College Matching Mechanisms and MatchingQuality: Evidence from a Natural Experiment in

China

Wei HaLe Kang

Yang Song∗

January 2019

Abstract

Matching mechanisms play a crucial role in the college admissions process,which in turn influence education and labor market outcomes. We exploit ge-ographical and temporal variations in the Chinese college admissions reformto provide new empirical evidence on how matching mechanisms affect match-ing quality. Consistent with theoretical findings by Chen and Kesten (2017),we show that switching from the Immediate Acceptance (IA) mechanism tothe Chinese parallel mechanism, a hybrid of IA and the Deferred Acceptancemechanism, improved matching stability, proxied by the level of stratificationprecision. This effect is stronger for provinces with wider first parallel choiceband and intensifies over time.

Keywords: College admissions; school choice; matching mechanism; matching qual-

ity; Immediate Acceptance mechanism; Chinese parallel Mechanism.

JEL Classification Numbers: C78, D61, I23, I28.

∗Wei Ha and Le Kang, Graduate School of Education and Institute of Economics of Education,Peking University. Yang Song, Department of Economics, Colgate University. 13 Oak Dr, Hamilton,NY, 13346. ysong@colgate.edu.

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1 Introduction

Matching mechanisms play a crucial role in the college admissions process, which

in turn influence education and labor market outcomes (Hoekstra, 2009; Li et al.,

2012; Dillon and Smith, 2018). Prior studies have shown that the widely adopted

Immediate Acceptance (IA), or Boston mechanism is vulnerable to manipulation and

have investigated ways to improve the school choice matching (Abdulkadiroglu and

Sonmez, 2003; Chen and Sonmez, 2006; Ergin and Sonmez, 2006; Pathak and Sonmez,

2008; Abdulkadiroglu et al., 2011b; Pathak and Sonmez, 2013).1 Although there

has been an increasing number of experimental studies comparing the IA mechanism

with the Deferred Acceptance (DA) mechanism or the Top Trading Cycle mechanism

(Chen and Sonmez, 2006; Calsamiglia et al., 2010; Chen et al., 2016; Basteck and

Mantovani, 2018), limited empirical evidence exists due to data limitations and the

lack of a proper setting to allow for causal inference.2

China’s college admissions reform offers a natural experiment to study the effects

of switching matching mechanism in a large-scale, centralized setting. Millions of

high school graduates compete for limited seats in colleges every year, among which

only around ten percent are seats in elite, Tier I universities. Provinces switched

from the IA mechanism to the Chinese parallel mechanism, a mechanism in between

the IA and DA mechanism, during the past two decades. Although theoretical and

experimental studies have shown that the parallel mechanism is more stable and less

manipulable than the IA mechanism(Chen and Kesten, 2017, 2018), few empirical

studies exist to provide a nation-wide and dynamic evaluation on the causal effects of

the matching mechanism reform on matching quality. This paper strives to do so by

constructing a comprehensive panel data on the college admission outcomes of almost

all Chinese Tier I universities over a time span of twelve years. The geographical and

temporal variations in the Chinese college admissions reform allows for a generalized

1For example, New York City high schools and the Boston Public School system have reformedtheir matching mechanisms (Abdulkadiroglu et al., 2005, 2009).

2To our knowledge, only a few studies empirically documented theoretical predictions usingempirical data in the US context(Abdulkadiroglu et al., 2017a,b) and in the Chinese context(Chenet al., 2015). Others may have studied the effects of mechanisms changes, but not in a way thatare connected to theoretical predictions per se.

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difference-in-differences design, as well as an event study design, to estimate the

policy effects on average and over time.

To compare the stability of matching outcomes across mechanisms, we construct

three outcome measures, a score range, a rank range, and a match index, for each

admission cell between 2005 and 2016 for over one hundred Tier I universities in

China. The admission cell is at the university-track-province-year level, and students

compete for seats in admission cells within each matching game at the track-province-

year level.3 The score range is the difference in test scores between the highest and

the lowest student admitted to each cell; the rank range is that difference in student

rankings based on test scores; the match index is defined as (rank range+1)/quota

in each cell. These three variables measure the level of overlap in the matching

outcomes across admission cells and proxy for matching instability.

Unique features of the Chinese college admissions setting allow us to connect

these three outcome measures closely to a commonly used theoretical measure for

stability, the number of blocking pairs.4 In China, colleges share homogeneous strict

preferences for students and students share similar preferences for colleges. In a

simplified theoretical setting, where colleges and students share homogeneous strict

preferences for each other, it is easy to see that any overlapping in the rank ranges of

admitted students across two schools would create justified envy. Thus, the unique

stable matching outcome assigns students to schools based on their ranking orders in

a perfectly stratified manner, with no overlapping in the rank ranges across schools.

Although there is no one-to-one correspondence between our outcome measures and

matching instability, we use several pieces of corroborating evidence to show that

they are reasonable proxies when realistic assumptions are imposed. In particular,

we analyze correlations between measures using millions of simulated matching game

outcomes, as well as repeat our analysis in a more restricted empirical setting where

the homogeneous strict preferences are more cleanly satisfied.

Using a generalized difference-in-differences design, we find that switching from

3There are STEM and Non-STEM track in college admissions.4When a student i would rather be assigned to a different school s than his own assignment,

and such school s would also rather admit this student i than some student j it admitted, then wehave a blocking pair.

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the IA Mechanism to the Chinese parallel mechanism significantly improved match

stability, manifested by more precise stratification in matching outcomes in this

setting. In particular, we find narrower score and rank ranges, and smaller match

indices among the admitted students after the reform. The effect is stronger for

STEM track students and admission cells with more available seats. Consistent with

the theoretical prediction of Chen and Kesten (2017), we also find that the effect

is stronger for provinces that adopted the parallel mechanism with more parallel

choices within the first choice band. The effect size increases in a non-linear fashion

with decreasing returns as the choice bandwidth widens. In addition, we use an

event study design to analyze whether and how the policy effect changes over time.

We show that the effects are doubled eight to ten years after implementation. These

results remain consistent in a series of robustness checks, such as excluding early

reformers, adding or excluding certain control variables, and conducting the analysis

with various weights or at different aggregated levels.

This paper is related to the large and growing school choice and college admis-

sions literature. There are numerous empirical studies on the effect of attending

a sought-after school on student outcomes,5 as well as a large body of theoretical

studies that characterize school choice reforms in various cities and countries.6 The

empirical evidence on the effect of the mechanisms adopted in school choice and col-

lege admissions remains scarce, perhaps due to data limitation and a lack of suitable

natural experiment.

Causal estimates from this study enable us to compare causal empirical estimates

with corresponding effect sizes found in prior experimental studies in the school choice

and college admissions setting. Although experimental studies allow better control

over important theoretical parameters and observations of true preferences and have

provided valuable insights in understanding matching mechanisms, how we generalize

these findings outside of the laboratories requires careful thinking (Levitt and List,

5See Cullen et al. (2006) in Chicago; Abdulkadiroglu et al. (2011a) in Boston; Dobbie andFryer Jr (2011) in Harlem; Zhang (2016) in China; Deming et al. (2014) in Charlotte for someexamples.

6See Table 1 in Fack et al. (forthcoming) for a list of over twenty papers studying school choice orcollege admissions mechanisms in different countries, the majority of which are theoretical studies.

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2007). First, most experimental studies on school choice are much smaller in the

number of participants and the number of schools per match.7 In addition to the

market size difference, people may also make different choices in an experimental

environment compared to a life-changing situation. This study provides a useful

comparison between the experimental results and the real scenario. The effect sizes

in our study are 30% of the mean for score range and rank range, and 54% of the

mean for match index. In the experimental study, Chen and Kesten (2018) find that

the parallel mechanism reduces the number of blocking pairs by 50% compared with

IA in the four-school environment, and by around 18% in the six-school environment.

Our estimates from a real world environment with hundreds of schools and hundreds

of thousands of students are similar to that found in the four-school environment and

slightly larger than that in the six-school environment.8 The fact that they are in

reasonably comparable sizes provides some external validity support for experimental

findings in the school choice and college admissions settings.

We also contribute to a growing body of literature on Chinese college admissions,

arguably the largest existing matching mechanism in the world. Several studies

theoretically characterize the Chinese college admissions mechanisms (Nie, 2007; Zhu,

2014; Chen and Kesten, 2017). In particular, Chen and Kesten (2017) show that

the parallel mechanism is more stable9 than the IA mechanism and that stability

improves as the number of parallel choices within a choice band increases.10 Similar

7Prior experimental studies on school choice and college matching typically have 3 to 36 par-ticipants per match. Even though the theoretical literature has suggested that the IA mechanismis not strategy proof in both small and large market, unlike some other mechanisms that performbetter in large markets (Azevedo and Budish, 2017; Kojima and Pathak, 2009), it is still importantto evaluate mechanisms in scenarios with market sizes that are closer to real life settings. Chen etal. (2018) recently overcome the small market size in laboratory experiments by introducing robotsas participants in the matching games.

8Although we do not directly observe the number of blocking pairs, we show with simulationsthat our outcome measures align closely with the variable.

9More stable is defined as following: for any matching problem, if the IA mechanism is stable,then the parallel mechanism is also stable; and that the converse is not always true, i.e. there is atleast one problem at which the parallel mechanism is stable but the IA mechanism is not.

10Chen and Kesten (2017) also show strategy proofness improves in the same direction as stabilityacross these mechanisms. However, this paper does not speak to the manipulability property ofmechanisms.

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to overall school choice mechanism literature, empirical research on how Chinese

college admissions reform affects matching quality is scarce. Wu and Zhong (2014)

use survey data from one top university to study how the reforms affect ex-ante

fairness.11 Other than a few qualitative studies12, Li et al. (2010) use individual-

level college admissions data from three provinces between 2004 and 2008 to study

how college admissions reform affected the probability of a student being accepted

to their top choice university. Their outcome variable may not be the best choice,

because one of the merits of the parallel mechanism is that it allows students to list

their true preferences as their top choice, while the IA mechanism induces students

to list a “safe” choice as their top choice university (Chen and Kesten, 2017). Chen

et al. (2015) use data on student choice listings and admission outcomes from a

county in Sichuan province in 2008 and 2009, and show that students list more

colleges in their rank-ordered list and those colleges are also more prestigious under

the parallel mechanism. Bo et al. (forthcoming) construct a measure for college-

student mismatch following Dillon and Smith (2018) and study the effect of the

college admissions reform on the mismatch using individual data between 2005 and

2011. Kang and Ha (2016) analyze the effects of the reform on score range at the

college-track-province-year level from 2005 to 2011 and find that the score range

decreased after the parallel reforms. However, score ranges are affected by variations

in examinations at the province by year level, which is the same level as the reforms.

We improve on these prior studies on several fronts. Rather than relying on data

from a specific university or county, or even several provinces, our data include all

provinces with over one hundred “211 Project” universities, allowing for a broader

understanding of the impact of the mechanism reform at the national level. Second,

11In addition to Wu and Zhong (2014), Zhong et al. (2004); Li et al. (2016); Lien et al. (2017)also theoretically or experimentally investigate the differences across the timing of college choice,especially concerning ex-ante and ex-post fairness. Li et al. (2016) conduct an experimental studysimulating Chinese college choice process and find that the Boston Mechanism with incompleteinformation performs better in ex-ante fairness than others when the score uncertainty is larger.

12Shen et al. (2008) compare the number of students who declined their assigned colleges inShanghai during the year of 2007 and 2008 and found that the number decreased in 2008, whichis the first year Shanghai adopted the parallel mechanism. Hou et al. (2009) use survey andinterview data in 2008 and present some qualitative evidence on improved satisfaction with the newmechanism.

