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Suppose $Y_{n \times 1} \sim N(X\beta,\sigma^2V)$ where $V_{n\times n}$ is invertible and $X_{n\times p}$ is of rank $p$ and $\beta_{p \times 1}$ is unknown and to be estimated by $Y$ and $X$. Which one of the following estimators is better?

1) $\hat{\beta_{LS}}=(X'X)^{-1}X'Y$

2) $\hat{\beta_{GLS}}=(X'V^{-1}X)^{-1}X'V^{-1}Y$

I was trying to show $Var(\hat{\beta_{LS})}-Var(\hat{\beta_{GLS})}$ is positive semi-definite(p.s.d). i.e. $(X'X)^{-1}X'VX(X'X)^{-1} - (X'V^{-1}X)^{-1}$ is p.s.d.

But could not do it. Also do we need normality in order to show the above? Can you please hint/help or provide appropriate study material? Thank you,

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  • $\begingroup$ $\beta$ is a $p \times 1$ vector $\endgroup$ – Robert Aug 13 '15 at 4:00

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