Suppose that you would like to estimate whether elite highschools add more value to their students than other high schools. Your data contains a test score on student competence measured at the end of high school, gender, parental education, and whether the student graduated from an elite high school.
a.    Write down a regression model (paying special attention to functional forms) that may enable you to estimate the effect of elite schools on students' achievement.
b.   State the assumptions under which OLS consistently estimates the effect in this regression. Do you think they are satisfied in this case? Derive the plim of the OLS estimator as if there were a single explanatory variable, and argue whether the asymptotic bias is zero, negative, or positive.
c.    Could you use elementary school grade point average as a proxy variable for unobservable? Would its inclusion lead to a consistent estimation of the effect?