Handling heteroskedasticity in the linear regression model

Abstract

This research project delves into the problem of heteroskedasticity in the linear regression model. Having defined the problem and its consequences for estimation and inference, a comprehensive literature review of existing methods for diagnosing and correcting for heteroskedasticity is undertaken, with special emphasis on heteroskedasticity tests. New theory on the statistical properties of the Ordinary Least Squares residuals is developed, leading to new models for estimating linear regression error variances. The most important of these models is the Auxiliary Linear Variance Model, which is further classified into sub-types (e.g., clustering, linear, penalised polynomial, spline). Model fitting techniques are discussed, which reduce to quadratic programming problems. An Auxiliary Nonlinear Variance Model is also developed, which can be fitted using a maximum quasi-likelihood method. Techniques for tuning of model hyperparameters and feature selection are discussed. Bootstrap methods of obtaining interval estimates for error variances are also proposed. A new heteroskedasticity test is constructed based on the auxiliary linear variance model. To make existing and new methods of handling heteroskedasticity more accessible to the practitioner, a new package called skedastic has been developed for R statistical software. Its functionality is described in detail.