Treatment Effect Estimators as Weighted Outcomes

TL;DR

This paper develops a comprehensive framework for deriving outcome weights in treatment effect estimators, revealing how implementation choices influence their properties and introducing novel weights for advanced machine learning methods.

Contribution

It provides a unified approach to obtain outcome weights, including new weights for double machine learning and generalized random forests, and analyzes their properties and implementation effects.

Findings

Outcome weights can be derived via a general framework.

Implementation choices affect the availability and properties of outcome weights.

Standard methods like causal forests may use outcome weights that do not sum to one.

Abstract

Estimators that weight observed outcomes to form effect estimates have a long tradition. Their outcome weights are widely used in established procedures, such as checking covariate balance, characterizing target populations, or detecting and managing extreme weights. This paper introduces a general framework for deriving such outcome weights. It establishes when and how numerical equivalence between an original estimator representation as moment condition and a unique weighted representation can be obtained. The framework is applied to derive novel outcome weights for the six seminal instances of double machine learning and generalized random forests, while recovering existing results for other estimators as special cases. The analysis highlights that implementation choices determine (i) the availability of outcome weights and (ii) their properties. Notably, standard implementations of partially linear regression-based estimators, like causal forests, employ outcome weights that do not sum to (minus) one in the (un)treated group, not fulfilling a property often considered desirable.