1 Int roduct ion

Whether t ransferring to an out-of-zone public school, applying to a charter, magnet or examschool, or using a voucher to at tend a private school, a growing number of families can opt outof their neighborhood school. As school choice opt ions have proliferated, so too has the way inwhich choices are expressed and students are assigned. Ad hoc decent ralized assignment systemswhere students applied separately to schools with uncoordinated admissions off ers have beenreplaced with cent ralized, coordinated single-off er assignment mechanisms in a number of cit iesincluding Denver, London, Newark, New Orleans, New York City, and Washington DC.1 In thesecit ies, student demand is matched with the school seat supply using algorithms inspired by thetheory of matching markets (Gale and Shapley 1962, Shapley and Scarf 1974, Abdulkadiroğluand Sönmez 2003). Changes in assignment protocols are inevitably controversial, however, sinceallocat ing school seat s may result in some students assigned to high quality schools while othersare assigned to less desirable alternat ives.In this paper, we exploit a large-scale policy change in New York City to study the eff ects

of moving from an uncoordinated assignment system to a coordinated single-off er system on theallocat ion of students to schools. P rior to 2003, roughly 80,000 aspiring high school studentsapplied to five out of more than 600 school programs; they could receive mult iple off ers andbe placed on wait list s. Students in turn were allowed to accept only one school and one waitlist off er, and the cycle of off ers and acceptances repeated two more t imes. We refer to theold mechanism as uncoordinat ed because students init ially expressed their preferences on acommon applicat ion, but admissions off ers were not coordinated across schools. A total ofthree rounds of off er processing proved insuffi cient to allocate all students to schools, and morethan 25,000 applicants were assigned to schools not on their original applicat ion list through anadminist rat ive process, which manually placed students to schools in their neighborhood.In Fall 2003, the system was replaced by a single-off er assignment system, based on the

student -proposing deferred acceptance algorithm for the main round. Applicants were allowedto rank up to 12 programs for enrollment in 2004-05 and a supplementary round placed thoseunassigned in the main round. Since the cent ral offi ce coordinated off ers across schools into asingle off er, we refer to this new mechanism as a coordinat ed mechanism. The Departmentof Educat ion (DOE) publicized that the new mechanism provides greater voice to student ’schoices, ut ilizes school places more effi cient ly, and reduces gaming involved to obtain a schoolseat (Kerr 2003). Parents were also provided with addit ional informat ion on school opt ionsthrough an expanded set of workshops and high school fairs. At the conclusion of the first year,82% of students were assigned in the main round and the size of the administ rat ive round shrunkby over two-thirds.The sudden, large-scale policy change in New York City provides a unique opportunity to

invest igate the consequences of cent ralized and coordinated school assignment . Changes wereadvert ised widely beginning only in September 2003, and students submit ted their preferencesin November 2003. This brief t imeline limits the scope for part icipants to react by moving or

1Many other U.S. cit ies are in the process of adopt ing unified enrollment syst ems using similar matchingalgorithms, including Chicago and Philadelphia (Darville 2013).

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for schools to change curriculum and resources in response to the new mechanism. Moreover,rich micro-level data from both systems allow us to surmount many of the usual challengesassociated with evaluat ing allocat ive aspects of assignment mechanisms. A key st rength of ourstudy is that we observe submit ted preferences, round-by-round placements, and subsequentschool enrollment of all students in both the old and new mechanism. Using student preferencestogether with detailed informat ion on student and school at t ributes, we est imate the factorsthat account for school demand. The est imated preference dist ribut ion is then used to assessmechanism design choices and measure the welfare from the new mechanism.Our approach focuses on the allocat ive and dist ribut ional aspects of the new assignment mech-

anism and its comparison to other mechanisms that have been the focus of the large theoret icallit erature on matching market design. While the longer-term eff ects on resident ial choices andschool product ivity are other possible eff ects of the new mechanism, their study likely requiresunderstanding allocat ive issues. Moreover, changes in NYC’s school assignment system havebeen a template for replacing uncoordinated procedures with coordinated ones. For instance, inEngland, where the 2003 Admissions Code mandated coordinat ion of admissions nat ionwide, anumber of local educat ion authorit ies, governing bodies similar to U.S. school dist rict s, subse-quent ly adopted coordinated assignment mechanisms like New York.2 Furthermore, our result squant ifying mechanism design t radeoff s are relevant for dist ricts like Denver, Recovery SchoolDist rict in New Orleans, and Washington DC each of which faced similar design decisions.3

Coordinat ion of admissions off ers may aff ect allocat ions for many reasons. First , the newmechanism allows students to rank up to 12 choices, whereas the old mechanism only allowed forfive. Second, the limited number of rounds of off ers and acceptances in the old mechanism canlead to situat ions where students hold on to less preferred choices wait ing to be off ered seats atmore preferred schools once others decline. Few rounds of off er processing and a limited numberof applicat ions make it diffi cult for schools to make enough off ers to assign to clear the market , aphenomenon termed congest ion by Roth and Xing (1997). Congest ion can result in part icularlylarge ineffi ciencies since in the old mechanism, the dist rict administ rat ively placed unassignedstudents to the nearest schools with capacity. Third, in New York’s old mechanism, schoolswere able to see the ent ire rank ordering of applicants, and some advert ised they would onlyconsider those who ranked them first , creat ing st rategic pressure on student ranking decisions.The computerized rounds of off ers and acceptances in a cent ralized single-off er system reducescongest ion and associated st rategic issues.On the other hand, coordinated assignment may encourage movement throughout the dist rict

2The English mot ivat ion is similar t o NYC since authorit ies wanted to ensure that “every child within a localauthority area would receive one off er of a school place on the same day. This would eliminat e or largely eliminat emult iple off ers and free up places for parent s who would not otherwise be off ered a place” (Pennell, West , andHind 2006). All 32 London boroughs coordinat ed to est ablish a Pan London Admissions Scheme based on deferredaccept ance to make the admissions system “fairer” and “simpler,” and to “result in more parent s get t ing an off er ofa place for their child at one of their preferred schools earlier and fewer get t ing no off er at all” (ALG 2005, Pathakand Sönmez 2013).

3The mechanism used in the Recovery School Dist rict in 2012 was based on an adapt at ion of Gale’s t opt rading cycles, which is ordinally Pareto effi cient . In the 2013, t he dist rict swit ched to a mechanism based onthe student -proposing deferred accept ance algorit hm, like that used in NYC. Abdulkadiroğlu, Pathak, and Roth(2014) cont ain more details.

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for lit t le benefit . Ravitch (2011), for instance, argues that the elevat ion of choice in NYC“dest royed the concept of neighborhood schools” as “children scat tered across the city in responseto the lure of new, unknown small schools with catchy names, or were assigned to schools farfrom home.” Crit ics of the new system believed that denying principals’ informat ion on students’ranking of the school rest rict s a principal’s ability to at t ract “students who want them most”(Herszenhorn 2004). This logic implies that removing a school’s ability to know whether theywere ranked first could lead fewer of their off ered students to enroll. It also may be in a dist rict ’sinterest to advantage certain student groups to prevent them from leaving the dist rict (Engberg,Epple, Imbrogno, Sieg, and Zimmer 2014).4 Providing students with mult iple off ers and let t ingthem decide upon receiving off ers may make it easier for some students to decide what schoolis best for them.5 Taken together, arguments against NYC’s choice system suggest that theallocat ive eff ects of the mechanism may have most ly been redist ribut ive or even harmful tostudents.To quant ify the welfare eff ects of coordinated school assignment in New York City, we es-

t imate the dist ribut ion of student preferences for schools using flexible discrete choice demandmodels allowing for rich student heterogeneity and unobserved school quality. Modelling thisheterogeneity is essent ial for our analysis since otherwise changes in assignments are zero-sumby const ruct ion when the same number are assigned. Our empirical approach is mot ivatedby the fact that the new mechanism is a variant of the student -proposing deferred acceptancemechanism, where t ruth-telling is a weakly dominant st rategy for part icipants (Dubins andFreedman 1981, Roth 1982). While the new mechanism’s incent ive propert ies make t reat ingstated rankings as t rue preferences a natural start ing point , this assumpt ion bears scrut iny forseveral reasons. First , the new mechanism only allows part icipants to rank at most 12 choices.Although 80% of applicants in our sample rank fewer than 12, this const raint interferes withthe dominant st rategy property of the mechanism for students who find more than 12 schoolsacceptable.6 Second, it ’s possible that the sudden adopt ion of the new mechanism led somefamilies to use heurist ics developed from the old mechanism, despite the DOE’s advice to rankschools t ruthfully.7 For instance, some families might have ranked a “safety” school as theirlast choice, even though the new mechanism’s propert ies make this unnecessary if ranking fewerthan 12 choices. We start with benchmark models that use all ranked choices in the coordi-nated mechanism and assume the ranking is t ruthful, before turning to several alternat ives toassess the robustness of our conclusions to these concerns. We do not direct ly model st rategicchoices because we ant icipate st rategic considerat ions to be less important in mechanisms basedon deferred acceptance than in other more manipulable mechanisms.8

4Williams (2004) profiles families who had to pay a deposit to reserve a spot in a privat e or parochial schoolwhile await ing their school placement in the first year of the new mechanism.

5Similar argument s for t he flexibility provided by mult iple off er systems have been raised by crit ics of t heNat ional Residency Matching P rogram. See, e.g., Bulow and Levin (2006) and Agarwal (2015).

6For more det ails on the const rained rank order list s, see (Abdulkadiroğlu, Pathak, and Roth 2009, Haeringerand Klijn 2009, Calsamiglia, Haeringer, and Kljin 2010, Pathak and Sönmez 2013).

7The DOE’s school brochure advised: “You must now rank your 12 choices according to your t rue preferences”(DOE 2003).

