We consider estimating a local average treatment effect given an endogenous binary treatment and two or more valid binary instruments. We propose a novel limited monotonicity assumption that is generally weaker than alternative monotonicity assumptions considered in the literature, and allows for a great deal of choice heterogeneity. Using this limited monotonicity, we define and identify the Combined Complier Local Average Treatment Effect (CC-LATE), which is arguably a more policy relevant parameter than the weighted average of LATEs identified by Two Stage Least Squares. We apply our results to estimate the effect of learning one’s HIV status on protective behaviors.
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