  There are many kinds of exogeneity assumptions. How should researchers choose
among them? When exogeneity is imposed on an unobservable like a potential
outcome, we argue that the form of exogeneity should be chosen based on the
kind of selection on unobservables it allows. Consequently, researchers can
assess the plausibility of any exogeneity assumption by studying the
distributions of treatment given the unobservables that are consistent with
that assumption. We use this approach to study two common exogeneity
assumptions: quantile and mean independence. We show that both assumptions
require a kind of non-monotonic relationship between treatment and the
potential outcomes. We discuss how to assess the plausibility of this kind of
treatment selection. We also show how to define a new and weaker version of
quantile independence that allows for monotonic treatment selection. We then
show the implications of the choice of exogeneity assumption for
identification. We apply these results in an empirical illustration of the
effect of child soldiering on wages.
