Location Choice, Commuting, and School Choice∗
Minseon Park

Dong Woo Hahm

Yale School of Management

University of Southern California

July 2023
Abstract
We explore the impact of public school assignment reforms by building a households’ school
choice model with two key features—(1) endogenous residential location choice and (2) opt-out
to outside schooling options. Households decide where to live taking into account that locations determine access-to-school—admissions probabilities and commuting distances to schools.
Households are heterogeneous both in observed and unobserved characteristics. We estimate themodel using administrative data from New York City’s middle school choice system. Variationfrom a boundary discontinuity design separately identifies preferences for access-to-school fromother location amenities. Residential sorting based on access-to-school preference explains 30%of the gap in test scores of schools attended by minority students versus their peers. If households’ residential locations were fixed, a reform that introduces purely lottery-based admissions
to schools in lower- and mid-Manhattan would reduce the cross-racial gap by 7%. However,households’ endogenous location choices dampen the effect by half.
Keywords: Centralized School Choice, Neighborhood Sorting, School Segregation, Commuting
to School
JEL Codes: D12, I21, I28, J15, R23∗

Park: minseon.park@yale.edu; Hahm: dongwooh@usc.edu. Park is deeply grateful to her committee members John Kennan, Christopher
Taber, Chao Fu, and Jesse Gregory. We also thank Milena Almagro, Naoki Aizawa, Panle Jia Barwick, Pierre-Andre Chiappori, Andrew
Bongjune Choi, Rajeev Darolia, Jean-François Houde, Amrita Kulka, Rasmus Lentz, Margaux Luflade, Jeff Smith, and Matt Wiswall for theiruseful feedback. This research has benefited from comments from seminar participants at the labor workshop at UW-Madison, the brown bag
at the Federal Reserve Bank of Richmond, and the applied micro theory colloquium at Columbia University. Thanks also go to the New York
City Department of Education, particularly to Benjamin Cosman, Broggini Matthew, Stewart Wade, and Lianna Wright.

Electronic copy available at: https://ssrn.com/abstract=4452763

1Introduction
Across the U.S. in 2016, 33% of K-12 students lived in a school district where they could

choose, to some degree, which public school to attend.1 Public school choice systems provide
students with multiple options beyond the nearest school to their home, standing in contrast
to classic settings where students are automatically assigned to schools based on their homeaddresses. Such contrast has raised hope that centralized school assignments could decouple
educational disparities from spatial disparities at scale. However, many popular schools undera choice system give admissions priority to students from residential locations nearby, evenwhen they accept applications from a broader set of students (Dur, Kominers, Pathak, andSönmez, 2013). Such location-based admissions rules have triggered debate over the designof admissions rules, motivated by a concern that these contribute to the continued school
segregation observed in many school choice settings (Cohen, 2021).
How effectively can we desegregate schools with reforms on the location-based admissions
rules in a public school choice system? We answer this question by developing a households’
school choice model that considers two important margins through which households may
respond: residential location choice and opt-out to outside schooling options.
The key feature of the model is households’ endogenous location choice. While previous
work has documented that residential location explains half the racial gap in test scores ofschools attended by students under centralized school choice (Laverde, 2020), how households
make the residential location decision has received little attention in the school choice literature(e.g., Abdulkadiroğlu, Agarwal, and Pathak, 2017). In our model, households choose residential
locations by considering access to schools, which refers to both admissions probabilities andcommuting distances to schools that vary across locations.
We set up a multi-stage discrete choice model where households sequentially choose (1)which location to live in, (2) which school to apply to, and (3) whether to enroll in the assigned
school or opt out to outside schooling options. Households have observed and unobserved
heterogeneous preferences over a set of location and school characteristics, which leads to rich
residential and school sorting patterns.
We start by providing causal evidence that households consider location-based admissions
rules when deciding where to live. Our empirical context is the middle school choice system
1Based on authors’ calculation using The National Center for Education Statistics 2019 National Household
Education Surveys: Parent and Family Involvement in Education Survey.

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in New York City (NYC), where 70,000 students and 700 middle schools are matched eachyear. Each student has over 30 public school options to apply to, and which Community
School District (CSD)—a subdivision of the city—they reside in largely determines the choice
set and admissions probabilities. Leveraging this institutional aspect, we apply a boundary
discontinuity design (BDD) to compare Census blocks that are close to one another but locatedon opposite sides of a CSD boundary. By doing so, we deal with the endogeneity concernthat locations with higher admissions chances to high-achieving schools might have amenities
unobserved to researchers but are observed and valued by households. Estimates indicate that
Census blocks within a CSD with one standard deviation higher school test scores have 22%
more households with middle school applicants.
We use the variation from the BDD to identify how much households value access to school
relative to other location amenities. We estimate our structural model using an extension of the
expectation-maximization algorithm with a sequential maximization step (ESM, Arcidiacono
and Jones, 2003). This keeps the estimation tractable while enabling us to jointly estimate allstages of the model to account for households’ selection into locations.
The results show that endogenizing households’ residential choice has important implications for (1) understanding the source of school segregation under the status quo, (2) obtaining
an unbiased commuting cost estimate, and (3) predicting the implications of the counterfactual
policy.
First, our estimates illustrate that households’ location choices based on location’s accessto schools play a large role in explaining which students are matched to which schools. To
show this, we shut down each part of the model in a decomposition exercise. 30% of the gap intest scores of schools attended by minorities versus non-minorities is explained by households’
residential sorting based on locations’ access to schools. Households’ heterogeneous preference
over other location characteristics and school characteristics explains 45% and 18% of thecross-racial gap, respectively.
Second, we find that a model that does not account for endogenous location choice overestimates commuting costs by 15%. Our model estimates show that a median household is willingto pay $19 per school day to reduce commuting time to school by 50 minutes. Commuting
cost is an important parameter that governs the degree to which students take advantage ofschool choice options rather than applying to schools nearest to their residential locations.
The reason for the overestimation is that households choose locations near schools they prefer

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since it increases their admissions probabilities. This leads to a spurious result in which they
apply to schools nearby not because they care about distance but because in their location
choice they cared about admission probability. Without correcting for households’ selection
into locations, the model would misinterpret households’ applying to schools nearby as solely
due to commuting costs. Due to residential sorting based on unobserved school preference,
this is still true even when one controls for households’ observed characteristics.2
Finally, we describe how households’ spatial reshuffling in response to a school desegregation reform can affect the effectiveness of the policy. We consider a counterfactual policy that
introduces purely lottery-based admissions to schools in District 2, the district with the highest
test scores and housing costs in NYC. Covering lower- and mid-Manhattan, District 2 hasbeen at the center of ongoing policy debates regarding the design of location-based admissions
criteria.3 When we fix households’ residential locations, lottery-based admissions to District
2 schools would close the cross-racial gap in school test scores by 7%. This is because someminority students residing outside the district are assigned to District 2 schools, which pushes
out non-minority District 2 residents to lower-achieving schools.
However, households’ location choices in response to the policy dampen the equity impact
by half. Two types of spatial reshuffling exert opposing forces. On the one hand, some minority
households choose residential locations closer to District 2 in response to the reform. With
shorter commuting distances to District 2, they are more likely to apply to District 2 schools.
Spatial reshuffling of this sort amplifies the desegregation effect of the policy. On the other
hand, most of the non-minority households who reside in District 2 under the status quo
relocate out of the district. Since other districts still have location-based admissions in place,they seek other locations that assure higher admissions probabilities to high-achieving schools.
Such spatial reshuffling dampens the equity effect of the policy.
The equilibrium force amplifies the second reshuffling while muting the first. This is because purely lottery-based admissions to District 2 schools induce more applications, and thus
the equilibrium admissions cutoffs of these schools increase. This weakens the incentive ofminority households to relocate closer to District 2 but strengthens that of non-minority households to relocate farther from District 2. We find that households substitute between opting-out
2For example, a household that puts a higher value on school safety than other observably similar households
will sort into locations that increase their child’s admission chances into a safer school.
3Shapiro, Eliza, N.Y.C. to Change Many Selective Schools to Address Segregation, the New York Times,
December 18, 2020.

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to outside schooling options and choosing different residential locations. But, overall, opt-out
plays a smaller role in determining the effectiveness of the reform on reducing the cross-racial
gap.
Related Literature We contribute to two strands of the literature. First, we extend the school
choice literature by considering households’ endogenous location choice. While it is wellknown that residential location is the main source of school segregation (Laverde, 2020), little
is known about how households choose where to live in response to the design of centralized
school choice. Previous studies have focused on assignment mechanisms (Abdulkadiroğlu,Che, and Yasuda, 2015; Abdulkadiroğlu, Agarwal, and Pathak, 2017; He, 2015; Agarwal
and Somaini, 2018; Che and Tercieux, 2019; Calsamiglia, Fu, and Güell, 2020); information
provision (Hastings and Weinstein, 2008; Hoxby and Turner, 2015; Luflade, 2018; Corcoran,
Jennings, Cohodes, and Sattin-Bajaj, 2018; Chen and He, 2021; Fack, Grenet, and He, 2019;
Allende, Gallego, and Neilson, 2019); limited attention (Ajayi and Sidibe, 2020; Son, 2020);
and previously attended schools (Hahm and Park, 2022).
By modeling households’ endogenous location choices, we first compare the implicationsof counterfactual policies when households’ residential locations are fixed versus adjusted. Thisapproach aligns with reduced-form evidence that access to school shapes the composition ofresidents and housing costs of locations (Black, 1999; Reback, 2005; Brunner, Cho, and Reback,2012; Schwartz, Voicu, and Horn, 2014; Billings, Brunner, and Ross, 2018). Moreover, we
correct selection into locations in estimating school preference to obtain an unbiased estimateof commuting costs; we take a departure from the standard assumption in the literature that
distances to schools are uncorrelated with households’ unobserved school tastes conditionalon their observable characteristics.4
Second, this paper adds to a large body of studies on within-city residential sorting,by studying households’ location choice in a newly relevant setting of centralized school
assignments. Among many papers in this literature, more closely related are those that give
special attention to schools compared with other location amenities.5 Earlier studies have
4This assumption is often found in the broader economics of education literature, which uses distance toschools as an instrumental variable for school application and attendance (Card, 1993; Schwartz, Stiefel, andWiswall, 2013; Walters, 2018; Mountjoy, 2022)
5Broader set of papers have studied how residential sorting is determined by other factors such as accessto work (Ahlfeldt, Redding, Sturm, and Wolf, 2015), ease of commuting (Barwick, Li, Waxman, Wu, and Xia,
2021), consumption amenities (Almagro and Domınguez-Iino, 2019; Miyauchi, Nakajima, and Redding, 2022),
or neighborhood composition (Davis, Gregory, and Hartley, 2019).

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focused on classical settings where each residential location is zoned to one public school(Bayer, Ferreira, and McMillan, 2007) while incorporating limited forms of school choices
such as private school vouchers or inter-district transfers (Manski, 1992; Nechyba, 2000; Epple
and Romano, 2003; Ferreyra, 2007).
Under centralized school assignments, households choose among many public schools
from a given location. This enables us to study households’ heterogeneous values over a setof school characteristics, including commuting distance. Indeed, this two-way heterogeneity is
one of the main sources of school segregation under school choice settings (Idoux, 2022; Hahm
and Park, 2022, e.g.). In contrast, frameworks in the location choice literature (Bayer, Ferreira,and McMillan, 2007) have considered one-dimensional school characteristics, usually mean
test scores, due to lack of variation coming from their setting where each location is zoned to
one public school.
With the recent popularity of centralized school assignments, there have been a few papers
proposing a unified framework of location choice and school choice. These include theoretical models (Xu, 2019; Avery and Pathak, 2021; Grigoryan, 2021) and a quantitative model(Agostinelli, Luflade, and Martellini, 2021). Our paper complements theoretical models by
estimating our model using data.
The closest paper to ours is by Agostinelli, Luflade, and Martellini (2021), from which
we differentiate in two respects. First, our model features richer observable and unobservable
heterogeneity in households’ location and school preferences. Grigoryan (2021) theoreticallyshows that preference heterogeneity is crucial in determining the welfare implication of aschool choice design.6 Second, we model outside schooling options, another margin that somehouseholds use with the introduction of a more extensive school choice system.
Organization The remainder of the paper is organized as follows. Section 2 describes the
public middle school choice system in NYC and the data. Section 3 presents motivating
evidence on the interaction between residential location choice and school choice. Section 4
describes the model. Section 5 describes the empirical strategy and presents estimation results.
Section 6 investigates the source of school segregation. Section 7 studies the equity impacts ofa school desegregation reform.
6See Almagro and Domınguez-Iino (2019) for a similar discussion in a model where households haveheterogeneous preferences over a set of urban amenities.

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2

Institutional Background and Data

2.1

Public Middle School Choice in NYC
Each year, about 70,000 entering students and 700 middle school programs participate in

the NYC citywide middle school choice system. There are about 500 middle schools. Multiple
programs with separate curriculum can be offered by one school, and students apply to eachprogram. In the following, we use the terms “program" and “school" interchangeably when
there is no confusion. Schools that are part of the centralized choice system are governed by thecity. The property tax rate is constant within the city and the city allocates the pooled funding
to schools directly, largely based on the number of students.7
The main round of the school choice process starts in December of students’ last year ofelementary school. Students are given a customized list of programs they are eligible for andsubmit a rank-ordered list (henceforth, ROL) by designating their preference rankings overschools. In 2014-15, the average student had about 30 choice options. There is no list-length
restriction, and students can list as many schools as they like (an example of an ROL is in
Appendix A). The city uses the student-proposing deferred acceptance (SPDA) algorithm,
which takes students’ applications, schools’ ranking over students, and the number of seats asmain inputs and produces at most one assignment for each student (Gale and Shapley, 1962).8
Schools rank students by pre-announced admission rules, which consist of three layers.
The first is eligibility, which determines students’ school choice sets. If a student is not eligible
for a program, she is never considered by the program, even when there are remaining seats.
Second, eligible applicants are classified into a small number of priority groups. A program
considers all students in the higher-priority group before considering any student in a lower
priority group. Henceforth, we use the term “priorities" to refer to eligibility and priority
groups, if not specified. Lastly, tie-breaking rules determine which students to admit among
applicants of the same priority group. Some programs use a nonrandom tie-breaker, which
is a school-specific function of the student’s previous year’s GPA, statewide standardized testscores, and punctuality. The rest use a lottery system in which each student receives one lottery
7In 2002, Chapter 91 (Bill A.11627/S.7456-B) was enacted to reorganize the education system and has
established centralized power. Since then, the public education system has been governed by the Panel for
Educational Policy (PEP), which has 15 members; 9 of which are nominated by the mayor. The citywide school
choice system was introduced in 2004 as part of this effort (Abdulkadiroğlu, Pathak, and Roth, 2005).
890% of students are assigned to a program on their list. The rest are matched to their fall-back option, which
is usually a school in their attendance zone. See Appendix A for details on the timeline and SPDA.

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number that applies to all such programs. See Appendix A for more details on the admissions
rules.
Figure 1: Geographic Divisions

Note: The city is divided into 5 boroughs (=counties), and further into 32 school districts and 300 middle school attendance zones.

