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Kernel balancing weights provide confidence intervals for average treatmenteffects, based on the idea of balancing covariates for the treated group anduntreated group in feature space, often with ridge regularization. Previousworks on the classical kernel ridge balancing weights have certain limitations:(i) not articulating generalization error for the balancing weights, (ii)typically requiring correct specification of features, and (iii) justifyingGaussian approximation for only average effects.  I interpret kernel balancing weights as kernel ridge Riesz representers(KRRR) and address these limitations via a new characterization of thecounterfactual effective dimension. KRRR is an exact generalization of kernelridge regression and kernel ridge balancing weights. I prove strong propertiessimilar to kernel ridge regression: population $L_2$ rates controllinggeneralization error, and a standalone closed form solution that caninterpolate. The framework relaxes the stringent assumption that the underlyingregression model is correctly specified by the features. It extends Gaussianapproximation beyond average effects to heterogeneous effects, justifyingconfidence sets for causal functions. I use KRRR to quantify uncertainty forheterogeneous treatment effects, by age, of 401(k) eligibility on assets.

Kernel Ridge Riesz Representers: Generalization, Mis-specification, and the Counterfactual Effective Dimension