An integral equation for the identification of causal effects in nonlinear models

TL;DR

This paper introduces an integral equation approach to identify causal effects in nonlinear models using instrumental variables, addressing the challenge of non-identifiability from (X,Y) distributions alone.

Contribution

It provides conditions under which causal effects can be identified via an integral equation involving distributions of (X,Z) and (Y,Z).

Findings

Causal effects are identifiable through the proposed integral equation.

The method applies to nonlinear models with instrumental variables.

Conditions for identification are explicitly characterized.

Abstract

When the causal relationship between X and Y is specified by a structural equation, the causal effect of X on Y is the expected rate of change of Y with respect to changes in X, when all other variables are kept fixed. This causal effect is not identifiable from the distribution of (X,Y). We give conditions under which this causal effect is identified as the solution of an integral equation based on the distributions of (X,Z) and (Y,Z), where Z is an instrumental variable.