Solution
= Solution
For $p=1$, part (c) gives
$$
\operatorname{Var}(\widehat\beta(\Sigma))\sim
\frac{\sigma^2}{n\mathbb E[X_1^2/v(X_1)]},
$$
$$
\operatorname{Var}(\widehat\beta(I))\sim
\frac{\sigma^2\mathbb E[v(X_1)X_1^2]}{n(\mathbb E[X_1^2])^2}.
$$
Consequently
$$
\boxed{\rho=\frac{(\mathbb E[X_1^2])^2}
{\mathbb E[X_1^2/v(X_1)]\,\mathbb E[v(X_1)X_1^2]}.}
$$