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TRANSCRIPT
ESSAYS ON LABOR ECONOMICS
by
Paola Bordon
A dissertation submitted in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
(Economics)
at the
UNIVERSITY OF WISCONSIN-MADISON
2014
Date of final oral examination: 6/27/2014
The dissertation is approved by the following members of the Final Oral Committee:
Christopher Taber, Professor of Economics, Chair
John Kennan, Professor of Economics
Chao Fu, Assistant Professor of Economics
James Walker, Professor of Economics
Nicholas Hillman, Assistant Professor of Education
To my parents
ii
Contents
List of Tables v
List of Figures vii
Abstract viii
1 Estimating the Returns of Attending a Selective University on Earnings
using Regression Discontinuity with Multiple Admission Cutoffs 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Chilean Higher Education System . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Estimating the Effect of Attending a selective university on Earnings: A Re-
gression Discontinuity Approach . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Model and Identification Strategy . . . . . . . . . . . . . . . . . . . . 7
1.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.2 Reconstructing the Application and Admission Process . . . . . . . . 14
1.4.3 Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.4 Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Validity of Regression Discontinuity Design: Robustness Analysis . . . . . . 18
1.5.1 Manipulation of the Assignment Variables . . . . . . . . . . . . . . . 18
1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
iii
1.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.8.1 Additional Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 The Effects of Private High Schools, University Rankings and Employer
Learning on Wages in Chile 28
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Higher Education System in Chile . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Statistical Discrimination and Employer Learning . . . . . . . . . . . . . . . 31
2.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.2 Statistical Discrimination on the basis of High School and University
Ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 The Employer Learning Model . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Policy Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.7 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.8.1 Formal Derivation of the Employer Learning Model . . . . . . . . . . 48
3 College-Major Choice to College-Then-Major Choice (with Chao Fu) 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Background: Education in Chile . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.1 Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.2 Student Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.3 Sorting Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.1 Data Sources and Sample Selection . . . . . . . . . . . . . . . . . . . 64
3.4.2 Aggregation of Academic Programs . . . . . . . . . . . . . . . . . . . 66
iv
3.4.3 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.5 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5.1 Target Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6.1 Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6.2 Model Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.7 Counterfactual Policy Experiments . . . . . . . . . . . . . . . . . . . . . . . 73
3.7.1 Overall Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.7.2 A Closer Look . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.9 Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.10 Appendix I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.10.1 Detailed Functional Form and Distributional Assumptions . . . . . . 89
3.10.2 Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.10.3 Estimation and Equilibrium-Searching Algorithm . . . . . . . . . . . 91
3.10.4 Additional Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.11 Appendix II: For Online Publication . . . . . . . . . . . . . . . . . . . . . . 95
3.11.1 Illustration: Gender Differences . . . . . . . . . . . . . . . . . . . . . 95
3.11.2 Counterfactual Model Details: Sys.S . . . . . . . . . . . . . . . . . . 96
3.11.3 A Closer Look at Sys.S: Gainers and Losers . . . . . . . . . . . . . . 101
3.11.4 Other Examples of Sys.J . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.11.5 Proof of existence in a simplified (baseline) model. . . . . . . . . . . . 105
3.11.6 Model Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Bibliography 108
v
List of Tables
1.1 University Programs Applications Example . . . . . . . . . . . . . . . . . . . 8
1.2 Descriptive Statistics for Chilean Universities . . . . . . . . . . . . . . . . . . 21
1.3 Descriptive Statistics for Chilean College Programs . . . . . . . . . . . . . . 21
1.4 Distribution of Students by Major and University . . . . . . . . . . . . . . . 22
1.5 Regression Discontinuity Design Estimates . . . . . . . . . . . . . . . . . . . 22
1.6 Regression Discontinuity Design Estimates: First Stage . . . . . . . . . . . . 27
2.1 Descriptive Statistics, Futuro Laboral, 1995-2005 . . . . . . . . . . . . . . . . 44
2.2 Descriptive Statistics for University Ranking . . . . . . . . . . . . . . . . . . 44
2.3 The effects of Standardized PSU, High School and University Ranking on
Wages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Aggregation of Colleges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2 Summary Statistics By Tier (All Students) . . . . . . . . . . . . . . . . . . . 83
3.3 Summary Statistics By Major (Enrollees) . . . . . . . . . . . . . . . . . . . . 84
3.4 Human Capital Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.5 Consumption Value (Major-Specific Parameters) . . . . . . . . . . . . . . . . 84
3.6 Enrollment by Tier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.7 Enrollee Distribution Across Majors . . . . . . . . . . . . . . . . . . . . . . . 85
3.8 Ability & Retention (by Tier) . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.9 Ability & Retention (by Major) . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.10 Enrollment, Retention & Welfare: Sys.S . . . . . . . . . . . . . . . . . . . . 86
vi
3.11 Enrollment, Retention & Welfare: Sys.S v.s. Hybrid . . . . . . . . . . . . . . 86
3.12 Enrollment, Retention & Welfare: Hybrid . . . . . . . . . . . . . . . . . . . 86
3.13 Enrollment, Retention & Welfare: Sys.S v.s. Rematch . . . . . . . . . . . . . 86
3.14 Enrollment and Retention . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.15 Distribution Across Majors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.16 Log Starting Wage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.17 Score Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.18 College-Major-Specific Cutoff Index . . . . . . . . . . . . . . . . . . . . . . . 92
3.19 College-Major-Specific Annual Tuition (1,000 Pesos) . . . . . . . . . . . . . . 93
3.20 Outside Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.21 Consumption Value (Major-Independent Parameters) . . . . . . . . . . . . . 94
3.22 College Cost (Major-Independent Parameters) . . . . . . . . . . . . . . . . . 94
3.23 Other Parameters in Log Wage Functions . . . . . . . . . . . . . . . . . . . . 94
3.24 Female Enrollee Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.25 Different Treatments Across Majors . . . . . . . . . . . . . . . . . . . . . . . 101
3.26 Welfare Gain and Student Characteristics . . . . . . . . . . . . . . . . . . . 102
3.27 Enrollment (Low Income) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.28 Enrollee Distribution Across Majors (Low Income) . . . . . . . . . . . . . . 107
3.29 Mean Test Scores Among Outsiders . . . . . . . . . . . . . . . . . . . . . . . 107
vii
List of Figures
1.1 Time-line of the University Application Process in Chile . . . . . . . . . . . 23
1.2 Descriptive Statistics for Chilean College Programs . . . . . . . . . . . . . . 23
1.3 Smoothed Application Scores Distribution at selective universities . . . . . . 24
1.4 Cutoffs Scores for Majors M1 and M2 at selective universities . . . . . . . . . 24
1.5 Fraction Enrolled at selective universities in Chile . . . . . . . . . . . . . . . 25
1.6 Discontinuity of Earnings for University Graduates in Chile . . . . . . . . . . 25
1.7 Discontinuity at Pre-treatment Outcomes . . . . . . . . . . . . . . . . . . . . 26
2.1 Higher Education Enrollment in Chile: 1983-2013 . . . . . . . . . . . . . . . 46
2.2 Higher Education Graduates per Year of Graduation . . . . . . . . . . . . . 46
2.3 The evolution on the effect of university ranking and PSU score on wages, the
learning parameter Θx, and the true productivity of PSU λx. . . . . . . . . . 47
3.1 Average Wage by Major and Experience . . . . . . . . . . . . . . . . . . . . 88
viii
Abstract
The first chapter estimates the effect attending a selective university has on earnings. I
show that using the admission cutoffs of two related majors offered by selective universities
is enough to identify the isolated “university effect” on wages. To do so, I propose a novel
identification framework using the cutoffs of two related majors in a regression discontinuity
design that distinguishes the effect of graduating from a selective university from the effects of
the confounding factors correlated with the student application and enrollment decision. This
design requires a system with strict cutoffs and simultaneous decision making of university
and major upon application. The university admission system in Chile has a strict admission
criteria based solely on the university selection test and high school grades, and has students
choose their school and major simultaneously. This admission system leads naturally to
a regression discontinuity design because we can compare the earnings of those above and
below the cutoff to estimate the “university effect”. My main findings suggest that, on
average, attending a selective university in Chile significantly increases earnings, that is,
selective universities not only select the best students, but also play a role increasing their
future earnings.
The second chapter studies the effects of attending a private high school, university
ranking and employer learning on wages. My empirical strategy is based on the Mincer-type
wage regressions. I carry out my analysis using individual-level data from Chile. I find a
large and significant effect on wages at the beginning of workers’ careers from attending a
private high school and from attending a highly ranked university. These findings can be
rationalized by the statistical discrimination and employer learning model since the effects
ix
of attending a private high school or highly ranked university decrease with experience. I
construct an employer learning model to explain these decreases and find employers decrease
by fifty percent the weight they place on university ranking when setting wages in three years.
My findings further indicate that incorporating university admission test percentile rankings
in employment applications can significantly improve the market’s ability to appropriately
assign wages by decreasing the information gap between potential employees and employers.
The third chapter explores the equilibrium effects of postponing student choice of major.
Many countries use college-major-specific admissions policies that require a student to choose
a college-major pair jointly. Motivated by potential student-major mismatches we develop
a sorting equilibrium model under the college-major-specific admissions regime, allowing
for match uncertainty and peer effects. We estimate the model using Chilean data. We
introduce the counterfactual regime as a Stackelberg game in which a social planner chooses
college-specific admissions policies and students make enrollment decisions, learn about their
fits to various majors before choosing one. We compare outcomes and welfare under the two
different regimes.
1
Chapter 1
Estimating the Returns of Attending a Selective University
on Earnings using Regression Discontinuity with Multiple
Admission Cutoffs
1.1 Introduction
Does it matter where you go to college? Data from many countries shows that students who
attend selective universities earn more on average than those who attend less selective ones.
There are several reasons for this. First of all, enrollees’ human capital should be higher
because selective universities are more likely to have a higher faculty quality and student
bodies (Becker, 1964). Second, selective institutions provide a signal to employers about the
graduate’s productivity; graduates of selective universities are believed to have higher innate
quality, in addition to the enhanced quality due to their superior college education (Spence,
1973). Third, selective universities tend to have alumni networks providing their graduates
better job opportunities. Therefore, graduates from more selective universities not only have
higher human capital, but also their admittance to such highly selective institutions signals
to the labor market that they are innately more productive. These graduates also have
greater access to high-quality jobs.
It becomes critical then for families, higher education institutions, and policy makers to
understand why graduates from selective universities perform better in the labor market.
2
Prospective college students and their families make enrollment choices. Universities make
choices about admission requirements, faculty, funding, and selection of their students. Fi-
nally, policymakers decide whether to invest additional resources in higher quality programs
based on several grounds, including how well college graduates perform in the labor market.
Estimating the economic returns to college prestige on earnings is difficult since there
are unobserved characteristics that affect both students attendance at a highly selective
college and his or her future earnings. That is, there are unobserved characteristics such
as ambition, family background, etc. that affect the student’s decision to attend a highly
selective college and also affect subsequent earnings. Therefore, the simple regression setting
would lead to biased estimates. Hence, researches have to use different estimation strategies
to reduce bias.1
Thus far, there are mixed findings about the effect of attending a selective university. On
one hand, there are studies who find a positive and significant effect of school selectivity on
earnings. See for example Loury and Garman (1995), Bowen and Bok (1998), among others
that use reduced form models that control for observed student characteristics, including
high school GPA, standardized tests scores, and family background. In a different attempt
to reduce bias, Behrman, Rozenzweig, and Taubman (1996) use information of twin pairs to
identify the college quality effect by comparing labor market outcomes of female twin pairs.
They find a positive payoff from attending private universities that provide Ph.D. degrees.
Lindahl and Regner (2005) using sibling data from Sweden find that the cross-sectional
estimates of college quality on earnings are twice as large as within-family estimates.
Brewer, Eide and Ehrenberg (1999) explicitly model high school students’ choice of college
type and estimate the effects of college quality on earnings and how this effect varies across
time. Even after controlling for selection effects, they find a significant economic return to
attending an elite private institution, and some evidence suggests this premium has increased
over time. A different approach is considered by Hoekstra (2009) who uses a discontinuity
1See Brand and Halaby (2006) for a review.
3
regression design. The idea is to estimate the causal effect of college quality by comparing the
earnings of students who were just above the admissions cutoff of a flagship state university
to those just below it; he finds a 20% premium on the wages for white men who attended a
flagship state university five to ten years after graduation.2
On the other hand, there are studies that find a small and statistically insignificant
effect of college quality on future earnings. Black and Smith (2004) estimate the returns
of college quality using propensity score matching. They point out that the assumption
of a linear functional form affects the estimates on the returns to college quality; that is,
using matching to control for selection the coefficients of college quality are small and not
significant. Dale and Krueger (2002, 2012) include the average SAT score of the colleges
that students went and of the colleges these students were admitted to. They find that
students who attended more selective colleges do not earn more than other students who
were admitted by comparable schools but attended less selective colleges; although, they find
that selective schools have a positive effect on earnings for less privileged students. Long
(2008) uses several methods, including instrumental variables to estimate the local average
treatment effect of college quality and did not find a consistent relationship between college
characteristics and earnings.
Most recently, Hershbein (2011) formalizes a model of ability signaling to explain the
return to college quality. He finds that the return on GPA is smaller at selective schools
than at less selective institutions and the return on selectivity itself declines as GPA, and
average ability, rise. These findings provide support for the signaling model.
Regarding the returns for college in Chile, most of the literature focuses on estimation
of the returns for college or the returns for particular majors. See, for example, Urzua et al
(2013), etc. Bordon and Braga (2013) estimate the wage premium for selective schools and
tests for statistical discrimination and employer learning. They find a 19% wage premium
of selective schools, and that employers statistically discriminate graduates as employers set
2See also Saavedra (2008) for a similar methodology and estimations for the Colombian case.
4
higher wages to recent graduates from selective schools, but this wage premium decreases
considerably over time. Hastings, Nielson and Zimmerman (2013) use regression discontinu-
ity and a unique data set to estimate the returns for different college majors and selectivity
controlling for family background and ability characteristics. They find positive and large
returns in health, law, social science, selective technology and business degrees; moreover,
they find that most selective schools (above the median average cutoff score) yield a return
of 14.8% in the business majors and 9% in science/technology majors.
Is important to keep in mind that different countries have different university systems. In
some countries like the U.S. and Canada, students are admitted to colleges without declaring
their major upon enrollment. In other countries, including China, Japan, Thailand, Turkey,
Spain, many in Latin America including Chile, among others, students apply to a major
and college simultaneously. In addition, students must take a university selection test and
are admitted if their score is above the admission cutoff. Therefore, how one estimates the
returns to selective universities depends also on the university system as students’ behavior
changes according to the admission system.
This chapter contributes to the college selectivity wage premium literature by proposing
a novel estimation strategy based on regression discontinuity design. More precisely, I show
that using the admission cutoffs of two different but related majors of selective universities is
enough to identify the isolated “university effect” on wages. To do so, I use the cutoffs of two
majors in this discontinuity regression design which distinguishes the effect of graduating
from a selective university in Chile from the effects of the confounding factors correlated
with the student application and enrollment decision. This design requires a system with
strict cutoffs and simultaneous decision making of an institution and a program in the
institution. The university admission system in Chile has a strict admission criteria based on
the university selection test and high school grades, and has students choose their school and
major simultaneously. This admission system leads naturally to a regression discontinuity
design approach as we can compare the earnings of those above and below the cutoff and
5
estimate the “major effect” and “university effect”.
The main findings of this chapter suggest that, on average, attending a selective university
increases earnings approximately 27%, whereas the most demanded majors increase earnings
approximately 23%. The “university effect” becomes more important for graduates within
fifty points from the cutoffs as the earnings of those above the cutoff are 33 percentage
points higher than the earnings of those below the cutoff, meanwhile the “major effect” for
this group becomes not significant. For business and accounting majors, I find that the
“university effect” is approximately 30% whereas the “major effect” (i.e. graduating from
business) is 13%, but statistically insignificant. On the other hand, for civil engineering and
engineering majors, the “university effect” is approximately 23% whereas the “major effect”
(i.e. graduating from civil engineering) increases earnings by 38%. These results suggest that
the “university effect” is more important than the “major effect” for business related majors,
whereas the “major effect” is more important than the “university effect” for engineering
related majors.
This chapter is organized as follows. Section 2 presents a description of the Chilean
higher education system. Section 3 gives an overview of regression discontinuity design and
explains the identification strategy and the model used in this paper. Section 4 describes
the estimation procedure and the data, reconstructs the application and admission process,
and provide the estimation results. Section 5 provides some robustness checks. Section 6
concludes the chapter.
