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 net social gains by providing additional school options, of charter school entry and school choice. In the model, and they benefit non-white, low-income, and middle- households choose among public, private, and char- school students the most. Further, policies that raise the ter schools, and a regulator authorizes charter entry and supply of prospective charter entrants in combination mandates charter exit. The model is estimated for Wash- with high authorization standards enhance social welfare. ington, D.C. According to the estimates, charters generate 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. Raviku- mar, 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 sem- inar 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 computa- tional 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 Wash- ington, 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, pri- vate and charter schools in the city including enrollment by grade, school demograph- ics, focus and proficiency rates in standardized tests. We supplement these data with neighborhood-level information on charter school attendance and travel distance to char- ter 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 pre- dictions. 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 con- sideration, 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 ef- fects 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 2013-2014.1 It has a single 1 As 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 re- search 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 curricu- lum. 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 exis- tence 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 (autho- rization) 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 achieve- ment effects,2 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 exist- ing 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 het- erogeneity 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 strate- gic 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. 2 For 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.3 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 qual- ity poses severe computational challenges. Hence, we recast our demand-side estimation as a mathematical programming with equilibrium constraints (MPEC) problem follow- ing Dube et al (2012) and Skrainka (2012). While these authors supply analytical gra- dients 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 differ- entiation 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, re- cently 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 ter- tiary 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. 3 Other 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 para- meters. 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 char- ter 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 opera- tional per-pupil funding as public schools. In addition, they receive a per-pupil facilities allowance. Since funding is fungible, henceforth “reimbursement” refers to total (oper- ational 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 atten- dance 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 Superinten- dent 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 compa- rability over time and across schools. In addition, 2007 marked the beginning of impor- tant changes in DCPS and hence constitutes a good endpoint for us.4 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.5 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 res- idential programs and early childhood centers. For each observation we have address, grade enrollment for kindergarten through 12th grade, percent of students of each eth- nicity (black, white and Hispanic),6 and percent of low-income (or “poor”) students, who qualify for free or reduced lunch. We also have the school’s thematic focus, which 4 In 2007, Michelle Rhee began her tenure as chancellor of DCPS. She implemented a number of re- forms, such as closing and merging schools, offering special programs and changing grade configurations, etc. The first such reforms took effect in Fall 2008. 5 A 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. 6 Since 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. Cur- ricular 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 state- ments, 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 char- ter data collection is the dispersion and inconsistencies of data sources, and their non- uniform 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 (cov- ering grades in the 9th-12th grade range), and elementary/middle, middle/high, and el- ementary/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 neigh- borhood’s fraction of children enrolled in charter schools relative to total public enroll- ment, 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 distrib- ution 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 this is not the case, we use 2000 Census data and other sources to estimate the joint distribution. Appendix A.3 provides further details. 3 Descriptive Statistics Population in Washington, D.C. peaked in the 1950s at about 802,000, declined steadily to 572,000 in 2000, and bounced back to 602,000 in 2010. Between 2003 and 2007 it grew from 577,000 up to 586,000, although school-age population declined from 82,000 to 76,000 according to the Population Division of the U.S. Census Bureau and American Fact Finder. The city’s racial breakdown has changed as well, going from 28, 65 and 5 percent white, black and Hispanic in 1990 to 32, 55 and 8 percent respectively in 2007. Despite these changes, the city remains geographically segregated by race and income, with large disparities among races. For instance, in 2013 median household income was $112,000 for whites, but only $38,000 for blacks and $51,000 for Hispanics. 3.1 Basic trends in school choice As Figure 1A shows, total enrollment declined by about 6,000 students over our sample period, yet charter enrollment grew by that amount. Since private school enrollment remained steady at about 21 percent of total enrollment, market share for public schools fell from 66 to 56 percent yet rose from 13 to 22 percent for charters. The number of charter school campuses more than doubled, from 27 to 59, whereas the number of public and private school campuses declined slightly due to a few closings and merg- ers (Figure 1B and Table 1). Over the sample period, 43 percent of private schools were Catholic, 24 percent belonged to other religions and 32 percent were nonsectarian. Average charter reimbursement grew from $7,900 in Fall 2003 to $9,600 in Fall 2007. Figure 1: Enrollment and Schools in Washington D.C. 8 Table 1: School Openings and Closings Student demographic and proficiency rates vary greatly among public schools (see Table 2). Nonetheless, on average public and charter schools have demographically similar students; more than 90 percent of them are non-white and about two-thirds are low-income. In contrast, about 60 percent of students in private schools are white and less than a quarter are low-income. Public and charter schools have similar average reading and math proficiency (about 41 percent). Table 2: Demographics and Achievement in Public, Charter and Private Schools Figure 2 shows the geographic distribution of schools and average household income. Charters are spread throughout the city except in the affluent Northwest. Even though private schools tend to be located in higher-income neighborhoods than public or charter schools, Catholic schools are located in less affluent neighborhoods than other private schools and enroll higher fractions of black and Hispanic students. At $7,800, their average tuition is lower than that at other religious or non-sectarian schools, whose average tuition equals $19,700 and $20,900, respectively. Panel a in Table 3 illustrates school choice by student race.7 About 70 percent of black and Hispanic children attend public schools, compared to only 27 percent of whites. Between 15 and 20 percent of blacks and Hispanics attend charters, relative to 3 percent of whites. Nearly three quarters of whites attend private schools, compared to less than 15 percent of blacks and Hispanics. As Appendix Figure 1 shows, children who live in the eastern portion of the city are more likely to attend charters. Regardless 7 Inthe absence of individual-level data, we use school-level data to approximate the distribution of school (and later focus) choice across students. 