Supplement to “ Understanding Instrumental Variables in Models with Essential Heterogeneity ” ∗

Definition of the counterfactual Dj . This all takes place at the level of hypothetical manipulations. The space may have a probability measure or just a measure. For some J (Z) = j, define Aj (z) = {z | J (z) = j}. For each element in the set, we “project” μD (z) onto the set. If we embed the set in L , this is clear. Otherwise, there may be other constructions (e.g., embed in space L and for each norm take a projection. Take the infimum of these norms.) Such projections do not necessarily define a unique Dj . Consider z = (z1, z2), μD (z) = γz and J (z1) = z1. Then Dj is undefined. To make a unique construction we need to supplement the model of hypothetical counterfactuals. One supplement is a “rational expectations” type supplement. We fix J (z) = j and draw a z at random from this hypothetical distribution, conditional on the realized J (z) = j. This entails constructing a distribution for z. One such construction is the actual distribution of z2, given z1 in the true population. With this z construction