= Solution
{c}
The full column rank makes $X^{\mathsf T}X$ invertible. The <least-squares normal equations> from minimizing squared residuals give
$$
\boxed{\widehat\theta=(X^{\mathsf T}X)^{-1}X^{\mathsf T}Y.}
$$
The errors have <covariance matrix> $\sigma^2I_n$ by <independence> and equal <variance>. Applying the <covariance> transformation rule to this linear estimator yields
$$
\operatorname{Var}(\widehat\theta)
=(X^{\mathsf T}X)^{-1}X^{\mathsf T}(\sigma^2I)X(X^{\mathsf T}X)^{-1}
=\boxed{\sigma^2(X^{\mathsf T}X)^{-1}.}
$$
The parameter estimator is unbiased because $\mathbb EY=X\theta$.
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