  We consider the setting in which a strong binary instrument is available for
a binary treatment. The traditional LATE approach assumes the monotonicity
condition stating that there are no defiers (or compliers). Since this
condition is not always obvious, we investigate the sensitivity and testability
of this condition. In particular, we focus on the question: does a slight
violation of monotonicity lead to a small problem or a big problem? We find a
phase transition for the monotonicity condition. On one of the boundary of the
phase transition, it is easy to learn the sign of LATE and on the other side of
the boundary, it is impossible to learn the sign of LATE. Unfortunately, the
impossible side of the phase transition includes data-generating processes
under which the proportion of defiers tends to zero. This boundary of phase
transition is explicitly characterized in the case of binary outcomes. Outside
a special case, it is impossible to test whether the data-generating process is
on the nice side of the boundary. However, in the special case that the
non-compliance is almost one-sided, such a test is possible. We also provide
simple alternatives to monotonicity.