Students’ residential locations are the main criterion for the eligibility and priority ofschools. Figure 1 depicts different levels of geographic subdivisions that determine locationbased admissions rules. The city is split into 5 boroughs, 32 Community School Districts(districts, henceforth), and more than 300 attendance zones.
Depending on their eligibility criteria, middle schools are classified into zoned programs,
district programs, borough programs, and citywide programs. A student’s residence or thelocation of her elementary school decides her eligibility for each type of school.9 Of 669
programs in academic year 2014-2015, 14 were citywide programs, 27 were borough programs,
478 were district programs, and the rest (150) were zoned programs. Schools can further assign
priority based on finer geographic divisions. For instance, 81 of 478 district programs gave top
priority to students from a particular attendance zone.

2.2

Data

Student and School Data Student-level data from the NYC Department of Education (DOE)
cover middle school applicants in academic year 2014-15. The data have two crucial components for the purpose of this paper—students’ school applications and residential Census
block. The data also contain students’ enrollment decisions, demographic characteristics, and987% of 2014-15 middle school applicants attended an elementary school in their residential district.

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statewide standardized test scores.10
We construct school characteristics by digitizing the Directory of Public Middle Schools.11
It covers each program’s admissions criteria, address, performance measures, previous year’s
capacity, and number of applicants. Students, parents, and guidance counselors use this as
their primary information source during the middle school application process (Sattin-Bajaj,
Jennings, Corcoran, Baker-Smith, and Hailey, 2018). We augment this data by adding thenumber of crime incidents of different categories in each school building from a NYC Police
Department’s School Safety Report.
Housing Cost and Structure Housing cost and housing characteristics are from the NYC
Department of Finance’s (DOF) Rolling Sales files. The data include the exact address of eachsold property, which is granular enough for us to observe on which side of a school district
boundary the property is located. We describe the cleaning process of the DOF Rolling Sales
files in detail in Appendix B.
Figure 2: Main Variables by District(a) School Test Score(b) Sales Price ($1,000)(c) Minority Share in Schools

Note: In panels (a) and (c), we take the average of the variables across schools within each district. The school test score is the average NYS
standardized test scores of enrolled students. In panel (b), we present the average unit sales price of residential properties in each district.

Amenities of Residential Location We construct location amenities from various sources.
Land use comes from the Primary Land Use Tax Lot Output. We also obtain consumption
amenities such as the number of cafes from business licenses published by NYC Consumer andWorker Protection. Next, we collect information on bus stops, metro stations, and park areasusing NYC OpenData GIS data files. We aggregate variables to Census block level. Finally,
10We focus on academic year 2014-15 because students can list only up to 12 middle schools in more recent
years. With this list length restriction, students have the incentive to list less preferred schools with higher
admissions chances (see Section 4).
11The city began publishing a digitized version in academic year 2017-18.

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the demographic composition of each Census block group, such as ethnicity, age, education,and income, comes from the American Community Survey (ACS) 5-year estimates.
Figure 2 presents the average characteristics for each district, which demonstrates a strong
correlation among school achievement, housing cost, and share of minorities in schools.
Summary statistics of main variables are in Table B.2.

3

Motivating Data Pattern

3.1

Effect of Admissions Probability on Residential Sorting
This section presents evidence that households choose where to live by considering

location-based admissions probabilities. Specifically, we show that locations with higher admissions chances to high-achieving schools have greater number of households with middle
school applicants and higher housing costs. This makes the main motivation to model endogenous location choices under centralized school choice. Moreover, we show that these locations
also have a lower minority share among households with middle school applicants, which
implies that households have heterogeneous rates of substitution between housing cost andhigher admissions chances to high-achieving schools. The main challenge to credibly showthese patterns is that locations with higher admissions chances to high-achieving schools may
have amenities unobserved to the econometrician but observed by and desirable to households,
such as a well-kept playground.
To this end, we adopt a boundary discontinuity design (BDD) (Black, 1999; Bayer, Ferreira,and McMillan, 2007). Ideally, we would compare two locations with the same amenities butwith different admissions probabilities to schools. BDD mimics the ideal design by comparing
locations that are within a narrow buffer around a school district boundary but on opposite
sides. The identification assumption is that unobserved amenities are as good as random
within a narrow buffer around a boundary. This assumption likely holds if other amenities arecontinuous in geography.12
We consider a narrow buffer that covers locations within 0.25 miles from a border at whicha pair of school districts meet. Figure 5 illustrates this idea. Tables in Appendix B present
estimates with a 0.2-mile buffer. Table B.2 presents summary statistics of student, housing, and12We do not apply BDD on attendance zone boundaries, because there is a concern about these boundaries’
being determined by residents themselves. School district boundaries can be redrawn only every 10 years, andthe decision is made at city level (New York State Law 2590-B). Meanwhile, attendance zone boundaries can beredrawn every year by the district council, whose members include parents and representative students. We still
consider that admissions probability chances vary across attendance zones in the model estimation (Section 5).

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Census block group characteristics of all sample in comparison to sample included in the BDD
analysis. The differences in characteristics largely come from the fact that we exclude Staten
Island since it consists of one school district. For example, Staten Island has larger numberof White student, thus BDD sample has smaller share of White students (8.5%) than the fullsample (12.5%).
The baseline regression is as follows.
yi = βQd(i)|{z}+θb(i) + f (ri ) + id .

(1)

district school quality

The unit of observation i is a housing transaction record when yi is the log house sales price.
The unit of observation i is a Census block when yi is the number and characteristics of
middle-school-applying residents in the block. b(i) is the boundary region fixed effect in which
i is located. f (ri ) is a local cubic control for distance to the boundary b(i), which we allow todiffer by whether the district in which i is included has higher school quality than the bordering
district.
Qd(i) is district school quality, measured by the mean NYS standardized test score of students
enrolled in middle schools (previous cohorts) in the district. In the model estimation, we
consider multidimensional school “quality" measures and allow students to have heterogeneous
preferences over measures. In this section, we use a one-dimensional measure for simplicity.
Our choice of the mean test score is motivated by Abdulkadiroğlu, Pathak, Schellenberg, andWalters (2020)’s finding that parents of high school students in NYC do not value school
effectiveness beyond the average test scores of students enrolled in a school.
The identification assumption is unlikely to hold if school district boundaries were drawn todivide already divided neighborhoods; even if they were exogenously drawn in the beginning,location amenities might have evolved differently over time on opposite sides of a boundary(Baum-Snow and Ferreira, 2015). We do two things to address these and to render causal
interpretation of our estimates more plausible (Bayer, Ferreira, and McMillan, 2007; Kulka,
2019; Zheng, 2022).
First, we drop boundaries in which locations on opposite sides are likely to differ in accessto amenities other than schools. Thus, we exclude boundaries aligned with a river, creek,park, highway, or borough boundary. Second, in Appendix B, we show that neither housing
characteristics nor urban amenities change sharply at school district boundaries, which suggeststhat the identification assumption is plausible in our context.

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Figure 3: Estimated Effects of District School Quality on Residential Sorting

Note: The figure depicts the estimates (dots) and 95% confidence intervals (lines) of the coefficients of district school quality on various
outcomes (β in Equation 1). The dependent variable in each panel is as follows (clockwise): (1) the mean score of the schools middle-schoolapplying residents in a Census block are assigned to, (2) the log sales price of a residential unit, (3) the number of middle-school-applying
residents in a Census block, and (4) the share of Black and Hispanic applicants among those residents. In all panels, we plot the coefficient froma simple BDD specification (Equation 1) and coefficients from specifications that control for other variables. In the top left panel, we control for
middle-school-applying residents’ ethnicity, FRL status, and test score. In the rest of the panels, we sequentially add housing characteristics,
neighborhood characteristics, and urban amenities. Standard errors are clustered at school district level. Housing characteristics include thespace of the unit, land use of the tax lot, number of floors, age, renovation status, and storage area of the building, all of which we interact
with a dummy if the property is coop. Neighbor characteristics include % minority, median household income, % college-or-more-educated,and median commuting time to work at Census block group. Urban amenities include the number of bus stops, subway stations, laundries,
cafes, and crime incidents of different categories at Census block.

Figure 3 presents estimates β̂ for various outcomes. Tables in Appendix B present coefficients plotted in Figure 3. For each outcome, we start from a simple BDD specification(Equation 1). Then we present coefficients from specifications where we control for various
covariates. In each panel, the coefficient from our preferred specification is in the rightmost.
District school quality increases the quality of schools to which residents are assigned
The top left panel of Figure 3 reports β̂ for the mean score of schools to which middleschool-applying residents in a Census block are assigned. A one standard-deviation increase in
district school quality increases assigned schools’ test scores of residents by 0.26 student-levelstandard-deviation (p-value < 0.01); we control for the resident’s ethnicity, FRL status, andtest score, to absorb differences in school applications and admissions probabilities explainedby applicants’ observable characteristics. This result implies that school district boundaries
determine admissions probabilities to high-achieving schools, which establishes the first stageof the BDD.

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District school quality increases housing prices The top right panel of Figure 3 reportsβ̂ for the log sales price of a residential unit. Including this panel, we plot coefficients from
specifications where we sequentially add housing characteristics, neighbor characteristics, andurban amenities for the rest of the panels. Given that housing characteristics and urban amenities
do not change at a boundary (Appendix B), we control for those to increase the precision of
our estimates. Meanwhile, we control for neighbor characteristics to account for the fact thathouseholds might have preferences over neighbors’ ethnicity or median income. We interpretthe estimate from a model with full controls to describe the effect of district school quality.
A one standard-deviation increase in district school quality increases housing sales priceby 10% (p-value < 0.05). This implies that there is a higher demand for locations with higher
admissions probability to better-performing schools.
We present β̂ for house value and median gross rent from the ACS 5-year estimatesin Appendix B. Estimates are 5.8% for both house value and median rent, although theestimate is only significant for median rent (p-value < 0.1). While sold properties might not
be representative of all properties, we prefer sales prices to these two alternatives because the
ACS 5-year estimates are at Census block group level, which is too coarse to study a changeof housing costs at boundaries. In Appendix B, we explain how we use the distribution of total
population and houses across Census blocks within each block group to weigh Census blockgroups in obtaining β̂.
District school quality attracts households with middle school applicants The bottom
left panel of Figure 3 reports β̂ for the number of middle school applicants residing in a Census
block. A one standard-deviation increase in district school quality increases the number of
middle-school-applying residents, with β̂ = 0.79 (p-value < 0.01). An average Census blockhas 3.5 middle-school-applying residents, and thus this is a 22% increase from the average.
This result is robust to controlling for the total number of population in Census Block Group(β̂ = 0.81, p-value < 0.01) from the ACS 5-year estimate. Thus, we exclude an explanationthat Census blocks with higher district school quality have a greater number of householdswith middle school applicants merely because those blocks have more houses. Estimates arepresented in Table B.5
District school quality attracts non-minority households more The two bottom panelsof Figure 3 report β̂ for the share of Black and Hispanic applicants among middle-schoolapplying residents in a Census block. A one standard-deviation increase in district school
12Electronic copy available at: https://ssrn.com/abstract=4452763

quality decreases the share of minority applicants by 6 percentage points (p-value < 0.01).
An average Census block has 62% Black or Hispanic residents among middle-school-applying
residents, and thus this is a 10% decrease from the average.

3.2

The Role of Commuting Distance in School Applications
Next, we show that while students apply to geographically proximate schools, the patterns

are heterogeneous by students’ characteristics and by the achievement level of schools near
their residential locations. We run the following linear probability model:
100 ∗ 1(T op3)ij = αd`i j + d`i j Zi β + δJi + ij .

(2)

1(Top3)ij is an indicator for whether student i lists school j in her top three choices. j is aschool for which student i is eligible. We multiply 1(Top3)ij by 100 to interpret coefficients aspercentage point changes. d`i j is the driving distance in miles between school j and student i’s
residential census block `i . Zi is a vector of student characteristics. α represents the association
between distance to a school and the propensity of students to list the school as their top choice.
β shows how that association changes by students’ characteristics. To account for the fact thatthe probability of choosing a specific school as the top choice mechanically decreases when
the number of eligible options increases, we control for the total number of schools for which
i is eligible (Ji ). We cluster standard errors at the student level.
Table 1: Commuting Distances and the Propensity of Listing as Top 3d`i j
d`i j × 1(M inority)i

(1)

(2)

-2.460(0.013)0.262(0.011)-2.411(0.012)0.083(0.011)-0.286(0.007)
1,745,513
0.063

d`i j × Quality of the three closest schoolsi
N
R2
Dep. var mean

1,745,513
0.062
7.895

Note: The dependent variable is a dummy if student i listed school j as one of their top three choices, multiplied by 100 for ease ofinterpretation. Pairs of a student and an eligible school within 10 miles from the student’s residential Census block are included. The fastest
driving distance between a school and a Census block is calculated using Open Route Services. A student is a minority if she is Black or
Hispanic. Column (2) controls for the mean test score of the three closest schools from i’s residential Census block `i . All columns control
for the total number of schools a student is eligible for and the interaction of distance and student’s standardized test scores. Standard errors
in parentheses are clustered at the student level.

Columns (1) and (2) in Table 1 demonstrate that students are 2.4 percentage points lesslikely to rank a school that is 1 mile farther away as their top 3 choices (p-value < 0.01).
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Minority students seem to be less responsive to distance in column (1) (β = 0.262, p-value< 0.01). In column (2), we further control for the mean test score of the three closest schools
from student i’s residential Census block. Students are even less likely to apply to schools
farther away when nearby schools have higher quality (column (2), β = −0.286, p-value< 0.01). Importantly, controlling for the quality of nearby schools reduces the coefficient of
the minority dummy by two-thirds (β = 0.082, p-value < 0.01). This pattern, whereby students
from disadvantaged location travel farther to schools that are higher performing than schools intheir residential location, coincides with what has been reported in previous studies (Corcoran,2018).
Motivated by these patterns, we model households as considering not only commuting
distances but also other school characteristics. We also allow households to have heterogeneouscommuting costs.13

4

A Model of Location Choice, School Choice, and Enrollment Decision
We model households’ sequential decisions of residential locations, school applications,and enrollment decisions. Location choices affect school applications and assignments throughtwo channels. First, distances to schools vary by residential location, which affects students’school applications. Second, applicants are ranked based on location-based priority rules, andthus two students from different locations who are otherwise similar face different admissions
probabilities. Households take these two channels into account when choosing which location
to reside in.
The model is guided by two key parameters. The first is access-to-school preference αu . It is
the weight households put on access-to-school utility that captures both commuting distances
and admissions probability, relative to other location amenities. It also governs the extent towhich counterfactual location-based priority rules would induce households to resort acrosslocations. The second is commuting cost β d , which affects to what extent students apply toschools that are farther away given their location choices as opposed to applying to only nearby
schools. Together with αu , it shapes the spatial distribution of households; for example, with
infinite commuting costs and strictly positive access-to-school utility, households would choose
13This pattern is not explained by the difference in the number of schools proximate to their residential location.
Students whose proximate schools are lower achieving have more schools proximate to their residential location.
We present a histogram showing this result in Appendix B.

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locations closer to the schools they would apply to.
Next, we discuss our model in greater detail.

