Conditional Influence Functions

TL;DR

This paper develops the theory of conditional influence functions, enabling improved estimation and inference for conditional distribution functionals, especially in high-dimensional settings.

Contribution

It introduces a general framework for conditional influence functions and Neyman orthogonal estimating equations for conditional objects of interest.

Findings

Derived conditional influence functions for various functionals.

Provided rate conditions for machine learning estimators to ensure valid inference.

Established a method for locally linear estimation of conditional objects.

Abstract

There are many nonparametric objects of interest that are a function of a conditional distribution. One important example is an average treatment effect conditional on a subset of covariates. Many of these objects have a conditional influence function that generalizes the classical influence function of a functional of a (unconditional) distribution. Conditional influence functions have important uses analogous to those of the classical influence function. They can be used to construct Neyman orthogonal estimating equations for conditional objects of interest that depend on high dimensional regressions. They can be used to formulate local policy effects and describe the effect of local misspecification on conditional objects of interest. We derive conditional influence functions for functionals of conditional means and other features of the conditional distribution of an outcome variable. We show how these can be used for locally linear estimation of conditional objects of interest. We give rate conditions for first step machine learners to have no effect on asymptotic distributions of locally linear estimators. We also give a general construction of Neyman orthogonal estimating equations for conditional objects of interest.