metadata: authors: 0: author: Ferreyra,Maria Marta
1: author: Kosenok,Grigory
count: United States
docna: 0: docna: Charter school entry and school choice: the
            case of Washington, D.C.
docty: Policy Research Working Paper
subtopic: Demographics,Economics of Education,Educational Sciences,Educational Institutions & Facilities,Education Finance,Educational Policy and Planning - Decentralization,Inequality
teratopic: Education
abstracts: cdata!: This paper develops and estimates an
            equilibrium model of charter school entry and school choice.
            In the model, households choose among public, private, and
            charter schools, and a regulator authorizes charter entry
            and mandates charter exit. The model is estimated for
            Washington, D.C. According to the estimates, charters
            generate net social gains by providing additional school
            options, and they benefit non-white, low-income, and
            middle-school students the most. Further, policies that
            raise the supply of prospective charter entrants in
            combination with high authorization standards enhance social welfare.
geo_reg_key: 119228,517191
repnme: repnme: Charter school entry and school choice: the
            case of Washington, D.C.
display_title: Charter school entry and school
            choice: the case of Washington, D.C.
listing_relative_url: /research/2015/07/24839593/charter-school-entry-school-choice-case-washington-dc
topicv3: Poverty m1326687 695,Education Districts m1327034 2141,Demographic Change m1327040 1448,Education m1327048 470,Private Education m1327271 1490
publishtoextweb_dt: 2015-07-29T00:00:00Z
historic_topic: Education
no_of_pages: 64
author_keys: 0: author_key: 1339928
model_extractions: WPS7383 Policy Research Working Paper 7383 Charter School Entry and School Choice The Case of Washington, D.C.

_Maria Marta Ferreyra Grigory Kosenok_ Latin America and the Caribbean Region Office of the Chief Economist July 2015
Policy Research Working Paper 7383

# **Abstract**

This paper develops and estimates an equilibrium model of charter school entry and school choice. In the model, households choose among public, private, and charter schools, and a regulator authorizes charter entry and mandates charter exit. The model is estimated for Washington, D.C. According to the estimates, charters generate net social gains by providing additional school options, and they benefit non-white, low-income, and middleschool students the most. Further, policies that raise the supply of prospective charter entrants in combination with high authorization standards enhance social welfare.

This paper is a product of the Office of the Chief Economist, Latin America and the Caribbean Region. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at mferreyra@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Charter School Entry and School Choice: The Case of Washington, D.C.

Maria Marta Ferreyra Grigory Kosenok The World Bank New Economic School mferreyra@worldbank.org gkosenok@nes.ru

> ∗We thank David Albouy, Joe Altonji, Stanislav Anatolyev, Steve Berry, Marcus Casey, Rajashri Chakrabarti, Estelle Dauchy, Pablo D’Erasmo, Dennis Epple, Fernando Ferreira, Steve Glazerman, Brett Gordon, Bryan Graham, Phil Haile, Justine Hastings, Dan McMillen, Aviv Nevo, Javier Penia, B. Ravikumar, Stephen Ryan, Tim Sass, Holger Sieg, Fallaw Sowell, Che-Lin Su, Chris Taber, Jacques-Francois Thisse and Patrick Wolf for useful conversations and comments. We benefitted from comments by seminar participants at Boston College, Carnegie Mellon, Cleveland Fed, Cornell, ECARES, High School of Economics, Maryland, McMaster, New Economic School, New York Fed, Notre Dame, Philadelphia Fed, St. Louis Fed, U. of Illinois-Urbana/Champaign, Yale and the World Bank, and by session participants at numerous conferences. Ferreyra thanks the World Bank and the Berkman Faculty Development Fund at Carnegie Mellon University for financial support. Kosenok acknowledges the support of the Ministry of Education and Science of the Russian Federation, grant No. 14.U04.31.0002, administered through the NES CSDSI. We are grateful to Steve Glazerman, the staff from FOCUS and 21st Century School Fund, and the charter school principals we interviewed for answering our questions on charter schools in D.C. We thank Jeremy Fox, Benjamin Skrainka and Che-Lin Su for conversations on MPEC and SNOPT. We owe special thanks to Michael Saunders for his assistance with the SNOPT and MINOS algorithms. For their help, we thank our research assistants Nick DeAngelis, Nathalie Gonzalez Prieto, Gary Livacari, Sivie Naimer, Djulustan Nikiforov, and Hon Ming Quek. Jeff Reminga assisted us with the computational aspects of the project, and Bill Buckingham from the Applied Population Lab at the University of Wisconsin-Madison provided Arc GIS assistance. All errors are ours.
# **1 Introduction**

The dismal academic performance of public schools in urban school districts has been a growing concern in recent decades. Charter schools provide families with additional school choices and are seen by many as a possible solution. Unlike traditional public schools, charter schools are run independently of school districts by private individuals and associations. They receive public funding in the form of a per-student stipend and do not have residence requirements; if oversubscribed, they determine admission by lottery. Charters are free from many regulations that apply to traditional public schools, but are subject to the same accountability requirements and are regulated by state laws. The first law passed in Minnesota in 1991 and has been followed by laws in 42 states and the District of Columbia, all of which differ widely in their permissiveness towards charters. Currently, the nation’s 6,200 charters serve about 2 million students, or 3.4 percent of the primary and secondary market. While seemingly small, this market share conceals large variation across states and districts.

A prospective charter entrant presents a proposal to the chartering entity. The proposal, akin to a business proposal, specifies the school’s mission, curricular focus (such as arts or language), grades served, teaching methods, anticipated enrollment, intended facilities, and financial plan. The decision to open a charter is similar to that of opening a firm in that both seek to exploit a perceived opportunity. For example, in a residence-based system, a low-income neighborhood with low-achieving public schools creates an opportunity for a charter entrant to serve households unsatisfied with the local public schools. Other example opportunities are middle-class families reasonably well served by local public schools but interested in different academic programs, or families attending private schools but willing to try charter schools to avoid tuition.

In this paper we investigate charter entry and household school choice for Washington, D.C. We document charter entry by geographic area, curricular focus and grade level to gain insight into the opportunities they exploit. We then explore how households sort among public, private and charter schools, and how the entry, exit or relocation of a school affects others. We also study the critical role of the chartering entity (henceforth, the regulator) in this market, quantify welfare gains from charters, and investigate how the educational landscape responds to regulatory changes.

Addressing these research questions is challenging. For example, when a student enrolls in a new charter school he affects the peer characteristics of both his new and former school. In other words, charter entry triggers equilibrium effects as students re-sort among schools. Although the entrant can specify some aspects of the school, like thematic focus and educational philosophy, the student body composition is largely beyond its control. The uncertainty about demand for charters and the regulator poses an additional research challenge. The uncertainty is more severe for new entrants, whose 2
ability to run the new enterprise may not be known.

Thus, we develop and estimate an equilibrium model of household school choice, charter school entry and school interaction in a large urban school district. In the model, a charter entry point is a combination of location (neighborhood), grade level and focus. For some entry points, prospective entrants submit entry applications to the regulator. Charter funding is connected to enrollment and prospective entrants must be financially viable. Hence, the regulator forecasts an applicant’s enrollment and peer characteristics as a function of its entry point and approves those expected to be financially viable.

We estimate the model using a unique and detailed data set from Washington D.C. from 2003 to 2007. The main data set contains information for all public, private and charter schools in the city including enrollment by grade, school demographics, focus and proficiency rates in standardized tests. We supplement these data with neighborhood-level information on charter school attendance and travel distance to charter and public schools. Lacking student-level data, we further augment the school-level data with the block-group level empirical distribution of child age, race, poverty status and family income, and draw from this distribution in order to calculate the model’s predictions. Since market shares for public, private and charter schools vary widely across grades, we define a market as a grade-year combination. We estimate the model in three stages corresponding to student demand, school supply and proficiency rates.

We model schools as differentiated products and estimate the demand side of the model using an approach similar to Berry et al (1995), henceforth BLP. We allow for a school-grade-year quality component (such as teacher quality) observable to households but not to the researcher. The ensuing correlation between school peer characteristics and the unobserved quality component is similar to the correlation between price and unobserved quality in BLP. Unlike price, which is determined by the firm under consideration, peer characteristics are determined by aggregate household choices and are similar to Bayer and Timmins’s (2007) local spillovers. Following Nevo (2000, 2001), we exploit the panel structure of our data and include school, grade and year fixed effects to capture some variation in the unobserved quality component. The school fixed effects are our estimates of school quality; they capture unmeasured factors in household choices such as school climate and culture, length of school day and year and facility characteristics. When estimating parameters of the proficiency rate function we estimate a separate set of school fixed effects that capture the ability of schools to raise passing rates in standardized math tests, and these constitute our measure of school value added.

A single, large urban district allows one to study the behavior of charters that confront the same institutional structure. We chose Washington, D.C. because it has a permissive, well-established 1996 charter law under which the charter school sector has grown to 44 percent of total public school enrollment as of 2013-2014.<sup>1</sup> It has a single

> 1As of 2013-2014, 12 districts had more than a 30 percent charter share. The five largest shares were

3
public school district, District of Columbia Public Schools (DCPS), which facilitates research design and data collection. Finally, it is relatively large with substantial variation in household demographics, which provides scope for charter entry.

Most charter entrants in D.C. have located in the most disadvantaged areas of the city, serve elementary and middle school students and offer a specialized curriculum. According to our estimates, household preferences over school characteristics are highly heterogenous, as are school quality and value added. On average, charter schools have a quality premium relative to public schools in middle school and in the District’s most disadvantaged areas. At the same time, fixed costs for charters are particularly high in those geographic areas and for high school. From a social standpoint, the existence of charter schools yields net benefits. Welfare gains from charters are highest for middle-school students, for whom charters contribute the most in quantity and quality of options, and for black, poor students in all school levels.

Given these benefits, in our counterfactuals we investigate alternative avenues for charter expansion in D.C., namely, a funding increase, a relaxation of approval (authorization) standards, and policies aimed at raising the supply of prospective entrants. Our results indicate that raising the supply of prospective entrants while maintaining strict approval standards is welfare-enhancing. Policies that facilitate the application process by aiding entrants in obtaining building facilities, developing business and instructional plans, learning from other charters and navigating bureaucratic processes can raise the supply of prospective entrants.

Throughout we make several contributions. First, we develop and estimate a rich yet tractable model of charter entry. While most charter school literature studies achievement effects,<sup>2</sup> relatively little research has focused on entry. In a reduced form fashion, Glomm et al (2005), Rincke (2007), Bifulco and Buerger (2012) and Imberman (2011) study charter entry while Henig and MacDonald (2002) study early charter location in Washington, D.C. Cardon (2003) models entrant quality choice when facing an existing public school. Closest to our approach is Mehta’s (2012) structural study of charter entry in North Carolina. We differ from Mehta in several ways: we model student heterogeneity in race, income and poverty status; we endogenize student body composition in these characteristics; and we include private schools in the student choice set. While we model charters as responsive to public schools, we do not model public school strategic response to charters given the lack of evidence for it - as explained below. In our model, as in reality, all charters in the economy are available to households regardless of their residential location, and this means that each public school competes against in New Orleans (91 percent), Detroit (55 percent), D.C. (44 percent), Flint (44 percent) and Cleveland (39 percent). Source: http://www.publiccharters.org.

> 2For a comprehensive review of the charter achievement literature, see Betts and Tang (2011). For recent studies, see Angrist et al (2013), Clark et al (2011) and references therein.

4
potentially many charters, and vice versa. Finally, we model charter heterogeneity in curricular focus, grade coverage and costs.<sup>3</sup> Second, we contribute to the empirical literature on school choice. While others have estimated school choice models with endogenous peer characteristics (Ferreyra 2007, Altonji et al 2010), we rely on the full choice set of private and charter schools, and model unobserved school quality. In addition to market shares, we match school peer characteristics, neighborhood fraction of children enrolled in charter schools and neighborhood average travel distance to public and charter schools.

Using the full school choice set in addition to modeling school unobserved quality poses severe computational challenges. Hence, we recast our demand-side estimation as a mathematical programming with equilibrium constraints (MPEC) problem following Dube et al (2012) and Skrainka (2012). While these authors supply analytical gradients and Hessians to optimization software, we combine two software solvers in such a way that requires only first-order derivatives, and for these we use a symbolic differentiation tool. We can therefore experiment with different model specifications without recoding derivatives. Thus, our research lies at the frontier of computational methods and estimation.

Third, we contribute to the literature on firm entry in industrial organization, recently reviewed by Draganska et al (2008). We develop a supply-side model of charters featuring the regulator’s key role. The model is realistic as well as tractable, and could be applied to other regulated industries such as child care provision and for-profit tertiary education. The entry literature typically uses reduced-form demand specifications, yet we specify a structural model of school choice and allow for unobserved school quality, as in Carranza et al (2011). A major focus of the entry literature is the strategic interaction between entrants and/or incumbents. We do not, however, model public or private school decisions because there is limited school entry and exit activity during our sample period, and this precludes the identification of a strategic decision-making model for them. Moreover, the six superintendents DCPS has had between 1998 and 2007, coupled with its financial instability, suggests that it may not have been able to react strategically to charters during our sample period.

The rest of the paper proceeds as follows. Section 2 describes the institutional framework and data sources. Section 3 describes basic data patterns. Section 4 presents the model. Section 5 describes the estimation strategy, and Section 6 presents estimation results. In Section 7 we discuss counterfactual results, and Section 8 concludes.

3Other related work includes Walters (2012) and Neilson (2013). Using data on charter school lotteries and individual-level school choice and achievement, Walters estimates preference and achievement parameters. Neilson (2013) uses Chilean student-level data to estimate achievement and BLP-style preference parameters. Neither Walters nor Neilson model school entry or endogenous peer characteristics.

5
# **2 Institutional Framework and Data Sources**

In 1995, Congress passed the DC School Reform Act allowing for the creation of charter schools in the District and instituting the DC Board of Education (BOE) as a charter authorizer. The Public Charter School Board (PCSB), created in 1996 as an additional, independent authorizer, has been the sole authorizing and supervising entity since 2006. Charters in D.C. are autonomous, non-profit institutions. They receive the same operational per-pupil funding as public schools. In addition, they receive a per-pupil facilities allowance. Since funding is fungible, henceforth “reimbursement” refers to total (operational plus facilities-related) per-student funding.

The Office of the Mayor has had direct authority over DCPS since 2007. DCPS includes multiple attendance zones for each grade level; middle- and high-school attendance zones are much larger than elementary school zones. At the “state” level, the overarching institution for public and charter schools is the Office of State Superintendent of Education (OSSE). In what follows, “total enrollment” refers to the aggregate over public, private and charter schools, and “total public enrollment” to the aggregate over traditional public and charter schools.

