Targeted Function Balancing
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This paper introduces Targeted Function Balancing (TFB), a covariate balancing framework for estimating the causal effect of a binary intervention on an outcome. TFB first regresses the outcome on observed covariates, and then selects weights that balance functions (of the covariates) that are probabilistically near the resulting regression function. This yields balance in the predicted values of the regression function and the covariates, with the regression function's estimated variance determining how much balance in the covariates is sufficient. Notably, TFB demonstrates that intentionally leaving imbalance in some covariates can increase efficiency without introducing bias, challenging traditions that warn against imbalance in any variable. TFB is shown here to be useful in high-dimensional settings, with particular focus given to kernel regularized least squares and the LASSO as potential regression estimators. With the former, TFB contributes to the literature of kernel-based weights. As for the LASSO, TFB uses the regression function's estimated variance to prioritize balance in certain dimensions of the covariates, a feature that can be exploited to great effects by choosing a sparse regression estimator. Most appealing is that TFB is entirely defined by the choice of regression estimator and an estimator for the resulting function's variance, turning the problem of how best to balance the covariates into how best to model the outcome. Additionally, this paper introduces a balance diagnostic, Targeted Function Imbalance, which TFB minimizes subject to constraints on its weights' variance, and may have other applications.
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