1.2 Chilean Higher Education System
Higher education in Chile comprises three types of institutions: Universities, Professional
Institutes (IPs), and Technical Formation Centers (CFTs). Universities can be divided into
two main categories: traditional and non-traditional or private institutions. Traditional
institutions include the oldest and most selective universities (created before 1981), and
those institutions that derived from the old universities (created after 1980). There are 25
6
traditional universities, fully autonomous and coordinated by the Council of Chancellors of
Chilean Universities (CRUCH). These institutions use a centralized and single admission
criteria: the University Selection Test (PAA)3. This test is made up of two compulsory
tests: language and mathematics; and optional tests: sciences (which may include advanced
mathematics, physics, chemistry or biology), and/or social sciences. These additional tests
are required for different majors.
The time-line of the admission process is shown in figure 1.1. First, students register and
take the PAA test. After receiving their score they make their application choices. Students
apply to a major and university (or program) simultaneously and can only apply to eight
programs, ranking them by preference. The only criterion for admission to the traditional
universities is the PAA score. Final admission scores consist of a weighted average of the
compulsory and major specific tests and high school GPA, with each program setting its
specific PAA weights4. The number of vacancies for each program is announced before
the application process and programs fill their vacancies solely based on the final weighted
scores. The admission score cutoff is defined by the score of the last student admitted into a
program and it is not known before the application decisions and therefore students cannot
manipulate which side of the cutoff on they fall on5.
Non-traditional, private universities were created after 1981 not necessarily use the PAA
score to select their incoming students. These universities that do not use the PAA score
generally soak up the excess local demand of students who either do not qualify or choose
not to go to the more selective traditional schools. Nevertheless, admissions data shows
evidence that the majority of students willing to attend higher education in Chile take the
PAA at the end of high school, regardless of the university they are planning to attend since
3In 2004 the were some changes to the university selection test and it is now called PSU.4For example, engineering in a selective university weights 30% mathematics, 10% language, 20% high
school GPA, 30% specific mathematics, and 10% physics. The final score to the same major in a differentuniversity might use different weights
5Students could use the admission score cutoff of previous years as a reference. Given the variation of theadmission cutoff overtime and the possibility to apply to eight different programs, we believe that studentswith marginal scores for admission to a selective university tend to apply to these competitive programs.
7
the test is relatively inexpensive and administered throughout the country.
Each program in an institution of higher education chooses how much to charge for tuition
and fees. However, for those students enrolled in one of the traditional universities, student
loans and scholarships are available6.
1.3 Estimating the Effect of Attending a selective university on Earn-
ings: A Regression Discontinuity Approach
I use the regression discontinuity (RD) design to estimate the wage premium of graduating
from the most selective universities in Chile. The idea is to identify the effect of university
selectivity on earnings by comparing the earnings of those just above the cutoff for admission
to the most selective universities to those of applicants who were barely below the admis-
sion cutoff and enrolled. The design will distinguish the effect of enrollment at a selective
university in a given major from other confounding factors as long as the determinants of
wages are continuous at the admission cutoff. Under this assumption, we can interpret the
discontinuous jump in earnings at the admission cutoff as the causal effect of admission to a
selective university. Notice here that it is possible that individuals with an higher score than
the cutoff of selective universities to have attended non-selective universities, which leads
to the fuzzy regression design. Hence, the results here are restricted to consider compliers
at the cutoff. Additionally, We also assume constant treatment effects, as I am looking at
students near the cutoff.
1.3.1 Model and Identification Strategy
As described in section 1.2 admission to a CRUCH college in Chile is based solely on the
University Selection Test (PAA) scores and GPA. Consequently, students apply for programs
(university and major) by listing their order of preference and then get admitted to the first
program their score qualifies for by being above the cutoff. The cutoff is determined by
6Nowdays, all higher education students can apply for student loans regardless of the institution type.
8
ranking the application scores in descending order until all vacancies are filled; students with
scores above the cutoff that fills the vacancies per program are admitted. Since students are
uncertain of the cutoff at the moment of application, they apply to the programs considering
their preferences and the probability of being admitted.
Students have different preferences for colleges, majors and their combinations. There-
fore, at the moment of application, students decide what program (college-major pair) to
apply to in order to maximize the probability of being admitted in their most preferred
choice. The application process leads to different student application behavior and appli-
cation outcomes. Consider for example the case where students can apply to business or
accounting in a selective or non-selective school as shown in table 1.1. Student A’s appli-
cation reveals a preference of a selective college, whereas student B’s application reveals a
preference for the business major. In the data, I see students attending different programs,
but not directly their application choices.
Table 1.1: University Programs Applications Example
Student A Student BRank Major University Rank Major University
1 Business Selective 1 Business Selective2 Accounting Selective 2 Business Non-selective
In order to tackle the university-major choice, I start with a simple model:
• Two types of universities: selective S1 and non-selective S0.
• Two majors: M1 and M2 (for example, Business and Accounting).
• 4 programs: (S1,M1), (S1,M2), (S0,M1), and (S0,M2).
Assumption 1: M1 M2 and S1 S0. I assume that most students would prefer to attend
a selective school S1 over S0 given the opportunity and enroll in major M1 instead of M2.
This assumption is based on the fact that in the data M1 is the most demanded major as
9
it has consistently higher cutoffs, therefore, by the revealed preferences argument, we have
that M1 M2. A similar argument holds for S1 S0 as the cutoffs for S1 S0 (see figure
1.4.
Besides assumption 1, students have different preferences about school and major com-
binations. For example, there are students who would prefer to attend a selective school no
matter what major, and others strictly prefer a major over the prestige of the school.
Lets define two types of students according to their preferences:
• Type A: (S1,M2) (S0,M1) with probability p
• Type B: (S0,M1) (S1,M2) with probability 1-p
Type A students would prefer to attend a selective school S1 even though they would have
to enroll in their less preferred major, M2. On the other hand, type B students would prefer
to enroll in major M1 even though they would have to attend a less selective school. As a
result of this student behavior we see enrollment in selective schools S1 paired with majors
M1 or M2. Students who enroll in program (S1,M1) are choosing the most preferred option,
whereas students enrolled in (S1,M2) are type A students. Therefore, in order to identify
the effect of attending a selective school we need two cutoffs: c1,1 for program (S1,M1) and
c1,2 for program (S1,M2).
The basic regression for estimating the effect of the school and a major on wages is7:
Y = βsS1 + βmM1 + ε (1.1)
where Y is the logarithm of real wages, S1 = 1 if students graduated from a selective school,
and M1 = 1 if the student graduated from major M1, the most preferred major.
Assumption 2: In order to isolate the effect on earnings of attending a selective school, I
assume that the effect of schools and majors is separable, so that I can capture the “school”
effect and “major” effect separately. Note that this assumption is needed because I only
7For simplicity of exposition I am omitting the constant and the controls from the regression.
10
have the cutoffs for the most selective schools, that is, I have only instruments for the most
selective schools. If I had the cutoff of the nonselective schools and the whole sample of
people who took the PAA tests, then this assumption is no longer needed.
Consequently, the probabilities of attending a selective school S1 are:
Pr(S1 = 1|X = x, c1,1) = γ1 + δ1T1 + g(x− c1,1) (1.2)
Pr(S1 = 1|X = x, c1,2) = γ2 + δ2T2 + g(x− c1,2) (1.3)
where T1 = 1[X ≥ c1,1] is an indicator function for (S1,M1), and T2 = 1[X ≥ c1,2] is an
indicator function for (S1,M2).
Using both cutoffs, the treatment effect of attending a selective school can be captured by
τ = τ1 + τ2, where
τ1 =limε↓0E[Y |X = c1,1 + ε]− limε↑0E[Y |X = c1,1 + ε]
limε↓0E[S1|X = c1,1 + ε]− limε↑0E[S1|X = c1,1 + ε](1.4)
τ2 =limε↓0E[Y |X = c1,2 + ε]− limε↑0E[Y |X = c1,2 + ε]
limε↓0E[S1|X = c1,2 + ε]− limε↑0E[S1|X = c1,2 + ε](1.5)
Notice that τ1 is capturing the effect on wages for those above c1,1 compared to those
below c1,1. Above the cutoff I observe graduates from (S1,M1). Below the cutoff I observe
graduates from (S1,M2), (S0,M1), and (S0,M2). On the other hand, τ2 is capturing the
effect on wages for those below c1,1 but above c1,2 compared to those below c1,2. That is,
between c1,1 and c1,2 I observe graduates from (S1,M2) and below c1,2 I see graduates from
(S0,M2). Since students have different preferences, this setup distinguishes those students
between c1,1 and c1,2.
Using both cutoffs (c1,1, c1,2), I identify the “effect of attending a selective school”, βs
or the “university effect”, where βm is the “major effect”. To see this, I estimate τ1 and τ2
11
using c1,1 and c1,2 simultaneously in this equation:
Y = α + βsS1 + βmM1 + f(x− c1,1) + f(x− c1,2) + ε (1.6)
Case 1: Cutoff for (S1,M1), c1,1:
Looking at those above and below c1,1,
τ1 = βs + βm − [pβs + (1− p)βm]
Notice that βs is the effect of graduating from S1 (the “university effect”) and βm is the
effect of graduating from M1 (the “major effect”). Graduates above the cutoff have both
effects, that by assumption work separately. Recall that we have type A students who prefer
(S1,M2) over (S0,M1), so p students graduated from (S1,M2) and (1 − p) graduated from
(S0,M1).
Case 2: Cutoff for (S1,M2), c1,2:
Looking at those above and below cutoff c1,2,
τ2 = [pβs + (1− p)βm]− βm
Notice that for case 2, graduates above the cutoff attended (S1,M2), whereas graduates
below the cutoff attended (S0,M2). Thus, p students graduated from (S1,M2) and (1 − p)
graduated from (S0,M1). On the other hand, βm captures the effect of graduating from M1.
As a result of the estimation, one can recover
τ = τ1 + τ2 = βs + βm − [pβs + (1− p)βm] + [pβs + (1− p)βm]− βm = βs (1.7)
Equation 1.7 shows that using both cutoffs I identify βs (the “university effect”), which is
the isolated effect of attending a selective school. Note that in order to obtain βs, I also need
12
to estimate the “major effect” βm.
1.4 Estimation
1.4.1 Data
I use a panel data set that follows graduates from higher education institutions in Chile
during the first 10 years they are in the labor market8. The data matches tax returns with
transcripts of students’ majors and the higher education institutions they graduated from in
Chile. The data only contains information for those who graduate; those who dropped out
from college or did not go to college are not in the sample. Income information is available
for four graduating classes (1995, 1998, 2000 and 2001) between the years 1996 and 2007.
The wage is measured annually and since I have people who worked less than 12 months in a
year, I compute the annual wage as their monthly wage times 12 to make it comparable with
the wages of 12-month workers. I use consumer price index (IPC) as a deflator to compute
real wages. I drop the highest 1% and the lowest 1% of the wage observations to remove
possible outliers. Experience is computed as the number of years an individual has been in
the labor market, that is, has positive income and paid taxes.
Moreover, the data contains information on the year a student took the PAA test, his
or her scores on each component of the test, the college he or she graduated from and the
major. I do not observe the full application set of programs applied to for each student. I
also obtained the PAA weights and the cutoff scores for college programs in the four most
selective universities9.
As of 2001, the Chilean higher education system consisted of 60 universities: 25 tradi-
tional and 35 non-traditional, private universities. I consider as a selective university any of
the 4 oldest, traditional, and most selective universities in Chile. I then aggregate all the
8The data is part of a project call Futuro Laboral run by the Chilean Ministry of Education.9I was only able to obtain PAA weights for years starting in the year 2000. Since most of the students
in my data took the PAA tests before 2000, I make the assumption that the PAA weights were the same inprior years.
13
remaining universities as non-prestigious, including traditional and non-traditional, private
institutions. Due to the model set up I need to use two related majors for the estimation as
students may consider applying to both majors in selective schools, I chose the most popular
pair of majors: business and accounting, and civil engineering and engineering10.
Table 1.2 shows some of the most important summary statistics for selective and non-
selective universities. The four prestigious, traditional schools were aggregated with a total
of 5,090 graduates. The 49 remaining, non-selective schools were aggregated with a total of
13,665 graduates. The traditional, selective institutions have higher capacity, therefore, they
provide considerably more vacancies than private schools. As expected, selective universities
have higher quality students as their PAA tests and GPA are significantly higher than non-
selective ones. Selective schools have a higher fraction of private high school graduates and
less women compared to non-selective universities.
Figure 1.2 shows the average PAA tests scores by program. It is easy to see that the
most demanded major in selective schools, S1,M1 has the highest average scores in all test
and GPA. On the other hand, the less demanded major in non-selective schools, S2,M2 has
the lowest average scores in all test and lowest GPA.
In summary, I am using the four most selective schools as a big “selective university”, and
just considering the pairs (business, accounting) and (6-year engineering, 4-year engineering)
for the estimation of ”university effect”11. Table 1.3 presents some summary statistics for
M1 (business and 6-year engineering) and M2 (accounting and 4-year engineering). In terms
of earnings, graduates from (S1,M1) have the highest earnings from all the graduates. It is
interesting to see how similar wages of (S1,M2) and (S0,M1) are, where the “major effect”
and “university effect” are definitely playing an important role. Finally, we see that (S0,M2)
graduates have the lowest wages. From the table we also see that PAA scores are in the
10In Chile, there are two types of engineering schools. The most prestigious and most demanded are thecivil engineering schools that provide 6-year engineering majors. On the other hand, there are the 4/5-yearengineering majors such as forestry engineering and the applied/executive engineering programs.
11Unfortunately, no other majors were able to use because in many cases the pairs are not easily distin-guishable as possible application pairs, or there were too few students below the cutoff score of a major.
14
expected order, where selective school have the students with the highest PAA scores and
major M1 enrollees have higher scores than M2 enrollees. Figure 1.3 shows the distribution
of application scores of the majors considered, that is Business and Civil Engineering (M1),
and Accounting and Engineering (M2).
One of the most important variables for the estimation of the “university effect” are the
cutoffs c1,1 and c1,2. Figure 1.4 shows the cutoffs for each year of application. It is easy to
see that for all years, c1,1 is higher than c1,2, assumption needed for the identification of βs,
the “university effect”.
1.4.2 Reconstructing the Application and Admission Process
I use the University Selection Test (PAA) for the computation of the application score
variable. Each program (university-major pair) has different PAA weights. For example, en-
gineering in a selective university weights mathematics at 30%, language at 20%, high school
GPA at 20%, science (advance mathematics and physics) at 30%; whereas a less selective
university may weight mathematics at 40%, language at 30%, high school GPA at 30% and
no specific tests. This implies that a student has different weighted application scores and
therefore choses the best college-major combinations in order to maximize his probability to
be admitted. As a result, I can reconstruct the application score for major j:
Scorej,i = wj,1(Math scorei) + wj,2(Language scorei) + wj,3(Social Sciences scorei)
+wj,4(GPAscorei) + wj,5(Sciences scorei)
where wj,1, ..., wj,5 are the specific weights required by selective schools for major j. Math
score is the score in the Mathematics test, Language Score is the score in the language test,
Social Sciences is the score in the history and social sciences test, GPA is score corresponding
to high school grade point average, and Sciences is the score in the specific science module
such as advance mathematics, biology, etc. The cutoff score at a selective universities S1 in
application year t for major j is the minimum score achieved by a student who attended that
15
program:
c1,j,t = mini∈S1scorej,it
where i ∈ S1 are all individuals from selective universities that applied in year t for major j.
I then define the average treatment effect at the discontinuity point:
τt = E[wit|score = 0+, t]− E[wit|score = 0−, t]
assuming that τt is non-increasing with t.
1.4.3 Estimation Procedure
1.4.3.1 Estimation of the “University Effect” and “Major Effect”
I use the following regression specification to estimate the effect of a selective university on
earnings:
wi = β0,i + βsS1,i + βmM1,i +∑
βl(f(Adjusted Score j,i)) + β′ΦΦi + εi
where
wi is logarithm of real wages for individual i,
S1,i = 1 if student i attended a selective school regardless of the major,
M1,i = 1 if student i graduated from major M1 regardless of the school type.
The running variable Adjusted Scorej,i is defined for each application cohort as the difference
between a student’s weighted score for major j (Scorej,i =∑
k ωj,kPAAk,i) and the admission
cutoff associated with major j at selective universities. f(Adjusted Scorej,i) is a polynomial
function of the Adjusted Score for major j.
