Inverting estimating equations for causal inference on quantiles

TL;DR

This paper introduces a unified method using inverse estimating equations to extend causal inference techniques from mean outcomes to quantiles, enabling richer analysis of potential outcomes.

Contribution

It develops a general framework for estimating quantiles of potential outcomes using inverse estimating equations and connects their efficient influence functions to existing mean-based methods.

Findings

Provides a construction of efficient influence functions for quantiles

Develops estimators with asymptotic properties under parametric and machine learning models

Illustrates the approach with analytical and numerical examples

Abstract

The causal inference literature frequently focuses on estimating the mean of the potential outcome, whereas quantiles of the potential outcome may carry important additional information. We propose a unified approach, based on the inverse estimating equations, to generalize a class of causal inference solutions from estimating the mean of the potential outcome to its quantiles. We assume that a moment function is available to identify the mean of the threshold-transformed potential outcome, based on which a convenient construction of the estimating equation of quantiles of potential outcome is proposed. In addition, we give a general construction of the efficient influence functions of the mean and quantiles of potential outcomes, and explicate their connection. We motivate estimators for the quantile estimands with the efficient influence function, and develop their asymptotic properties when either parametric models or data-adaptive machine learners are used to estimate the nuisance functions. A broad implication of our results is that one can rework the existing result for mean causal estimands to facilitate causal inference on quantiles. Our general results are illustrated by several analytical and numerical examples.