We focus on the 2003-2007 period in order to maximize data quality and comparability over time and across schools. In addition, 2007 marked the beginning of important changes in DCPS and hence constitutes a good endpoint for us.<sup>4</sup> Our data include school-level information on every public, charter and private school in Washington, D.C. for 2003-2007, neighorhood level information on school choice and distance traveled to school for 2003-2006, and block group-level information on child age, race, poverty status, and family income. Appendix A provides further details on the data.

While public and private schools have one campus each, many charters have multiple campuses. Hence, our unit of observation is a campus-year; “campus” is the same as “school” for single-campus schools.<sup>5</sup> We have 700, 228 and 341 campus-year observations for public, charter and private schools respectively. Our dataset includes regular schools; it excludes special education and alternative schools, schools with residential programs and early childhood centers. For each observation we have address, grade enrollment for kindergarten through 12th grade, percent of students of each ethnicity (black, white and Hispanic),<sup>6</sup> and percent of low-income (or “poor”) students, who qualify for free or reduced lunch. We also have the school’s thematic focus, which

> 4In 2007, Michelle Rhee began her tenure as chancellor of DCPS. She implemented a number of reforms, such as closing and merging schools, offering special programs and changing grade configurations, etc.

> 5A campus is identified by its name and not its geographic location. For instance, a campus that moves but retains its name is still considered the same campus.

> 6Since students of other races (mostly Asian) constitute only 2.26 percent of the total K-12 enrollment, for computational reasons we folded them into the white category.

6
we classify into Core, Language, Arts, Vocational and Other (math and science, civics, etc.). In addition, for public and charter schools we have reading and math proficiency rates (i.e., the fraction of students who pass D.C.’s reading and math tests); for charter schools we have per-student reimbursement by grade and year; and for private schools we have school type (Catholic, other religious and non-sectarian) and tuition.

Enrollment and proficiency for public and charter schools come from OSSE. Public school addresses and student demographics come from the Common Core of Data (CCD) from the National Center for Education Statistics (NCES) and OSSE. Curricular focus (henceforth, focus) for public schools come from Filardo et al (2008). Charters’ student body composition and proficiency rates come from OSSE and the School Performance Reports (SPRs). Charters’ focus comes from schools’ own statements, SPRs and Filardo et al (2008). Charter reimbursement rates come from D.C.’s Office of the Chief Financial Officer. Further information comes from past Internet archives and from Friends of Choice in Urban Schools. A complicating factor in charter data collection is the dispersion and inconsistencies of data sources, and their nonuniform treatment of multi-campus charters.

NCES’s Private School Survey (PSS) is our main source of private school data. Since PSS is biennial, we use the 2003, 2005 and 2007 waves. We impute 2004 data by linear interpolation of 2003 and 2005, and similarly for 2006. Average school tuition for school year 2002-2003 comes from Salisbury (2003). All dollar amounts are expressed in dollars of year 2000.

We follow DCPS’s criteria and classify schools into the following grade levels: elementary (covering grades in the K-6 range, which was the typical range for public elementary schools in 2003-2007), middle (covering grades 7th and/or 8th), high (covering grades in the 9th-12th grade range), and elementary/middle, middle/high, and elementary/middle/high. Mixed-level categories (such as middle/high) are quite common among charters. Note that a grade level is a _set_ of grades rather than a single grade.

Our neighborhood-level data comes from Filardo et al (2008)’s data appendices. Local urban planning agencies use the concept of “neighborhood cluster” to proxy for a neighborhood, and group D.C.’s Census tracts into 39 clusters. We observe each neighborhood’s fraction of children enrolled in charter schools relative to total public enrollment, and average distance traveled to public or charter schools. An alternative (but larger) measure of neighborhood is given by wards. The District has eight wards; ward 3, in the Northwest, is the most advantaged, and wards 7 and 8, in the Southeast, are the most disadvantaged. When convenient we split the city into three regions: West (ward 3 and some parts of ward 2), Southeast (wards 7 and 8) and Northeast.

For the sake of demand estimation, ideally we would observe the joint distribution of child grade, race, poverty status and parental income at the block group level (there are 433 block groups in D.C.) for every year between 2003 and 2007. Since 7
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for opening a charter school or not. The prospective entrant receives a random draw of the nonpecuniary net benefit from operating a charter, based on whether it decides to submit an entry application to the regulator. To decide whether to authorize the entry, the regulator forecasts the prospective entrant’s enrollment by predicting household _equilibrium_ choice of school. In addition, the regulator decides whether incumbent charters can remain open based on their financial and academic viability.

Thus, our model has multiple stages of charter, regulator and household actions. We start with the household choice stage.

# **4.1 Household Choice of School**

We use _J_ to denote total number of schools in the economy. Households in the economy have one child each. In what follows, we use “household”, “parent”, “child” and “student” interchangeably. Student _i_ is described by variables ( _D_ ,l, _I_ , _g_ , _ε_ ), where: • _D_ is a row vector describing student demographics. In our data this vector contains � _D_ = 3 binary elements indicating whether the household is white, Hispanic (omitted category is black) and non-poor, respectively.

• l, l ∈{1,..., _L_ } is household location in one of the economy’s _L_ neighborhoods.<sup>12</sup>

- _I_ is annual income of the student’s family.

• _g_ is the child’s grade, ranging from _g_ = 0 (kindergarten) to _g_ = 12 (12th grade).

• _ε_ is a vector that describes the student’s idiosyncratic preference for each school.

Subscripts _j_ and _t_ denote a school campus and school year respectively. We treat multiple campuses of the same school as separate units because they are often run as such. In what follows, we use “school” and “campus” interchangeably as well as “school year” and “year”. When making its choice for year _t_ , household _i_ takes into account the following characteristics for school _j_ :

- _κ jt_ is the set of grades served by the school, or “grade level.”

- _xi jt_ is the geographic distance from the household’s residence to the school.

• _y j_ is a row vector with time-invariant school characteristics such as type (public, charter, Catholic, other religious, nonsectarian) or focus (Core, Language, Arts, Vocational, Other). For brevity we refer to _y j_ as “focus.”

- _p jgt_ is tuition. It is always equal to zero for public and charter schools.

• _D_<sup>~~�~~</sup> _jt_ is the row vector of households’ beliefs about the school’s peer (or student body) composition at _t_ . As explained below, in equilibrium these beliefs are consistent with the realized schools’ peer composition. Empirically we use another variable, _D jt_ , which

> 12A child’s location determines her travel distance to each school. We take student location as given and do not model household residential choice. For models of joint residential and school choice, see Nechyba (2000) and Ferreyra (2007). In estimation we measure distance as network distance, expressed in miles. We use Census block groups as household locations for demand estimation, and neighborhood clusters as the locations used to define entry points for supply estimation.

15
is the school’s actual percent of white, Hispanic and non-poor students, calculated by averaging _D_ s for all students in _j_ .

- _ξ jgt_ is an unobserved (to us) characteristic of the school and grade.

We define a market as a (grade _g_ , year _t_ ) pair. Market size _Mgt_ is the number of students who are eligible to enroll in grade _g_ in year _t_ .

The household indirect utility function is: where _δ jgt_ and _μijgt_ are defined below in (2) and (3). As explained in Appendix D.1, (1) incorporates both household preference over school characteristics as well as the contribution of those characteristics to expected student achievement. Lack of achievement data at the student level prevents us from separately identifying those two aspects. The baseline utility component _δ jgt_ is equal to: where vector _β_ captures student preferences for school focus as well as focus impact on achievement, and vector _α_ captures household preferences over peer characteristics as well as the impact of these on expected achievement. Thus, the model captures the potential tension between enhancing productivity and attracting students. For example, an Arts curriculum may not enhance achievement, but parents may like it. Similarly, _ξ jgt_ captures elements such as teacher characteristics that may reflect a similar tension. The student-specific component of (1) is: The household may choose the outside good ( _j_ = 0) with normalized utility _Ui_ 0 _gt_ = _εi_ 0 _gt_ . The outside good may represent home schooling, dropping out of school, etc.

As in Nevo (2000, 2001), we decompose the demand shock as follows: _ξ jgt_ = _ξ j_ + _ξg_ + _ξt_ + ∆ _ξ jgt_ , where _ξ j_ captures school-specific elements such as culture and educational philosophy; we refer to _ξ j_ as “school quality.” Component _ξg_ captures grade-specific elements, while _ξt_ captures time-varying elements common to all schools and grades, such as city-wide income shocks. We normalize as follows: _E_ (∆ _ξ jgt_ ) = 0. Hence, _ξ j_ + _ξg_ + _ξt_ is the mean of _ξ jgt_ , and ∆ _ξ jgt_ is a deviation from it.

Among schools offering grade _g_ in year _t_ , the school choice set for child _i_ encompasses all public schools (as if there were open enrollment),<sup>13</sup> all charter schools,

> 13Data limitations motivate this “open enrollment” assumption, innocuous for the development of the model. Using GIS software we established the public schools assigned to children in each block group depending on their grade level. However, based on the resulting assignment and other sources (Filardo et al 2008, and phone conversations with DCPS staff), we concluded that the _actual_ assignment mechanism in D.C. during the sample period was based on residential location only to a limited extent. For instance, Filardo et al document that approximately half of the children enrolled in public schools attend an out-ofboundary shool. Moreover, the mechanism was seemingly not systematic across the District. Hence, we simplified by assuming public school open enrollment.

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and all private schools affordable to the household. We assume that _i_ can afford private school _j_ if tuition _p jgt_ does not exceed a certain share of the household’s annual income _Ii_ . Let _Jgt_<sup>_i_be the number of schools in</sup><sup>_i_’s choice set.Student</sup><sup>_i_chooses a school in order</sup> to maximize utility. Assuming that the error terms in (1) are i.i.d. type I extreme value, the probability that household _i_ chooses school _j_ for year _t_ is: where vector _θ_<sup>_d_</sup> refers to the collection of demand-side parameters to be estimated and _ygt_ is the vector that describes focuses of the schools offering _g_ at _t_ . Vectors _D_<sup>~~�~~</sup> _gt_ , _ξgt_ and _pgt_ have similar meaning. _Xigt_ denotes the observable variables that are either specific to _i_ or to its match with the schools: _Di_ , _Ii_ , and _xi jt_ .

Let _h_ ( _D_ , _I_ ,l| _g_ ) be the joint distribution of student demographics, income and locations conditional on grade. Given (4), school _j_ ’s expected market share for grade _g_ in _t_ is:

The expected number of students choosing school _j_ for grade _g_ at _t_ is: � _N jgt ygt_ , _D_<sup>~~�~~</sup> _gt_ , _ξgt_ , _pgt_ ; _θ_<sup>_d_�</sup> = _Mgt S_<sup>�</sup> _jgt_ (·) . (6) � The total expected number of students in school _j_ at year _t_ is hence equal to where _yt_ , _D_<sup>~~�~~</sup> _t_ , and _pt_ are vectors that describes focuses, household beliefs on demographic compositions and prices respectively of all operating schools in _t_ , and Ξ _t_ is the vector that stores the _ξ jgt_ s of all operating schools.

For computational tractability, in demand estimation we replace _D_<sup>~~�~~</sup> by the observed peer characteristics _D_ in the right hand side of equations (4)-(8).<sup>14</sup> Lack of individual-level data on achievement precludes the identification of the achievement function (see (25) in Appendix D.3). Nonetheless, we can derive the following equation for a school’s expected proficiency rate (see Appendix D.3 for details): _q jt_ = _y jα_<sup>_q_</sup> + _D jtφ_<sup>_q_</sup> +[ _y j_ ⊗ _D jt_ ] _ω_<sup>_q_</sup> + _ξ j_<sup>_q_+</sup><sup>_ξ_</sup> _t_<sup>_q_+∆</sup><sup>_ξ q_</sup> _jt_ (9) where _ξ j_<sup>_q_is the school’s productivity shock (later referred to as “value added”) while</sup><sup>_ξ_</sup> _t_<sup>_q_</sup> captures shocks that affect proficiency rates in all schools and grades in _t_ . The error term

> 14This exploits the fact that the difference between _D_ and _D_<sup>~~�~~</sup> is due to sampling (or measurement) error.

17
∆ _ξ jt_<sup>_q_is a function of</sup><sup>_j_’s idiosyncratic productivity shocks and the mean of idiosyncratic</sup> components of students’ performance.

# **4.2 School Supply**

Below we present a game among charter schools, the regulator and households. The game reflects the institutional aspects of regulator behavior and charter entry, exit and relocation in Washington, D.C.<sup>15</sup> Let **M** _t_ be a market structure in year _t_ . It describes operating schools’ characteristics (type, focus, location, grades served and tuition) observed by students when making their choice with the exception of Ξ _t_ .

## **Payoffs of the agents.**

The agents that participate in the game receive the following payoffs: _Household’s payoff:_ For any ( **M** _t_ , Ξ _t_ ) households obtain utility as described in Section 4.1.

_Entrant’s payoff:_ When a charter operates, the corresponding entrant receives nonpecuniary net benefit _B_ and 0 otherwise.<sup>16</sup> We assume that _B_ is a random draw from a distribution with cdf _FB_ (·). It is independent of the charter’s expected profit and of other prospective entrants’ _B_ .

The _regulator_ receives charter applications and decides whether to approve them based on charters’ expected profit, as explained below. It also decides whether incumbent charters can remain open based on their profits and proficiency.

## **Consistent household beliefs.**

For a given market structure **M** _t_ and schools’ qualities Ξ _t_ , each household forms beliefs _D_<sup> ̃</sup> ( **M** _t_ , Ξ _t_ ) about schools’ demographic composition and chooses a school accordingly. From now on, we use _D_<sup>~~�~~</sup> to denote a _set of consistent household beliefs_ which satisfy the following system of equations: where (10) is based on (8), and _Pjgt_<sup>_i_(</sup> _D_<sup>~~�~~</sup> ), _N_<sup>�</sup> _jgt_ ( _D_<sup>~~�~~</sup> ) and _N_<sup>�</sup> _jt_ ( _D_<sup>~~�~~</sup> ) are calculated as in (4), (6) and (7) correspondingly.

Generally, (10) has multiple solutions and hence the model has multiple equilibria.<sup>17</sup> The issue does not affect demand-side estimation because we use observed values, _D_ , to calculate predicted market shares in demand estimation. However, supplyside estimation and counterfactuals might be affected. Hence, based on (10) we use 15The model is based on conversations with staff from PCSB and charter-advocacy organizations, and with charter founders. Appendix D.2 provides further institutional details.

