FITTING HETEROSCEDASTIC REGRESSION-MODELS

In heteroscedastic regression models assumptions about the error distr ibution determine the method of consistent estimation of parameters. F or example. consider the case where the model specifies the regression and dispersion functions for the data but robustness is of concern an d one wishes to use least absolute error regressions. Except in certai n special circumstances, parameter estimates obtained in this way are inconsistent. In this article we expand the heteroscedastic model so t hat 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 paramet ers carry over into this setting. Careful attention is given to the qu estion of when one can adapt for heteroscedasticity when estimating th e regression parameters. We find that in many cases such adaption is n ot possible. This complicates inference about the regression parameter s but does not lead to intractable difficulties. We also extend regres sion quantile methodology to obtain consistent estimates of both regre ssion and dispersion parameters. Regression quantiles have been used p reviously 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 resu lts.