Heteroscedastic Regression Models for the Systematic Analysis of Residual Variances

Abstract

Conventional linear regression models assume homoscedastic error terms. This assumption often is violated in empirical applications. Various methods for evaluating the extent of such violations and for adjusting the estimated model parameters if necessary are generally available in books on regression methodology. Recent developments in statistics have taken a different approach by examining the data to ascertain whether the estimated heteroscedastic residuals (from a first-stage regression model of the conditional mean of an outcome variable as a function of a set of explanatory variables or covariates) are themselves systematically related to a set of explanatory variables in a second-stage regression. These extensions of the conventional models have been given various names but, most generally, are heteroscedastic regression models (HRMs). Instead of treating heteroscedasticity as a nuisance to be adjusted out of existence to reduce or eliminate its impact on regression model parameter estimates, the basic idea of HRMs is to model the heteroscedasticity itself. This chapter systematically reviews the specification of HRMs in both linear and generalized linear model forms, describes methods of estimation of such models, and reports empirical applications of the models to data on changes over recent decades in the US income distribution and in self-reported health/health disparities. A concluding section points to similarities and complementarities of the goals of the counterfactual approach to causal inference and heteroscedastic regression models.