Consider the linear probability model (Y_{i}=beta_{0}+beta_{1} X_{i}+u_{i}), and assume that (Eleft(u_{i} mid X_{i} ight)=0). a. Show that

Consider the linear probability model \(Y_{i}=\beta_{0}+\beta_{1} X_{i}+u_{i}\), and assume that \(E\left(u_{i} \mid X_{i}\right)=0\).

a. Show that \(\operatorname{Pr}\left(Y_{i}=1 \mid X_{i}\right)=\beta_{0}+\beta_{1} X_{i}\).

b. Show that \(\operatorname{var}\left(u_{i} \mid X_{i}\right)=\left(\beta_{0}+\beta_{1} X_{i}\right)\left[1-\left(\beta_{0}+\beta_{1} X_{i}\right)\right]\). [Hint: Review Equation (2.7).]

c. Is \(u_{i}\) heteroskedastic? Explain.

d. Derive the likelihood function.