Note that the OLS estimation will lead to biased estimates due to the endogeneity since
there are unobserved characteristics that affect the student’s decision to attend a selective
college and also affect subsequent earnings. Therefore, I am using two instruments for the
16
two endogenous variables in the model, namely, S1,i and M1,i. The instruments are the
dummy variables: AboveAdmissionCutoff = 1 if (Adjusted Score j,i) ≥ 0 for at least one
major, and AboveAdmissionCutoff Major 1 = 1 if (Adjusted Score 1,i) ≥ 0 for M1. Since
the discontinuity in the probability of enrollment is less than one, all estimates of βs and βm
need to be reweighed by the discontinuity in enrollment in order to calculate the effect of
enrolling at a selective university.
For the estimations I need to use two related majors with similar weights and specific
test requirements where one major, M1 is more demanded, therefore, having a higher cutoff
than a less demanded major, M2. I use business and accounting as M1 and M2 respectively.
Similarly, I use 6-year engineering (a.k.a civil engineering) and 4-year engineering as M1 and
M2 majors. For each group of related majors, I compute the application score for each major
in the group at selective schools.12
Table 1.4 shows the distribution of students by major and university. From the table
we see that 50.19% of the students have a score above cutoff of M1 whereas 59.29% of the
students have a score above the cutoff of M2. From the total of students in the sample,
15.48% enrolled in M1 and 11.15% enrolled in M2. For those above the cutoffs 30.84%
enrolles in M1 and 18.81% enrolles in M2. For those students whose score is above M2 but
below M1, 8.69% enrolls in M2, which tell us than only a 8.69% prefers a school over a major.
In practice, the estimation procedure is the following:
1. Estimate the first stage
2. Estimate the reduced form
3. Estimation using 2SLS, which uses instrumental variables to estimate the four different
local average treatment effects.
12I use the application score of the 4th most selective school in the country in terms of cutoff and PAAweight requirements.
17
1.4.4 Estimation Results
First of all, we need to see if there is a discontinuity at the probability of enrollment is
discontinuous at the admission cutoff. Figure 1.5 shows the probability of enrollment and
graduation from selective universities as a function of the number of points of the adjusted
entry score above or below the cutoff. The open circles are the estimated local averages.
From the figure we see that at score 0, the cutoff, there is a discrete jump in the probability
of enrollment as students cannot be enrolled if they are not admitted (that is, have score
below the cutoff).
The discontinuity of enrollment is approximately 19.4% near the cutoff (t=58.49). This
probability increases considerably as we move towards the right (highest adjusted scores).
Notice here that the probability of 19.4% is the probability of being enrolled in a selective
university no matter what major, therefore, there are many students (type B in the model)
who prefer to be enrolled at M1 in a non-selective university even though they have score
high enough to be enrolled in M2 at a selective university. Nevertheless, it is easy to see
that being just above the admission cutoff causes a significant increase in the probability of
attending a selective university.
Regarding the effect of attending selective universities on earnings, we need to see a
discontinuity on earnings at the admission cutoff as well. Figure 1.6 shows earnings on a cubic
polynomial of adjusted PAA score. It is clear from the figure that there is a discontinuity
on earnings at the cutoff. Assuming that the other factors causing increases of earnings
are continuous at the cutoff, we can interpret the discontinuity in earnings as the cause of
attending a selective university in earnings.
Table 1.5 presents the discontinuity estimates from the estimation 2SLS estimation form
using different functional forms, bandwidths, experience, gender and majors.13 Overall,
the results reveal statistically significant earnings discontinuities. On average, attending
13See table 1.6 at the Appendix section for the first stage estimates. The F-statistic of these regressionsare over 120.
18
a selective university increases earnings approximately 27%, whereas the most demanded
majors increase earnings 23%. The “university effect” becomes more important for graduates
within 50 points from the cutoff as the earnings of those above the cutoff are 33 percentage
points higher than the earnings of those below the cutoff, meanwhile, the “major effect” for
these group becomes not significant.
By experience, we see that the estimates suggests that the major studied becomes more
important than the school for the first 3 years in the labor market, but with time the “major
effect” decreases and is not significant, whereas the “university effect” increases from 3 to
6 years of experience and then decreases from 6 to 10 years in the labor market. It is
also interesting to see that the “university effect” and “major effect” is considerably larger
for females than males. This result may be explained by the fact than less women choose
business or engineering majors, but those who do perform well enough in the labor market
that the signaling and/or human capital effects of studying these major are larger for women.
For business and accounting majors, I find that the “university effect” is approximately
30% whereas the “major effect” (i.e. graduating from business) is 13%, but not statistically
significant. On the other hand, for civil engineering and engineering majors, the “university
effect” is approximately 23% whereas the “major effect” (i.e graduating from civil engi-
neering) increases earnings 38%. These results suggest that the “university effect” is more
important than the “major effect” for business related majors, whereas the “major effect”
is more important than the “university effect” for engineering related majors.
1.5 Validity of Regression Discontinuity Design: Robustness Analysis
1.5.1 Manipulation of the Assignment Variables
Given the university selection test, the PAA, is administered only once a year, students
cannot manipulate the PAA scores. If students could manipulate the score, then the score
19
is invalid as a running variable14. Once the scores are sent to the test takers, they can
compute the application score for each major they are interested in applying to, but they
cannot retake the test within the same admission process. Students can only retake the test
and re-apply to university programs in later year cohorts. In other words, students have
no direct control over each of the components of the PAA score. Although in theory it
seems really unlikely students can manipulate the score I present further evidence for the
validity of the regression discontinuity design. The idea is to test the continuity of the pre-
treatment variables around the cutoff that should not be affected by the treatment. That is,
if the design is well specified and the probability of being above or below the cutoff is truly
random then the treatment should have no effect on the pre-treatment variables such as the
probability of being female or graduating from a private high school. Figure 1.7 shows that
there is little if any discontinuity around the cutoff. The estimated discontinuity is so small
it is unlikely to have an effect on the outcome.
1.6 Conclusions
This study proposes a new identification strategy to estimate the effect of attending a selec-
tive university on earnings. I build a model that accounts for simultaneous decision making
of university and major upon application, students preferences for colleges, majors and their
combinations; thus, at the moment of application, students decide what program (college-
major pair) to apply to in order to maximize the probability of being admitted in their
most preferred choice. Furthermore, the identification strategy requires a system with strict
cutoffs that lead to regression discontinuity designs. I show that using the admission cut-
offs of two related majors offered by selective universities is enough to identify the isolated
“university effect” on wages.
I estimated the model for four different cohorts of universities graduates from business
and engineering related majors in Chile using earnings up to the first ten years in the labor
14See McCrary (2008) for details.
20
market. The estimates suggest that attending a selective university increases earnings ap-
proximately 27% (“university effect”) whereas the most demanded majors increase earnings
approximately 23% (“major effect”), although the latter is not always significant.
Finally, it is important to extend this work including different majors and universities in
order to understand the effect of college selectivity on earnings for all university programs.
Moreover, it would be interesting to disentangle the signaling effect from the human capital
effect that is attributed to the wage premium of attending a selective university.
21
1.7 Tables and Figures
Table 1.2: Descriptive Statistics for Chilean Universities
Selective universities Non-selective universitiesVariable Mean Std. Dev. Mean Std. Dev.Number of Universities 4 0 49 0Number of Graduates 5,090 0 13,665 0PAA Math 734.9 51.2 643.4 87.4PAA Language 650.1 66.9 580.7 77.9High School GPA 638.1 77.5 578.2 67.0PAA Science (math) 666.6 63.9 575.1 67.0Private High School 0.17 0.37 0.11 0.31Female 0.30 0.46 0.39 0.49
Table 1.3: Descriptive Statistics for Chilean College Programs
Selective Universities Non-selective UniversitiesMajor: M1 Major: M2 Major: M1 Major: M2
Variable Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. Mean Std. Dev.Log of annual wages 16.08 0.65 15.71 0.67 15.75 0.75 15.45 0.75PAA Math 753.6 42.3 706.1 55.2 675.5 78.4 617.9 85.7PAA Language 665.7 64.17 623.4 66.6 600.6 76.2 564.8 75.6High School GPA 657.8 72.3 6.14.1 81.0 597.2 89.4 563.1 89.2PAA Science (math) 690.9 58.6 624.7 49.5 598.3.4 63.5 555.1 63.3
22
Table 1.4: Distribution of Students by Major and University
% Above Cutoff % EnrollmentM1 M2 M1 M2
Total 50.19 59.29 15.48 11.15Above Cutoff – – 30.84 18.81Below Cutoff M1 – 27.87 – 8.69
Table 1.5: Regression Discontinuity Design Estimates
Regression Sample Function of OLS 2SLS OLS 2SLSSpecification Points from (Major (Major (University (University
Cutoff Effect,βm) Effect,βm) Effect, βs) Effect, βs)
(1) One Cutoff All Cubic 0.305 0.292 0.164 0.320(0.005)*** (0.008)*** (0.006)*** (0.074)***
(2) Two Cutoffs All Linear 0.306 0.313 0.173 0.299(0.005)*** (0.061)*** (0.008)*** (0.028)***
(3) Two Cutoffs All Quadratic 0.304 0.193 0.182 0.256(0.005)*** (0.084)** (0.008)*** (0.022)***
(4) Two Cutoffs All Cubic 0.308 0.238 0.179 0.262(0.005)*** (0.087)*** (0.009)*** (0.036)***
(5) Two Cutoffs Bandwith Cubic 0.324 0.287 0.214 0.33550 points (0.007)*** (0.397) (0.019)*** (0.138)**
(5) Two Cutoffs Experience less Cubic 0.282 0.326 0.189 0.287than 3 years (0.007)*** (0.156)* (0.015)*** (0.063)***
(6) Two Cutoffs Experience 3-6 Cubic 0.328 0.149 0.172 0.321years (0.007)*** (0.128) (0.014)*** (0.059)***
(7) Two Cutoffs Experience greater Cubic 0.325 0.175 0.173 0.141than 6 years (0.011)*** (0.185) (0.019)*** (0.062)**
(8) Two Cutoffs Males Cubic 0.312 0.120 0.152 0.225(0.006)*** (0.119) (0.011)*** (0.044)***
(9) Two Cutoffs Females Cubic 0.269 0.353 0.227 0.312(0.008)*** (0.151)** (0.016)*** (0.064)***
(10) Two Cutoffs Business & Accounting Cubic 0.291 0.133 0.218 0.299(0.008)*** (0.165) (0.018)*** (0.075)***
(11) Two Cutoffs Civil & Engineering Cubic 0.322 0.380 0.115 0.234(0.008)*** (0.192)** (0.013)*** (0.092)**
Robust, cluster standard errors in parentheses, *** p<0.01, ** p<0.05, * p<0.1.
All specifications include cubic experience polynomial, standardize PAA tests for mathematics, language, GPA
and dummies for science test, experience, years, test years, cohorts of graduation, female and economic sectors.
All regressions include the interaction for points from cutoff with the dummy above cutoff allowing the
estimated coefficients to differ on each side of the admission cutoff.
23
Figure 1.1: Time-line of the University Application Process in Chile
July -‐ August
• PAA test registra1on. • Informa1on on prior year cutoffs and and weights given for each program.
1st week of December
• PAA test administered.
3rd week of December
• PAA test results delivered. • Applica1on score major m at school S:
4th week of December
• Applica1on to school-‐major program pairs. • Applica1on allows up to 8 programs, ordered by preference.
2nd week of January
• Applica1on results are delivered. • Students choose whether or not to enroll. • Each program has a cutoff at each university.
Figure 1.2: Descriptive Statistics for Chilean College Programs
020
040
060
080
0M
ean
PAA
Scor
e
GPA Language Math Science
S1M1 S1M2S2M1 S2M2
24
Figure 1.3: Smoothed Application Scores Distribution at selective universities
0.0
02.0
04.0
06.0
08.0
1
500 600 700 800Application Score
Major M2 Major M1
Figure 1.4: Cutoffs Scores for Majors M1 and M2 at selective universities
25
Figure 1.5: Fraction Enrolled at selective universities in Chile
Figure 1.6: Discontinuity of Earnings for University Graduates in Chile
26
Figure 1.7: Discontinuity at Pre-treatment Outcomes
27
1.8 Appendix
1.8.1 Additional Tables
Table 1.6: Regression Discontinuity Design Estimates: First Stage
Regression Sample Function of Points First Stage First StagePoints from Cutoff (University Admission) (Major Admission)
(1) All Linear 0.152 0.131(0.003)*** (0.005)***
(2) All Quadratic 0.153 0.095(0.003)*** (0.005)***
(3) All Cubic 0.075 0.065(0.003)*** (0.005)***
(4) Bandwith Cubic 0.085 0.01950 points (0.005)*** (0.009)**
(5) Experience less Cubic 0.060 0.062than 3 years (0.004)*** (0.009)***
(6) Experience 3-6 Cubic 0.058 0.066years (0.004)*** (0.009)***
(7) Experience greater Cubic 0.140 0.061than 6 years (0.007)*** (0.010)***
(8) Males Cubic 0.062 0.058(0.003)*** (0.007)***
(9) Females Cubic 0.103 0.077(0.005)*** (0.009)**
(10) Business & Accounting Cubic 0.130 0.110(0.004)*** (0.009)***
(11) Civil & Engineering Cubic 0.092 0.054(0.007)*** (0.014)***
Robust, cluster standard errors in parentheses, *** p<0.01, ** p<0.05, * p<0.1.
All specifications include cubic experience polynomial, standardize PAA tests for mathematics, language,
GPA and dummies for science test, private high school,experience, years, test years, cohorts of graduation,
female and economic sectors
All regressions include the interaction for points from cutoff with the dummy above cutoff allowing the
estimated coefficients to differ on each side of the admission cutoff.
28
Chapter 2
The Effects of Private High Schools, University Rankings
and Employer Learning on Wages in Chile
2.1 Introduction
It is widely believed in Chile that too much emphasis in the hiring process is given to the
high school and college an applicant attended. Explanations for these hiring differences range
from actual inequality in education to variation in networking capabilities from attending
different institutions to statistical discrimination. Statistical discrimination refers to the fact
that when there is incomplete information, more educated individuals are expected to be,
on average, more productive than less educated ones.1 Therefore, firms use easy-to-observe
variables such as schooling to predict workers productivity and set wages. As time passes,
firms learn about their workers true productivity, a process known as employer learning.
Farber & Gibbons (1996) and Altonji & Pierret (2001, hereafter AP) pioneered the study
of statistical discrimination and employer learning. They investigated the fact that more
able individuals have greater wage growth with experience and the schooling effect reduces
in time. In fact, AP find that the wage coefficients on the variables that affect productivity
but cannot be directly observed by firms, such as AFQT scores, increase with experience,
whereas the coefficients on schooling, which firms can directly observe, decrease with time.
1Spence (1973) and Weiss (1995) developed models of ability signaling where education is a mechanismfor individuals to sort into groups (education levels) that are correlated with ability.
29
The authors explain that these findings are evidence of statistical discrimination on the basis
of schooling.
Lange (2007) shows how to estimate the speed of employer learning using the observed
returns to schooling and ability measures at different experience levels. He finds that U.S.
employers learn fast. Mansour (2009) further studies employers learning across different
occupations and Pasche (2008) includes multiple cognitive and noncognitive ability measures
simultaneously to test statistical discrimination and estimate the speed of employer learning
finding that employers learn about all kinds of abilities, and discriminate on the basis of
schooling.2
There are few papers looking at the return to college, PSU and type schools in Chile. See
for example, Contreras, Rodriguez and Urzua (2013), Reyes, Rodriguez and Urzua (2013),
among others. Bordon and Braga (2013) test the statistical discrimination hypothesis using
regression discontinuity design estimating a 19% wage premium for recent graduates of the
most selective universities in Chile. However, they find that this premium decreases by 3
percentage points per year of labor market experience.
This chapter studies the effects of attending a private high school, university ranking
and employer learning on wages in the Chilean labor market. This is the first study to my
knowledge that extends the model to incorporate key features of Chilean resumes: high school
and college attended. Employers have this information in the resumes of Chilean applicants,
but it is not usually available to researchers. I find a large and significant effect on wages at
the beginning of workers careers from attending a private high school and from attending a
highly ranked university. These findings can be rationalized by the statistical discrimination
and employer learning model since the effects of attending a private high school or highly
ranked university decrease with experience. From the employer learning model estimation
I conclude that employers learn fast and decrease by fifty percent the weight they place on
university ranking when setting wages over three years.
2See Schonberg (2007) and Arcidiacono et.al (2000) for more general employer learning models.
