  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.
