When should we (not) interpret linear IV estimands as LATE?

In this paper I revisit the interpretation of the linear instrumental variables (IV) estimand as a weighted average of conditional local average treatment effects (LATEs). I focus on a practically relevant situation in which additional covariates are required for identification while the reduced-form and first-stage regressions implicitly restrict the effects of the instrument to be homogeneous, and are thus possibly misspecified. I show that the weights on some conditional LATEs are negative and the IV estimand is no longer interpretable as a causal effect under a weaker version of monotonicity, i.e. when there are compliers but no defiers at some covariate values and defiers but no compliers elsewhere. The problem of negative weights disappears in the overidentified specification of Angrist and Imbens (1995) and in an alternative method, termed "reordered IV," that I also develop. Even if all weights are positive, the IV estimand in the just identified specification is not interpretable as the unconditional LATE parameter unless the groups with different values of the instrument are roughly equal sized. I illustrate my findings in an application to causal effects of college education using the college proximity instrument. The benchmark estimates suggest that college attendance yields earnings gains of about 60 log points, which is well outside the range of estimates in the recent literature. I demonstrate that this result is driven by the existence of defiers and the presence of negative weights. Corrected estimates indicate that attending college causes earnings to be roughly 20% higher.