Instrument Validity for Heterogeneous Causal Effects

This paper provides a general framework for testing instrument validity in heterogeneous causal effect models. The generalization includes the cases where the treatment can be multivalued (and ordered) or unordered. Based on a series of testable implications, we propose a nonparametric test which is proved to be asymptotically size controlled and consistent. Because of the nonstandard nature of the problem in question, the test statistic is constructed based on a nonsmooth map, which causes technical complications. We provide an extended continuous mapping theorem and an extended delta method, which may be of independent interest, to establish the asymptotic distribution of the test statistic under null. We then extend the bootstrap method proposed by Fang and Santos (2018) to approximate this asymptotic distribution and construct a critical value for the test. Compared to the tests in the literature, our test can be applied in more general settings and may achieve power improvement. Evidence that the test performs well on finite samples is provided via simulations. We revisit the empirical study of Card (1993) and use their data to demonstrate application of the proposed test in practice. We show that a valid instrument for a multivalued treatment may not remain valid if the treatment is coarsened.