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we analyze new outcome variables, the match index and the rank range, which are

shown to proxy for match instability under reasonable assumptions in the Chinese

college admissions context. Third, we not only look at the average effect of the

parallel reform but also examine the theoretical prediction with regard to the parallel

choice bandwidth. In particular, we study whether provinces with more parallel

choices in the first choice band experienced stronger improvement after switching

from the IA mechanism to the parallel mechanism. Finally, our data spans more

than ten years post-reform, which allows us to evaluate the effect of mechanism

reforms over a longer period.

The remaining of this paper is organized as follows. Section 2 introduces some

basic theoretical background of the college admissions and the institutional back-

ground in China. Section 3 discusses the data and the outcome measures, and how

these measures serve as reasonable proxies for instability under certain assumptions.

Section 4 presents the empirical strategies, and Section 5 reports the results. Section

6 discusses a variety of robustness checks, and Section 7 concludes.

2 Background

In this section, we first offer a brief theoretical background to this paper by describ-

ing three key matching mechanisms and relevant theoretical findings, which draws

heavily from Chen and Kesten (2017). Then we discuss the institutional background

of the Chinese college admissions system.

2.1 A Brief Theoretical Background

A college admissions problem includes a number of students, denoted by I =

i1, i2, ...in, ...iN , to be assigned to one of the colleges, S = s1, s2, ...sm, ...sM ∪∅, where

J ≥ 2 and ∅ denotes a student’s outside option, or the null school. Each college has

a limited number of available seats, or quota qm. Each student has strict preferences

over all colleges, and each college has strict preferences over all students. We denote

the preference order for student i by �i and the preference order for college s by �s.

For example, if student i prefers college A than college B, then A �i B.

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A matching m : I → S is a list of assignments that assign each student to a

school and allows no school to admit more students than its quota. Denote M the

set of all matchings m. A matching is non-wasteful if no student prefers a college

with any unfilled quota to his own assignment. The preference order of student i for

college s is violated at a given matching f : I → S if i would rather be assigned to

school s, where some student j who has lower priority than i, is assigned. In other

words, student i justifiably envies student j for school s. In this case, student i and

school s are a blocking pair. A matching is stable if it is non-wasteful and justifiable

envy-free.

Three mechanisms are of central relevance to this paper, the Immediate Accep-

tance (IA) mechanism, the Deferred Acceptance (DA) mechanism, and the parallel

mechanism. Next, we will describe how each mechanism operates. First, in the IA

Mechanism,

Step 1. For each school s, consider only those students who have listed it as their

first choice. Up to qs students among them with the highest s preference orders

are assigned to school s.

Step k, k ≥ 2. Consider the remaining students. For each school s with qs

available seats, consider only those students who have listed it as their kth

choice. The qs students among them with the highest preference orders by

school s are assigned to school s.

The algorithm continues until all students are assigned to a school. The IA

mechanism has been criticized by its lack of strategy-proofness, and students have

incentives to strategically misreport their preferences.

Second, in the student-optimal Deferred Acceptance (DA) Mechanism (Gale and

Shapley, 1962),

Step 1. For each school s, up to qs applicants who listed school s as their top choice

and have the highest preference orders by school s are tentatively assigned to

school s. The remaining applicants are rejected.

Step k, k ≥ 2. Each student rejected from a school at step k − 1 is considered

at the next school listed. For each school s, up to qs students who have the

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highest preference orders by school s among both the new applicants and those

tentatively on hold from the earlier step are tentatively assigned to school s.

The remaining applicants are rejected.

The algorithm continues until all students are assigned to a school. The DA

mechanism provides incentive for students to state their true preferences and is sta-

ble (Gale and Shapley, 1962). Because of its nice theoretical properties, it has been

widely adopted in the school choice and college admissions mechanism reforms, in-

cluding a dozen of cities and countries.13

Lastly, in the Chinese parallel mechanism, students select several “parallel” col-

leges within each choice band. Within each choice band, it operates like DA; across

choice bands, it operates like IA. Chinese parallel mechanism is somewhere between

IA and DA: In IA, every choice is final; in DA, every choice is temporary until all

seats are filled. We will describe a parallel mechanism with two parallel choices in

each choice band to illustrate how it operates.

Step 1. For each school s, up to qs applicants who listed school s as their top choice

and have the highest preference orders by school s are tentatively assigned to

school s. The remaining applicants are rejected.

Step 2. Each student rejected from a school at step 1 is considered at the second

choice school. For each school s, up to qs students who have the highest pref-

erence orders by school s among both the new applicants and those tentatively

on hold from step 1 are assigned to school s. The remaining applicants are

rejected. All assignments are final after all students’ two parallel choices in the

first choice band are considered.

Step 3. Step 1 and Step 2 are repeated using the two parallel choices in the second

choice band for students who have not yet been assigned to any school. Seats

assigned during the previous choice band consideration are no longer available.

The algorithm continues until all students are assigned to a school.

Chen and Kesten (2017) parametrized these three types of mechanisms into a

family of application-rejection mechanisms, φe, e ∈ {1, 2, ...,∞}. When e = 1, φe is

13See Table 1 in Fack et al. (forthcoming) for the list of places adopting the DA mechanism,including Boston, NYC, Paris, Turkey, Finland, Ghana, Hungary, and Spain.

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equivalent to the IA (Boston) mechanism. When 2 ≤ e <∞, φe is equivalent to the

Chinese parallel mechanism. When e = ∞, φe is equivalent to the DA mechanism.

Among other findings, Chen and Kesten (2017) show that first, the Chinese parallel

mechanism is less manipulable and more stable than the IA mechanism;14 second,

as the parallel choice band increases by an integer multiplier of k, it becomes less

manipulable and more stable.15 These results provide the theoretical foundation for

the two main testable hypotheses in our empirical analysis.

2.2 College Admissions in China

In China, high school students usually decide between two tracks at the end of 10th

grade: STEM or Non-STEM.16 Both tracks contain Chinese, math, and English

classes, but their College Entrance Exams (CEE) and the college admissions process

will be track-specific. STEM track students continue to study physics, chemistry, and

biology in 11th and 12th grade and will be tested on these three subjects in addition

to Chinese, math, and English; for the Non-STEM track, students continue to take

history, politics, and geography classes and will be tested on these three subjects

in addition to Chinese, math, and English. The admission process is track-specific

not only because the exams are different between tracks, resulting in incomparable

test scores, but also because some universities only admit STEM-track students into

certain STEM majors.

Students take the CEE, administered annually over two to three days in early

June. They then submit their preference listing including several college choices

14More stable is defined as for any matching problem, if the IA mechanism is stable, then parallelmechanism is also stable; and that the converse is not always true, i.e. there is at least one problemat which the parallel mechanism is stable but the IA mechanism is not. Similarly for the definitionof less manipulable.

15Chen and Kesten (2017) also hypothesize a more general result where the manipulability andstability changes monotonically with the parallel choice band and provide simulation analysis tosupport it.

16A small percentage of students who specialize in arts or athletics tracks go through a differentcollege admissions process than the STEM (Li Ke) and Non-STEM (Wen Ke) tracks. We do notconsider these cases in our analysis.

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within each tier17 to the college placement offices centralized at the province level.18

The timing of the college choice submission varies across provinces and years. There

are three types: ex-ante, ex-interim, and ex-post submission, which correspond to

students submitting college choices before the exam takes place, submitting choices

after the exam but before the students are notified of their scores, and submitting

choices after students are notified of their scores, respectively. Prior to the reforms,

almost all provinces used ex-ante submission, whereby students have to make their

choices based on their expected performance in the exams, instead of the actual test

scores. When a student’s test score turns out to be significantly below expectation

and they have chosen highly competitive colleges beforehand, the student may end

up not getting admitted into any of the Tier I universities.

There has been a gradual policy shift from ex-ante to ex-interim and ex-post pref-

erence submission since the 1990s as shown in Figure 1. The right-hand panel plots

the number of provinces adopting ex-ante, ex-interim and ex-post preference submis-

sion in each year from 2005 to 2016. Approximately half of all provinces had already

adopted the ex-post preference submission before 2005. For the remaining half, the

number of provinces opting for ex-post preference submission steadily increased after

2005. All such provinces have since switched to ex-post preference submission.

Finally, the provincial placement offices make centralized matching assignments

based on student CEE scores, submitted choices, and the admission quotas of each

17Generaly speaking, there are four tiers of Chinese universities in descending order of prestigeand admission sequence: Tier I National Key Universities, Tier II other National Universities andProvincial Universities, Tier III Local Universities, and Tier IV Technical and Vocational Colleges.We only focus on Tier I universities in this paper.

18In addition to college choice, students have to list several major choices for each college. Thecollege-level matching is first-stage; a student has to meet the lowest admission score cutoff to beconsidered in any major by a college. Within students who meet the score cutoff and are in the poolfor potential admission by a college, the admission scores differ across majors. If all majors listedare popular and have high cutoff scores, this student will be assigned to any major with remainingseats, if she indicated her willingness to be assigned a random major in case all her major choicesfail. However, if she indicated that she would rather get rejected by the university than beingassigned to an unlisted major, then she will be rejected by this college even though her score is highenough to meet the cutoff. Students are advised to say yes to the major flexibility option. Fromanecdotal evidence from Wu and Zhong (2014), very few students were rejected by a top universityin China because they said no to major flexibility. Our data is at the college level and does notpermit us to conduct major-level analysis.

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university. Before 2003, all provinces used the IA (Boston) mechanism. Under the

IA mechanism, each school in the first round only considers students who rank them

as the top choice. Colleges admit students immediately up to their quota. Students

who are rejected in the first round will have their scores submitted to their second

choice. The procedure continues until all seats in Tier I universities are filled. In

fact, Tier I universities were almost always oversubscribed and filled up after the first

choice is considered, rendering students’ second choice and further choices redundant.

Therefore, if a student aims too high in their first choice in Tier I, they may end

up not being admitted to any Tier I university and fall straight to Tier II, even if

their test score is high enough to have entered one of the Tier I universities. On

the other hand, if a student overreacts to the risk, they may significantly undershoot

and regret not choosing a better university later on. The prevalence of this issue was

such that the Ministry of Education posted advice on how to strategize in the IA

mechanism on its college admissions website.19

In recent years, many provinces have switched from the IA mechanism to the

parallel mechanism, where students choose several parallel schools within choice-

bands. Under the parallel mechanism, families do not have to agonize as much over

whether to take a risk or to play it safe with the first choice on the listing. Students

can list their true preference as the first choice and still have insurance in listing a

“safe” school as the last parallel choice. As the number of parallel choices in the

first choice band increases to three or more, students can not only state their true

preference as the top parallel choice and list a safe choice as the last parallel choice,

but can also attempt to apply to more feasible universities in between.

Because of the positive feedback received by the provinces first adopted the par-

allel mechanism, the number of provinces adopting the IA or parallel mechanism

increased over time from 2005 to 2016. The left panel in Figure 1 plots the timing of

this trend. With the exception of three early-adopting provinces (Hunan, Jiangsu,

and Zhejiang), the majority of provinces switched from the IA to the parallel mech-

anism after the endorsement by the Ministry of Education in 2008, which creates

19See https://gaokao.chsi.com.cn/gkxx/zytb/201106/20110616/214533410-2.html, re-trieved on 10/31/2018.

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a steep increase as seen in Figure 1 and provides us with a plausibly exogenous

policy change.20 21 provinces adopted the parallel mechanisms within four years

of the endorsement. In September 2014, the State Council further emphasized its

recommendation towards ex-post preference submission and parallel mechanism in a

landmark document entitled “Guidance on the Deepening of the Reform of Admis-

sion Policies of Educational Institutions.”21 As of 2018, Inner Mongolia is the only

province that has not yet adopted the parallel mechanism.22 We focus on mechanism

reform in this paper, although we do include controls for the changes in the timing

of submission in all our analyses.