9 Figure 2: Geographic Location of Schools in 2007 10 Table 3: School Choice by Student Race of their residential location, children travel longer to charter than to public schools. Median distance traveled to public schools is equal to 0.33, 0.64 and 1.47 miles for elementary, middle and high school respectively, whereas median distance traveled to charter schools is equal to 1.42, 1.66 and 2.37 miles respectively (Filardo et al 2008). 3.2 Variation by grade level and focus As Appendix Table 1 shows, most public schools are elementary. While the average public elementary school has about 280 students, the average public middle and high school has almost 400 and 640 students, respectively. Charters tend to be smaller than public schools, and private schools tend to be smaller than charters. Although public schools rarely mix grade levels, charters and private schools often do. Figures 3A-3C show that market share for each school type varies across grade levels. Public school shares peak for elementary school grades; charter school shares peak for middle school grades and private school shares peak for high school grades. Consistent with this, panels b and c of Table 3 show that students from all races are less likely to choose public schools after 6th grade - whites tend to switch into private schools; blacks tend to switch into charters (and Catholic schools, to a lesser extent), and Hispanics tend to switch into Catholic and charter schools.8 Figure 3 also shows that during the sample period public schools lost market share in all grades but particularly in middle school. Charters, in contrast, gained market share in all grades, particularly in grades 6-8. This gain might partly relate to the fact that 6th and 7th grades are natural entry points into charters, since students must change 8 It is possible that white parents would leave the District once their children finish elementary school. As a simple test of this conjecture we calculate the fraction of white children per age. The fraction declines from 19 to 13 percent between ages 0 and 4, but stabilizes around 10 percent between ages 5 and 18. Thus, white parents appear to leave the District before their children start school. 11 Figure 3: Aggregate Enrollment Share by Grade schools when finishing elementary school. But the gain might also relate to the fact that the charter sector seems to have increased its supply of seats the most for middle school. As Figure 4A shows, during the sample period the number of charters falls well below that of public schools for grades K-6 but is almost the same for grades 7 and 8 by the end of the sample period. Although average class size (a proxy for the number of seats per grade offered in the school) is lower for charters than public schools after grade 6 (Figure 4B), the difference is relatively small for grades 7 and 8. In other words, charters seem to have expanded the choice set for middle school students relatively the most. Figure 4 Moreover, proficiency rates vary across school types, grade levels and neighbor- hoods (see Table 4).9 Relative to public schools, charters tend to have slightly lower proficiency at the elementary level. Nonetheless, on average they surpass public schools in the upper (particularly middle-school) grades. For instance, outside ward 3 charter middle schools attain an additional 17 percentage points relative to public schools. Fur- thermore, in wards 7 and 8 charters attain an additional 22 and 24 percentage points in middle school and high school grades, respectively. The vast majority of public and private schools offer a Core (i.e., non-specialized) curriculum, yet more than half of charters offer a specialized curriculum (see Appendix Table 2). Overall, 80, 11, 4, 3 and 2 percent of students attend a Core-curriculum, Other, 9 Since reading shows similar patterns, we focus on math from now on. 12 Table 4: Average Math Proficiency Rates in Public and Charter Schools Language, Vocational or Arts school respectively (see Appendix Table 3). Blacks are more likely than other students to choose Arts or Vocational; whites and non-poor stu- dents are more likely than black or low-income students to choose Other, and Hispanics are more likely than whites or blacks to choose Language. 3.3 Entries, relocations and closings Public and private schools experienced relatively few openings, closings and relocations during the sample period (see Table 1), particularly when measured against the number of schools of each type that existed by the end of 2002.10 In contrast, openings and relocations were quite frequent among charters. Charters often open with a subset of their intended grades and add grades over time, moving from small, temporary facilities to larger, permanent facilities. Our charter sample includes 63 campuses and 45 schools. Ten schools run multi- ple campuses, mostly to serve different grade levels. Appendix Table 4 displays charter school entry patterns between 2004 and 2007 (the years used for supply-side estimation, as explained below). Of the 33 entrants, 19 offer elementary grades, 9 middle grades, 4 elementary/middle or middle/high, and one high school. Two-thirds of entrants offer a specialized curriculum; Other is the most popular focus choice. The West, Northeast and Southeast of the District are home to 16, 47 and 37 percent of all school-age chil- dren and have received 15, 61 and 24 percent of all charter entries during our sample period, respectively (see Figure 5). In other words, the Northeast has received dispro- portionately large entry, and the Southeast disproportionately little. Southeast entrants are more likely than others to offer elementary grades and a Core curriculum. Of the four charter closings in our sample, two were due to academic reasons and one to mismanagement.11 The average charter relocation distance is 3.47 miles (median 10 Since 2000, DCPS has engaged in efforts to “rightsize” the public school system through school renovations, openings, mergers and closings. Declining student population and enrollment were the main driver of closings (Filardo et al 2008). Most school moves were temporary and due to renovations. Most private school closings during the sample period affected schools with fewer than 30 students. 11 The fourth closing involved a campus from a multi-campus organization. The campus existed only 13 Figure 5: Location of Charter School Entrants in Washington D.C., 2004-2007 = 3.09 miles), and 5 of the 20 moves happened within the same cluster. 4 Model In this section we develop our model of charter schools, household school choice, reg- ulator actions and equilibrium. In the model, the economy is Washington, D.C. Public, private and charter schools exist in the economy and serve various grade levels. The economy is populated by households that live in different locations within the city and have children who are eligible for different grades. Given its budget constraint, each household chooses among the schools offering its required grade. Although the model includes public, private and charter schools, we only model charter behavior. Since entry, exit and relocation are more common among charters than public or private schools, as explained above, we would not be able to identify a strategic decision-making model for public or private schools. Hence, we assume that in any given period these schools make decisions first, and the regulator and charters take these decisions as given. Since public and private schools might react to changes in the environment at some point, in our counterfactuals we implement a simple rule whereby they close if enrollment falls below a specific threshold, and remain open otherwise. A charter entry point is a combination of location, grade level and focus. At a given time, in each entry point there is a prospective entrant deciding whether to apply for a year and then re-assigned students to the two other campuses in the organization. 