Interacting Treatments with Endogenous Takeup

We study causal inference in randomized experiments (or quasi-experiments) following a $2\times 2$ factorial design. There are two treatments, denoted $A$ and $B$, and units are randomly assigned to one of four categories: treatment $A$ alone, treatment $B$ alone, joint treatment, or none. Allowing for endogenous non-compliance with the two binary instruments representing the intended assignment, as well as unrestricted interference across the two treatments, we derive the causal interpretation of various instrumental variable estimands under more general compliance conditions than in the literature. In general, if treatment takeup is driven by both instruments for some units, it becomes difficult to separate treatment interaction from treatment effect heterogeneity. We provide auxiliary conditions and various bounding strategies that may help zero in on causally interesting parameters. As an empirical illustration, we apply our results to a program randomly offering two different treatments, namely tutoring and financial incentives, to first year college students, in order to assess the treatments' effects on academic performance.