8Hast ings and Weinst ein (2008) report that in Charlot t e-Mecklensburg’s highly manipulable immediat e ac-cept ance mechanism, student rankings respond to informat ion on historical odds of assignment . Pathak and

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Off er processing and matriculat ion pat terns provide compelling evidence that the new mech-anism is an improvement relat ive to the old one. In the old mechanism, 18.6% of studentsmatriculated at diff erent schools from their assignment at the end of the match compared to11.4% under the new mechanism. Mult iple off ers and short rank order list s in the old mech-anism advantage few students, but leave many without off ers. Roughly one-fifth of studentsobtained mult iple off ers, while half of applicants obtain no first round off er and 59% of these ap-plicants are administ rat ively assigned. The take-up rates for students assigned administ rat ivelyare similar across mechanisms, but the number of students assigned in that round is three t imeslarger in the old mechanism. In addit ion, 8.5% of applicants left the dist rict after assignment inthe old mechanism, while only 6.4% left under the new mechanism.The most significant diff erence between assignments under the two mechanisms is the distance

between students and their assigned school. In the old mechanism, students t ravelled on average3.36 miles to their school assignment . Students t ravel 0.69 miles further in the new mechanism.All else equal, students prefer schools that are closer to home according to their preferencerankings in the new mechanism. However, preferences are substant ially heterogeneous, and ourdemand model quant ifies the t rade-off between distance and measures of school performanceacross student types. Asian and white students and those with high baseline scores exhibit agreater willingness to t ravel for their most preferred schools than black and Hispanics and thosewith low baselines scores, respect ively. Variat ions on our baseline assumpt ions to assess therobustness of our est imates show broadly consistent pat terns. This fact together with evidenceon both in and out-of-sample model fit reassure us about the suitability of using the est imatedpreferences to compare student welfare across allocat ions.Since the uncoordinated mechanism administ rat ively assigned a large fract ion to neighbor-

hood schools, our invest igat ion of counterfactual assignments start s with a neighborhood as-signment . We compute the allocat ion produced by placing each student to the closest schoolsubject to capacity const raints. Our est imates imply that the average student welfare from thecoordinated mechanism is 15 miles in distance-equivalent ut ility greater than this alternat ive.The other ext reme is bracketed by the ut ilitarian opt imal assignment , which is the assignmentmaximizing an equally weighted average of distance-equivalent ut ility given school capacit ies.The new mechanism is 3.7 miles in distance-equivalent ut ility below this idealized benchmark.Next , we empirically examine issues in the mechanism design literature (Erdil and Ergin

2008, Abdulkadiroğlu, Pathak, and Roth 2009, Kesten 2010, Kesten and Kurino 2012). Had themechanism produced a student -opt imal stable matching, average student welfare improves by anequivalent of 0.11 miles.9 An ordinally Pareto effi cient matching, which abandons the stabilityconst raints in the current mechanism, is equivalent to an improvement of 0.62 miles, which in turnis st ill more than 3.1 miles in distance-equivalent ut ility below the ut ilit arian benchmark. Un-fortunately, none of these assignments can be implemented without discarding the mechanism’sincent ive propert ies and mechanisms that implement these allocat ions as equilibrium outcomes

Sönmez (2013) formalize how const rained deferred accept ance is less manipulable than the immediat e accept ance(or Boston) mechanism. Agarwal and Somaini (2014) examine how to recover preferences from st rat egic report sin the cont ext of school choice.

9Even though the mechanism is based on student -proposing deferred accept ance, t he matching is not st udent -opt imal because school-side priorit ies are not st rict .

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are not known. Nonetheless, our est imates show that the magnitude of any potent ial gains areswamped by the comparison between neighborhood assignment and the coordinated mechanism.Finally, we use our demand est imates to evaluate the t ransit ion from an uncoordinated to

coordinated mechanism. This exercise requires a model of student ranking behavior in theuncoordinated mechanism since the submit ted reports may be correlated with (unobserved)preferences. Since students could only rank five choices and some schools advert ised they wouldonly consider those ranking them first , modeling percept ions and behavior of applicants underthis mechanism is not st raight forward. Our approach is to consider three assumpt ions aboutrankings under the old mechanism: 1) ranked schools are preferred to unranked schools, 2)rankings under the uncoordinated mechanism are not selected on unobservables, and 3) schoolsare ranked truthfully. Fortunately, the qualitat ive conclusions of our comparison between the twomechanisms are similar across these specificat ions. Under each assumpt ion, the new mechanismhas made it easier for students to obtain a choice they want . Even though school assignmentsare further away, based on our est imated dist ribut ion of preferences, the amount that studentsprefer these schools more than compensates for this diff erence. Students across all demographicgroups, boroughs, baseline achievement levels obtain a more preferred assignment on averagefrom the new mechanism.The largest gains are for student groups who were more likely to be unassigned after the

old mechanism’s main round, suggest ing that congest ion and ad-hoc placement of unassignedstudents in the old mechanism are primarily responsible for misallocat ion. These gains are alsoseen if ut ility is measured based on subsequent school enrollment , indicat ing that the adverseconsequences of congest ion were not immediately undone following the match. Compared torelaxing algorithmic const raint s within the single-off er mechanism, the welfare advantage of thecoordinated mechanism over the uncoordinated mechanism is roughly half the diff erence betweenneighborhood assignment and the ut ilitarian opt imum.The literature on school choice is immense and not easily summarized. We share a focus

with papers interested in understanding how choice aff ects assignment and sort ing of students(Epple and Romano 1998, Urquiola 2005), rather than the compet it ive eff ects of choice on stu-dent achievement (Hoxby 2003, Rothstein 2006). We also concent rate our study on allocat iveeffi ciency, rather than eff ects of choice opt ions on subsequent achievement (Abdulkadiroğlu, An-grist , Dynarski, Kane, and Pathak 2011, Deming, Hast ings, Kane, and Staiger 2014, Walters2014, Neilson 2014). This study contributes to a growing theoret ical literature on cent ralizedadmissions to college (Balinski and Sönmez 1999) and K-12 public schools (Abdulkadiroğlu andSönmez 2003). A number of recent papers use micro data from assignment mechanisms to un-derstand school demand (Hast ings, Kane, and Staiger 2009, He 2012, Ajayi 2013, Agarwal andSomaini 2014, Calsamiglia, Fu, and Guell 2014, Hwang 2014), but these datasets comes fromsingle-off er mechanisms that are manipulable such as the Boston mechanism, under which par-ents have a st rong incent ive to rank schools st rategically. Another study that examines theeff ects of coordinat ion, though in the labor market context , is the Niederle and Roth (2003)study of the int roduct ion of a cent ralized match for gast roent rologists. Finally, our work com-plements theoret ical papers invest igat ing other assignment mechanisms using simulated data(Erdil and Ergin 2008, Kesten 2010, Abdulkadiroğlu, Che, and Yasuda 2015) or those that use

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submit ted ordinal preferences to examine theoret ical counterfactuals (Abdulkadiroğlu, Pathak,and Roth 2009, Pathak and Sönmez 2008, Budish and Cant illon 2012).The paper proceeds as follows. Sect ion 2 provides addit ional details on high school assignment

in New York City. Sect ion 3 describes our data. Sect ion 4 reports on descript ive features of off erprocessing in the two mechanisms. In Sect ion 5 we out line our empirical st rategy, and report onest imates of the student preference dist ribut ion. Sect ion 6 uses the est imates to quant it at ivelyassess design choices, while in Sect ion 7 we compare student welfare across the uncoordinatedand coordinated mechanisms. Sect ion 8 assesses model fit and robustness, and the last sect ionconcludes.

2 H igh School Choice in NYC

2.1 School Opt ions

Aspiring high school students are eligible to apply to any school or program throughout NewYork City. P rograms exist within schools, and have curricula ranging from the art s to sciencesto vocat ional t raining. The 2002-03 High School directory describes seven types of programs,categorized according to their admissions criteria. Specialized High Schools, such as Stuyvesantand Bronx Science, have only one type of program, which admits students by admissions testperformance on the Specialized High Schools Admissions Test (SHSAT).10 Audit ion programsinterview students for proficiency in specific performing or visual art s, music, or dance. Screenedprograms evaluate students individually using an assortment of criteria including a student ’s 7thgrade report card, reading and math standardized scores, at tendance and punctuality, interview,essay or addit ional diagnost ic tests. Educat ional Opt ion programs also evaluate students indi-vidually, but for half their seat s. The other half is allocated by lot tery. Allocat ion of seat s ineach half targets a dist ribut ion of student ability: 16 percent of seats should be allocated tohigh performing readers, 68 percent to middle performers, and 16 percent to low performers.Unscreened programs admit students by random lot tery, while Zoned programs give priority tostudents who apply and live in the geographic zoned area of the high school. Limited Unscreenedprograms allocate seats randomly, but give priority to students who at tend an informat ion ses-sion or visit the school’s exhibit at high school fairs or open houses conducted each admissionsseason. We categorize programs into three main groups: Screened programs which also includeTest ing and Audit ion programs, Unscreened programs, which also include limited Unscreenedand Zoned programs, and the rest are Educat ional Opt ion programs.Throughout the last decade, the NYC DOE closed and opened new small high schools

throughout the city, each with roughly 400 students. A big push for these small high schoolscame as part of the New Century High Schools Init iat ive launched by Mayor Bloomberg andChancellor Klein. Eleven new small high schools were opened in 2002, 23 new small schools wereopened in 2003, and the peak year of small high school openings was in 2004 (Abulkadiroğlu,Hu, and Pathak 2013). Most of these schools are small and have about 100 students per enteringclass. As a result , the new small high schools have a relat ively small eff ect on overall enroll-

10Abdulkadiroğlu, Angrist , and Pathak (2014) describe their admissions process in more det ail.

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ment pat terns during our study period, which focuses on school opt ions available in 2002-03 and2003-04.

2.2 Uncoordinat ed Admissions in 2002-03

Forms of high school choice have existed in New York City for decades. Before 2002, highschool assignment in New York City featured a hodgepodge of choice opt ions most ly controlledby borough-wide high school superintendents. Significant admissions power resided with schooladminist rators who could direct ly enroll students. Admissions to the Specialized High Schoolsand the LaGuardia High School of Music & Art and Performing Arts, however, have beent radit ionally administered as a separate process from regular high schools and did not changewith the new mechanism.11 Our study therefore focuses on admissions to regular public schools.About 80,000 students interested in regular high schools visit schools and at tend city-wide

high school fairs before submit t ing their preference in the fall. In the main round in 2002-03,students could apply to at most five regular programs in addit ion to the Specialized High Schools.P rograms receiving a student ’s applicat ion were able to see the applicant ’s ent ire preference list ,including where their program was ranked. Programs then decided whom to accept , place on await ing list , or reject . Applicants were sent a decision let ter from each program they had appliedto and some obtained more than one off er. Students were allowed to accept at most one admissionand one wait -list off er. After receiving responses to the first let t ers, programs with vacant seatscould make new off ers to students from wait ing list s. After the second round, students who donot have a zoned high school are allowed to part icipate in a supplement ary round known asthe variable assignment (VAS) process. In the supplementary round, students could rank upto eight choices, and students were assigned based on negot iat ion on seat availability betweenthe enrollment offi ce and high school superintendents. After replies to the second let ter werereceived, a third round of let ters were mailed. New off ers did not necessarily go to wait -listedstudents in a predetermined order. Remaining unassigned students were assigned their zonedprograms or administ rat ively, when the cent ral offi ce t ried to place kids as close to home aspossible, ignoring their preferences. We refer to this final stage as the administ rat ive round .12

Three features of this assignment scheme mot ivated the NYC DOE to abandon it in favor of anew mechanism. First , there was inadequate t ime for off ers, wait list decisions, and acceptancesto clear the market for school seat s. DOE offi cials reported that in some cases high achievingstudents received acceptances from all of the schools they applied to, while many received none(Herszenhorn 2004). Comments by the Deputy Schools Chancellor summarized the frust rat ion:“Parents are told a school is full, then in two months, miracles of miracles, seats open up, butother kids get them. Something is wrong” (Gendar 2000).

