2. Private Schools You are assigned to evaluate the effects of a voucher for attending private schools. The voucher was distributed in an experiment to families in a single school district. The voucher is structured as follows: Family Income (as Voucher Size Probability of Getting Voucher in a reported on tax return) Random Lottery Conducted among Families in the Income Group 0-15k 12,000 | 0.75 l5k-3 0k 1 0,000 I 0.60 Economics of Education, Econ 4180-01 Fulya Y. Ersoy, Ph.D. 3 0k-45k 9 ,000 0.45 45k-60k 8 000 0.30 60k-80k 3' 000 0.15 80k plus 6,000 0.05 Families participated in the experiment if they had a child who was scheduled to attend kindergarten in year t and if they expressed an interest in participating. Families who won obtained a voucher that could be used to offset tuition payments if they sent their child to a private school. If they used the voucher at a private school in kindergarten, they could continue to get vouchers for grades 1 through 6, but they could leave all or any vouchers unused and attended their neighborhood public school instead. You may assume that richer families tend to live in neighborhoods with other richer families. The voucher lotteries took place in the summer before kindergarten when a "pre-test" was administered to every child who participated in a lottery. You have data on the achievement of students on the pre-test and also at the end of grade 6. (a) Write down a regression equation that estimates the effect of winning a voucher lottery on 6th grade scores and that assumes that the effect is the same for families of all income groups. Use notation that is stande (similar to what you see in class). (b) The estimates you have estimated in part (a) are intention-to-treat effects. Explain how to estimate treatment-on-the-treated effects (the effect of attending private school on grade 6 test scores). Use clear notation and write down every equation that you need to run. Hint: You need to do an instrumental variables estimation here. (c) Write down a regression equation that estimates the effect of winning a voucher lottery on 6th grade scores and that allows that the effect to be different based on families' income groups. One reason the win-voucher effect might vary by income group that the size of the voucher depends on family income. What are some other reasons the win-voucher effect might vary by family income? (d) How could you use this setting to distinguish between the effect of voucher size and other reasons why the win-voucher effect might vary by family income? (Hint: you might want to use only a subset of your data.)