Identification of suitable explanatory variable in goldfeld-quandt test and robust inference under heteroscedasticity and high leverage points

Violation of the assumption of homogeneity of variance of the errors in the linear regression model, causes heteroscedasticity. In the presence of heteroscedastic errors, the ordinary least squares (OLS) estimates are unbiased and consistent, but their covariance matrix estimator is biased and not consistent. As a consequence, this problem negatively affects inference made with biased standard errors. As such, before making any inferences from the OLS, it is very important to check whether or not heteroscedasticity is present. It is now evident that Goldfeld-Quandt (GQ) test is a very powerful test of heteroscedasticity among its competitors. The GQ test requires ordering observations of one explanatory variable in increasing order such that arrangement of observations from the other explanatory variables and the dependent variable in the model follows. When the model involves more than one explanatory variables, identifying suitable variable to be used in the ordering becomes problem when there is no prior knowledge of which variable causes the heteroscedasticity problem. This study has developed an algorithm of identifying this variable prior to conducting the Goldfeld-Quandt test in multiple linear regression model. To overcome the heteroscedasticity problem, many adjustment methods have been proposed in the literature to correct the biased covariance matrix estimator . These heteroscedasticity correcting estimators are known as heteroscedasticity-consistent covariance matrix estimators (HCCME) which include among others HC0, HC1, HC2, HC3, HC4, HC4m and HC5. However, HC4 and HC5 were designed to take into account, the combined problems of heteroscedasticity and high leverage points in a data. For the same purpose, Furno (1996) used a weighted least squares approach and Lima et al. (2009) extended the idea to HC4 and HC5. This study has modified the weighted HCCME used by Furno and Lima et al. to come out with two new weighted HCCME that perform well in quasi t inference, in the presence of heteroscedasticity and high leverage points in small to moderate sample size.

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