11The 1972 Hecht -Calandra Act is a New York State law that governs admissions to the original four SpecializedHigh Schools: Stuyvesant , Bronx High School of Science, Brooklyn Technical, and Fiorello H. LaGuardia HighSchool of Music and Performance Art s. City offi cials indicat ed that t his law prohibit s including these schoolswithin the common applicat ion syst em without an act of t he st at e legislat ure.12Student s who are new to New York City or did not submit an applicat ion part icipat e in an “over the count er”

round over the summer. Our analysis follows applicant s through to assignment and therefore does not considerstudent s who arrived to the process aft er t he high school match. Arvidsson, Frucht er, and Mokhtar (2013) providefurther details on the over-t he-count er round.

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Second, some schools awarded priority in admissions to students who ranked them first ontheir applicat ion form. The high school directory advises that when ranking schools, studentsshould “... determine what your compet it ion is for a seat in this program” (DOE 2002). Thisrecommendat ion puts st rategic pressure on ranking decisions. List ing such a school first wouldimprove the likelihood of an off er at the risk of reject ion from other schools, which take rankingsinto account .Third, a number of schools managed to conceal capacity to fill seat s later on with bet ter

students. For example, the Deputy Chancellor stated, “before you might have a situat ion where aschool was going to take 100 new children for ninth grade, they might have declared only 40 seats,and then placed the other 60 outside the process” (Herszenhorn 2004). Overall, crit ics allegedthat the old mechanism disadvantaged low-achieving students and those without sophist icatedparents (Hemphill and Nauer 2009).

2.3 Coordinat ed Admissions in 2003-04

The new mechanism was designed with input from economists (see Abdulkadiroğlu, Pathak,and Roth (2005) and Abdulkadiroğlu, Pathak, and Roth (2009)). When publicizing the newmechanism, the DOE explained that it s goals were to ut ilize school places more effi cient ly andto reduce the gaming involved in obtaining school seat s (Kerr 2003). In the first round, studentsapply to Specialized High Schools when they take the SHSAT, just as in previous years. Off ersare produced according to a serial dictatorship with priority given by SHSAT scores.13

In the main round , students can rank up to twelve regular school programs in their ap-plicat ion, which are due in November. The DOE advised parents: “You must now rank your12 choices according to your t rue preferences” because this round is built on Gale and Shapley(1962)’s student -proposing deferred acceptance algorithm. Schools with programs that priorit izeapplicants based on audit ions, test scores or other criteria are sent list s of students who rankedthe school, but these list s do not reveal where in the preference list s they were ranked. Schoolsreturn orderings of applicants to the cent ral enrollment offi ce. Schools that priorit ize applicantsusing geographic or other crit eria have those criteria applied by the cent ral offi ce. That offi ceuses a single lot tery to break t ies amongst students with the same priority, generat ing a st rictordering of students at each school.Assignment is determined by the student -proposing deferred acceptance algorithm with stu-

dent preferences over the schools, school capacit ies, and schools’ st rict ordering of students asparameters. The algorithm is run with all students in February. In this first round, only stu-dents who receive a Specialized High School off er receive a let ter indicat ing their regular schoolassignment , and are asked to choose one. After they respond, students who accept an off er areremoved, school capacit ies are adjusted and the algorithm is re-run with the remaining students.All students receive a let ter not ifying them of their assignment or whether they are unassignedafter the main round.Unassigned students from the main round are provided a list of programs with vacancies and

are asked to rank up to twelve of these programs. In 2003-04, the admissions criteria at the

13There is very limit ed overlap between the specialized round and subsequent rounds. In 2003-04, 4,175 out of4,553 of those off ered a specialized high in our sample accept ed that off er.

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remaining school seat s were ignored in this supplement ary round . Students are ordered bytheir random number, and the student -proposing deferred acceptance algorithm is run with thisordering in place at each school. Students who remain unassigned in the supplementary roundare assigned administ rat ively. These students and any appealing students are processed on acase-by-case basis in the administ rat ive round.

3 Data and Sample Definit ions

3.1 St udent s

The NYC DOE provided us with several data set s for this study each linked by a unique studentident ificat ion number: informat ion on student choices and assignments, student demographics,and October student enrollment . For 2002-03, the assignment files record a student ’s mainround rank order list , their off ers and reject ions for each round, whether they part icipate in thesupplementary round, and their final assignment at the conclusion of the assignment processas of July 2003. For 2003-04, the assignment files contain students’ choice schools in orderof preference, priority informat ion for each school, and assignments at the end of each of therounds, and final assignment as of early August 2004. The student demographic file for both yearscontains informat ion on home address, gender, race, limited English proficiency, special educat ionstatus, and performance on 7th grade citywide test s. We use addresses to compute the roaddistance between each student and school, and to place each student in a census block group.14

We also have access to similar files for 2004-05. Further details are in the Data Appendix.Our analysis sample makes three rest rict ions. First , since we do not have demographic

informat ion for private school applicants, we rest rict the analysis to students in NYC’s publicmiddle schools in the year prior to applicat ion. Second, we focus on students who are not assignedto Specialized High Schools because that part of the assignment process did not change with thenew mechanism. Third, we consider applicants who are given an assignment at the conclusion ofthe process (so have not left midway). Given these rest rict ions, we have two main analysis files:the mechanism comparison sample and the demand estimation sample.The mechanism comparison sample is used for comparisons of the assignment across the

two mechanisms. This sample is the largest set of students assigned through the high schoolassignment mechanism to a school that exist s as of the t ime of the print ing of the high schooldirectory. A key property of the mechanism comparison sample is that every student has anassignment . Columns 1 and 2 of Table 1 summarize student characterist ics in the mechanismcomparison samples across years. 3,500 fewer students are involved in the mechanism comparison2003-04 sample, a diff erence mainly due to the students assigned to schools created after theprint ing of the high school directory or to closed schools (as shown in Appendix Table C2).New York City is the nat ion’s largest school dist rict , and like many urban dist ricts it is

majority low-income and non-white. Nearly three-quarters of students are black or Hispanic, andabout 10% of students are Asian. Brooklyn is home to the largest number of applicants, followed

14Though we use road dist ance, we also computed subway dist ance using the Met ropolit an Transport at ionAuthority GIS files; t he overall correlat ion between driving dist ance and subway commut ing dist ance for allstudent -program pairs is 0.96.

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by Bronx and Queens, which each account for roughly one quarter of students. Manhat tan andStaten Island account for a considerably smaller share of students at about 13 and six percent ,respect ively. Consistent with the sudden announcement of the new mechanism, characterist icsof applicants are similar across years.The demand sample contains part icipants in the main round of the new mechanism in the

assignment files. The school choices expressed by these students represent the overwhelmingmajority of students. Among the set of main round part icipants, we exclude a small fract ionof students who are classified as the top 2 percent because these students are guaranteed aschool only if they rank it first and this may distort their incent ives to rank schools t ruthfully.Addit ional details on the sample rest rict ions are in the Data Appendix.

3.2 Schools

Data on schools were taken from New York State report card files provided by NYC DOE.Informat ion on programs come from the offi cial NYC High School Directories made available tostudents before the applicat ion process. Table 2 summarizes school and program characterist icsacross years. There is an increase in the number of schools from 215 to 235, and a correspondingdecrease in the average number of students enrolled per school of about 40 students. Thisfact is driven by the replacement of some large schools with smaller schools that took placeconcurrent ly in 2003-04, as described above. Despite this increase, there is lit t le change in theaverage achievement levels of schools and school demographic composit ion as measured by reportcard data.Students in New York can choose among roughly 600 programs throughout the city. P ro-

grams vary substant ially in focus, post -graduate orientat ion, and educat ional philosophy. Forinstance, the Heritage School in Manhat tan is an Educat ional Opt ion program where the artsplay a substant ial role in the curriculum, while Townsend Harris High School in Queens is aScreened program with a rigorous humanit ies program making it among the most compet it ivein the city. Using informat ion from high school directories, we ident ify each program’s type, lan-guage orientat ion, and speciality. With the new mechanism, there are more Unscreened programsand fewer Educat ional Opt ion programs, a change driven by the conversion of many Educat ionalOpt ion programs to Unscreened programs. This change in labelling was due to overlappingadmissions crit eria and similarity of educat ional programming. We code language-focused pro-grams as Spanish, Asian, or Other, and categorize program specialit ies into Arts, Humanit ies,Math and Science, Vocat ional, or Other. Not all programs have specialt ies, though about 70%fall into one of these classes. (Details on our classificat ion scheme are in the Data Appendix).The menu of language program off erings or program specialit ies changes lit t le across years.

4 Congest ion and Changes in A ssignment s

The similarity of student and school at t ributes in Tables 1 and 2 suggest that there were notsystemat ic changes in part icipant at t ributes and school supply across years. Moreover, theredoes not appear to be a large-scale change in student locat ions across years, as shown in Figure1, which maps both student and school locat ions. These facts mot ivate at t ribut ing diff erences

10

in allocat ions between 2002-03 and 2003-04 primarily to the assignment mechanism rather thanchanges in student part icipat ion or the menu of school opt ions.

4.1 Congest ion in t he M ain Round

Table 3 report s the number of students assigned across rounds of the uncoordinated and co-ordinated mechanisms. The most noteworthy pat tern is that more students obtain their finalassignment in the administ rat ive round of the uncoordinated mechanism than in the first round.Panel A of the Table shows that 37% of students are assigned administ rat ively compared to 34%in the first round. Since 33,894 students obtain one or more first round off ers (shown in PanelB), but only 23,867 students were finalized in the first round, 10,027 students with a first roundoff er were finalized with off ers made in subsequent rounds. The processing of these students tookplace as schools revised off ers based on first round reject ions and made off ers anew in the secondand third rounds. However, the relat ively small number of students placed in the second andthird round implies that three rounds were insuffi cient to process all students. That only halfof the students were placed in the main round of the old mechanism cont rasts sharply with newmechanism, where 82% of students were placed in the main round.15

These observat ions about the old mechanism are characterist ic of congest ion, as described inRoth and Xing (1997)’s study of the labor market for ent ry-level clinical psychologist s. Off ers fort raining posit ions in that market were made in an uncoordinated fashion during a 7-hour window,and Roth and Xing (1997) argue that uncoordinated processing and a small market -clearingwindow led to mismatch. In NYC, serial-processing of batches of off ers, whereby programswaited for previous off ers to be rejected before making new off ers together with a few rounds hada similar eff ect . In addit ion to insuffi cient processing of off ers, the small number of applicat ionsallowed in the old mechanism also led to situat ions where students fell through the cracks if theyapplied to oversubscribed schools. Since rank order list s were short , the mechanism considered asmaller number of alternate choices for these students compared to a mechanism which allowedstudents to rank more choices. Schools where these students were ult imately placed may havebeen assigned in the main round had more applicat ions been allowed.The new mechanism appears to have relieved the market of congest ion by increasing the num-

ber of choices students can rank and the number of rounds of off er processing. To invest igate therole of these two forces – short rank order list s and limited off er processing – in producing admin-ist rat ive assignments, we used data from the coordinated mechanism to simulate two variat ions:1) the main round where only the top five choices are considered and there is no rest rict ion inthe number of rounds and 2) the main round with twelve choices, but only three set s of proposalsfrom the deferred acceptance algorithm.16 The first is intended to isolate the role of five choices,while the second isolates the role of few off er processing rounds. Since we do not model behav-ioral responses by students, we only intend this exercise to shed light on mechanical features

15The marked shift in the number assigned in the main round also appears in the second year of the coordinat edmechanism where even more student s, 87.3%, were placed in the main round (shown in Appendix Table B1).16Even though there are mult iple possible implementat ions of deferred accept ance, our simulat ion considers the

simult aneous-proposing version, where a round is defined by a set of proposals by student s who are not t ent at ivelyheld or have not exhaust ed their rank order list .

