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This is a homework question.

Consider the following two regression models
$$mathbf y=mathbf X_1mathbf {beta_1} + mathbf {varepsilon}$$
$$mathbf y=mathbf X_1mathbf {beta_1} + mathbf X_2mathbf {beta_2} + mathbf {varepsilon}$$

Where $operatorname{Disp} (mathbf {varepsilon}) =sigma^2mathbf I$

Now, if the first model is correct, show that for any vector $mathbf u$ one has

$$operatorname{Var} left( mathbf {u'}hat {mathbf {beta_1}}^{(2)}right) ge operatorname{Var} left( mathbf {u'}hat {mathbf {beta_1}}^{(1)}right)$$

Where $hat{mathbf beta_1}^{(1)}$ and $hat {mathbf beta_1}^{(2)}$ are the Least-Square estimators of $mathbf beta_1$ from the first and second models respectively.

My attempt: The Least-Square (LS) estimators of $mathbf beta_1$ obtained from the first and second models are
$$hat {mathbf {beta_1}}^{(1)}=mathbf {(X_1'X_1)^{-1}X_1'y}$$
$$hat {mathbf {beta_1}}^{(2)}=mathbf {(X_1'X_1)^{-1}X_1'}( mathbf y - mathbf {X_2hat {beta_2}^{(2)}})$$
Now since $mathbf {hat beta_2^{(2)}}$ is the LS estimator of $mathbf beta_2$, it is a linear function of $mathbf y$. So we may write $mathbf {X_2hat beta_2^{(2)}}= mathbf {Sy}$. Hence

$operatorname{Var} left( mathbf {u'}hat {mathbf {beta_1}}^{(2)}right) - operatorname{Var} left( mathbf {u'}hat {mathbf {beta_1}}^{(1)}right)$

$= sigma^2Bigl( mathbf {u'left(X_1'X_1right)^{-1}X_1'left(I - Sright)'left(I - Sright)X_1left(X_1'X_1right)^{-1}u - u'left(X_1'X_1right)^{-1}X_1'X_1left(X_1'X_1right)^{-1}u}Bigr)$

$= sigma^2Bigl(mathbf {z'(I - S)'(I - S)z - z'z}Bigr)$

$Bigl($Where $mathbf z = mathbf {X_1left(X_1'X_1right)^{-1}u}$ $Bigr)$

$= sigma^2Bigl(mathbf {z'SS'z - z'S'z -z'Sz}Bigr)$

Now this is where I am stuck. I know that the first term in the brackets is non-negative as the matrix $mathbf {SS'}$ is non-negative definite, but the two negative terms are something that I am completely unable to comment on. Any help would be appreciated. Thanks.