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 = β0 + β1 choice + β2 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 famine 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 famine in the equation.
(i) Even with famine 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 «,?
(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.