Constrained Design of a Binary Instrument in a Partially Linear Model

We study the question of how best to assign an encouragement in a randomized encouragement study. In our setting, units arrive with covariates, receive a nudge toward treatment or control, acquire one of those statuses in a way that need not align with the nudge, and finally have a response observed. The nudge can be seen as a binary instrument that affects the response only via the treatment status. Our goal is to assign the nudge as a function of covariates in a way that best estimates the local average treatment effect (LATE). We assume a partially linear model, wherein the baseline model is non-parametric and the treatment term is linear in the covariates. Under this model, we outline a two-stage procedure to consistently estimate the LATE. Though the variance of the LATE is intractable, we derive a finite sample approximation and thus a design criterion to minimize. This criterion is convex, allowing for constraints that might arise for budgetary or ethical reasons. We prove conditions under which our solution asymptotically recovers the lowest true variance among all possible nudge propensities. We apply our method to a semi-synthetic example involving triage in an emergency department and find significant gains relative to a regression discontinuity design.