ANALYTICAL COMPARISON OF HCCMEs

Abstract

This thesis considers the issue of evaluating heteroskedasticity consistent covariance matrix estimators (HCCME) in linear heteroskedastic regression models. Several HCCMEs are considered, namely: HC0 (White estimator), HC1 (Hinkley estimator), HC2 (Horn, Horn & Duncan estimator) and HC3 (Mackinnon & White estimator). It is well known that White estimator is biased in finite samples; see e.g. Chesher & Jewitt and Mackinnon & White. A number of simulation studies show that HC2 & HC3 perform better than HC0 over the range of situations studied. See e.g. Long & Ervin, Mackinnon & White and Cribari-Neto & Zarkos. The existing studies have a serious drawback that they are just based on simulations and not analytical results. A number of design matrices as well as skedastic functions are used but the possibilities are too large to be adequately explored by simulations. In the past, analytical formulas have been developed by several authors for the means and the variances of different types of HCCMEs but the expression obtained are too complex to permit easy analysis. So they have not been used or analyzed to explore and investigate the relative performance of different HCCMEs. Our goal in this study is to analytically investigate the relative performance of different types of HCCMEs. One of the major contributions of this thesis is to develop new analytic formulae for the biases of the HCCMEs. These formulae permit us to use minimax type criteria to evaluate the performance of the different HCCMEs. We use these analytical formulae to identify regions of the parameter space which provide the ranges for the best and the worst performance of different estimators. If an estimator performs better than another in the region of its worst behavior, then we can confidently expect it to be better. Similarly, if an estimator is poor in area of its best performance, than it can be safely discarded. This permits, for the first time, a sharp and unambiguous evaluation of the relative performance of a large class of widely used HCCMEs. We also evaluate the existing studies in the light of our analytical calculations. Ad hoc choices of regressors and patterns of heteroskedasticity in existing studies resulted in ad hoc comparison. So there is a need to make the existing comparisons meaningful. The best way to do this is to focus on the regions of best and worst performance obtained by analytical formulae and then compare the HCCMEs to judge their relative performance. This will provide a deep and clear insight of the problem in hand. In particular, we show that the conclusions of most existing studies change when the patterns of heteroskedasticity and the regressor matrix is changed. By using the analytical techniques developed, we can resolve many questions: 1) Which HCCME to use 2) How to evaluate the relative performance of different HCCMEs 3) How much potential size distortion exists in the heteroskedasticity tests 4) Patterns of heteroskedasticity which are least favorable, in the sense of creating maximum bias. ii Our major goal is to provide practitioners and econometricians a clear cut way to be able to judge the situations where heteroskedasticity corrections can benefit us the most and also which method must be used to do such corrections. Our results suggest that HC2 is the best of all with lowest maximum bias. So we recommend that practitioners should use only HC2 while performing heteroskedasticity corrections.