3 Data and Proxies for Match Quality

3.1 Data

We obtain college matching outcome data at the university-track-province-year level

for over one hundred “211 Project” universities between 2005 and 2016. There are

116 “211 Project” universities, handpicked by the Chinese government in 1995 and

given more resources in order to be transformed into world-class universities by the

early 21st century (Yu et al., 2012). They constitute the majority of Tier I universities

in China. We focus on the “211 Project” universities for three reasons. First, these

universities admit students from most provinces in significant numbers, which allows

us to exploit the difference in reforms across provinces. Second, the main argument

for admission reforms has been high-scoring students failing to be admitted into

any good university, and “211 Project” universities are the target schools for these

students. Last but not least, an assumption for our outcome variables to better

proxy for matching stability is that students have homogenous preferences towards

20See http://old.moe.gov.cn/publicfiles/business/htmlfiles/moe/s3258/201001/xxgk_

79906.html, retrieved on 10/06/2018.21See http://www.moe.edu.cn/publicfiles/business/htmlfiles/moe/moe_1778/201409/

174543.html, retrieved on 10/06/2018.22Inner Mongolia switched from IA mechanism to a dynamic adjustment mechanism that dif-

fers from the typical parallel mechanism. Students can observe the dynamics of matching onlineand change their choices after the initial submission of choice listings in this dynamic adjustmentmechanism.

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universities (see more details in the next subsection). This assumption is most likely

to be satisfied in the prestigious sector in the Chinese higher education system.

The data include aggregate admission information for each college in a specific

track (STEM vs. Non-STEM) in a given province during a given year. The variables

include the highest, average, and lowest scores23 and ranks of the admitted students,

as well as the admission quota, for each college by track-province-year. For exam-

ple, an observation in our data describes the admission outcomes for STEM-track

students admitted to Hunan University in STEM-track in Liaoning Province dur-

ing 2010. There were 90 students admitted to this admission cell. Among these 90

students, the highest CEE score was 613, which ranked number 1,806 in Liaoning

province within the STEM-track in 2010; the lowest CEE score was 558 with a rank

of 11,195.

We exclude three military academies because they are subject to an independent

admission procedure.24 Due to missing data, we also exclude the Central Conserva-

tory of Music and Central South University. Our final sample, therefore, consists of

111 universities. Among the “211 Project” universities in our sample, 38 universities

belong to a more elite group, called “985 Project” universities. These universities

are higher ranked and carry a much stronger signaling value to employers (Wang

2014). We use the membership of a school to the “985 Project” university group as

a unanimously agreed-upon prestige proxy in parts of our analysis.

3.2 Outcome variables

3.2.1 A Simplified College Admissions Problem

Consider a college admissions problem including a number of students, denoted by

I = i1, i2, ...in, ...iN , ordered by their test scores from highest to lowest, to be assigned

to one of the colleges, S = s1, s2, ...sm, ...sM ∪ ∅, ordered by their rankings from

highest to lowest. J ≥ 2 and ∅ denotes a student’s outside option, or the null school.

23Since CEE in most provinces has a maximum score of 750, we scale CEE scores proportionatelyfor provinces with maximum exam scores different from 750.

24They are the National University of Defense Technology, the Second Military Medical Universityand the Fourth Military Medical University.

14

Each college has a limited number of available seats, or quota qm. In addition, the

total number of available seats is smaller than the number of students competing

for college admissions:∑M

m=1 qm < N . Each student has strict preferences over

all colleges based on their rankings, and each college has strict preferences over all

students based on their test scores. We denote the preference order for student i by

�i and the preference order for college s by �s. For example, if student i prefers

college A to college B, then A �i B; if college s prefers student a to student b, then

a �s b.

Assumption 1. Homogeneous strict preferences by schoolsIf n < n′, in �s in′ for all s ∈ S.

This assumption states that there exists a strict preference order: student i1

is preferred than i2, i2 is preferred than i3, and so on, by all schools. Colleges

share homogeneous strict preferences over students. This assumption includes two

elements. First is the homogeneity in preferences across colleges: if one college prefers

student A over student B, then all other colleges also prefers student A over student

B. This is almost universally true in the Chinese college admissions process because

all colleges rank students by the centrally organized College Entrance Examination

scores. One exception is “zizhu zhaosheng” policy, or college-specific bonus policy,

where a small number of students can earn a few bonus points (usually 10 to 20 out

of 750 points) that would apply when they are considered by a specific college. In

these cases, students who have bonus points may end up with a higher rank when

being considered by that specific college than their rank when considered by other

colleges. However, this policy is normally restricted by the Ministry of Education

to be less than 5% of the admission quota.25 The second element in Assumption 1

is the strictness in preferences. It is easy to see the strict preferences over students

with different CEE scores. When there are ties in CEE scores, they are broken by

25See http://www.moe.gov.cn/srcsite/A15/moe_776/s3110/201111/t20111115_127339.

html, retrieved on 10/31/2018.

15

the examination scores in individual subjects in a pre-determined order.26

Assumption 2. Homogeneous strict preferences by studentsIf m < m′, sm �i sm′ for all i ∈ I.

This assumption states that there exists a strict preference order: s1 is preferred

than s2, s2 is preferred than s3, and so on, by all students. Two obvious challenges

are faced by this assumption. First, even though ranking or prestige is an important

factor in student preferences towards colleges, colleges have other dimensions that

students may care about, such as proximity to home, employment prospects, and

campus amenities. The fact that the college admissions are not centralized at the

country level but at the province level helps alleviate some of the concerns over

this first challenge. This is because heterogeneity in geographic proximity and local

reputation is minimized within provinces. In addition, we control for province fixed

effects, college fixed effects, and province and university two-way fixed effects in our

analysis. This way, we are comparing the matching outcomes within province-college

pairs. The second challenge to this assumption is that, despite a general consensus on

which colleges are better than others, students may have different beliefs on college

quality, especially among a few similarly ranked schools. To test this in a cleaner

environment as a robustness check, we estimate the policy effect in a setting where

the student-side homogeneous preference assumption is more strictly satisfied. We

collapse our unit of observation from schools into two groups: “985” universities

and Non-“985” universities, so that across the group there exists a unanimously

agreed-upon strict preference. This is further discussed in Section 6.

As discussed above, we argue that these two assumptions overall suit the charac-

teristics of the Chinese college admissions problem reasonably well. Under these two

assumptions, the unique stable matching outcome should be one that is “perfectly

stratified.” The lowest ranked student assigned to a college should have a higher pref-

erence order than the highest ranked student assigned to an inferior college. More

26For STEM track, students with the same CEE score will be ranked based on their math scores;if tied in math again, then they will be ranked by science scores; then by Chinese; lastly, by English.The sequence of tie-breaking procedure for non-STEM track is Chinese, social sciences, math, andEnglish.

16

formally, the stable matching is described below.27 Denote in → sm if student in

is assigned to sm in a matching. In the unique stable matching, student in will be

assigned based on its preference order n within the matching game:

1 ≤ n ≤ N1, in → s1

N1 + 1 ≤ n ≤ N1 +N2, in → s2

...M−1∑m=1

qm ≤ n ≤M∑

m=1

qm, in → sJ

n >

M∑m=1

qm, in → ∅

Imagine a line with all students ordered from high-scoring to low-scoring

(i1, i2, ..., iN), this line would be neatly cut into M + 1 groups (M schools plus ∅),resulting in the unique stable matching. There should be zero overlap in the admit-

ted student rank coverages (a bracket drawn between the highest student admitted

and lowest student admitted) between any two colleges in the stable matching. This

is because any overlap in the student rank coverages by two colleges implies the ex-

istance of justified envy. For example, say among the top school’s admitted student,

i1 is the highest and i8 is the lowest ranked student. If any other school admits a

student in, where n < 8, i.e. ranked higher than 8, then in justifiably envy i8 because

all students prefer the top school to any other school, and the top school prefers the

higher-scoring student in to i8.

3.2.2 Constructing the Instability Measures

With this “perfectly stratified” stable matching outcome in mind, we use the fol-

lowing outcome variables to proxy for match instability. We first construct the

score range, i.e. the difference between the highest College Entrance Exam (CEE)

score and the lowest among the students admitted within each admission cell at the

university-track-province level. When the matching outcome is perfectly stratified,

27See proof in Appendix B.

17

the score ranges would be the smallest for both high- and low-rank colleges; when

high-scoring students end up being mismatched to low-ranked colleges or vice versa,

the score ranges in those colleges would be larger. One concern in using the score

range is that the CEE varies across years within a province. If the difficulty of the

exams somehow systematically declined contemporaneously with the parallel reform,

resulting in compression in the score distributions, that in turn could generate smaller

score ranges on average.

To alleviate this concern, we introduce a second outcome variable, the rank range,

i.e. the difference between the highest and the lowest rank within each admission

cell. Since admission cells vary in quota for a variety of reasons, such as provincial

population size and college size, it may be hard to compare rank ranges across obser-

vations. The third outcome variable is a standardized version of the rank range, the

match index, defined by (highest rank admitted - lowest rank admitted +1)/quota.

This variable would equal one for all admission cells in the stable matching outcome,

where students are perfectly stratified by test scores into colleges.

Ceteris paribus, we should observe narrower score ranges, narrower rank ranges,

and smaller match indices when matching stability improves.

3.2.3 An Illustrative Example

To give a more concrete example, consider the following simple case of two universities

U1, U2 and four students S1, S2, S3, S4. The test scores of the four students are 700,

500, 400, and 200, respectively. Both universities prefer students with higher test

scores. U1 is preferred to U2 by all students. Each university has two available seats.

Table 1 lists all six possible matchings, among which the unique stable match is

Matching 1, where S1 and S2 are matched to U1, and S3 and S4 are matched to U2.

In this case, U1’s match index is calculated by the highest rank admitted minus the

lowest rank admitted plus one, divided by its quota, i.e. (2-1+1)/2=1. All other

matchings have justifiable-envy case(s) and are unstable.

The unique stable match Matching 1 has the smallest match indices, rank ranges

and score ranges for both colleges. Now consider Matching 6, where S3 and S4 are

matched to U1, and S1 and S2 are matched to U2. Although it is obviously an

18

unstable matching, the match index is 1 for both colleges, and rank and score ranges

are also identical to the stable matching. This occurs when two student clusters are

mismatched, and the quotas of both colleges are the same. Therefore, it is important

that better schools admit better quality students on average, that is, there is “no

admission reversal.”

Fortunately, these “admission reversal” matchings almost never occur in reality.

We compare the mean quality of admitted students (score and rank average) across

high and low ranked colleges. It is only in 0.79% of observations where a higher-

ranked college, based on the Alumni College Ranking, admitted a group of students

with a lower average score or average rank than the lower-ranked college.28 When we

categorize these schools into two groups (“985 Project” universities and other “211

Project” universities) and compare their mean student quality across groups, we find

zero incidences of a higher-ranked group of schools admitting lower quality students

on average, supporting our “no admission reversal” assumption.

3.2.4 Other Corroborating Evidence to Support Outcome Measures

The data only allow us to observe three moments (maximum, minimum, and mean)

of a given matching outcome. Therefore, we are unable to map our outcome measures

directly to stability in a closed analytical form. However, we provide the following

two pieces of evidence to support the suitability of our outcomes as a proxy for

stability.

First, we run simulations of matching outcomes without imposing any preference

assumptions. After eliminating the matching outcomes that violate the “no admis-

sion reversal” assumption, we compare how closely match index and rank range align

with the number of blocking pairs, wherein a student justifiably envies another stu-

dent. Figure 2 includes a set of graphs charting the relationship between the match

index and the number of blocking pairs. Since match index and rank range are lin-

ear transformations for each other in these cases, we do not present the duplicated

graphs for the relationship between rank range and the number of blocking pairs.

