Exclusion and exogeneity are core assumptions in instrumental variable (IV) analyses, but their empirical validity is often debated. This paper develops new sensitivity analyses for these assumptions. Our results accommodate arbitrary heterogeneity in treatment effects and do not impose any monotonicity requirements on the first stage. Specifically, we derive identified sets for the marginal distributions of potential outcomes and their functionals, like average treatment effects, under a broad class of nonparametric relaxations of the exclusion and exogeneity assumptions. These identified sets are characterized as solutions to linear programs and have desirable theoretical properties. We explain how to estimate these solutions using computationally tractable methods even when the linear program is infinite-dimensional. We illustrate these methods with an empirical application to peer effects in movie viewership, using weather as a potentially imperfect instrument.

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