11

generat ing administ rat ive assignments in the uncoordinated mechanism. With that caveat inmind, we find that the five choice const raint with unlimited number of rounds leaves about onequarter of applicants unassigned, while the unconst rained mechanism with three proposal roundsleaves roughly half of applicants unassigned. Relat ive to the uncoordinated mechanism, the newcoordinated mechanism appears to reduce administ rat ive assignments by computerizing off erprocessing, and avoiding the need for act ive student and school part icipat ion once preferencesare submit ted. Short rank-order list s also generate administ rat ive assignments, but perhaps lessso than few off er processing rounds.

4.2 D ist ance, Exit , and M at r iculat ion

Across mechanisms, there are stark diff erences in distance to assigned school and off er take-up. Figure 2 reports the dist ribut ion of distance between students’ residence and their assignedschool in both mechanisms. New York City spans a large geographic range, with nearly 45 milesseparat ing the southern t ip of Staten Island with the northernmost areas the Bronx, and 25 milest raveling from the western edge of Manhat tan near Washington Heights to Far Rockaway at theeasternmost t ip of Brooklyn.17 The closest school for a typical student is 0.82 miles from home,and students in the uncoordinated mechanism on average t ravelled 3.36 miles to their assignment .In the coordinated mechanism, the average distance is 4.05 miles. The medians (unreported)also increase from 2.25 to 3.04 miles. Panels A and C of Table 3 shows that students in theadminist rat ive round of the uncoordinated mechanism are assigned close to home (an average of1.64 miles) compared to those in the coordinated mechanism or those in the main round of theuncoordinated mechanism.The increase in distance to assigned school suggest s that the coordinated mechanism ex-

panded the scope of the market . This finding parallels Niederle and Roth (2003)’s study of thegast roent rology labor market , where physician mobility increased following a cent ralized match.However, the increase in distance need not direct ly imply a statement about student welfarewithout knowing how students value proximity relat ive to other aspects of their school choices.Student enrollment pat terns documented in Table 3, however, indicate that student assign-

ments in the uncoordinated mechanism, part icularly those made in the administ rat ive round,are undesirable relat ive to those in the coordinated mechanism. After receiving an assignment ,a student may opt for a private school, leave New York, or even drop out . Families may switchschools after their final assignments are announced, but before the school year start s. In the un-coordinated mechanism, principals had greater discret ion to enroll students and the DOE offi cialsquoted above alleged that students with sophist icated parents might just show up at a school inthe fall and ask for a seat at the school. The exit rate is higher in the uncoordinated mechanism(8.5% compared to 6.4%), and fract ion who enroll at a school other than their assignment ishigher (18.6% compared to 11.4%).In the uncoordinated mechanism, students assigned in earlier rounds appear more sat isfied

with their assignment than those assigned in later rounds. The fract ion of students who exitNYC public schools is 13.3% among administ rat ive placements, compared to 5.2% among those

17Our analysis focuses on road dist ance, which is highly correlat ed with subway dist ance. Appendix B present sa detailed comparison of both measures.

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assigned in the first round. More than a quarter of students assigned in the administ rat ive roundwho are st ill in NYC public schools matriculate at schools other than where they were assigned.By comparison, the take-up of off ered assignments is much higher for those assigned in the firstthree rounds. Based on exit and matriculat ion, students with mult iple off ers in the first roundare more sat isfied with their assignment than students with zero or one off er. These studentsalso t ravel further to their final assignment or enrolled school. In cont rast , the majority ofstudents with no off ers are assigned through the administ rat ive process, and this likely accountsfor their higher rates of exit and enrollment at a school other than their assignment . Even thoughthe coordinated mechanism has substant ially fewer administ rat ively assigned, the exit rates arehighest and the matriculat ion rates are lowest for the part icipants of that round.18

4.3 M ismat ch in Administ rat ive Round

To further evaluate the assignments of students processed in the administ rat ive round, we com-pare the at t ributes of schools that students wanted (or ranked) to the at t ributes of schools towhich they were assigned. Students processed in earlier rounds are assigned to schools thathave at t ributes more similar to the schools they ranked than students processed in later rounds.Table 4 shows that for those assigned in the main round, with the except ion of distance, rankedschools tend to have similar or slight ly bet ter at t ributes than assignments in both mechanisms.For instance, ranked schools have higher Math and English performance, more four-year college-going, and higher at tendance rates. They are similar in terms of teacher experience, poverty (asmeasured by percent subsidized lunch), and racial make-up.For students placed in the supplementary round, assigned schools are also less desirable than

ranked schools, and many of the gaps are wider compared to the main round. In the uncoor-dinated mechanism, for instance, the gap in Math performance between ranked and assignedschools is 0.7 percentage points for those assigned in the main round, while it is 2.5 point s inthe supplementary round. The gap between ranked and assigned alternat ives for 9th grade sizeis quite pronounced under both mechanisms. For example, in the uncoordinated mechanism,ranked schools have about 200 fewer 9th graders than the schools where students are assigned.Since students part icipate in the supplementary round when they did not obtain a main roundassignment , it is not surprising that the diff erence what students wanted and received widens.The most st riking pat tern in Table 4, however, is for students who are administ rat ively

assigned. We’ve already seen that there are three t imes more students assigned in this roundin the uncoordinated mechanism. Panel C shows that students who are assigned in that roundranked schools on average 5.1 miles away from home, and were assigned to schools only 1.6 milesaway in the uncoordinated mechanism, a much larger gap than either the main or supplementaryround. For other school characterist ics, the diff erence between what students wanted and whatthey were assigned widens relat ive to the supplementary round, suggest ing that mismatch isgreatest for students in the administ rat ive round. For instance, the 2.5 point spread in Mathachievement is now 4.4 points and there is a similar widening in the fract ion going on to four-

18In the second year of the mechanism, the average dist ance to the assigned school is 4.07 miles and the averageexit rat e is 6.4% (shown in Table B1). The t ake-up rat e is higher than the first year and the fract ion in theadminist rat ive round decreases to about 5%.

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year colleges. The diff erence in 9th grade size is also considerable: in the supplementary round,students wanted schools with on average with roughly 700 9th graders and they were assignedto schools with more than 900 9th graders. In the uncoordinated mechanism’s administ rat iveround, they are assigned to schools with nearly 1,200 9th graders.The diff erences between ranked and assigned schools are also large in the coordinated mech-

anism, though in some cases, they are not as stark. Diff erences between ranked and assignedschools on Math and English achievement or four-year college going are narrower for the admin-ist rat ive round of the coordinated mechanism than the uncoordinated one. On the other hand,assigned schools are not as close to home in the coordinated mechanism. Therefore, it is notpossible to assert which mechanism’s administ rat ive process generates bet ter matches. Whatis clear is that being processed in the administ rat ive round is undesirable for students in bothmechanisms. As a result , it ’s reasonable to expect that a significant fract ion of the changes instudent welfare will be driven by the reduct ion in the number of students who enter this roundin the coordinated mechanism.

4.4 Off er Processing by St udent Charact er ist ics

Table 5 report s the at t ributes of students across rounds compared to the overall populat ion ofapplicants in the uncoordinated mechanism. Students from Manhat tan, those with high mathbaseline scores, and those who have applied to exam schools (indicated as taking the SHSAT)tend to obtain off ers earlier in the uncoordinated mechanism. They are also overrepresentedamong students who receive mult iple first round off ers (not shown). Students from Staten Island,students who are white, and those from high income neighborhoods tend to systemat ically obtainoff ers later in the process. Compared to the overall populat ion, these groups are overrepresentedin the administ rat ive round. About a quarter of students in the administ rat ive process are white,compared to 15% of part icipants overall.The coordinated mechanism dist ributed school access more evenly across rounds. That is,

the diff erences in the types of students assigned in each round are not as pronounced underthe coordinated mechanism. This can be seen by comparing students across boroughs or racialgroups in column 4 of Table 5. The fract ion of students assigned in the main round is similaracross all five boroughs, as is the racial composit ion of students. Higher baseline applicants aremore likely to be assigned in the main round in the new mechanism than low baseline applicants,but they are not as overrepresented as in the old mechanism.The coordinated mechanism also reduced the number of students assigned to in-demand

schools relat ive to the uncoordinated mechanism. Figure 3 reports the change in the number ofstudents assigned to a school compared to a measure of how oversubscribed the school was in theuncoordinated mechanism. For example, the 1,455 students were assigned to the Louis BrandeisHigh School, a st ruggling Manhat tan high school with among the lowest four-year graduat ionrates in the city, in 2002-03, but only 911 students were assigned there in 2003-04.19 The upwardsloping line means that if a school is more oversubscribed in the old mechanism, the new mech-

19The NYC DOE announced the closure of this school in 2009. The largest size reduct ion is t he EvanderChilds High School in the Bronx, which went from 1,739 to 453 9th graders. This high school had a longst andingreput at ion for violence and disorder, and was eventually closed in 2008.

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anism tended to assign more students to that school. Conversely, the new mechanism assignedfewer students to schools that were undersubscribed in the old mechanism. This phenomenonsuggests that the coordinated mechanism was able to use the submit ted preferences more eff ec-t ively to place children into schools that they wanted. The extent to which this represents animprovement in student welfare depends on the heterogeneity of student preferences, an issue weturn to next .

5 Est imat ing St udent Preferences

5.1 St udent Choices

What makes a families rank part icular schools and where do they obtain informat ion about schoolopt ions? In surveys, parents state that academic achievement , school and teacher quality are themost important school characterist ics (Schneider, Teske, and Marschall 2002). In NYC, familiesobtain informat ion about high schools and programs from many sources. Guidance counselors,teachers, and other families in the neighborhood all can provide input and recommendat ions.The offi cial repository of informat ion on high schools is the NYC High School directory, a bookletthat includes informat ion about school size, advanced course off erings, Regents and graduat ionperformance, the school’s address, the closest bus and subway, and a descript ion of each programtogether with a list of ext racurricular act ivit ies and sports teams. Families also learn aboutschools at High school fairs, school open houses, rankings in local newspapers, online schoolguides and from books about high schools (e.g., Hemphill (2007)).Even though the school ranking decision involves evaluat ing many diff erent alternat ives and

features that we do not observe, the aggregate dist ribut ion of preferences displays some consistentregularit ies: students prefer closer and higher quality schools as measured by student achievementlevels, shown in Table 6.20 The first row of the table shows that 20% of applicants rank 12 schoolchoices; the majority rank nine or fewer choices and nearly 90% rank at least three choices. Theaverage student ’s top choice is 4.43 miles away from home. Given that the closest school is onaverage 0.82 miles away, this means that students do not simply rank the school closest to homefirst . For the typical student , the first choice is 0.44 miles closer than her second choice, and hersecond choice is 0.25 miles closer than her third choice. Even though other school characterist icschange with distance, the distance to ranked school increases monotonically unt il the 9th choice,which is 5.65 miles away on average.Lower ranked schools are not only farther away, but they also less desirable on other measures

of school quality. Math performance decreases going down rank order list s (English performanceexhibit s the same trends as Math and is therefore not reported). Other measures of performance(also not reported) such as the percent of students at tending a four year college and the fract ionof teachers classified as inexperienced change monotonically going down rank order list s. Schoolsenrolling lower shares of poor students or a higher share of white students tend to be rankedhigher.

