  The hypothesis of homogeneous treatment effects is central to the
instrumental variables literature. This assumption signifies that treatment
effects are constant across all subjects. It allows to interpret instrumental
variable estimates as average treatment effects over the whole population of
the study. When this assumption does not hold, the bias of instrumental
variable estimators can be larger than that of naive estimators ignoring
endogeneity. This paper develops two tests for the assumption of homogeneous
treatment effects when the treatment is endogenous and an instrumental variable
is available. The tests leverage a covariable that is (jointly with the error
terms) independent of a coordinate of the instrument. This covariate does not
need to be exogenous. The first test assumes that the potential outcomes are
linear in the regressors and is computationally simple. The second test is
nonparametric and relies on Tikhonov regularization. The treatment can be
either discrete or continuous. We show that the tests have asymptotically
correct level and asymptotic power equal to one against a range of
alternatives. Simulations demonstrate that the proposed tests attain excellent
finite sample performances. The methodology is also applied to the evaluation
of returns to schooling and the effect of price on demand in a fish market.
