Exercise 7.11 Take a regression model with i.i.d. observations $\left(Y_{i}, X_{i} ight) $ with $X \in \mathbb {R} $ $$ \begin{aligned} Y &=X \beta+e \mathbb{E}[e \mid X] &=0 W \Omega &=\mathbb{E}\left[X^{2} e^{2} ight] . \end{aligned} $$ $$ Let $\widehat{\beta}$ be the OLS estimator of $\beta$ with residuals $\widehat{e}_{i}=Y_{i}-X_{i} \widehat{\beta}$. Consider the estimators of $\Omega$ \begin{array}{1} \widetilde{ Omega)=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2} e_{i}^{2} \ \widehat{\Omega}=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2} \widehat{e}_{i}^{2} . \end{array} $$ (a) Find the asymptotic distribution of $\sqrt{n}(\widetilde{\Omega}- Omega)$ as $n ightarrow \infty$. (b) Find the asymptotic distribution of $\sqrt{n}(\widehat{\Omega}- \Omega)$ as $n ightarrow \infty$. (c) How do you use the regression assumption $\mathbb{E}\left[e_{i} \mid X_{i} ight ]=0$ in your answer to (b)? SP.JG.417