This paper develops a specification test for the instrument validity conditions in the heterogeneous treatment effect model with a binary treatment and a discrete instrument. A necessary testable implication for the joint restriction of instrument exogeneity and instrument monotonicity is given by nonnegativity of point-identifiable complier's outcome densities. Our specification test infers this testable implication using a Kolmogorov-Smirnov type test statistic. We provide a bootstrap algorithm to implement the proposed test and show its asymptotic validity. The proposed test procedure can apply to both discrete and continuous outcome cases.
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