28The Alumni College Ranking is an annual college ranking list in China, similar to the U.S.News College Rankings.

19

Each graph is labeled by the number of schools times the quota in each of these

schools. For the purpose of concise demonstration, each school has the same quota

and we only present the 3 by 3, 4 by 4, and 5 by 5 cases here.29 The match index

closely aligns with the number of blocking pairs in all cases, suggesting that the

match index can serve as a proxy for stability.

Second, our main outcome measures utilize the maximum and the minimum

values of scores and ranks but do not incorporate the mean values. In a separate

analysis, we also test whether the mean moment confirms the hypothesis that the

parallel mechanism leads to more precisely stratified matching outcomes. We test

whether more prestigeous colleges would admit students with higher average rankings

after the parallel reform, and vice versa. More details are discussed in Section 6,

alongside other robustness checks.

3.3 Explanatory variables

The matching mechanism a particular province adopts in a given year is a key ex-

planatory variable of interest. We collect and code this information from official

documents published on provincial educational bureau websites.30 Table 2 summa-

rizes the timing of policy reforms for each province. Since our data starts in 2005, we

use “before 2005” to denote a policy reform that occurred prior to 2005. “-” indicates

that a province never adopted this type of submission timing or matching mecha-

nism. It is worth mentioning that some province-years used hybrid mechanisms in

place where they experimented with the parallel mechanism in Tier II admissions

but kept the IA mechanism in Tier I admissions. Since the focus of our analysis is

211 universities in Tier I, we only consider the mechanism adopted by the Tier I

admissions in those hybrid cases.31

29The total numbers of possible matching outcomes in the 4 by 4 and 5 by 5 settings are toolarge, so we randomly keep around half a million matchings for these two cases.

30In a few cases where such documents are not readily available from official sources, we consultwebsites including gaokao.chsi.com.cn, www.huaue.com, www.eol.cn, and other news outlets.

31According to Li et al. (2010), at least nine other provinces have used similar hybrid admissionmechanisms at some point including Beijing, Chongqing, Gansu, Henan, Hubei, Jilin, Shaanxi,Shandong, Sichuan, and Xinjiang.

20

We also collect information on the number of parallel choices in the first parallel

choice band for all provinces that adopted the parallel mechanism. Some provinces

adopted the parallel mechanism initially by only allowing parallel choices in the

second choice band and still restricting one choice in the first choice band in Tier I

admission. In these scenarios, the parallel nature of the second choices barely affect

the matching outcomes among the Tier I universities, as their seats are almost always

filled after the only choice in the first choice band is considered. Therefore, we focus

our attention on the number of parallel choices in the first choice band in Tier I

admission. Figure 3 shows the frequencies of admission cells that adopted different

numbers of parallel choices in the first choice band. Earlier in the reform process,

the number of parallel choices ranged from two to six. By 2016, provinces had

increased this number to four to ten, with five and six being modal and representing

the practice of two-thirds of the provinces.

3.4 Summary Statistics

Table 3 presents summary statistics of our data at the university-track-province-year

level. Altogether, we have 62,396 admission cells with score-related variables in our

dataset. The mean and standard deviation of the score range across university-track-

province-year admission cells are 35.44 and 26.4. Due to data availability, we observe

fewer observations for rank-related variables.32 After excluding extreme values where

the rank range is greater than 20,000, which is less than five percent in the long upper

tail, we are left with 51,670 observations, which we use to construct the rank range

and the match index.33 The mean and standard deviation of the rank range are 4,261

and 4,509. The mean of the match index is 181.1, with a standard deviation of 304.2.

In particular, the mean under the IA mechanism is 211.15, much higher than that

under the parallel mechanism, 125.75. Note that the mean of post-reform match

indices is still far from the theoretical stable value of one, which could mean two

32The rank-related variables are missing for all cells in the year of 2012, which account for themajority of the missing data.

33If we keep the extreme values, results on match index are still robust at one percent significancelevel. Estimates of rank range are in similar magnitudes but statistically insignificant, with largerstandard errors due to the extreme values.

21

things. It could be that the parallel mechanism still does not rule out all blocking

pairs. Alternatively, the homogeneous strict preferences assumptions could be too

strict and the post-reform matchings could be stable, despite having some remaining

overlaps in rank ranges. Most likely, it is a combination of both reasons.

On average, each admission cell has a quota of 67.8 seats, with a large standard

deviation. This is due to the varying size of universities and graduating high school

cohorts between provinces, as well as the fact that some universities admit more

students from their home province. We control for the quota in all our regressions

and explore heterogeneous effects along this dimension. 35.2% of the admission cells

are with “985 Project” universities, and 44% of the observations are in the Non-

STEM track.

Table 4 summarizes the data for the STEM and Non-STEM tracks. STEM track

admission cells have 97 slots on average, much larger than the average of 30 slots

among Non-STEM track cells. The means of the highest score admitted are 592.9

for the Non-STEM track and 613 for the STEM track. Score ranges of STEM track

admission cells are also around 20 points greater than that of Non-STEM tracks.

Rank ranges of STEM track cells are also significantly larger: 6,422 versus 1,622.

The third outcome measure, match index, is also larger for STEM track cells (189.4)

than for Non-STEM track cells (133.7). In Section 5, we will present results showing

that the effect size for the STEM track is also larger, potentially because there is

more scope to improve the existing outcome measures.

4 Empirical Strategies

Our empirical analysis exploits the natural experiment in policy reforms across

provinces and years to investigate whether the match quality improves from the

IA to parallel mechanism. Recall two theoretical results by Chen and Kesten (2017),

that the parallel mechanism is more stable than the IA mechanism; and that as

the number of parallel choices increases by an integer multiplier of k, the parallel

mechanism becomes more stable. We will test the following two hypotheses:

Hypothesis 1.

22

Parallel reform results in narrower score ranges, narrower rank ranges, and lowermatch indices.

Hypothesis 2.

Provinces that adopted the parallel mechanism with more parallel choices experiencestronger effects.

4.1 Difference-in-Differences Model

First, we consider the following generalized difference-in-differences with fixed effects

model:

Yijst = β0 + β1Paralleljt + β2T1jt + β3T

2jt + β4Quotaijst

+φi + ηj + πs + σt + αij + γit + εijst (1)

where the dependent variable Yijst is the outcome variable, either score range,

rank range, or match index of college i in province j for track s in year t. T 1jt and T 2

jt

are two indicators for ex-interim and ex-post preference submission timing adopted

in province j in year t. The ex-ante preference submission is the omitted comparison

group. They control for the effects the preference submission timing reform may

have on matching outcomes. We should expect that the ex-post preference submis-

sion have a positive impact on the matching stability, which translates to negative

coefficients for β2 and β3. Quotaijst is the total number of available seats allocated

by college i to province j in year t for track s. Admission cells with larger quotas

are expected to have the wider the score ranges and rank ranges.

The model also includes a set of college, province, track, and year fixed effects.

The college fixed effects φi capture colleges’ unobserved time-invariant predisposition

to attract a more homogenous or heterogeneous group of applicants. The province

fixed effects ηj account for the cross-province differences in the distribution of out-

come variable. For example, if the rank range or score range in Beijing has always

been narrower or wider than in Shanghai, the province fixed effects would control for

it. The track fixed effect πs controls for the difference between STEM and Non-STEM

track that is constant across colleges, provinces, and years. The year fixed effects

23

σt account for the variations of the outcome variable across all provinces over time.

For example, as the number of test-takers of CEE continues to rise in China, the

academic preparedness of prospective applicants may become more heterogeneous

over time.

In a given province, students may have stronger preferences for a particular college

due to geographic proximity or local legacies. To control for that, we include two-way

fixed effects between college and province αij. When a college expands and admits

more students in some years than others, it may lead to changes in the outcomes

as well. The two-way fixed effects between college and year γit can account for this

local preference. Lastly, εijst is clustered at the provincial level.

Paralleljt is an indicator for adopting the parallel mechanism in province j in

year t , the main explanatory variable of interest. The omitted comparison group

is the IA mechanism. If Hypothesis 1 is true, that the parallel mechanism improves

the match stability, we would expect the coefficients of Paralleljt to be negative and

significant.

To test Hypothesis 2, we replace the parallel dummy in equation (1) with a

continuous variable for the number of parallel choices:

Yijst = β0 + β1No.Paralleljt + β2T1jt + β3T

2jt + β4Quotaijst

+φi + ηj + πs + σt + αij + γit + εijst (2)

If β1 is statistically significant and negative, it suggests that the more parallel

choices there are, the stronger policy effects the parallel mechanism brings.

4.2 Event Study Model

While it is useful to know whether the reforms improve match quality, we are also

interested in when the effects take place and whether they persist. An event-study

specification (Jacobson et al., 1993) formalizes this analysis by mapping the effect

on the CEE score range to a set of dummy variables indicating the number of years

before or after the parallel mechanism reform to replace the single dummy variable in

the fixed effects model. As illustrated in Equation (2), Dmjt is a dummy variable set

24

to 1 if province j experienced the reform in year t−m and 0 otherwise. If province

j shifted from the IA to parallel mechanism m years before year t , m is positive.

Alternatively, if m is negative, province j switched from the IA to parallel mechanism

m years later. The omitted period is m = 0, i.e. the last year province j was under

the IA admission mechanism. For those provinces who had not switched to parallel

mechanism, Dmjt=0 for all m. The other control variables in the Equation (2) are the

same as in Equation (1).

Yijst = β0 ++14∑

m=−9

φmDmjt + +β2T

1jt + β3T

2jt + β4quotaijst

+φi + ηj + πs + σt + αij + γit + εijst (3)

The identifying assumption of common time trend is similar to the difference-in-

differences model, and it offers a systematic check within the model. If φm is not

statistically significant for all m < 0, we can confirm that the time trend prior to the

mechanism change was the same across the treatment and control groups.

5 Empirical Results

5.1 Average Effects and Some Heterogeneity Results

Table 5, Table 6, and Table 7 present the generalized difference-in-differences results

with fixed effects specified by Equation (1). First, column 1 in Table 5 shows that the

parallel reform led to a 10.55 decrease in the score range among admitted students for

a university in a province within a given track and a given year, which is equivalent

to 30 percent of the mean and 40 percent of the standard deviation. We also find

that relative to the base group of ex-ante submission, both the ex-interim submission

and ex-post submission have a substantially smaller score range. However, they are

less precisely estimated as the standard errors are five times larger than those of

the parallel reform dummy. The ex-post submission has a statistically significant

effect of 20.14 points, which is around three-quarters of a standard deviation of

the score range. The coefficient for ex-interim submission is half the size compared

25

to the coefficient for ex-post and is statistically insignificant. The results seem to

suggest that making choices after taking the exams without knowing scores does not

significantly help with the sorting process since students may not recall their answers

or may not accurately estimate how their answers to subjective questions, such as in

essays, would be graded. On the other hand, when students know their test scores,

they better understand their relative place in the distribution, which results in a

much more precise sorting pattern and overall smaller score ranges. However, we take

these results on submission timing, especially ex-interim, with some caution, given

that our data (2005-2016) captures no within-province variations in the ex-interim

reform. Provinces either never adopted ex-interim submission before jumping into

ex-post submission, or adopted ex-interim submission before the start of our sample.

Therefore, we rely on cross-province variations when estimating the effect of the

ex-interim submission. As for the ex-post reform, our sample time range also only

covers 15 provinces, less than half of all provinces, limiting our ability to obtain a

more complete picture.

As for other control variables, the score range is larger for admission cells with

larger quotas. A one standard deviation change in the quota translates into a 3.94

point or 1/7 increase in score range. This is rather intuitive because when more

students are taken from the same distribution, the score range is bound to be larger.