30
The chapter is organized as follows. Section 2 presents a description of the Chilean higher
education system. Section 3 gives an overview of statistical discrimination and employer
learning, describes the data and presents the main results of the estimation for the Chilean
college labor market. Section 4 develops the employer learning model and the estimation
results. Section 5 provides some policy implications and section 6 concludes the paper.
2.2 Higher Education System in Chile
Enrollment in higher education in Chile has increased significantly since the early 1980s.
In 1983 there were 175,250 students in the system, while in 1990 enrollment increased to
249,482 students. Most of the participation growth took place in the 1990s when enrollments
increased dramatically. By 1995, nationwide enrollment reached approximately 345,000 stu-
dents, in 2000 there were 452,325 students and in 2013, enrollment was 1,075,668 (see figure
2.1). This huge expansion is the result of a process of privatization and diversification of the
education system in order to generate permanent economic growth and provide stability to
the country.
Universities can be divided into two main categories: traditional and non-traditional
institutions. Traditional institutions comprise the oldest and most prestigious universities
created before 1981, and those institutions that derived from the old universities (created af-
ter 1980). There are 25 fully autonomous traditional universities coordinated by the Council
of Chancellors of Chilean Universities (CRUCH) , which are eligible to obtain partial fund-
ing from the state. They employ a single admission process: the University Selection Test
(PSU).3 This test is made up of two compulsory tests: language and mathematics, and
optional tests: sciences and/or social sciences. Each university sets their minimum entry
scores on the PSU and fill up their vacancies with the students applying with the highest
PSU scores. The score that will determine the current year’s cutoff and fills the total number
3Until 2004 the university selection test was called PAA, but in 2004 the PAA was modified and it is nowcalled PSU.
31
of vacancies is uncertain at the moment of application. Students apply for a university and
a major simultaneously. Non-traditional or private universities were created after 1981, have
no state financial support and do not necessarily use the PSU score to select their incoming
students. All higher education institutions charge tuition and fees.
The increasing enrollment in higher education has led to an increasing number of grad-
uates in the last two decades. In 1995, 24,400 graduates entered the labor market, whereas
in 2000 around 42,000 graduates did, and in 2011, around 139,500 new graduates were en-
tering the labor market. This means that in fifteen years the number of graduates has more
than quintupled (see figure 2.2). Traditional universities have more than quadrupled the
number of graduates they produce, but private universities have increased their number of
graduates by 17 times. Since most private universities are relatively new, that is, they were
mostly created since the beginning of 1990, employers do not fully know how productive their
graduates are on average. In contrast, employers have been hiring workers who graduated
from the oldest universities for many years and they may have an idea about their average
productivity. Thus, employers could, in principle, statistically discriminate workers from
traditional universities, but could not discriminate workers from new private universities.
2.3 Statistical Discrimination and Employer Learning
AP (2001) develop a model to test the employer learning and statistical discrimination
(EL-SD) hypothesis. The main idea of the EL-SD model is that employers cannot observe
prospective workers’ productive characteristics directly. Therefore, employers must use easily
observable correlates of productivity such as schooling and university to predict prospective
workers’ productivity. Over time more information about an employee’s productivity is
available to employers, a process known as employer learning. More formally, the model is
the following:
χix = αsi + βqi + γzi + ηi + H(xi) (2.1)
32
where χix is the log labor market productivity of worker i with x years of experience. si
are the easy-to-observe variables, such as high school and university; qi are the variables
observed by the employer but not available to the researcher, for instance, information from
job interviews and recommendation letters; zi are variables not observed by the employers
but available to the researcher, such as tests scores; ηi are variables that are observed by
neither the employer nor the researcher, or the error term. H(xi) summarizes the experience
profile of log-productivity, which may be due to a process of investment over the life cycle.
In what follows I will explore the implications of the EL-SD model in the Chilean market.
The first implication is that the linear regression coefficients on unobserved ability z increase
with experience. In other words, with experience more information about workers true
ability is revealed and consequently wages increasingly reflect productivity, augmenting the
correlation between wages and ability. Thus, the returns from the z variables are expected
to increase with experience. A second implication is that the returns to university prestige
should decrease with experience since experience reduces employers reliance on university
prestige to predict the productivity of its workers. University quality and prestige raises
earnings early in the worker’s career because quality universities raise productivity, and
also because of statistical discrimination. However, over time the contribution of statistical
discrimination on wages declines. Consequently, allowing the z variable to interact with
experience lowers the university prestige coefficients for experienced workers more than the
less-experienced workers.
In this section, we estimate a log earnings equation that allows for linear interactions
between easy-to-observe variables and the Standardized PSU scores with experience:
wi,t = β0 +∑j
βsjsji,t+∑j
βzzi,t+∑j
βsj ,x(sji,t×xi,t)+βz,x(zi,t×xi,t)+f(xi,t)+β′ΦΦi,t+υi,t
(2.2)
where wi is the log wage of individual i in period t, s1i is a private school dummy, s2i up to
33
s4i are dummies for university rankings as a proxy for university prestige, zi is the ability
measured by the standardized university admission test PSU, xi is experience in the labor
market, and Φi is the vector of controls. The interesting feature of this regression is that it
shows how the relationship between wages and ability, wages and private schools, and wages
and university ranking change with experience.
I use the language and mathematics tests of the PSU as a measure of ability that are not
easily observed by firms for the following reasons. First, these tests are the components of
the PSU formulated to measure inherent abilities of applicants. These tests are administered
so that prospective college students show their knowledge and ability to succeed in college.4
Second, there is evidence that employers do not have access to the PSU score at the moment
of setting wages. It is not common practice for Chilean employers to ask for the PSU score
in the application process (Swett, 2011). An explanation for this could be because they do
not think it provides relevant information beyond high school and college, or in contrast to
universities, employers do not have access to the full distribution of PSU scores, therefore,
the absolute value of the PSU score for a single worker might be not very informative to a
firm.
From equation 2.2 we can test statistical discrimination on the basis of easy-to-observe
variables if the interactions between high school and college to experience are non-increasing,5
and the interaction between abilities and experience are non-decreasing.6 Obviously, if em-
ployers have full information on the ability and productivity of prospective workers, then
βsj ,x = βz,x = 0.7
4Repetto (2003) shows that high school grades and specific physics and math tests of the PSU areimportant predictors for outcomes in the engineering school.
5βsj ,x ≤ 0.6βz,x ≥ 0.7The test that AP developed could be expressed as:
∂βsj ,x
∂x= −λzj ,sj
∂βsj ,x
∂x(2.3)
where λz,sj is the regression coefficient of z on sj . Equation (2.3) claims that as sj is part of the initialinformation the employers have about new worker’s productivity, the decline in βsj ,t comes only from therelationship between sj and z.
34
2.3.1 Data
The data used in the study comes from Futuro Laboral, a project of the Ministry of Edu-
cation of Chile that follows individuals over the first years of labor market experience after
graduating from higher education programs. This is a panel data set that matches tax re-
turns with transcripts of students’ majors and the institutions they graduated from. There
are only data for those who graduate; those who dropped out or never enrolled in college are
not in the sample. Income information is available between the years 1996 and 2005.
The Internal Revenue Service (SII) provides data on subjects’ annual income reported
on tax returns, city or cities of employment, number of employers, and economic sector.
Demographic data including age, sex, name of the institution where students studied, major,
and the year of graduation were provided by the Ministry of Education (MINEDUC). A
subsample is used with additional data gathered by MINEDUC on PSU score, high school
grades and the high school from which the subject graduated. Therefore, I run the analysis
using the subsample of the data.
The wage measure in the sample is the annual income that comes from jobs and services
provided by the individual in the formal sector. I use consumer price index (IPC) as a
deflator to compute real wages. The ability variable, PSU score is constructed as a simple
average of the score of the language and mathematics tests. In order to facilitate the inter-
pretation, I standardize the PSU scores to have a mean of zero and a standard deviation of
1. The experience variable is computed as the number of years an individual has income and
paid taxes. Economic sectors are divided in twelve categories, including agriculture, mining,
manufacturing, construction, business, services, real estate, transportation and communica-
tions, education, and health. Public administrators are excluded as they belong to the public
sector with a different wage setting structure. Self-employed workers are also excluded from
the sample.
Finally, for the university ranking variables I rank all the universities using the Que
Pasa magazine ranking, which uses a similar methodology as the U.S. News college ranking.
35
Then, I cluster universities in tiers according to their ranking and type, that is, there are
two tiers for the traditional universities and two tiers for the private universities, where tier
1 includes the oldest and most prestigious universities in each group. Table 1 shows the
summary statistics of the data. Table 2 provides some statistics for ranking clusters. It
is interesting to see from this table that workers from tier 1 traditional universities have
significantly higher PSU scores than the workers from other universities.8 Tier 1 private
institutions have the higher percentage of graduates from a private high school, 24%.
2.3.2 Statistical Discrimination on the basis of High School and University Ranking
I first estimate equation (1) without incorporating the PSU scores interaction with experi-
ence.910 This restriction implies that the effect of ability on log earnings is constant over
the life cycle. Table 2.3 displays the estimation results using different specifications. Fe-
males earn considerably less even controlling for major, region, economic sector, high school,
university and PSU scores. In fact, females show an average of 25.6% lower wages than men.
Column 1 in Table 2.3 reports the results of the estimation when the interaction of ability
(PSU score) to experience is excluded. We see that one standard deviation increase in the
standardized score on the PSU tests leads to an average of a 9.3% increase in wages, all
else equal. Attending a private high school has a positive and important effect on earnings
for college graduates, increasing wages 16%. This result is interesting because we are only
looking at college graduates, but it is not surprising for the Chilean labor market given the
common practice of including high school information on resumes. In principle, this might
allow employers to discriminate workers based on high school. Here, the private school
dummy may capture other effects such as the socioeconomic level of the parents which may
be and important factor in the quality of a persons social networks for finding better jobs.
8Since I standardized PSU scores to have mean 0 and standard deviation of 1, it is posible to have PSUscore that are less than 1, that is, scores below the mean.
9βz,x = 0.10Throughout the analysis the variables private high school (s1), the university ranking dummies (s2− s4)
and standardized PSU (z) are interacted with experience divided by 10. Thus, the coefficients of theseinteraction terms capture the change in the wage slope between x = 0 and x = 10.
36
The coefficient on private high school interacted to experience that allows us to see the effect
of attending a private high school with time is -0.024, meaning that the effect of attending
a private high school on wages declines with experience.
In column 2 the interaction between PSU score and experience is included. In this case,
we are adding a control for ability over time. The effect of one standard deviation shift on
PSU scores increases wages 6.15% right after graduation. Attending a private high school
increases log wages by 0.17, that is, when accounting for ability (PSU score), those who
attended a private high school have an average of 18.5% higher wages. The coefficient on
private high school interacted with experience in this second specification decreases sharply
to -0.0565. Thus, the implied effect of going to a private high school instead of a public high
school decreases from 0.170 to 0.1135 in ten years. These findings support the statistical
discrimination hypothesis since employers also base their prior assumptions about ability on
information of the high school that workers attended to.11 It is important to keep in mind
that these results apply for college graduates; hence, we find that even after going to college,
the high school an individual attended is an important determinate of earnings, in particular
at the beginning of the worker’s career.12
In column 3 I introduce the interaction between university ranking dummies and experi-
ence while excluding the effect of the interaction between ability (PSU score) and experience.
University rankings provide a signal of college quality to employers, which in turn is used to
make priors about workers ability. Therefore, we expect graduates from high ranked institu-
tions to be more productive, thus, their earnings should be higher at least at the beginning of
their careers, relative to graduates from low ranked institutions. As time passes, employers
should rely less and less on the university the worker graduated from to infer productivity
since they observe the worker’s performance. The results from the regression suggest that,
for the universities belonging to the cluster of the top old traditional universities, the effect
11Private high schools are much more selective and expensive than public schools in Chile.12See Nunez and Gutierrez (2004) for similar results, although they use data from only one university.
37
on earnings is positive, large and statistically significant.13
The important result of table 3 comes from column 4 where I include the interaction
between university ranking dummies and experience, and the interaction between PSU score
and experience. The effect of one standard deviation shift on PSU scores increases wages
4.4% right after graduation. Note that the PSU interacted with the experience coefficient
is large and statistically significant. In fact, one standard deviation shift on PSU scores
increases wages 13% after ten years in the labor market. This result captures the idea of
employers learning in the Chilean labor market as they learn about their worker’s abilities,
which is reflected on wages.
The estimates for attending a private high school and its interaction with experience
increase with respect of the estimates from column 2. That is, the effect of attending a private
high school increases wages approximately 20% for inexperienced workers and decrease to
only 9.8% for experienced workers (after ten years in the labor market).
Regarding the effect of university prestige on wages, column 4 shows that just the tier
1 traditional universities matter for wages, as it is the only coefficient that is large and
statistically significant. That is, the premium on earnings from attending a tier 1 traditional
university compared to a new private university is approximately 24% at the beginning of the
worker’s career, on average, but ten years later the effect decreases to 8.7%. Therefore, the
effect on earnings of attending a tier 1 traditional university decreases over time, supporting
the theory of the SD-EL model.14 The coefficients on tier 2 traditional universities and
tier 1 private universities are smaller and not statistically significant. Consequently, there
is no evidence of statistical discrimination nor employer learning for the tier 2 traditional
universities or the tier 1 private universities compared to the newest private universities.
13The omitted ranking dummy is “New Private Universities”.14The fact that the coefficient of Universities×experience/10 is negative while the coefficient of stan-
dardized PSU*experience/10 is positive decreases the possibility that the latter is associated with training.Human capital models and on-the-job training empirical evidence suggest that education and ability makeworkers most likely to be trained and that more educated and more able workers receive more training. Ifthis is the case, then we expect that the effect of going to college and the PSU score on wages increase overtime.
38
Finally, the coefficient of 0.0436 on PSU score and the coefficient of 0.123 on PSU score
interacted to experience suggest that a one-standard-deviation shift in the standardized PSA
score increases in effect from 4.5% to 17% in ten years, consistent with the employer learn-
ing thesis that wages increasingly reflect productivity, augmenting the correlation between
wages and ability. One might argue that the positive coefficient on the interaction of PSU
scores and experience could be attributed to an association between PSU score and training.
However, it is hard to reconcile this argument with the fact that the coefficient on the inter-
action of traditional universities and experience is negative. In conclusion, the results of the
regression exercise indicate that, in the Chilean labor market, there is evidence of statistical
discrimination on the bases of high school and university ranking, in particular if the worker
graduated from a tier 1 traditional university in Chile.
2.4 The Employer Learning Model
The objective of this section is to study the wage profiles assuming that the observed wage
differences are driven entirely by statistical discrimination on the basis of university ranking.
The results from section 2.3.2 suggest that getting a degree from a tier 1 traditional university
increases wages considerably compared to graduates from tier 2 private universities. That
is, graduating from a top university allows workers to revel their ability to employers at the
time of entry into the labor market. Therefore, I develop and estimate an employer learning
model incorporating this insight.
The model builds on that of Altonji and Pierret (2001), Lange (2007) and Arcidiacono
et al (2010).15 Wages can be described by the following equation for each year of experience
x:
wx = λx(1−Θx)PSU + ΘxPSU+ kx (2.4)
15See Appendix for the formal derivation of the model.
39
The model presented here yields an estimating equation that relates log wages (w) to
a linear function of the worker’s own ability (PSU score, which is initially not observed),
and the mean PSU score of the group she/he belongs to. Employers use workers ability to
set wages, but since ability is revealed gradually, employers give considerably more weight
to the average ability of graduates from different tiers of universities. At the beginning of
the worker’s career individual ability is not initially observable, but over time employers
give less importance to the average ability of graduates from a tier of universities and more
importance to individual ability as it becomes observable.
From equation (2.4) we see the weights that the employer places on the worker’s own
ability (PSU score) at time x is Θx and the weight on the average PSU score of the graduates
of a tier of universities is (1 − Θx). For simplicity of understanding employer learning
dynamics, I divide the sample of university graduates into two groups: graduates from tier
1 traditional universities and graduates from all other universities. The goal is to determine
the learning parameter Θx. Note that the weight employers put on the individual’s ability
measure may increase over time as employers learn about the individual’s ability, captured
by Θx, and because the true productivity of ability increases over time, captured by λx. At
time x = 0, if employers do not observe anything related to the PSU score, then they rely on
group averages to set wages and Θx = 0. Over time, employers observe more signals about
the worker’s productivity and Θx will increase while (1−Θx) will decrease. The other part of
the weight placed on average PSU and individual PSU score come from the true productivity
value of ability, λx. Suppose that ability is less important for productivity for initial jobs
than for jobs acquired later in the career path. Then, λx will have a low value initially
increasing over time, implying that employers place additional weight on both average PSU
and individual PSU.16
Following Lange (2007) and Arcidiacono et.al.(2010), I could estimate equation (2.4)
directly by regressing log wages on mean PSU score for each university ranking group and
16Notice that if λx increases rapidly enough, the weight on PSU would increase with time even thoughlearning would tend to reduce it, which happens if λx(1−Θx) > λx−1(1−Θx−1).
