  When a researcher combines multiple instrumental variables for a single
binary treatment, the monotonicity assumption of the local average treatment
effects (LATE) framework can become restrictive: it requires that all units
share a common direction of response even when separate instruments are shifted
in opposing directions. What I call vector monotonicity, by contrast, simply
assumes treatment uptake to be monotonic in all instruments. I characterize the
class of causal parameters that are point identified under vector monotonicity,
when the instruments are binary. This class includes, for example, the average
treatment effect among units that are in any way responsive to the collection
of instruments, or those that are responsive to a given subset of them. The
identification results are constructive and yield a simple estimator for the
identified treatment effect parameters. An empirical application revisits the
labor market returns to college.
