$1.(20$p$)$ Consider a simple model to measure the effects of taking a preparatory course $($a binary

variable, course$)$ on eventual score on a college admissions exam:

$=0+1+$

$($a$)(5$p$)$ Why might course be correlated with u$?$

$($b$)(5$p$)$ Is course likely to be related to parents$$ income? If so$,$ does this mean parental

income is a good IV for course? Explain.

$($c$)(5$p$)$ Suppose that $20\%$ of students at every school were randomly given tuition

waivers for the course. Carefully explain how you would use this information to

construct an IV for course.

$($d$)(5$p$)$ What are the two assumptions that the tuition waiver must satisfy to be a valid

IV$?$ Do these seem to be reasonable assumptions? How would you test these in the

data?