16These benefits represent the net present value of the founder’s satisfaction from contributing to society through the charter school, net of the effort and time cost of submitting the charter entry application.

17For instance, white households may choose school A if they believe that other white households will attend A, yet they may choose school B if they believe that other white households will attend B.

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a _tatonnement_ -type of algorithm. We choose an initial value for _D_<sup>~~�~~</sup> , _D_<sup>~~�~~</sup> <u>0,</u> and calculate the sequence _D_<sup>~~�~~</sup> _<u>k</u>_ ( _k_ = 0, 1,...). We obtain _D_<sup>~~�~~</sup> _<u>k</u>_ from _D_<sup>~~�~~</sup> _<u>k</u>_ −1 by substituting the latter into right hand side of (10) for _D_<sup>~~�~~</sup> until convergence.<sup>18</sup> To address multiplicity we choose the equilibrium attained by iterating from a specific starting point. The starting point is observed school demographics for incumbents, and a linear function of neighborhood demographic and school characteristics for new entrants.

# **Entry points, expected charter entrant enrollment and profit.**

In our model we assume that there is one potential charter entrant for each entry point in each year. Entry points result from all possible combinations of _L_ locations, _Y_ focuses and _K_ grade levels, for a total of _E_ = _L_ × _Y_ × _K_ entry points or potential entrants. Each potential entrant may enter the market or not. Entering the market requires the submission of an entry application as well as the regulator’s approval of the application. Indicator _d_<sup>_e_</sup> _jt_<sup>denotes whether</sup><sup>_j_enters in</sup><sup>_t_.19</sup> Consider potential entrant _j_ for entry point (l, _y_ , _κ_ ). When receiving applications in school year _t_ − 1, the regulator knows which schools will be operating in _t_ .<sup>20</sup> . It also observes _ξg_ s, _ξt_ , and the demand shocks of all operating schools in _t_ , _ξ_ − _jgt_ s. However, the regulator does not observe potential entrants’ _ξ j_ s or ∆ _ξ j_ s. We assume that shocks _ξ j_ and ∆ _ξ jg_ for potential entrants are independently distributed with cdfs ϝ _ξ j_ (.) and ϝ∆ _ξ_ (.), respectively. These distributions are common knowledge.

The regulator decides whether to approve _j_ ’s application based on _j_ ’s expected profit. Hence, the regulator predicts equilibrium household choices for a market structure **M** _t_ that includes includes _j_ . In this equilibrium, _j_ ’s expected enrollment for _g_ is: � _Eξ N_<sup>�</sup> _jgt_ ( **M** _t_ ; _θ_<sup>_d_</sup> ) = _N jgt_ ( _ygt_ , _D_<sup>~~�~~</sup> _gt_ , { _ξ jgt_ , _ξ_ − _jgt_ }, _pgt_ ; _θ_<sup>_d_</sup> ) _d_ ϝ _j_ ( _ξ jgt_ ) (11) � where _N_<sup>�</sup> _jgt_ (·) is given by (6); _ygt_ and _pgt_ correspond to the schools in **M** _t_ ; _D_<sup>~~�~~</sup> _gt_ are consistent beliefs on peer characteristics given **M** _t_ and { _ξ jgt_ , _ξ_ − _jgt_ } for schools that offer _g_ ; and ϝ _j_ (.) (which can be derived from ϝ _ξ j_ (.) and ϝ∆ _ξ_ (.)) stands for the demand

> 18Given that _D_ ~~�~~ is defined on a compact set (all values in _D_<sup>~~�~~</sup> are between 0 and 1) and (10) describes a continuous mapping, such iterations always converge. In addition, in the iterations we only allow for changes in _D_<sup>~~�~~</sup> within a specified range (±0.06). In the absence of capacity constraints, this restriction prevents unrealistic changes in enrollments and student body compositions.

> 19We do not model the behavior of multi-campus charters. Given the sample low number of entries, identifying a separate entry model for them would not be possible. In addition, the computational burden of such a model would be substantially higher than the current model’s. Nonetheless, when estimating utility function parameters we include an indicator for multi-campus charters (see Table 5). This purges the estimates of school quality _ξ j_ s of possible multi-campus biases. Since these estimates are used to form the empirical distribution of entrants’ quality ϝ _ξ j_ (.), eliminating such biases is consistent with modeling entry of single-campus charters.

> 20As explained below, the regulator receives entry applications in early Spring. It is reasonable to assume that at that point the regulator knows the set of operating schools and time-specific shock for the coming school year.

19
shock distribution for grade _g_ in year _t_ . For the whole school,<sup>21</sup> the regulator calculates _j_ ’s expected profits as follows ̄ _π_<sup>_e_</sup> _jt_ � _d_<sup>_e_</sup> _jt_<sup>= 1,</sup><sup>**M**</sup><sup>_t_;</sup><sup>_θ d_�</sup> = ∑ _g_ ∈ _κ Eξ N_<sup>ˆ</sup> _jgt_ ( **M** _t_ ; _θ_<sup>_d_</sup> )( _Rgt_ − _V_ ) − _ζ_ − _Fκ_ l + _σν ν_<sup>_e_</sup> _jt_ (12) where _ζ_ is entry cost, _Rgt_ is per-child reimbursement in _g_ , _V_ is per-child variable cost, _Fκ_ l is the fixed cost of operating in location l and grade level _κ_ , and _ν_<sup>_e_</sup> _jt_<sup>is the entrant’s</sup> type,<sup>22</sup> unobserved to us. We assume that _ν_<sup>_e_</sup> _jt_<sup>followsani.i.d.typeIextremevalue</sup> distribution, and _σν_ is a scale parameter. Given the extensive information submitted by charter applicants, we assume that the regulator observes _σν ν_<sup>_e_</sup> _jt_<sup>.</sup>

# **Timing of entry-exit-relocation events.**

In this game we model market activity as a multiperiod interaction. The game period encompasses Spring semester and Summer of school year _t_ −1, and Fall semester of school year _t_ (i.e., the game period is a calendar year). During the game period market participants interact in the sequence of actions of the following stage game:

**Step 1 (** **_Submission of applications by the potential entrants_ )** . For each entry point, prospective entrant _j_ privately learns the value of its _B j_ . All draws are independent. Based on its draw, _j_ decides whether to submit an application to start operating in Fall _t_ + 1 (i.e., in the next game period). Prospective entrants that submit an application learn their type _ν_<sup>_e_</sup> _jt_<sup>while others do not.In reality this step takes place in Spring</sup><sup>_t_−1.</sup>

**Step 2** **_(Relocation opportunities for incumbent charters)._** For each incumbent charter _j_ , located in l _jt_ −1 in _t_ − 1, a new location l, l = l _jt_ −1, may become available with probability Pr(l) = exp{  ̆ _α_ − _β_<sup> ̆</sup> _d_ ll _jt_ −1}/(1 + ∑<sup>_L_</sup> l<sup>′</sup> =1:l<sup>′</sup> =l _jt_ −1<sup>exp{</sup><sup>_α_ ̆−</sup><sup>_β_ ̆</sup><sup>_d_l′l</sup> _jt_ −1<sup>}),</sup> where _d_ l′l _jt_ −1 is the distance between l<sup>′</sup> and l _jt_ −1. If a new location becomes available, the charter moves and reports the move to the regulator. Relocations become public knowledge. In reality, the regulator is informed of the move in Spring _t_ − 1, and the actual move usually takes place in Summer _t_ − 1.

**Step 3** **_(Public and private schools)._** Public and private schools make their entry, exit and relocation decisions, and these become public knowledge. In reality this step takes place in Spring _t_ − 1 and the decisions become effective in Fall _t_ .

**Step 4 (** **_Households’ school choice_ ).** The set of operating schools in Fall _t_ constitutes the market structure **M** _t_ . This includes all incumbent public, private and charter schools that remain open in Fall _t_ , and the new charters previously authorized in Spring

> 21We assume for simplicity that the entrant offers its full set of grades since entering. Although in reality charters add grades over time until reaching their full grade coverage, the regulator does consider expected profits for the full grade coverage when deciding on authorization. Also for simplicity we do not model capacity choice, although the model naturally precludes the entry of very small charters that

> 22This shock encompasses aspects of the school’s financial performance that are observed to the regulator but not to the econometrician, such as revenue sources besides student reimbursement, or the charter’s deviation from average costs in the chosen location.

20
_t_ − 2 that start operating in Fall _t_ . The _ξ jgt_ s of all the schools in **M** _t_ become public knowledge. Thus, in Spring _t_ − 1 households choose a school for year _t_ .

**Step 5** **_(Processing of entry applications)._** The regulator decides whether to authorize the entry applications submitted in Step 1. If approved, a new charter starts operating in Fall _t_ + 1. The regulator makes its decision based on market structure **M** _t_<sup>_j_, which</sup> augments **M** _t_ with new charter _j_ , as if _j_ were to open in Fall _t_ . The regulator learns _ν_<sup>_e_</sup> _jt_ of _π_  ̄<sup>_e_</sup> _jt_ each<sup>(·,·) is given</sup> applicants.<sup>by (12),</sup> Applicant<sup>and</sup><sup>_σ_</sup> _j_<sup>_νν_</sup> is<sup>_e_</sup> _jt_ approved0<sup>is an entry threshold.</sup> iff _π_  ̄<sup>_e_</sup> _jt_ � _d_<sup>_e_</sup> _jt_<sup>= 1Component,</sup><sup>**M**</sup> _t_<sup>_j_;</sup><sup>_θ d_�</sup> ≥<sup>_ν_</sup> _σ_<sup>_e_</sup> _jtν_ 0 _ν_<sup>follows an</sup><sup>_e_</sup> _jt_ 0<sup>,where</sup> i.i.d. type I extreme value distribution. Applications are processed independently. Authorized charters carry out several activities, such as student and teacher recruiting and building renovations, during the remainder of the current game period and the beginning of the next one. We assume that all parties learn entrants’ demand shocks _ξ jgt_ +1s through these activities. In reality this step takes place in Spring _t_ − 1.

**Step 6** **_(Closing of charter incumbents)._** The regulator determines whether incumbent charters can continue operating. For public and charter schools, academic performance is measured as proficiency in the tests taken at the beginning of the current game period; test results become public information at the end of Step 4. Let _<u>q jt</u>_ stand for charter _j_ ’s proficiency rate in the tests. Charter _j_ must close by the end of _t_ if either its financial mis-performance index _a_<sup>_π_</sup> + _b_<sup>_π_�</sup> ∑ _g_ ∈ _κ N jgt_ ( _Rgt_ − _V_ ) − _Fκ_ l� or its proficiency mis-performance index _a_<sup>_q_</sup> + _b_<sup>_q_</sup> _<u>q</u> jt_ (adjusted for school tenure) exceed a certain threshold. In these indices, _a_<sup>_π_</sup> and _b_<sup>_π_</sup> are parameters related to charters’ financial standing while _a_<sup>_q_</sup> and _b_<sup>_q_</sup> are related to academic performance. Hence, _j_ is closed iff

max � _a_<sup>_π_</sup> + _b_<sup>_π_�</sup> ∑ _g_ ∈ _κ_<sup>_N_</sup> _jgt_<sup>(</sup><sup>_R_</sup> _gt_<sup>−</sup><sup>_V_)−</sup><sup>_F_</sup> _κ_ l� , _a_<sup>_q_</sup> + _b_<sup>_q_</sup> _<u>q jt</u>_ + _c_<sup>_q_</sup> _e jt_ � + _ε jt_ 1 ≥ _ε jt_ 0 (13) where _e jt_ is a closing eligibility variable, equal to 1 if the charter has at least five years of operation and 0 otherwise.<sup>23</sup> Shock _ε jt_ 1 captures elements of the charter’s performance observed to the regulator (not to us), and _ε jt_ 0 is the threshold that the school must surpass. Shocks follow i.i.d. type I extreme value distributions. Charters are evaluated independently. All closings become public knowledge. In reality closings decisions are made during Fall _t_ , and school closings take place at the end of _t_ . **Solution of the game.** The solution of the game is a Perfect Bayesian Equilibrium. It requires consistent beliefs by agents and expected payoff maximizing behavior based on the beliefs.

Agents make choices only in steps 1 and 4. We analyze the equilibrium of the stage game backwards. In Step 4, households exhibit the equilibrium behavior and formation of beliefs that we described previously. In Step 1, the equilibrium strategy on the part of the applicant is to submit an application only when _B_ > 0. The probability of

> 23This dummy is included for empirical purposes, as closings due to academic reasons rarely happen before the school’s fifth year.

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a positive _B_ is equal to _b_ = 1 − _FB_ (0). An increase in _b_ raises the supply of prospective entrants. Such increase could be due to an influx of socially motivated individuals in the economy; to an increase in social appreciation for charters’ contribution; or to a reduction in the cost of preparing an application, locating facilities, developing a business and instructional plan, learning about successful charters, etc. For brevity we refer to an increase in _b_ as a reduction in entrants’ application costs.

# **5 Data and Estimation**

Estimation proceeds in three stages, in which we estimate demand-side parameters _θ_<sup>_d_</sup> , supply-side parameters _θ_<sup>_s_</sup> and proficiency rate parameters _θ_<sup>_q_</sup> respectively. We describe the data and estimation below.

## **5.1 Data**

Our data include 65 markets (13 grades times 5 years) and _J_<sup>_S_</sup> =281 campuses, for a total of _J_<sup>_D_</sup> =1,269 school-year observations and _J_<sup>_X_</sup> =8,112 school-grade-year observations. It also includes _J_<sup>_C_</sup> =153 neigborhood-year observations. Recall that we observe the following school characteristics: type (public, charter, Catholic, other religious, private non-sectarian), location, grades covered, focus, student body composition by race and poverty status, tuition (for private schools) and proficiency rates (for public and charter schools). In the data some characteristics (such as student body composition) change over time. Similarly, household choice sets change as well as some schools enter, exit or change grade coverage. Since we have tuition only for school year 2002, we assign the same value to all years. Lacking direct information on the number of children eligible for each grade, we estimate market sizes _Mgt_ s as described in Appendix A.2. Then, for each market we make _ns_ = 100 draws (each one corresponding to a hypothetical child) from the joint distribution of child race, poverty status, income and location.<sup>24</sup>

## **5.2 Demand Estimation**

To estimate _θ_<sup>_d_</sup> we use Generalized Method of Moments (GMM) and match market shares at the school-grade-year level (“share moments”), student demographic composition at the school-year level (“demographic moments”), and average fraction of students attending charter schools, average distance traveled to public schools, and average distance traveled to charter schools at the neighborhood-year level (“neighborhood moments”). While typical BLP consists of share moments, we augment the GMM objective function with the other moments.

