Consider the one-variable regression model \(Y_{i}=\beta_{0}+\beta_{1} X_{i}+u_{i}\), and suppose it satisfies the least squares assumptions in Key Concept 4.3. The regressor \(X_{i}\) is missing, but data on a related variable, \(Z_{i}\), are available, and the value of \(X_{i}\) is estimated using \(\widetilde{X}_{i}=E\left(X_{i} \mid Z_{i}\right)\). Let \(w_{i}=\widetilde{X}_{i}-X_{i}\).

a. Show that \(\widetilde{X}_{i}\) is the minimum mean square error estimator of \(X_{i}\) using \(Z_{i}\). That is, let \(\hat{X}_{i}=g\left(Z_{i}\right)\) be some other guess of \(X_{i}\) based on \(Z_{i}\), and show that \(E\left[\left(\hat{X}_{i}-X_{i}\right)^{2}\right] \geq E\left[\left(\widetilde{X}_{i}-X_{i}\right)^{2}\right]\).

b. Show that \(E\left(w_{i} \mid \widetilde{X}_{i}\right)=0\).

c. Suppose that \(E\left(u_{i} \mid Z_{i}\right)=0\) and that \(\widetilde{X}_{i}\) is used as the regressor in place of \(X_{i}\). Show that \(\hat{\beta}_{1}\) is consistent. Is \(\hat{\beta}_{0}\) consistent?