A Convexified Matching Approach to Imputation and Individualized Inference

We introduce a new convexified matching method for missing value imputation and individualized inference inspired by computational optimal transport. Our method integrates favorable features from mainstream imputation approaches: optimal matching, regression imputation, and synthetic control. We impute counterfactual outcomes based on convex combinations of observed outcomes, defined based on an optimal coupling between the treated and control data sets. The optimal coupling problem is considered a convex relaxation to the combinatorial optimal matching problem. We estimate granular-level individual treatment effects while maintaining a desirable aggregate-level summary by properly constraining the coupling. We construct transparent, individual confidence intervals for the estimated counterfactual outcomes. We devise fast iterative entropic-regularized algorithms to solve the optimal coupling problem that scales favorably when the number of units to match is large. Entropic regularization plays a crucial role in both inference and computation; it helps control the width of the individual confidence intervals and design fast optimization algorithms.