information about the
distribution of the missing variables conditional on proxy variables that are
observed in both the primary and the auxiliary database, when such distribution
is common to the two data sets. The auxiliary sample can be independent of the
primary sample, or can be a subset of it. For both cases, we derive bounds when
the probability of missing data given the proxy variables is unknown, or known,
or belongs to a correctly specified parametric family. We find that the
conditional probability is not ancillary when the two samples are independent.
For all cases, we discuss efficient semiparametric estimators. An estimator
based on a conditional expectation projection is shown to require milder
regularity conditions than one based on inverse probability weighting.
