Heteroscedastic Normal Regression Models

This chapter presents the heteroscedastic normal linear regression models under frequentist and Bayesian perspectives. The chapter begins with an introduction to homoscedastic regression models from frequentist and Bayesian perspectives, including the Bayesian methods used to obtain parameter estimates. When there is variance heterogeneity, that is, when the homoscedasticity hypothesis does not hold, it is sometimes still possible to satisfy this hypothesis by transforming the response variable. After that, the chapter proposes normal heteroscedastic linear regression models, with explicit modeling of the variance heterogeneity through the explanatory variables. This means that appropriate functions of the variance follow linear regression structures. From a maximum likelihood perspective, an alternate iterative Fisher scoring algorithm is developed and applied. Based on the results obtained from this maximum likelihood algorithm, Bayesian methods to fit Bayesian heteroscedastic models are described, obtained by assuming normal prior distributions for the mean and variance regression parameters. Finally, a natural extension is proposed to study lognormal heteroscedastic regression models.