Solution
= Solution
Independence and common error <variance> give $\operatorname{Var}(\epsilon)=\sigma^2I_n$. For the fixed matrix $A=(X^TX)^{-1}X^T$, the <covariance> transformation rule gives
$$
\begin{aligned}
\operatorname{Var}(\widehat\theta)&=A\operatorname{Var}(\epsilon)A^T\\
&=\sigma^2(X^TX)^{-1}X^TX(X^TX)^{-1}\\
&=\boxed{\sigma^2(X^TX)^{-1}}.
\end{aligned}
$$
Thus the <covariance> is determined by the error <variance> and the inverse <Gram matrix> of the design columns.