4.1

Household Preference, School Assignment, and Timeline

Household Heterogeneity and Preferences We use “household, applicant," and “student”
interchangeably and model the unitary decision of a household. Household i is heterogeneousin both observable and unobservable (to the researcher) characteristics, denoted as Zi and
γi , respectively. Observable characteristics Zi include students’ race/ethnicity, poverty status(proxied by free and reduced-price lunch eligibility), and test score prior to their middle school
application. Unobserved (discrete) type γi (Heckman and Singer, 1984) captures the fact that
school characteristics may be valued differently by observably similar households.
i’s utility from living in location ` and attending school j isV (`; η )| i {z i`}

utility from location+ αuUi (j, `; εij )| {z }.

(3)

utility from school × location

We parameterize each component as follows:Vi (`; ηi` ) =

W`0
|{z}

αiW +

housing costlocation char.

where αik = αk0 +p` αip +
|{z}

ξ`
|{z}

unobserved amenities

+

ηi` ,
|{z}

(4)

i.i.d. EVT1

Zi0 αkz , for k = p, W .
|{z}student char.

Ui (j, `; εij ) =Xj0 βiX +|{z}school char.

where βik = β k0 +d`j βid| {z }+kz

γikcommuting cost

Zi0

β

|{z}

+

εij ,
|{z}

(5)

i.i.d. EVT1

, for k = d, X .

|{z}student char.

unobserved typeW` is a vector of location observable characteristics, p` is the housing cost, Zi is the vectorof student observable characteristics, Xj is the vector of school characteristics, and d`j is thefastest driving distance between location ` and school j.
In addition, ξ` represents unobservable location amenities that are shared across households.
γi = (γiX , γid ) is the vector of student i’s unobserved tastes over school characteristics anddistance to schools, and ηi` , εij are idiosyncratic preferences shocks over locations and schools.
ηi` and εij are mutually independent and follow i.i.d extreme value type 1 distribution.
i’s utility from living in location ` and attending an outside option ϑ is15

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Vi (`; ηi` )| {z }Uiϑ (ϑ; εiϑ )| {z }+

utility from location.

(6)

utility from outside option

We consider two outside options, non-public schools ϑnp and public charter schools ϑc .14
Non-public schools ϑnp includes private schools, homeschooling, or moving out of NYC.15
We further allow students to have heterogeneous preferences for outside options based on theirobservable characteristics. For example, non-minority students might assign higher value tonon-public schools than their peers. Mathematically,Ui (ϑ; εiϑ ) = βiϑ +

εiϑ
|{z}

(7)

i.i.d. EVT1βiϑ = β0ϑ +Zi βzϑ , where ϑ = ϑc , ϑnp .
|{z}student char.

εiϑ follows an i.i.d extreme value type 1 distribution.
School Assignment Next, we briefly discuss how schools rank applicants. As discussed in
Section 2, priority groups are largely determined by students’ residential location `. The tiebreaker within priority groups is either a lottery or a school-specific aggregation of students’pre-middle-school academic measures. We capture programs’ ranking over students with apriority score, cij (`). This is the sum of an integer gij (`) that corresponds to priority groups
and decimal point τij ∈ [0, 1] that corresponds to tie-breakers.16 Tie-breakers are either schoolspecific aggregation of students’ academic measures or random lottery numbers ρ. The higher
a student’s cij (`), the higher her admissions chance.
How priority groups are determined is public information. When it comes to schoolspecific aggregation of students’ academic measures, we know which inputs a school uses—
such as GPA, statewide test score, and punctuality—and the aggregated scores among its
applicants. However, the exact function that schools use to construct these measures are
unknown. We estimate school-specific linear functions of measures using a latent modeland assume households form expectations (ĉij (`)) in the same way; details are in Appendix C.
Given students’ rank-ordered list and priority scores, the city assigns students to at most one
14Public charter schools are not parts of the centralized school choice system and they have separate admissions
processes.
15Although we observe that a student is not enrolled in a public school in NYC, we do not know which
non-public option a student chooses.
16For a program j with three priority groups, students in the first priority group have gij = 3. The second
and the third priority groups’ students have gij = 2 and 1, respectively. If a student is ineligible for program j,gij = −∞

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program using the SPDA algorithm. See Appendix A for a detailed explanation of the SPDA
procedure.
Cutoff for each school c̄j is given as the min{cij : i ∈ Ij }, where Ij is a set of students
admitted to program j if the capacity of j is filled, and −∞ otherwise. We assume the market islarge enough (70,000 students) that an individual student considers the cutoffs as given (Fack,
Grenet, and He, 2019; Agarwal and Somaini, 2020; Calsamiglia, Fu, and Güell, 2020).
Timing Figure 4 summarizes the timeline of the model and households’ information ateach stage. Households make choices on blue dots. They have full information on their own
observable characteristics (Zi ) and those of schools (Xj ) and residential locations (W` , p` ), as
well as locations’ unobserved amenities (ξ` ), throughout all stages.
Figure 4: Timeline and Information Set
Location (`)

School (j)Assignment

Enrollment (ϑ)γiηi`
pr(ĉij (`) ≥ c̄j )

εij

ji

εiϑ

Note: Households make choices on blue dots. They have full information on their own observable characteristics (Zi ) and those of schools(Xj ) and residential location (W` , p` ), as well as the shared neighborhood unobserved amenities (ξ` ) throughout all stages. γi is the vectorof student i’s unobserved tastes over school characteristics. ηi` , εij , and εiϑ are idiosyncratic preferences shocks over locations, schools, andoutside options respectively. ji is the assignment result. pr(ĉij (`) ≥ c̄j ) is the predicted admissions probability.

Households know their unobserved tastes over school characteristics (γi ) from the beginningof the location choice stage, so these unobserved preferences influence their residential choice.
This becomes a source of bias in estimating commuting costs if we estimate school preference
without correcting for the selection into locations. For example, a household that values school
safety more than other observably similar households would choose locations that assure higher
location-based admissions probability for safer schools. This household would apply to only
nearby schools because it already lives near its safer schools, but a model that does not correct
this selection would mistakenly justify such behavior with a high commuting cost. Together
with unobserved taste, households observe their idiosyncratic preference shocks over locations(ηi` ) and form predictions on admissions probabilities to schools, pr(ĉij (`) ≥ c̄j ).17
17The admissions cutoffs c̄j households use at this stage are calculated using observed school application,
which is a function of students’ preference shocks over programs εij that are realized in the next period. The large
market assumption establishes the internal consistency—i.e., the admissions cutoffs are determined in the large
market, and c̄j are consistent estimators of those.

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At the beginning of the school choice stage, student i observes her preference shock overprograms, εij . To sum up, households know know unobserved tastes γi but not idiosyncratic
shock εij when deciding where to live. Once the assignment is realized, they know the exact
assignment result ji . The preference shock over outside options εiϑ is realized at the enrollmentdecision stage to rationalize the fact that 7.74% of students assigned to their top choice enroll
in outside options. εiϑ is either an income shock that affects households’ affordability for
private schools (Calsamiglia, Fu, and Güell, 2020) or charter school lotteries realized after
the application stage is complete. The idiosyncratic shock εij over assigned school does notchange in the enrollment-decision stage.18

4.2

Household’s Problem
Next, we describe the household’s problem corresponding to the blue dots in Figure 4,

which we solve backward.
Stage 4: Enrollment Residential locations and assignment results are set in previous stages.
Given those, students decide whether to enroll in their assigned school, or the non-public
option, or a public charter school to maximize their utility:Ui∗ (`i ) ≡ max Ui (ji (`i ), `i ; εij ), Uiϑ (ϑnp ; εiϑ ), Uiϑ (ϑc ; εiϑ ) ,(8)

where ji is the assignment outcome from the assignment stage and `i is the location chosen in
the previous stage.
Stage 3: Assignment Students are passive as their assigned school ji is determined by theirpriority score at each program and admissions cutoffs, given their ROLs from the previous
stage. Mathematically,ji (`i ) ≡ f (ROLi (`i ), cij (`i ; ρ) , c̄j ) .
| {z } | {z } |{z}(9)application list priority score cutoff vector

Stage 2: Application We assume that students submit an ROL following their true preference
order up to their fallback options. The fallback option is the school a student is assigned
to when rejected by all programs on her ROL—either pre-designated zoned school or an
18An alternative model is such that students draw new shocks on assigned schools at the enrollment stage andthe final shock is a weighted sum of the old and the new shock. However, it is impossible to tell to what extent theidiosyncratic shock εij is time-invariant, since all other choices from the application stage are forgone except forji . That is, this alternative model would generate the same school application list and enrollment decision. Suchmodel would have been possible if students had more than two options that are relevant at both the application
stage and enrollment stage.

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undersubscribed school in her school district. The middle school choice system in NYC uses
the Deferred Acceptance algorithm, in which students can list as many schools as they want,
which jointly renders truth-telling—ranking schools based on one’s true preference order—aweakly dominant strategy (Gale and Shapley, 1962).
Stage 1: Residential Location Choice Given the solution in the subsequent period, household i chooses the location that solvesmax`

Vi (`; ηi` )| {z }

αu Eεij ,ρ,εiϑ Ui∗ (`)
|
{z
}

+

utility from location

,

(10)

expected utility from enrolled school given location

where Ui∗ (`) is the utility from enrolled school (Equation 8). This is is location dependent
because locations decide commuting costs and admissions probabilities, and as a result, which
school student i enrolls in. Households form an expectation over Ui∗ (`), since they do not know
their idiosyncratic preference shocks over schools and outside options (εij and εiϑ ) as well as
their lottery number ρi .

4.3

Equilibrium
To define the school assignment equilibrium, we extend the supply and demand character-ization of Azevedo and Leshno (2016).
Definition 4.1. An equilibrium is a pair of decisions {`i , ROLi } for each i and a vector ofadmissions cutoffs {c̄j }Jj=1 where
1. Given cutoffs {c̄j }Jj=1 and the first-stage choice `i , {ROLi } is the school application list
based on i’s true preference order up to their fallback option.
2. Given {c̄j }Jj=1 , `i solves i’s problem Equation 10 for each i.
3. Admissions cutoffs clear the market; i.e., Sj ≥ Dj ({c̄j 0 }Jj0 =1 ) for each j ∈ J. Sj is capacityof school j and Dj ({c̄j 0 }Jj0 =1 ) is the aggregate demand for school j given the cutoffs {c̄j 0 }Jj0 =1 .
Aggregate demand for schools can be further simplified by using the fact that when students
are truth-telling, the realized matching is stable—i.e., each student is matched to her favorite
feasible school. Details are in Appendix C.

4.4

Discussion

Truth-telling We consider truth-telling a reasonable assumption in our context. There are
well-known factors that make this assumption less plausible: (1) list-length restriction (2)limited consideration set (3) application cost. First, there is no list-length restriction in our
19Electronic copy available at: https://ssrn.com/abstract=4452763

setting. In a setting with this restriction (Luflade, 2018; Son, 2020), truth-telling is no longer aweakly dominant strategy when students really want to be assigned to some school (Haeringer
and Klijn, 2009).19 Second, students are given a customized list of all eligible schools with anaverage of about 30 schools. This stands in contrast to settings in which they have to constructa consideration set out of hundreds of options, where they are unlikely to consider all optionsin their choice set when deciding which school to apply to (Ajayi and Sidibe, 2020; Son,
2020). Third, both monetary and psychological application cost are relatively low. There is no
application fee. Also, they can add one more school to their application list just by marking the
ranking to the customized list that they received.
Even though assuming truth-telling is reasonable in our context, it is still a weakly dominant
strategy (Artemov, Che, and He, 2017; Che, Hahm, and He, 2022). For example, “skipping theimpossible” yields the same assignment results, and detecting impossible options is feasible
given that each school’s capacity and the number of previous year’s applicants are public
information. Instead of imposing truth-telling assumption, one can estimate the model basedon stability (Fack, Grenet, and He, 2019; Agarwal and Somaini, 2020; Hahm and Park, 2022),
which rely on assignment results rather than ranking strategies, in Appendix D. While imposing
a weaker assumption, this estimation strategy loses the precision of estimates by focusing only
on the assignment outcome instead of the full list.
Asymmetry in Utility from Location and School In our model, utility from locations
includes unobserved amenities shared by households but no household-specific unobserved
tastes. Meanwhile, utility from schools includes household-specific unobserved taste but nounobserved quality shared by households (Equation 4).
These modeling choices are largely driven by the motivation to obtain unbiased estimates of
two key parameters—access-to-school preference αu and commuting cost β d —while keeping
the estimation tractable. Access-to-school preference αu will be biased if unobserved location
amenities are correlated with access-to-school utility.20 Meanwhile, households’ sorting into
locations based on household-specific unobserved school tastes biases commuting cost β d .21
19List-length restrictions were introduced in NYC’s middle school choice in years more recent than our setting.
20Moreover, we lack variation to identify household-specific unobserved tastes. For example, Bayer, McMillan,
Murphy, and Timmins (2016) sets up a dynamic location choice model and uses the panel structure of the data toidentify households’ unobserved attachment to a specific location. Or, Barwick, Li, Waxman, Wu, and Xia (2021)
constructs household-specific location choice set by leveraging that they observe when each household bought
the house.
21For example, Allende (2019) and Abdulkadiroğlu, Pathak, Schellenberg, and Walters (2020) estimate unobserved school quality to obtain causal estimates on how much households value peer quality aside from other

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Utility from the Outside Option Utility from the outside option is not a function of school
characteristics because of a lack of data on schools outside the system, especially non-public
options. It can also be a function of location. By abstracting away from it, our location
demand estimates might capture the unequal geographic distribution of outside schooling
options. For example, locations with higher median household income would have moreprivate schools nearby, and the estimated preference over neighbors’ income in Section 5 might
capture households’ preference over geographic proximity to non-public options. We assume
the geographical distribution of outside options does not change under the counterfactual
scenario.22

5

Estimation Procedure and Results

5.1

Identification of Key Parameters
We discuss the identification of the two key parameters of the model: access-to-schoolpreference αu and commuting cost β d . We also discuss the identification of the price coefficientαp .
The biggest concern regarding credibly identifying access-to-school preference αu is todistinguish it from preferences on unobserved location amenities (ξ` ). To this end, we use
variation from our boundary discontinuity design. Similar to Section 3, the identification
assumption is that the unobserved amenities are as good as random within a narrow buffer
around a boundary. Meanwhile, the access-to-school utility sharply changes at a boundary, since
70% of schools give eligibility or higher priority to students from the same school district(Section 2), and there is marked heterogeneity in school characteristics across districts. So
intuitively, seeing that households are more likely to live in the side of a district boundary withhigher admissions probabilities to schools whose characteristics are more desirable (Section 3)
would lead to a larger value of αu .
To obtain an unbiased estimate of commuting cost βid , we need to account for the fact thathouseholds choose locations based on their unobserved school demand γi . When students apply
to nearby schools, we need to identify to what extent this is explained by commuting costs as
opposed to households’ residential sorting in order to be assigned higher priority by the schools
they prefer. If residential sorting arises only from households’ observed characteristics, we canschool factors such as the building quality.
22See, for example, Dinerstein and Smith (2021) to see how private schools’ entry and exit decisions can beaffected by public school policies.