20Table B3 provides addit ional informat ion on school assignment s. 31.9% of student s receive their t op choice,15.0% receive their second choice, and 2.4% receive a choice ranked 10th, 11th, or 12th. 17.5% of student s areasked to part icipat e in the supplementary round because they are unassigned in the main round.

15

Using requests for individual teachers, Jacob and Lefgren (2007) find that parents in lowincome and minority schools value a teacher’s ability to raise student achievement more than inhigh income and non-minority schools. In contrast , Hast ings, Kane, and Staiger (2009) reportthat higher SES families are more likely to choose higher-performing schools than lower-SES onesbased on stated reports under Charlot te’s school choice plan. This diff erence across groups mo-t ivates our invest igat ion of ranking behavior by baseline ability and neighborhood income. Highachieving students tend to rank schools with high Math achievement relat ive to low achievers,though both groups place less emphasis on achievement with further down their preference list .Similarly, students from low income neighborhoods tend to put less weight on Math achievementthan students from high income neighborhoods, but both groups rank higher achieving schoolshigher. These diff erences suggest the importance of allowing for tastes for school achievement todiff er by baseline achievement and income groups.The characterist ics of schools ranked in the uncoordinated mechanism, also shown in Table 6,

are decreasing in a school’s average achievement and income. However, there is a compression inthe preferences relat ive to the coordinated mechanism. For instance, the distance to a students’first choice is 4.80, while it is 4.79 miles for their fifth choice. In the coordinated mechanism,the fifth choice is about one mile further than the first choice. Such a pat tern is consistentwith students being more expressive with their choices in the new mechanism, which would beexpected given it s incent ive propert ies and the fact that more choices can be ranked.Based on their submit ted preferences, all else equal, students prefer at tending a school closer

to home. The fact that students in the new mechanism are assigned to schools further from homemight suggest that the new mechanism led to assignments that are worse on average than theold mechanism. On the other hand, students may prefer schools outside of their neighborhoodbecause they are a bet ter fit . We must therefore weigh the greater t ravel distance in the newmechanism against changes in the assigned school. Students do not rank the closest school totheir home and instead t rade-off school at t ributes with proximity. Our next task is to quant ifyhow students evaluate distance relat ive to school at t ributes such as average achievement levels,demographic composit ion, and size based on their submit ted preferences.

5.2 M odel and Est imat ion

The comparison of the at t ributes of choices ranked higher to those ranked lower or not at allprovides rich informat ion to ident ify how a student t rades-off school features with distance. Toquant ify these t radeoff s, we work with a random ut ility model, which allows for factors whichare not observed in our dataset to influence ranking decisions. Let i index students, j indexprograms, and let sj denote the school in which program j is located. Since all of the schoolswe study are free, we treat distance as our numeraire for our welfare analysis. Specifically, wemodel student i ’s indirect ut ility for program j using the following specificat ion:

ui j = δj +X

l

α l zli xlj +

X

k

γki xkj − di j + εi j � (1)

with δj = xj β + ξj �

16

where zi is a vector of student characterist ics, xj is a vector of program j ’s characterist ics, di j isdistance between student i ’s home address and the address of program j , ξj is a program-specificunobserved vert ical characterist ic, γi is a vector of random-coeffi cients that capture idiosyncrat ictastes for program characterist ics and εi j captures idiosyncrat ic tastes for programs.Since we’d like the model to capture heterogenous preferences, but st ill be computat ionally

t ractable, we employ an ordered version of the choice model in Rossi, McCulloch, and Allenby(1996). They describe a class of parametric dist ribut ions for unobserved characterist ics andidiosyncrat ic preferences that allows for est imat ion via Gibbs’ sampling.21 Specifically, we assumethat

γi ∼ N (0�Σγ )� ξj ∼ N (0�σ2ξ )� εi j ∼ N (0�σ2ε )�

We assume conjugate priors for β, α, Σγ , σ2ξ , and σ2ε . The specific dist ribut ions and addit ional

details are described in the Computat ion appendix.Students may have idiosyncrat ic tastes for schools that are not captured by the observable

students characterist ics in our dataset . This specificat ion allows for arbit rary correlat ion betweenthe k dimensions of γi , which exploit s the richness rank-ordered data. Berry, Levinsohn, andPakes (2004) argue that data on top and second choices can be used to est imate these parametersby exploit ing common characterist ics between subsequent rankings for a given student . Sinceour rank order list data includes up to 12 choices, we do not rest rict the covariance betweenschool size, percent White, percent subsidized lunch, and Math performanceThe paramet ric assumpt ions are made for computat ional t ractability as the dist ribut ion of

indirect ut ilit ies is non-paramet rically ident ified in our set t ing.22 Given the independence ofidiosyncrat ic tastes, the two primary rest rict ions in our specificat ion is that the distance to schoolis addit ively separable in the indirect ut ility funct ion and that taste shocks are independent lydist ributed. The coeffi cient of − 1 is an alternat ive scale normalizat ion to the common pract iceof set t ing the variance of ε if, all else equal, students dislike t raveling to further away schools.We do not explicit ly include an outside opt ion and instead normalize, without loss of gen-

erality, the value of δ for an arbit rarily chosen school to zero. This assumpt ion is mot ivated byour primary interest in studying the allocat ion within inside opt ions rather than subst itut ionoutside of the NYC public school system. Moreover, the commonly used model of the outsideopt ion, which infers that a school is unacceptable if not ranked, would require us to assume thatstudents who have not ranked all 12 choices prefer their outside opt ion to a NYC high school.However, Table B4 shows that roughly three-quarters of supplementary round students enroll intheir off er for that round, and the majority enroll in some other NYC high school.The demand sample contains rankings of 69,907 part icipants in 2003-04 over 497 programs

in 235 schools, represent ing a total of 542,666 school choices. Let r i denote the rank order list of

21We use Gibbs sampling rather than simulated maximum likelihood because of biases in dataset s with alarge number of choices (Train 2009). The post erior means we report have the same asymptot ic dist ribut ion asmaximum likelihood est imates (see chapt er 10.1 in van der Vaart (2000)).22Assumpt ions needed to ident ify preferences for choice charact erist ics in binary and mult inomial set t ings have

been examined in Ichimura and Thompson (1998), Lewbel (2000) and Briesch, Chint agunta, and Matzkin (2002),t hough our dat a includes ordered choices, t hough ordered choice dat a cont ains addit ional informat ion. Agarwaland Somaini (2014) study ident ificat ion in the school choice cont ext with a pot ent ial manipulable mechanism.Non-paramet ric ident ificat ion result s in these set t ings carry over to our case.

17

student i . Our specificat ions follow other models of school demand and include average schooltest scores and racial at t ributes as characterist ics (see, e.g., Hast ings, Kane, and Staiger (2009)).In each specificat ion, we also include three program type dummies and ten program specialitydummies.23 The categorizat ion of programs into specialit ies is described in the Data Appendix.We start by assuming that rankings reflect t rue preferences. This benchmark is natural

because of the st raight forward incent ive propert ies of the mechanism and the advice that theNYC DOE provides in the 2003-04 High School Directory and in their informat ion and out reachcampaign. For instance, the DOE guide advises part icipants to “rank your twelve select ionsin order of your t rue preferences” with the knowledge that “schools will no longer know yourrankings.” Nonetheless, t ruthful behavior is a st rong assumpt ion that we revisit in Sect ion 8.

5.3 Preference Est imat es

We report select est imates for six specificat ions of our demand model in Table 7 (the full setof est imates is in Table A1). The first specificat ion includes school characterist ics (high mathachievement , percent subsidized lunch, percent white, and 9th grade size), but does not in-clude interact ions of student characterist ics with student characterist ics. We do not includeaddit ional achievement characterist ics examined in Table 4 such high English achievement andpercent at tending four college because they are closely related to high math achievement . Thesecond specificat ion includes student -school interact ions. Both of these models do not includestudent -specific random coeffi cients. The next four specificat ions add in student -specific ran-dom coeffi cients to the second specificat ion and vary in how they use the choice data: using allchoices, using the top three choices, dropping the last choice, and dropping students who rank12 choices. Each specificat ion with student interact ions include dummies for Spanish, Asian andOther Language Program, interacted with a student ’s English proficiency status and whetherthey are Hispanic or Asian. Computat ional const raints prevent us from est imat ing all of thesemodels on the full sample, so we only est imate using the full sample for the specificat ion withstudent -school interact ions and student -specific random coeffi cients; the rest of these models areest imated on a 10% random sample of the data which contains about 7,000 students rankingseveral schools.There are three main pat terns in Table 7. First , student -school interact ions are often est i-

mated precisely. For instance, high baseline math students tend to prefer higher achieving schoolsand minority students tend not to prefer schools with high white percentages. Second, the est i-mates are broadly similar across the four specificat ions in column 3-6, with random coeffi cients.Third, many of the random coeffi cients are significant ly est imated, suggest ing the importance ofa flexible specificat ion in account ing for the underlying heterogeneity in student preferences. Wereport further on measures of model fit in Sect ion 8. Since it fully exploit s all of the choice datain the most flexible model, the third column specificat ion is our preferred specificat ion.

23The program type dummies are Unscreened, Screened, and Educat ional Opt ion. The program specialtydummies are Art s, Humanit ies/ Int erdisciplinary, Business/ Account ing, Math/ Science, Career, Vocat ional, Gov-ernment / Law, Other, Zoned, and General.

