Proximal Causal Inference without Uniqueness Assumptions.

We consider identification and inference about a counterfactual outcome mean when there is unmeasured confounding using tools from proximal causal inference. Proximal causal inference requires existence of solutions to at least one of two integral equations. We motivate the existence of solutions to the integral equations from proximal causal inference by demonstrating that, assuming the existence of a solution to one of the integral equations, n-estimability of a mean functional of that solution requires the existence of a solution to the other integral equation. Solutions to the integral equations may not be unique, which complicates estimation and inference. We construct a consistent estimator for the solution set for one of the integral equations and then adapt the theory of extremum estimators to find from the estimated set a consistent estimator for a uniquely defined solution. A debiased estimator is shown to be root-n consistent, regular, and semiparametrically locally efficient under additional regularity conditions.