Non-STEM tracks have much smaller score ranges than STEM tracks by a difference

of 18 points, consistent with the summary statistics in Table 4.

As mentioned earlier, we cannot control for the interaction terms of province and

year in Equation (1) as these are perfectly collinear with our key policy variable of

interest. Therefore, score ranges may become smaller if the difficulty of the CEE

in a province in a certain year declines at the same time as the reform from IA to

parallel mechanism, resulting in compression in the score distributions. To address

this concern, we present results using rank range as the dependent variable. Column

1 of Table 6 shows that the parallel reform resulted in a decrease in rank range by

around 1,266, which is around 0.30 of the mean or 0.28 standard deviations. A one

standard deviation change in quota translates into a 946 or 0.21 standard deviations

increase in rank range. Non-STEM tracks once again have much smaller rank ranges.

26

Whether or not a student submits their preferences after they take the exam or after

they know their scores still negatively correlates with the rank range but is not

statistically significant.

While the measure of rank range is better than score range in comparing test

results within a province across years, it can still be difficult to compare rank ranges

across universities, some of which have larger quotas and therefore larger absolute

changes in terms of rank range. Column 1 in Table 7 shows that the parallel reform

led to an 89.2 decrease in the match index, which is equivalent to around half of

the mean or one-third standard deviations. Here, ex-interim and ex-post reforms

did not significantly improve the match index, although their point estimates are

substantial and in the expected negative direction. Consistent with Table 5 and

Table 6, Non-STEM tracks have a smaller match index. Lastly, unlike score range and

rank range, which are positively correlated with quota, match index is constructed

to be comparable across admission cells with different quota sizes.

Heterogeneous Effects across STEM, Eliteness, Local, and Admission Cell Size

Column 2 through 4 in Table 5, Table 6, and Table 7 present some explorations

on heterogeneous effects by interacting the parallel mechanism reform dummy with

several admission cell characteristics, Parallel#Xjt, including the interaction terms

with the track type, the prestige of university, and admission quota.

Yijst = β0 + β1Paralleljt + β2Parallel#Xijst + β3T1jt + β4T

2jt

+β5Quotaijst + φi + ηj + πs + σt + αij + γit + εijst (4)

Column 2 shows that STEM tracks experienced a stronger policy effect than

Non-STEM tracks. For STEM track admission cells, the effect size is captured by

the coefficient of parallel, i.e. admission range narrowed by 14.36 points in score

or 1,842 in rank and the match index decreased by 106.40. For Non-STEM tracks,

the effect is the sum of the coefficient of parallel and the interaction term, i.e. the

admission range narrowed by only 5.57 points in score or 593 in rank and the match

index only decreased by 69.16.

27

Column 3 shows the results of interacting parallel with an indicator for “985

Project” university, which is a subset of more prestigious universities among “211

Project” universities. We find that the more prestigious universities experienced a

weaker effect on the match index compared to the overall effect for the “211 Project”

universities. This heterogeneity is statistically insignificant when it comes to score

range and rank range. Finally, column 4 presents results for the interaction between

parallel and the admission quota. It suggests that the parallel mechanism has a

stronger impact on the score range and rank range for admission cells with more

students admitted.

5.2 Heterogeneity across Parallel Mechanisms with Differ-ent Numbers of Parallel Choices

One important theoretical result from Chen and Kesten (2017) is that as the number

of parallel choices of the parallel mechanism increases, it becomes more stable. Em-

pirically, provinces that adopted the parallel mechanism with more parallel choices in

the first choice band should experience stronger effects in Tier I admission stability

after the parallel reform. We go through the government documents and collect the

number of parallel choices between 2005 and 2016 to test whether provinces that

adopted the parallel mechanism with more parallel choices experience stronger ef-

fects. Table 8 presents these results. For each additional parallel choice, the score

range decreases by 1.81 (1/15 standard deviations), the rank range decreases by

249.65 (1/18 standard deviations), and the match index decreases by 17.56 (1/15

standard deviations). All three measures give us policy effects in similar magnitude

in terms of standard deviations.

To further understand the potential non-linear relationship between the number

of parallel choices and stability improvement, we substitute the continuous measure

for the parallel choice bandwidth with a set of dummy variables, one for provinces

and years that adopted two to three parallel choices in the first choice band, and

three more for those with four, five, and six-to-ten choices in the first choice band.

These four categories are divided based on the frequency of adoption to allow for

28

enough observations in each category (see Figure 3 for a distribution of the number

of parallel choices). Results are shown in Table 9. We can see a general pattern of

increasing effects as the number of parallel choices go up. However, the effect does

not grow in a linear fashion. The effect size for six to ten parallel choices is not

triple the size of that for two to three parallel choices, as a linear effect would have

predicted. Most of the matching quality improvement is harvested when moving from

one to two or three parallel choices, as the coefficient in the first dummy variable

“2 to 3 parallel choices” is 70.52, around four times as large as the coefficient in the

linear model.

5.3 Effects Over Time: Event Study Results

Figure 4, Figure 6, and Figure 8 present the event study results from Equation 3

on three outcomes: score range, rank range, and match index. The dots represent

the point estimates, and the extension lines show their corresponding 95 percent

confidence intervals. First, we observe no pre-trends prior to the parallel mechanism

reform in any of the figures since all the 95 percent confidence intervals include zero.

As shown in Figure 4, the effect size for score range is around 12 points (1/2 standard

deviations) in the first five years. The score range continues to narrow by around

15 points in year 6 through 8, and further narrow by around 25 points in year 12

to 14, which is almost an entire standard deviation of the score range (standard

deviation = 26.4). Figure 6 presents the event study results for rank range. The

parallel reform narrowed the rank range by around 2,000 (1/2 standard deviations)

in the first seven years and further narrowed by around 4,000-6,000 in year 8 through

14, or 0.9 to 1.3 standard deviations. Lastly, Figure 8 presents the effects on match

index. Relative to the IA mechanism, the parallel mechanism decreased the match

index by around 100 (1/3 standard deviations) in the first eight years and further

decreased by around 200 (2/3 standard deviations) in year 8 through 12.

In summary, the match quality improved over the years, and the estimated effect

is visibly larger for provinces and years that have switched to the parallel mechanism

for at least nine years. Although the effect sizes for score range and rank rage are

29

slightly greater than those for the match index, the relative increases from the short

run to the long run are similar. These effects approximately doubled after 8-10 years

of policy implementation.

Event Study Results by track and by college ranking

To explore whether the event study results differ by track, we run the analysis

on the sample of Non-STEM and STEM track students separately. The left panels

in Figure 5 present the comparisons between the two tracks for the outcome variable

of score range. The effect sizes are smaller for the Non-STEM track sample. We

observe similar patterns for the other two outcome variables, the rank range and the

match index in Figure 7 and Figure 9. The right panels in Figure 5 further restrict

the sample to “985 Project” universities. For score range, the event study results

do not differ when college eliteness is compared using the “985 Project” universities

as a proxy for eliteness, which is consistent with the difference-in-differences results.

However, when we look at the rank range and match index outcome variables, the

comparisons suggest that the “985 Project” Universities experience a smaller im-

provement. Overall, the effect sizes increase over time in all event study results.

6 Robustness Checks

Testing Results in a Setting with Homogeneous Student Preference Over Schools

One assumption we made to derive the perfectly stratified stable outcome is that

students share homogeneous strict preferences over universities. In reality, students

may disagree on whether they prefer the 12th ranked school or the 14th ranked

school. To test the results in a setting where the homogeneous student preferences

are more clearly satisfied, we collapse our unit of observation from individual colleges

into two groups, “985 Project” Universities versus the rest of the “211 Project”

Universities, so that across these two groups there exists a unanimously agreed-upon

strict preference. We calculate the highest and lowest ranks and scores among the

admitted students within each group at the track-province-year level. Although

the number of observations shrinks significantly in this robustness check, Table 10

30

shows that the results in this homogeneous preference setting are consistent with our

main results. The coefficients themselves may seem significantly smaller compared

to the main results, but when interpreted in terms of standard deviations, they are

in comparable magnitudes with the main results.

Using average rank to crosscheck the results

If the parallel reform indeed led to more stratification, we would also expect

an increasing gap between student quality of top colleges and that of lower ranked

colleges. To test this empirically, we regress the average rank among the admitted

students of each college with an interaction term between the parallel reform and

“985 Project” university dummy that serves as a proxy for school prestige, along

with various other control variables as specified below.

Yijst = β0 + β1Paralleljt + β2Parallel#985Univjt + β3T1jt + β4T

2jt

+β5Quotaijst + φi + ηj + πs + σt + αij + γit + εijst (5)

We would expect β2, the coefficient of the interaction term, to be negative, as the

985 universities would admit students with higher rank (lower in numerical value)

after the parallel reform. Table 11 presents the stratification results. The coefficients

for the interaction terms are negative and significant, consistent with our hypothesis.

In addition to the average rank admitted by each school, we also run the regressions

on the highest and the lowest admitted student’s rank, and the results are consistent.

The parallel reform allows elite colleges to be matched with students of even higher

average rank than before, increasing the ability sorting in college admission.

Excluding early reformers from the sample

It is possible that the reform is endogenous to the policy response and early re-

formers drove most of the results. If there is endogenous policy adoption, we may

observe some key characteristics showing pre-policy trends. As shown in Figure A1,

we find no observable pre-trend or post-trend in the size of their high school graduat-

ing class or college enrollment. We also conduct the robustness checks for three key

outcome measures using a subsample that excludes the early reformers and present

31

the event study results in Figure 10. Results are consistent with our main analysis,

rejecting the hypothesis that early reformers were the main drivers of the result.

Another aspect worth noting is that we rely on the early reformers to obtain

the long-term effect estimates. If the early reformers benefit more than the later

reformers, it could confound the increasing trend in the effect size that we observe

and change our interpretation for the event study result. Assuming that the early

reformers have an overall higher average effect that is constant over time, we expect

to see a decline in the coefficients when excluding them. However, the coefficients

are consistently similar to the main results for all three outcome measures.

Is the Event Study Result Driven by Increasing Numbers of Parallel Choices?

One possible explanation for the event study result is that some provinces in-

creased the number of parallel choices over time. As we show in Subsection 5.2,

increases in parallel choices could result in stronger effects. To test whether the

event study result is entirely driven by the provinces that increased the number of

parallel choices after the initial parallel reform, we exclude those provinces and re-

peat the regression specified in equation (2). Results are similar to the main results,

as shown in Figure A2.

Controlling for dosage effect of the ex-post submission reform

Since provinces that switched to the ex-post submissions earlier also tend to

be those who switched to the parallel mechanism earlier, the event study design

may capture the dosage effect of the ex-post submission. To separate the potential

cumulative effect of submission timing reform, we add a control variable for the years

following ex-post reform and rerun the analysis. Results are robust and are presented

in Panel A of Table A2 in the Appendix.

Controlling for partial parallel reform

Some provinces adopted the parallel element in the second choice of the Tier I

college admissions process or in the Tier II college admissions process before they

adopted it for the first choice of the Tier I college admissions process. Since essentially

all Tier I colleges fill up after considering students’ first choices, the partial parallel

32

mechanism that only allows students to list several parallel colleges as their second

choice does not change the incentive to manipulate their true preference in their first

choice and has limited impact on the matching quality. Nevertheless, to account for

the potential effect of this partial parallel element before the full implementation, we

include a dummy variable in all regressions for provinces and years when the partial

parallel mechanism was used. Results are robust and are presented in Panel B of

Table A2 in the Appendix.