14 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 equilib- rium 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 econ- omy have one child each. In what follows, we use “household”, “parent”, “child” and “student” interchangeably. Student i is described by variables (D; `; I ; g; ε ), where: D is a row vector describing student demographics. In our data this vector contains e = 3 binary elements indicating whether the household is white, Hispanic (omitted D category is black) and non-poor, respectively. `, ` 2 f1; :::; Lg is household location in one of the economy’s L neighborhoods.12 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, Voca- tional, 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. b D is the row vector of households’ beliefs about the school’s peer (or student body) jt 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 12 A 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 Ds 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: Ui jgt = δ jgt + µi jgt + εi jgt (1) where δ jgt and µi jgt 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 con- tribution 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: δ = y β +D b α p ϕ +ξ (2) jgt j jt jgt jgt 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: ˜ + [Di D b ]α µi jgt = Di ω + [y j Di ]β jt ˜ + xi jt γ : (3) The household may choose the outside good ( j = 0) with normalized utility Ui0gt = εi0gt . 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 en- compasses all public schools (as if there were open enrollment),13 all charter schools, 13 Data 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-of- boundary shool. Moreover, the mechanism was seemingly not systematic across the District. Hence, we simplified by assuming public school open enrollment. 16 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 i be the number of schools in i’s choice set. Student i chooses a school in order Ii . Let Jgt 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: b ;ξ ; p ;X ;θd = exp(δ jgt + µi jgt ) Pi y ; D jgt gt gt gt gt igt (4) Ji 1 + ∑kgt =1 exp(δkgt + µikgt ) d where vector θ 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 b , ξ and gt gt 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 ; `jg) 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: Z Z Z b b d S jgt ygt ; Dgt ; ξgt ; pgt ; θ = i Pjgt ( )dh(D; I ; ` j g) (5) D I ` The expected number of students choosing school j for grade g at t is: b ;ξ ; p ;θd = M S b jgt ygt ; D b N gt gt gt gt jgt ( ) : (6) The total expected number of students in school j at year t is hence equal to b ;Ξ ; p ;θd = b jt yt ; D N b jgt ( ) t t t ∑g2κ N jt (7) b ; and p are vectors that describes focuses, household beliefs on demo- where yt , D t t graphic compositions and prices respectively of all operating schools in t , and Ξt is the vector that stores the ξ jgt s of all operating schools. The resulting expected demographic 8 composition for school j is thus 9 equal to 0. The probability of 23 This dummy is included for empirical purposes, as closings due to academic reasons rarely happen before the school’s fifth year. 21 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 re- duction 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 d , supply-side parameters θ s and proficiency rate parameters θ q respectively. We describe the data and estimation below. 5.1 Data Our data include 65 markets (13 grades times 5 years) and J S =281 campuses, for a total of J D =1,269 school-year observations and J X =8,112 school-grade-year observations. It also includes JC =153 neigborhood-year observations. Recall that we observe the fol- lowing 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 char- ter 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.24 5.2 Demand Estimation To estimate θ d we use Generalized Method of Moments (GMM) and match market shares at the school-grade-year level (“share moments”), student demographic compo- sition at the school-year level (“demographic moments”), and average fraction of stu- dents attending charter schools, average distance traveled to public schools, and average distance traveled to charter schools at the neighborhood-year level (“neighborhood mo- ments”). While typical BLP consists of share moments, we augment the GMM objective function with the other moments. 24 We 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 “b” for predicted values. Consider the ns draws for children eligible to attend grade g in ˆ jgt = Mgt ∑ns year t . Predicted enrollment for ( j; g; t ) is N i d i=1 Pjgt ygt ; Dgt ; ξgt ; pgt ; Xigt ; θ , ns where Pi ( ) is given by (4) and we use D to approximate D b in the r.h.s. of (4). Predicted jgt enrollment share for ( j; g; t ) is b Sbjgt = N jgt (14) Mgt and S jgt denotes its observed counterpart. Predicted school characteristics are equal ˆ N jgt ∑g2κ jt ∑ns i d i=1 Di Pjgt (ygt ;Dgt ;ξgt ; pgt ;Xigt ;θ ) b = to D ns b be the C . Finally, let C e 1 vector jt ˆ jgt ∑g2κ jt N kt of predicted average values for neighborhood k in year t , with the following C e=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 jX = D b and E C jX = C b , and that D and C are different from their jt t jt kt t kt jt kt expected values due to sampling (or measurement) error: = D jt D b and uC = uD jt jt kt b Ckt Ckt . Following BLP and Nevo (2000, 2001), we assume that ∆ξ jgt is mean- independent of the corresponding instruments: E (∆ξ jgt jZ X jgt ) = 0. In addition, we as- D D C C sume E (u jt jZ jt ) = 0 and E ukt jZkt = 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 d . 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: 2 30 2 32 3 λX (∆ξ ) VX O O λX (∆ξ ) min 4 λD (∆ξ ; θ d ) 5 4 O VD O 5 4 λD (∆ξ ; θ d ) 5 s:t : S = S ˆ(∆ξ ; θ d ) (15) ∆ξ ; θ d λC (∆ξ ; θ d ) O O VC λC (∆ξ ; θ d ) where λX , λD and λC are sample interactions of the shocks ∆ξ , uD and uC with the corresponding instruments (see details in Appendix E.1); S ˆ( ) 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 d ; 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 ˆ. Given the decomposition of the demand shock ξ jgt ;we include school-, grade- and time-fixed effects in the utility function. Since the school fixed effects capture both the value of time-invariant school characteristics, y j β , 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 zξ and z∆ξ : 5.3 Supply estimation n o Supply-side parameters are θ s = b; ζ ; V; F; σν ; α ˘; β˘ ; aπ ; bπ ; aq ; bq ; cq . In our application, the economy includes L = 39 locations (neighborhood clusters), Y = 5 fo- cuses (Core curriculum, Arts, Language, Vocational, Other) and K = 5 grade levels (elementary, middle, high, elementary/middle, middle/high), for a total of E = 975 po- tential 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 Mt 1 as described in the game’s Step 4; b) charter closings; c) charter relocations.25 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 ˜t be the number of incumbent charters that remain in the same location as in t 1. Let ` jt be j’s location in t , and let ` jt be its observed counterpart. Variable d e jt 2 f0; 1g indicates whether there is a new entrant in entry point j in year t ; variable d x jt 2 f0; 1g indicates whether incumbent j e x closes at the end of t 1; and variables d jt and d jt are the observed counterparts of d e jt and d x jt respectively. The likelihood function is:26 8 9 T < E ˜ Ct 1 Ct 1 = e x = dx ˜ (θ s ) = ∏ ∏ Pr d e L jt = d jt jMt 1 ∏ Pr d jt jt ∏ Pr ` jt = ` j` jt jt 1 t =2 : j=1 j=1 x j=1: d =0 ; jt where the first product inside L ˜ ( ) stands for new entries in school year t , the second product corresponds to charter closings, and the third product corresponds to reloca- tions. The closed-form formulas for these probabilities are described in Appendix E.2. Infrequent entry and exit in our sample complicate the estimation of θ s . Nonethe- less, 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 ˆ = 71 / (975 entry points * 4 years) = 0.018.27 in b 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 25 Closing 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. 