<p>I think the problem is that you are mixing up terminology. The statment about an estimator being BLUE states <em>given the Gauss-Markov assumption, OLS is the best linear unbiased estimator</em>. But as your model is heteroskedastic the Gauss-Markov assumptions no longer hold and the proof of OLS being BLUE is no longer true. </p>

<p>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. </p>

CC BY-SA 3.0
2015-08-02T17:09:53.007Z
2015-09-02T19:41:37.173Z
2015-09-02T19:41:37.173Z
Rud Faden
<p>$\hat{\beta}_{OLS}=(x'x)^{-1}(x'y)$. 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 $\Omega=\left(\begin{array}{cccc}
e_{1}^{2} &amp; 0 &amp; . &amp; 0\\
0 &amp; .\\
. &amp;  &amp; .\\
0 &amp; . &amp; 0 &amp; e_{N}^{2}
\end{array}\right)$ 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. </p>

CC BY-SA 3.0
2015-09-02T15:07:16.797Z
2015-09-02T15:07:16.797Z
ChinG
<p>If I have a model with heteroskedasticity issue, can I tell the bias direction of the coefficents variance estimator?</p>

<p>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.</p>

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

CC BY-SA 3.0
2015-08-02T11:31:19.587Z
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.
Jamzy
CC BY-SA 3.0
2015-08-02T11:31:34.563Z
also welcome to econ SE!
Jamzy
CC BY-SA 3.0
2015-08-02T14:50:18.277Z
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
user5344
<econometrics>
heteroskedasticity variance estimator bias direction
user5344
# heteroskedasticity variance estimator bias direction

(Asked by: username_0 on 2015-08-02)

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?

**Tags:** econometrics
> **Comments:**
> **username_1**: 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.  
> **username_1**: also welcome to econ SE!  
> **username_0**: 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  

---

## 2 answers

### Answer by username_2 on 2015-08-02. Score: 3

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.


---

### Answer by username_3 on 2015-09-02. Score: 1

$\hat{\beta}\_{OLS}=(x'x)^{-1}(x'y)$. 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.


---