40
PSU score for each experience level:
wi,x = βx,ss+ βx,PSUPSUUniv.ranking + βx,PSUPSU + β′ΦΦi,t + βx + εx (2.5)
In this specification, βx captures the effect of the variables only observed by the employer
denoted earlier by kx in the general model. Φi,t is the vector of demographic controls. Notice
that for simplicity of the estimation, one could write βx,PSUPSUUniv.ranking as:
βx,PSUPSUUniv.ranking = βx,PSU(PSUUniv − PSUTop−Univ)Univ + βx,PSUPSUTop−Univ
= βx,Univ(Univ) + βx,PSUPSUTop−Uni (2.6)
Notice that PSUUniv and PSUTop−Uni are the same for everyone. Let
β∗x = βx + βx,PSUPSUTop−Uni (2.7)
Then, it follows that equation (2.5) could be written as:
wi,x = βx,ss+ βx,UnivUniv + βx,PSUPSU + β′ΦΦi,t + β∗x + εx (2.8)
The estimation of this equation is straight forward since it includes a dummy variable
Univ that takes the value of one if the worker did not graduate from a tier 1 traditional
university. Employers place weight on university ranking mainly from two sources: the term
(1 − Θx) that captures learning about ability, and the term λx that captures the changing
value of productivity of ability.17
Once we have estimated equation (2.8), we could solve for λx and Θx:
17Note that the size and value of the coefficient on Univ depends entirely on the experience profile of λxand Θx.
41
λx = βx,PSU − βx,Univ/(PSUUniv − PSUTop−Univ) (2.9)
Θx =βx,PSU
βx,PSU − βx,Univ/(PSUUniv − PSUTop−Univ)(2.10)
(2.11)
The results from the estimation could be seen in Figure 2.3. The first two plots on the top
show the estimated coefficients on Univ (non-tier 1 traditional universities graduates) and
standardized PSU score for each experience level including the 95 percent confidence interval
(in grey). The initial wage difference due to university ranking is 16 percent, vanishing almost
completely in 9 years. The effect of a 1 standard deviation increase in the PSU score starts
at 5 percent initially, increasing up to 13.5 percent after 6 years, and then decrease.
The last two plots of the bottom present the learning parameter, Θx, and the true pro-
ductivity parameter λx. I use the experience profiles to compute how much of the change
in the returns to university ranking, standardized PSU and the productive value of PSU
can be attributed to employer learning. The graph of the left shows that the parameter Θx
reaches 0.5 in 3 years, meaning that in 3 years, employers observe about 50 percent of a
worker’s ability captured by the standardized PSU score. Relating to the speed of learning,
one could say that Chilean employers learn fast.18 On the other hand, the true productivity
of a worker’s ability, captured by λx, initially increases with experience, but then decreases
(after 5 years).
The weight employers place on Univ in the wage regression due to employer learning is
(1− Θ), which starts at 1.0 and decreases to 0.5 after 3 years. Since employers initially do
not observe ability directly, they rely on the average PSU score of the university from which
the worker graduated. With time, employers learn about the worker’s productivity and have
less incentive to statistically discriminate, putting less weight on average PSU score and
18Lange (2007) shows that employers reduce the initial expectation error by 50% over the first 3 years.
42
more weight on observed ability.
2.5 Policy Implications
The belief that, in Chile, too much emphasis in the hiring process is given to high school
and university rankings is confirmed by my analysis. In fact, high school and university
rankings have an outsized effect on wages even after controlling for ability, major, gender,
geographical region, and economic sector. This is because an employer does not have perfect
information about a potential employee’s productivity and uses high school and university
rankings as a signal of ability. Therefore, it would be useful for potential employers to have
more information which will help them assess a potential employee’s productivity.
The findings of this paper support a conclusion that a PSU score can provide significant
information on a potential employee’s productivity which is not captured by high school
and university rankings or any of the other control variables. Consequently, incorporating
the PSU score into workers’ resumes could reduce the information gap between workers
and employers, decreasing the amount of statistical discrimination based on high school
and university ranking. However, since the PSU score is not informative without adequate
information on the distribution of the scores, prospective employees and employers would be
best served by having PSU percentile rankings on resumes.
Statistical discrimination in Chile based on high school and university ranking will not
disappear by incorporating PSU percentile rankings. There are other characteristics which
can be signaled by high school and university ranking which may be valued in the labor
market such as ambition, religious affiliation, or social class which cannot be measured by
the PSU.
Although the PSU percentile ranking is not a perfect measure of an applicant’s ability,
its inclusion on a resume could decrease the information gap between prospective employees
and employers. Other measures could be found which could be more useful in measuring
ability which could have a greater effect on decreasing statistical discrimination in Chile.
43
These other measures might provide a better proxy for specific abilities which are valued in
the labor market such as ambition or manual and other non-cognitive skills.
2.6 Conclusions
This paper studies the effects on earnings of university ranking and attending a private high
school in the Chilean labor market for college graduates as well as the effects of statistical
discrimination and employer learning on wages. Employers use university ranking and high
school to form assumptions about the workers’ productivity. With time, employers gather
information on workers’ output and adjust their initial assumptions, a process known as
employer learning. As the learning process goes on, the model predicts that wages should
be less determined by university ranking and high school and more by ability.
According to traditional models of ability signaling, education allows individuals to sort
themselves into groups correlated with ability. Hence, providing the university from which
the individual graduated should be a strong signal of ability, however, we see in the data that
ability is gradually revealed. Therefore, it is economically relevant to know how long it takes
for college graduates to reveal their true ability. From the employer learning model estimation
I conclude that employers decrease by 50 percent the weight they place on university rankings
when setting wages in 3 years. These results suggest that incorporating the percentile of the
PSU score (Chile’s standardized college entrance test) into workers’ resumes could reduce
the informational gap between workers and employers, reducing the amount of statistical
discrimination on the basis of high school and university ranking in the Chilean labor market.
It is noteworthy that incorporating PSU percentile rankings in employment applications
can significantly reduce the information gap between potential employees and employers and,
consequently the amount of statistical discrimination on the basis of university ranking that
workers face in the Chilean labor market for college graduates. This measure could improve
the market’s ability to appropriately assign wages.
44
2.7 Tables and Figures
Table 2.1: Descriptive Statistics, Futuro Laboral, 1995-2005
Variable Mean Standard Minimum MaximumDeviation
Real Hourly wagesa 2,412 2,255 62 34,793Log of annual wages 15.16 1.0797 11.9497 18.2796Female 0.53 0.50 0 1Experience 3.89 2.36 0 9Standardize PSU score 0.01 .99 -4.33 2.59Private High School .08 0.26 0 1a In Chilean pesos, base year 1995. This is equivalent to US$4.5 per hour.
Table 2.2: Descriptive Statistics for University Ranking
University Cluster Observations Mean PSU Standard PrivateDeviation PSU School
Top Old Traditional 10,594 0.96 0.63 0.12Traditional 30,664 -.087 0.84 0.03Old Private 5,180 -0.21 0.94 0.23New Private 7,522 -1.02 0.91 0.08
45
Table 2.3: The effects of Standardized PSU, High School and University Ranking on Wages.Dependent Variable Log Wage; OLS estimates.
(1) (2) (3) (4)
Female -0.256*** -0.256*** -0.256*** -0.256***(0.0187) (0.0187) (0.0187) (0.0186)
Standardized PSU 0.0938*** 0.0597*** 0.0939*** 0.0436**(0.0128) (0.0173) (0.0128) (0.0205)
Private High School 0.158*** 0.170*** 0.168*** 0.183***(0.0296) (0.0300) (0.0303) (0.0308)
Tier 1 Traditional Universities 0.160*** 0.161*** 0.125*** 0.214***(0.0368) (0.0369) (0.0474) (0.0571)
Traditional Universities 0.0449 0.0458 0.0185 0.0697(0.0289) (0.0289) (0.0418) (0.0462)
Tier 1 Private Universities 0.0798*** 0.0806*** 0.00731 0.0400(0.0297) (0.0298) (0.0448) (0.0455)
Private High School×Experience/10 -0.0240 -0.0565 -0.0499 -0.0897(0.0503) (0.0514) (0.0530) (0.0557)
Tier 1 Traditional Universities× 0.0848 -0.131Experience/10 (0.0764) (0.103)Traditional Universities× 0.0637 -0.0571Experience/10 (0.0700) (0.0814)Tier 1 Private Universities× 0.175** 0.0973Experience/10 (0.0868) (0.0873)Standardized PSU×Experience/10 0.0838*** 0.123***
(0.0255) (0.0359)Constant 12.21*** 12.22*** 12.24*** 12.20***
(0.199) (0.199) (0.201) (0.203)
Cubic Experience Yes Yes Yes YesObservations 38,482 38,482 38,482 38,482R-squared 0.353 0.354 0.353 0.354
Experience is modeled with a cubic polynomial. The specifications also include controls
for majors, geographical regions, economic sectors, and time effects.
Standard errors (in parenthesis) are White/Huber in order to account for possible
correlation at individual level. ***p<0.01, **p<0.05, *p<0.1
46
Figure 2.1: Higher Education Enrollment in Chile: 1983-2013
0
400,000
300,000
200,000
100,000
1983 1986 1989 1992 1995 1998 2001 2004 2007 2010 2013
Year
Traditional Universities Private UniversitiesProfessional Institutes Technical Formation Centers
Figure 2.2: Higher Education Graduates per Year of Graduation
0
10000
20000
30000
40000
50000
1998 2000 2002 2004 2006 2008 2010
Year
Traditional Universities Private UniversitiesProfessional Institutes Technical Formation Centers
47
Figure 2.3: The evolution on the effect of university ranking and PSU score on wages, thelearning parameter Θx, and the true productivity of PSU λx.
−.2
−.1
5−
.1−
.05
0.0
5
2 4 6 8 10Experience
95% CI
Coefficient on Univ
−.0
50
.05
.1.1
5
2 4 6 8 10Experience
95% CI
Coefficient on PAA
−.5
0.5
11.
5
2 4 6 8 10Experience
95% CI
Learning parameter theta
.05
.1.1
5.2
.25
2 4 6 8 10exper
95% CI
True Productivity of PAA
48
2.8 Appendix
2.8.1 Formal Derivation of the Employer Learning Model
An individual is log labor market productivity χix with x years of experience could be
specified as:
χix = f(si) + λx(qi + zi + ηi) + H(xi) (2.12)
where si, qi, zi and ηi are defined as in section 2.3. The effect of (qi, zi, ηi) on log-productivity
is captured by the parameter λx. H(xi) summarizes the experience profile of log-productivity,
which may be due to a process of investment over the life cycle.
The model needs three important assumptions. First, employers predict productivity
based on the variables si, qi and the signals they get over time. Second, the unobserved
part of ability and the information that employers have initially cannot be used to predict
zi. Third, all employers have the same information, that is, symmetric information.
The idea of the model is to interpret the variation in the experience gradient as a result
of an employer learning process. Consequently, we can focus on the problem faced by firms
trying to predict the workers productivity based on s and q and other information that
reveals as workers spend time in the labor market. In other words, the firms solve a signal-
extraction problem. This problem varies over time as more information is available. Due
to data restrictions, this paper does not consider other variables besides innate ability that
affect productivity with time, like on-the-job training.
Employers do not observe z, but we assume that they observe the average ability of
the group the worker belongs to z = E(z|s, x, Univ, .ranking). In this case, I assume
that employers know the average PSU score for each university ranking.19 Consequently,
employers predict z = z + ε.
19In Chile, the cutoff PSU scores are public, for each major in each university. Hence, employers at leasthave perfect information of the lower bound of the PSU score for each university.
49
Firms do not observe χ, but they observe a noisy signal of workers productivity, yx = z+
η+εx. The number of additional signals available comes from the experience of workers in the
labor market, Y x = (y0, y1, ..., yx−1). In each period x the employer observes yx = z+ η+ εx,
so that the number of additional measures available comes from the experience of individuals.
In the first period, x = 0, the mean of the prior of employers believes about z+ η is µ0 =
z+α1s+α2q. After x number of years, the posterior distribution is µx = (1−θx)µx−1 +θxyx
where θx is the optimal Bayesian weight that employers use on the calculation of the prior
mean. At time x, employers expectations about the worker productivity is:
Ex[χ|z, s, q, Y x] = rs+ λxq + λx[(1− θx)µx−1 + θxyx] + H(x). (2.13)
With time, employers learn about the workers true productivity, the term [(1−θx)µx−1 +
θxyx] converges to (q + z + η) and the expectation error colapses to zero.
Assuming perfect competition, symmetric information, risk neutral firms, and a spot
market for labor, we have that wages equal the expected productivity conditional on the
information at time x:
Wx(z, s, q, Yx) = Ex[exp(χ)|z, s, q, Y x] = exp
(E[χ|z, s, q, yx] +
1
2σ2x
)(2.14)
the distribution of (χ|s, q, Y x) is normal,20. Taking logs we obtain the log wage process:
wx(z, s, q, Yx) = λx[(1− θx)µx−1 + θxyx] + Cx (2.15)
Cx = rs+ λxq + H(x) + 12σ2x
Equation 2.15 links the log wages and the information (z, s, q, Y x) available to firms. It
also measures the log productivity signals Y x that will be accessible over the life cycle of the
worker. Notice however, that in the data we have (z, z, s, x) instead of (z, s, q, Y x), thus, we
define q = γ1s+ υ1 and η = γ2s+ υ2. We need also to define a linear projection of log wages
20Recall that if X is normally distributed with mean µ and variance σ2, then E[etX ] = eµt+12σ
2t2
50
on s and z at different experience levels x. Using the same notation as Lange (2007), we
have that the linear projection of log wages conditional on (z, z, s, x) is21
E∗[wx|s, z] = λx[(1− θx)E∗[µx−1|z, s] + θxE∗(yx|z, s)] + cx (2.16)
where cx = rs+ λx(γ1s+ υ1) + H(x) + 12σ2x
Substituting µx and q, we obtain the following expression for the log wages at x = 1
w1 = λ1[(1− θ1)z + θ1z] + k1 (2.17)
where k1 = λ1(1− θ1)[α1s+ α2(γ1s+ υ1)] + c1
Therefore, log wages at period x = 1 could be seen as a weighted average of the mean
group PAA and the PAA score plus a constant. Notice that the components of k1 indicate
that employers priors about workers’ ability depend on mean ability z, on schooling variables
s and the information q. For some period x > 1, wages behave as follows:
wx = λx
x∏i=1
(1− θi)z +
[1−
x∏i=1
(1− θi)
]z
+ kx (2.18)
where kx = λx∏x
i=1(1− θi)[α1s+ α2(γ1s+ υ1)] + cx
Finally, we can rewrite equation 2.18 as wx = λx(1−Θx)z + Θxz+ kx.
21The linear projection of x on y is written as E∗(x|y).
51
Chapter 3
College-Major Choice to College-Then-Major Choice (with
Chao Fu)
3.1 Introduction
In countries such as Canada and the U.S., students are admitted to colleges without declaring
their majors until later years in their college life.1 Peer students in the same classes during
early college years may end up choosing very different majors. In contrast, many (if not most)
countries in Asia, Europe and Latin America use college-major-specific admissions rules. A
student is admitted to a specific college-major pair and attends classes with peers (mostly)
from her own major upon enrollment. We label the first system where students choose majors
after enrollment by Sys.S (for sequential), and the second system where students have to
make a joint college-major choice by Sys.J (for joint).
Which system is more efficient for the same population of students? This is natural and
policy-relevant question, yet one without a simple answer. To the extent that college educa-
tion is aimed at providing a society with specialized personnel, Sys.J may be more efficient: it
allows for more specialized training, and maximizes the interaction among students with sim-
ilar comparative advantages. However, if students are uncertain about their major-specific
fits, Sys.J may lead to serious mismatch problems. Efficiency comparisons across these two
1With the exception of Quebec province.
