[Paper Review] On the Testability of Causal Models with Latent and Instrumental Variables

Judea Pearl|arXiv (Cornell University)|Feb 20, 2013

Advanced Causal Inference TechniquesMathematics11 references94 citations

TL;DR

This paper establishes a general formula to test causal models involving latent variables and instrumental variables by deriving observable inequality constraints on the joint distribution of observed variables. It enables researchers to determine whether a given variable can act as an instrument and whether a causal model is compatible with empirical data, even when unmeasured confounders are present.

ABSTRACT

Certain causal models involving unmeasured variables induce no independence constraints among the observed variables but imply, nevertheless, inequality contraints on the observed distribution. This paper derives a general formula for such instrumental variables, that is, exogenous variables that directly affect some variables but not all. With the help of this formula, it is possible to test whether a model involving instrumental variables may account for the data, or, conversely, whether a given variables can be deemed instrumental.

Motivation & Objective

• To address the challenge of testing causal models when unmeasured (latent) variables and instrumental variables are present.
• To identify conditions under which causal models with latent variables can be empirically tested despite lacking conditional independence constraints.
• To develop a formal criterion for determining whether a variable qualifies as an instrumental variable in a causal model.
• To provide a systematic method for testing the compatibility of a causal model with observed data, even when confounding variables are unobserved.
• To extend the testability framework of causal models beyond conditional independence to include inequality constraints derived from instrumental variable structures.

Proposed method

• Derives a general formula for instrumental variables in causal models with latent confounders, based on structural equations and d-separation principles.
• Uses the concept of instrumental variables as exogenous variables that directly affect only a subset of endogenous variables.
• Applies the do-calculus framework to derive conditional distributions and identify observable implications of the model.
• Identifies inequality constraints on the joint distribution of observed variables that must hold if the model is valid.
• Employs a transformation of the structural model into a set of observable moment inequalities that can be tested empirically.
• Leverages the d-separation criterion to identify conditional independence relations and their violations in the presence of latent variables.

Experimental results

Research questions

• RQ1 Under what conditions can a causal model with latent variables be tested using only observed data?
• RQ2 How can one determine whether a given variable qualifies as an instrumental variable in a model with unmeasured confounders?
• RQ3 What observable inequality constraints arise from instrumental variable structures in the presence of latent confounders?
• RQ4 Can a causal model involving instrumental variables be rejected based on empirical data, even when no conditional independence constraints are present?
• RQ5 What is the general form of the inequality constraints that characterize the testability of such models?

Key findings

• The paper derives a general formula for instrumental variables that enables the identification of observable inequality constraints in models with latent variables.
• It establishes that even without conditional independence constraints, causal models with instrumental variables can be tested via inequality constraints on the observed distribution.
• The method allows for the validation or rejection of a causal model based on empirical data, even when confounding variables are unobserved.
• The framework provides a criterion to determine whether a variable can be considered an instrumental variable by testing its implications on the observed distribution.
• The approach generalizes the testability of causal models beyond the classical conditional independence framework to include inequality-based constraints.
• The results demonstrate that instrumental variable structures induce testable implications in the form of inequalities, which can be used to falsify or support causal models.