Here's a working paper that observes that assignments of places to pre-kindergarten students remains strongly influenced by where the children live, even in the presence of a school choice system. I.e. many pre-kindergarten students go to neighborhood schools.
Distance to Schools and Equal Access in School Choice Systems, by Mariana Laverde
Abstract: This paper studies the limits of school choice policies in the presence of residential segregation. Using data from the Boston Public Schools choice system, I show that white prekindergarteners are assigned to higher-achieving schools than minority students, and that cross-race school achievement gaps under choice are no lower than would be generated by a neighborhood assignment rule. To understand why choicebased assignments do not reduce gaps in school achievement, I use data on applicants’ rank-order choices to estimate preferences over schools, and consider a series of counterfactual assignments. I find that half of the gap in school achievement between white and Black or Hispanic students is explained by minorities’ longer travel distance to high-performing schools. Differences in demand parameters explain a smaller fraction of the gap, while algorithm rules have no effect."
From the conclusions:
"The salience of travel costs shows a first-order channel for why neighborhoods matter, highlighting how the effective provision of public goods can be affected by geography at very granular levels. In some way these results are not surprising. Most, if not all, of the papers that study school demand agree that distance is a key factor in parental choices. This paper takes this observation one step further and quantifies how much this cost limits the effectiveness of school choice policies in equalizing access to high-achieving schools. The results show that even in a generous choice environment where parents face minimal restrictions to their choices and free transportation is provided, distance can contribute greatly to inequity and that the design of the assignment algorithm can do little to break structural place-based inequities. This finding is not only relevant for the pre-kindergarten population. Not only we know that early investments can have lasting impacts on adult outcomes, but also, choice systems are typically designed to grandfather students into subsequent grades within a school. Then, even if travel costs are lower for older children, early assignments are held for several years after. In consequence, inequities in pre-kindergarten extend well after that period."
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