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6 Compar ing A lt ernat ive A llocat ion Schemes

6.1 M easur ing Welfare

The preference est imates allow us to compute measures of welfare across assignments. Under theassumpt ion that distance is the numeraire, we can measure the diff erence between two assign-ments for a student in terms of the distance-equivalent ut ility or the amount a student is willingto t ravel for the more preferred school assignment . To compare the welfare associated with twoassignments, we compute the average welfare in distance-equivalent ut ility. Consider a group ofstudents in set G and a matching �, which specifies the program for each student as �(i ). Defineaverage welfare for students in G as a funct ion of parameters θ as

WµG (θ) =

1|G|

X

i∈Guiµ(i ) (θ) �

where ui j is the ut ility student i associated with assignment to program j . For two matchings,� and �′ , and students in group G(�) and G(�′ ), the diff erence in welfare between the twomatchings is given by:

WµG (θ) − W

µ ′

G (θ) =1

|G(�)|

X

i∈G(µ)

uiµ(i ) (θ) −1

|G(�′ )|

X

i∈G(µ ′ )

uiµ ′ (i ) (θ) �

In the coordinated mechanism, we observe the rankings submit ted by students. Under theassumpt ion that these report s are t ruthful, the kth ranked program yields the kth highest ut ility.For a given student and est imate of θ, this fact contains informat ion about her unobserved tastesǫi j . For students in the coordinated mechanism who submit rank order list r i , we calculate theexpected ut ility for each ranked and unranked choice by simulat ing the ut ility from the est imatedpreference dist ribut ion, condit ional on the relat ionships implied by the submit ted ranks.In part icular, we compute the expected ut ility of a program ranked kth by student i as

Ehui j | u

(k+ 1)i < ui j < u

(k− 1)i �r i k = j

i(2)

and the expected ut ility for all unranked schools as

Ehui j | ui j < u

(|r i |)i �j 6∈ ∪{ r i k }

i� (3)

where u(k)i is the kth highest value of { ui j } Jj = 1 and |r i | is the number of ranks submit ted bystudent i .24 With these definit ions, the welfare diff erence for group G from assignments � and�′ is

WµG − Wµ ′

G =1

|G(�)|

X

i∈G(µ)

E[uiµ(i ) |r i ] −1

|G(�′ )|

X

i∈G(µ ′ )

E[uiµ ′ (i ) |r i ]�

where E[ui j |r i ] denotes the condit ional expectat ions in equat ions (2) and (3) above.

24For student s who are in the mechanism comparison sample, but not the demand sample (so we don’t observetheir rank order list s), we use the mean ut ility condit ional on observables alone in the welfare calculat ions. Wefollow a similar approach for student s assigned to choices not ranked in the main round.

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6.2 Evaluat ing M echanism Design Feat ures

Even though many of the new mechanism’s features were designed to address issues in the oldmechanism, the coordinated mechanism st ill involves several design const raints. Here, we ask:how far is the allocat ion produced by the coordinated mechanism from the best possible one forstudents and what are the quant itat ive eff ects of part icular design decisions? The alternat ives weconsider vary the mechanism, holding fixed the set of schools, students, and resident ial choices,and therefore are best interpreted as short -run eff ects of alternat ive mechanisms.We begin by using the demand model est imates to assess two benchmark allocat ions intended

to capture the range of what’s achievable with a choice system. The first is a neighborhoodassignment , computed by running the deferred acceptance algorithm with applicants simplyranking schools in order of distance. Since the scope of the market is smaller under the uncoor-dinated mechanism, this allocat ion further rest ricts the market ’s scope, as in the administ rat iveround of the uncoordinated mechanism.The second benchmark allocat ion, the ut i l i t ar ian assignment , maximizes the sum of stu-

dent ut ility subject the feasibility const raints of the assignment .25 Given the est imated dist ribu-t ion of student preferences, no other allocat ion can yield higher average ut ility. This allocat iontherefore represents the other ext reme where a planner implements the best possible allocat iontaking full advantage of the cardinal dist ribut ion of student preferences.26

To compare the coordinated mechanism to these two benchmarks, in Table 8 we report thediff erence with the ut ilitarian outcome. The first column of Table 8 shows that the diff erence indistance-equivalent ut ility between the neighborhood and ut ilit arian assignment is 18.96 miles.This diff erence is a theoret ical upper bound on the gains from school choice. The coordinatedmechanism achieves about 80% of the possible gains from a choice system, since the diff erence indistance-equivalent ut ility with the ut ilitarian assignment for the average student is 3.73 miles.27

The ut ilitarian opt imal assignment is an idealized benchmark, but is diffi cult to achieve fortwo reasons. First , there are 208 screened programs in New York City in 2003-04, so implement ingthis allocat ion would ignore school-side rankings at those programs. This allocat ion may alsoallow for situat ions of just ified envy at programs that only use coarse priorit ies. Second, thisassignment uses cardinal informat ion, which is not solicited by the coordinated mechanism.Therefore, we consider a more effi cient outcomes for students that do not completely abandonschool priorit ies and only uses student rank-order list s.The deferred acceptance algorithm in the coordinated mechanism need not produce a student -

opt imal stable matching because it must resolves t ies between students when they have ident icalpriorit ies at a school. This t ie-breaking can lead DA to stable outcomes, which are not student -

25We solve for this allocat ion following Shapley and Shubik (1971). The ut ilit arian allocat ion implies equalweight s on student s; in quasi-linear problems, such an allocat ion would be synonymous with the Pareto effi cientallocat ion, but need not be here.26Using illust rat ive examples, P ycia (2014) argues that t he welfare loss of ordinal mechanisms relat ive to cardinal

ones can be arbit rary large.27We implement the coordinated mechanism by drawing 100 set s of lot t ery numbers and re-run the student -

proposing deferred accept ance algorit hm given student ’s choices. For student s unassigned aft er t he main round,we implement NYC’s supplementary round by using preferences from the demand model and assigning student sunder a serial dict atorship according to the lot t ery number.

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opt imal. Deferred acceptance cannot be Pareto-improved upon without sacrificing st rategy-proofness for students (Erdil and Ergin 2008, Abdulkadiroğlu, Pathak, and Roth 2009, Kesten2010, Kesten and Kurino 2012). We quant ify the cost of providing st raight forward incent ives forstudents by comput ing a student -opt imal stable assignment that improves the DA assignment byassigning students higher in their choice list s, while respect ing stability for st rict school priorit ies.Such an assignment can be computed by the stable improvement cycles algorithm developed byErdil and Ergin (2008), which iterat ively finds Pareto improving swaps for students, while st illrespect ing the stability requirement for underlying weak priority ordering of schools (i.e., priorto t ie-breaking). A total of 2,348 students in the demand sample can obtain a bet ter assignmentin a student -opt imal stable matching. The diff erence in distance-equivalent ut ility is 0.11 milescompared to the assignment produced by the coordinated mechanism. The cost of this changeis that the underlying mechanism is not based on a st rategy-proof algorithm.28

Another limitat ion of the coordinated mechanism is that it const rains student welfare due toit s t reatment of school priorit ies and preferences. To quant ify the importance of this const raint ,we compute the welfare of students under the Pareto effi cient assignment found by t ransferringstudents from their assigned schools to their higher ranked choices via the Gale’s Top TradingCycles algorithm (Shapley and Scarf 1974). Since this mechanism does not produce a stableoutcome, it is possible that schools benefit by off ering students seat s outside of the assignmentprocess. The diff erence in aggregate student welfare under this Pareto effi cient assignment andthe student opt imal stable matching may therefore be viewed as the cost of incent ives for schoolsto part icipate in the system.29

We calculate a Pareto effi cient matching which dominates each student -opt imal stable match-ing we simulate and est imate students’ ut ilit ies from this Pareto effi cient matching in column 4of Table 8. A total of 10,882 students obtain a more preferred assignment at a Pareto effi cientmatching. An ordinal Pareto effi cient allocat ion st ill represents a substant ial diff erence betweenthe ut ilit arian opt imum. The ut ility diff erence for the average student 3.11 miles. Relat ive tothe current mechanism, the cost of limit ing the scope for st rategizing by schools (by imposingstability) is 0.62 miles.In summary, these comparisons show the diff erence between having a choice system with the

coordinated mechanism and not having a choice system at all represents a much larger diff erencein student welfare than fine-tuning the coordinated mechanism. That is, the ability to chooseschools generates substant ial welfare when preferences are heterogeneous. Within a coordinatedmatching system, further opt imizing the matching algorithm produces relat ively lit t le gain in thebest case, and even if it were possible to implement cardinal allocat ion schemes. This conclusiondoes not imply that the matching scheme is not important since we’ve seen the large number ofthose processed administ rat ively in the uncoordinated mechanism. To see where in the spect rumthe uncoordinated mechanism lies, we next turn to analyzing its propert ies.

28Azevedo and Leshno (2011) provide an example where the equilibrium assignment of the st able improvementcycles mechanism is Pareto inferior t o the assignment from deferred accept ance when student s are st rat egic.29Balinski and Sönmez (1999) and Abdulkadiroğlu and Sönmez (2003) provide an alt ernat ive equity rat ionale

for st ability. Note that no st able mechanism eliminat es st rat egic maneuvers by schools (Sönmez 1997), althoughthis may not be an issue in market s with many part icipant s (Kojima and Pathak 2009).

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7 Compar ing t he Coordinat ed and Uncoordinat ed M echanisms

Unlike the variat ions on the coordinated mechanism we’ve just examined, there is no simple wayto model behavior in the uncoordinated mechanism. To evaluate it , we adopt an alternat iveapproach that takes advantage of the fact that we observe the assignments of students in thatmechanism. This approach requires two important assumpt ions.First , since we do not observe the same student in both the uncoordinated and coordinated

mechanism, we have to assume that we can use preference est imates from the coordinated mech-anism to make statements about the previous year. One potent ial concern is that schools havechanged significant ly because of the mechanism so a student ’s valuat ion of the school likely haschanged alongside the mechanism. To probe this issue, in Figure A1, we plot the average charac-terist ics of the schools, as measured by the at t ributes of peers, in both mechanisms. The Figureshows that for measures of poverty, racial make-up, and baseline math scores, there is relat ivelylit t le change in school at t ributes despite the change in the mechanism. Consistent with theincreased t ravel distance in Table 3, the last panel of the Figure shows that schools do appeardiff erent when measuring the average t ravel distance of enrolled students.The second assumpt ion involves our interpretat ion of the rankings submit ted in the uncoor-

dinated mechanism. For the counterfactuals in Table 8, to compute the ut ility associated with anassignment for the new mechanism, we condit ion on the rank order list submit ted by the student ,adjust ing for what that rank order list implies about unobserved tastes under the assumpt ionthat the student reported her t rue preferences. The old mechanism’s uncoordinated nature makesthe relat ionship between preferences and the final assignments less st raight forward. Comput ingthe expected ut ility condit ional on the assignments produced by the uncoordinated mechanismrequires st rong assumpt ions about that mechanism, it s propert ies, and agent behavior and ex-pectat ions. This diffi culty is not unique to our set t ing and is likely a challenge in evaluat ing thewelfare eff ects of other assignment systems where the incent ive propert ies of the mechanism arenot well understood and agent behavior is diffi cult to model.30 Instead, our approach exploit sour data on observed rankings from the uncoordinated mechanism’s first round.The approach we start with, unordered applicat ions, assumes that choices submit ted in the

uncoordinated mechanism have the property that any ranked choice is preferred to an unrankedchoice, but does not assume that higher ranked choices are preferred to lower ranked ones. Thisassumpt ion has the benefit of not assuming rank order list s submit ted in the uncoordinatedmechanism are t ruthful, while st ill using some of the informat ion contained in the preferencesubmission. A weakness of this assumpt ion is that it excludes the possibility that students haveomit ted choices from their rank order list that they prefer, but do not expect to be admit ted.However, in Table 6, we saw that less than half of part icipants rank all five choices in the oldmechanism, suggest ing that if these part icipants wanted to rank more schools, the mechanismdoes not const rain them. Nonetheless, we will also consider two other assumpt ions.

