Fitting Heteroscedastic Regression Models

Abstract

Abstract In heteroscedastic regression models assumptions about the error distribution determine the method of consistent estimation of parameters. For example, consider the case where the model specifies the regression and dispersion functions for the data but robustness is of concern and one wishes to use least absolute error regressions. Except in certain special circumstances, parameter estimates obtained in this way are inconsistent. In this article we expand the heteroscedastic model so that all of the common methods yield consistent estimates of the major model parameters. Asymptotic theory shows the extent to which standard results on the effect of estimating regression and dispersion parameters carry over into this setting. Careful attention is given to the question of when one can adapt for heteroscedasticity when estimating the regression parameters. We find that in many cases such adaption is not possible. This complicates inference about the regression parameters but does not lead to intractable difficulties. We also extend regression quantile methodology to obtain consistent estimates of both regression and dispersion parameters. Regression quantiles have been used previously to test for heteroscedasticity, but this appears to be their first application to modeling and estimation of dispersion effects in a general setting. A numerical example is used to illustrate the results.