  We provide a novel characterization of augmented balancing weights, also
known as automatic debiased machine learning (AutoDML). These popular doubly
robust or de-biased 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 and coefficients
from an unpenalized ordinary least squares (OLS) fit on the same data. We see
that, under certain choices of regularization parameters, the augmented
estimator often collapses to the OLS estimator alone; this occurs for example
in a re-analysis of the Lalonde 1986 dataset. 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.
