Estimation of Parameters in Heteroscedastic Linear Models

SUMMARY This paper deals with linear regression models with non-homogeneous error variances. The common situation where the error variance is a smooth function of the values of the regressor variables is considered. The error variance function is represented by a function of known form, but its expression involves the vector of unknown regression coefficients β and an additional vector parameter θ. Estimation of β by iterative weighted least squares (IWLS) therefore also requires updating the estimate of θ at each iteration. When the error variance function is misspecified, different methods of updating θ would yield IWLS estimators of varying degrees of efficiency. Under such situations, a method of updating θ is proposed, resulting in an iterative procedure for estimating β that is asymptotically optimal in the class of IWLS estimators. When the form of the error variance function is correctly specified, the estimation procedure proposed is asymptotically equivalent to weighted least squares with known optimal weights. A real example of its application is given and the extension to general non-linear models is briefly indicated.