Excluding Beijing and Shanghai

Beijing and Shanghai are the two regions with the highest concentration of elite

universities and job opportunities. Since universities normally allocate dispropor-

tionally larger quotas to the local region, students in Beijing and Shanghai have a

better chance of being admitted to these elite universities. We exclude these two

cities from our analysis and find similar results, as shown in Panel C of Table A2 in

the Appendix.

Analysis at the Matching Game Level

Our data and main analysis have mostly been at the school-track-province-year

level, while the matching game is at the track-province-year level. We collapse the

analysis at the game level, by calculating the averages of score range, rank range,

and match index as the outcome variables, and dropping the tens of thousands

of interaction fixed effects between school and province, and between school and

year. Not surprisingly, results are consistent with our main result (see Table A3 and

Table A4).

Analysis with Importance Weight

In our main analysis and in the game-level result, we treat each school’s matching

outcome as equally important. However, if we assign them an importance weight by

the eliteness of the school, as measured by the inverse of the university’s ranking, or

by the number of available seats, the results may be different. Table A5 shows the

robustness check results using importance weights by school prestige and by school

quota, which are consistent with our main results.

33

7 Conclusions

Matching mechanisms play a crucial role in the college admissions process, which

in turn influence education and labor market outcomes (Hoekstra, 2009; Li et al.,

2012; Dillon and Smith, 2018). Although there exists a large theoretical literature

on school choice and college admissions mechanism design, the empirical evidence

remains scarce. The Chinese college admissions reform provides a unique empirical

setting to study how switching away from the IA mechanism affect match quality. We

compile a novel dataset of over one hundred “211 Project” universities between 2005

and 2016 and the timing of parallel reform and numbers of parallel choices and test

theoretical predictions by Chen and Kesten (2017). We show that switching from the

IA mechanism to the parallel mechanism significantly improves the match stability

and that more parallel choices have stronger effects. The parallel reform reduced the

dispersion of incoming student quality by 0.28 to 0.40 standard deviations, or 0.29 to

0.55 of the mean values, depending on the outcome used. Our estimates from a real

world environment with hundreds of schools and hundreds of thousands of students

are reasonably close to the experimental findings by Chen and Kesten (2018). These

results are robust to alternative sample cuts, different specifications, and the level at

which we measure the match quality.

Are ex-post stability and a more stratified matching outcome desirable for so-

ciety? In terms of its effect on student outcomes, prior literature on tracking has

mixed conclusions on whether tracking generates net benefits (Betts, 2011; Garlick,

2018). This question is certainly important but is beyond the scope of our study.

In terms of its impact on the higher education system, the increased concentration

of highly talented students in Tier I universities in China could help facilitate the

Chinese government’s efforts to enhance its top universities, since a crucial element

of world-class universities is a critical mass of top students and faculty (Salmi, 2009).

One caveat of our paper is that our outcome measures do not perfectly correspond

to theoretical properties, such as the number of blocking pairs or the number of stu-

dents with justifiable envy. We provide simulation results and show our outcome

measures align closely with those variables. Although the assumption of homoge-

34

neous strict preferences over colleges by students is somewhat stringent in the overall

sample with over one hundred colleges, results using a sample collapsed to two tiers

of schools, where this assumption is more clearly satisfied, still provide the same

story. We also show another piece of corroborating evidence for stratification, that

the average quality of students admitted by schools that are more prestigious became

even higher after the reform.

Another limitation is that the outcome measures constructed based on test scores

do not guarantee ex-ante matching efficiency. If there are large idiosyncratic factors

that result in student test scores deviating from true abilities, then it may not result

in the best outcome. For example, Cai et al. (forthcoming) show that the gender

gap in the College Entrance Exams is 0.15 standard deviations larger than mock

exams, suggesting that females perform worse under pressure and are less likely

to be admitted to better universities since the admissions are entirely based on

these isolated test scores. As pointed out by Wu and Zhong (2014) and Lien et

al. (2017), only high performers who consistently do well are willing to take the risk

of submitting top colleges as their first choices under ex-ante preference submission

and the IA mechanism. Individual-level data on student ability, college application

choices, and matching assignments would allow for a better understanding of student

strategy and ex-ante and ex-post efficiency.

Acknowledgment

We would like to thank Yan Chen, Sue Dynarski, Isla Globus-Harris, Brian Jacob,

Zhuan Pei, Yun Wang, Binzhen Wu, Lily Yang, and seminar participants at Colgate

University and at the University of Michigan for their helpful feedback. Wei Ha

would like to acknowledge financial support by the National Science Foundation of

China (Grant Number: 71673013) and the Chinese Ministry of Education (Grant

Number: 14JZDW004).

35

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39

Figure 1: College Admissions Reform Over Time

Notes: The left panel shows the number of provinces that used the Boston or IAmechanism versus the parallel mechanism between 2005 and 2016. The right panel showsthe number of provinces that adopted choice submission after knowing test scores(Ex-post), choice submission after the exam without knowing test scores (Ex-interim),and choice submission before the exam (Ex-ante) across years.

40

Figure 2: Relationship Between Match Index and # of Blocking Pairs

Notes: These graphs present the relationship between match index and the number ofblocking pairs in simulated matching outcomes. The left graph portrays the case of 3schools, each with a quota of 3. The middle and right graphs portray the case of 4schools, each with 4 seats, and the case of 5 schools, each with 5 seats. For the 3*3 case,we generate all possible matchings; for the 4*4 and 5*5 cases, because the total numbersof matchings are too large (around 5.8e+7 and 6.3e+14), we randomly generate aroundhalf a million (580,664 and 625,891) matchings. We take these matchings and delete thematchings that violate the “no admission reversal” criteria, calculate. For the remainingmatchings, we calculate the sum of match indices and the number of blocking pairs andgenerate the scatter plots. Darker dots indicate more observations. The red lines are thelinear fitted lines.

41

Figure 3: Number of Parallel Choices in the First Choice Band

Notes: This figure plots the frequencies of admission cells that adopted different numbersof parallel choices in the first choice band. Most provinces adopted the parallelmechanism with four to six choices in the first choice band.

42

Figure 4: Event Study Results: Effects of Switching from the Immediate Acceptanceto the Chinese Parallel Mechanism on Score Range

Notes: This figure presents the event study coefficients of Equation 3, controlling foryears treated by ex-post preference submission reform. The outcome is the admissionscore range (highest score admitted - lowest score admitted) at theuniversity-track-province-year level. The dots show the point estimates and theextensions show their corresponding 95 percent confidence intervals. It shows that thereis no pre-trend prior to the parallel reform and the score range significantly narrows afterthe reform.

43

Figure 5: Event Study Results on Score Range: Subsample Analysis

Notes: The top (bottom) figures present the event study coefficients of Equation 3 usinga subsample of STEM (Non-STEM) track students, controlling for years treated byex-post preference submission reform. The right graphs further restrict the sample to“985 Project” universities, a group of 38 higher ranked universities. The outcome is theadmission score range (highest score admitted - lowest score admitted) at theuniversity-track-province-year level. The dots show the point estimates and theextensions show the corresponding 95 percent confidence intervals.

44

Figure 6: Event Study Results: Effects of Switching from the Immediate Acceptanceto the Chinese Parallel Mechanism on Rank Range

Notes: This figure presents the event study coefficients of Equation 3, controlling foryears treated by ex-post preference submission reform. The outcome is the admissionrank range (highest rank admitted - lowest rank admitted) at theuniversity-track-province-year level. The dots show the point estimates and theextensions show their corresponding 95 percent confidence intervals. It shows that thereis no pre-trend prior to the parallel reform and the rank range significantly narrows afterthe reform.

45

Figure 7: Event Study Results on Rank Range: Subsample Analysis

Notes: The top (bottom) figures present the event study coefficients of Equation 3 usinga subsample of STEM (Non-STEM) track students, controlling for years treated byex-post preference submission reform. The right graphs further restrict the sample to“985 Project” universities, a group of 38 higher ranked universities. The outcome is theadmission rank range (highest rank admitted - lowest rank admitted) at theuniversity-track-province-year level. The dots show the point estimates and theextensions show the corresponding 95 percent confidence intervals.

46

Figure 8: Event Study Results: Effects of Switching from the Immediate Acceptanceto the Chinese Parallel Mechanism on Match Index

This figure presents the event study coefficients of Equation 3, controlling for yearstreated by ex-post preference submission reform. The outcome is the match index(highest rank admitted - lowest rank admitted)/quota at theuniversity-track-province-year level. The dots show the point estimates and theextensions show their corresponding 95 percent confidence intervals. It shows that thereis no pre-trend prior to the parallel reform and the rank range significantly narrows afterthe reform.

47

Figure 9: Event Study Results on Match Index: Subsample Analysis

Notes: The top (bottom) figures present the event study coefficients of Equation 3 usinga subsample of STEM (Non-STEM) track students, controlling for years treated byex-post preference submission reform. The right graphs further restrict the sample to“985 Project” universities, a group of 38 higher ranked universities. The outcome is thematch index (highest rank admitted - lowest rank admitted)/quota at theuniversity-track-province-year level. The dots show the point estimates and theextensions show the corresponding 95 percent confidence intervals.

48

Figure 10: Results Excluding Early Reformers

Notes: These figures present the event study coefficients of Equation 3 using a subsampleof provinces that reformed to parallel mechanism after 2008. The outcomes are rankrange, score range, and match index at the university-track-province-year level. The dotsshow the point estimates and the extensions show their corresponding 95 percentconfidence intervals. It shows that there is no pre-trend prior to the parallel reform andscore range significantly narrows after the reform.

49

Table 1: Illustrative Example

Matching 1 Stable Match Index Rank Range Score Range

U1 S1 score = 700 (2-1+1)/2=1 1 200S2 score = 500

U2 S3 score = 400 (4-3+1)/2=1 1 200S4 score = 200

Matching 2 Unstable Match Index Rank Range Score Range

U1 S1 score = 700 (3-1+1)/2=1.5 2 300S3 score = 400

U2 S2 score = 500 (4-2+1)/2=1.5 2 300S4 score = 200

Matching 3 Unstable Match Index Rank Range Score Range

U1 S1 score = 700 (4-1+1)/2=2 3 500S4 score = 200

U2 S2 score = 500 (3-2+1)/2=1 1 100S3 score = 400

Matching 4 Unstable Match Index Rank Range Score Range

U1 S2 score = 500 (3-2+1)/2=1 1 100S3 score = 400

U2 S1 score = 700 (4-1+1)/2=2 3 500S4 score = 200

Matching 5 Unstable Match Index Rank Range Score Range

U1 S2 score = 500 (4-2+1)/2=1.5 2 300S4 score = 200

U2 S1 score = 700 (3-1+1)/2=1.5 2 300S3 score = 400

Matching 6 Unstable Match Index Rank Range Score Range

U1 S3 score = 400 (4-3+1)/2=1 1 200S4 score = 200

U2 S1 score = 700 (2-1+1)/2=1 1 200S2 score = 500

50

Table 2: Timing of policy reform in preference submission schemes

Type of Reform Timing of Choice Matching Mechanism

Province Ex-interim Ex-post Partial parallel Parallel

Beijing - 2015 2008 2014Tianjin 2005 or earliera 2011 - 2010Hebei - 1999 2007 2009Shanxi 2005 or earlier 2012 2008 2012Inner Mongolia - 2002 - -Liaoning 2005 or earlier 2014 2003 or earlier 2008Jilin - 2008 2006 2009Heilongjiang 2005 or earlier 2013 - 2013Shanghai - 2017 - 2008Jiangsu - 2003 2004 or earlier 2005Zhejiang - 1999 - 2007Anhui 2005 or earlier 2007 2002 or earlier 2008Fujian - 2005 - 2009Jiangxi 2005 or earlier 2007 - 2009Shandong - 1998 - 2013Henan 2005 or earlier 2010 2002 or earlier 2010Hubei - 2004 - 2011Hunan - 2001 - 2003Guangdong - 2008 - 2010Guangxi - 2004 - 2009Hainan - 2002 - 2009Chongqing - 2006 2004 or earlier 2010Sichuan - 2005 2008 2009Guizhou 2005 or earlier 2008 2004 or earlier 2009Yunnan - 2004 2007 2009Tibet - 1996 - 2010Shaanxi 2005 or earlier 2010 - 2010Gansu 2005 or earlier 2008 2005 or earlier 2015Qinghai - 1996 - 2018Ningxia - 1999 - 2009Xinjiang 2005 or earlier 2015 2005 or earlier 2011

Notes: The table shows the year of submission timing reform or the year of match-ing mechanism reform. All provinces used to use ex-ante preference submissionand IA (Boston) mechanism . Many provinces phased into the ex-post and parallelmechanism, by adoping the ex-interim and partial parallel first. “-” indicates thatthis province never adopted this type of timing of choice or matching mechanism.

aSome provinces adopted certain submission timing or partial parallel mechanism prior to ouranalysis time frame (2005-2016), but we could not find the exact year of reform.