24We assume two ages per grade (for instance, ages 5 and 6 in kindergarten), and draw an equal number of children of each age per grade. Given the low fraction of white and hispanic students in the population, we stratify our sample by year, grade and race.

22
We first calculate moments’ predicted values. We abuse notation and use symbol “�” for predicted values. Consider the _ns_ draws for children eligible to attend grade _g_ in year _t_ . Predicted enrollment for ( _j_ , _g_ , _t_ ) is _N_<sup>ˆ</sup> _jgt_ =<sup>_M_</sup> _ns_<sup>_<u>gt</u>_∑</sup> _i_<sup>_ns_</sup> =1<sup>_P_</sup> _jgt_<sup>_i_</sup> � _ygt_ , _Dgt_ , _ξgt_ , _pgt_ , _Xigt_ ; _θ_<sup>_d_�</sup> , where _Pjgt_<sup>_i_(·) is given by (4) and we use</sup> _D_ to approximate _D_<sup>~~�~~</sup> in the r.h.s. of (4). Predicted enrollment share for ( _j_ , _g_ , _t_ ) is andto _D_<sup>~~�~~</sup> _S jtjgt_ = denotes∑ _g_ ∈ _κ jt_ <u>�</u> _N_ ˆ _nsjg_ its _t_ <u>�observed∑</u><sup>_ns_</sup> _i_ =1∑<sup>_D_</sup> _g_<sup>_i_</sup> ∈<sup>_P_</sup> _κjgt_<sup>_i_</sup> _jt_<sup>(</sup> _N_<sup><u>ˆ</u></sup><sup>_y_</sup> counterpart.<sup>_gt_</sup> _jgt_<sup>,</sup> _Dgt_ , _ξgt_ , _pgt_ , _Xigt_ Predicted; _θ_<sup>_d_</sup> ). Finally,schoollet characteristics _C_<sup>~~�~~</sup> _kt_ be the _C_<sup>�</sup> ×are 1 vectorequal of predicted average values for neighborhood _k_ in year _t_ , with the following _C_<sup>�</sup> = 3 elements: (i) percent of children enrolled in charter schools, (ii) average travel distance for children enrolled in public schools, and (iii) average travel distance for children enrolled in charter schools. Let _Ckt_ denote this vector’s observed counterpart.

Denote by _Xt_ the union of the _Xigt_ sets over all households. We assume that _E_ � _D jt_ | _Xt_ � = _D_<sup>~~�~~</sup> _jt_ and _E_ ~~�~~ _Ckt_ | _Xt_ � = _C_<sup>~~�~~</sup> _kt_ , and that _D jt_ and _Ckt_ are different from their expected values due to sampling (or measurement) error: _u_<sup>_D_</sup> _jt_<sup>=</sup> _D jt_ − _D_<sup>~~�~~</sup> _jt_ and _u_<sup>_C_</sup> _kt_<sup>=</sup> _Ckt_ − _C_<sup>~~�~~</sup> _kt_ . Following BLP and Nevo (2000, 2001), we assume that ∆ _ξ jgt_ is meanindependent of the corresponding instruments: _E_ (∆ _ξ jgt_ | _Z_<sup>_X_</sup> _jgt_<sup>) = 0.Inaddition,weas-</sup> sume _E_ ( _u_<sup>_D_</sup> _jt_<sup>|</sup><sup>_ZD_</sup> _jt_<sup>) = 0 and</sup><sup>_E_</sup> � _u_<sup>_C_</sup> _kt_<sup>|</sup><sup>_Z_</sup> _kt_<sup>_C_</sup> � = 0.

To estimate the BLP model, researchers typically rely on a nested-fixed point algorithm. This finds the baseline utilities _δ_ that equate predicted and observed market shares for each value of _θ_<sup>_d_</sup> . Since this algorithm is slow and potentially inaccurate, we follow Dube et al (2012) and formulate our estimation as a mathematical programming with equilibrium constraints (MPEC). Our MPEC problem, more complex than Dube et al’s given the inclusion of demographic and neighborhood moments, is as follows: where _λX_ , _λD_ and _λC_ are sample interactions of the shocks ∆ _ξ_ , _u_<sup>_D_</sup> and _u_<sup>_C_</sup> with the corresponding instruments (see details in Appendix E.1); _S_<sup>ˆ</sup> (·) is given by (14); and _VX_ , _VD_ and _VC_ are positive definite matrices. The MPEC algorithm simultaneously searches over values of ∆ _ξ_ (and hence _δ_ ) and _θ_<sup>_d_</sup> ; given values for these, it calculates moments’ predicted values. The constraints of the MPEC problem ensure that observed enrollment shares _S_ match predicted shares _S_<sup>ˆ</sup> .

Given the decomposition of the demand shock _ξ jgt_ ,we include school-, gradeand time-fixed effects in the utility function. Since the school fixed effects capture both the value of time-invariant school characteristics, _y jβ_ , and of school quality _ξ j_ in (2), 23
we apply a minimum-distance procedure as in Nevo (2000, 2001) to estimate _β_ and _ξ j_ separately (see Appendix E.1 for details). Finally, we use our estimates of ∆ _ξ jgt_ s for all schools and of _ξ j_ s for entrants to obtain the empirical counterparts of ϝ _ξ_ and ϝ∆ _ξ_ .

# **5.3 Supply estimation**

̆ Supply-side parameters are _θ_<sup>_s_</sup> = _b_ , _ζ_ , _V_ , _F_ , _σν_ , _α_ , _β_<sup> ̆</sup> , _a_<sup>_π_</sup> , _b_<sup>_π_</sup> , _a_<sup>_q_</sup> , _b_<sup>_q_</sup> , _c_<sup>_q_�</sup> . In our � application, the economy includes _L_ = 39 locations (neighborhood clusters), _Y_ = 5 focuses (Core curriculum, Arts, Language, Vocational, Other) and _K_ = 5 grade levels (elementary, middle, high, elementary/middle, middle/high), for a total of _E_ = 975 potential entrants per year. For each school year _t_ we observe the schools operating in the market. We also observe the following: a) new charter entries, authorized in _t_ − 2 based on **M** _t_ −1 as described in the game’s Step 4; b) charter closings; c) charter relocations.<sup>25</sup> Let _Ct_ be the total number of charters operating in _t_ , including incumbents from _t_ − 1 that remain open in _t_ as well as new entrants. Let _C_<sup> ̃</sup> _t_ be the number of incumbent charters that remain in the same location as in _t_ − 1. Let l _jt_ be _j_ ’s location in _t_ , and let l _jt_ be its observed counterpart. Variable _d_<sup>_e_</sup> _jt_<sup>∈{0,1}indicateswhetherthereisanew</sup> entrant in entry point _j_ in year _t_ ; variable _d_<sup>_x_</sup> _jt_<sup>∈{0,1}indicateswhetherincumbent</sup><sup>_j_</sup> closes at the end of _t_ − 1; and variables _d_<sup>_~~e~~_</sup> _jt_<sup>and</sup> _d_<sup>_~~x~~_</sup> _jt_<sup>are the observed counterparts of</sup><sup>_de_</sup> _jt_ and _d_<sup>_x_</sup> _jt_<sup>respectively.The likelihood function is:26</sup> Infrequent entry and exit in our sample complicate the estimation of _θ_<sup>_s_</sup> . Nonetheless, we can calibrate some parameters. According to Buckley and Schneider (2007), there were 71 charter entry applications between Fall 2004 and Fall 2007, which results in _b_<sup>ˆ</sup> = 71 / (975 entry points * 4 years) = 0.018.<sup>27</sup> In order to calibrate _V_ and _F_ , we use budget data for school year 2009-2010, which is the closest to our sample period with publicly available financial data. We use data for the charters in our sample that were still open in 2009, including the new

> 25Closing year is the first year that the school is not in the data. Relocation year is the first year that the school operates in its new location.

> 26Note that the likelihood for 2003 cannot be calculated as we lack data on market structure, charter locations and charter proficiency in 2002.

> 27In practice we assume that the distribution of entries across grade levels is the same as the distribution of entry applications, and use a different value of _b_ for each grade level. This feature improves the performance of our MLE estimator.

24
campuses they had added by then. We run the following regression: � _TC_ = _F_ 0 + _Ve_ · _Enr_ + _FW_ · _W_ + _FSE_ · _SE_ + _FH_ · _H_ + _FM_ · _M_ (16) where _TC_ is school total cost; _Enr_ is enrollment; _W_ and _SE_ indicate whether the school is located in the West or Southeast respectively; _H_ is a high school indicator, and _M_ is an indicator for whether the school offers mixed levels (such as middle/high). The estimate of _Ve_ is our calibrated value for _V_ , and we use the appropriate combination of _F_ 0, _FW_ , _FSE_ , _FH_ and _FM_ estimates to calibrate _F_ by charter location and grade level. We thus impute expected costs to each entrant. We impute expected revenues based on actual reimbursements and predicted enrollment, and obtain expected profits per entrant. We ̆ then use MLE to estimate remaining parameters { _ζ_ , _σν_ , _α_ , _β_<sup> ̆</sup> , _a_<sup>_q_</sup> , _b_<sup>_q_</sup> , _c_<sup>_q_</sup> }.<sup>28</sup>

# **5.4 Proficiency Rate Estimation, and Summary**

According to (9) the observed proficiency rate _<u>q</u> jt_ is given by: _<u>q</u> jt_ = _y jα_<sup>_q_</sup> + _D jtφ_<sup>_q_</sup> +[ _y j_ ⊗ _D jt_ ] _ω_<sup>_q_</sup> + _ξ j_<sup>_q_+</sup><sup>_ξ_</sup> _t_<sup>_q_+∆</sup><sup>_ξ q_</sup> _jt_<sup>+</sup><sup>_vq_</sup> _jt_ (17) where the error term is the addition of the a school-year unobserved shock on proficiency ∆ _ξ jt_<sup>_q_and sampling or measurement error</sup><sup>_vq_</sup> _jt_<sup>.Since ∆</sup><sup>_ξ q_</sup> _jt_<sup>may be correlated with ∆</sup><sup>_ξ jgt_,</sup> _D jt_ is likely to be correlated with _ξ jt_<sup>_q_, thus requiring the use of instrumental variables.</sup> To estimate _α_<sup>_q_</sup> separately from the school fixed effects _ξ j_<sup>_q_,weuseasimilar</sup> procedure to the demand-side estimation. We first run a 2SLS regression of passing rate on campus and year fixed effects, _D jt_ and [ _y j_ ⊗ _D jt_ ]. Then we regress the campus fixed effects estimates on time-invariant school characteristics; the residuals from this regression are our estimates of _ξ j_<sup>_q_,or“valueadded”.Weuseourestimatesof</sup><sup>_ξ q_</sup> _j_<sup>for</sup> entrants to obtain the empirical distribution of value added.

The estimation of proficiency rate parameters is straightforward, and so is Maximum Likelihood estimation of supply side parameters once expected enrollments for potential entrants have been computed.<sup>29</sup> However, GMM estimation of the demand-side parameters is computationally involved as it requires solving the large-scale constrained optimization problem in (15). This problem has 8,436 unknowns – 324 parameters in _θ_<sup>_d_</sup> (including 281 campus fixed effects) and 8,112 elements in the ∆ _ξ_ vector – and 8,112 equality constraints. Through a creative use of solvers, we avoid coding first- or secondorder derivatives and attain great speed, despite our problems’ complicating features

> 28We cannot estimate _aπ_ and _bπ_ because there are no closings for financial reasons in the sample.

> 29In order to calculate entrants’ expected profits for the entry probabilities in the likelihood function, we first compute expected enrollment in (11) for each entry point and year given our estimates of _θ_<sup>_d_</sup> . Recall our assumption that the regulator observes demand shocks _ξ jgt_ s for all schools in the market at _t_ but not for potential entrants. Hence, in order to calculate entrants’ expected enrollments we integrate over the distribution of _ξ jgt_ for potential entrants by using Monte Carlo simulations based on our estimates of ϝ _ξ_ and ϝ∆ _ξ_ . Although this calculation takes multiple days given our number of entry points, for a given set of demand estimates it takes place only once, before the likelihood estimation.

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relative to the typical BLP problem. See Appendix E.3 for computational details.

# **5.5 Instruments**

For the identification of the demand-side and proficiency rate parameters, the main concern is the endogeneity of peer characteristics in (1) and (17). Thus, we instrument for _D jt_ using local demographics of the school’s neighborhood as of year 2000. To the extent that these are correlated with demand shocks _ξ jgt_ , this correlation is absorbed by the campus fixed effect _ξ j_ , and we expect ∆ _ξ jgt_ to be mean-independent of local demographics. We instrument using the following local neighborhood characteristics: percent of school-age children of each race and poverty status, average family income, average house value, percent of owner-occupied housing units, average number of children per family, number of public, private and charter schools, percent of families in each income bracket, ward indicators, and interactions between some of these variables with school type and grade level. Additional instruments include campus, grade and year dummies.

The instruments for the sampling error in school-year student demographics include school type, focus, and interactions of school type with ward. The instruments for sampling error in neighborhood-level variables are neighborhood-level number of public and charter schools, average family income, racial composition of school-age children, age distribution of school-age children, and ward dummies. Finally, the instruments used for proficiency rate estimation contains local demographics for the schools’ neighborhoods, similar to ones for ∆ _ξ jgt_ . They also include campus and year dummies.

# **5.6**

On the demand side, variation in school type, focus, grade level, location, tuition and peer characteristics vis-a-vis variation in household demographics and location helps identify preference parameters. A sufficient condition for identification is that the matrix of derivatives of sample moments with respect to the parameters have full column rank. Evaluated at our parameter estimates, this matrix indeed has full column rank. Although most parameters affect the predicted value of multiple moments, parameters ( _α_ , _β_ , _φ_ ) ̃ mainly affect predicted enrollment shares, and parameters _ω_ , _α_ , _β_<sup> ̃</sup> , _γ_ mainly affect � � predicted demographic and neighborhood moments. Proficiency rate parameters in (17) are identified by variation in school type, focus, grade level and peer characteristics.

On the supply side, conditional on the calibrated _b_ , _V_ and _F_ , parameters’ main effects on model predictions are as follows. Increasing _ζ_ lowers overall predicted entry, and increasing _σν_ changes the predicted distribution of entry across entry points, making it less sensitive to predicted enrollment and costs. Increasing _α_ ̆ raises predicted relocations’ frequency, and increasing _β_<sup> ̆</sup> lowers average predicted distance, respectively. Increasing _a_<sup>_q_</sup> , _b_<sup>_q_</sup> and _c_<sup>_q_</sup> raises the number of predicted closings, their sensitivity to proficiency rates and their predicted probability after the first five years, respectively.

