i). You have a sample of individuals who applied for a private school voucher. Winners were awarded randomly among the applicants. You then estimate the following regression: Y =0 + 1W + , where W is a dummy variable indicating whether the student won the lottery (W = 1 if a student wins the lottery), Y is test scores in standardized units (so each one-unit change is a standard deviation)and is an error term. Conditional on winning, the probability of enrolling is 50%. a. The estimated coefficient on 1 is 0.35. Interpret this coefficient in words. b. Calculate the causal effect of attending a private school on test scores. c. Explain the difference between an intent-to-treat estimator and a treatment-effect-on-thetreated estimator. Which one is the estimator in part a and which one is the estimator in part b? Is one estimator more important for policy than the other?

ii). No-excuses charter schools in New York City and Boston have been shown to have significant effects on student outcomes. Why can't this finding be generalized to all charter schools?