52
admissions systems depend critically on the degree of uncertainty faced by students, the rel-
ative importance of peer effects, and student sorting behavior that determines equilibrium
peer quality. Simple cross-system comparisons are unlikely to be informative because of the
potential unobserved differences between student populations under different systems. The
fundamental difficulty, that one does not observe the same population of students under
two different systems, has prevented researchers from conducting efficiency comparison and
providing necessary information for policy makers before implementing admissions policy
reforms. We take a first step in this direction, via a structural approach.
We develop a model of student sorting under Sys.J, allowing for uncertainties over
student-major fits and endogenous peer quality that affects individual outcomes. Our first
goal is to understand the equilibrium sorting behavior among students in Sys.S. Our second
goal is to examine changes in student welfare and the distribution of educational outcomes if,
instead of college-major-specific, a college-specific admissions regime is adopted. We apply
the model to the case of Chile, where we have obtained detailed micro-level data on college
enrollment and on job market returns. Although our empirical analysis focuses on the case
of Chile, our framework can be easily adapted to other countries with similar admissions
systems.
In the model, students differ in their (multi-dimensional) abilities and educational pref-
erences, and they face uncertainty about their suitability to various majors. The cost of and
return to college education depend not only on one’s own characteristics, but may also on
the quality of one’s peers. In the baseline case (Sys.J), there are two decision periods. First,
a student makes a college-major enrollment decision, based on her expectations about peer
quality across different programs and about how well suited she is to various majors. The
choices of individual students, in turn, determine the equilibrium peer quality. In the second
period, a college enrollee learns about her fit to the chosen major and decides whether or
not to continue her studies.
In our first set of counterfactual policy experiments (Sys.S), a planner chooses optimal
53
college-specific, rather than college-major-specific, admissions policies; students make enroll-
ment decisions and postpone their choices of majors until after they learn about their fits
to various majors. Taking into account the externality of peer effects, the planner’s optimal
admissions policy guides student sorting toward the maximization of their overall welfare.
Several factors are critical for the changes in equilibrium outcomes as Sys.J switches to
Sys.S. The first factor is the degree of uncertainty students face about their major-specific
fits, which we find to be nontrivial. Indeed, postponing the choice of majors increases the
overall college retention rate from 75% in the baseline to 90% in the counterfactual.
Second, in contrast to Sys.J, where peer students are from the same major upon col-
lege enrollment, Sys.S features a much broader student body in first-period classes. While
students differ in their comparative advantages, some students have advantages over others
in multiple majors, and some majors have superior student quality. With the switch from
Sys.J to Sys.S, on the one hand, the quality of first-period peers in ”elite” majors will de-
cline; on the other hand, ”non-elite” majors will benefit from having ”elite” students in their
first-period classes. The overall efficiency depends on, among other factors, which of the two
effects dominates. Our estimation results show that for ”elite” majors, own ability is more
important than peer ability in determining one’s market return, while the opposite is true
for ”non-elite” majors. Combining this fact with the improvement in student-major match
quality, we find that the average productivity of college graduates improves in all majors
when Sys.S is adopted.
Finally, as students spend time trying out different majors, their specialized training is
delayed. Welfare comparisons vary with how costly this delay is. Average student welfare
will increase by 5%, if delayed specialization under Sys.S does not reduce the amount of
marketable skills one obtains in college compared to Sys.J. At the other extreme, if the first
period in college contributes nothing to one’s skills under Sys.S, and if a student has to make
up for this loss by extending her college life accordingly, a 0.9% loss in mean welfare will
result.
54
As there are both pro’s and con’s to each system, we also explore a hybrid policy that
combines the merits of Sys.S and Sys.J: college enrollees are allowed to choose either to
specialize upon enrollment or to postpone the choice of major until after they learn about
their fits to various majors. With this extra flexibility, the hybrid policy leads to even higher
welfare gains than Sys.S does: the higher the cost of postponing specialization, the greater
the advantage of the hybrid policy.
Our paper is closely related to studies that treat education as a sequential choice made
under uncertainty and emphasize the multi-dimensionality of human capital.2 For example,
Altonji (1993) introduces a model in which college students learn their preferences and
probabilities of completion in two fields of study. Arcidiacono (2004) estimates a structural
model of college and major choice in the U.S., where students learn about their abilities via
test scores in college before settling down to their majors. As in our paper, he allows for
peer effects.3 Focusing on individual decisions, he treats peer quality as exogenous.4
While this literature has focused on individual decision problems, our goal is to study
the educational outcomes for the population of students, and to provide predictions about
these outcomes under counterfactual policy regimes. One cannot achieve this goal without
modeling student sorting in an equilibrium framework, because peer quality may change as
students re-sort themselves under different policy regimes.
In its emphasis on equilibrium structure, our paper is related to Epple, Romano and Sieg
(2006) and Fu (2013). Both papers study college enrollment in a decentralized market, where
colleges compete for better students.5 Given our goal of addressing efficiency-related issues,
2Examples of theoretical papers include Manski (1989) and Comay, Melnick and Pollachek (1973).3There is a large and controversial literature on peer effects. Methodological issues are discussed in
Manski (1993), Moffitt (2001), Brock and Durlauf (2001), and Blume, Brock, Durlauf and Ioannides (2011).Limiting discussion to recent research on peer effects in higher education, Sacerdote (2001) and Zimmerman(2003) find peer effects between roommates on grade point averages. Betts and Morell (1999) find thathigh-school peer groups affect college grade point average. Arcidiacono and Nicholson (2005) find no peereffects among medical students. Dale and Krueger (1998) have mixed findings.
4Stinebrickner and Stinebrickner (2011) use expectation data to study student’s choice of major. Altonji,Blom and Meghir (2012) provides a comprehensive survey of the literature on the demand for and return toeducation by field of study in the U.S.
5Epple, Romano and Sieg (2006) model equilibrium admissions, financial aid and enrollment. Fu (2013)models equilibrium tuition, applications, admissions and enrollment.
55
and the fact that colleges in Sys.J countries are often coordinated, we study a different type
of equilibrium, where the players include students and a single planner. In this centralized
environment, we abstract from the determination of tuition, which is likely to be more
important in decentralized market equilibria studied by Epple, Romano and Sieg (2006) and
Fu (2013).6 Instead, we emphasize some other aspects of college education that are absent in
these two previous studies but are more essential to our purpose: the multi-dimensionality
of abilities and uncertainties over student-major fits. Moreover, we relate college education
to job market outcomes, which is absent in both previous studies.
Studies on the comparison across different admissions systems are relatively sparse. Ofer
Malamud has a series of papers that compare the labor market consequences between the
English (Sys.J) and Scottish (Sys.S) systems.7 Malamud (2010) finds that the average earn-
ings are not significantly different between the two countries, while Malamud (2011) finds
that individuals from Scotland are less likely to switch to an unrelated occupation compared
to their English counterparts, suggesting that the benefits to increased match quality are
sufficiently large to outweigh the greater loss in skills from specializing early. With the caveat
that students in two countries may differ in unobservable ways, his findings contribute to our
understanding of the relative merits of the two systems. Our paper aims at pushing the fron-
tier toward comparing the relative efficiency of alternative systems for the same population
of students.
The rest of the paper is organized as follows: Section 2 provides some background in-
formation about education in Chile, which guides our modeling choices. Section 3 lays out
the model. Section 4 describes the data. Section 5 describes the estimation followed by the
empirical results. Section 7 conducts counterfactual policy experiments. The last section
concludes the paper. The appendices in the paper and online contains additional details and
6College-provided financial aid and scholarships are rare in Chile.7“English students usually follow a narrow curriculum that focuses on the main field and allows for little
exposure to other fields. Indeed, most universities in England require students who switch fields of study tostart university anew (though several do allow for some limited switching across related fields).” (Malamud(2010)).
56
tables.
3.2 Background: Education in Chile
There are three types of high schools in Chile: scientific-humanist (regular), technical-
professional (vocational) and artistic. Most students who want to pursue a college degree
attend the first type. In their 11th grade, students choose to follow a certain academic track
based on their general interests, where a track can be humanities, sciences or arts. From
then on, students receive more advanced training in subjects corresponding to their tracks.
The higher education system in Chile consists of three types of institutions: universities,
professional institutes, and technical formation centers. Universities offer licentiate degree
programs and award academic degrees. In 2011, total enrollment in universities accounts for
over 60% of all Chilean students enrolled in the higher education system.8 There are two
main categories of universities: the 25 traditional universities and the over 30 non-traditional
private universities. Traditional universities comprise the oldest and most prestigious two
universities, and institutions derived from them. They are coordinated by the Council of
Chancellors of Chilean Universities (CRUCH), and receive partial funding from the state.
In 2011, traditional universities accommodated about 50% of all college students pursuing a
bachelor’s degree.
The traditional universities employ a single admission process: the University Selection
Test (PSU), which is very similar to the SAT test in the U.S. The test consists of two
mandatory exams, math and language, and two additional specific exams, sciences and
social sciences. Taking the PSU involves a fixed fee but the marginal cost of each exam is
zero.9 Students following different academic tracks in high school will take either one or both
specific exam(s). Together with the high school GPA, various PSU test scores are the only
components of an index used in the admissions process. This index is a weighted average
8Enrollments in professional institutes and technical formation centers account for 25.7% and 13.7%respectively.
9In 2011, the fee was 23, 500 pesos (1 USD is about 484 Chilean pesos).
57
of GPA and PSU scores, where the weights differ across college programs. A student is
admitted to a specific college-major pair if her index is above the cutoff index required by
that program. That is, college admissions are college-major specific. A student must choose
a college-major pair jointly.
In our analysis, colleges refer only to the traditional universities for several reasons. First,
we wish to examine the consequences of a centralized reform to the admissions process.
This experiment is more applicable to the traditional universities, which are coordinated
and state-funded, and follow a single admissions process. Second, non-traditional private
universities are usually considered inferior to the traditional universities; and most of them
follow (almost) open-admissions policies. We consider it more appropriate to treat them as
part of the outside option for students in our model. Finally, we have enrollment data only
for traditional universities.
Transfers across programs are rare in Chile. Besides a minimum college GPA requirement
that differs across programs, typical transfer policies require that a student have studied at
least two semesters in her former program and that the contents of her former studies be
comparable to those of the program she intends to transfer to. In reality, the practice is even
more restrictive. According a report by the OECD, ”students must choose an academic field
at the inception of their studies. With a few exceptions, lateral mobility between academic
programmes is not permitted, even within institutions. This factor, combined with limited
career orientation in high school, greatly influences dropout rates in tertiary education.”10
The same report also notes that the highly inflexible curriculum design further limits the
mobility between programs.11 If a student dropped out in order to re-apply to other programs
in traditional universities, she must re-take the PSU test.12
It is worth noting that the institutional details in Chile are similar to those in many
10Reviews of National Policies for Education: Tertiary Education in Chile (2009) OECD, page 146.11”A review of the curricular grid shows a rigid curriculum with very limited or no options (electives
classes) once the student has chosen an area of specialisation. In some cases, flexibility is incorporated bymaking available a few optional courses within the same field of study.” page 143.
12This was true for cohorts in our sample. A new policy was announced recently that allows students touse one-year-old PSU test results for college application.
58
other countries, such as many Asian countries (e.g., China and Japan) and European coun-
tries (e.g., Spain and Turkey), in terms of the specialized tracking in high school, a single
admissions process and rigid transfer policies. The online Appendix 3.11.4 provides further
descriptions of the systems in these other countries.
3.3 Model
This section presents our model of Sys.J, guided by the institutional details described above.
A student makes her college-major choice, subject to college-major-specific admissions rules.
After first period in college, she learns about her fit to her major and decides whether or not
to continue her studies.
3.3.1 Primitives
There is a continuum of students with different gender (g), family income (y), abilities (a) and
academic interests. There are J colleges, each with M majors. Let (j,m) denote a program.
Admissions are subject to program-specific standards. An outside option is available to all
students.
3.3.1.1 Student Characteristics
A student comes from one of the family income groups y ∈ low, high , 13 has multi-
dimensional knowledge in subjects such as math, language, social science and science, sum-
marized by s = [s1, s2, ..., sS], the vector of test scores. Various elements of such knowledge
are combined with the publicly known major-specific weights to form major-specific (pre-
college) ability,
am =S∑l=1
ωmlsl,
13y = low if family income is lower than the median among Chilean households.
59
where ωm = [ωm1, ..., ωmS] is the vector of major-m-specific weights and∑
l = 1Sωml = 1.
ωm’s differ across majors: for example, an engineer uses math knowledge more and language
knowledge less than a journalist. Notice that abilities are correlated across majors as multi-
dimensional knowledge is used in various majors.
Given the different academic tracks they follow in high school, some students will consider
only majors that emphasize knowledge in certain subjects, while some are open to all majors.
Such general interests are reflected in their abilities.14 Let Ma be the set of majors within
the general interest of a student with ability vector a.15 Denote student characteristics that
are observable to the researcher, i.e., the vector of abilities, family income and gender by the
vector x ≡ [a, y, g] , and its distribution by Fx (·).
3.3.1.2 Skills and Wages
Skill attainments in college depends on a student’s major-specific ability (am), peer quality
(Ajm), and how efficient/suitable she is for the major.16 In particular, Ajm is the average
major-m ability of enrollees in (j,m) .17 A student faces uncertainty in making enrollment
decisions because major-specific efficiency is revealed to her only after she takes courses in
that major. Denote one’s major-specific efficiency as ηmm ˜i.i.d.Fη(·). The human capital
14Without increasing the test fee, taking both the science and the social science exams will only enlarge astudent’s opportunity set. A student who does not take the science exam will not be considered by programsthat require science scores, but her admissions to programs that do not require science scores will not beaffected even if she scores poorly in science. However, some students only take either the science or the socialscience exam, we view this as indication of their general academic interests. We treat students’ preferencesand abilities as pre-determined.
15Letting am = n/a if a student does not take the subject test required by major m, Ma is given by
Ma = m ∈ 1, ...,M : am 6= n/a.
16Peer quality may affect market returns via different channels, such as human capital production, statis-tical discrimination, social networks, etc. Our data do not allow us to distinguish among various channels.For ease of illustration, we describe peer quality in the framework of human capital production.
17Arguably, the entire distribution of peer ability may matter. For feasibility reasons, we follow thecommon practice in the literature and assume that only the average peer quality matters.
60
production function is18
hm (am, ηm, Ajm) = aγ1mm Aγ2mjm ηm.
Wages are major-specific stochastic functions of one’s human capital (hence of am, ηm, Ajm),
work experience (τ) and one’s observable characteristics besides ability, where the random-
ness comes from a transitory wage shock ζτ . Denote the wage rate for a graduate from
program (j,m) by wm (τ, x, ηm, Ajm, ζτ ) .19
3.3.1.3 Consumption Values and Costs
The non-pecuniary consumption value of a particular major enters one’s utility both in
college and in the workforce. This value depends on one’s ability: an individual with higher
ability am may find it more enjoyable (less costly) to study in major m and work in major-m
related jobs. We also allow preferences to differ across genders: on average, some majors
may appeal more to females than to males.20 In addition, each student has idiosyncratic
tastes for majors, represented by a random vector ε1 = ε1mm .21 In sum, the per-period
consumption value of major m is
vm(x, ε1m) = υmI(female) + λ1mam + λ2ma2m + ε1m,
where the mean major-specific consumption values for males are normalized to zero, and
υm is the mean major-m value for females. λm’s measure how consumption values in major
m change with major-specific abilities.22 Besides the consumption value from her major, a
student also derives consumption value from her program while in college. Let ε2jm be one’s
taste for program (j,m), and Fε (·) be the joint distribution of the unobserved idiosyncratic
18Notice that hm (·) represents the total amount of marketable skills. As such, hm (·) may be a combinationof pure major-specific skill and general skill.
19Recall that a ∈ x.20Gender-specific preferences may arise from not only individual tastes, but also social norms and other
channels. We label the combination of all these potential factors as ”gender-specific tastes.”21We will adopt the convention that ε1m = −∞ if m is not in one’s general interest.22In the estimation, we restrict λ2m to be the same across majors.
61
tastes for major and for programs ε = [ε1, ε2] . An individual student’s tastes are correlated
across majors within a college, and across colleges given the same major.
The monetary and effort costs of attending program (j,m), governed by the cost function
Cjm (x,Ajm) , depend on student characteristics x and peer quality Ajm. In particular, we
allow the same tuition level to have different cost impacts on students from different family
income groups, so as to capture possible credit constraints.23 The cost also depends on one’s
own ability a ∈ x, as well as peer quality Ajm. For example, it may be more challenging
to attend a class with high-ability peers because of direct peer pressure and/or curriculum
designs that cater to average student ability. In sum, the per-period net consumption value
of attending (j,m) is
vjm(x, ε, Ajm) = vm(x, ε1m) + ε2jm − Cjm(x,Ajm).