26
<!-- Start of picture text --> Utility Characteristics<br>a(0.357)<br>FA a<br>(0.152) | (0.221) | (0.150)<br>eae aie(0.090) | (0.248) lst | (0.149) | (0.084)<br>remoerms(0.392) | (0.245)<br>Tage esal |<br>sai<br>(0.527) | (0.422) | (0.298)<br>oe Tae(0.436) | (0.299)gen [al<br>a<br>Ar |<br>| (0.961) (0.195)<br>paar Tae (0.15 30 ) | (0.481) | a<br>(0.092)<br>ramen Tat(0.161) | (0.258)TL| (0.250) | (0.137)<br>aE(0.197)<br>A<br>cearremrers<br>eet tT<br>(0.114)<br>career [ett(0.122 |<br>Peers<br>a et |<br>rere Te(0.165) |[aco (0.339) lee]<br>Teo| (1.066) | (0.760)<br>emt (2. 635018 ) | (0.449) |[ate (0.272) |<br>aa<br>4 A<br>(0.034)<br>poem et(1.615) tT (0.312)<br>paneer(0.066)<br>[eT TT<br>La(0.061)<br>Notes: Based on 8,112 on 8,112 8,112 observations for share share moments; 1,269 observations for demographic demographic moments and 153 observations for neighborhood neighborhood<br>moments. Except where noted, parameters are GMM GMM estimates including campus, grade and and year fixed effects. Asymptotic standard errors are<br>given in parentheses. “Baseline utility” corresponds to parameters from 6, except for the coefficients on distance, which correspond to p.<br>Coefficients marked with (*) are significant at the 5% 5% significance level and (**) denotes significance at the 10% level. Estimates and standard<br>errors in Italics were obtained through minimum-distance estimation of campus campus fixed effects on time-invariant school characteristics (number<br>of observations observations in this regression = 281 281 campuses).<br><!-- End of picture text -->

Notes: Based on 8,112 on 8,112 8,112 observations for share share moments; 1,269 observations for demographic demographic moments and 153 observations for neighborhood neighborhood moments. Except where noted, parameters are GMM GMM estimates including campus, grade and and year fixed effects. Asymptotic standard errors are given in parentheses. “Baseline utility” corresponds to parameters from 6, except for the coefficients on distance, which correspond to p. Coefficients marked with (*) are significant at the 5% 5% significance level and (**) denotes significance at the 10% level. Estimates and standard errors in Italics were obtained through minimum-distance estimation of campus campus fixed effects on time-invariant school characteristics (number of observations observations in this regression = 281 281 campuses).
schools yet not with the same intensity. Poverty raises the likelihood of choosing charters. Low-income blacks are less likely to choose a single-campus charter than a public school, but slightly more likely to choose a multi-campus charter than a public school. Hispanics have a stronger preference for charters than blacks. Although non-whites are more likely to choose a public than a Catholic school, Hispanics have a stronger Catholic school preference. Whites are more likely to choose a Catholic over a public school and have a stronger preference than non-whites for private MHS. Our estimated preferences match school choices well (see Appendix Table 5), overall and by grade level.

According to the estimates of focus preferences, households generally prefer Core over non-Core, although whites prefer Other over Core. Hispanics’ preference for Core over Language seems inconsistent with their relatively high attendance of Language-focused schools. Yet Hispanics exhibit a strong same-race preference, as described below, which suggests that they might choose Language schools because they attract other Hispanics and not necessarily because of the language curriculum. As Appendix Table 6 shows, our focus preference estimates capture students’ focus choices **.** Coefficients on ward dummies (not shown) indicate that parents place a high value on characteristics of the school’s neighborhood. Travel disutility is high when attending a public school but not otherwise. Nonetheless, since approximately 50 percent of children in public schools attend their assigned neighborhood school, we expect the estimated travel disutility to public schools to be upward biased in absolute value. Note, also, that the coefficient on tuition is negative and significant. It implies that a $1000decline in private school tuition would raise attendance probability by 28 percent.

Students of all races prefer to attend a school with more white (and hence fewer black) students. Nonetheless, whites have the strongest preference for a school with other white students, and are willing to pay approximately $5,500 for an extra 10 percentage point white students. Moreover, they have the highest ability to pay for such a school. Hispanics have strong same-race preferences as well and are willing to pay approximately $2,400 for an extra 10 percentage points Hispanic. These preferences are consistent with the fact that students are quite segregated by race across schools.<sup>30</sup> Recall that our school quality estimates capture unmeasured school characteristics such as culture, proximity to transportation, and facilities’ characteristics. Thus, in Appendix Table 7 we compare average school quality in public and charter schools. The average quality difference outside ward 3 makes a family 32 and 46 percent more likely to choose a charter school for non-MHS and MHS grades, respectively. In wards 7 and 8, the difference makes the family 36 and 272 percent more likely to choose the

> 30In our data, while 74, 17 and 9 percent of the students are black, white, Hispanic and non-poor, respectively, the average black student attends a school that is 89 percent black, the average white student attends a school that is 69 percent white, and the average Hispanic student attends a school that is 37 percent Hispanic. Filardo et al (2008) finds similar racial segregation.

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average charter for non-MHS and MHS grades, respectively. Importantly, the average quality premium commanded by charters relative to public schools is particularly large for MHS grades. Outside ward 3, a family would be willing to pay around $1,500 for it, an amount that would rise to about $5,300 in wards 7 and 8.<sup>31</sup>

# **6.2 Supply Side**

Appendix Table 8 shows estimates for (16), used to calibrate variable and fixed costs. According to the estimates, fixed costs are higher for high schools and mixed-level schools than for elementary or middle schools. They are also higher for schools located in the West (due to high real estate prices) or Southeast (due to buildings’ poor condition and high security and insurance costs) than in the Northeast. These estimates are consistent with the fact that most entry has taken place in the Northeast and has served elementary or middle school students.

Table 6 presents the MLE estimates. As Appendix Table 9 shows, our estimates capture observed entry patterns. They also match the number of relocations and the distribution of relocation distance (see Appendix Figure 2). The estimated entry fee, equal to $36,000, captures set-up costs such as building renovations; fees from legal, accounting and real estate services; cost of student and teacher recruiting, etc. The estimate is reasonable, as it is of the same order of magnitude as charters’ average and median profits (equal to $21,000 and $34,000 respectively). Intuitively, if the entry fee were much higher, charters would have incurred a loss when entering the market; forecasting this loss, the regulator would not have authorized their entry. The 95 percent confidence interval for the estimate is wide, reflecting the infrequent entry, limited cost data and wide variation in actual set-up costs (related, for instance, to initial facilities expenses).

The estimated standard deviation of profits _σν_ is approximately equal to $295,000, which is of the same order of magnitude as the standard deviation of charters’ profits (equal to $165,000). The estimate indicates potential entrants’ heterogeneity in financial aspects observed by the regulator but not by us, such as actual costs, business plan ̆ quality and additional revenue sources. The estimates for _α_ and _β_<sup> ̆</sup> indicate that schools are averse to moving and seek close destinations when moving. Estimates for _b_<sup>_q_</sup> and _c_<sup>_q_</sup> indicate that charters are more likely to be closed if their academic performance is

> 31School capacity constraints might bias the parameter estimates of the utility function. Consider, for instance, the estimated negative coefficient on the charter indicator. If neither public nor charter schools faced capacity constraints, this negative coefficient would indicate that parents prefer public over charter schools, all else equal. Yet in the presence of charter capacity constraints, the negative coefficient could also indicate lack of space in charters even if families preferred them over public schools. Distinguishing between these possibilities requires excess demand data for all schools, not just charters. Unfortunately these data are not available. To the extent that capacity is a problem, it would mostly affect baseline utility parameters. Most likely estimates of school quality would be downward-biased for schools facing excess demand, and upward-biased for the schools absorbing the excess demand. Hence, in the public-charter comparisons conducted above we likely understate the quality premium of the best charter schools.

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|(in $1,000)|(123.0)|
|---|---|
|(in $1,000)|(174.7)|
||(0.453)|
|Relocation sensitivity to distance (B)<br>(distance inmiles)|3.182*<br>(1.109)|
||(1.793)|
|(proficiency<br>between 0 and 100)|(0.114)|
||(1.667)|

Notes: Maximum Likelihood Estimates based on 3,900 observations for entry probabilities, 169 observations for closing probabilities and 165 observations for relocation probabilities.
<!-- Start of picture text --> (0.267,<br>(0.059)<br>(0.195)<br>(0.126)<br>(0.040)<br>(0.161<br>(0.091)<br>(0.034)<br>(0.053)<br>(0.015)<br>(0.004)<br>(0.009)<br>Mean of Dependent Vble. -0.340<br>Mean of Passing Rate (%) 42.29<br>Std. Error of Regression 0.574<br>Pseudo-R? 0.810<br><!-- End of picture text --> Notes: Based on 871 school-year observations corresponding to schools with at least 2 years of data. Passing rate is expressed between 0 and 100. Parameters are 2SLS estimates including campus and year fixed effects. Observations are weighted by total school enrollment. Standard errors are clustered at the school level. Coefficients marked with (*) are significant at the 5% significance level. Estimates and standard errors in Italics were obtained through minimum-distance estimation (regression of campus fixed effects on time-invariant school characteristics; number of observations = 193). Pseudo-R? equal to the squared correlation between observed and predicted values of the dependent variable.
||Baseline <br>1<br><br>|No<br>|Charters |<br>2<br>|Raise<br>Funding<br>3|Approve <br>All<br>4||LowerApp. |<br>Costs<br>5|Close<br>Public<br>6|
|---|---|---|---|---|---|---|
||CharterEntry<br>|<br><br>||<br>||||||
||Numberofcharterentrants<br><br><br>|9.08<br><br>||<br>T|88|72||<br>18.04<br>|9.08|
|<br><br>|Charterentrantsbyregion<br><br>|<br><br>|<br>|||<br>|||
||___FractionenteringinNortheast__—<br> <br>||_—0.53,<br>||<br>|048<br>|TAT<br>|053<br><br>|053<br><br>|
|<br>|__FractionenteringinSoutheast<br><br> <br><br>|<br>[|_0.29<br>||<sup>T</sup><br>|<br><sup>3</sup><br>|<br><sup>0</sup><br>|<br><sup>2</sup><br><br>|<sup>9</sup><br><br>|
|<br><br> <br><br><br><br>|<br><br>||||||
|<br><br>Non-CharterClosings<br><br><br> <br>||<br>||<br>|<br>|<br>|
|<br><br> <br><br><br> <br><br> <br>|<br><br> <br>|<br> <br><br><br>|<br><br>|<br><br>|<br><br>|<br><br>|
|<br><br> <br> <br>|<br><br>|<br><br><br><br>|||||
|<br><br> <br>Clsings<br>|<br><br>|<br><br>S<br><br>|<br><br><br><br>|<br><br>|<br><br>|<br><br>T|<br><br>|
|<br> <br><br>[MarketShares<br>|<br>|<br><br><br>|<br>|<br><br><br>|<br><br>|<br><br>|<br><br>|<br><br>|
||___Fraction inincumbentcharters<br><br>|0.220,<br>||<br>|<br> <br>|0.217<br>|0.214 <br>|||<br>|0.259<br>|
|<br>|___Fractioninnew charters<br>|<br>|<br> 0.033, <br>||<br><br>|<br>T0044 <br>|<br>| 0.057 <br>|<br>| 0.065<br><br>|0.039 |<br>|
||___FractioninCatholicschools||0.101_—|_0.118||0.100|||0.099||<br>0.099||0.13|

Note: Each column reports average over 1,000 draws of (&, §4).
eerBlack Hisp. White Black Hisp. White Baseline Note: “Baseline”Noshows Chartersaverage Fitignproficiency index[520for students|of each009demographic| 2mgroup[028and grade level[| (inonpercent).|“NoonCharters”| shows the difference in average proficiency index between Baseline and No Charters (in percentage points). “Elementary” refers to K-6 grades; “Middle” to grades 7 and 8, and “High” to 9-12 grades.

eeBlack Hisp. White Black Hisp. White Baseline No Charters Note: “Baseline” shows the average welfare for students of each demographic group and grade level. “No Charters” shows average compensating variation from the counterfactual for student each demographic group and grade level (negative values are in parentheses). “Elementary” refers to K-6 grades; “Middle” to grades 7 and 8, and “High” to 9-12 grades.
|d)<br>Q)|(3)<br>(4)|(5)<br>(6)|
|---|---|---|

Note: Total Benefits is aggregate willingness to pay over households. Total (and average) Costs I assumes that per-student expense is that same in public and charter schools, and that per-student expense in Catholic schools equals tuition. Total (and average) Costs II assumes that per-student expense is 20 percent higher in public than charter schools, and that Catholic school tuition covers only 2/3 of per-student expense. Per-student averages are obtained by dividing totals by number of students, approximately equal to 80,000. Row 1 shows baseline costs and benefits; Rows 2-6 show changes in costs and benefits relative to the baseline (negative changes in parentheses). Dollar amounts in columns (1)-(3) are rounded to the nearest thousand.
Charters” panel. On average all student groups lose welfare due to the loss of school options, but losses are the greatest for those previously most likely to attend charters. Middle school students, who gain much from the quantity and quality of options offered by charters, are particularly hurt. Further, poor blacks in middle school experience a loss of about 15 percent of their baseline welfare. For the population as a whole, losses represent about 8 percent of household income on average. The 25 percent of students most hurt by charter removal are non-white, have an average household income of $27,000 and experience an average welfare loss equivalent to 19 percent of their income. They lose by having less access to specialized curricula, traveling longer to school, and losing school quality for the equivalent of $1,100.

From a social standpoint, Table 11’s second row indicates that total social benefits fall by about $77,000,000 when the 59 charters are removed. Whether total costs rise or fall with charter removal depends on cost assumptions; regardless of these, total social _net_ benefit falls by at least $62,000,000, equal to about $1,000 per student and $1,000,000 per charter. Thus, eleven years after their inception in Washington, D.C. charter schools seemed to have been generating substantial net social benefits.

# **7.3 Charter Expansionary Policies**

Given the gains from charters, we now turn to three possible avenues for charter expansion: an increase in charter reimbursement, the elimination of entry selectivity, and a reduction of application costs. Thus, we compute the Fall 2007 equilibrium for three policies. The first (“Funding Increase”) is a 20 percent increase in charter reimbursements, approximately similar to the one requested by charters in ongoing litigation. In the second (“Approve All”), the regulator approves all charter applications, regardless of expected profits. The third (“Lower Application Costs”) implements a reduction in entrant application costs, operationalized by doubling _b_ .

