A heteroskedasticity-consistent covariance matrix estimator and a direct test

TL;DR: In this paper, a covariance matrix estimator which is consistent even when the disturbances of a linear regression model are heteroskedastic is presented, but does not rely on a (possibly incorrect) specific formal model of the structure of the heter-kedasticity.

Abstract: This paper presents a parameter covariance matrix estimator which is consistent even when the disturbances of a linear regression model are heteroskedastic. This estimator does not depend on a formal model of the structure of the heteroskedasticity. By comparing the elements of the new estimator to those of the usual covariance estimator, one obtains a direct test for heteroskedasticity, since in the absence of heteroskedasticity, the two estimators will be approximately equal, but will generally diverge otherwise. The test has an appealing least squares interpretation. IT IS WELL KNOWN that the presence of heteroskedasticity in the disturbances of an otherwise properly specified linear model leads to consistent but inefficient parameter estimates and inconsistent covariance matrix estimates. As a result, faulty inferences will be drawn when testing statistical hypotheses in the presence of heteroskedasticity. If the investigator has a formal model of the process generating the differing variances, these difficulties are easily eliminated by performing an appropriate linear transformation on the data, based on this model. However, even when such a model is available, it may be incorrect. Often, several models are considered (e.g., Griliches [10]), but still without the certain knowledge that any of them is correct. In this situation one can test each of the alternative transformed models for remaining heteroskedasticity (using any of several available tests), and eliminate those which fail. But what is one to do if all fail the heteroskedasticity test? Although the investigator will have a fairly good idea of the parameter values of the linear model, there remains a considerable difficulty in assessing the precision of the parameter estimates and testing hypotheses due to the possible inconsistency of the usual covariance matrix estimator. In this paper I resolve this difficulty by presenting a covariance matrix estimator which is consistent in the presence of heteroskedasticity, but does not rely on a (possibly incorrect) specific formal model of the structure of the heteroskedasticity. Thus, even when heteroskedasticity cannot be completely eliminated, proper inferences can be drawn. Under appropriate conditions, a natural test for heteroskedasticity can be obtained by comparing the consistent estimator to the usual covariance matrix estimator; in the absence of heteroskedasticity, both estimators will be about the same-otherwise, they will generally diverge. The test shares the advantage of the covariance estimator, in that no formal structure on