# heteroskedasticity variance estimator bias direction

If I have a model with heteroskedasticity issue, can I tell the bias direction of the coefficents variance estimator?

I would think that because I would correct it with WLS then I get BLUE (Best Linear, Unbiased Estimator) which means the variance estimator is the smallest for unbiased coefficient but i am wrong.

Can someone please explain to me if I can know the direction of the bias and how?

• If I understand your question correctly, you have a model with heteroskedasticity and you want to know which direction the bias direction is. As far as I am aware, you need to use your intuition. Think about what omitted variables there may be the impact these may have. Aug 2 '15 at 11:31
• also welcome to econ SE! Aug 2 '15 at 11:31
• thanks you for your answer :) i'll rephrase my question. if i estimate a model using Ordinary least squares and i ignored the existance of heteroskedasticity. does the bestimated variance of the coeffiecent in this model would be greater than in the same model i would do a WLS correction? thank you Aug 2 '15 at 14:50

$\hat{\beta}_{OLS}=(x'x)^{-1}(x'y)$. Under homoskedasticity, the (estimated) variance of this estimator is $var(\beta)=\sigma^{2}(x'x)^{-1}$. Under heteroskedasticity, it becomes $(x'x)^{-1}(x'\Omega x)(x'x)^{-1}$. The $\Omega$ is the variance-covariance matrix of the error terms. OLS is a nested case of GLS under homoskedasticity. $\hat{\beta}\,_{GLS}=(x'\Omega^{-1}x)^{-1}(x'\Omega^{-1}y)$. As you can see, when the variance-covariance matrix is a scalar, the omegas drop out. WLS is a special form of GLS where $\Omega=\left(\begin{array}{cccc} e_{1}^{2} & 0 & . & 0\\ 0 & .\\ . & & .\\ 0 & . & 0 & e_{N}^{2} \end{array}\right)$ and the $e's$ stand for the residuals. The HCCME (White) standard errors use a model estimated by OLS but correct for the standard errors by using the squared residuals obtained from OLS estimation in the sandwich form of the estimate of the variance. One can never know the true form of heteroskedasticity and therefore one can never know if the standard errors of the OLS estimates are biased upwards or downwards. The problem with using the robust standard errors is that this is an asymptotic result and can therefore be biased in small samples. Personally, I would use the small-sample correction of the HCCME.