26 Note that the likelihood for 2003 cannot be calculated as we lack data on market structure, charter locations and charter proficiency in 2002. 27 In 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: c = F0 + Ve Enr + FW W + FSE SE + FH H + FM M TC (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 F0 ; 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 fζ ; σν ; α ˘; β˘ ; aq ; bq ; cq g.28 5.4 Proficiency Rate Estimation, and Summary According to (9) the observed proficiency rate q jt is given by: q q q q q jt = y j α q + D jt φ q + [y j D jt ]ω q + ξ j + ξt + ∆ξ jt + v jt (17) where the error term is the addition of the a school-year unobserved shock on proficiency q q q ∆ξ jt and sampling or measurement error v jt . Since ∆ξ jt may be correlated with ∆ξ jgt , q D jt is likely to be correlated with ξ jt , thus requiring the use of instrumental variables. q To estimate α q separately from the school fixed effects ξ j , we use a similar 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 q q regression are our estimates of ξ j , or “value added”. We use our estimates of ξ j for entrants to obtain the empirical distribution of value added. The estimation of proficiency rate parameters is straightforward, and so is Maxi- mum Likelihood estimation of supply side parameters once expected enrollments for po- tential entrants have been computed.29 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 d (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 second- order derivatives and attain great speed, despite our problems’ complicating features 28 We cannot estimate aπ and bπ because there are no closings for financial reasons in the sample. 29 In 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 d . 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 zξ and z∆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. 25 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 con- cern 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 ex- tent 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 demo- graphics. 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 in- clude 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 instru- ments 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 Identification 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˜ , 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 en- try, 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 lowers average predicted distance, respectively. Increasing aq ; bq and cq raises the number of predicted closings, their sensitivity to pro- ficiency rates and their predicted probability after the first five years, respectively. 26 6 Estimation Results 6.1 Demand Side Table 5 presents our preference parameter estimates. Most of them are statistically sig- nificant and of the expected sign. The “baseline utility” column displays the parameter estimates for (2), and represents the preferences of black, low-income households. Re- maining columns present parameter estimates for (3), with differences in the preferences of white, Hispanic and non-poor households with respect to those of black, low-income households. We interpret the estimates in terms of the choice probability difference they would induce between two schools that only differ in a given characteristic. In what follows, “middle and/or high schools” (MHS) refers to middle, middle/high, high, elementary/middle/high, and “non-MHS” to the remaining levels. Table 5: Parameters Estimates: Utility Function Our estimates show heterogeneity in school type preferences across races, poverty status, and school level. Among non-MHS, most households prefer public over charter 27 schools yet not with the same intensity. Poverty raises the likelihood of choosing char- ters. 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 Ap- pendix 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 at- tending 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 $1000- decline 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 per- centage 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.30 Recall that our school quality estimates capture unmeasured school character- istics 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 30 In 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. 28 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 8.31 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 lo- cated 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 dis- tribution 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, account- ing 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 finan- cial 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 ˘ indicate that schools are averse to moving and seek close destinations when moving. Estimates for bq and cq indicate that charters are more likely to be closed if their academic performance is 31 School 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. 29 Table 6: Parameter Estimates: Supply Side low, particularly if they have been open for more than five years. 6.3 Academic Proficiency Table 7 presents estimates of the passing rate function for math. Since we have at most five annual observations per school, these estimates should be taken with caution. We interpret our estimates in terms of how a particular school characteristic affects the rel- ative odds of passing the math test, holding everything else constant. Among public schools, the relative odds of passing are lower in MHS than in non-MHS. When com- paring a single-campus charter with a public school, the relative odds of passing are 79 percent lower for the charter at non-MHS but 34 percent higher at MHS. The relative odds of passing are 53 percent lower for multi-campus charters than for public schools at non-MHS, but 205 percent higher at MHS.32 Schools offering Other and Arts raise the relative odds of passing relative to those offering Core. Coefficients on percent white and non-poor students are not significantly different from zero. Recall that our value added estimates estimates capture unmeasured school char- acteristics affecting proficiency such as leadership and culture, instructional style, teacher recruiting practices, length of school day and year, etc. As Appendix Table 10 shows, outside Ward 3 average value added in charters is higher than in public schools for el- ementary, elementary/middle and middle schools, with a particularly large premium in elementary/middle and middle schools. Furthermore, in wards 7 and 8 charters have higher value added at all levels. 32 This “middle school advantage” in math is consistent with Betts and Tang (2011), Clark et al (2011) and Dobbie and Fryer (2013). 30 Table 7: Parameters Estimates: Math Proficiency Rate 7 Policy Analysis and Counterfactuals In this section we quantify the social value of charter schools and analyze counterfac- tual policies. “Baseline” denotes the Fall 2007 benchmark equilibrium, calculated as follows. Starting from the set of actual operating schools in Fall 2007 except for ac- tual entrants,33 we generate 1000 market structures through Monte Carlo simulation of potential entrants and steps of the stage game. For each simulated entrant we draw (ξ , ξ q ). We average over simulations to obtain the baseline. We calculate the equilibrium for each counterfactual similarly. 