  In instrumental variable (IV) settings, such as in imperfect randomized
trials and observational studies with Mendelian randomization, one may
encounter a continuous exposure, the causal effect of which is not of true
interest. Instead, scientific interest may lie in a coarsened version of this
exposure. Although there is a lengthy literature on the impact of coarsening of
an exposure with several works focusing specifically on IV settings, all
methods proposed in this literature require parametric assumptions. Instead,
just as in the standard IV setting, one can consider partial identification via
bounds making no parametric assumptions. This was first pointed out in
Alexander Balke's PhD dissertation. We extend and clarify his work and derive
novel bounds in several settings, including for a three-level IV, which will
most likely be the case in Mendelian randomization. We demonstrate our findings
in two real data examples, a randomized trial for peanut allergy in infants and
a Mendelian randomization setting investigating the effect of homocysteine on
cardiovascular disease.
