1. Following the previous question, now suppose that the State of Missouri mandates that all high school students above a certain height (which differs by gender and school) will be given an offer to play on the basketball team (for boys) or basketball and volleyball teams (for girls). No student who is shorter than the gender- and schoolspecific mandated height is allowed to play on these teams. You imagine that this might be an ideal application for a regression discontinuity design. After crunching some numbers, you find that boys above the cutoff are 80% more likely to play on the team than those who are below the height cutoff, while girls above the cutoff are 65% more likely to play on one of the teams. a. Identify the forcing, treatment, and outcome variables in this RD. b. Is this a fuzzy or sharp RD? How can you tell? c. Do you think that this RD is subject to manipulation? Why or why not? d. Suppose that you verify that no one below the height cutoff is found on either the basketball or volleyball teams. Suppose also that you find that, at the cutoff, college graduation rates are 10 percentage points higher for both genders. Compute the local ATE for high school sports participation among the population of male Missouri high school students. What is the local ATE for female students? e. What can you conclude from this analysis about the college graduation prospects of students who are very short or very tall relative to the cutoff?

2.You are interested in whether there is a causal impact of high school sports participation on one's likelihood of graduating from college. A number of researchers have examined this claim using instrumental variables.1 One popular instrument is a student's height as an adolescent. a. Identify the treatment, outcome, and instrument for this research setting. b. What is first condition of validity for instrumental variables as described in Lovenheim and Turner? Do you think this instrument satisfies it? Why or why not? c. What is second condition of validity for instrumental variables as described in Lovenheim and Turner? Do you think this instrument satisfies it? Why or why not? d. If you answered that the height does not satisfy either of the requirements in (b) and (c), how do you think that assuming validity of the instrument will affect the conclusions of the research?