## IV with Heterogeneous Potential Outcomes

The discussion of IV up to this point postulates a constant causal effect. In the case of a dummy variable like veteran status, this means Y^—Yoi = p for all i, while with a multi-valued treatment like schooling, this means Ysi — YS-1,i = p for all s and all i. Both are highly stylized views of the world, especially the multi-valued case which imposes linearity as well as homogeneity. To focus on one thing at a time in a heterogeneous-effects model, we start with a zero-one causal variable. In this context, we’d like to allow for treatment-effect heterogeneity, in other words, a distribution of causal effects across individuals.

Why is treatment-effect heterogeneity important? The answer lies in the distinction between the two types of validity that characterize a research design...

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