In the simulations we assume that the corresponding policy change takes place just before step 1 of the stage game in Spring 2006 and hence affects entry in Fall 2007. We then compare the resulting equilibrium for each policy change with the baseline. Our analysis focuses on short-run effects. We assume that costs of Approve All or Lower Application Costs are negligible relative to charter reimbursements.

Columns (3)-(5) in Table 8 illustrate these effects. The three policies increase charter entry. Approve All and Lower Application Costs have almost the same expansionary effect, an effect larger than that of Funding Increase. Although entry patterns by focus and grade level are similar to those in the baseline, Funding Increase and Approve All lead to a (slightly) greater fraction of entrants in the Southeast and West. Since these entrants face relatively high fixed costs, Funding Increase raises their expected profits and hence approval probability. By approving all applications, Approve All raises entry probability disproportionately in those regions. Nonetheless, the limited expansion 35
of charters where they are most beneficial (i.e., upper grades and/or in the Southeast) suggests that only a targeted and larger funding increase might induce that entry.

All three policies increase charter market share. New entrants attract students from all incumbent schools, including charters, and away from the outside option. Funding Increase raises the profitability of all charters, including incumbents, and hence (slightly) lowers charter closings. In contrast, Approve All and Lower Application Costs raise charter closings as the additional entrants cannibalize incumbent charters.

The three expansionary policies have small, if any, proficiency effects (see Appendix Table 12). Because of the additional school options, average welfare effects are positive for all student groups (see Appendix Table 13); they are greatest for those most likely to attend charters, and for poor or middle-school students. Among the top-25 percent winners from these policies are poor, non-white families with low-quality nearby public schools for whom welfare gains amount to 3 percent of income on average.

Since Raise Funding leads to the entry of about three additional schools and an increase in total social benefits of $4,086,000 (see row 3 of Table 11), the social benefit per additional entrant is about $1,450,000. Nonetheless, the increase in total social costs from Raise Funding is larger than the increase in benefits. As lack of data prevents us from modeling the relationship between funding and school quality, we do not capture the likely quality improvements that all charters (including incumbents) might attain with greater funding. We do, however, capture cost increases. Thus, we can only provide a lower bound on social welfare gains from Raise Funding.

In contrast, Approve All and Lower Application Costs raise net social benefits because they almost double entry without reimbursement increases (see rows 4 and 5 of Table 11). Although these two policies expand charters at about the same rate, total social benefits are larger for Lower Application Costs (by $2,697,000). Further, based on the most conservative estimates for net charter gains (associated with Table 11’s column 2), Lower Application Costs produces a _net_ social gain of $1,043,000 per entrant relative to $872,000 from Approve All. The reason is that the selective regulator of Lower Application Costs only authorizes the entry of charters with a sufficiently large expected enrollment (and hence social value). These charters, in addition, are expected to be financially more robust and less likely to be closed in the future. Thus, our counterfactuals indicate that the combination of selective charter approval and policies that encourage the supply of potential charter entrants (for instance, by minimizing charter application costs) can deliver sizable social gains. Current charter advocacy in Washington, D.C. points to the importance of lowering application costs - for instance, by facilitating charters’ access to unused public school buildings.<sup>37</sup>

> 37Access to adequate facilities is a main challenge for charters. Although by law charters have the right of first offer on vacant DCPS buildings, the law is often not enforced (www.focusdc.org/advocacy). Greater enforcement of the law exemplifies a reduction in application costs.

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## **7.4 Public School Closings**

Unlike public schools, charter schools must close if they are unable to cover their costs. To investigate the effect of this rule on public schools, in the absence of public school cost data we simulate a counterfactual (“Close Public”) whereby a public school must close if its enrollment loss is greater than 40 percent during the sample period. As column 6 of Table 8 shows, this leads to closing approximately 49 public schools, affecting about 14 percent of all K-12 students. Note that DCPS actually closed 33 schools between 2008 and 2012; most closings affected elementary or K-8 schools in the Northeast or Southeast. Our counterfactual captures these aspects.

In the simulations, about 64, 29 and 7 percent of the displaced students switch into (other) public, charter of private schools respectively, consistent with the observed reallocation of students after the 2008 closings (documented in www.21csf.org). Because the schools that are predicted to close lag behind in proficiency, quality and value added, most student groups enjoy proficiency gains (see Appendix Tables 11 and 12). For instance, middle-school black students attain an average proficiency gain of 3.17 percentage points. Nonetheless, Appendix Table 13 shows that most students suffer welfare losses from the elimination of school options. Thus, although the closings lower total social costs (see Table 11’s row 6), they lower total social benefits even more. Our estimated per-school net social loss from these closings is between $603,000 and $755,000 depending on cost assumptions. This loss, however, is lower than the per-school loss of closing charter schools (approximately $1,000,000).

By simulating a closing rule strictly based on enrollment rather than the multiple factors actually considered by DCPS for closings (such as student travel time and facility condition), we likely overpredict closings and hence welfare losses. Nonetheless, the counterfactual suggests in the short run students would be less hurt by the closing of a low-enrollment public school than of an average charter. Further, losses from public school closings might be reversed in the long run through the entry of new charters serving the displaced students.

# **8 Conclusion**

In this paper we develop and estimate a rich yet tractable model of charter school entry and household school choice. We model the equilibrium sorting of households across schools as well as regulator behavior. We estimate the model using data on the full choice set of schools in Washington, D.C. between 2003 and 2007. Our estimates indicate that charter schools have generated net welfare gains in the city by providing new options to serve a heterogeneous population. Welfare gains have been particularly large for middle school- and for non-white, low-income students, whose options before charters were quite limited. According to our counterfactuals, raising the supply 37
of prospective charter entrants (for instance, by lowering application costs and providing information on charter best practices) while applying tight admission and oversight standards is a welfare-enhancing mechanism for charter expansion.

While informative, our counterfactuals must be taken with caution for several reasons. First, they only reflect short-run policy effects and ignore strategic responses from non-charter schools. Second, our results reflect the institutional environment for charters in D.C., where funding is relatively high vis-a-vis other states and the charter law is permissive. Third, we do not model the relationship between school funding and quality. Nonetheless, the counterfactuals stress the role of private initiative (facilitated by low application costs) in the charter sector. They also stress the regulator’s role (through its approval and oversight activies) in attaining a high-quality charter sector.

Our findings are informative for D.C., particularly as the city adjusts to a rising number of families with school-age children and highly educated parents. They are also informative for school reform in other large, U.S. cities. Finally, they are relevant for countries with private operation of public schools, such as Colombia, the UK and Spain (Patrinos et al 2009), and for nascent charter efforts in developing countries such as Uganda and Morocco.

# **References**

- [1] Altonji, J., C. Huang and C. Taber. 2010. “Estimating the Cream Skimming Effect of Private School Vouchers on Public School Students.” _Nation Bureau of Economic Research_ , Working Paper 10-12.

- [2] Angrist, J. D., P. A. Pathak, and C. R. Walters. 2013. “Explaining Charter School Effectiveness.” _American Economic Journal: Applied Economics_ , 5(4): 1-27.

- [3] Bayer, P. and C. Timmins. 2007. “Estimating Equilibrium Models of Sorting Across Locations.” _Economic Journal,_ 117(518): 353-374.

- [4] Berry, S., J. Levinsohn, and A. Pakes 1995. “Automobile Prices in Market Equilibrium,” _Econometrica,_ 63(4): 841-890.

- [5] Betts, J. R., & Tang, Y. E. 2011. “The Effect of Charter School on Student Achievement: A Meta-Analysis of the Literature.” _Center on Reinventing Pub. Education._

- [6] Bifulco, R. and C. Buerger. 2012. “The Influence of Finance and Accountability Policies on Charter School Locations.” _Syracuse University,_ Working Paper.

- [7] Booker K., Sass T., Gill B. and Zimmer R. 2011.“The Effects of Charter High Schools on Educational Attainment.” _Journal of Labor Economics,_ 29(2):377-415.

- [8] Buckley, J. and M. Schneider. 2007. “Charter Schools: Hope or Hype?” _Princeton and Oxford: Princeton University Press_ .

- [9] Cardon, J.H. 2003. “Strategic Quality Choice and Charter Schools.” _Journal of Public Economics_ , 87 (3-4): 729-737.

- [10] Carranza, J. E., J. Houde and R. Clark. 2011. “Dynamic Entry and Firm Reconfiguration in Canadian Gasoline Markets.” Working Paper, ICESI University.

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- [11] Clark M., P. Gleason, C. Clark Tuttle, M. K. Silverberg. 2011. “Do Charter Schools Improve Student Achievement? Evidence from a National Randomized Study” _Mathematica Policy Research._

- [12] Dobbie, W. and R. G. Fryer. 2013. “Getting beneath the Veil of Effective Schools: Evidence from New York City.” _American Economic Journal: Applied Economics,_ 5(4): 28-60.

- [13] Draganska, M., S. Misra, V. Aguirregabiria, P. Bajari, L. Einav, P. Ellickson, & T. Zhu. 2008. “Discrete Choice Models of Firms’ Strategic Decisions.” _Marketing Letters_ , 19(3-4), 399-416.

- [14] Dubé, J., J. Fox, and C. Su. 2012. “Improving the Numerical Performance of BLP Static and Dynamic Discrete Choice Random Coefficients Demand Estimation.” _Econometrica_ , 80 (5): 2231-2267.

- [15] Ferreyra, M. M. 2007. “Estimating the Effects of Private School Vouchers in MultiDistrict Economies.” _American Economic Review_ , 97 (3): 789-817.

- [16] Filardo, M., M. Allen, N. Huvendick, P. Sung, D. Garrison, M. Turner, J. Comey, B. Williams and E. Guernsey. 2008. “Quality Schools and Healthy Neighborhoods: A Research Report.” _Sponsored by the OSSE_ . Available at www.brookings.edu.

- [17] Henig, J. R. and J. A. MacDonald. 2002. “Locational Decisions of Charter Schools: Probing the Market Metaphor.” _Social Science Quarterly,_ 83(4): 962–980.

- [18] Glomm, G., D. Harris, and T. Lo. 2005. “Charter school location.” _Economics of Education Review,_ 24(4): 451–457.

- [19] Imberman, S. A. 2011. “Achievement and Behavior of Charter Students: Drawing a More Complete Picture.” _Review of Economics and Statistics,_ 93 (2): 416 - 435.

- [20] Mehta, N. 2012. “Competition in Public School Districts: Charter School Entry, Student Sorting, and School Input Determination.” Working Paper, _Western Ontario University._

- [21] Nechyba, T. J. 2000. “Mobility, Targeting and Private School Vouchers.” _American Economic Review,_ 90 (1): 130–146.

- [22] Neilson, C. 2013. “Targeted vouchers, competition among schools, and the academic achievement of poor students. Working Paper, _Yale Univesity_ .

- [23] Nevo, A. 2000. “Mergers with Differentiated Products: The Case of the Ready-ToEat Cereal Industry.” _RAND Journal of Economics,_ 31 (3): 395-421.

- [24] Nevo, A. 2001. “Measuring Market Power in the Ready-To-Eat Cereal Industry.” _Econometrica,_ 69 (2): 307-342.

- [25] Patrinos, H., F. Barrera-Osorio and J. Guaqueta. 2009. “The Role and Impact of Public-Private Partnerships in Education.” Washington, D.C.: _The World Bank_ .

- [26] Rincke, J. 2007. “Policy Diffusion in Space and Time: The Case of Charter Schools in California School Districts.” _Regional Science and Urban Economics,_ 37(5): 526-541.

- [27] Salisbury, D. 2003. “What Does a Voucher Buy? A Closer Look at the Cost of Private Schools.” CATO I _nstitute Policy Analysis_ Paper No. 486, August.

- [28] Skrainka, B. 2012. “A Large-Scale Study of the Small Sample Performance of Random Coefficient Models of Demand.” Working Paper, _University of Chicago._

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- [29] Walters, C. 2012. “A Structural Model of Charter School Choice and Academic Achievement.” Working Paper, _U. of California-Berkeley_ .

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# For Online Publication: Appendices

## **A Data**

### **A.1 School-Level Data**

#### **A.1.1 Public Schools**

The starting point for this dataset is audited enrollments from the District of Columbia Office of State Superintendent of Education (OSSE), available at http://osse.dc.gov. This gives us the list of public schools and their grade-level enrollment. From the original list we exclude alternative schools, special education schools, early childhood centers (as long as they never include any of grades 1 through 12 during the sample period) and schools with residential programs. Our data pertain to grades kindergarten through 12th. For each school we collect the information listed below:

- Address: from the Common Core of Data (CCD). We geocode all addresses.

- School enrollment: total school enrolment excluding ungraded and adult students. Source: own calculations based on OSSE.

- Grade-level enrollment: for each grade between kindergarten and 12th. Source: OSSE.

- Focus: Source: Filardo et al (2008).

- Percent of white students: calculated based on CCD. For cases in which CCD data are not available, we use the demographics reported to OSSE in order to fulfill No Child Left Behind (NCLB) requirements (see http://www.nclb.osse.dc.gov/). Note, however, that the NCLB requirements pertain to students enrolled in the grades tested by law, not to the entire student body. We include “other ethnicities” (such as Asian students) among white students.

- Percent of black students, percent of Hispanic students: constructed similarly to percent of white students.

- Percent of low income students: calculated as the percent of students who receive free or reduced lunch. Source: own calculations based on CCD. For the few cases in which the CCD data are not available we use the demographics reported in fulfillment of NCLB requirements.

- Reading proficiency: percent of students who are proficient in reading. Source: OSSE. In 03 and 04, proficiency levels were determined according to the Stanford9 assessment. To be considered proficient, a student was supposed to score at the national 40th percentile or higher. Since 05, proficiency has been determined according to DC CAS (Comprehensive Assessment System).

Prior to 05, grades tested were 3, 5, 8 and 10 (according to the School Performance Reports for PCBS-authorized charter schools, and according to our own calculations comparing grade-level enrollment with number of students tested). Since 05, grades tested have been 3, 4, 5, 6, 7, 8 and 10 (according to OSEE, School Performance Reports and our own calculations comparing grade-level en41
rollment with number of students tested). For some schools and years, proficiency is not available for one of the following three reasons: 1) the school only includes early childhood enrollment; 2) the school only includes grades that are not tested; 3) the school includes grades that are tested but enrollment in those grades is below the minimum threshold for reporting requirements. The last reason is the most prevalent cause of missing proficiency. See below for the imputations made in those cases.

