Augmented balancing weights as linear regression

Area of Science:

• Statistics
• Machine Learning
• Econometrics

Background:

• Doubly robust estimators are popular in causal inference and machine learning.
• Augmented balancing weights combine outcome modeling with direct covariate balancing.
• Existing literature lacks a unified framework for understanding these estimators.

Purpose of the Study:

• To provide a novel characterization of augmented balancing weights.
• To unify existing results on doubly robust and undersmoothed estimators.
• To offer new insights into the performance of augmented balancing weights.

Main Methods:

• We analyze augmented balancing weights under linear outcome and weighting models.
• We extend the analysis to kernel ridge regression and lasso-penalized weighting models.
• The study involves theoretical analysis and mathematical derivations.

Main Results:

• The augmented estimator is equivalent to a single linear model when outcome and weighting models are linear.
• Under specific regularization, it collapses to ordinary least-squares (OLS).
• Using kernel ridge regression leads to an undersmoothed kernel ridge regression estimator.
• Lasso-penalized weighting models exhibit a 'double selection' property.

Conclusions:

• The framework demystifies augmented balancing weights, bridging theory on undersmoothed and doubly robust estimators.
• Provides novel analysis of undersmoothing in augmented balancing weights.
• Offers new understanding of the performance of these popular debiased machine learning estimators.