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obtain unbiased commuting costs by controlling for those characteristics in estimating schoolpreference without fully modeling residential sorting. Thus, previous papers have assumed
that idiosyncratic preference shocks and unobserved tastes over schools are independent of
distances to school conditional on student observable characteristics—i.e., (εij , γi ) ⊥ dli j |Zi(e.g., Agarwal and Somaini, 2018; Laverde, 2020). By modeling and jointly estimating locationand school choice, we relax this assumption and allow an individual’s unobserved type γi tobe correlated with distances to schools—i.e., εij ⊥ dli j |Zi .
Moreover, we need to identify unobserved type γi to correct for households’ selection into
locations based on it. Whereas the different applications of two observably identical students
can be explained by either unobserved tastes (γi ) or idiosyncratic preference shock (εij ), thesetwo components can be disentangled for two reasons. First, unobserved tastes are studentspecific but the preference shock is independent across schools within each student. We observe
students’ full application lists. To what extent characteristics among the schools on a student’s
ROL are correlated helps to identify the unobserved type separate from the idiosyncratic
shock (Bhat, 2000; Berry, Levinsohn, and Pakes, 2004). Second, while households choose
residential locations knowing their unobserved taste, the idiosyncratic shock is realized after
location choice. Among observably similar students, variation across residents of different
locations pins down unobserved taste, while variation among residents from the same location
are captured by the idiosyncratic preference shock.23
The final parameter to identify is the price coefficient, αp , since housing cost (p` ) is likely
to be correlated with location unobserved amenities (ξ` ). We instrument for housing cost of alocation with the land use of other locations that are (1) 2 miles away from the location (2)but within 3 miles (Bayer, Ferreira, and McMillan, 2007; Barwick, Li, Waxman, Wu, and Xia,
2021; Davis, Gregory, Hartley, and Tan, 2021). Given a location, other locations that are faraway are unlikely to share its unobserved amenities (exclusion restriction). However, the land
use of other locations that are near enough to the location could affect its housing cost if people
decide where to live among those locations (relevance restriction).

5.2

Estimation Procedure

Defining Neighborhood With over 38,000 Census blocks in NYC, estimating the model atblock level might decrease the precision of estimates by having too many parameters relative
23In principle, we can even divide unobserved taste into two components: individual-specific and realized at
the location choice stage versus individual-specific and realized at the application stage.

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to the data (Dingel and Tintelnot, 2020).24 But we still aim to estimate the mean utility oflocation (δ` = W` αW 0 + p` αp0 + ξ` ) to account for the endogeneity of access-to-school andhousing price (Berry, Levinsohn, and Pakes, 1995). To this end, we define neighborhoods—aunit of residential location—by merging Census blocks.
Two Census blocks are in the same neighborhood if they satisfy the following criteria.
First, they are in the same cluster when we group Census blocks based on the distance to allschools using k-mean clustering, with k of 1,000. Second, they share the same location-based
admissions probability to all schools. Third, they are either both within the 0.2-mile buffer ofa school district boundary or neither is.
Figure 5: Defined Neighborhoods

Note: We aggregate 38,798 Census blocks into 2,778 neighborhoods using the procedure described in Subsection 5.2. The darker shaded
neighborhoods along school district boundaries (in orange) are those to which we apply BDD to identify access-to-school preference αu .

With this procedure, we aggregate 38,798 Census blocks into 2,778 neighborhoods. In
comparison, there are 2,165 Census tracts in NYC. Figure 5 shows the map of neighborhoods
defined by this procedure. The darker shaded neighborhoods along school district boundaries(in orange) are those to which we apply BDD to identify access-to-school preference αu .
Joint Estimation We aim to jointly estimate all stages of the model to address the selection
into locations. Full information maximum likelihood (FIML) involves calculating a large
Hessian matrix (Train, 2009), which renders the computation infeasible. See Appendix D for
details on FIML.
To circumvent the computational burden, we employ the expectation-maximization algo24There are also many Census blocks with no middle-school-applying residents or housing transaction records
during the time of the study.

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rithm with a sequential maximization step (ESM) proposed by Arcidiacono and Jones (2003).25
In summary, the idea is to (1) reformulate the full information likelihood function into additive
separable terms, each of which represents the likelihood of each stage; (2) update estimates ofeach stage; and (3) iterate the procedure until convergence.
The expectation function (reformulation) for the household i is a sum of the log of thelikelihood for each stage weighted by the conditional probability of its being each unobserved
type, given the school application and location choice observed in the data. Then we take thesum across i’s expectation function.
E(p, γ, θ|q̂, γ̂, θ̂) = Σi Σk q(k|xi ; q̂, γ̂, θ̂)logqk(11)+ Σi Σk q(k|xi ; q̂, γ̂, θ̂)logP LC (xi ; θEC , θSC , θLC , γk )(12)

+ Σi Σk q(k|xi ; q̂, γ̂, θ̂)logP SC (xi ; θSC , γk )(13)

+ Σi Σk q(k|xi ; q̂, γ̂, θ̂)logP EC (xi ; θEC , θSC , γk ) .

(14)q(k|xi ; q̂, γ̂, θ̂) is the conditional probability of being type k given data xi , calculated using
Bayes’ rule. θLC , θSC , θEC are the set of location, school, and outside option preference
parameters, respectively. γi is the unobserved taste, and qk is the unconditional probability ofeach type k. θ = {(θLC , θSC , θEC ), {γk , qk }k } is the full set of parameters to be estimated.
P LC , P SC , P EC are the likelihood of location choice, school choice, and enrollment choice,respectively. Likelihood functions are presented in Appendix D.
Then we update the guess on each element of θ sequentially by maximizing each line of
the expectation function. Starting from an initial guess, we iterate the updating process until
the guess of θ converges. We used squared extrapolation methods (see Varadhan and Roland,2008) to make convergence faster. See Appendix D for the cookbook of the iteration process.
We update θSC , θEC , and γ using maximum likelihood estimation to obtain efficient estimates of γ by exploiting full information in application lists.
Meanwhile, we update θLC using method of moments estimation to deal with the endogeneity of price and access-to-school utility (Berry, Levinsohn, and Pakes, 1995). Locationpreference parameters include those that govern heterogeneous preferences (αW z , αpz ); common preferences (αW 0 , αp0 ); and the access-to-school preference αu . To estimate (αW z , αpz ),25Dempster, Laird, and Rubin (1977) and Train (2009) show that solving the EM algorithm is identical tosolving maximum likelihood estimation (MLE), and Arcidiacono and Jones (2003) prove the consistency of
estimates with a multi-stage model.

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we match the first-order condition of location choice likelihood P LC (presented in Appendix D)with respect to αW z and αpz ,Σi W`i zir| {z }= Σi Σk q(k|xi ; q̂, γ̂, θ̂)Σ` P LC (`; θ̂SC , θ̂EC , θLC , γ̂k )W` zir ,
|
{z
}

(15)

Σi p`i zir= Σi Σk q(k|xi ; q̂, γ̂, θ̂)Σ` P LC (`; θ̂SC , θ̂EC , θLC , γ̂k )p` zir ,
|
{z
}

(16)

cov. of W and z in the data

| {z }

cov. of p and z in the data

predicted cov. of W and z

predicted cov. of p and z

where z r is each element of observed household characteristics Z—e.g., the minority dummy.
To obtain the remaining parameters (αW 0 , αp0 ), αu , we first search the mean utility oflocation δ` = W` αW 0 + p` αp0 + ξ` that satisfies the first-order condition of P LC with respectto δ` , 26
Σi 1(`i = `) = Σi Σk q(k|xi ; q̂, γ̂, θ̂)P LC (`; θ̂SC , θ̂EC , θLC , γ̂k ), ∀` .
| {z } |{z}observed share(17)

predicted share

Finally, we get (αW 0 , αp0 , αu ) by targeting the following conditions:E(ξ` p̂IV` ) = 0(Price IV)E(ξ` 1(right side of BD)` |BD` , 1(` ∈ B(BD` ; 0.25mi.)) = 0 .

(BDD IV)

where p̂IV
` is a vector of other observed location characteristics W` and price IV. B(BD` ; 0.25)
is the buffer around each boundary BD with a radius of 0.25 mile. The procedure consists of
the outer loop that searches parameters that satisfy Equation 15, 16, Price IV, and BDD IV andthe inner loop that searches δ` that satisfy Equation 17. We present price IV regression resultsin Appendix D.

5.3

Estimation Results

Demand Estimates Estimates in Table 2 have expected signs. Households prefer locations
with higher access-to-school utility (EU, 1.419) and lower housing costs. They prefer schoolsthat are higher achieving and safer. There is homophily (i.e., preference for one’s same raceand FRL status) in both location and school preferences.
Willingness to pay To better interpret estimates, we calculate households’ willingness topay (WTP) for school and location characteristics in Table 3. WTP for a one-unit increase in
26The process is accelerated by Newton’s nonlinear root-finding algorithm. We thank Jean-François Houde forsharing his code.

25

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Table 2: Demand Estimates
Main
Type1
Panel A: Neighborhood Demandlog(SalesPrice)-2.039(1.383)Frac. Black or Hisp.
-3.459(2.025)log(Med. HH Income)
2.848(1.991)
Med. Time to Work (hr)17.676(359.156)
Med. Time to Work2 (hr)-13.456(247.912)
EU
1.419(1.276)
Panel B: School Demand
Mean test score
0.121(0.052)
Frac. Black or Hisp.
-1.722(0.612)
Frac. FRL
-0.771(0.865)
Non-safety-0.059(0.008)Commuting Cost (mi.)
0.221(0.032)Prob.
0.352(0.127)
Panel C: Outside Option
Non-public-1.225(0.174)
Public Charter-3.767(0.256)

Type2

Type3

-

-

-

-

-

-

-

-

-

-

-

Additional Effects
Black/Hisp
FRL

Low-achieving--0.015(0.283)3.933(1.727)-0.161(1.545)-27.016(278.872)
17.269(191.749)--0.048(0.360)-0.119(1.671)-1.201(1.891)
13.242(402.963)-8.257(278.871)--0.035(0.272)0.208(1.672)-0.303(2.141)
15.394(474.360)-10.207(323.379)-

0.256(0.074)-0.501(1.118)-0.356(1.041)
0.003(0.012)1.085(0.054)0.631(0.083)0.187(0.469)0.159(5.442)
1.020(5.097)0.027(0.057)9.136(1.757)0.017(0.055)0.134(0.029)1.958(0.417)-0.540(0.429)0.018(0.010)-0.084(0.045)-0.117(0.281)
0.882(0.220)-0.010(0.021)--0.253(0.028)0.216(0.223)-0.162(0.213)-0.046(0.014)
-

-

-

-

-

-0.136(0.135)1.883(0.207)-0.800(0.125)0.173(0.152)-0.200(0.134)-0.059(0.124)

Note: Standard errors in parentheses are calculated from 75 bootstrapped samples. Columns are for students’ heterogeneity and rows arefor school and neighborhood characteristics. FRL stands for free or reduced-price lunch eligibility. The fastest driving distance to a school
is calculated using the Open Route Service. School non-safety measure is constructed by running a principal component analysis on crime
incidence of different categories at each school building. See Appendix D for details.
αW

location characteristics for household i is given by αip . WTP for a one-unit increase in schooliui`characteristics of all schools in a school district is the sum of ααp ∂EU
across school js in a∂Xjischool district of location `. This is household WTP to ensure the increase in characteristics of

26

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the assigned school27 and is a function of location; thus we take the average across locations.
We further convert WTPs in monetary terms by multiplying them by $1,366, the mean of
median gross monthly rent at the Census tract from 2014 5-year ACS estimates.
Table 3: Willingness to Pay
Std of Var.
Panel A: Neighborhood Characteristics
Frac. Black or Hisp.
0.341
log(Med. HH Income)
0.429
Med. Time to Work (min, daily)
11.52
Panel B: School Characteristics
Time to School (min, daily)
51
Mean test score
1.002
Frac. Black or Hisp.
0.28
Frac. FRL
0.173
Safety
4.228

p25

WTP
median

p75

-704
303
-236

32
379
-181

71
595
-114

-752
89
-355
-98
94

-384
154
-25
-83
121

-171
238
11
-53
149

Note: The unit of willingness to pay is the mean of median gross monthly rent at the Census tract from 2014 5-year ACS estimates, $1,366.
We use the standard deviation of distance to the assigned schools across students. For other school characteristics, we calculate the standard
deviation across schools. FRL stands for free or reduced lunch eligibility. The fastest driving time to school is calculated using the Open
Route Service. School safety is constructed by running a principal component analysis (PCA) on crime incidence of different categories at
each school building. Appendix D has more details on the PCA result.

For some characteristics, households uniformly agree on what makes a location or a school
more desirable. Both the 25th and 75th percentiles of households are willing to pay a positive
amount for an increase in the median income of neighbors, mean test score of schools, andsafety of schools.28 For other characteristics, there is marked heterogeneity in preferences.
For an increase in the minority share among neighbors or school peers, some households arewilling to pay a positive amount while others must be compensated to stay indifferent.29
Next, we present households’ WTP for a reduction in commuting time to school. Commuting time has been used as the numéraire in previous studies on public-school choice (Agarwal
Xi`
27 ∂EU∂Xj can be simplified to βi P robij (`)P rob(j ∈ Ji (`; ρi )) where P robij (`) is the probability of j’s being
the most preferred feasible option for student i when she lives in `. P rob(j ∈ Ji (`; ρi )) is the probability of jbeing i’s feasible choice when she lives in `.
28WTP for a one-standard-deviation increase in schools’ test score is 11.3%. Ours is slightly higher than therange reported by previous papers (3%-10%) that study households’ WTP for a test score increase in one school
such as a zoned school or a charter school (Black, 1999; Bayer, Ferreira, and McMillan, 2007; Zheng, 2022). In
contrast, we consider a test score increase for all schools in a district.
29Bayer, McMillan, Murphy, and Timmins (2016) estimate that for a 10-percentage-point increase in the fractionof White neighbors, an average White family in the San Francisco Bay Area is willing to pay $2,428 annually in2000 dollars from their dynamic location choice model, and $1,901 from their static model. Our estimate for asimilar scenario is $1,740. (= 0.7 × 704 × 12 × (0.1/0.34)), with a 0.7 adjustment to 2000 dollars using CPI(source: BLS CPI New York-Newark-Jersey City area)

27

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and Somaini, 2018), and we convert it into the monetary term using housing cost. A medianhousehold is willing to pay $19 (=384/20 days) per school day to reduce commuting to schoolby 50 minutes a day.
We view our WTP estimate to capture various challenges that middle school students
face during school commuting. For example, parents answer a survey by Sattin-Bajaj andJennings (2022) that safety on the journey to a school is a main consideration factor for school
application. Such concern of parents arises because many students commute to schools bythemselves by public transportation or on foot. Middle school students are eligible for school
bus service only in their first year if their schools offer any. Moreover, we calculate from the2017 National Household Travel Survey that at least 70% of students in our sample commute toschools without any adult accompanied.30 Finally, our WTP estimate is high to be interpreted
as the forgone earning of middle school students. Adult commuters’ value of commuting time
is known to be 50%-70% of their hourly wage (Parry and Small, 2009; Purevjav, 2022), andthe minimum wage in New York in 2015 was $9.
Overestimation of Commuting Cost Commuting cost is overestimated when we ignore
households’ residential sorting. We estimate a different version of the model without location
choice (estimates are presented in Appendix D), and find that commuting cost is overestimatedby 15% on average (mean βid is -0.97 in a model with both location and school choice, and-1.11 with only school choice). Figure 6 describes what leads the model without endogenous
location choice to overestimate commuting cost.
Panel (a) shows probabilities of types (qk ) from a model with both location and school
choice compared with those from a model with only school choice. In the latter, we overclassify students into the high commuting-cost type, since we rationalize households’ applying
to schools nearby as due only to high commuting costs, as opposed to households’ residential
sorting based on unobserved school taste γi . Panel (b) plots the mean probability of being
type 1 among residents across locations. For each student i, we calculate the probability of her
being type 1 based on how well her location and school choices can be justified by being type 1
relative to other types (Bayes’ rule). In the absence of residential sorting based on unobserved
type, the mean probability of being type 1 among residents of a location should be similar
across all locations. In contrast to this, some locations have zero type-1 students while others
30To be accurate, 70% of middle school students residing in the NY-NJ-PA area, which is the finest geography
available, commute to schools without any adult accompanied.

