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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?
I think the problem is that you are mixing up terminology. The statment about an estimator being BLUE states given the Gauss-Markov assumption, OLS is the best linear unbiased estimator. But as your model is heteroskedastic the Gauss-Markov assumptions no longer hold and the proof of OLS being BLUE is no longer true.
Unfortunalty there is no general result that shows whether a estimater is the "best" in the case of hetroskedacity. In theory if you know the exact functional form of the heteroskedascity you can perfectly correct for the hetroskadicity and you WLS is just as effective as OLS when the Gauss-Markov assumptions hold. But in reality this is never the case, and unless you have a small sample or a very strong argument of why you know the functional form of the hetroskedacity then you are better of using Whites robust standart errors.
$\hat{\beta}_{OLS}=(x'x)^{-1}(x'y)$. 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.
