Cross-Balancing for Data-Informed Design and Efficient Analysis of Observational Studies

Causal inference starts with a simple idea: compare groups that differ by treatment, not much else. Traditionally, similar groups are constructed using only observed covariates; however, it remains a long-standing challenge to incorporate available outcome data into the study design while preserving valid inference. In this paper, we study the general problem of covariate adjustment, effect estimation, and statistical inference when balancing features are constructed or selected with the aid of outcome information from the data. We propose cross-balancing, a method that uses sample splitting to separate the error in feature construction from the error in weight estimation. Our framework addresses two cases: one where the features are learned functions and one where they are selected from a potentially high-dimensional dictionary. In both cases, we establish mild and general conditions under which cross-balancing produces consistent, asymptotically normal, and efficient estimators. In the learned-function case, cross-balancing achieves finite-sample bias reduction relative to plug-in-type estimators, and is multiply robust when the learned features converge at slow rates. In the variable-selection case, cross-balancing only requires a product condition on how well the selected variables approximate true functions. We illustrate cross-balancing in extensive simulations and an observational study, showing that careful use of outcome information can substantially improve both estimation and inference while maintaining interpretability.