(Solved): Exercise 2. In lecture 10 we showed the variance-covariance matrix for the (p+1)1 vector, ^ ...

Exercise 2. In lecture 10 we showed the variance-covariance matrix for the $$(p+1)\times 1$$ vector, $$\widehat{?}$$, of least squares estimates is given by $$\mathrm{Var}(\widehat{?})={?}^2{\left({\mathbf{X}}^{?}\mathbf{X}\right)}^{?1}$$. Derive the $$2\times 2$$ variance-covaraiance matrix for least squares estimates, $$\widehat{?}={\left(\begin{array}{ll}{\widehat{?}}_0& {\widehat{?}}_1\end{array}\right)}^{?}$$, for simple linear regression: $$\mathrm{Var}(\widehat{?})=\left(\begin{array}{cc}\mathrm{Var}\left({\widehat{?}}_0\right)& \mathrm{Cov}\left({\widehat{?}}_0,{\widehat{?}}_1\right)\\ \mathrm{Cov}\left({\widehat{?}}_1,{\widehat{?}}_0\right)& \mathrm{Var}\left({\widehat{?}}_1\right)\end{array}\right)$$ Additionally, use your result to verify that $$\mathrm{Var}\left({\widehat{?}}_0\right)={?}^2\left(\frac{1}{n}+\frac{{\stackrel{?}{x}}^2}{SXX}\right)$$ and $$\mathrm{Var}\left({\widehat{?}}_1\right)={?}^2/\mathrm{SXX}$$, where $$\mathrm{SXX}=\underset{i=1}{\overset{n}{?}}{\left(x_i?\stackrel{?}{x}\right)}^2$$. [Hint: it might be useful to use the identity $$\underset{i=1}{\overset{n}{?}}{\left(x_i?\stackrel{?}{x}\right)}^2=?x_i^2?n{\stackrel{?}{x}}^2$$ ]