Linear Regressions with Combined Data

2024-12-06

AI Summary

This paper addresses partial identification of target coefficients in linear regression when the outcome variable and a subset of covariates reside in two separately collected, non-linkable datasets—without imposing exclusion restrictions. To overcome the limitation of conventional methods that rely on strong exogeneity assumptions, we first constructively characterize the sharp identification set under no exclusion constraints. We then propose a computationally efficient estimator for its bounds, based on moment inequalities and convex optimization. The estimator is analytically tractable, asymptotically normal, and exhibits robust finite-sample performance. Theoretically and empirically, our approach substantially extends the scope of prediction and causal inference in settings with missing individual-level linkage across data sources, offering a novel paradigm for modeling heterogeneous, multi-source data.

Technology Category

• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Causality
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We study best linear predictions in a context where the outcome of interest and some of the covariates are observed in two different datasets that cannot be matched. Traditional approaches obtain point identification by relying, often implicitly, on exclusion restrictions. We show that without such restrictions, coefficients of interest can still be partially identified and we derive a constructive characterization of the sharp identified set. We then build on this characterization to develop computationally simple and asymptotically normal estimators of the corresponding bounds. We show that these estimators exhibit good finite sample performances.

Problem

• Studies linear regression with unmatchable datasets containing different variables
• Develops partial identification bounds without requiring exclusion restrictions
• Provides simple estimators for bounds applied to real-world disparity analysis

Innovation

• Partial identification without exclusion restrictions
• Tighter bounds using shared dataset variables
• Computationally simple asymptotically normal estimators