On the Identifying Power of Generalized Monotonicity for Average Treatment Effects

In the context of a binary outcome, treatment, and instrument, Balke and Pearl establish that the monotonicity condition of Imbens and Angrist has no identifying power beyond instrument exogeneity for average potential outcomes and average treatment effects in the sense that adding it to instrument exogeneity does not decrease the identified sets for those parameters whenever those restrictions are consistent with the distribution of the observable data. This article shows that this phenomenon holds in a broader setting with a multi-valued outcome, treatment, and instrument, under an extension of the monotonicity condition that we refer to as generalized monotonicity. We further show that this phenomenon holds for any restriction on treatment response that is stronger than generalized monotonicity provided that these stronger restrictions do not restrict potential outcomes. Importantly, many models of potential treatments previously considered in the literature imply generalized monotonicity. We show through a series of examples that restrictions on potential treatments can provide identifying power beyond instrument exogeneity for average potential outcomes and average treatment effects when the restrictions imply that the generalized monotonicity condition is violated. In this way, our results shed light on the types of restrictions required for help in identifying average potential outcomes and average treatment effects. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.