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I am trying to answer the following question:

Given $\vec{\epsilon} \sim N(0,\Sigma)$ vector where $\epsilon_i$ are NOT iid. Find the estimator $\hat{\beta}$ that minimises $(Y − X\beta)^T\Sigma^{−1}(Y − X\beta)$. Find the mean squared error of this estimator.

My approach:

$(Y − X\beta)^T\Sigma^{−1}(Y − X\beta)$ is $N(0,I_n)$ in distribution. Where $\beta$ is the usual ordinary least squares estimator.

In that case, the estimator does NOT change in case of heteroscedasticity, does it?

If $X^T$ is full rank, then

$\hat{\beta}=(X^TX)^{-1}X^TY$,from where $\hat{\beta} \sim N(\beta, (X^TX)^{-1}X^T\Sigma X(X^TX)^{-1})$

And the bias of $\hat{\beta}$ is $0$. So I find:

$$\begin{eqnarray} \text{Quadratic risk} &=& \text{Variance} + \text{bias}^2 \\ &=& (X^TX)^{-1}X^T\Sigma X(X^TX)^{-1} \\ \end{eqnarray}$$ Is this right?

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  • $\begingroup$ The OLS does not minimize the mean squared error when data are heteroscedastic. See the Gauss-Markov theorem. $\endgroup$ – AdamO May 15 '18 at 17:15
  • $\begingroup$ Multiply out the function you want to minimize and then find the $\hat{\beta}$ which minimizes it by taking the first derivative, setting it equal to 0, and solving. It's pretty straightforward. $\endgroup$ – klumbard May 15 '18 at 17:29
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In case of heteroskedasticity the estimator actually changes:

If $Var$($\varepsilon$$\mid$$X$) = $\Sigma$ $\ne$ $\sigma^{2}$$I_{n}$, despite heteorskedasticity, we still have that $\Sigma$ is positive defined $n$ $\times$ $n$ matrix.

This implies that $\Sigma$ can be factorized as follows:

$\Sigma$ = $C$$\Lambda$$C'$

where $\Lambda$ is an $($$n$$\times$$n$$)$ diagonal matrix with the main diagonal elements all strictly positive and $C$ is an $n$$\times$$n$ matrix such that $C$$C'$ = $C'$$C$ = $I$.

Such decompostion of $\Sigma$ is useful since it canbe proved that:

$\Sigma^{-1}$ = $C$$\Lambda^{-1}$$C'$

and:

$\Sigma^{-1/2}$ = $C$$\Lambda^{-1/2}$$C'$.

The Genarlized Least Squares transformed model is defined as:

$\Lambda^{-1/2}$$C'$$y$ = $\Lambda^{-1/2}$$C'$$X$$\beta$ + $\Lambda^{-1/2}$$C'$$\varepsilon$

from which we finally find the $\hat{\beta_{Gls}}$ applying to it the OLS:

$\hat{\beta_{Gls}}$ = $($$X'$$C$$\Lambda^{-1/2}$$\Lambda^{-1/2}$$C'$$X$$)^{-1}$$X'$$C$$\Lambda^{-1/2}$$\Lambda^{-1/2}$$C'$$y$ = $($$X'$$\Sigma^{-1}$$X$$)^{-1}$$X'$$\Sigma^{-1}$$y$

with:

$Var$$($$\hat{\beta_{Gls}}$$\mid$$X$$)$ = $($$X'$$\Sigma^{-1}$$X$$)^{-1}$.

Note that sometimes, for heteroskedastic $\varepsilon$, the Variance matrix is further decomposed as follows:

$\Sigma$ = $\sigma^{2}$$\Omega$

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