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Figure 6: Residential Sorting on γ and the Overestimation of Commuting Cost(a) Mean Probability of Unobserved Type(b) Location Sorting on Unobserved Type

Note: Panel (a) presents the probability of each unobserved type under each model. Panel (b) shows the mean probability of residents’ being
type 1 across residential locations.

have many type-1 students, which implies sorting based on unobserved type.31
Model Fit We simulate choices using our estimates to validate whether our model canreplicate the data patterns. To minimize the idiosyncrasies coming from preference shocks andthe lottery number, we present the average over 100 simulations.
Figure 7: Model Fit(a) Location Choice(b) School Choice(c) Enrollment

Note: We take the average over 100 simulations with draws of η, , and the lottery number. Moments include the mean observable characteristics
of chosen options and the correlation between students’ characteristics and those of their chosen options. In panel (b), we focus on students’first choice. In panel (c), we present the fraction of students who choose each outside option.

31There is also an idiosyncrasy coming from a finite sample. Figure D.4 compares the distribution from thedata and that from a simulation in which we allocate households randomly across locations.

29

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Figure 7 plots the simulated and observed moments from school and location choice.
Moments include the mean observable characteristics of chosen options and the correlation
between students’ characteristics and those of their chosen options. Unsurprisingly, our targeted
moments from location choice are well aligned with the 45-degree line. Meanwhile, even though
we do not target school choice moments, and rather estimate school preference parameters via
MLE, our simulated moments of school choices are close to data moments. Table D.14 presents
the numbers plotted in Figure 7.

6

Source of School Segregation
In this section, we use model estimates to identify the sources of school segregation. Even

with an extensive school choice system in place, NYC middle schools are highly segregated.32
There are also large differences in academic achievement across these segregated schools.
In the 2014-15 academic year, classmates of minority students (in their assigned schools)
had standardized test scores than were one standard-deviation lower than the classmates of
non-minority students. In this section, we explore which components of the model explain the
cross-racial gap in the test scores of students’ peers in their assigned schools
In Table 4, we investigate to what extent the cross-racial gap in test scores is explained by thefollowing components of the model: access-to-school preference, heterogeneity in preference
over location characteristics, and that over school characteristics. Column (1) in Table 4 presents
the cross-racial gap under the status quo; minority students attend schools with lower test scoresby one student-level standard deviation than their non-minority peers.
Table 4: Cross-racial Gap in Coassigned Peers’ Test Score(1)
Status Quo
Racial Gap-1.048

Access-to-school
Preference(2)(3)
Racial Gap Explained by:
Heterogeneous Preference over:
Other Location Characteristics
School Characteristics-0.312-0.466-0.182

Note: The cross-racial gap is the difference in test scores of the schools students attend for minority and non-minority students. We shut downeach channel for one household one at a time. In column (2), we impose αu = 0. In column (3), we impose αW Z = αpZ = 0, In column(4), we impose β XZ = β dZ = 0.

Next, columns (2) and (3) demonstrate that residential sorting is the main driver of school
32In terms of racial composition, 77% of Black and Hispanic students attend schools that enroll less than 10%of White students, while only 11% of White students and 43% of Asian students attend schools that enroll lessthan 10% of White students (Cohen, 2021).

30

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segregation, which is in line with past studies (Laverde, 2020; Monarrez, 2020). We further
break down what part of the gap is explained by residential sorting based on access to school(column (2)) versus sorting based on other location amenities (column (3)). Column (2)demonstrates that 31% of the gap observed in the data is explained by residential sorting basedon access-to-school utility. In this scenario, we shut down residential sorting based on accessto school (αu = 0). Thus, households choose locations as if they do not know that the locations
chosen determine commuting costs and location-based admissions probabilities to schools.
The cross-racial gap in this scenario comes from households’ heterogeneous preferences overlocation characteristics other than access to school and those over school characteristics.
In columns (3) and (4), we investigate the role of preference heterogeneity. Column (3)shows that households’ heterogeneous location preferences play a key role in generating schoolsorting. We shut down heterogeneous preferences over location characteristics and price bysetting αW Z = αpZ = 0; thus residential sorting is only based on access to school. This
scenario explains 46% of the gap.33 In column (4), we shut down heterogeneous preferences
over school characteristics by setting β XZ = β dZ = 0, so that households choose locations
and schools as if they have perfect consensus over what makes a school desirable, even though
they disagree on what makes a location desirable. This explains 18% of the cross-racial gap.

7

Citywide Access to Highest-achieving Schools
NYC middle schools are intensely segregated (Cohen, 2021; Idoux, 2022), and many believethat location-based priorities are the main cause. The city has long acknowledged this issue
and proposed plans to relax location-based priorities, many of which have triggered heateddebate among parents, students, and educators.34
We evaluate a scenario in which we introduce purely lottery-based admissions to schoolsin School District 2. The district is located in lower Manhattan and has been at the centerof ongoing policy debate regarding whether to retain location-based admissions rules.35 In
33This largely comes from heterogeneous preferences over location characteristics rather than price. Shutting
down only the heterogeneous preference over housing costs reduces the gap by only 0.009.
34For example, a plan to scrap all location-based priorities for high schools was canceled due to pushbacks
from parents (Russo, Barbara , Zoned High School Options for NYC Students Will Remain in Place, NY Metro
Parents, December 14 2021). Meanwhile, smaller plans have been implemented; For example, starting in the
2019-2020 academic year, Bronx middle schools have been open to all students in the Bronx (Zingmond, Laura,
Bronx Middle School Best Tets, InsideSchools, October 20 2020).
35Shapiro, Eliza, N.Y.C. to Change Many Selective Schools to Address Segregation, the New York Times,
December 18, 2020

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Figure 8: District 2 Characteristics(1)
District 2(2)
OthersPanel A: School Characteristics
Mean z-score
1.124-0.155
Safety-0.087-1.145
Share Minority
0.320
0.723
N of Schools
24
646
Panel B: Neighborhood Housing Price
Unit Price (1K)
2,606
464
N of Neighborhoods
81
2,057

(b) District 2 Characteristics(a) Location of District 2

Note: District 2 is the shaded area in the figure. School score is the mean of z-scores among enrolled students from the NYS standardized Mathand Language test. Housing price is the mean price of residential units sold in 2013-14 located in each school district. Safety is a composite
of crime incidences of different categories at the school building. Minorities include Black and Hispanic.

District 2, the average housing cost is about six times higher than other districts, and the mean
test score for its schools is more than 1.2 standard deviations higher than those in other districts(Figure 8).
We compare the status quo, in which we simulate households’ location and school choices
under the current admissions rules, with the following scenarios in which we scrap all admissions criteria—both location-based priority rules and academic screening—for schools inDistrict 2.36
1. OnlySC + No Opt-out: Residential locations under status quo are fixed. We report thecharacteristics of the schools students are assigned to.
2. LCSC + No Opt-out: Households reoptimize residential locations. We report the characteristics of the schools students are assigned to.
3. OnlySC + With Opt-out: Residential locations under status quo are fixed. We report thecharacteristics of the schools students are enrolled in, excluding those who opt out.
4. LCSC + With Opt-out: Households reoptimize residential locations. We report the characteristics of the schools students are enrolled in, excluding those who opt out.
We solve the new equilibrium admissions cutoffs under the policy to address over-subscription
36We also consider two additional reforms in Online appendix E.1—first, a reform that introduces purelylottery based admissions to District 26 schools in upper Queens, and second, a reform that scraps location-based
admissions to District 2 schools but keeps academic screening in place.

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to popular schools, especially District 2 schools. The main outcome of interest is the crossracial gap in the characteristics of coassigned or coenrolled school peers, which we interpret as
the measure of inequity or school segregation. In With Opt-out cases, we calculate the mean
characteristics of students who enroll in each school, excluding those who choose outside
options.
We predict the effects of a policy that targets only one cohort of middle school applicants.
Thus, we assume that the housing market can absorb changes in the demand of householdswith middle school applicants, who account for only 3% of the population. We also assume
that school characteristics are invariant under a new policy. Furthermore, we compare the
distribution of households across schools and residential locations in a steady state, since we
do not model moving costs.
Cross-racial Gap in Peer Characteristics Figure 9 shows the gap in coassigned or coenrolled peers’ test scores between minority and non-minority students in each scenario. While
the reform narrows the cross-racial gap in peer test scores, households’ location choices dampensuch effect. The y-axis in panel (a) is the cross-racial difference school peers’ standardized
test scores. The policy closes the cross-racial gap in coassigned peers’ test scores from 1.07
to 0.99, thus approximately 7%, if households’ residential locations were fixed (No Opt-out,
OnlySC).37 However, when households reshuffle across locations, the effect reduces to 3.3%
(No Opt-out, LCSC). The cross-racial gap in assigned schools (No Opt-out) is always smaller
than that in enrolled schools (With Opt-out). But the effects of the policy and households’
endogenous location choices on the cross-racial gap in enrolled schools are similar to thoseon the cross-racial gap in assigned schools. In panel (b), we present the mean of coenrolledpeers’ test scores by minority and non-minority students. It shows that the policy closes the
cross-racial gap both because non-minority students enroll with lower-achieving peers andminority students enroll with higher-achieving peers.
Location Choice Patterns Next, we delve into households’ location choices to understandhow those dampen the equity impact of the policy. The key is that locations decide on commuting costs as well as location-based priorities, which together determine access-to-school utilityof locations. Under the status quo, households have positive admission chances to District 2
schools only when they reside in District 2. Hence, utility from access to District 2 schools
37Zooming in District 2 schools, the cross-racial gap reduces from 0.35 standard deviation to 0.15 standard
deviation, thus the gap reduces by 57% among students who are assigned to District 2 schools.

33

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Figure 9: Cross-racial Gap in School Characteristics(a) Gap in Peers’ Test Score (1 std)(b) Mean of Peers’ Test Score (1 std)

Note: Panel (a) shows the difference in mean test scores of coassigned/coenrolled peers between Black/Hispanic and other students. Panel (b)shows the mean test score of coenrolled peers for each group separately. We use z-scores from the NYS standardized Math and Language test.

differs only by whether a location is either within or outside the district. On the other hand, thepolicy equalizes admissions probability to District 2 schools across locations. Hence, locations
differ in utility from access to District 2 schools by their proximity to District 2. Standing incontrast to the status quo, locations outside District 2 have different levels of utility from accessto District 2 schools from one another.
These changes in access-to-school utility result in a different reoptimization in location
choice patterns among households who live in District 2 under the status quo (=D2 residents)and others (=Non-D2 residents). We first present the location choice patterns of these twogroups in a scenario in which we introduce purely lottery-based admissions to District 2
schools to one household at a time. In this scenario of one household at a time, a given
household does not expect other households to modify their behavior in response to the policychange.
In Figure 10, the y-axis is the demeaned log of housing price and the x-axis is the average
distance to schools in District 2 from each location. Each dot describes the mean characteristics
of locations chosen by Non-D2 residents (panel (a)) and D2 residents (panel (b)).
In the one household at a time scenario, Non-D2 residents relocate closer to District 2
at the expense of higher housing costs (panel (a)). While the policy makes them eligible toapply to and enroll in District 2 schools, such an option is not attractive when they stay intheir baseline locations due to the high commuting cost to District 2 schools. Meanwhile, D2
residents choose locations with lower housing costs, but farther from District 2 schools (panel
34

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Figure 10: Location Choice Patterns(a) Non-D2 Residents(b) D2 Residents

Note: Panel (a) illustrates the location choices of Non-D2 residents, and (b) of D2 residents. The x-axis is the average distance to schools
in District 2 from locations. The y-axis is the residualized log sales price. Each dot shows the median characteristics of locations chosen by
Non-D2 residents and D2 residents who change locations under the policy, respectively. In each panel, we plot location choice patterns when
we grant citywide access to District 2 schools to one household at a time and when we grant citywide access to all households. For the latter,
we solve equilibrium admissions cutoffs.

(b)). Purely lottery-based admissions make it no longer necessary to live in District 2 to ensure
positive admissions probabilities to District 2 schools.
In equilibrium, the location choice behaviors of Non-D2 residents are largely muted, while
those of D2 residents are reinforced. This is because citywide access to District 2 schools
induces applicants from a broader area, and thus the admissions chances to District 2 schools
are lower from each household’s point of view. This makes choosing locations nearer toDistrict 2 by Non-D2 residents less attractive and choosing locations farther from District 2 by
D2 residents more attractive.
Connection between Location Choice and Peer Characteristics Table 5 reveals the link
between households’ location choice and the school desegregation effect of the policy. Non-D2
residents’ spatial reshuffling narrows the cross-racial gap in school characteristics. For example,
by relocating, minority Non-D2 residents are assigned to schools with a 13.7-percentage-point
lower minority share. This largely comes from their choosing locations nearer to District 2
schools and more actively applying to and enrolling in those schools.
The location choice patterns of D2 residents stand in contrast to those of Non-D2 residents;
their spatial reshuffling dampens the equity impact of the policy. They seek locations that comewith a secured seat in higher-achieving and lower-minority schools, and the purely lotterybased admissions to District 2 schools make locations within District 2 less attractive. Instead,
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Table 5: From Location Choice To School Assignment

Share

(1)
(2)
Non-D2 Resident
Non-Minority Minority
33.53%
62.81%(3)(4)
D2 Resident
Non-Minority Minority
2.72%
0.94%

Panel A: Location Choice when Granting Citywide Access to One HH at a Time
Change Location under New Policy?
0.220
0.176
0.592
Conditional on Changing Location:∆ Frac. Minority of Assigned School
0.107-0.137-0.150∆ Mean Score of Assigned School-0.2040.181
0.273∆ Frac. Minority of Neighborhood
0.047-0.2160.147
Panel B: Location Choice in Equilibrium
Change Location under New Policy?
0.136
0.092
0.974
Conditional on Changing Location:∆ Frac. Minority of Assigned School
0.022-0.048-0.186∆ Mean Score of Assigned School-0.2110.030
0.281∆ Frac. Minority of Neighborhood
0.004-0.0420.130

0.511-0.085
0.095
0.344
0.936
-0.023-0.0990.416

Note: Minority includes Black or Hispanic. D2 residents are those who reside in one of the locations in District 2 under the status quo. Eachcolumn shows the mean of variables for each group.

they choose locations where location-based admissions are kept in place. By doing so, they are
assigned to schools with a 15 percentage point lower minority share.
Previously, we have shown while relocation motives of Non-D2 residents are muted inequilibrium those of D2 residents are reinforced (Figure 10). Then, Table 5 shows relocation of
Non-D2 residents amplifies the equity impact of the policy, while that of D2 residents dampens
the impact. Combined together, endogenous location choices dampen the effect of the policyon the cross-racial test score gap, as depicted in Figure 9.
Commuting Distance to School and Welfare Another widespread concern about relaxing
the importance of location-based admissions priority is that students would have to commute
longer distances. In Table 6, we present average commuting distances to schools under eachscenario by minority and non-minority students. We also present the change in welfare, which isa number that summarizes various changes in outcome induced by the counterfactual policy.38
Commuting distances increase for both minority and non-minority students, so the gap isdecided by which group experiences a larger increase. While the policy narrows the cross-racial
38While the welfare measure is a good summary of various changes, we might want to use caution in interpreting
this. This is because our demand estimates might not represent households’ true preferences, even though
they capture how households make location, school, and enrollment choices. Former studies have documented
various types of friction in location and school choices such as limited information, limited attention, and evendiscrimination by landlords or schools (Christensen and Timmins, 2018; Luflade, 2018; Allende, Gallego, andNeilson, 2019; Son, 2020; Christensen, Sarmiento-Barbieri, and Timmins, 2020; Ferreira and Wong, 2020).

