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These popular doubly robust estimators combine outcome modelling with balancing weights\u2014weights that achieve covariate balance directly instead 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 those of the original outcome model with those from unpenalized ordinary least-squares (OLS). Under certain choices of regularization parameters, the augmented estimator in fact collapses to the OLS estimator alone. We then extend these results to specific 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\u2014implying a novel analysis of undersmoothing. When the weighting model is instead lasso-penalized, we demonstrate a familiar \u2018double selection\u2019 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.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkaf019","type":"journal-article","created":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T07:53:58Z","timestamp":1743494038000},"page":"699-723","source":"Crossref","is-referenced-by-count":10,"title":["Augmented balancing weights as linear regression"],"prefix":"10.1093","volume":"88","author":[{"given":"David","family":"Bruns-Smith","sequence":"first","affiliation":[{"name":"Graduate School of Business, Stanford University, Stanford, CA 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