7.1 Baseline Table 8’s column 1 reports the baseline. On average, 9.1 schools enter the market and capture 3.3 percent of K-12 enrollment. These predictions are consistent with the ob- served annual average number of entries in 2004-2007. Projected to Fall 2013, the predictions are consistent with the observed number of regular charters (equal to 93 ac- cording to www.dcpcsb.org) and charter market share (equal to 35 percent) in Fall 2013, 33 We allow for a response on the part of public and private schools when computing baseline and counterfactual equilibria. We assume that in Step 3 of the stage game (Spring of calendar year 2007), public and private schools close if their predicted enrollment (computed to take into account entry of approved charters in Fall 2007) falls below 20 percent of their lowest enrollment during the sample period. With this threshold choice we approximately match the observed closing rate for non-charter schools in the sample. 31 Table 8: Equilibrium in Baseline and Counterfactuals as explained in Appendix F. The model matches the actual distribution of entrants by region and focus. It predicts almost no public or private closings, since such closings were mostly idiosyncratic during our sample period, and predicts the closing of three charters, consistent with the observed charter closing rate of about 1/3 between 2007 and 2013. The model also replicates observed market share by school type, and fraction of students not enrolled in school. In addition, the model replicates the observed distribution of proficiency, school quality and value added (see Appendix Table 11). The model predicts that charter en- trants tend to have lower quality, value added and proficiency than charter incumbents.34 Nonetheless, in the baseline charter entrants have higher quality than public schools out- side ward 3 and in the SE, and higher value added in the SE. For each student we construct a proficiency index equal to the weighted average predicted proficiency of the schools attended by the student; each school is weighted by the student’s corresponding choice probability. The first row of Table 9 shows the percent of students in each (race, poverty status) demographic group, and the “Baseline” panel shows average proficiency index per student group and grade level. The index is lower for blacks than non-blacks and for poor than non-poor students, and shows a large gap between white and non-white students. For each student i we calculate the willigness to pay for its choice set relative to Ji the outside option (or welfare, for brevity) as Wi = lnf1 + ∑kgt =1 exp(δkgt + µikgt )g=ϕ . 34 The reason is that charter incumbents include recent entrants (entering in 2003-2007) as well as early entrants (entering before 2003, and from whose distribution we draw entrants’ ξ s and ξ q s). Incumbents that entered early are the survivors among all early entrants. Their estimated school quality and value added is higher than that of the average entrant, as we would expect based on Step 6 of the stage game. 32 Table 9: Average Proficiency Index by Student Group and Grade Level Table 10: Average Welfare and Welfare Gain by Student Group and Grade Level The “Baseline” panel of Table 10 shows average welfare for each demographic group. Welfare varies substantially depending on student race, poverty status and grade level. On average, black and poor students attain lower welfare than others because they have access to fewer school options, particularly among private schools. They also tend to live far from public schools with a high percent of white students or of high quality. Hispanics attain greater welfare than blacks because on average they live closer to desir- able public schools, enjoy strong same-race peer effects, and value non-public schools more than blacks. Whites, in turn, have higher welfare than non-whites because they have access to more school options and live closer to desirable ones. They also en- joy strong same-race peer effects and derive substantial utility from attending private schools, particularly at the MHS level. Average welfare is lower for students in MHS than non-MHS, mostly because of the lower number of options. Table 11 presents a cost-benefit analysis for the economy. Column 1 presents total willingness to pay for schools, or total social benefits relative to the outside option (equal to households’ total Wi s). Column 2 presents total educational costs assuming that public school per-student cost is equal to charter reimbursement and that private school per-student cost is equal to tuition. Costs in Column 3 assume that public school per-student cost is 20 percent higher than in charters, as D.C. charters contend that a 20-percent reimbursement increase would equalize their funding with public schools.35 35 InNovember 2014, the DC Association of Charter Schools, along with two DC charters filed a suit in federal court against the Mayor and the Chief Financial Officer requesting approximately this funding increase. See http://dcschoolfundingequity.org/ for additional information. The plaintiff contends that 33 Table 11: Cost-Benefit Analysis In addition, column 3 assumes that Catholic schools’ tuition covers about two-thirds of expenses (www.ncea.org). Columns 4, 5 and 6 present per-student costs and benefits. Regardless of cost assumptions, total social benefits relative to the outside option exceed total costs by a factor of about 2, reflecting the net social value attached to schools. 7.2 The Social Value of Charter Schools In order to quantify the net social gains generated by charter schools, we run a coun- terfactual consisting of not having charters at all in 2007. Thus, we compute the equi- librium for the actual 2007 market structure excluding charters. When discussing coun- terfactuals, for brevity we often use “now” and “before” to refer to counterfactual and baseline respectively, and “switch” to describe school choices that differ in both equi- libria, even though these are not consecutive equilibria. Column 2 in Table 8 depicts the resulting no-charter equilibrium. Most students previously enrolled in charters switch into public schools, though a small proportion chooses private (particularly Catholic) schools. An additional 2 percent of students choose the outside option, including dropping out of school.36 As suggested by Ap- pendix Tables 7 and 11, charter students who switch into public schools outside Ward 3 experience lower proficiency, quality and value added than before. Proficiency losses are quite severe at the middle school level and for poor black students, who on average lose 6.4 and 5.3 percentage points out of their baseline average proficiency of 35.1 and 30.3 percent in middle school and high school, respectively (see Table 9). To estimate the welfare impact of eliminating charters, we compute child i’s compensating variation CVi = WiE WiB , where WiB and WiE are i’s welfare in the base- line and counterfactual experiment, respectively. Table 10 presents the results in the “No DCPS receives additional, off-formula funding that puts charter schools at a disadvantage. 36 Booker et al (2011) finds positive effects of charter school attendance on high school graduation rates. As of 2013, graduation rate in D.C. is 79 and 58 percent for charter and public school students, respectively. Source: www.focusdc.org. 34 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 rep- resent 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 ben- efits 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 expan- sion: 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 reimburse- ments, 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 expan- sionary 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 en- try 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. Fund- ing 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 Ap- pendix 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 per- cent 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 col- umn 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 coun- terfactuals 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 Wash- ington, D.C. points to the importance of lowering application costs - for instance, by facilitating charters’ access to unused public school buildings.37 37 Access 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. 