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. Internal validity is the question of whether a given design successfully uncovers causal effects for the population being studied. A randomized clinical trial or, for that matter, a good IV study, has a strong claim to internal validity. External validity is the predictive value of the study’s findings in a different context. For example, if the study population in a randomized trial is especially likely to benefit from treatment, the resulting estimates may have little external validity. Likewise, draft-lottery estimates of the effects of conscription for service in the Vietnam era need not be a good measure of the consequences of voluntary military service. An econometric framework with heterogeneous treatment effects helps us to assess both the internal and external validity of IV estimates.