Developing a General Consistent Standard Error Estimator under Varying Strengths of Heteroscedasticity

Abstract

In econometric studies involving cross-sectional data, the assumption of a constant variance for the disturbance term is a fluke. In consumer budget studies (micro consumption function), the residual variance about the population regression function is very likely to increase with income. Also, in cross-sectional studies of firms the residual variance probably increases with the size of the firm. In a simple linear regression model, the dependent variable Y is explained by Z. Thus we assume  y=f(z)+e and postulate that var(y)=var(e)=sigma squared Z. Let z_i = X_i  . By implication we formulate the assumption about var(e_i)=sigma squared Z  in a rational and fairly general manner. In general, to validate this assumption, it is convenient and quite plausible to specify the form of association var(e_i)=sigma squared Z_i^g , where g is the strength of heteroscedasticity and the lower the strength (magnitude) of g, the smaller the difference between the individual variances. Except when g =0, the model is homoscedastic otherwise |g|<=2 generally. This paper developed a general heteroscedasticity consistent standard error (HCSE) estimator using weight related to regressors that characterizes the random error term denoted by HC5. Comparative studies of the developed estimator with the existing HCSE estimators using various strengths of heteroscedasticity on a continuum scale at sample sizes 25, 30, 35, 40, 45, and 50 were implemented. The OLS estimator remains unbiased and the results showed that the developed estimator is indeed a generalization of all the existing HCSE estimators and proved to be consistent and asymptotically efficient.