36 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 col- umn 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 be- tween 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). Be- cause 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 in- dicate 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 be- fore charters were quite limited. According to our counterfactuals, raising the supply 37 of prospective charter entrants (for instance, by lowering application costs and provid- ing 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 char- ter law is permissive. Third, we do not model the relationship between school funding and quality. Nonetheless, the counterfactuals stress the role of private initiative (facili- tated 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 Eco- nomic 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. 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Houde and R. Clark. 2011. “Dynamic Entry and Firm Reconfig- uration in Canadian Gasoline Markets.” Working Paper, ICESI University. 38 [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 Multi- District 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 On- tario 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 acad- emic achievement of poor students. Working Paper, Yale Univesity. [23] Nevo, A. 2000. “Mergers with Differentiated Products: The Case of the Ready-To- Eat 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 Institute 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. 39 [29] Walters, C. 2012. “A Structural Model of Charter School Choice and Academic Achievement.” Working Paper, U. of California-Berkeley. 40 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 Stanford- 9 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 Per- formance 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 en- 41 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 be- low 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 enroll- ment. 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 regres- sion 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 con- sistency 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. How- ever, 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 sam- ple 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 sam- ple 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 stu- dents. 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 avail- able; 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 43 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 or- ganization 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. 44 Type: Catholic, other religious or non-sectarian. Source: PSS. School enrollment: total school enrollment, excluding ungraded and adult stu- dents. 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 per- cent 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 receiv- ing 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;38 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 distri- bution 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 38 Perfederal 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) combi- nation 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 remain- ing 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 three- quarters 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 stu- dent 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 probability- weighted, with the weights being equal to the measure of the household type and age in the corresponding year. 48 B Tables Table 1: Grade Levels in Public, Charter and Private Schools Table 2: Program Focus by School Type Table 3: Focus Choice by Student Race and Poverty Status 49 Table 4: Charter School Entry Patterns, 2004 - 2007 Table 5- Goodness of Fit: School Choice 50 Table 6- Goodness of Fit: Focus Choice Table 7: Average Public v. Charter School Quality Table 8: Charter School Costs 51 Table 9- Goodness of Fit: Number of Entries Between 2004-2007 Table 10: Public v. Charter School Value Added 52 Table 11- Average proficiency (in percent), value added and school quality 53 Table 12: Average Proficiency Index by Student Group and Grade Level (in per- cent) Table 13: Average Welfare and Welfare Gain by Student Group and Grade Level 54 C Appendix Figures Figure 1: Neighborhood Percent of Children in Charter School in 2006 55 Figure 2: Goodness of Fit - Relocations D Model D.1 Derivation and interpretation of utility function Below we describe the derivation of the utility function presented in the model, as en- compassing both elements of preference and achievement. The household indirect utility function is: p p Ui jgt = δ jgt + µi jgt + εi jgt (18) p where δ jgt is the baseline utility enjoyed by the children enrolled in grade g at school j in p p school year t , µi jgt is a student-specific deviation from δ jgt , and εi jgt is an idiosyncratic preference. Baseline utility depends on school, expected peer characteristics and tuition as follows: p b αp p ϕ +ξ p : δ jgt = y j β p + D (19) jt jgt jgt p p p Here, α and β are parameter vectors and ξ jgt is an unobserved (to us) charac- teristic of the school and grade, such as the teacher’s responsiveness to parents and her enthusiasm in the classroom. The household-specific component of utility is given by: p ˜ p + [Di D b ]α µi jgt = E Ai jgt φ + [y j Di ]β jt ˜ + xi jt γ : (20) This component depends on the expected achievement of the student E (Ai jgt ), which is explained below. It also depends on the interaction of student demographics Di with focus y j and D b , which captures the utility variation from y and D b across students jt j jt of different demographic groups. In addition, it depends on the distance between the household’s residence and the school. Student achievement Ai jgt depends on a school-grade factor common to all students 56 Q jgt , student characteristics Di , the interaction [y j Di ], and a zero-mean idiosyncratic achievement shock wi jgt , which parents do not observe at the time of choosing a school: ˜ a + wi jgt : Ai jgt = Q jgt + Di ω a + [y j Di ]β (21) The school-grade factor Q jgt , depends on the school’s thematic focus y j , peer char- acteristics D jt , and a productivity shock ξ jgta : a Q jgt = y j β a + D jt α a + ξ jgt (22) a where ξ jgt is an unobserved (to the econometrician) characteristic of the school and grade that affects children’s achievement. This captures, for instance, the teachers’s effectiveness at raising achievement. In contrast with ξ jgt a , ξ p affects household satis- jgt faction for reasons other than achievement. Substituting (22) into (21), and taking parents’ expectation of (21) we obtain: b α a + D ω a + [y D ]β E Ai jgt = y j β a + D ˜a +ξa : (23) jt i j i jgt Substituting (23) into (20), we obtain: µ p = y β aφ + D b α a φ + D ω + [y ˜ + [Di b ]α a i jgt j jt i j Di ]β D jt ˜ + xi jt γ + φ ξ jgt (24) where ω a = ω φ. The coefficient of the interaction of y j and Di is β˜ =β ˜ +φβ p ˜ . a Next, we substitute (19) and (24) into (18) and regroup terms to obtain expressions (1), (2) and (3). In (2), the coefficient of y j captures both household preference for school focus and impact of focus on achievement: β = β p + φ β a . For instance, parents may like a focus on Arts even though this focus does not raise achievement. Similarly, the coefficient of D b captures both household preference for peer characteristics and