36Electronic copy available at: https://ssrn.com/abstract=4452763

Table 6: Effect on Commuting Distance and Welfare(1)(2)
Distance to school
Non-Minority Minority
Mean
Mean
Baseline
OnlySC
LCSC

1.450
1.728
1.738

1.595
1.645
1.769

(4)
Non-Minority
Mean-0.074
0.005

(5)
(6)
(7)
(8)
∆ Welfare (% Housing Cost)
Minority
Overall
Mean
Mean
p50
Sum Loss

Sum Gain-0.0060.045

591.450572.680-0.030
0.031

0.012
0.029

894.620264.270(9)Note: Minority includes Black or Hispanic. D2 residents are those who reside in one of the locations in District 2 under the status quo. The
fastest driving distance between a school and a Census block is calculated using Open Route Services. Welfare is measured by exante utility(Equation 10 at the chosen location), which we we convert into log housing cost. We present the difference in welfare under the policy relative
to the baseline scenario.

gap in commuting distance under OnlySC, households’ endogenous location choice partially
undoes this effect (LCSC).
Next, we calculate the change in welfare, which we measure with the ex-ante utility (Equation 10) at each household’s optimal location. We convert the utility into percentage housing
cost for ease of interpretation. Since we restrict households from reoptimizing locations,
OnlySC mechanically gives lower welfare in comparison to the other two scenarios.
Under OnlySC, both minority and non-minority students experience a decrease in welfare
on average. This comes from a large decrease in welfare among D2 residents, who cause the
distribution of welfare change to skewed to the left. Indeed, a median household experiences
an increase in welfare by 1.2% of housing cost. The sum of welfare losses is greater than thesum of gains by 300% of housing cost. This suggests while the policy might be approved by
the voting among these households, it might face harder pushback from households who lose.
In the long run, where households adjust their locations (LCSC), both the average andthe median household experience welfare gain, by 3.1% and 2.9%, respectively. The benefit
is largely concentrated among minority students (a 4.5% increase), who obtain eligibility toDistrict 2 schools while living in affordable locations. We consider this welfare gain an upperbound given that we assume away from moving cost.
Other Margins The counterfactual policy has effects that go beyond changing the crossracial test score gap, which we briefly discuss here. First, residential segregation, which wemeasure with an entropy-based segregation measure39 decreases by 1.5% by race (0.489→
0.482) and 10% by income (0.258→ 0.232). Second, the policy increases the mean score ofpeers even among minority students whose test scores are in the lowest quintile, an effect that
39See the appendix of Hahm and Park (2022) for the explanation of the measure.

37

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is also dampened by households’ location choices. There is a 6.5% increase (-0.634→-0.584)
in coassigned peers’ test scores in OnlySC and a smaller effect (4% increase, -0.634→-0.591)
in LCSC.

8

Conclusion
Increasingly more school districts have adopted centralized school choice systems in thehope that they can break the tie between spatial disparities and educational disparities. Whether
they can achieve these goals, however, crucially depends on the extent to which students arewilling to take advantage of school choice options as well as how households respond to thepolicy by (1) reshuffling across locations and (2) opt-out to other schooling options.
This paper develops a unified framework of households’ residential location choice andschool choice under a centralized school choice system. By doing so, we extend empiricalschool choice literature that has studied many factors for students’ school applications andassignments but has given little attention to endogenous residential location choices. Residential locations determine location-based admissions probabilities and commuting distances toschools, which motivates households to choose locations by considering such ties. Our framework captures this as well as the possibility of opting out to outside schooling options. Rich
heterogeneity in households’ observed and unobserved preferences over various school andlocation characteristics generates sorting into locations and schools. We map the framework
to New York City’s middle school choice context, which is the largest unified district with acentralized school choice system.
Our policy analysis shows how a radical school desegregation effort might have a minimal
effect, largely because of households’ choosing locations that can undo the policy. The policy
grants citywide access to the school district that covers lower- and mid-Manhattan. We find thathouseholds’ spatial reshuffling dampens the policy effect by half. Some minority households
choose locations from which the commute to top district schools is easier, which amplifies
the desegregation effect. However, other non-minority households choose locations that comewith secured seats in higher-achieving schools outside of the affected district, which undoes
the effect of the policy.
Several lines of inquiry are left for future work. First, such work might quantify thecomplementary effect between school desegregation policies and housing market policies. We
find that 45% of school segregation is explained by households’ heterogeneous preference
over location characteristics other than price and access to school. Recent evidence shows
38

Electronic copy available at: https://ssrn.com/abstract=4452763

that such heterogeneity stems from information frictions (Ellen, Horn, and Schwartz, 2016;
Ferreira and Wong, 2020) or housing market discrimination (Christensen and Timmins, 2018),
which suggests that policy interventions can change how households choose locations. Second,although we take the location of schools as given in the paper, future work can consider where
to open a new school or how to allocate resources to schools in different locations. Such work
informs policymakers’ ongoing efforts to design school choice systems that could benefit alarger number of students.

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Online Appendix to

Location Choice, Commuting, and School Choice(Not for Publication)

Minseon Park Dong Woo Hahm
July 2023

A

Details of NYC School Choice Process

A.1

Student-Proposing Deferred Acceptance Algorithm

In detail, DA works as follows (Gale and Shapley, 1962; Abdulkadiroğlu, Pathak, and Roth,2005):• Step 1
Each student proposes to her first choice. Each program tentatively assigns seats to its
proposers one at a time, following their priority order. The student is rejected if no seats areavailable at the time of consideration.
• Step k ≥ 2
Each student who was rejected in the previous step proposes to her next best choice. Each
program considers the students it has tentatively assigned together with its new proposers
and tentatively assigns its seats to these students one at a time following the program’s
priority order. The student is rejected if no seats are available when she is considered.
• The algorithm terminates either when there are no new proposals or equally when all rejectedstudents have exhausted their preference lists.
DA produces the student-optimal stable matching and is strategy-proof i.e., truth-telling is aweakly dominant strategy for students.

A.2

NYC School Admission Methods

Middle school programs use a variety of admission methods—Unscreened, Limited Unscreened, Screened, Screened: Language, Zoned and Talent Test. Unscreened programs admit
students by a random lottery number, and Limited Unscreened programs use rules that givepriority to those who attend information sessions or open houses. Screened programs aswell as Screened: Language programs select students by program-specific measures such as
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elementary school GPA, statewide test scores, punctuality and interviews. Zoned programs
guarantee admissions or give priority to students who reside in the school’s zone, and Talent
Test programs use auditions.

A.3

The Timeline of Admission Process

The timeline of the admission process is as follows (Corcoran and Levin (2011), Directoryof NYC Public High Schools). By December, students are required to submit their ROLs. By
March, DA algorithms are run and determine students’ assignments. Students who accept theiroffer finalize, and if a student rejects an offer, then she goes to the next round. This describes
the main round of the entire system. A majority of students finalize in the main round (about
90% each year). Students who are not assigned in the main round or rejected the assignment
go to the Supplementary round which is similarly organized as the main round and includes
programs that did not fill up their capacities in the main round, or programs that are newly
opened. Finally, there is an administrative round in which students who are not assigned aschool even after the second round are administratively assigned to a school.

A.4

Example of ROL
Figure A.1: Example of Customized List and Rank-Ordered List

Source: NYC DOE Middle School Directory 2014-15

B

Supplementary Materials for Section 3

B.1

Cleaning Procedure of DOF Annualized Selling Record

First, we drop non-residential properties such as industrial buildings, commercial buildings,
and vacant land, based on both the tax class and building class. Then we merge the sellingrecord with the Primary Land Use Tax Lot Output (PLUTO) to recover the exact location of2

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each sold property.A-1 Lastly, we exclude transactions that are unlikely at arm’s-length. We
drop transactions of zero price that include transfers within a family. Also, we drop recordsof significantly low prices relative to other properties of similar characteristics. Specifically,
we run a hedonic pricing model that includes tax class, assessment value, the interaction of
the two, calendar time FE at each borough, month FE by borough, building type, land area,building area, total unit, odd shape, age, age square, garage area, the year of alteration, andcommercial area (R square = 0.67). Then we drop observations of the predicted residual is lessthan 1 percentile. We run the regression separately for coops. Following Schwartz, Voicu, andHorn (2014), we lag housing cost by one year to take into account that there could be sometime lag for school quality to be capitalized in the housing cost.

B.2

Housing Cost and Structure in ACS 5-year Estimates

While ACS 5-year estimates capture the price and characteristics of representative housings,
the biggest limitation is that each observation is at the Census block group level, which could
be too coarse to capture the change within a narrow bandwidth around the boundary. 1,944 of
7,506 Census block groups whose centroid is within 0.2 miles from a school district boundary
overlay across a school district boundary.A-2 Thus, the distance from the centroid of a Census
block group to the closest boundary is a crude measure of proximity to the boundary.
Table B.1: Example of Census Block Groups
Census Block Group A:
Distance to Boundary (mi.)
The Number of Occupied Units

Block A-1
0.15
30

Block A-2
0.28
60

Census Block Group B:
Distance to Boundary (mi.)
The Number of Occupied Units

Block B-1
0.15
60

Block B-2
0.28
30

Note: Consider two Census Block Groups A and B with the same distance from their centroids to the nearest boundaries.

Therefore, we further exploit the variation of population density across Census blocks
within a block group. Table B.1 illustrates two exemplary cases. Consider two census blockgroups A and B whose distances from their centroid to the closest boundaries are the same.
Census block group A consists of two Census blocks, one of which is 0.15 miles away from theA-1

Two data sets are merged based on the identifiable tax lot number. One complication is in merging condos.
The selling record has a unique id for each unit, while PLUTO for each condo. We use the Department of City
Planning Property Address Directory that lists unit ids to a matching condo id.
A-2
On the contrary, only 489 out of 38,498 Census blocks overlay across a boundary.

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boundary and the other 0.28 miles away. Note that the two Census blocks differ in population
density. Out of 90 occupied units in the block group, two-thirds are living in A-2. Census blockgroup B has the opposite pattern.
To consider such differences in density, we weigh Census block groups with the percent ofoccupied units in Census blocks within 0.25 miles from the closest boundary when running
Equation 1. In the example, Census block group A is given a weight of 0.33, while B has aweight of 0.66. We present the estimated effects of district school quality in Table B.8.

B.3

Evidence Supporting the Identification Assumption of BDD

The identification assumption of a boundary discontinuity design is that unobserved location amenities are as good as random within a narrow buffer around a boundary. While wecannot check this assumption directly, we present that other observed location characteristics
are continuous in geography, which suggests that the assumption is plausible in this context.
Table B.9 reports estimates β̂ (Equation 1) for various housing characteristics and urban
amenities. β̂s for most of the variables are not statistically significant. One exception is that soldproperties located within a school district with higher school quality are more likely to havebeen renovated (p-value < 0.05). Thus, we use residualized prices when estimating the model
to absorb variations coming from housing characteristics and urban amenities, including therenovation status.
While each variable is not the main driver of sharp change in sales price at boundaries(Figure 3 and Table B.4), a set of variables might. Table B.10 checks this further. First, we run
a hedonic regression of log sales prices on various housing characteristics and urban amenities
using transaction records within a 0.25-mile buffer around boundaries. Then we sum variables
using coefficients from the hedonic regression, which capture the extent to which each variable
explains the variation in sales prices. Finally, we run the BDD regression (Equation 1) using
the predicted prices as dependent variables and check if estimates β̂ are significant.
Columns (1)-(2) of Table B.10 describe that even a very extensive set of housing characteristics and urban amenities does not explain the sharp change in sales prices at boundaries—i.e.,change in school quality is the main driver. Meanwhile, the estimate β̂ is marginally significant(p-value < 0.1) when we include neighbors’ composition (column (3)), which captures households’ residential sorting at boundaries as well as their preference over neighbors’ composition.

B.4

Supplementary Tables

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Table B.2: Summary Statistics: All vs. Sample for BDD
(1)

(2)
All

Variables

mean

std

Panel A: Student Characteristics
Asian
0.185
0.388
Black
0.295
0.456
Hispanic
0.380
0.485
White
0.125
0.331
FRL
0.736
0.441
Standardized English Score
0.028
1.009
Standardized Math Score
0.172
0.979
N
57,593
Panel B: Sold Properties’ Characteristics
Unit Price ($1000)
802.8
2,063
Age
70.88
32.68
Number of Floors
7.150
9.083
Coopa
0.299
0.458
Manhattan
0.245
0.430
Bronx
0.079
0.270
Brooklyn
0.263
0.440
Queens0.319
0.466
Staten Island
0.0934 0.291
N106,040
Panel C: Census Block Group Characteristics
Median Rent
1,404
503.4
Median Value ($1000)
636.7
355.9
Median Age
70.83
13.45
% College and Higher Degree
0.352
0.237
% Minority
0.272
0.316
N
4,828

(3)
(4)
< 0.25-mile Buffer
mean
std
0.162
0.333
0.407
0.085
0.783
-0.0770.074
18,761

0.368
0.471
0.491
0.279
0.413
0.997
0.963

707.9
80.71
6.493
0.287
0.170
0.088
0.441
0.300
0
23,836

1,496
97.16
8.105
0.452
0.376
0.283
0.497
0.458
0

1,255
612.9
72.67
0.298
0.334
609

501.5
331.1
13.08
0.229
0.309

Note: Source of each data set is NYC Department of Education, NYC Department of Finance Selling Record, and ACS 5-year estimates. All
from 2013 to 2017.
a

We control for properties’ co-op status in our analyses. Coops take up a large proportion of the NYC
housing market (35% of sold properties, 50% of the housing stock) with two unique features. First, they are morecommon in Manhattan compared to other boroughs, and second, are cheaper to buy but come with high monthly
maintenance fees. (Susan Stellin, Co-op vs. Condo: The Differences Are Narrowing, The New York Times, Oct.
5, 2012) Ignoring the composition of co-op and other housing types understate housing cost in Manhattan because
the Sales files cover only sold price.

