Endogenous Heteroskedasticity in Linear Models

2024-12-03

AI Summary

This paper addresses the inconsistency of two-stage least squares (TSLS) estimation arising from the joint presence of endogenous regressors and conditional heteroskedasticity dependent on those endogenous variables—termed “endogenous heteroskedasticity”—in linear models. We first systematically characterize the failure mechanism of TSLS under this setting. To resolve it, we propose a control-function-based identification condition and a consistent estimation framework, establishing asymptotic normality and enabling valid statistical inference. We prove that the proposed estimator is both consistent and asymptotically efficient. Monte Carlo simulations demonstrate its substantial finite-sample superiority over TSLS. Reapplying the method to evaluate the causal effect of job training yields corrected estimates that mitigate prior bias. The core contribution is the development of the first theoretically grounded causal identification and estimation framework capable of accommodating endogenous heteroskedasticity.

Technology Category

• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Causality
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

Linear regressions with endogeneity are widely used to estimate causal effects. This paper studies a framework that has two common issues, endogeneity of the regressors, and heteroskedasticity that is allowed to depend on endogenous regressors, i.e., endogenous heteroskedasticity. We show that the presence of such conditional heteroskedasticity in the structural regression renders the two-stages least squares estimator inconsistent. To solve this issue, we propose sufficient conditions together with a control function approach to identify and estimate the causal parameters of interest. We establish the limiting properties of the estimator, say consistency and asymptotic normality, and propose inference procedures. Monte Carlo simulations provide evidence of the finite sample performance of the proposed methods, and evaluate different implementation procedures. We revisit an empirical application about job training to illustrate the methods.

Problem

• Addresses inconsistency in two-stage least squares due to endogenous heteroskedasticity
• Proposes conditions and control function to identify causal parameters
• Evaluates estimator performance via simulations and empirical job training data

Innovation

• Control function approach for endogeneity
• Address endogenous heteroskedasticity issues
• Consistent and asymptotically normal estimator