  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 outline. 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, however, that
this result is driven by the presence of negative weights. Corrected estimates
indicate that attending college causes earnings to be roughly 20% higher.
