  In a nonparametric instrumental regression model, we strengthen the
conventional moment independence assumption towards full statistical
independence between instrument and error term. This allows us to prove
identification results and develop estimators for a structural function of
interest when the instrument is discrete, and in particular binary. When the
regressor of interest is also discrete with more mass points than the
instrument, we state straightforward conditions under which the structural
function is partially identified, and give modified assumptions which imply
point identification. These stronger assumptions are shown to hold outside of a
small set of conditional moments of the error term. Estimators for the
identified set are given when the structural function is either partially or
point identified. When the regressor is continuously distributed, we prove that
if the instrument induces a sufficiently rich variation in the joint
distribution of the regressor and error term then point identification of the
structural function is still possible. This approach is relatively tractable,
and under some standard conditions we demonstrate that our point identifying
assumption holds on a topologically generic set of density functions for the
joint distribution of regressor, error, and instrument. Our method also applies
to a well-known nonparametric quantile regression framework, and we are able to
state analogous point identification results in that context.