- Math proficiency: percent of students who are proficient in math. Constructed similarly as reading proficiency.

- Year of Opening: year the school opened if it was open after 2003. Using the CCD “status” variable and web searches we verify the school’s initial year, which is the first year for which we have records.

- Year of Closing: first year that the school is no longer open. We verify the content of this variable using the CCD “status” variable and web searches.

- Year of Merge: year the school merges with another school. The variable stores the first year that the school no longer operates separately, which is the first year for which we have joint records.

Ethnic composition and low-income status of the student body are missing for 2 and 4 (out of 701) observations respectively; these are schools with very low enrollment. To the cases of missing ethnic composition we impute the school’s average ethnic composition over the years for which we do have data. When possible, we impute the predicted value coming from a school-specific linear trend. Achievement is missing in 16 out of 701 observations. To these observations, we impute the predicted achievement coming from the regression of school-level proficiency rates on year dummies, ethnic composition variables, percent of low income students, enrollment, and school fixed effects. In cases in which we have no proficiency data at all, we run a similar regression excluding school fixed effects and including dummies for school level and use the resulting predicted values for our imputations.

# **A.1.2 Charter Schools**

As with public schools, the starting point for this dataset is audited enrollments from the District of Columbia Office of State Superintendent of Education (OSSE). This gives us the list of charter schools along with their grade- and school-level enrollment. For consistency with public schools, we exclude alternative schools. We also exclude schools in which ungraded or adult students constitute the majority of the student body, and schools with residential programs.

By law, charter schools cannot serve special education students exclusively. However, they can offer services targeted to specific populations. We exclude schools whose services target special ed students. We exclude early childhood schools only if they never add regular grades during our sample period. We also exclude online campuses. Some non-early childhood charters opened an early childhood campus during the sample period; we only included these campuses if at some point during the sample period 42
they add regular grades.

Below is the list of variables for charter schools:

- Address: geographic location of the campus. For PCSB-authorized charters, the main source is the School Performance Reports (SPRs). For BOE-authorized charters, the main source is the CCD. We supplement these sources with web and Internet archive searches. We geocode all addresses. Several schools moved in the middle of the school year, temporarily relocated, or closed during the sample period. We consult the SPRs and various web sites to handle these cases. If the school moved in the middle of a school year, the address variable contains the more recent address. Some schools relocated some students for a few months during renovations. Since this was a temporary, anticipated arrangement, we do not consider these cases as address changes.

- Statement: the school’s mission statement. Source: schools’ web sites, FOCUS , SPRs.

- Focus: the school’s curricular focus. Source: school statements, and Filardo et al (2008).

- School enrollment: total school enrollment, excluding ungraded and adult students. Source: own calculations based on OSSE.

- Grade-level enrollment (grades kindergarten through 12). Source: OSSE.

- Percent of white students: for PCSB charters, the source is the SPRs when available; if not, we use data reported for NCLB purposes. For BOE charters in 2007, the source is the SPRs (in 2007, the PCSB began including the BOE-authorized charters in its reports). For BOE charters before 2007, the source is the NCLB web site. If necessary, we supplement these sources with CCD data. When the school has multiple campuses but we only have one set of ethnic composition data, we impute it to all campuses.

- Percent of black students, percent of Hispanic students: constructed similarly to percent of white students.

- Percent of low income students: for PCSB charters, the source is the SPRs. For BOE charters, the source is NCLB information in http://www.nclb.osse.dc.gov/. These sources are supplemented by CCD when necessary and possible. When the school has multiple campuses but we only have one set of low-income variables, we impute it to all campuses.

- Reading proficiency: sources are the same as for public schools. Some schools span elementary as well as secondary grades and, for some years, have separate proficiency rates per grade level. In these cases we combine the rates into a single proficiency indicator for comparability with years for which we have a single indicator. For multi-campus charters we usually have proficiency data per campus. When we do not, we impute the available data to all the campuses. As with public schools, we do not have proficiency data for some campuses and years for the reasons described above. In the case of PCSB schools for which the NCLB web site does not report test scores due to low enrollment, we obtain proficiency rates from the SPRs. This is not possible for BOE schools with low enrollment since

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the SPRs only cover PCSB schools before 07. In cases in which we cannot find proficiency data, we make imputations (see below).

- Math proficiency: constructed similarly as reading proficiency.

- Year of Opening: source: SPRs, FOCUS, web searches. The variable stores the Fall of the first academic year that the school is open.

- Year of Closing: first year that the school is no longer open. The sources are the Center for Education Reform , current SPRs and PCSB listings of charter schools, current NCLB reports, and web searches.

- Reason for Closing: classified as academic, financial or mismanagement. Source: Center for Education Reform, SPRs, web searches.

• Multi-campus: an indicator variable that equals 1 if the school belongs to an organization that has multiple campuses by the end of the sample period. Percent of low-income students is missing for 9 out of 228 observations. These schools have low enrollments. To most of these cases we impute the school’s average percent of low-income students, calculated over the years for which the school does have data. In the case of missing proficiency rates (36 out of 228 observations), we make imputations similar to those described for public schools, the only difference being that we used school-fixed effects (as opposed to campus-fixed effects) in the predicting regressions.

# **A.1.3 Private Schools**

The starting point is the list of private schools from the Private School Survey (PSS). Since PSS is biennial, we use the 2003, 2005 and 2007 waves. PSS classifies schools as regular, vocational, special ed, and other/alternative. For the years of interest, 92 percent of the schools in Washington, D.C. are classified as regular, and the remaining schools are classified as other/alternative. Although an alternative public school usually serves students with behavioral problems, an alternative private school is usually a regular school with a specialized curriculum. Hence, we keep most alternative schools. We eliminate vocational schools because they enroll ungraded students exclusively. We also eliminate special ed schools, early childhood centers (as long as they never have enrollment in regular grades during the sample period), and other schools that only teach ungraded students.

Since PSS does not have a 2004 wave, we assign 2004 values to the variables through linear interpolation of 2003 and 2005, and similarly for 2006. For instance, we calculate percent white for 2005 as the average between percent white in 2003 and in 2005. In cases in which a school does not report to the survey in a particular year, we make imputations based on the school’s reported data for the other years. If a school does not appear again in PSS after a particular wave, we assume that the last year of operation is the year of the last wave in our data.

For the list of variables below, PSS is the main source of data:

- Address: if needed, we supplement PSS with web and Internet archive searches. We geocode all addresses.

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- Type: Catholic, other religious or non-sectarian. Source: PSS.

- School enrollment: total school enrollment, excluding ungraded and adult students. Source: own calculations based on PSS.

- Grade-level enrollment: number of students in each grade between K and 12.

- Percent of white students: source: own calculations based on the reported number of white students and the total enrolment. “White” includes other ethnicities as well.

- Percent of black students, percent of Hispanic students: constructed similarly to percent of white students.

- Percent of low income students: since PSS does not collect this information, we impute it based on the following logistic regression. Using data for public and charter schools, we regress percent low income on school percent white and percent Hispanic, school enrollment, and average household income of the school’s tract. We use this regression to predict percent low income in private schools. We check our predictions by comparing OSSE data on the percent of students receiving free and reduced lunch in private schools. Our predictions compare favorably with OSSE data.

- Tuition: annual tuition for the 2002/03 school year. Source: Salisbury (2003). Tuition is expressed in dollars of 2000.

# **A.2 Market Size and the Outside Good**

Since a market is a grade-year combination, market size _Mgt_ is equal to the number of children eligible for grade _g_ in year _t_ between 2003 and 2007. This number is not available, and neither is the number of children by age. Hence, we estimate market size based on the following, available data for Washington, D.C.:

1. The 2000 Census count of children by age;

2. The intercensal estimates of the number of children in the 5-13 and 14-17 year old brackets; 3. The 2000 Census count of enrolled and not enrolled children by age group. The resulting percent of children not enrolled in school is our best proxy for the outside good share in 2000 .

4. Observed enrollment for each grade and year between 2003 and 2007. Use _Ngt_ to denote aggregate enrollment in grade _g_ and year _t_ .

We estimate market size as _Mgt_ = _Ngt_ ∗ _θgt_ , which implies an outside good share equal to _Ogt_ = 1 − _Ngt_ / _Mgt_ . Adjustment factors _θgt_ are chosen so that _Ogt_ matches the 2000 Census fractions of children not enrolled in school. We adjust these fractions slightly to account for the fact that our enrollment data is based only on regular schools and excludes early childhood centers. Thus, we use fractions equal to 3 percent for ages 5-14 (corresponding to grade K-8) and 10 percent for ages 14-17 (grades 9 through 12). For computational reasons we impose the following constraint: _Ogt_ ≥ 0.01 (see Appendix E.3).

An appealing feature of our solution is the consistency of _Mgt_ with the growth rate for child population implied by intercensal Census estimates. In particular, the 45
estimated _Mgt_ grows at the following rates between 2003 and 2007: -13 percent for grades K through 8, and 7 percent for grades 9 through 12. These rates line up with the Census growth rates for the corresponding age groups (equal to -13 percent and 13 percent , respectively) for the same period.

# **A.3 Household Characteristics**

We describe the estimation of the joint distribution of household location, child age and race, parental income and child poverty status for year 2000, and the adjustments made to this distribution for years 2003-2007. The term “demographic type” refers to a combination of race (black, white and Hispanic), income (16 values for income, each one representing the midpoint of the corresponding Census income bracket), and poverty status (eligible for free- or reduced-lunch, or not eligible). This yields a total of 96 demographic types. For each of the 13 grades and 433 locations (i.e., block groups) in our data, we estimate the number of children of each demographic type.

## **A.3.1 Household Types for Year 2000**

We do not observe the joint distribution of child age, race, household income and poverty status at the block group level. Instead, the Census provides us with the following information:

- tract-level joint distribution of age and race;

- tract-level joint distribution of age bracket, race and poverty status;<sup>38</sup>

- tract-level joint distribution of family income (by brackets) and race;

- block group-level joint distribution of age brackets and race.

We use this information to calculate the number of children in each demographic type, grade and location. Recall that Washington, D.C. includes 433 block groups and 188 Census tracts. The calculations described below apply to the 185 tracts (and the corresponding block groups) that include children aged 5-18. We proceed as follows:

1. Assuming the same distribution of age among the block groups of a given tract, we estimate the number of children of each age and race by block group.

2. For each block group, race and age we impute the poverty distribution that prevails for the corresponding tract, race and age. For each block group we obtain _narp_ , which is the number of children of each age _a_ , race _r_ and poverty status _p_ .

3. The tract-level income distribution for families is not adjusted by family size and hence does not reflect income per child. In the absence of data on the joint distribution of family income and size, we calculate tract-level average family income and average family size by race, and construct the city-level joint distribution of average family income and size. Then we reweight the original tract-level family

> 38Per federal guidelines, in order to qualify for free (reduced) lunch, a child must live in a household whose income is below 130 (185) percent of the Federal poverty guidelines for that household size. We pool children eligible either for free or reduced lunch into a single category. Thus, “poverty status” is a binary variable that describes whether the child is eligible for free- or reduced-lunch or not.

46
income distribution to reflect differences in family size by income bracket, in an attempt to reflect the distribution of income per child.

4. To determine how many of the _narp_ children in the corresponding ( _a_ , _r_ , _p_ ) combination a given block group fall in each income bracket, we assign a low-income status to the lowest incomes. For instance, consider a block group in which 20 percent of 5-year old white children are poor, and the remaining 80 percent are not. In the corresponding tract, 5 percent of white families have incomes below $20,000; 15 percent have incomes between $20,000 and $40,000, and the remaining 80 percent have incomes between $40,000 and $60,000. Thus, we assign a family income of $10,000 (i.e., the midpoint for the $0-$20,000 income bracket) to a quarter of the 5-year old white children, where 1/4 = 0.05 /0.20, and a family income of $30,000 (midpoint for the $20,000-$40,000 income bracket) to threequarters of those children. To the 80 percent of 5-year old white children who are not poor, we assign an income of $50,000 (midpoint of the $40,000-$60,000 income bracket).

5. Based on step (4), for each block group _l_ we calculate the number _μlam_ of children of each age and demographic type _m_ .

Calculating the number of children aged 18-years old in each demographic type and location is challenging for D.C. because the number of 18-year olds is much higher than the number of 17-year olds. The age bracket 18-24 has different demographics than the age bracket 12-17, most likely because it reflects college-age students, many of whom are not originally from Washington, D.C. Thus, we set the block-group level number of 18-year olds equal to the average number of children by age in the 12-17 year-old bracket. We assign to 18-year olds the average demographics of the 12-17 year-old bracket.

# **A.3.2 Household Types for Years 2003-2007**

Recall our assumption that each grade draws equally from the two most frequent ages in the grade, and only from those ages (for instance, 50 percent of second graders are 6 years old, and 50 percent are 7 years old). We also assume that while the marginal distribution of child age may change over time, the distribution of demographic types conditional on age remains constant. We assume, then, that all demographic types of a given age grow at the same rate _θat_ .

Based on these assumptions, for year _t_ we calculate the number of children of each age, _Nat_ , as follows. Let _θat_ = _Na_ , _t_ / _Na_ ,2000 be the proportional growth for age _a_ between year 2000 and year _t_ , _t_ = 2003,... 2007. The household type measure _μlamt_ for _t_ = 2003,... 2007 is then equal to _μlamt_ = _μlam_ ,2000 _θat_ , where _μlam_ ,2000 is the Census 2000 measure described above.

For each year, these measures imply a breakdown of the children eligible for each grade by race and poverty status. When compared with the observed breakdown of student enrollment by race and poverty status, two discrepancies arise. The first is that the resulting fraction of white children is lower than the fraction of the student body that is 47
white (perhaps because the fraction of white children in the population grew at a higher rate during the sample period). The second is that the resulting fraction of low-income children is lower than the fraction of the student body that receives free or reduced lunch. Hence, we apply upward adjustments to the measures of the corresponding household types in order to minimize discrepancies and faciliate the fit of the data.

# **A.3.3 Sampling from the Distribution of Household Characteristics**

Given the distribution of households by location, demographic type and age for each year, we draw 100 children for each grade and year, 50 for each of the grade’s two most frequent ages. For each age, we stratify the sample by race. The sample is probabilityweighted, with the weights being equal to the measure of the household type and age in the corresponding year.

