Testing for Endogeneity: A Moment-Based Bayesian Approach

2026-03-08

AI Summary

This study addresses the violation of the exogeneity assumption—commonly required in Bayesian inference for linear regression—caused by endogeneity between regressors and error terms. The authors propose the first Bayesian factor test based on exponentially tilted empirical likelihood, which assesses the presence of endogeneity by comparing a baseline model correctly specified only under exogeneity against an expanded model that remains correctly specified regardless of endogeneity. Integrating Bayesian model comparison with moment condition-based inference, the method achieves both theoretical rigor and computational tractability, while also ensuring frequentist consistency. Simulation studies and empirical applications—including the effects of car prices on demand and airfare on passenger volume—demonstrate that the approach effectively detects endogeneity and exhibits strong theoretical properties and practical performance.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Game Theory and Economic Paradigms: Behavioral Game Theory

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

A standard assumption in the Bayesian estimation of linear regression models is that the regressors are exogenous in the sense that they are uncorrelated with the model error term. In practice, however, this assumption can be invalid. In this paper, using the exponentially tilted empirical likelihood framework, we develop a Bayes factor test for endogeneity that compares a base model that is correctly specified under exogeneity but misspecified under endogeneity against an extended model that is correctly specified in either case. We provide a comprehensive study of the log-marginal exponentially tilted empirical likelihood. We demonstrate that our testing procedure is consistent from a frequentist point of view: as the sample grows, it almost surely selects the base model if and only if the regressors are exogenous, and the extended model if and only if the regressors are endogenous. The methods are illustrated with simulated data, and problems concerning the causal effect of automobile prices on automobile demand and the causal effect of potentially endogenous airplane ticket prices on passenger volume.

Problem

• endogeneity
• exogeneity
• Bayesian testing
• linear regression
• causal inference

Innovation

• Bayes factor
• endogeneity testing
• exponentially tilted empirical likelihood
• model misspecification
• consistency