3.3.1.4 Timing
There are three stages in this model.
Stage 1: Students make college-major enrollment decisions.
Stage 2: A college enrollee in major m observes her major-specific efficiency ηm, and chooses
to stay or to drop out at the end the first period in college. Student choice is restricted to be
between staying and dropping out, which is consistent with the Chilean practice mentioned
in the background information section. Later in a counterfactual experiment, we explore the
gain from more flexible transfer policies.24
Stage 3: Stayers study one more period in college and then enter the labor market. The
following table summarizes the information at each decision period.
23If we had information on how students financed their college education, we would have modeled creditconstraints more explicitly.
24We also assume that an enrollee fully observes her efficiency in her major by the end of Stage 2 (2 yearsin college). It will be interesting to allow for gradual learning. Given the lack of information on studentperformance in college, we leave such extensions to future work.
62
Information Set: Sys.J
Stage Student Researcher
1: Enrollment x, ε x
2: Stay/Drop out x, ε, ηm x
3.3.2 Student Problem
This subsection solves the student’s problem backwards.25
3.3.2.1 Continuation Decision
After the first college period, an enrollee in (j,m) observes her major-m efficiency ηm, and
decides whether to continue studying or to drop out. Let Vd (x) be the value of dropping out,
a function of student characteristics.26 Given peer quality Ajm, a student’s second-period
problem is
ujm(x, ε, ηm|Ajm) =
max
vjm(x, ε, Ajm) +
T∑τ ′=3
βτ′−2 [Eζ (wm(τ − 3, x, ηm, Ajm, ζ)) + vm(x, ε)] , Vd (x)
.
If the student chooses to continue, she will stay one more period in college, obtaining the
net consumption value vjm(x, ε, Ajm), and then enjoy the monetary and consumption value
of her major after college from period 3 to retirement period T = 45, discounted at rate β.
Let δ2jm(x, ε, ηm|Ajm) = 1 if an enrollee in program (j,m) chooses to continue in Stage 2.
25To ease the notation, we present the model as if each period in college lasts one year. In practice, wetreat the first two years in college as the first college period in the model, and the rest of college years asthe second period, which differs across majors. Students’ value functions are adjusted to be consistent withthe actual time framework. See the Appendix 3.10.2.1 for details.
26Ideally, one would model the dropout and the outside options in further detail, by differentiating variouschoices within the outside option: working, re-taking the PSU test and re-applying the next year, or attendingan open admissions private college. Unfortunately, we observe none of these details. In order to make themost use of the data available, we model the values of the dropout and the outside options as functions ofstudent characteristics. These value functions, hence student welfare, are identified up to a constant because1) we have normalized the non-pecuniary value of majors to zero for males and 2) a student’s utility ismeasured in pesos and we observe wages. See Appendix 3.10.1 for functional forms.
63
3.3.2.2 College-Major Choice
Under the Chilean system, program (j,m) is in a student’s choice set if only if am ≥ a∗jm,
the (j,m)-specific admissions cutoff. Given the vector of peer quality in every program
A ≡ Ajmjm, a student chooses the best among the programs she is admitted to and the
outside option with value V0 (x), i.e.,
U(x, ε|a∗, A) = max
max
(j,m)|am≥a∗jmβEηm(ujm(x, ε, ηm|Ajm)) + vjm(x, ε, Ajm), V0 (x)
.
Let δ1jm(x, ε|a∗, A) = 1 if program (j,m) is chosen in Stage 1.27
3.3.3 Sorting Equilibrium
Given cutoffs a∗, a sorting equilibrium consists of a set of student enrollment and continuation
strategiesδ1jm(x, ε|a∗, ·), δ2
jm(x, ε, ηm|·)jm, and the vector of peer quality A = Ajmjm ,
such that28
(a) δ2jm(x, ε, ηm|Ajm) is an optimal continuation decision for every (x, ε, ηm);
(b)δ1jm (x, ε|a∗, A)
jm
is an optimal enrollment decision for every (x, ε) ;
(c) A is consistent with individual decisions such that, for every (j,m) ,
Ajm =
∫x
∫εδ1jm(x, ε|a∗, A)amdFε (ε) dFx(x)∫
x
∫εδ1jm(x, ε|a∗, A)dFε (ε) dFx(x)
.
Finding a sorting equilibrium can be viewed as a classical fixed-point problem of an
equilibrium mapping from the support of peer quality A to itself. Online Appendix 3.11.5
proves the existence of an equilibrium in a simplified model. Appendix 3.10.3 describes our
27For a student, the enrollment choice is generically unique.28A sorting equilibrium takes the admissions cutoffs as given. We choose not to model the cutoff rules
under the status quo (Sys.J) because our goal is to consider a different admissions regime (Sys.S) andcompare it with the status quo. For this purpose, we need to understand student sorting and uncover theunderlying student-side parameters, which can be accomplished by estimating the sorting equilibrium model.We also need to model how the admissions policies are chosen under Sys.S, which we do in the counterfactualexperiments.
64
algorithm to search for equilibria, which we always find in practice.29
3.4 Data
3.4.1 Data Sources and Sample Selection
Our first data source is the Chilean Department of Evaluation and Educational Testing
Service, which records the PSU scores and high school GPA of all test takers and the college-
major enrollment information for those enrolled in traditional universities. Besides multiple
years of macro data, we also obtained micro-level data for the 2011 freshmen cohort. There
were 247, 360 PSU test takers in 2011. We focus on the 159, 365 students, who met the
minimum requirement for admission to at least one program and who were not admitted
based on special talents such as athletes.30 From the 159, 365 students, we draw 10, 000
students as our final sample due to computational considerations.31,32
Our second data source is Futuro Laboral, a project by the Ministry of Education that
follows a random sample of college graduates (classes of 1995, 1998, 2000 and 2001). This
panel data set matches tax return information with students’ college admissions information,
so we observe the worker’s annual earnings, months worked, high school GPA, PSU scores,
college and major. For each cohort, earnings information is available from graduation until
2005. We calculated the monthly wage as annual earnings divided by the number of months
worked, and the annual wage as 12 times the monthly wage, measured in thousands of
deflated pesos. For each major, we trimmed wages at the 2nd and the 98th percentiles.
The two most recent cohorts have the largest numbers of observations and they have very
29Uniqueness of the equilibrium is not guaranteed. Our algorithm deals with this issue using the fact thatall equilibrium objects are observed in the data.
30Ineligible students can only choose the outside option and will not contribute to the estimation.31For each parameter configuration, we have to solve for the equilibrium via an iterative procedure as
discussed in the appendix. Each iteration involves numerically solving the student’s problem and integratingout unobserved tastes. This has to be repeated for every student in the sample, since each of them has adifferent x.
32Some options are chosen by students at much lower frequency than others. To improve efficiency, weconduct choice-based sampling with weights calculated from the distribution of choices in the population of159, 365 students. The weighted sample is representative. See Manski and McFadden (1981).
65
similar observable characteristics. We combined these two cohorts to obtain our measures
of abilities and wages among graduates from different college-major programs. We also use
the wage information from the two earlier cohorts to measure major-specific wage growth at
higher work experience levels. The final wage sample consists of 19, 201 individuals from the
combined 2000-2001 cohorts, and 10, 618 from the 1995 and 1998 cohorts.
The PSU data contains information on individual ability, enrollment and peer quality, but
not the market return to college education. The wage data, on the other hand, does not have
information on the quality of one’s peers while in college. We combine these two data sets
in our empirical analysis. We standardized the test scores according to the cohort-specific
mean and standard deviation to make the test scores comparable across cohorts. Thus, we
have created a synthetic cohort, the empirical counterpart of students in our model.33
The wage data from Futuro Laboral contains wage information only in one’s early career.
To obtain information on wages at higher experience levels, we use cross-sectional data from
the Chilean Characterization Socioeconomic Survey (CASEN), which is similar to the Cur-
rent Population Survey in the U.S. We compare the average wages across different cohorts of
college graduates to obtain measures of wage growth at different experience levels. Although
they are not from panel data, such measures restrict the model from predicting unrealistic
wage paths in one’s later career in order to fit other aspects of the data.
Our last data source is the Indices database from the Ministry of Education of Chile.
It contains information on college-major-specific tuition, weights (ωml) used to form the
admission score index, the admission cutoffs(a∗jm), and the numbers of enrollees in con-
secutive years.
33Given data availability, we have to make the assumption that there exists no systematic difference acrosscohorts conditional on comparable test scores. This assumption rules out, for example, the possibility thatdifferent cohorts may face different degrees of uncertainties over student-major match quality η.
66
3.4.2 Aggregation of Academic Programs
For both sample size (of the wage data) and computational reasons, we have aggregated spe-
cific majors into eight categories according to the area of study, coursework, PSU require-
ments and average wage levels.34 The eight aggregated majors are: Business, Education,
Arts and Social Sciences, Sciences, Engineering, Health, Medicine and Law.35 We also ag-
gregated individual traditional universities into three tiers based on admissions criteria and
student quality.36 Thus, students have 25 options, including the outside option, in making
their enrollment decisions.37
Table 3.1 shows some details about the aggregation. The third column shows the quality
of students within each tier, measured by the average of math and language scores. Treating
each college-level mean score as a variable, the parentheses show the cross-college standard
deviations of these means within each tier. The last two columns show similar statistics for
total enrollment and tuition. Cross-tier differences are clear: higher-ranked colleges have
better students, larger enrollment and higher tuition.
3.4.3 Summary Statistics
This subsection provides summary statistics for the aggregated programs based on our final
sample. Table 3.2 shows summary statistics by enrollment status. Both test scores and
graduate wages increase with the ranking of tiers. Over 71% of students in the sample were
not enrolled in any of the traditional universities and only 5% were enrolled in the top tier.38
34Although we can enlarge the sample size of the PSU data by including more students, we are restrictedby the sample size of the wage data. Finer division will lead to too few observations in each program.
35All these majors, including law and medicine, are offered as undergraduate majors in Chile. Medicine andhealth are very different majors: medicine produces doctors and medical researchers while health producesmainly nurses.
36The empirical definitions of objects such as program-specific retention rates are adjusted to be consistentwith the aggregation, see Appendix 3.10.2.2 for details.
37As a by-product of the aggregation of programs, the assumption that students cannot transfer becomeseven more reasonable because any transfer across the aggregated programs will involve very different pro-grams.
38For students not enrolled in the traditional universities, we have no information other than their testscores.
67
Compared to average students, females (53% of the sample) are less likely to enroll in college
(25% v.s. 28%) and a larger fraction of female enrollees are enrolled in the lowest tier (35%
v.s. 32%).
Table 3.3 shows enrollee characteristics by major. The majors are listed in the order
of the observed average starting wages.39 This rank is also roughly consistent with the
rank of average test scores across majors. Medical students score higher in both math
and language than all other students, while education students are at the other extreme.
Comparative advantages differ across majors. For example, law and social science majors
have clear comparative advantage in language, while the opposite is true for engineering and
science majors. The last two columns show the fraction of students in each major among,
respectively, all enrollees and female enrollees. Females are significantly more likely to major
in education and health but much less so in engineering.
3.5 Estimation
The model is estimated via simulated generalized method of moments (SGMM). For a
given parameter configuration, we solve for the sorting equilibrium and compute the model-
predicted moments. The parameter estimates minimize the weighted distance between
model-predicted moments (M (Θ)) and data moments(Md):
Θ = arg minΘ
(M (Θ)−Md
)′W(M (Θ)−Md
),
where Θ is the vector of structural parameters, and W is a positive-definite weighting ma-
trix.40 Θ includes parameters governing the distributions of student tastes, the distribution
39See Figure 1 in the online appendix for wage paths by major.40In particular, W is a diagonal matrix, the (k, k)
thcomponent of which is the inverse of the variance of
the kth moment, estimated from the data. To calculate the optimal weighting matrix, we would have tonumerically calculate the derivatives of the GMM objective function, which may lead to inconsistency due tonumerical imprecision. So we choose not to use the optimal weighting matrix. Under the current weightingmatrix, our estimates will be consistent but less efficient. However, as shown in the estimation results, theprecision of most of our parameter estimates is high due to the relatively large sample size.
68
of major-specific efficiency shocks, the human capital production function, the wage func-
tion, the consumption values and costs of colleges and majors, and the values of the outside
and the dropout options.
Given that the equilibrium peer quality is observed and used as target moments, we
have also estimated the parameters without imposing equilibrium conditions, which boils
down to an individual decision model. We deem model consistency critical for the empirical
analysis we do, so we focus on the first approach because it favors parameters that guarantee
equilibrium consistency over those that may sacrifice consistency for better values of the
SGMM objective function.41
3.5.1 Target Moments
The combined data sets contain information on various predictions of the model, based on
which we choose our target moments. Although the entire set of model parameters work
jointly to fit the data, one can obtain some intuition about identification from considering
various aspects of the data that are more informative about certain parameters than others.
The PSU data contains information that summarizes the sorting equilibrium: program-
specific enrollment and peer quality (Moments 1 (a) and 2 (a) as listed below). It also
provides information critical for the identification of student preferences and costs. The
different enrollment choices made by students with different demographics (Moments 1 (a))
reveal information on the effects of these characteristics on preferences and costs. Students
also differ in their unobservables. Among similar students who pursued the same major,
some chose higher-ranked colleges and others lower-ranked colleges (Moments 1 (b)). This
informs us of the dispersion in tastes for colleges. Similar students within the same college
made different major choices (e.g., more lucrative majors v.s. less lucrative ones), reflecting
the dispersion of their tastes for majors (Moments 1 (c)). Together with student enrollment
41Differences between the estimates from these two estimation approaches exist but are not big enoughto generate significant differences in model fits or in counterfactual experiments. The results from thealternative estimation approach are available upon request.
69
choices (Moments 1), the distribution of abilities within a program (Moments 2 (a) and 2 (b))
is informative about the relationship between peer quality and effort costs. For example,
if the relationship is too weak (strong), then more (fewer) students who are eligible will be
drawn to programs with better peers in order to benefit from the positive peer effects on
wages, which will increase (reduce) the dispersion of abilities in these programs.
In the wage data, the relationship between wages and student’s observable characteris-
tics (Moments 4 (b) and 4 (c)) provides key information about major-specific human capital
production and wage functions. College retention rates (Moments 2 (c)), the ability dif-
ference between enrollees and those who stayed (Moments 2 (a) and 3), together with the
dispersion of wages among workers with similar observables (Moments 4), inform us of the
dispersion of major-specific efficiency shocks. For example, lower dispersion in those shocks
would lead to higher retention rates and lower wage dispersion; moreover, since pre-college
ability is relatively more important in this case, conditional on retention rate, the ability
difference between enrollees and graduates should be larger. Finally, Moments 5 inform us
of wage growth over the life cycle. In total, we estimate 88 free parameters by matching 448
moments.42
1. Enrollment status:
(a) Fractions of students across tier-major (j,m) overall, among females and among
low-family-income students.
(b) Fractions of students enrolled in (j,m) with am ≥ a∗j′m where j′ is a tier ranked
higher than j and am ≥ a∗jm guarantees that the student can choose (j′,m) .
(c) Fractions of students enrolled in j with am ≥ a∗jm by (j,m) .
2. Ability by enrollment status:
42We have also conducted Monte Carlo exercises to provide some evidence of identification. In particular,we first simulated data with parameter values that we choose, treated as the ”truth” and then, using momentsfrom the simulated data, started the estimation of the model from a wide range of initial guesses of parametervalues. In all cases, we were able to recover parameter values that are close to the ”truth.”
70
(a) First and second moments of major-m ability (am) by (j,m) .
(b) Mean test scores among students who chose the outside option.
(c) Retention rates by (j,m) calculated from enrollments in the college data.
3. Graduate ability: First and second moments of major-m ability among graduates by
(j,m) .
4. Starting wage:
(a) First and second moments of log starting wage by (j,m).
(b) First moments of log starting wage by (j,m) for females.
(c) Cross moments of log starting wage and major-specific ability by (j,m) .
5. Wage growth:
(a) Mean of the first differences of log wage by major for experience τ = 1, ..., 9.
(b) From CASEN: first difference of the mean log wage at τ = 10, ..., 40.