jt the impact of peer characteristics on student achievement: α = α p + φ α a . The error term (or demand shock) impounds both a preference and a productivity shock: ξ jgt = p a φ . For instance, parents may not like a teacher’s strict policies even if they ξ jgt + ξ jgt raise achievement. D.2 Charter entry: institutional details If a charter wishes to open in the Fall of calendar year X, it must submit its applica- tion no later than February of (X-1). According to the Washington, D.C. charter law, the application must include a description of school focus and philosophy, targeted stu- dent population (if any), educational methods, intended location, strategies for student recruiting and enrollment projections. In addition, the applicant must file letters of com- munity support and specify two potential parents for the school board. The application must also contain a plan for growth – what grades will be added, at what pace, etc. When submitting its application, the school must provide reasonable evidence of its ability to secure a facility. The authorizer evaluates the enrollment projections by con- sidering elements such as enrollment in nearby public schools and incumbent charters, the size of the school’s intended building, and expected fixed costs. The applicant learns whether it was approved in the Spring of (X-1). If the charter is approved and has already secured a building, then the charter and the regulator begin negotiating on a number of issues. The school then uses the following twelve months 57 to hire and train prospective school leaders and teachers, conduct building renovations, recruit students, and finalize preparations before formally openings its doors. Charters are very aggressive when recruiting students. They contact parents directly, advertise in churches, contact parents directly, post flyers in public transportation stops and local shops, advertise in local newspapers and in schools that are being closed down or re- constituted, and host open houses. PCSB conducts a “recruitment expo” in January and charters participate in it. Based on projected enrollment, a charter opening in Fall of X receives its first installment in July of X. An enrollment audit is conducted in October of X and install- ments are adjusted accordingly. Charters can run surpluses, as is the case of charters that plan to expand in the future. Although they can also run deficits, PCSB only tolerates them temporarily and provided the school is academically in good standing. D.3 Expected Proficiency Rate As described in Appendix D.1, student i’s achievement in school j, grade g and year t , represented by Ai jgt , is a function of school time-invariant characteristics y j , student body composition D jt , student own characteristics Di , and the student-school interac- tion [y j Di ]. We assume that the probability qi jgt that i passes the proficiency test is monotonically related to her achievement, and is given by the following specification: qi jgt = y j β q + D jt α q + Di ω q + [y j Di ]β ˜ q + ξ q + vq ; (25) jgt i jgt q where vi jgt is a zero-mean idiosyncratic shock. A student who passes the test is deemed “proficient”. The school-level expected share of proficient students is N jgt ∑g2κ jt ∑i=1 qi jgt q jt = ; N jt where N stands for the number of enrolled students. Averaging qi jgt over students and grades we obtain the following: q jt = y j β q + D jt φ q + [y j D jt ]β ˜q +ξq; (26) jt N jgt ∑g2κ jt ∑i=1 Di where the identity D jt = N jt is applied. The following new variables are intro- N jgt q q q ∑g2κ jt ∑i=1 (ξ jgt +vi jgt ) 39 duced: φq = αq + ωq and = ξ jt N jt : q In (26), we decompose ξ jt as follows q q q q ξ jt = ξ j + ξt + ∆ξ jt : Substituting the above expression into (26) we obtain the following expression for the expected share of proficient students: ˜ q + ξ q + ξ q + ∆ξ q : q jt = y j β q + D jt φ q + [y j D jt ]β (27) j t jt Let q jt be observed proficiency rates. They are related to expected proficiency rates 39 This averaging is possible because the probability of passing the test is specified as a linear function. 58 in the following way: q q q q ¯ jt = y j α q + D jt φ q + [y j D jt ]ω q + ξ j + ξt + ∆ξ jt + v jt q (28) q where v jt = q jt q jt incorporates sampling and measurement errors, and is conditionally mean-independent of all the explanatory variables in (28). However, it is possible that q q ∆ξ jt is correlated with ∆ξ jgt , in which case D jt is correlated with ∆ξ jt . Hence, we use IV estimation for (28). E Estimation E.1 Moment conditions Recall that we have J X =8,112 school-year observations for share moments, J D =1,269 school-year observations for demographic moments and JC =153 neigborhood-year ob- servations for neighborhood moments. In total we have J S = 281 campuses in our data. We use GMM to estimate the following 324 utility function parameters: 3 coefficients on peer composition (coefficients on fraction white, fraction Hispanic and fraction non- poor), 21 coefficients on interactions between student and household characteristics, 3 distance-related parameters, 281 campus fixed effects, 12 grade fixed effects and 4 year fixed effects. Below we describe how to form the GMM objective function in (15). Let Z X jgt be a X D D row vector of L instruments for share moments, Z jt be a row vector of L instruments for demographic moments and Zkt C be a row vector of LC instruments for neighborhood moments. In our preferred specification, LX = 310, LD = 102 and LC = 54. Vertically stacking all observations yields matrices Z X (dimension J X by LX ), Z D (dimension J D by LD ) and ZC (dimension JC by LC ). h i h i Recall our mean-independence assumptions E ∆ξ jgt j Z X jgt = 0 ; E u D j ZD = 0 jt jt C C D and E u j Z = 0: Also, recall that vector u has D e = 3 elements, and u has C C e= 3 kt kt jt jt elements. The mean-independence assumptions yield the following + DL e C e D + CL LX moment conditions: h 0 i h 0 i 0 X D d E Z jgt ∆ξ jgt = 0; E Z jt u jt = 0 and E Zkt ucC kt = 0; (29) where ud c jt and u jt indicate the sampling error in a specific demographic characteris- tic d (for instance, in percent white students) and neighborhood-level variable c (for instance, percent of children in charter schools). Vertically stacking all observations and rearranging elements yields column vectors ∆ξ , uD and uC with J X , DL e D and CLe C rows respectively. The first set of J D rows in vector uD correspond to the first demo- graphic characteristic; the second set set to the second demographic characteristic, and so forth for the e D demographic characteristics. Vector uC has a similar structure for neighborhood-level variables. In order to interact the sampling error for each demographic characteristic with every instrument in Z D we introduce matrix Z eD , which is block diagonal and repeats Z D along the diagonal for a total of De times. Similarly, block-diagonal matrix Z eC repeats 59 e times. ZC along the diagonal for a total of C The sample analogs of (29) are the following vectors: 1 0 1 eD 0 D 1 eC 0 C λX (∆ξ ) = X Z X ∆ξ ; λD (∆ξ ; θ d ) = D Z u and λC (∆ξ ; θ d ) = C Z u ; J J J e C elements respectively. e D and CL with LX , DL 1 1 In the GMM weighting matrix, we use VX = Z X 0 Z X , VD = Z eD eD0 Z and 1 eC eC0 Z VC = Z . Our standard errors are robust to arbitrary within-school correlation of ∆ξ (across grades and over time), arbitrary correlation of sampling errors ud within a school-year, and arbitrary correlation of sampling errors uc within a neighborhood-year. The weighting matrix has block structure because we assume that errors uD C jt and ukt are independent. Further, they are independent of the elements upon which students base their choices, including ∆ξ jgt . Finally, we use Minimum Distance Estimation to obtain separate estimates for the coefficients on time-invariant school characteristics (β ) and school qualities (ξ j s). De- note by Θ the J S 1 vector of campus fixed effects estimated