Estimating Continuous Treatment Effects with Two-Stage Kernel Ridge Regression

2026-04-14

AI Summary

This study addresses selection bias induced by covariates in the estimation of continuous treatment effects by proposing a two-stage kernel ridge regression approach. In the first stage, the method models the joint dependence of the outcome on both the treatment variable and covariates; in the second stage, it constructs pseudo-outcomes that correct for distributional shifts to estimate the average treatment effect. The core innovation lies in an estimator that adaptively exploits potential structural simplifications in the treatment effect function, coupled with a fully data-driven model selection procedure that requires no prior knowledge. This framework simultaneously adapts to unknown degrees of overlap and kernel eigenvalue decay rates, achieving theoretical guarantees for adaptivity to both overlap conditions and function smoothness, thereby yielding more accurate estimates of continuous treatment effects.

Technology Category

• Machine Learning: Kernel Methods
• Game Theory and Economic Paradigms: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We study the problem of estimating the effect function for a continuous treatment, which maps each treatment value to a population-averaged outcome. A central challenge in this setting is confounding: treatment assignment often depends on covariates, creating selection bias that makes direct regression of the response on treatment unreliable. To address this issue, we propose a two-stage kernel ridge regression method. In the first stage, we learn a model for the response as a function of both treatment and covariates; in the second stage, we use this model to construct pseudo-outcomes that correct for distribution shift, and then fit a second model to estimate the treatment effect. Although the response varies with both treatment and covariates, the induced effect function obtained by averaging over covariates is typically much simpler, and our estimator adapts to this structure. Furthermore, we introduce a fully data-driven model selection procedure that achieves provable adaptivity to both the unknown degree of overlap and the regularity (eigenvalue decay) of the underlying kernel.

Problem

• continuous treatment effects
• confounding
• causal inference
• selection bias
• effect function

Innovation

• continuous treatment effect
• two-stage kernel ridge regression
• confounding adjustment
• data-driven model selection
• distribution shift correction