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[| Stud. | Enr. Enr. Stud. | Enr. Enr. Stud. | Enr. Enr. ee Pate Schoots Schoots [Charter Schools | | Private Schools] Schools] Notes: Avg. Enr. is average enrolment and % % Stud. is percentage of students. students. The unit of observation observation is a campus-year. For instance, on average during the sample period 68.57 percent of public schools are elementary, 4.29 are elementary/middle, etc. Among public school students, on average 55.02 percent attend elementary schools, 4.97 attend elementary/middle schools, etc. “n/a” indicates there are no public schools of this this grade level.

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[|<br>Stud. | Enr. Enr. Stud. | Enr. Enr. Stud. | Enr. Enr.<br>ee Pate Schoots Schoots [Charter Schools | | Private Schools] Schools]<br>Notes: Avg. Enr. is average enrolment and % % Stud. is percentage of students. students. The unit of observation observation is a<br>campus-year. For instance, on average during the sample period 68.57 percent of public schools are<br>elementary, 4.29 are elementary/middle, etc. Among public school students, on average 55.02 percent<br>attend elementary schools, 4.97 attend elementary/middle schools, etc. “n/a” indicates there are no public<br>schools of this this grade level.<br><!-- End of picture text -->

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Notes: the unit of observation is a campus-year. For instance, among charter campuses, on average 47.37 percent offer a core curriculum, 9.65 percent offer an arts curriculum, etc.

at_[ [27 | sa7_| 266 [iFocus_| Note: Each row indicates the fractiontowne.of students |of 8055 the corresponding217race or poverty| 453status |enrolled31sin each|type of96 school. For each row, sum across columns equals 100. Data from all years has been pooled for the table.
<!-- Start of picture text --> Region Level Core Non-Core | Total<br>| | Middle<br>Ee<br>| Miixea | io | mw | 2 |<br>|_| Forat Northeast<br>—__[Miaat<br>| Mixea | oo | o | o |<br>| Total Southeast| 5 | ss<br>West |Elementary | 0 | 1 |<br>| Mixed | a | a | @ |<br>|| Total West<br>Totl | | tt |<br><!-- End of picture text -->

Note: For each row, sum across columns equals 100. For a given group of students, a cell denotes the percent of students of that group that attend a particular kind of school.
<!-- Start of picture text --> Focus Choice by Student Race and Poverty Status — Observed and Predicted Values, All Years<br>| | Observed Values(%) | Predicted Values(%)<br>Focus Focus<br>[LowInc. _|80.56{ 2.17] 4.53] 3.14] 9.60] 79.98] 2.33] 4.04] 3.49] 10.16 |<br>Note: For each row, sum across columns equals 100. For a given group of students, a cell denotes the percent of students of that group that attend a<br>school with a particular focus.<br><!-- End of picture text -->

<!-- Start of picture text --> All Wards Wards 7 and 8<br>Except Ward 3<br>[School Level | Public_| Charter _| Public | Charter |<br><!-- End of picture text -->

Note: school quality is the residual of the minimum-distance estimation regression of campus fixed effects on timeinvariant school characteristics. MHS= | if school level is middle, middle/high, high, or elementary/middle/high. Ward corresponds to the school’s location. Statistics are weighted by enrollment in the last year of the school in the sample.

<!-- Start of picture text --> (0.232)<br>(180.0)<br>(82.77)<br>(122.8)<br>eineeee’Mixed levels (=1 if school serves mixed levels) | an |88.90<br>(88.66)<br>Number of observations 64<br>R? 0.950<br>s.e. of regression 296.8<br>Mean of dependent variable 2526<br><!-- End of picture text -->

Note: Unit of observation: charter school campus. Dependent variable: total costs at the campus level, expressed in $1,000. OLS; standard errors in parentheses. Data: charter schools budgets for school year 2009-2010 from www.depesb.org. Before running the regression, the dependent variable was multiplied by (0.82)*(7/10) to incorporate the following: 1) charter reimbursements account for about 82 percent of charter revenues (which also include sources not modeled in this paper, such as federal grants for specific programs); 2) charter average reimbursements in 2003 and 2009 were approximately equal to $7,000 and $10,000 respectively
|Pe|ObservedValues||PredictedValues|
|---|---|---|
|All Entries<br>||<br>35.31|
|Level<br>|||
|Elementary<br>||19.37<br><br>|
|Middle|a<br>ee|ee<br>ee|
|jFous|||
|||27.56<br>|
|Region<br>|||
|Northeast<br>||18.15<br>|
|Southeast|a|10.67|

||Ss|AWards E|xceptWard 3|| Wards|7 and 8|
|---|---|---|---|---|
|School Level|||||
||0.074|||0.084|
|Elementary/Middle<br>|||-0.547||
|Middle<br>|-0.011|0.614<br>||0.078<br>|
|Middle/Highand High||-0.653|-0.878|-0.618|

Note: school value added is the residual of the mininmmm-distance estimation regression of campus fixed effects on time-invariant school characteristics. Statistics weighted by Fall 2007 enrollment.
<!-- Start of picture text --> Baseline and Counterfactuals (weighted by expected enrollment)<br>aBaseline |No Charters | Public School Cosmgs<br>Proficiency-NonMBSSchook<br>[Avg School +4bara—*4|er«4d erSSsC—CSCSCS<br>[GarermnabeatsSSCSCSC~C~C~“C~SCSCS~i<br>|__Charterentrants sosang PCS [ics<br>a<br>[—“Ghanterincumbents<br>SSC sts | i Cd<br>Se Oy<br>Lmse<br>[__Gharermcunbeats<br>|__Charterentrants SSCS nt SCSC*iCSCSCSCOTB<br>[Avg School Froficiency-MHS Schoos 30SCS SSS| Cid832CSCS<br>[Guatermcumbents——OSOC—~—“—~—S*S*~“‘*S*S*~*~drC<br>SCS<br>["Giarterentrants a “P| Cd CC“ C*d<br>SSC~—“—sSsSC“C~C~‘dTSC(‘sYSSSCSC‘idYSC<br>[Guide Wards ssf SSC~idSCSCSCS~—sSCS a —*d<br>[“Ghanermcunbeats———SSSOSCSCSOSOSCCCCCS<br>|__Charterentrants Tasat] [363SSC<br>2<br>Po Pubic tt | ts toe<br>|___Charterincumbents8639 TTT<br>[“Gharterentants———SOS—CSCSSSCS<br>ss | SSC~idSCtC‘“‘*‘“~*‘sCSC*~C‘*d<br>[AvgaSchoolQualty SCS Cd Cd(COC<br>Se<br>|_Charterentrants Yc<br>SSSSCSC—CSSSSCCS<br>[Beate Ts Ts<br>CSCS“<br>[ouside[Pubic Wads CT 0855]co | tot00" =PSTOCS St<br>Se<br>|__Charterentrants Toe Tt<br>SC<br>OSCSCSCSSCCCCCCCCS<br>[me os [on [| oe |<br>[—~Gharterentrants|____CharterincumbentsSCS|| 2.1]ts To+4) ir<br>a<br>[Avg SchoolVaeAdded SCS] XS SCS SSCSC~idT CCS<br>OCOC~CSCSOSOSOCOC‘“‘“(LW<br>[hte OS CCtSSC~*d<br>[Gharermeumbents————SSOSCSCSCSCSSCCCCCC<br>|_Charterentrants co | SCS SSC<br>SSSSCSCSCSOSOSCSCCYYS<br>[Outside Wards ToteSYTtCrd CY<br>|__Charterincumbents<br>SSCS]<br>[—“Gharterentrants | oto | SS—C~iS—ts<br>PSE<br>[_“thanerncunbeats<br>[Chartereatrants SCSSC S . «|107 | — S SCSCCSC~ id SS CSC—ZSC— CC”<br><!-- End of picture text -->

|Baselineand Counterfactuals <br><br>|(weighted by ex<br><br><br>|pected enroll<br><br><br>|ment)<br> <br><br>|
|---|---|---|---|
|[Avg<br><br>|Baseline <br><br>bara<br><br><br>|No Charters <br><br>er<br><br><br>|Public SchoolCosmgs<br><br>er<br><br><br>|
|School<br>Proficiency-<br><br><br>|<br>Non<br>+4<br>—*4<br><br>|<br>MBS<br>|<br>«4<br><br>|Schook<br>d<br>SSsC—CSCSCS<br><br>|
|[GarermnabeatsSSC<br><br>|CSC~C<br> <br>|~C~“<br> <br>|C~SCSCS~i<br> <br>|
||<br>|<br>|<br>|
||<br> <br><br><br>|<br> <br><br><br><br>|<br> <br><br><br><br>|
|[—“Ghanter<br><br>|in<br><br>|um<br><br><br>|bents<br><br><br>|
|Se|<br>|||
|Lm<br><br><br>|||e<br><br>|
|[__Gharer<br>SSC<br>|__Charterentrants<br><br> <br>|mc<br>nt<br><br>|unb<br>SCSC*<br><br>|eats<br>CSCSCSCOTB<br><br><br>|
|[Avg SchoolFroficiency-MHSSchoos<br>S<br>|<br>0 <br>S SSS<br><br>|Cid<br>|832<br><br>CSCS<br> <br>|
|[Guatermcumbents——OSOC—<br><br>|~—“—~—S<br><br><br>|*S*~“‘<br><br><br>|*S*S*~*~drC<br> <br><br> <br>|
|S<br> <br>|||<br><br>S<br> <br><br><br>|
|Wards<br>SSC~—“—sSsSC“C~C~‘<br><br>|dTSC(‘<br><br>|<br><br><br>sYSSS<br><br>|<br><br> <br><br>CSC‘idYSC<br>SCS<br><br>|
|[“Ghanermcunbeats——<br><br>|—SSSO<br><br>|SCSCS<br><br>|OSOSCCCCCS<br><br>|
|T|s|[|63|
||<br><br><br>|<br> <br><br>|<br><br>|
|Po<br>Pubic<br>t<br><br>|t<br><br><br>||ts<br><br>|toe<br><br>|
||___Charte<br>8<br>|rin<br>39 <br><br>|<br>cum<br>T<br><br><br>|bents<br>TT<br><br>|
|[“Gharterentant<br><br>|<br>s———<br>ss <br><br>|<br>OS—<br>|<br>SSC~idS<br>|<br>SCSSSCS<br>CtC‘“‘*‘“~*‘sCSC*~C‘*d<br><br>|
|[AvgSchoolQualtySCS|Cd|Cd|(<br><br>COC|
|<br>S<br>|_Charterentrants<br><br>|||Yc<br>|
|T<br>[Beate<br>SSSSCSC—<br>[ouside <br><br>|s <br>CS<br>0855<br>|T<br>SS<br>00"=<br>|s<br>SCCS<br> St<br><br>|
|Wads<br>CS<br><br>|<br><br>C<br>|<br> <br>S<br>|<br> <br>“<br><br>|
|<br><br>C|co|<br>tot|<br><br>ST|
|S<br><br>|<br><br>|<br><br>||
||__Charterentrants<br>T<br>|e <br>|T<br>|t<br>|
|S<br>|<br><br>|<br><br><br>|<br><br><br>|
|[me<br>OSCSCSCSS<br><br><br>|CC<br>os<br><br>|CCC<br>[on<br><br><br>|CCCS<br>|<br>oe|<br><br>|
||____Charte<br><br><br>|<br>rin<br><br>|cum<br><br><br>|<br>bents<br><br>|
|[—~Gharterentrants<br>SCS<br><br><br>|ts<br>| 2.1<br><br>|+4<br><br>|o<br> ir<br><br>|
|[AvgSchoolVaeAdded<br>SCS<br>|]XSSCS<br>|SSCSC~id<br>|TCCS<br>|
|<br><br>[hte<br>OCOC~CSCSOSO<br>|<br>SOC<br><br>|<br>OC‘<br>S<br>|<br>“‘“(LW<br>CCtSSC~*d<br>|
|[Gharermeumbents———<br><br><br>|—SSOS<br><br>|CSCSC<br><br>|SCSSCCCCCC<br><br>|
|To<br>[OutsideWards<br>SSSSCSCSCS|<br>te<br>OSO<br>S|<br>T<br>SCS<br><br>Crd|<br>t<br>CCYYS<br><br>CY|
|<br><br>|__Charter<br><br>|inc<br>|um<br><br>|bents<br><br>|
|SS|C|S<br>SS—|]<br>~iS—|
|P|<br><br>||E|
|[_“thaner<br>SSC**S**<br>[Chartereatrants|nc<br>.107<br>|unb<br>—**S**SCSC**i**<br><br>|eats<br>**SS**SC—<br>|
|SC|«|CSC~|CSC—ZCC”|
<!-- Start of picture text --> SSBlack Hisp. SSSWhite Black Hisp. White<br>Baseline<br>| secs [ Highilies| 3029 =| 471ioe fer| sais | ator |Pa417s | 60.02_|<br>[High | 5.29 | 009 | 202 | 023 | ota | 020<br>Raise Funding [Elementary | 025 | -o39 | 033 | o1 | 025 | 019 |<br>Approve All<br>[High | ota | 0.32 | 0.02 | 005 | ot | _<br>Costs [Middle | -020 | -063 | -106 [| on | 083 | 089 |<br>Close Public<br>Note: “Baseline” shows average|Highproficiency| index0.07for students| 000of each demographic| oorgroup[| andoos_grade level |(in-oo1percent). |“No Charters”0.00 _|<br>shows the difference in average proficiency index between Baseline and No Charters, and similarly for the other counterfactuals (in<br>percentage points). Statistics weighted by students’ weights. “Elementary” refers to K-6 grades; “Middle” to grades 7 and 8, and “High”<br>to 9-12 grades.<br><!-- End of picture text -->

<!-- Start of picture text --> Black Hisp. White Black Hisp. White<br>ee 80-9<br>Baseline<br>[High | sit.oos_| 323,717 | 526,122 | sizs24 | s30,160 | $44.19_|<br>No Charters<br>[High | (8139) | (s6)_| (S058) _| (s4ss)_| (866) | (858) _|<br>Raise Funding<br>[High | sss | sia | sai | sto | se | 826<br>Approve All [High | ssi_—| sis | sso | soz | sa] sto.<br>Lower App.<br>ee<br>Costs [High | sis | s30s__| sizi_—[| suis] sts | so4__<br>Close Public<br>Note: “Baseline” showsHighthe average |welfare fors23)_|students ofsis)each demographic| ss)_|group and gradesis)level. _|“No iyCharters”|shows3)average _|<br>compensating variation from the counterfactual for student each demographic group and grade level, and similarly for the other<br>counterfactuals (negative values are in parentheses). “Elementary” refers to K-6 grades; “Middle” to grades 7 and 8, and “High” to 9-<br>12 grades.<br><!-- End of picture text -->