Many centralized school admissions systems use lotteries to ration limited seats at oversubscribed schools. The resulting random assignment is used by empirical researchers to identify the effects of schools on outcomes like test scores. I first find that the two most popular empirical research designs may not successfully extract a random assignment of applicants to schools. When do the research designs overcome this problem? I show the following main results for a class of data-generating mechanisms containing those used in practice: The “first-choice” research design extracts a random assignment under a mechanism if the mechanism is strategy-proof for schools. In contrast, the other “qualification instrument” research design does not necessarily extract a random assignment under any mechanism. The former research design is therefore more compelling than the latter. Many applications of the two research designs need some implicit assumption, such as large-sample approximately random assignment, to justify their empirical strategy.