by GMM; by y the J S Y matrix of time-invariant characteristics β , and by ξ the J S 1 vector of school-specific demand shocks. From (2) and ξ jgt = ξ j + ξg + ξt + ∆ξ jgt , the content of the campus dummies is Θ = yβ + ξ . We assume that E ξ j j y j = 0, which allows us to recover the estimates of β and ξ as β b and ξ ˆ = (y0 y) 1 y0 Θ b yβ ˆ =Θ ˆ respectively. The standard b: ˆ are corrected to account for the estimation error of Θ errors of β E.2 Probabilities of entry, exit and relocation In the likelihood function the probabilities for charter entries in school year t are: 8 j > expfEν π¯e e jt 1 (d jt 1 =1;Mt 1 ;θ )=σν g d e > < b j if d jt = 1 e 1+ exp fE π¯ e (d e =1 ;M ;θ d ) σ g Pr d e ν jt 1 jt 1 t 1 ν jt = d jt j Mt 1 = ¯e e j d )=σ g > > expfEν π ( d jt 1 jt 1 = 1 ;M t 1 ;θ ν e : 1 b e ¯ jt e j d if d jt = 0 1+expfEν π 1 (d jt 1 =1;Mt 1 ;θ )=σν g j where Mt 1 is the market structure of year t 1 adjusted to include entrant j; and expected profit is given by formula (12). closings are: The probabilities for charter 8 q q q < expfa +b q jt 1 +c e jt g if d x = 1 x x 1+expfaq +bq q jt 1 +cq e jt g jt Pr d jt = d jt = x : q 1 q q if d jt = 0 1+expfa +b q jt 1 +c e jt g where q jt 1 is the proficiency rate of charter j in year t 1 and e jt is a closing eligibility variable, equal to 1 if the charter has completed at least five years of operation and 0 otherwise. 60 8 relocations are The probabilities for charter given by: ˘d > > exp f α˘ β ` jt ` jt 1 g < ˘d if ` jt 6= ` jt 1 1+∑L`0 =1:`0 6=` jt 1 expfα ˘ β ` jt ` jt g Pr ` jt = ` jt j` jt 1 = 1 > > 1 if ` jt = ` jt : 1+∑L0 0 expfα ˘ β ˘d 1 ` =1:` 6=` jt 1 ` jt ` jt 1 g E.3 Computational Considerations We code the MPEC problem in MATLAB using the code from Dube et al (2011) as a starting point. Rather than code analytical first-order and second-order derivatives for the MPEC problem, we use the automatic differentiation capabilities in TOMLAB’s TomSym package (included in the Base module). This enables us to experiment with different model specifications and instruments by only modifying the objective function and the constraints, and leaving TomSym to recompute the derivatives. Automatic dif- ferentiation can be memory intensive, especially for second-order derivatives, but our problem size and our choice of the SNOPT and MINOS solvers available from TOM- LAB makes it efficient and easy. SNOPT and MINOS require only analytic first order derivatives (which were computed by TomSym in our case). In contrast, Dube et al (2012) supply second-order derivatives to the KNITRO solver and use the Interior/Direct algorithm. Avoiding the provision of analytical first- or second-order derivatives greatly facilitates our use of MPEC. We use both the SNOPT and MINOS solvers in the following manner: we run a few hundred major iterations of SNOPT to establish the basis variables (the variables of interest for the optimization problem) and to approach a local minimum, and then hand over the problem to MINOS in a “warm-start” fashion to converge to the local optimum. This combination allows us to exploit the virtues of each solver and solve the problem in the most efficient way. Broadly speaking, SNOPT is better suited for a large numbers of unknowns, but makes progress only by changing its limited-memory approximation of the full Hessian of the Lagrangian between major iterations. Once it gets to the point at which it no longer updates the Hessian approximation, it stops making progress. In contrast, MINOS works with the exact Lagrangian and can also make many updates to a full quasi-Newton approximation of the reduced Lagrangian. Hence, MINOS can make progress even when SNOPT cannot provided the size of the problem is not too large. At the same time, MINOS only works well if started sufficiently close to a local minimum. Hence, SNOPT starts the problem with the full set of unknowns, quickly solves for ∆ξ and establishes θ d as the basis variables. After having reduced the size of the problem, it hands the optimization problem over to MINOS. This approach proved fast and accurate, allowing us to obtain results with 5 or 6 decimal digits of precision.40 For our preferred specification, SNOPT-MINOS takes 40 The precision is determined by a combination of the algorithm’s optimality tolerance, the condition number of the Jacobian at the optimum, and the size of the dual variables. We use an optimality tolerance of 1e-6 and re-scale the problem as needed to ensure that the dual variables had order unity. The output logs report the Jacobian’s condition number, and these are checked. SNOPT and MINOS work best if the objective function gradients, the Jacobian of the constraints, and the dual variables are of order unity. 61 10.5 hours for the first stage MPEC problem, and 3.5 hours for the second stage MPEC problem on a workstation with a 2.8 GHZ AMD Opteron 4280 processor with 64GB of RAM.41 The computational time compares favorably with that reported by Dube et al (2011) and Skrainka (2012) for BLP problems, particularly taking into account that our problem has complicating features relative to straightforward BLP. The first is that our objective function includes demographic and neighborhood moments in addition to share moments. The second is that we have a relatively large number of products (schools) relative to the number of markets (grade-years). In a typical industrial organi- zation context there are many markets relative to products. This gives rise to a sparser Jacobian, which in turn speeds up performance (see Dube et al 2011 for a discussion of how the speed advantage of MPEC declines as the sparsity of the Jacobian falls). The third complicating feature is the presence of some very small market shares, an issue related to the large number of schools relative to the number of students. This issue motivates the constraint of a minimum outside share, as described in Appendix A.2. F Market projections The 59 charters that were open in Fall 2007 captured a market share of 21.5 percent. Of these schools, 17 had closed by Fall 2013, for a charter exit rate of approximately 0.3. Hence, we can expect 6.4 (=0.7*9.1) of the 9.1 baseline entrants to last in the long run, capturing 2.3 (=0.7 * 3.3) percent of the market. Assuming the same entry rate for the following six years yields an increase in charter share due to entry approximately equal to 13.8 (=2.3*6) percentage points. In addition, 33 public schools (or 24 percent of all public schools open in school year 2007) were closed between 2007 and 2014, and about 1/6 of the displaced students shifted to charter schools (source: http://www.21csf.org/csf-home/publications/ Memo- ImpactSchoolClosingsMarch2009.pdf). Assuming that the closed schools captured 24 percent of public school enrollment (equal to 57 percent of total K-12 enrollment in 2007), we estimate that 2.3 (=1/6*0.24*57) percent of all students switched from the closed public schools to charters – enough to support 6.4 additional charters in the long run. Thus, the total charter share for 2013 would be equal to 21.5 + 13.8 + 2.3 = 37.6 percent, close to the observed 2013 charter share of 35 percent. Similarly, the predicted number of charters for 2013 is the addition of the surviving charters from our sample, entries between 2008 and 2013, and additional entries asso- ciated with public school closings, or 42 + 6.4*6 + 6.4 = 87 charters, close to the actual 93 regular charters in Fall 2013. This is easily achieved by multiplying the objective function and constraints by constant factors. We find that the solvers are 3-5 times faster by employing this scaling. 41 The workstation has many cores, but the SNOPT-MINOS solvers are single-threaded and so use only one core. The solvers have a peak memory consumption of 10GB when the derivatives are symbolically computed, and then work with 5GB of RAM. On our 64GB workstation we can therefore run multiple jobs at once from multiple starting points. 62