Solved – Proving efficiency of OLS over GLS

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I'm trying to prove the efficiency of OLS over GLS when the covariance matrix of the error $varepsilon$ is mistakenly assumed to be $sigma^2Sigma$ instead of $sigma^2 I$. After deriving the variances, what I have so far is: $Var(beta^{ols}) = sigma^2(X'X)^{-1}$ and
$Var(beta^{gls}) = sigma^2(X'Sigma^{-1} X)^{-1}$.

Want to show $Var(beta^{gls}) – Var(beta^{ols})$ is psd.
$$implies sigma^2(X'Sigma^{-1} X)^{-1} – sigma^2(X'X)^{-1} geq 0\
implies (X'Sigma^{-1} X)^{-1} – (X'X)^{-1} geq 0 \
iff X'X – X'Sigma^{-1} X geq 0
$$
But this is where I don't know how to proceed. First I tried $X'(I-Sigma^{-1})X geq 0$ but got stuck. Any hints on how to continue?

Update:
After one of the comments, I want to double check the variance of $beta^{gls}$:
$$Var(beta^{gls}) = Var[(X'Sigma^{-1}X)^{-1} X'Sigma^{-1}varepsilon]\
=(X'Sigma^{-1}X)^{-1} X'Sigma^{-1} Var(varepsilon) Sigma^{-1} X(X'Sigma^{-1}X)^{-1}
$$
Here's is where I think I made the mistake, I replaced the variance of $varepsilon$ as the one that we think is "right" (i.e. $sigma^{2}Sigma) $. Should I replace it with the real one (i.e. $sigma^2I$)?

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Best Answer

This is what I ended up doing: $$Var(beta^{gls}) = Var[(X'Sigma^{-1}X)^{-1} X'Sigma^{-1}varepsilon]\ =(X'Sigma^{-1}X)^{-1} X'Sigma^{-1} Var(varepsilon) Sigma^{-1} X(X'Sigma^{-1}X)^{-1}\ = sigma^2 (X'Sigma^{-1}X)^{-1} X'Sigma^{-1} Sigma^{-1} X(X'Sigma^{-1}X)^{-1}\ $$

Want to show $Var(beta^{gls}) – Var(beta^{ols})$ is psd. $$implies sigma^2 (X'Sigma^{-1}X)^{-1} X'Sigma^{-1} Sigma^{-1} X(X'Sigma^{-1}X)^{-1} – sigma^2(X'X)^{-1} geq 0\ iff X'X – X'Sigma^{-1}X( X'Sigma^{-1} Sigma^{-1} X)^{-1}X'Sigma^{-1}X geq 0\ implies X'(I-Sigma^{-1}X( X'Sigma^{-1} Sigma^{-1} X)^{-1}X'Sigma^{-1})X geq 0\ implies X'MXgeq0 $$ Where $M$ is the residual maker matrix, and since $M$ is psd, the statement holds.

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