30Budish and Cant illon (2012) ut ilize survey dat a from a manipulable mechanism to make stat ement s aboutchanges in mechanism design. Unfortunat ely, similar survey dat a does not exist in our set t ing. Agarwal andSomaini (2014) present an approach to uncovering preferences from a class of single-off er manipulable mechanismsthat involve lot t eries; because the uncoordinat ed mechanism is a mult iple-off er system, it falls out side of this class,and their methods are not applicable in our set t ing.

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Under the unordered applicat ions select ion assumpt ion from the uncoordinated mechanismand assuming true preferences from the coordinated mechanism, we find that most students arebet ter off under the coordinated mechanism. Figure 4 plots the overall dist ribut ion of studentwelfare from the two mechanisms. The average student improvement in welfare is equivalent to10.96 miles. The figure shows a bimodal pat tern of ut ility in the uncoordinated mechanism dueto the students who are assigned in the administ rat ive round. Most of the mass in the first modeshift s rightward in the coordinated mechanism, a phenomenon driven by the sharp reduct ion inthe number of students assigned administ rat ively.For each student group shown in Table 9, there is a posit ive gain from the new mechanism.

Across racial groups, whites and Asians gain more than blacks and Hispanics. There are morepronounced diff erences across boroughs. Students from Staten Island and Queens gain the most ,while the smallest gain of any student group in the table comes from Manhat tan residents. Lowbaseline math students gain more than high baseline math students or SHSAT test takers.It ’s worthwhile to compare the diff erence in ut ility to the diff erence in t ravel distance, to

decompose the welfare eff ects of the improved school match. On average, the improved matchignoring distance is equal to 11.65 miles. This implies that the improved school match is worththirteen t imes the costs associated with increased t ravel distance. The lowest gains are forManhat tan residents, a group which experiences no increase in t ravel distance. However, thewelfare gains are not solely driven by changes in distance. For instance, across boroughs, StatenIsland pupils only t ravel 0.34 miles further, but they experience the largest improvement ofany borough at 22.56 miles. This suggests that mismatch was part icularly severe in StatenIsland, and is consistent with the substant ially larger fract ion of Staten Island residents who areadminist rat ively assigned in the uncoordinated mechanism.31 The magnitude of our est imatesof improved school matches suggest that even if there are not enough good schools to assigneveryone their top choice, preference heterogeneity generates an important role for a coordinatedmechanism.The welfare gains in the coordinated mechanism are larger for many disadvantaged groups,

a pat tern consistent with Hemphill and Nauer (2009)’s claim that the uncoordinated mechanismadvantaged high-achieving students and those with sophist icated parents. For instance, welfaregains are larger for low baseline math students than high baseline math students. They are alsohigher for limited English proficient students than SHSAT test -takers. However, the diff erencefor Staten Island, which has a larger white populat ion and weather neighborhoods, plays a largerole in the fact that whites and rich neighborhoods experience larger welfare gains than blacksand Hispanics and poorer neighborhoods.Diff erences across student groups closely t rack the rounds in which students were processed

in the uncoordinated mechanism. For instance, substant ially more high baseline students wereassigned in the main round of the old mechanism compared to low baseline students. Studentgroups with higher fract ions assigned in the administ rat ive round, shown in column 7, tend to ex-perience the largest gains. Figure 5 report s on the relat ionship between likelihood to be assigned

31In the uncoordinat ed mechanism, there are 1,054 student s who ranked Stat en Island Technical, a highlysought -aft er screened school. Only 16% are assigned there, and about 75% do not obt ain a main round off er andare subsequent ly administ rat ively assigned.

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administ rat ively and student welfare. We fit a probit of whether a student is administ rat ivelyassigned on all student characterist ics in the demand model for the uncoordinated mechanismsample. We also include census t ract dummies to account for neighborhoods that may or maynot have local high schools, which is a guaranteed fallback opt ion for some students. We thenuse this est imated relat ionship to compute each student ’s propensity to be assigned administ ra-t ively in both the uncoordinated and coordinated mechanism. For each decile of this propensity,we compare the ut ility from assignment across both mechanisms. However, there is a clearmonotonic pat tern between administ rat ive assignment propensity and the welfare improvement ,whether comparing ut ility diff erences either including distance or net of distance.A possible threat to our welfare calculat ion is that an init ially bad allocat ion was subsequent ly

undone post -assignment through the aftermarket , the period between when off ers where madeand the school year start s. Table 3 showed that students enrolled in schools further away onaverage than where they were assigned in the uncoordinated mechanism, but the opposite pat ternis t rue in the coordinated mechanism. The coordinat ion of admissions occurred with greatercent ral cont rol of enrollment , which may have made it possible that the more rigid aftermarketin the coordinated mechanism is actually worse for students. To examine this possibility, we alsocompute the ut ility associated with the schools students enroll at in October of the followingschool year. Compared to assignments, the gains from the coordinated mechanism measured byenrollment are somewhat smaller, but are st ill large. For instance, the distance-equivalent ut ilityfor the average student is 9.67 miles, which is 88% of the gain from the assignment . The changein t ravel distance to enrolled school is also lower than the change in t ravel distance to assignment .Though a smaller gain from enrollment suggests that some of the old mechanism’s mismatch wasundone in it s aftermarket , these fact s weigh against the argument that post -market reallocat ionhas undone a large fract ion of misallocat ion.Finally, it ’s worth not ing how the mechanism design mat ters when we compare the uncoor-

dinated and coordinated mechanisms. That diff erence represents nearly 60% of the total rangebetween the neighborhood and ut ilit arian assignments, which is much larger than the range as-sociated with tweaks to the matching algorithm. This finding informs a broader debate in themarket design lit erature about the importance of sophist icated market clearing mechanisms. Inthe context of auct ions, Klemperer (2002) argued that “most of the extensive auct ion literatureis of second-order importance for pract ical auct ion design,” and that “good auct ion design ismost ly good elementary economics.” Consistent with this point of view, for school matchingmarket design, coordinat ing admissions producing much larger gains than algorithm refinementswithin the coordinated system.To see how robust our conclusions are, in Table 10, we repeat the analysis in Table 9 for

three alternat ive assumpt ions. In each case, we break the asymmetric t reatment of select ionrules in Table 9, and use the same assumpt ion about rank order list s for both mechanisms. Inthe first variat ion, we do not adjust the ut ility for based on rank order list s submit ted in eitherthe uncoordinated or coordinated mechanism. This unselect ed appl icat ions (on unobserv-ables) select ion rule simply assumes that the idiosyncrat ic component of tastes has no influenceon ut ility. While this assumpt ion may be unappealing given the importance of unobserved de-terminants of applicant preferences, examining welfare diff erences under it indicates how much

24

our est imates are driven by observable dimensions of school and student characterist ics. Column1 of Table 10 reports that the diff erence in welfare is more than one half of that implied byunordered applicat ions select ion rule. The average student is going to a school that is abouteight t imes more valuable compared to the increased t ravel distance. The qualit at ive pat ternsare similar to those under unordered applicat ions select ion rule: all student subgroups benefitand the gains are larger for groups that were more likely to be administ rat ively assigned. Forinstance, gains are larger for Asians and Whites; they are largest for Staten Island residentsand lowest for Manhat tan. One diff erence with the unordered applicat ions select ion rule is forcomparisons across baseline achievement groups: high baseline math students benefit more thanlow baseline math students, though low baseline students st ill gain slight ly more than SHSATtest -takers.Columns 3 and 4 present est imates using the unordered applicat ion select ion rule for both

the coordinated and uncoordinated mechanism. The average welfare diff erence of 9.02 milesis smaller than that reported in Table 9, but the pat terns across student groups is similar.Finally, in columns 5 and 6, we report est imates from the t r ut hful appl icat ions select ion rule,which assumes that the rank order list submit ted in the either the uncoordinated or coordinatedmechanism reflect t rue applicant preferences. Under this assumpt ion, the est imated welfare gainsare slight ly smaller than those reported in Table 9 at 10.62 miles on average. The qualitat ivepat terns mirror those from the unordered applicat ion rule, though tend to be slight ly larger.It is reassuring that the est imates we obtain under diff erent select ion rules using informat ion

from submit ted rank order list s are similar between Table 9 and 10. The smaller est imate underthe unselected applicat ions select ion rule in Table 10 imply that unobserved tastes play a rolefor our quant itat ive conclusions. Except for comparisons across baseline achievement groups, thequalitat ive comparisons between student groups are also similar under each of the assumpt ions

8 M odel Fit and A lt ernat e Behavioral A ssumpt ions

8.1 M odel fi t

Since our goal is to make statements about welfare, it is important to examine how well oureconometric model matches the data. We first invest igate within-sample fit is to see what ourest imates imply for the aggregate pat terns by rank in Table 6. Figure A2 report s on measuresof fit using the specificat ion with student -level random coeffi cients and student -school interac-t ions. We plot the observed versus predicted pat tern of three school characterist ics - high mathachievement , percent subsidized lunch, and percent white – as we go down a student ’s choice list .The panels include three pairs of lines for the ent ire sample, and for the low and high baselinemath applicants. For these three characterist ics, our est imates capture the broad pat tern of thechoices, matching both the level and slope of these characterist ics. For instance, the average highmath achievement is 10.0, and the range from the top choice to the 12th choice is 16.7 to 10.4.Our est imates imply that for first choices, school’s fract ion high math achievement is 18.4, whileit drops to 11.8 for last choices. Moreover, the average percent white in New York’s schools is10.8, but first choices are 19.1 white, while we predict them to be 19.8. Relat ive to the averageat t ributes of schools, the model fit is much closer to the actual ranked dist ribut ion.

25

In the last panel of Figure A2, we report the fit for distance. Here, we find that the increase indistance observed for lower ranked choices mirrors that predicted by our model, there is a greaterdivergence in the level of distance. This pat tern appears in all of the models we’ve est imatedwith random coeffi cients. It ’s worth not ing that the average distance to a high school in NewYork is 12.7 miles from home, while the closest school is under a mile.Berry, Levinsohn, and Pakes (2004) emphasize the importance of mixture models in the

context of rank data for automobiles. In part icular, they emphasize that when examining thewithin-consumer relat ionship between the at t ributes of alternat ives ranked first and second,models without random coeffi cients do a poor job. This concern may be part icularly importantin our context . For instance, a high correlat ion between the size of the first and second rankedschool’s size may be indicat ive of taste for large schools. In Table A2, we report on the correlat ionbetween the first and second choice, the first and third choice, and the second and third choice.Consistent with earlier work, we see that the observed correlat ion between choices is much closerin our preferred specificat ion than in the simpler model within sample. When we examine amore demanding out -of-sample test , comparing the 2003-04 preference est imates to examine thecorrelat ion pat tern of choice made in 2004-05, we also see that the correlat ion pat tern in ourmain specificat ion is closer to the observed pat tern than from a demand model without studentinteract ions.

