Generalized Optimal Matching Methods for Causal Inference

We develop an encompassing framework and theory for matching and related methods for causal inference that reveal the connections and motivations behind various existing methods and give rise to new and improved ones. The framework is given by generalizing a new functional analytical characterization of optimal matching as minimizing worst-case conditional mean squared error given the observed data based on specific restrictions and assumptions. By generalizing these, we obtain a new class of generalized optimal matching (GOM) methods, for which we provide a single theory for tractability and consistency that applies generally to GOM. Many commonly used existing methods are included in GOM and using their GOM interpretation we extend these to new methods that judiciously and automatically trade off balance for variance and outperform their standard counterparts. As a subclass of GOM, we develop kernel optimal matching, which, as supported by new theory, is notable for combining the interpretability of matching methods, the non-parametric model-free consistency of optimal matching, the efficiency of well-specified regression, the judicious sample size selection of monotonic imbalance bounding methods, the double robustness of augmented inverse propensity weight estimators, and the model-selection flexibility of Gaussian-process regression. We discuss connections to and non-linear generalizations of equal percent bias reduction and its ramifications.