Identification in Multiple Treatment Models under Discrete Variation
We develop a marginal treatment effect based method to learn about causal effects in multiple treatment models with discrete instruments. We allow selection into treatment to be governed by a general class of threshold crossing models that permit multidimensional unobserved heterogeneity. An inherent complication is that the primitives characterizing the selection model are not generally point-identified. Allowing these primitives to be point-identified up to a finite-dimensional parameter, we show how a two-step computational program can be used to obtain sharp bounds for a number of treatment effect parameters when the marginal treatment response functions are allowed to satisfy only nonparametric shape restrictions or are additionally parameterized. We demonstrate the benefits of our method by revisiting Kline and Walters' (2016) empirical analysis of the Head Start program. Our approach relaxes their point-identifying assumptions on the selection model and marginal treatment response functions, allowing us to assess the robustness of their conclusions.