Linear Regression: Inference Based on Cluster Estimates

2026-02-04

AI Summary

This study addresses the bias in regression coefficient inference that arises when within-cluster dependence is ignored in clustered data. The authors propose a novel estimator that explicitly models the intra-cluster dependence structure, accommodating both fixed and diverging cluster sizes. The framework is further extended to random-coefficient models to enable inference on average effects and testing of linear hypotheses. Key contributions include a robust estimation approach that accounts for within-cluster correlation, a new Wald-type test designed for improved stability in high-level hierarchical parameters, and a theoretical demonstration of the inconsistency of pooled ordinary least squares (POLS) under random-coefficient settings. The validity and practical utility of the proposed methodology are substantiated through rigorous theoretical analysis, simulation studies, and empirical applications.

Technology Category

• Machine Learning: Clustering
• Reasoning under Uncertainty: Causality
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• 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: Incentives in network design for Web infrastructures and ecosystems

Abstract

This article proposes a novel estimator for regression coefficients in clustered data that explicitly accounts for within-cluster dependence. We study the asymptotic properties of the proposed estimator under both finite and infinite cluster sizes. The analysis is then extended to a standard random coefficient model, where we derive asymptotic results for the average (common) parameters and develop a Wald-type test for general linear hypotheses. We also investigate the performance of the conventional pooled ordinary least squares (POLS) estimator within the random coefficients framework and show that it can be unreliable across a wide range of empirically relevant settings. Furthermore, we introduce a new test for parameter stability at a higher (superblock; Tier 2, Tier 3,...) level, assuming that parameters are stable across clusters within that level. Extensive simulation studies demonstrate the effectiveness of the proposed tests, and an empirical application illustrates their practical relevance.

Problem

• clustered data
• linear regression
• parameter stability
• random coefficients
• inference

Innovation

• clustered data
• random coefficients model
• Wald-type test
• parameter stability
• asymptotic properties