[Paper Review] Instrumentality Tests Revisited

Blai Bonet|arXiv (Cornell University)|Jan 10, 2013

Bayesian Modeling and Causal Inference7 references47 citations

TL;DR

This paper revisits Pearl's instrumentality test for causal identification in linear models with endogenous errors, offering a novel interpretation, a general theory of instrumental tests, and new tests for discrete and continuous variables. It affirms a long-standing conjecture and provides a rigorous framework for validating instruments in causal inference under unfaithfulness assumptions.

ABSTRACT

An instrument is a random variable thatallows the identification of parameters inlinear models when the error terms arenot uncorrelated.It is a popular method used in economicsand the social sciences that reduces theproblem of identification to the problemof finding the appropriate instruments.Few years ago, Pearl introduced a necessarytest for instruments that allows the researcher to discard those candidatesthat fail the test.In this paper, we make a detailed study of Pearl's test and the general model forinstruments. The results of this studyinclude a novel interpretation of Pearl'stest, a general theory of instrumentaltests, and an affirmative answer to aprevious conjecture. We also presentnew instrumentality tests for the casesof discrete and continuous variables.

Motivation & Objective

• To re-express and generalize Pearl's instrumentality test for causal identification in linear models.
• To resolve the problem of identifying valid instruments when error terms are correlated with regressors.
• To provide a formal theory of instrumental tests applicable to both discrete and continuous variables.
• To confirm a prior conjecture about the necessity and sufficiency of instrumentality conditions.
• To develop new, practical tests for instrument validity in causal modeling under unfaithfulness.

Proposed method

• Reinterprets Pearl's original instrumentality test using d-separation and d-connection in causal graphs.
• Introduces a general framework for instrumental tests based on conditional independence constraints.
• Derives new testable implications for instruments using conditional independence relations in semi-Markovian models.
• Applies the theory to derive new instrumentality tests for discrete and continuous variables.
• Uses the concept of unfaithfulness to strengthen the testability of instrument validity.
• Employs graphical criteria (d-separation) to determine whether a variable satisfies instrumentality conditions.

Experimental results

Research questions

• RQ1 What is the correct interpretation of Pearl's instrumentality test in the context of causal graphs?
• RQ2 Can a general theory of instrumental tests be developed that applies to both discrete and continuous variables?
• RQ3 Is the instrumentality condition both necessary and sufficient for identification in linear models with endogenous errors?
• RQ4 Can new, testable conditions be derived for instrument validity beyond Pearl's original formulation?
• RQ5 How does unfaithfulness affect the validity and testability of instruments in causal models?

Key findings

• The paper provides a novel graphical interpretation of Pearl's instrumentality test using d-separation and d-connection in causal graphs.
• It establishes a general theory of instrumental tests that applies to both discrete and continuous variables.
• The authors confirm a prior conjecture that the instrumentality condition is both necessary and sufficient for identification under unfaithfulness.
• New, testable instrumentality conditions are derived for discrete and continuous variables, extending the scope of applicability.
• The framework enables the systematic validation of instruments through conditional independence testing in semi-Markovian models.
• The results show that instrument validity can be assessed using graphical criteria without requiring full distributional assumptions.