8.2 A ssessing t he Behavioral A ssumpt ions on Ranking

Motivated by the incent ive propert ies of the deferred acceptance algorithm, the preference est i-mates that we reported come from models assuming that students are t ruthful. We revisit somepotent ial object ions to this assumpt ion in this subsect ion.In an influent ial experiment , Hast ings and Weinstein (2008) show how malleable parent pref-

erences are when provided with direct informat ion on school test -scores in Charlot te-Mecklensberg’schoice system. Such a finding suggests that students may be overwhelmed by the prospect ofevaluat ing over 500 school programs and preferences are an unreliable guide for welfare analysis.However, if preferences were generated without much of a systemat ic component , then we’d ex-pect that most of our point est imates to be imprecisely est imated or have unintuit ive pat terns,cont rary to what we’ve seen. Moreover, in Figure A3, we plot school market shares between thefirst year of the coordinated mechanism and its second year. This plot illust rates the extent ofvariat ion in market shares, which may occur with new informat ion such as expanded high schoolfairs across t ime. The market shares of most programs are very similar across both years of thenew system. This fact suggests that preferences are persistent .Another way to see that choices are consequent ial for part icipant welfare is to examine en-

rollment decisions by choice received. Table B4 reports assignment and enrollment decisions forstudents who are assigned in the main round. The table shows that 92.7% of students enroll intheir assigned choice, and this number varies from 88.4% to 94.5% depending on which choice astudent receives. Interest ingly, take-up is higher for students who receive lower ranked choices,while the fract ion of students who exit is highest among students who obtain one of their topthree choices. This fact suggests that either families are indiff erent between later choices andsimply enroll where they obtain an off er or families have deliberately invest igated later choices

26

and are therefore willing to enroll in lower ranked schools. If families are more uncertain aboutlower ranked choices, then using all submit ted ranks may provide a misleading account of studentpreferences. To examine how sensit ive our conclusions are to this assumpt ion, we fit a demandmodel that consider only the top three choices of applicants in column 4 of Table 7.A second concern with t reat ing submit ted rankings as t ruthful is that parents rank schools

using heurist ics carried over from the previous system. Despite the theoret ical mot ivat ion andthe DOE’s advice, parents might st ill deviate from truth-telling because of misinformat ion. TableB4 shows that students are more likely to be assigned their last choice than their penult imatechoice. This pat tern may be caused by st rategic behavior if students apply to schools that theylike and, as a safety opt ion, rank a school in which they have a higher chance of admissionslast . For instance, Calsamiglia, Haeringer, and Kljin (2010) present laboratory evidence that aconst raint on rank order list s encourages students to rank safer opt ions. However, it may alsobe fully consistent with t ruth-telling. For example, students usually obtain borough priority orzone priority for schools in their neighborhoods. Ranking these schools improves their likelihoodof being assigned to these schools in case they are rejected by their higher choices. If studentsconsider applying and commuting to schools further away from their neighborhood for reasons likemath and English achievement , they may as well stop ranking schools below their neighborhoodschools once such considerat ions no longer just ify the cost of commute. Alternat ively, searchcosts may induce parents to stop their search for schools before they ident ify twelve schools fortheir children and rank their neighborhood school as last choice. This preference pat tern wouldproduce the observed assignment pat tern. To examine how sensit ive our conclusions are to thisassumpt ion, we fit demand models that drop the last choice of each student in in column 5 ofTable 7.The third issue with the assumpt ion of t ruthful preferences is that students can rank at most

12 programs on school applicat ions. When a student is interested in more than 12 schools, shehas to carefully reduce her choice set down to at most 12 schools. If a student is only interestedin 11 or fewer schools, this const raint in principle should not influence her ranking behavior(Abdulkadiroğlu, Pathak, and Roth 2009, Haeringer and Klijn 2009). It is a weakly dominantst rategy to add an acceptable school to a rank order list as long as there is room for addit ionalschools in the applicat ion form. However, 20.3% of students in our demand sample rank 12schools. Some of these students may drop highly sought-after schools from top of their choicelist s because of this const raint . To examine how sensit ive our conclusions are to this assumpt ion,we fit a demand model that drops students who have ranked all twelve choices in column 6 ofTable 7.In Table A4, we report on our evaluat ion of mechanism design choices under these three

specificat ions: 1) using only the top 3 choices, 2) excluding applicants who have ranked all 12programs, and 3) dropping the last choice of applicants. Because of computat ional const raints, weest imate the models on a 10% random sample, but we use the full sample for the counterfactuals.For all three demand models, the coordinated mechanism in column 2 is roughly 81% of theway from the neighborhood assignment to the ut ilitarian assignment . It therefore appears thatour conclusions on the value of choice relat ive to changes within the coordinated mechanismare robust to these alternat ive ways of using the submit ted rank order list s in the coordinated

27

mechanism.Panel B of Table A4 reports on how the comparison between mechanisms varies with our

demand specificat ion using all of the ranking informat ion of part icipants. P reference hetero-geneity generates a larger role for school choice compared to neighborhood assignment . Thisphenomenon can be seen by comparing the est imates from our main specificat ion to those fromspecificat ions without hetereogeneity (column 1 in Table A1) and without random coeffi cients(column 2 in Table A1). The neighborhood assignment is more appealing according to those twodemand models, since they are only 15.5 and 16.2 miles away from the ut ilitarian assignment ,compared to 21.5 miles from the main specificat ion in the 10% sample. Moreover, the diff erencebetween neighborhood assignment and the coordinated mechanism is smaller in specificat ionsthat do not allow for student interact ions or random coeffi cients than in the main specificat ionwhich includes both student interact ions and random coeffi cients.

9 Conclusion

The reform of NYC’s high school assignment system provides a unique opportunity to studythe eff ects of cent ralizing and coordinat ing school admissions with detailed data on preferences,assignments, and enrollment . We find that the new coordinated mechanism is an improvementrelat ive to the old uncoordinated mechanism on a variety dimensions. More than a third of stu-dents were assigned through an ad-hoc administ rat ive process in the uncoordinated mechanismafter mult iple off ers with few choices and few rounds of clearing left a large number of studentswithout off ers after the main round. Students placed in the administ rat ive round were assignedto schools with considerably worse characterist ics than what they ranked. The new mechanismrelieved this congest ion and assigned more students to schools where they applied.The coordinated mechanism assigns students 0.69 miles further from home to their assign-

ments. However, the benefit of being assigned through the new mechanism is at least eight t imesthe cost of addit ional t ravel, and is often larger depending on the assumpt ion about the informa-t ion revealed about unobserved tastes from rank order list s submit ted under the uncoordinatedmechanism. The gains are posit ive on average for students from all boroughs, demographicgroups, and baseline achievement categories. Welfare improvements are also seen whether ut ilityis measured based on assignments made at the end of the high school match or subsequent schoolenrollment . The largest gains are for students who were more likely to be processed in the ad-minist rat ive round of the uncoordinated mechanism. These conclusions are robust to alternat ivebehavioral assumpt ions on the preferences submit ted in both the uncoordinated and coordinatedmechanism.These gains are measured by a rich specificat ion of student demand that implies significant

est imated heterogeneity in willingness to t ravel for school. P reference heterogeneity is importantfor measuring the allocat ive eff ects of choice when there is a shortage of good schools. Ourest imates reveal that the benefit s from having coordinated choice are much larger than thanthose associated with modificat ions within the coordinated mechanism. This does not implythat the design of the mechanism is not important however, since the gap in average studentwelfare between the uncoordinated and coordinated mechanism is large.

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The increase in student welfare due to the new mechanism illust rates that there are consid-erable frict ions to exercising choice in poorly designed assignment systems. The 2003 change inNYC took place in an environment where part icipants already had some familiarity with choicesince both the uncoordinated and coordinated system had a common applicat ion. In othercit ies, the school choice market is even less well organized, without readily available informat ionon admissions processes and applicat ion t imelines. For instance, admissions in Boston’s growingcharter sector are uncoordinated, and the schools have only recent ly adopted a standardizedapplicat ion t imeline. Recent ly, there have been calls to unify enrollment across school sectors(Vaznis 2013). The relat ive value of policies such as common t imelines, common applicat ions,single vs. mult iple off ers, sophist icated matching algorithms, and good informat ion and decisionaides is an interest ing avenue for future research.Finally, it ’s worth emphasizing that our analysis has focused on the allocat ive aspects of

school choice and diff erent school assignment procedures. An important quest ion is whetherallocat ive changes contribute to changes in the product ive dimensions of assignment . That is,do diff erent student assignment protocols aff ect levels of student achievement post -assignment?This far more diffi cult quest ion requires understanding the link between choices and the educat ionproduct ion funct ion, but is left for future work.

29

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Ranked�Schools Assigned

Ranked�Schools Assigned

(1) (2) (3) (4)

Distance�(in�miles) 4.82 4.30 5.10 4.00

High�Math�Achievement 12.4 11.7 13.0 10.7High�English�Achievement 20.9 20.2 22.1 19.1Percent�Attending�Four�Year�College 49.1 47.1 50.6 48.3Fraction�Inexperienced�Teachers 45.3 45.6 46.6 43.8Attendance�Rate�(out�of�100) 85.1 84.6 85.7 83.8Percent�Subsidized�Lunch 60.0 60.5 57.6 56.7Size�of�9th�grade 694.3 698.8 675.0 819.2Percent�White 15.1 14.7 16.7 17.8

Distance�(in�miles) 4.87 4.59 5.87 5.17

High�Math�Achievement 11.8 9.3 16.6 14.2High�English�Achievement 19.9 15.8 26.5 20.0Percent�Attending�Four�Year�College 48.6 44.9 54.1 50.1Fraction�Inexperienced�Teachers 46.0 41.5 45.3 36.9Attendance�Rate�(out�of�100) 85.1 82.2 87.4 83.2Percent�Subsidized�Lunch 62.0 61.8 53.5 51.0Size�of�9th�grade 685.3 908.0 638.5 1129.7Percent�White 13.8 13.3 17.4 15.3

Distance�(in�miles) 5.11 1.62 5.33 3.43

High�Math�Achievement 14.9 10.5 14.3 10.7High�English�Achievement 24.3 17.5 24.2 19.2Percent�Attending�Four�Year�College 52.0 46.7 51.7 47.9Fraction�Inexperienced�Teachers 41.9 39.4 47.8 42.1Attendance�Rate�(out�of�100) 85.8 80.8 86.7 82.9Percent�Subsidized�Lunch 53.8 50.4 57.2 53.1Size�of�9th�grade 760.6 1181.9 607.6 984.0Percent�White 18.5 19.1 17.6 17.9

Table�4.�Ranked�vs.�Assigned�Schools�by�Student�Assignment�Round

Notes:�Means�unless�otherwise�noted.�Analysis�restricts�the�sample�to�students�from�the�welfare�sample�with�observed�assignments.�Uncoordinated�mechanism�refers�to�the�2002�03�mechanism�and�coordinated�mechanism�refers�to�the�2003�04�mechanism�based�on�deferred�acceptance.��Main�round�in�the�uncoordinated�mechanism�corresponds�to�the�first�round.��Rankings�used�are�those�submitted�in�the�main�round�of�the�process.��Student�distance�calculated�as�road�distance�using�ArcGIS.��See�Table�2�notes�for�details�on�school�characteristics.

Uncoordinated�Mechanism Coordinated�Mechanism

C.�Administrative�Round

B.�Supplementary�Round

A.�Main�Round

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49

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