Regularizing Extrapolation in Causal Inference

Many causal inference and machine learning estimators are linear smoothers, where the prediction is a weighted average of training outcomes. Whether weights are constrained to be non-negative creates a key tradeoff: non-negative weights (e.g., inverse propensity weighting, random forests) limit extrapolation but can worsen covariate imbalance, while unconstrained weights (e.g., OLS, kernel ridge regression) improve balance but increase dependence on parametric assumptions. We propose a unified framework that directly penalizes extrapolation via a soft constraint on negative weights, replacing the standard hard non-negativity restriction. We derive a worst-case error bound and introduce a novel "bias-bias-variance" tradeoff among distributional imbalance, model misspecification, and estimator variance; this tradeoff is especially pronounced in high dimensions with poor positivity. We develop a convex optimization procedure that regularizes this bound and outline how to use the extrapolation penalty as a sensitivity analysis for parametric assumptions. We demonstrate our approach on synthetic data and a real-world application generalizing randomized trial estimates to a target population.