51

Table 3: Summary Statistics

(1) (2) (3) (4) (5)VARIABLES mean sd min max N

score range of admittedstudents

35.44 26.40 0 297 62,396

highest score admitted 604.2 40.47 394 750 62,462lowest score admitted 568.8 42.13 250 719.7 62,918average score admitted 564.2 91.41 0.224 730.9 62,155rank range of admittedstudents

4,261 4,509 0 19,996 51,670

highest rank admitted 2,307 3,647 1 51,253 51,670lowest rank admitted 6,568 6,811 1 62,952 51,670average rank admitted 4,233 5,282 0.333 57,229 51,441match index 164.1 264.8 0.000689 7,818 51,670quota 67.80 197.1 1.000 5,172 63,695year 2,011 3.457 2,004 2,017 63,779parallel 0.583 0.493 0 1 63,699# of parallel choices 2.808 2.552 0 10 63,699# of parallel choices (reform only) 4.814 1.228 2 10 37,174ex-interim 0.170 0.376 0 1 63,699ex-post 0.757 0.429 0 1 63,699start year of parallel 2,010 2.427 2,003 2,015 60,397start year of expost 2,006 5.372 1,996 2,017 63,699“985 Project” university 0.352 0.478 0 1 63,699non-STEM 0.438 0.496 0 1 63,699

Notes: The data are at the university-track-province-year level from 2005-2016.The matching game is at the track-province-year level.

52

Table 4: Subsample Summary Statistics by High School Track

Panel A. Summary Statistics for STEM Track

(1) (2) (3) (4) (5)VARIABLES mean sd min max N

score range of admitted students 44.14 27.78 0 297 35,109highest score admitted 613.0 42.19 394 750 35,144lowest score admitted 568.8 46.68 250 709 35,398average score admitted 564.0 101.1 0.224 714.1 35,032rank range of admitted students 6,422 4,878 0 19,996 28,206highest rank admitted 3,303 4,439 1 51,253 28,206lowest rank admitted 9,726 7,477 1 62,952 28,206average rank admitted 6,234 6,159 1 57,229 28,139match index 189.4 310.5 0.000689 7,818 28,206quota 97.37 239.2 1.000 5,172 35,806# of high school graduates (in10,000)

26.49 16.89 0.770 74.98 35,808

Panel B. Summary Statistics for Non-STEM Track

(6) (7) (8) (9) (10)VARIABLES mean sd min max N

score range of admitted students 24.24 19.46 0 223.3 27,287highest score admitted 592.9 35.07 427 750 27,318lowest score admitted 568.8 35.43 427 719.7 27,520average score admitted 564.4 77.17 1.359 730.9 27,123rank range of admitted students 1,662 1,948 0 19,866 23,464highest rank admitted 1,109 1,725 1 51,253 23,464lowest rank admitted 2,771 2,922 1 55,899 23,464average rank admitted 1,816 2,262 0.333 54,054 23,302match index 133.7 192.0 0.00323 5,551 23,464quota 29.83 112.6 1.000 4,672 27,889# of high school graduates (in10,000)

28.06 16.98 0.770 74.98 27,891

Notes: The data is at university-track-province-year level from 2005-2016.

53

Table 5: Diff-in-diff and Heterogeneous Effects of Parallel Reform on Score Range

(1) (2) (3) (4)score range

Parallel (0/1) -10.55*** -14.36*** -10.31*** -9.85***(1.79) (1.97) (1.77) (1.81)

Parallel * Non-STEM (0/1) 8.79***(1.51)

Parallel * 985 Univ (0/1) -0.68(0.88)

Parallel * Quota -0.01***(0.00)

Quota 0.02*** 0.02*** 0.02*** 0.02***(0.00) (0.00) (0.00) (0.00)

Non-STEM (0/1) -18.03*** -23.11*** -18.03*** -18.11***(1.21) (1.55) (1.21) (1.20)

Ex-interim (0/1) -10.97 -10.97 -10.96 -10.87(9.81) (9.86) (9.82) (9.66)

Ex-post (0/1) -20.14** -20.27** -20.14** -20.05**(9.70) (9.75) (9.70) (9.55)

Year FE Y Y Y YUniversity FE Y Y Y YProvince FE Y Y Y Y

University by Year FE Y Y Y YProvince by University FE Y Y Y Y

Observations 62,396 62,396 62,396 62,396R-squared 0.62 0.63 0.62 0.62

Notes: This figure presents the coefficients from OLS regressions based on equa-tion (1) and equation (4). The outcome is the admission score range (highest scoreadmitted - lowest score admitted) at the track-university-province-year level. Ro-bust standard errors clustered at the province level in parentheses. *** p<0.01, **p<0.05, * p<0.1.

54

Table 6: Diff-in-diff and Heterogeneous Effects of Parallel Reform on Rank Range

(1) (2) (3) (4)rank range

Parallel (0/1) -1,266.20*** -1,841.81*** -1,303.13*** -1,160.27***(369.89) (455.46) (355.36) (360.09)

Parallel * Non-STEM (0/1) 1,248.62***(372.32)

Parallel * 985 Univ (0/1) 103.37(225.91)

Parallel * Quota -1.82***(0.56)

Quota 4.80*** 4.93*** 4.80*** 6.00***(0.58) (0.60) (0.58) (0.85)

Non-STEM (0/1) -4,952.85*** -5,643.29*** -4,952.89*** -4,949.97***(277.02) (403.84) (277.02) (278.76)

Ex-interim (0/1) -484.40 -474.49 -485.14 -484.51(551.59) (568.59) (552.09) (548.58)

Ex-post (0/1) -816.51 -816.04 -816.90 -821.99(513.12) (529.55) (513.55) (510.14)

Year FE Y Y Y YUniversity FE Y Y Y YProvince FE Y Y Y Y

University by Year FE Y Y Y YProvince by University FE Y Y Y Y

Observations 51,670 51,670 51,670 51,670R-squared 0.66 0.66 0.66 0.66

Notes: This figure presents the coefficients from OLS regressions based on equation(1) and equation (4). The outcome is the rank range, defined by (highest rankadmitted - lowest rank admitted) at the track-university-province-year level. Ro-bust standard errors clustered at the province level in parentheses. *** p<0.01, **p<0.05, * p<0.1.

55

Table 7: Diff-in-diff and Heterogeneous Effects of Parallel Reform on Match Index

(1) (2) (3) (4)Match Index

Parallel (0/1) -89.23*** -106.40*** -104.72*** -96.98***(16.39) (17.77) (18.08) (17.16)

Parallel * Non-STEM (0/1) 37.24***(11.98)

Parallel * 985 Univ (0/1) 43.36***(9.55)

Parallel * Quota 0.13***(0.03)

Quota -0.21*** -0.21*** -0.21*** -0.30***(0.03) (0.03) (0.03) (0.04)

Non-STEM (0/1) -78.69*** -99.28*** -78.71*** -78.90***(8.04) (13.76) (8.04) (7.91)

Ex-interim (0/1) -39.32 -39.02 -39.63 -39.31(40.04) (40.65) (40.15) (40.29)

Ex-post (0/1) -42.24 -42.23 -42.41 -41.84(36.99) (37.51) (37.09) (37.15)

Year FE Y Y Y YUniversity FE Y Y Y YProvince FE Y Y Y Y

University by Year FE Y Y Y YProvince by University FE Y Y Y Y

Observations 51,670 51,670 51,670 51,670R-squared 0.47 0.47 0.47 0.47

Notes: This figure presents the coefficients from OLS regressions based on equa-tion (1) and equation (4). The outcome is the match index (highest rank admitted- lowest rank admitted +1)/quota at the track-university-province-year level. Ro-bust standard errors clustered at the province level in parentheses. *** p<0.01,** p<0.05, * p<0.1.

56

Table 8: Effects of Parallel Mechanisms with More Parallel Choices

(1) (2) (3)VARIABLES score range rank range match

# of parallel choices -1.81*** -257.20*** -17.57***(0.39) (66.27) (3.11)

Ex-interim -11.49 -503.72 -40.68(10.12) (564.10) (41.38)

Ex-post -20.80** -780.87 -40.67(10.14) (512.72) (37.87)

Quota 0.02*** 4.79*** -0.21***(0.00) (0.58) (0.03)

Non-STEM -18.05*** -4,954.28*** -78.77***(1.20) (276.88) (7.97)

University FE Y Y YProvince FE Y Y YYear FE Y Y YUniversityXYear FE Y Y YProvinceXUniversity FE Y Y YObservations 62,396 51,670 51,670R-squared 0.62 0.66 0.47

Notes: This table presents the coefficients from OLS regressionsbased on equation (2). Robust standard errors clustered at theprovince level in parentheses. *** p<0.01, ** p<0.05, * p<0.1.

57

Table 9: Parallel Mechanisms with Different Choice Bands: Non-linear Relationship

(1) (2) (3)VARIABLES match rank range score range

2 to 3 parallel choices -70.52*** -1,122.59* -10.85***(16.17) (555.59) (2.94)

4 parallel choices -97.30*** -1,183.46** -10.91***(27.90) (440.34) (2.29)

5 parallel choices -96.13*** -1,695.42*** -8.71***(13.49) (412.39) (2.19)

6 to 10 parallel choices -103.94*** -1,425.74*** -10.51***(20.99) (460.51) (2.44)

Ex-interim -41.10 -454.84 -10.94(40.77) (614.04) (9.33)

Ex-post -40.08 -699.58 -20.47**(37.55) (566.41) (9.32)

Quota -0.21*** 4.80*** 0.02***(0.03) (0.58) (0.00)

Non-STEM -78.78*** -4,953.22*** -18.03***(8.01) (277.19) (1.21)

University FE Y Y YProvince FE Y Y YYear FE Y Y YUniversityXYear FE Y Y YProvinceXUniversity FE Y Y Y

Observations 51,670 51,670 62,396R-squared 0.47 0.66 0.62Notes: This table presents the coefficients from OLS regressionsbased largely on equation (2), substituting the continuous variablewith four dummy variables to understand the non-linear effects asthe number of parallel choices increases. The effect size for six toten parallel choices is not triple the size of that for two to threeparallel choices, as a linear relationship between choice bandwidthand matching quality improvement would have predicted. Robuststandard errors clustered at the province level in parentheses. ***p<0.01, ** p<0.05, * p<0.1.