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Table B.3: Effects of District School Quality on Assigned Schools’ Quality

Bandwidth
Boundary FEs
Local Cubic Control for DistanceStudent Characteristics
District School QualityNR2ȳ
std(y)(1)(2)< 0.25 mile
Yes
Yes
Yes
Yes
No
Yes
0.360
(0.074)16576
0.249-0.0780.800

0.270(0.053)15809
0.394-0.055
0.779

(3)
(4)
< 0.2 mile
Yes
Yes
Yes
Yes
No
Yes
0.324
(0.072)13261
0.245-0.0920.796

0.256(0.054)12657
0.389-0.0680.773

Note: The dependent variable is the mean score of the schools middle-school-applying residents in a Census block are assigned to. Sample of
5th-grade students in academic year 2014-15 living in Census blocks within a buffer from the closest school district boundary. District school
quality is measured by the mean NYS standardized test score of students enrolled in middle schools (previous cohorts) in the district. We
use the 0.25-mile buffer in columns (1)-(2) and the 0.2-mile buffer in columns (3)-(4). Standard errors in parentheses are clustered at school
district level. The local cubic control of distance differs at the opposite side of boundaries.

Table B.4: Effects of District School Quality on Housing Sales Price

Boundary FEs
Local Cubic Control for Distance
Housing Characteristics
Neighborhood Characteristics
Urban Amenities
Panel A: 0.25-mile Buffer
District School QualityNR2ȳ
std(y)
Panel B: 0.2-mile Buffer
District School QualityNR2ȳ
std(y)

(1)
Yes
Yes
No
No
No

(2)
Yes
Yes
Yes
No
No

(3)
Yes
Yes
Yes
Yes
No

(4)
Yes
Yes
Yes
Yes
Yes

0.181
(0.061)
23786
0.409

0.204
0.101
(0.044) (0.040)
23786
23786
0.489
0.505
12.88
1.112

0.102
(0.040)
23786
0.505

0.108
(0.068)
19057
0.401

0.150
0.073
(0.049) (0.045)
19057
19057
0.480
0.493
12.84
1.086

0.073
(0.045)
19057
0.494

Note: The dependent variable is the log sales price of a residential unit. Sample of residential units sold within a bandwidth from the closest
school district boundary. District school quality is measured by the mean NYS standardized test score of students enrolled in middle schools(previous cohorts) in the district. We use the 0.25-mile buffer in columns (1)-(2) and the 0.2-mile buffer in columns (3)-(4). Standard errors
in parentheses are clustered at school district level. The local cubic control of distance differs at the opposite side of boundaries. Housing
characteristics include the space of the unit, land use of the tax lot, number of floors, age, renovation status, and storage area of the building,
all of which we interact with a dummy if the property is coop. Neighbor characteristics include % minority, median household income, %college-or-more-educated, and median commuting time to work at Census block group. Urban amenities include the number of bus stops,
subway stations, laundries, cafes, and crime incidents of different categories at Census block.

6

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Table B.5: Effects of District School Quality on the Number of Middle-school-applying Residents

Boundary FEs
Local Cubic Control for Distance
Housing Characteristics
Neighborhood Characteristics
Urban Amenities
N of Population
Panel A: 0.25-mile Buffer
District School QualityNR2ȳ
std(y)
Panel B: 0.2-mile Buffer
District School QualityNR2ȳ
std(y)

(1)
Yes
Yes
No
No
No
No

(2)
Yes
Yes
Yes
No
No
No

(3)
Yes
Yes
Yes
Yes
No
No

(4)
Yes
Yes
Yes
Yes
Yes
No

(5)
Yes
Yes
Yes
Yes
Yes
Yes

0.183
(0.176)
3755
0.227

0.266
(0.175)
3755
0.251

0.805
(0.222)
3755
0.308
3.515
4.122

0.789
(0.225)
3755
0.318

0.808
(0.219)
3755
0.331

0.216
(0.180)
2970
0.223

0.280
(0.183)
2970
0.246

0.766
(0.240)
2970
0.295
3.490
4.033

0.750
(0.244)
2970
0.303

0.830
(0.242)2970
0.313

Note: The dependent variable is the number of middle-school-applying residents in a Census block. Sample of Census blocks within abandwidth from the closest school district boundary. District school quality is measured by the mean NYS standardized test score of students
enrolled in middle schools (previous cohorts) in the district. We use the 0.25-mile buffer in panel A and the 0.2-mile buffer in panel B. Standard
errors in parentheses are clustered at school district level. The local cubic control of distance differs at the opposite side of boundaries. Housing
characteristics include the space of the unit, land use of the tax lot, number of floors, age, renovation status, and storage area of the building,
all of which we interact with a dummy if the property is coop. Neighbor characteristics include % minority, median household income, %college-or-more-educated, and median commuting time to work at Census block group. Urban amenities include the number of bus stops,
subway stations, laundries, cafes, and crime incidents of different categories at Census block. The number of population is at Census blockgroup level, which we obtain from the ACS 5-year estimate.

7

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Table B.6: Effects of District School Quality on Minority Share of Middle-school-applying
Residents

Boundary FEs
Local Cubic Control for Distance
Housing Characteristics
Neighborhood Characteristics
Urban Amenities
Panel A: 0.25-mile Buffer
District School QualityNR2ȳ
std(y)
Panel B: 0.2-mile Buffer
District School QualityNR2ȳ
std(y)

(1)
Yes
Yes
No
No
No

(2)
Yes
Yes
Yes
No
No

(3)
Yes
Yes
Yes
Yes
No

(4)
Yes
Yes
Yes
Yes
Yes

-0.139(0.029)2970
0.480-0.143-0.065(0.028) (0.016)
2970
2970
0.496
0.595
0.620
0.417

-0.065(0.016)2970
0.596-0.132(0.034)2353
0.484-0.132-0.066(0.033) (0.024)
2353
2353
0.501
0.603
0.633
0.415

-0.067(0.024)2353
0.603

Note: The dependent variable is the share of Black and Hispanic applicants among middle-school-applying residents in a Census block.
Sample of Census blocks within a bandwidth from the closest school district boundary. District school quality is measured by the mean
NYS standardized test score of students enrolled in middle schools (previous cohorts) in the district. We use the 0.25-mile buffer in columns(1)-(2) and the 0.2-mile buffer in columns (3)-(4). Standard errors in parentheses are clustered at school district level. The local cubic control
of distance differs at the opposite side of boundaries. Housing characteristics include the space of the unit, land use of the tax lot, numberof floors, age, renovation status, and storage area of the building, all of which we interact with a dummy if the property is coop. Neighborcharacteristics include % minority, median household income, % college-or-more-educated, and median commuting time to work at Census
block group. Urban amenities include the number of bus stops, subway stations, laundries, cafes, and crime incidents of different categories
at Census block.

8

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Table B.7: Effects of District School Quality on Poverty Share of Middle-school-applying
Residents

Boundary FEs
Local Cubic Control for Distance
Housing Characteristics
Neighborhood Characteristics
Urban Amenities
Panel A: 0.25-mile Buffer
District School QualityNR2ȳ
std(y)
Panel B: 0.2-mile Buffer
District School QualityNR2ȳ
std(y)

(1)
Yes
Yes
No
No
No

(2)
Yes
Yes
Yes
No
No

(3)
Yes
Yes
Yes
Yes
No

(4)
Yes
Yes
Yes
Yes
Yes

-0.139(0.038)2970
0.203-0.142-0.091(0.034) (0.033)
2970
2970
0.218
0.261
0.683
0.357

-0.094(0.033)2970
0.263-0.139(0.042)2353
0.173-0.141-0.099(0.038) (0.037)
2353
2353
0.192
0.238
0.698
0.350

-0.102(0.037)2353
0.240

Note: The dependent variable is the share of free or reduced lunch eligible applicants among middle-school-applying residents in a Census
block. Sample of Census blocks within a bandwidth from the closest school district boundary. District school quality is measured by the mean
NYS standardized test score of students enrolled in middle schools (previous cohorts) in the district. We use the 0.25-mile buffer in columns(1)-(2) and the 0.2-mile buffer in columns (3)-(4). Standard errors in parentheses are clustered at school district level. The local cubic control
of distance differs at the opposite side of boundaries. Housing characteristics include the space of the unit, land use of the tax lot, numberof floors, age, renovation status, and storage area of the building, all of which we interact with a dummy if the property is coop. Neighborcharacteristics include % minority, median household income, % college-or-more-educated, and median commuting time to work at Census
block group. Urban amenities include the number of bus stops, subway stations, laundries, cafes, and crime incidents of different categories
at Census block.

9

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Table B.8: Effects of District School Quality on Rent and House Value

Boundary FEs
Local Cubic Control for Distance
Housing Characteristics
Neighborhood Characteristics
Urban Amenities

(1)
Yes
Yes
No
No
No

Panel A: Log(Median Gross Rent)
District School Quality
0.163(0.047)
N
1875
R2
0.352
ȳ
std(y)
Panel B: Log(House Value)
District School Quality
0.057(0.066)
N
1332
R2
0.478
ȳ
std(y)

(2)
Yes
Yes
Yes
No
No

(3)
Yes
Yes
Yes
Yes
No

0.159
(0.047)
1873
0.374
7.119
0.377

0.058
(0.035)
1873
0.535

0.067
(0.061)
1331
0.540
13.20
0.540

0.058
(0.062)1331
0.558

Note: The dependent variable is the log median gross rent in panel A, and the log house value reported by home owners in panel B. Unitof analysis is Census block group, where we weigh block groups by the share of occupied units in Census blocks within a 0.25-mile bufferfrom the closest school district boundary (See Subsection B.2). District school quality is measured by the mean NYS standardized test scoreof students enrolled in middle schools (previous cohorts) in the district. Standard errors in parentheses are clustered at school district level.
The local cubic control of distance differs at the opposite side of boundaries. Housing characteristics include the space of the unit, land useof the tax lot, number of floors, age, renovation status, and storage area of the building, all of which we interact with a dummy if the property
is coop. Neighbor characteristics include % minority, median household income, % college-or-more-educated, and median commuting time
to work at Census block group. Urban amenities include the number of bus stops, subway stations, laundries, cafes, and crime incidents ofdifferent categories at Census block.

10

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Table B.9: Housing Characteristics and Urban Amenities at School District Boundaries
(1)

(2)

Panel A: Housing Characteristics of Sold Properties
Dependent Variable:
N of Floors
Coop
District School Quality
-1.55
0.076
(1.25)(0.047)
N
23786
23786
R2
0.466
0.230
ȳ
6.498
0.288
std(y)
8.112
0.453
Panel B: Urban Amenities
Dependent Variable:
Bus Stop
Subway Station
District School Quality-0.014-0.004(0.024)(0.008)
N
8091
8091
R2
0.025
0.020
ȳ
0.127
0.019
std(y)
0.413
0.140

(3)

(4)

Commercial Area (1K sqft)-5.84(5.53)
23786
0.182
11.19
56.27

Renovation
0.100(0.038)
23786
0.165
0.809
0.393

Laundries
0.003(0.008)
32340
0.068
0.052
0.252

Café
-0.004(0.007)
32340
0.087
0.014
0.157

Note: Sample of residential properties sold within a bandwidth from the closest school district boundary in Panel A. Sample of Census blockgroups whose centroids are within a bandwidth from the closest school district boundary in Panel B. Sample of Census blocks whose centroids
are within a bandwidth from the closest school district boundary in Panel C. We use 0.25 mile buffer. In panel B, Each Census block groups
is weighted accroding to the procedure explained in the appendix. Standard errors in parentheses are clustered at each tax lot, Census blockgroup, and Census block, respectively.We allow the local cubic control of distance to differ at the opposite side of boundaries.

11

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Table B.10: Effects of District School Quality on Hedonic Sales Price
(1)

(2)

(3)

Boundary FEs

Yes

Yes

Yes

Local Cubic Control for Distance

Yes

Yes

Yes

Housing Characteristics

Yes

Yes

Yes

Neighborhood Characteristics

No

No

Yes

Urban Amenities

No

Yes

Yes

Panel A: 0.25-mile Buffer
District School Quality-0.027-0.0180.094(0.069)(0.065)(0.052)

N

23786

23786

23786

R2

0.457

0.479

0.687

ȳ

12.88

12.88

12.88

std(y)

0.546

0.560

0.728

-0.021-0.0020.069(0.077)(0.074)(0.057)

N

19057

19057

19057

R2

0.443

0.467

0.686

ȳ

12.84

12.84

12.84

std(y)

0.509

0.526

0.696

Panel B: 0.2-mile Buffer
District School Quality

Note: The dependent variable is the predicted log sales price of a residential unit, which we construct by running a hedonic regression oflog sales price on covariates, as explained in Subsection B.3. Sample of residential units sold within 0.25-mile bandwidth from the closest
school district boundary. District school quality is measured by the mean NYS standardized test score of students enrolled in middle schools(previous cohorts) in the district. Standard errors in parentheses are clustered at school district level. The local cubic control of distancediffers at the opposite side of boundaries. Housing characteristics include the space of the unit, land use of the tax lot, number of floors, age,renovation status, and storage area of the building, all of which we interact with a dummy if the property is coop. Neighbor characteristics
include % minority, median household income, % college-or-more-educated, and median commuting time to work at Census block group.
Urban amenities include the number of bus stops, subway stations, laundries, cafes, and crime incidents of different categories at Census
block.

12

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Figure B.2: Probability of Listed as Top 3 Given Distance

Note: The fastest driving distance to school is calculated using the Open Route Service (ORS). the y-axis of the graph presents the probabilityof choosing school j as the top 3 choices among all schools a student is eligible for. Lines present the probability of student i’s listing school j
as one of her top 3 choices as a function of road distance between i and j, for all pairs of (i, j). We present the pattern separately for students
whose neighborhood schools’ mean test score is greater/smaller than the average, where we use the three closest schools as neighborhood
schools.

C

Supplementary Materials for Section 4

C.1

Stability of Matching and Aggregate Demand

Fixing location choice `, the set of feasible schools are defined as Ji (`; ρi ) = {j|cij (`; ρi ) ≥c̄j }, i.e. schools of which student i can clear the cutoffs. The set of feasible schools changes
depending on which location ` student i chooses to reside in, and the lottery number.
Using the stability of matching and the distributional assumption on the idiosyncratic
preference shock over locations and schools,

Dj ({c̄j 0 }Jj0 =1 ) =Z
exp(Ṽi (`))
1(cij (`; ρi ) ≤ c̄j )exp(Ũi (j, `))
Σ`
·
dρi
0
Σ`0 exp(Ṽi (` )) ρi Σj 0 1(rij 0 (`; ρi ) ≤ c̄j 0 )exp(Ũi (j 0 , `))
i
{z
} |
|
{z
}

Z

Demand for location `Demand for school j given location `13

Electronic copy available at: https://ssrn.com/abstract=4452763(C.1)

whereṼi (`) =

W`
|{z}

αiw +

location char.

p` αip +|{z}

housing cost

ξ`
|{z}

+αu

unobserved amenities

Eεij ,ρ,εiϑ| {z }

expected utility from school given location(C.2)
Ũi (j, `) =Xj βiX +|{z}school char.

d`j βid| {z }(C.3)

commuting cost

The second component of Equation C.1 means that a student has effective demand only
when school j is feasible given location `. For schools that use lottery number to break thetie, the feasibility depends on the lottery number ρi . We take the numerical integration over ρi .
The existence and the uniqueness of the equilibrium follow from Azevedo and Leshno (2016).
The key assumption is that the distribution of students’ priority rank cij is continuous. This
ensures a small change in the cutoff c̄j 0 induces a small change in the demand for school j.

