Evaluating the Application of Weighted Least Squares in Handling Heteroscedastic Regression Data Models

Posted: May 05, 2009

• Jasmine Reed
• Jason Powell
• Jeremy Cox

Abstract

Regression analysis stands as one of the most fundamental and widely applied statistical methodologies across scientific disciplines, providing a framework for understanding relationships between variables and making predictions. The classical linear regression model, typically estimated using Ordinary Least Squares (OLS), rests upon several key assumptions, including linearity, independence, normality, and homoscedasticity of errors. Among these, the assumption of homoscedasticity—that the variance of errors remains constant across all observations—frequently proves untenable in practical applications. Heteroscedasticity, the condition where error variances differ across observations, manifests commonly in real-world datasets spanning economics, biology, engineering, and social sciences. The presence of heteroscedasticity violates the Gauss-Markov theorem assumptions, leading to inefficient parameter estimates, biased standard errors, and invalid hypothesis tests, thereby compromising the reliability of statistical inferences. Weighted Least Squares (WLS) emerges as the conventional remedy for heteroscedastic regression models, operating on the principle of assigning weights inversely proportional to the variance of each observation. Traditional WLS implementations, however, face significant practical limitations. These approaches typically require either prior knowledge of the variance structure or a correctly specified variance model, conditions rarely met in empirical research. Furthermore, conventional WLS methods often assume simplistic variance patterns that fail to capture the complex heteroscedastic structures present in modern datasets. The existing literature provides limited guidance on diagnosing the specific nature of heteroscedasticity and selecting appropriate weighting schemes for diverse variance patterns. This research addresses these limitations through a comprehensive investigation of WLS methodology, introducing several innovative contributions. We develop an adaptive weighting framework that dynamically responds to varying error structures without presupposing the form of heteroscedasticity. Our approach incorporates machine learning techniques for variance estimation, creating a more robust and accurate method for handling heteroscedastic data.