58

Table 10: Results Collapsing to 985 Versus 211 Universities

(1) (2) (3)VARIABLES score range rank range match

Parallel -6.557* -1,218** -0.570***(3.774) (564.2) (0.192)

985 University 25.95*** -303.3 1.332***(1.770) (728.2) (0.325)

Ex-interim -29.90 -477.0 0.167(21.83) (795.8) (0.353)

Ex-post -37.29* -1,433** -0.279(21.69) (558.2) (0.268)

Quota 0.00160** 2.561*** -0.000124**(0.000671) (0.144) (6.07e-05)

Non-STEM -35.54*** -3,655*** 0.753***(3.977) (579.1) (0.235)

Year FE Y Y YProvince FE Y Y YObservations 1,416 1,236 1,236R-squared 0.707 0.872 0.428

Notes: This table presents the coefficients from OLS regres-sions based on equation (1) using a sample that is collapsedto (985 vs other 211) * track * province* year level, as op-posed to the sample at the university-track-province-yearlevel used in the main results. This exercise provides a set-ting where students have unambiguously homogeneous pref-erence across the two groups of universities. Robust stan-dard errors clustered at the province level in parentheses.*** p<0.01, ** p<0.05, * p<0.1.

59

Table 11: Does Parallel Mechanism Result in More Stratified Matching?

(1) (2) (3)VARIABLES average rank highest rank lowest rank

Parallel * 985 Univ (0/1) -584.78** -518.37* -642.67***(241.22) (274.87) (203.74)

parallel (0/1) 523.95 -654.97 670.62***(384.07) (463.18) (234.91)

ex-interim (0/1) -975.21** -801.47 -319.00*(424.85) (549.72) (181.92)

ex-post (0/1) -850.51** -684.86 124.91*(319.81) (519.55) (71.85)

quota 1.19** 3.45*** -1.33***(0.48) (0.62) (0.20)

non-STEM (0/1) -4,798.18*** -7,417.15*** -2,457.42***(369.23) (449.40) (195.21)

Year FE Y Y YUniversity FE Y Y YProvince FE Y Y YUniversityXYear FE Y Y YProvinceXUniversity FE Y Y YObservations 50,680 50,909 50,909R-squared 0.73 0.75 0.70

Notes: This table reports coefficients from OLS regressions as specified inequation 5. Robust standard errors clustered at the province level. ***p<0.01, ** p<0.05, * p<0.1. Controls include timing reform, parallelreform, quota, and a battery of fixed effects specified in equation 2.

60

Appendix A: Supplementary Figures and Tables

61

Figure A1: Related Characteristics Trends

Notes: These graphs show pre- and post-reform trends of some key provincialcharacteristics, including standardized high school enrollment (enrollment/population)and standardized college enrollment (enrollment/population).

62

Figure A2: Robustness Check: Excluding Provinces with Increasing Numbers ofParallel Choices

Notes: The left panel shows the number of provinces that adopted the IA mechanism(Boston Mechanism) versus the parallel mechanism over time. The right panel shows thenumber of provinces that adopted choice after knowing test scores (Ex-post), choice afterthe exam without knowing test scores (Ex-interim), and choice before the exam(Ex-ante).

63

Table A1: Results Excluding Early Reformers

(1) (2) (3)VARIABLES match rank range score range

parallel -100.90*** -1,716.76*** -12.29***(15.75) (249.21) (1.67)

exinterim -38.47 -475.53 -10.95(40.12) (540.53) (9.87)

expost -43.88 -973.69* -20.52**(37.11) (500.43) (9.75)

quota -0.20*** 4.53*** 0.02***(0.03) (0.54) (0.00)

Observations 47,193 47,193 56,734R-squared 0.47 0.66 0.62

Notes: Robust standard errors clustered at the province level inparentheses. *** p<0.01, ** p<0.05, * p<0.1.

64

Table A2: Robustness checks

Control for years of ex-post reform instead of dummy variable

(1) (2) (3)VARIABLES match rank range score range

parallel -92.85*** -1,336.26*** -12.32***-16.2 -356.33 -1.86

Observations 51,670 51,670 62,396R-squared 0.47 0.66 0.61

Control for partial parallel

(4) (5) (6)VARIABLES match rank range score range

parallel -95.42*** -1,267.27*** -10.73***-21.5 -442.68 -1.76

Observations 51,670 51,670 62,396R-squared 0.47 0.66 0.62

Exclude Shanghai and Beijing

(7) (8) (9)VARIABLES match rank range score range

parallel -81.12*** -1,233.51*** -10.40***-14.19 -392.6 -1.94

Observations 48,806 48,806 58,894R-squared 0.48 0.65 0.61

Year FE Y Y YUniversity FE Y Y YProvince FE Y Y YUniversity by Year FE Y Y YProvince by University FE Y Y Y

Notes: This figure presents the coefficients from OLS regressionsbased on equation (1) with additional or alternative control vari-ables, or using a different subsample. Robust standard errors clus-tered at the province level in parentheses. *** p<0.01, ** p<0.05,* p<0.1.

65

Table A3: Game-level Results

(1) (2) (3) (4)VARIABLES game-level score range

parallel -16.48*** -9.11*** -10.04*** -9.61***(1.08) (1.44) (1.78) (1.30)

Observations 708 708 708 708R-squared 0.79 0.82 0.85 0.85

(5) (6) (7) (8)VARIABLES game-level rank range

parallel -1,768.11*** -901.49** -1,124.03*** -1,272.29***(314.36) (349.65) (379.30) (264.25)

Observations 618 618 618 618R-squared 0.81 0.82 0.82 0.85

(9) (10) (11) (12)VARIABLES game-level match

parallel -100.12*** -77.95*** -87.78*** -75.36***(17.03) (17.52) (16.99) (12.32)

Observations 618 618 618 618R-squared 0.73 0.74 0.75 0.81Year Trend N Y N NYear FE N N Y NProvince Specific Trend N N N Y

Robust standard errors clustered at the province level in parentheses.*** p<0.01, ** p<0.05, * p<0.1. This table presents OLS regressionresults with controls for ex-interim, ex-post reform, quota, and STEMtrack, as well as the trend and fixed effects specified in the correspondingcolumn.

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Table A4: Game-level Results: Increasing Parallel Choices

(1) (2) (3) (4)VARIABLES game-level score range

N parallel choices -3.07*** -1.72*** -1.82*** -1.53***(0.23) (0.30) (0.32) (0.48)

Observations 486 486 486 486R-squared 0.78 0.81 0.84 0.85

(5) (6) (7) (8)VARIABLES game-level rank range

N parallel choices -285.05*** -162.81** -188.83** -223.82***(79.74) (72.58) (75.07) (47.50)

Observations 419 419 419 419R-squared 0.81 0.82 0.82 0.86

(9) (10) (11) (12)VARIABLES game-level match

N parallel choices -15.26*** -14.36*** -14.09*** -14.81***(3.16) (3.55) (3.45) (3.80)

Observations 419 419 419 419R-squared 0.83 0.83 0.84 0.87Year Trend N Y N NYear FE N N Y NProvince Specific Trend N N N Y

Robust standard errors clustered at the province level in parentheses.*** p<0.01, ** p<0.05, * p<0.1. This table presents OLS regressionresults with controls for ex-interim, ex-post reform, quota, and STEMtrack, as well as the trend and fixed effects specified in the correspondingcolumn.

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Table A5: Robustness Check: Results with Importance Weight

Panel A. Importance weight = quota

(1) (2) (3)VARIABLES score range rank range match

parallel -10.09*** -1,507.03*** -37.30***(2.21) (508.31) (7.51)

Observations 62,396 51,670 51,670R-squared 0.8 0.77 0.6

Panel B. Importance weight = (1/university rank)

(1) (2) (3)VARIABLES score range rank range match

parallel -8.85*** -887.86** -49.43***(2.09) (345.79) (10.07)

Observations 61,454 50,909 50,909R-squared 0.58 0.59 0.49

Notes: Each coefficient comes from a separate OLS regression withcontrol variables including ex-interim, ex-post, quota, and non-STEM,as well as a battery of fixed effects and interaction terms specified inequation (1). The only change here is adding an importance weightspecified in the panel title. Robust standard errors clustered at theprovince level in parentheses. *** p<0.01, ** p<0.05, * p<0.1

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Appendix B: A Simple Theoretical Framework on

Match Index

Consider a college admissions problem including a number of students, denoted by

I = i1, i2, ...in, ...iN , ordered by their test scores from highest to lowest, to be assigned

to one of the colleges, S = s1, s2, ...sm, ...sM ∪ ∅, ordered by their rankings from

highest to lowest. J ≥ 2 and ∅ denotes a student’s outside option, or the null school.

Each college has a limited number of available seats, or quota qm. In addition, the

total number of available seats is smaller than the number of students competing

for college admissions:∑M

m=1 qm < N . Each student has strict preferences over

all colleges based on their rankings, and each college has strict preferences over all

students based on their test scores. We denote the preference order for student i by

�i and the preference order for college s by �s. For example, if student i prefers

college A than college B, then A �i B.

A matching m : I → S is a list of assignments that assign each student to a

school and allows no school to admit more students than its quota. Denote M the

set of all matchings m. A matching is non-wasteful if no student prefers a college

with any unfilled quota to his own assignment. The preference order of student i for

college s is violated at a given matching f : I → S if i would rather be assigned to

school s, where some student j who has lower priority than i, is assigned. In other

words, student i justifiably envies student j for school s. In this case, student i and

school s are a blocking pair. A matching is stable if it is non-wasteful and justifiable

envy-free.

Assumption 1. Homogeneous strict preferences by schoolsIf n < n′, in �s in′ for all s ∈ S.

This assumption states that there exists a strict preference order, student i1 is

preferred than i2, i2 is preferred than i3, and so on, by all schools.

Assumption 2. Homogeneous strict preferences by studentsIf m < m′, sm �i sm′ for all i ∈ I.

This assumption states that there exists a strict preference order, s1 is preferred

than s2, s2 is preferred than s3, and so on, by all students.

69

Denote in → sm if student in is assigned to sm in a matching.

Proposition 1. There is one and only one stable matching f : I → S in the collegeadmissions problem described above:

1 ≤ n ≤ N1, in → s1

N1 + 1 ≤ n ≤ N1 +N2, in → s2

...M−1∑m=1

qm ≤ n ≤M∑

m=1

qm, in → sJ

n >M∑

m=1

qm, in → ∅

Proof. i. Stability: First, it is easy to see that f is non-wasteful. Next, suppose thatstudent in1 ’s priority for school sm1 is violated. Then, there must be a student in2

whose priority is lower in sm1 but is matched to sm1 . Since all schools prefer studentswith higher test scores, n2 > n1. Meanwhile, in1 must have been assigned to sm2 ,where m2 > m1. However, in the matching outcome f described above, for any twoschools sm1 and sm2 , m2 > m1, students who are admitted to school sm1 have ahigher preference order, thus smaller rank order n than those who are admitted toschool sm2 . Since student n2 is admitted to school m1 and student n1 is admitted toschool m2, then there must be n2 < n1, which contradicts above.

ii. Uniqueness: Let there be an alternative stable matching f ′ : I → S. Supposein f ′, in where 1 ≤ n ≤ N1 is not assigned to s1. There are two possible alternativescenarios: First, no one else is assigned to s1. In this case, this matching is wasteful.Second, student in′ is assigned to s1. In this case, in justifiably envy in′ , since inprefers s1 to any other school, and s1 prefers in to in′ . Same reasoning applies for inassigned to the 2nd, 3rd, ... Mth school as specified in f , and we can always findthe alternative matching either wasteful or non-stable or both.

Denote rm the highest rank among admitted students by school m, and rm the

lowest rank among school m’s admitted students. A match index M for school s is

defined as Ms =rm−rm+1

qm.

Proposition 2. In a stable match, we should have Ms = 1 ∀ s ∈ S =s1, s2, ...sm, ...sM .

70

Proof. In the stable matching f : I → S described above, Ms = 1 ∀ s ∈ S =s1, s2, ...sm, ...sM . Since this is the unique stable matching, the above is true for anystable match.

71