The following is a simple model to measure the effect of a school choice program on standardized test performance [see Rouse (1998) for motivation]: score = beta0 + beta1 choice + beta2 faminc + u1
where score is the score on a statewide test, choice is a binary variable indicating whether a student attended a choice school in the last year, and faminc is family income. The IV for choice is grant, the dollar amount granted to students to use for tuition at choice schools. The grant amount differed by family income level, which is why we control for faminc in the equation.
(i) Even with faminc in the equation, why might choice be correlated with u1?
(ii) If within each income class, the grant amounts were assigned randomly, is grant uncorrelated with u1?
(iii) Write the reduced form equation for choice. What is needed for grant to be partially correlated with choice?
(iv) Write the reduced form equation for score. Explain why this is useful. (Hint: How do you interpret the coefficient on grant?)