On Measuring Model Complexity in Heteroscedastic Linear Regression

TL;DR

This paper extends the concept of model complexity, specifically degrees of freedom, to heteroscedastic linear regression, revealing how weights influence model complexity and providing insights for risk estimation and model selection.

Contribution

It develops a new measure of model complexity for heteroscedastic linear regression, accounting for weights used in fitting and evaluation, and analyzes its properties.

Findings

Optimal weights reduce degrees of freedom compared to equal weights.

Model complexity depends on both fitting and evaluation weights.

Heteroscedastic data modeling benefits from tailored weighting schemes.

Abstract

Heteroscedasticity is common in real world applications and is often handled by incorporating case weights into a modeling procedure. Intuitively, models fitted with different weight schemes would have a different level of complexity depending on how well the weights match the inverse of error variances. However, existing statistical theories on model complexity, also known as model degrees of freedom, were primarily established under the assumption of equal error variances. In this work, we focus on linear regression procedures and seek to extend the existing measures to a heteroscedastic setting. Our analysis of the weighted least squares method reveals some interesting properties of the extended measures. In particular, we find that they depend on both the weights used for model fitting and those for model evaluation. Moreover, modeling heteroscedastic data with optimal weights generally results in fewer degrees of freedom than with equal weights, and the size of reduction depends on the unevenness of error variance. This provides additional insights into weighted modeling procedures that are useful in risk estimation and model selection.