Augmented balancing weights as linear regression

We provide a novel characterization of augmented balancing weights, also known as automatic debiased machine learning (AutoDML). These popular doubly robust or double machine learning estimators combine outcome modeling with balancing weights -- weights that achieve covariate balance directly in lieu of estimating and inverting the propensity score. When the outcome and weighting models are both linear in some (possibly infinite) basis, we show that the augmented estimator is equivalent to a single linear model with coefficients that combine the coefficients from the original outcome model coefficients and coefficients from an unpenalized ordinary least squares (OLS) fit on the same data; in many real-world applications the augmented estimator collapses to the OLS estimate alone. We then extend these results to specific choices of outcome and weighting models. We first show that the augmented estimator that uses (kernel) ridge regression for both outcome and weighting models is equivalent to a single, undersmoothed (kernel) ridge regression. This holds numerically in finite samples and lays the groundwork for a novel analysis of undersmoothing and asymptotic rates of convergence. When the weighting model is instead lasso-penalized regression, we give closed-form expressions for special cases and demonstrate a ``double selection'' property. Our framework opens the black box on this increasingly popular class of estimators, bridges the gap between existing results on the semiparametric efficiency of undersmoothed and doubly robust estimators, and provides new insights into the performance of augmented balancing weights.