  It is common in instrumental variable studies for instrument values to be
missing, for example when the instrument is a genetic test in Mendelian
randomization studies. In this paper we discuss two apparent paradoxes that
arise in so-called single consent designs where there is one-sided
noncompliance, i.e., where unencouraged units cannot access treatment. The
first paradox is that, even under a missing completely at random assumption, a
complete-case analysis is biased when knowledge of one-sided noncompliance is
taken into account; this is not the case when such information is disregarded.
This occurs because incorporating information about one-sided noncompliance
induces a dependence between the missingness and treatment. The second paradox
is that, although incorporating such information does not lead to efficiency
gains without missing data, the story is different when instrument values are
missing: there, incorporating such information changes the efficiency bound,
allowing possible efficiency gains. This is because some of the missing values
can be filled in, based on the fact that anyone who received treatment must
have been encouraged by the instrument (since the unencouraged cannot access
treatment).