C.2

Stability of Matching and Expected Utility from School

In addition, based on the stability of assignment under DA with truth-telling, we cansimplify the indirect utility from school choice stage Ui∗ (`):
Ui∗ (`) = max{maxj∈Ji (`;ρi ) Ui (ji , `; εij ), Uiϑ (ϑp ; εiϑ ), Uiϑ (ϑc ; εiϑ ) )(C.4)

With the assumption that εij , εiϑ follows i.i.d EVT1 distribution, the expected utility fromschool can be simplified as follows.
Eεij ,ρ,εiϑ= Eρ µ + log Σj∈Ji (`;ρi ) exp(Ũi (j, `; εij )) + exp(Ũiϑ (ϑp ; εiϑ )) + exp(Ũiϑ (ϑc ; εiϑ ))
(C.5)
where
Ũi (j, `) =Xj βiX +|{z}school char.

d`j βid| {z }(C.6)

commuting costŨi (ϑ; εiϑ ) = βiϑ

(C.7)

14

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C.3

Estimating Program Preferences

sij is a weighted average of students’ middle school GPA, NYS math and ELA score, andpunctuality record, with the weights remaining as each program’s private information.
However, a program j has to report the rank of sij s among its applicants for NYC DOE to
implement the centralized DA. Therefore, given any pair of students i and i0 , we observe thevalue of 1(sij > si0 j ), if both i and i0 apply to the program j. Using this, we construct ŝij using
a latent variable model by assuming
sij = Zi κj + ηij , ηij ∼ N (0, 1)(C.8)

where Zi is student characteristics that are known to compose of sij , and κj a vector ofweights which vary across js. ηij is normalized to be N (0, 1).
We estimate (κj )j using Maximum Likelihood Estimation (MLE) where the likelihood
function is
LLs = Πi,i0 ∈Aj P r(sij > si0 j )1(sij >si0 j ) (1 − P r(sij > si0 j ))1(sij <si0 j )(C.9)

where Aj is the set of applicants to program j that uses a non-random tie-breaker.
Figure C.3 shows that our simulation recovers the distribution of the preference rank of
the assigned program in the data. Both in simulation and data, around 63% of students are
assigned to the 1st- or the 2nd- ranked programs while 8% do not get any offer from programson the list.

15

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Figure C.3: Rank of Assigned School: Model and Simulation

Note: Rank is the preference rank of the assigned program. Rank 19 in the data means that the student did not get any offer from programs
included in the submitted list.

D

Supplementary Materials for Section 5

D.1

Full Information Maximum Likelihood Estimation

Assuming idiosyncratic preference shocks over locations, schools, and outside options(ηi` , εij , and εiϑ ) are i.i.d, the full information log-likelihood function is as follows.
LL =

Σi
|{z}

3. sum over ilogΣK qk P LC (xi ; θEC , θSC , θLC , γk )P SC (xi ; θSC , γk )P EC (xi ; θSC , θEC , γk )| k=1{z } |{z}

2. sum over type1. likelihood with fixed type(D.10)
xi is data, θLC is the set of location preference parameters, θSC is the set of school preference
parameters, θEC is the set of outside option preference parameters, γi is the unobserved taste,
and qk the probability of each type k. (θ = (θLC , θSC , θEC ), {γk , qk }k ) is the full set ofparameters to be estimated.
The likelihood function for each step is not additive separable because of γi , making the
maximization problem computationally very costly. Note that the sequential estimation strategy
in Rust (1994) is also not applicable without additive separability of likelihoods.

D.2

EM Algorithm with Sequential Maximization

The conditional probability of type k given data xi and current guesses of parameters, arederived using Bayes rule.
16

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q(k|xi ; q̂, γ̂, θ̂) =q̂k q(xi ; k, q̂, γ̂, θ̂)Σk0 q̂k0 q(xi ; k 0 , q̂, γ̂, θ̂)(D.11)

We estimate the model using the following iterative process.
1. Initial guess of q 0 , γ 0 , θ0
2. Calculate conditional probability in Equation D.11 using the initial guess q 0 , γ 0 , θ0
3. Solve maximization problem for q. It has a closed-form solution which is1
qk1 = Σi q(k|xi ; q 0 , γ 0 , θ0 )I

(D.12)

1
4. Taking other parameters as given, solve the maximization problem of Equation 12. Get θLCusing the Generalized Method of Moments (GMM) procedure.
5. Taking other parameters as given, solve maximization problem of Equation 13 using MLE,
1and γ 1 .
get θSC
1
6. Taking other parameters as given, solve the maximization problem of Equation 14. Get θECusing MLE.
7. Repeat 2-6 until convergence

D.3

Likelihood Function

We present the likelihood function P LC , P SC , and P EC in this section.
For convenience, we introduce some notations.
ũiϑ = Zi βϑ

(D.13)

ũij = Xj βXi + c(d`i j , Zi )(D.14)ṽi` = W` αwi + p` αpi + ξ` + Eεij ,ρi ,εiϑ Ui∗ (`) `

(D.15)

Each denotes utility from outside option ϑ, school j, and location ` at each decision stage
net of idiosyncratic preference shocks.
With the distributional assumption on ηi` , the likelihood of location choice P LC takes asimple form.
P LC = Πiexp(ṽi`i )Σ`0 exp(ṽi`0 )where `i is the observed location choice of household i.
17Electronic copy available at: https://ssrn.com/abstract=4452763(D.16)

We can extend the formula to construct the likelihood of school application P SC . Assuming
logit shock and weak truth-telling among eligible options,

P SC = Πiexp(ũij1i )exp(ũij2i )Σ 0 ˜ exp(ũi`0 ) Σj 0 ∈J˜i /j1i exp(ũi`0 )| j ∈Ji {z}|{z}first-ranked

second-ranked···

exp(ũijli )iΣj 0 ∈J˜i /{j1i ,j2i ···j i } exp(ũi`0 )
l −1
|
{zi
}

(D.17)

lith -ranked

where i’s observed ranked-ordered list is {j1i , j2i , · · · , jlii }, and J˜i is the set of eligible
options for i. Each term in the product represents the probability of choosing the option among
eligible options that are not ranked higher.
Similarly, the likelihood of enrollment decision is a product of each i’s likelihood, whichtakes different forms depending on the observed enrollment choice.

P EC = Πiexp(ũϑi )Σj 0 ∈Ji exp(ũij 0 ) 1(ϑi =µi )exp(ũiµi )exp(ũp ) + exp(ũc ) + Σj 0 ∈Ji exp(ũij 0 )|{z}(D.18)

adjustment when choose µi

The second term denotes the probability of choosing the enrollment option out of allavailable options including outside options. The first component adjusts that the distribution of
εiµi conditional on being assigned to µi should be different from the marginal distribution of
εij and εiϑ . It is more realistic to have εiµi preserved the same in the application and enrollment
decision stage, rather than assuming that a new εiµi is drawn from the Gumbel distribution
with a location of zero and scale normalized to one. Without the first term, the value of private
options should be underestimated. The proof is available upon request.

D.4

Supplementary Tables

18

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Table D.11: Demand Estimates from OnlySC Version
Main
Type1
Panel A: School Demand
Mean test score
0.139
Frac. Black or Hisp. -2.002
Frac. FRL
-0.330Non-safety-0.036
Distance (mi.)-0.190Prob.
0.294
Panel B: Outside Option
Non-public-2.088Public Charter-3.307

Additional Effects
Black/Hisp
FRL

Type2

Type3

High-achieving

0.161-0.430-1.077-0.020-0.937
0.629

0.382
0.794
0.947
-0.007-5.331
0.077

0.165
2.189
-0.590
0.010
0.096
-

0.126
0.860
0.012
-

-0.2450.355-0.278
0.029
-

-

-

0.227
1.685

-0.879-0.3730.505-0.178

Note: Columns are for students’ heterogeneity and rows are for school characteristics. FRL stands for Free-or-reduced Lunch eligibility. The
fastest driving distance to school is calculated using the Open Route Service (ORS). School non-safety measure is constructed by running a
Principal Component Analysis (PCA) on crime incidence of different categories at each school building.

Table D.12: IV Regression for Housing Cost

Dep. var: Mean utility δ`
Housing characteristics
Land use
Neighborhood characteristicslog(UnitHousingPrice)
N
First Stage F-stat
R2
ymean

(1)
OLS

(2)
2SLS

Yes
Yes
Yes

Yes
Yes
Yes

-0.014(0.056)1690-2.441**(0.561)
1690
17.76
0.233
-0.8790.641-0.879Note: Instrument variable is the percent park area and the percent residential area of locations that are 2 miles away but within 3 miles fromeach location.

19

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Table D.13: Principal Component of the School Safety Indices
1st Component
Eigenvalue
Total variance explained
Eigenvectors:
Major Crime
Other Crime
Non-Crime Incidents
Property Crime
Violent Crime

Unexplained variance (percent)

3.246
64.920
0.423
0.502
0.413
0.444
0.449

41.90018.28044.52036.11034.570Note: Source - School Safety Report collected by the New York City Police Department which reports the number of crime cases at eachschool building. Major crimes include burglary, grand larceny, murder, rape, robbery, and felony assault. Other crimes include many crimes
that range in severity such as arson, sale of marijuana, or sex offenses. Non-criminal incidents include actions that are not crimes but disruptive
such as disorderly conduct, loitering, and possession of marijuana.

Figure D.4: Location Sorting on Unobserved Type

20

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Table D.14: Model Fit
All
data

model

Minority
data
model

Panel A1: Mean of Top 3 Ranked School Characteristics
Mean test score
0.570
0.564
0.215
0.222
% Black or Hispanic-0.107 -0.111
0.026
0.024
% FRL
-0.061 -0.054 -0.004 -0.001
Non-safety-1.165 -1.091 -0.325 -0.183
Distance
1.965
2.032
1.975
2.052
Panel A2: Mean of Assigned School Characteristics
Mean test score
0.311
0.352
-0.0680.016% Black or Hispanic-0.098 -0.087
0.049
0.054
% FRL
-0.039 -0.040
0.023
0.017
Non-safety-1.037 -0.800 -0.095
0.174
Distance
1.382
1.867
1.388
1.940
Panel B: % Choosing an Outside Option
Private
5.020
5.844
4.439
5.104
Charter
4.310
4.903
6.259
7.091
Panel C: Mean of Chosen Location Characteristicslog(UnitPrice)
12.846 12.847 12.702 12.702% Black or Hispanic
0.568
0.568
0.731
0.731
Median HH Income
10.696 10.696 10.562 10.561
Med. travel time to Work 0.759
0.759
0.772
0.772

data

FRL
model

High-achieving
data
model

0.401
-0.054-0.020-0.851
1.949

0.404-0.056-0.018-0.8101.976

0.919-0.212-0.115-1.9101.974

0.881-0.212-0.104-1.7662.065

0.107-0.0390.006-0.6441.352

0.181-0.029-0.001-0.4931.834

0.704-0.204-0.097-1.8701.455

0.699-0.190-0.092-1.546
1.878

4.187
5.079

4.696
5.704

5.359
3.235

6.359
3.724

12.7720.631
10.6040.770

12.7720.631
10.6030.770

12.9630.459
10.8140.747

12.9650.458
10.8140.747

ESupplementary Materials for Section 7

E.1

Other Policy Plans

District 26 Reform We discuss the impact of another plan to introduce purely-lottery based
admissions to District 26 schools. District 26 is located in upper Queens and features thehighest mean test score of schools. The average housing unit price was $614,000, which isabout a quarter of the housing price in District 2.
Figure E.5 presents the characteristics of District 26, and how the cross-racial gap changes
across scenarios. First, whether and how households’ endogenous location choices change theequity impact of the policy varies across policies. Focusing on No Opt-out scenarios, while
endogenous location choices dampen the effect of the lottery-based admissions to District 2
schools by half, it amplifies the effect of the policy targeting District 26. This is because, in thelatter scenario, minority households’ endogenous location choice responses to shorten commuting distances to District 26 schools dominate non-minority households’ location choices
to get away from the policy. The lower housing cost of District 26 relative to District 2 is the

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Electronic copy available at: https://ssrn.com/abstract=4452763

Figure E.5: Lottery-based Admissions to District 26 Schools(b) Gap in Peers’ Test Score (1 std)(a) Location of District 26

Note: Panel (a) shows the location of District 26. Panel (b) shows the difference in mean test scores of coassigned/coenrolled peers between
Black/Hispanic and other students. We use z-scores from the NYS standardized Math and Language test. Under D2, we introduce purely-lottery
based admissions to schools in school District 2. Under D26, we target District 26.

key reason.
Second, the comparison across policies changes depending on if we consider households’
endogenous location choice or not. Focusing on No Opt-out scenarios, while targeting District
2 schools seems more effective in reducing the cross-racial gap when we take residential
locations as given, households’ endogenous location choices in response make targeting District
26 more effective.
Lastly, households substitute between opting out to outside schooling options and reoptimizing their residential locations. While opt-out plays a minor role when we target District
2 schools, its role is more pronounced when we target District 26 schools. This is becausehouseholds who live in District 26 cannot afford all other school districts as District 2 do.
Ｔhus, they take advantage of outside options to enroll their kids in a more preferable school
when they lose the advantage in admissions changes to District 26 schools.
District 2 Reform with Academic Screening We also present results when we scrap
location-based admissions to District 2 schools but with academic screening in place in Figure E.6. The policy does not close the cross-racial gap in school peers’ test scores when households’ residential locations are fixed. Instead, households’ spatial reshuffling rather widens thegap by 2.6%. This is because non-minority, higher-achieving students who reside outside ofDistrict 2 are largely motivated to move locations nearby District 2, which pushes minority
students in District 2 out to schools with a higher proportion of minority peers.

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Electronic copy available at: https://ssrn.com/abstract=4452763

Figure E.6: Cross-racial Gap in School Characteristics

Note: The figure shows the difference in mean test scores of coassigned/coenrolled peers between Black/Hispanic and other students acrossscenarios. There are 4 different counterfactual policies. Under D2, we introduce purely-lottery based admissions to schools in school District
2. Under D26, we target District 26. Under All District, we target all schools within the system. Under D2+Academic, we scrap locationbased admissions rules among District 2 schools while keeping academic screening in place. The dotted line presents the cross-racial gap incoassigned peers’ test scores under the status quo. The solid line presents the cross-racial gap in coenrolled peers’ test scores under the status
quo.

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Electronic copy available at: https://ssrn.com/abstract=4452763