= Solution
Full column rank of $X$ makes $X^T:\mathbb R^n\to\mathbb R^p$ surjective. Given $x\in\mathbb R^p$, choose $u$ with $x=X^Tu$. The fitted values for the unrestricted and restricted regressions are
$$
X\widehat\beta=PY,
\qquad
X\widetilde\beta=P_0Y.
$$
Therefore
$$
x^T\widehat\beta=u^TPY,
\qquad
x^T\widetilde\beta=u^TP_0Y.
$$
Using $\operatorname{cov}(Y)=\sigma^2I_n$ and idempotence of the projections,
$$
\operatorname{var}(x^T\widehat\beta)
=\sigma^2u^TPu,
\qquad
\operatorname{var}(x^T\widetilde\beta)
=\sigma^2u^TP_0u.
$$
Part (d) now gives
$$
\boxed{\operatorname{var}(x^T\widetilde\beta)
\le\operatorname{var}(x^T\widehat\beta)}.
$$
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