Question 10:
(a) You are told that students who get above a 93 (out of 100) on an admissions test are admitted a magnet program. The program is amazing, and everyone above the cutoff enrolls (no one below the cutoff can go). Draw a picture of what the relationship between test scores and participation in the program looks like (let participation rate in the program be the y-axis and entry test score be the x-axis).
(b) Remember the magnet program is amazing. Given that draw a hypothetical graph of how average student test scores the following year relate to the entry test score. Think about the Chicago summer school example.
(c) What would be a regression model to capture this relationship? For full credit--write two separate models—one where the slope is the same on both sides of the cutoff and another model where the relationship between entry test and the posttest can be different for the treatment and control groups.
(d) What do we mean when we say a regression discontinuity estimates a local average treatment effect? How does this connect to the external validity of our estimate?
(e) Now, you find out that the relationship at the cutoff is fuzzy. No one below cutoff attends, and only 70 percent of those above line attend. Draw a picture of what this would look like with the forcing variable on the x-axis and the probability of enrolling in the magnet program on the y-axis. Why would you expect the regression based on the model above would understate the impact of the program?
(f) How could you potentially correct for this problem using a two-stage process?
(g) However, even after the correction, you only have estimated based on compliers. If a high fraction of those who would get the most out of the program opt for private school, would you expect the estimate overstates or understates the impact of the program compared to an estimate where all students complied? Why?
(h) What if you found out that some students randomly are allowed to attend immediately after taking the exam and can begin the program in 9th
grade while others must wait until 10thgrade. As part of our evaluation of the impact of the magnet program, what would we want to see in relation to John Stuart Mill’s level of cause and level of effect?
(i) Now, the program changes its rules and allows anyone with a 3.7 GPA into the program. Why might you be more suspicious of a regression discontinuity based on this set up?