3.6 Results
3.6.1 Parameter Estimates
This section reports the estimates of parameters of major interest. Tables A2.1-A2.4 in the
appendix report the estimates of other parameters. Standard errors (in parentheses) are
calculated via bootstrapping.43
The first eight rows of Table 3.4 show the estimates of the parameters in the human
capital production function, which also measure the elasticities of wages with respect to peer
43Calculating standard errors via standard first-order Taylor expansions might be problematic because wehave to use numerical method to calculate the derivatives of our GMM objective function. We took 200bootstrap iterations. Given the sample size (10,000) and the sampling scheme described in footnote 30, theprecison of most of our estimates is high.
71
ability and own ability. The left panel shows significant differences in the importance of
peer ability across majors: the elasticity of wage with respect to peer quality is over 1.4 in
business and science, while only 0.01 in medicine.44 Considering both the left and the right
panels of Table 3.4, we find that the relative importance of peer ability versus own ability
differs systematically across majors although no restriction has been imposed in this respect.
In majors with the highest average wages, the elasticity of wage with respect to peer ability
is at most half of that with respect to own ability, while the opposite is true for education,
the major with the lowest average wage.45 This finding has major implications for welfare
analysis as Sys.J switches to Sys.S, because the quality of first-period peers will decline for
”elite” majors, while increase for ”non-elite” majors. Table 3.4 suggests that the former
negative effect is likely to be small, while the latter positive effect may be significant.
The last row of Table 3.4 shows the dispersion of major-specific efficiency shocks. To
understand the magnitude of this estimate (0.6) , imagine two counterfactual scenarios: first,
if a student’s fit to her major were improved by one standard deviation, her starting wage
would increase by about 40% ceteris paribus. Second, if the dispersion of these shocks were
reduced by 25% from 0.6 to 0.45, the overall college retention rate would increase from
75% in the baseline to about 85%. Clearly, students face non-trivial uncertainties over their
major-specific fits.
Table 3.5 reports parameter estimates for major-specific consumption values. The first
two columns show how these values vary with own ability and peer ability. The three majors
with highest average wages and social science major are the most satisfying for high ability
44As mentioned earlier, our model is silent about why peer ability affects one’s market return. Thesereasons are likely to differ across majors. For example, the high elasticity of wage with respect to peerquality in business may arise because the social network one forms in college is highly valued in the businessprofession. It may be surprising to see small effects of both own ability and peer ability in medicine. Onepossible reason is that compared to their counterpart from lower-tier medical schools who have lower pre-college ability, a higher fraction of graduates from top medical schools work in research/education-relatedjobs and/or in the public sector, where wages are lower than those in the private sector.
45One possible reason for this finding is statistical discrimination on the labor market. For example, in lawand medicine, the practice of licensing and residency/intership reduces the need for statistical discrimination,while the opposite may be true for the education major since the productivity of a potential teacher is difficultto judge from one’s own characteristics.
72
individuals. Except for engineering, effort costs in these majors are also the most responsive
to peer abilities. This is especially true for law programs, which constantly put students
in competitive situations such as those in case studies. Empirically, the high cost helps
to explain why some law-eligible students chose other majors despite of the expected high
wage for law students. Similarly, the consumption value in education is found to be low
because the low wages in education are not sufficient to explain why most students who were
eligible for the education major (the least selective major) chose other majors. The last
column of Table 3.5 shows that on average, females have higher tastes for the conventionally
”feminine” majors: health and education, but lower tastes for all the other majors. In the
online Appendix 3.11.1, we show that when females are endowed with the same preferences
as males, there will no longer exist majors that are obviously dominated by one gender.
However, the difference in comparative advantages across genders also plays a nontrivial role
in explaining their different enrollment patterns.46
3.6.2 Model Fit
Overall, the model fits the data well. Table 3.6 shows the fits of enrollment by tier, for
all students and for females.47 The model slightly underpredicts the fraction of students
enrolled in the top tier.
Table 3.7 shows the distribution of enrollees across majors. The fit for the distribution
among all enrollees is very close. For female enrollees, the model underpredicts the fraction
in social sciences and overpredicts that in education.
Table 3.8 (Table 3.9) shows the fits of average student ability and retention rates by tier
(major).48 All ability measures are closely matched. The retention rate is over-predicted for
Tier 3 in Table 3.8 and for science in Table 9.
46The importance of gender-specific preferences has been noted in the literature. For example, Zafar (2009)finds that preferences play a strong role in the gender gap of major choices in the U.S.
47The fits of enrollment patterns for students with low family income are in the online appendix.48The retention rates reported seem to be high for two reasons. First, we focus on the traditional colleges,
which are of higher quality than private colleges. Second, consistent with our data aggregation, a student issaid to be retained in (j,m) if she stays in any specific program within our (j,m) category.
73
3.7 Counterfactual Policy Experiments
We first introduce two counterfactual admissions regimes, Sys.S and a hybrid of Sys.S and
Sys.J, providing overall cross-system comparisons. Then, we conduct a milder policy change
that allows students one chance to switch programs within Sys.J. Finally, we examine the
effects of admissions systems in detail, focusing on the contrast between Sys.J (the baseline)
and Sys.S.
3.7.1 Overall Comparison
3.7.1.1 Sys.S
Under Sys.S, students choose their majors after they learn about their fits. We solve a plan-
ner’s problem, one who aims at maximizing total student welfare by setting college-specific,
rather than college-major-specific, admission policies.49 The constraints for the planner in-
clude: 1) a student admitted to a higher-tier college is also admitted to colleges ranked lower,
and 2) the planner can use only ability a to distinguish students. These two restrictions keep
our counterfactual experiments closer to the current practice in Chile in dimensions other
than the college-specific versus college-major-specific admissions. Restriction 1 prevents the
planner from assigning a student to the college that the planner deems optimal, which is
both far from the current Chilean practice and also may lead to mismatches due to the
heterogeneity in student tastes. Restriction 2 rules out discrimination based on gender or
family income.
There are four stages in this new environment:
Stage 1: The planner announces college-specific admissions policies.
Stage 2: Students make enrollment decisions, choosing one of the colleges they are admitted
to or the outside option.
49The planner takes into account tuition and effort costs for the student in her optimization problem.To maximize social welfare, one would also include other costs of college education, for example, costs forcolleges that are not fully covered by tuition revenue. This will be a relatively straightforward extension yetone that requires information that is unavailable to us.
74
Stage 3: An enrollee takes courses in majors within her general interests and learn her
efficiency levels in these majors. Then, she chooses one of these majors or drops out.50
Stage 4: Stayers spend one more period studying in the major of choice and then enter the
labor market.
Information and Decision: Sys.S
Stage 1 Stage 2 Stage 3
Info Planner Choice Info Student Choice Info Student Choice
a Admissions x, ε College (j) x, ε, ηmm∈MaOne major/Dropout
The planner acts as the Stackelberg leader in this game. Instead of simple unidimensional
cutoffs, optimal admissions policies will be based on the whole vector of student ability a. To
calculate the benefit of admitting a student of ability a to a certain set of colleges, the planner
has to first form expectation of the student’s enrollment and major choices, integrating out
the student’s characteristics and tastes that are unobservable to the planner, and the major-
specific efficiency shocks. Then, the planner calculates the expected value for this individual
and her effect on peer quality. Peer quality matters both because it affects the market return
and because it affects student effort costs. Overall, the planner’s optimal admissions policies
lead student sorting toward the maximization of total student welfare.51 Online Appendix
3.11.3 contains formal theoretical details.
To compare welfare, one factor that deserves special attention is the potential loss of
major-specific human capital due to the delay in specialized training.52 The data we have
does not allow us to predict the exact change in human capital associated with the shift
of admissions regimes because we do not observe the return to partial college education or
student performance in college. However, it is still informative to provide bounds on welfare
gains under Sys.S by considering various possible scenarios. In this paper, we explore two
50In this section, students are free to choose majors. Section 7.2 will explore a case where additionalrestrictions are imposed.
51As a caveat, our policy experiments hold the wage functions unchanged. A more comprehensive modelwould consider the reactions of labor demand to the new regime, which is beyond the scope of this paper.
52On the other hand, if the labor market values the width of one’s skill sets, one would expect greatergains from the new system than those predicted in this paper.
75
different sets of scenarios, one in this section and another in Section 7.2. In each set, we
conduct a series of experiments, solving for new equilibria to compare with the baseline.
In this section, we assume that to make up for the first period (2 years) of college spent
without specialization, students have to spend, respectively, 0, 1 and 2 extra year(s) in
college. Table 3.10 shows the equilibrium enrollment, retention and student welfare under
the baseline and under Sys.S with different lengths of college life. In all cases, postponing
major choices increases the overall retention rate from 75% to around 90% : a significant
fraction of dropouts occur in the current system because of student-major mismatches.53 In
the first counterfactual case, enrollment increases from 29% to 39%, and the mean student
welfare increases by about 4.6 million pesos or 5%. When one has to spend extra time
in college, college enrollment decreases sharply. In the last case where the first period of
college contributes only to a student knowledge about her major-specific fits but not to
her marketable skills, the new system causes a 0.9% welfare loss relative to the baseline.
However, we believe the last case to be overly pessimistic: one is likely to obtain at least
some basic skills by taking first-year courses even without specialization.
3.7.1.2 Hybrid of Sys.J and Sys.S
Although Sys.S allows students the opportunity to better learn about themselves before
choosing their majors, the extra time cost may outweigh the benefit for some students.54 We
therefore consider a hybrid system that combines the merits of Sys.J and Sys.S by allowing
students the choice between early and postponed specialization. This hybrid framework in-
volves the following stages:
Stage 1: The planner announces college-specific admissions policies, subject to the same
constraints as in Sys.S.
Stage 2: Students make enrollment decisions. An enrollee chooses between claiming a major
53If students face uncertainties other than major-specific efficiency shocks, for example, shocks that changethe value of college in general, then we might over-predict the retention rate in the new system.
54A discussion about gainers and losers will be provided in the next section.
76
upon enrollment (specializer), or taking courses in majors within her general interests (di-
versifier).
Stage 3: A specializer learns about her fit in her major and decides whether or not to drop
out. A diversifier learns her fits in various majors she has been exposed to and chooses one
of these majors or drops out.
Stage 4: Specializers who chose to stay spend one more period specializing before entering
the labor market. Diversifiers who chose to stay spend n more years specializing before
entering the labor market, where n may be longer than one period.
Information and Decision: Hybrid
Stage 1 Stage 2 Stage 3
Info Planner Choice Info Student Choice Info Student Choice
Non-Enrollee x, ε -
a Admissions x, ε Specializer (j,m) x, ε, ηm Stay/Dropout
Diversifier (j) x, ε, ηmm∈MaOne major/Dropout
Again, we consider three cases under the hybrid policy, where a diversifier has to spend
0, 1 or 2 extra year(s) in college, compared to a specializer. The results are shown in Table
3.11. For comparison, we list results from the baseline, the hybrid, as well as the results
under Sys.S. Combing the merits of both systems, the hybrid policy leads to greater welfare
gains compared to Sys.S: the higher the time cost for diversifiers, the larger the advantages
of the hybrid policy. As more students are attracted to colleges under the hybrid policy,
retention rates are lower than those under Sys.S in all cases.
How many students choose to specialize early? As the first row of Table 3.12 shows,
it depends critically on the cost of diversification. When diversification involves no extra
time, only 15% enrollees will give up the free opportunity to learn more about themselves.
At the other extreme, when two extra years are at stake, 97% of enrollees will choose early
specialization. Although it may be costly, diversification does improve a student’s chance
77
to find the right match: college retention rates for diversifiers are significantly higher than
those for specializers in all cases (Row 2 of Table 3.12).
3.7.1.3 Rematch Under Sys.J
Although the same rigid transfer policies are practiced in quite some countries like Chile
and those described in Appendix 3.11.4, some other countries (e.g., England) with the same
admissions system are more flexible in terms of transfers. To explore how much can be
gained from such flexibility, the following policy experiment allows students under Sys.J one
chance to rematch after the first period in college. The timing under this policy is:
Stage 1: Students make college-major enrollment decisions, subject to college-major-specific
admissions policies.55
Stage 2: A college enrollee in major m observes her major-specific efficiency ηm, and chooses
to stay, to transfer to a different college-major pair, or to drop out at the end the first period
in college. To prevent arbitrage, we impose the same admissions standards on transfers.
Stage 3: Students who chose to stay in Stage 2 stay one more period in college and then
enter the labor market. Transfer students observe their major-specific efficiency in their new
majors and decide whether to stay and later enter the labor market or to drop out.
We consider three cases where a transfer student has to spend 0, 1 or 2 extra year(s)
in college, compared to a non-transfer student. Under the 0 and 1 extra year scenarios,
rematch policy leads to smaller improvement over the baseline than Sys.S does. When two
extra years are required, the rematch policy still improves welfare over the baseline, while
Sys.S leads to a welfare loss.
55On the student side, we impose the equilibrium peer quality conditions. On the admissions side, weimpose the same admissions policies used in the current Chilean system. Results from this experiment aresubject to these exogenous admissions policies.
78
3.7.2 A Closer Look
We make a more detailed investigation into the impacts of changes in admission policies,
focusing on the contrast between the baseline Sys.J and Sys.S.
3.7.2.1 Gainers and Losers
Who is likely to gain/lose when Sys.J switches to Sys.S? To make a more informative com-
parison, we hold the average student welfare equalized between the two systems. To do so,
instead of extending college life for all, we take an arguably more realistic approach and treat
majors differently.56 For the two most specialized majors, law and medicine, students have
to spend more time in college in order to make up for the early non-specialization period. For
other majors, the lengths of studies are unchanged at the cost of potential losses of human
capital, the production of which becomes (1− φ)hm(am, Ajm, ηm). Thus, φ is the fraction
of human capital lost ceteris paribus. Given this framework, we seek the combinations of
extra year (in law and medicine) and φ (in other majors) under Sys.S that yield the same
average student welfare as Sys.J. The results are shown in online Appendix 3.11.3.57 We
find that males and students from low income families are more likely to be gainers than
their counterparts, and that when a student already has a clear comparative advantage as
reflected in her pre-college abilities, the cost of delayed specialization is likely to outweigh
its benefit.
3.7.2.2 Enrollment and Major Choice Distribution
To compare the distributions of student choices, we hold the total enrollment equalized
between Sys.S and Sys.J, which happens when medical and law students spend one more
56For example, in the U.S., for most majors, students receive specialized training only in upper collegeyears. For law and medicine, specialization usually starts after one has received more general college trainingand lasts another 3 to 6 years.
57Each of the following combinations will equalize the welfare: 1) law and medicine majors extend for 1year, and φ = 23% for other majors; or 2) law and medicine majors extend for 2 years, and φ = 19.5% forother majors.
79
year in college and φ = 8.5% for other majors. Tables 14-16 are based on this configuration.
Table 3.14 displays enrollment and retention rates by tier. Compared to the baseline case,
Sys.S features more students enrolled in both the top tier (Tier 1) and the bottom tier. What
explains the growth of Tier 1 relative to Tier 2? Under the baseline, a nontrivial fraction
of students were eligible to enroll in Tier 1 but only for majors other than their ex-ante
most desirable ones. Among these students, some opted for their favorite majors in Tier 2
rather than a different major in Tier 1. Under Sys.S, the planner still deems (some of) these
students suitable for Tier 1, and some of them will matriculate.58 This is because, regardless
whether or not these students eventually choose their ex-ante favorite majors, given their
relatively high ability, enrolling them in Tier 1 does not have a significant negative effect on
peer quality, while the improved match quality significantly increases the benefit of doing
so.
What explains the growth of Tier 3 relative to Tier 2? Although the total enrollment
remains the same, the composition of enrollees changes as the system shifts. On the one
hand, some former outsiders choose to enroll given the prospect of a better match. A large
fraction of them are students with relatively low ability, whom are deemed suitable only for
hence admitted only to Tier 3 by the planner. On the other hand, some former enrollees
choose the outside option because of the potential loss of either time or human capital
(φ = 8.5%). Since one’s outside value increases with one’s ability, a lot of students in this
group are former Tier 2 enrollees who have middle-level abilities.
Table 3.14 also shows that retention rates in all three tiers improve significantly with the
change of the system. In fact, even the worst case under the new system (Tier 3) features a
retention rate that is 10% higher than the best case under the old system (Tier 1).
Table 3.15 displays the distribution of students across majors in the first and second
period in college.59 Focusing on the first four columns, we see that without major-specific
58Some of these students will opt for a lower-ranked tier due to tastes.59For the first period in college, the distribution across majors is defined only for the baseline case, since
in the new system students do not declare majors until the second period.
80
barriers to enrollment, the fraction of students increases significantly in law and medicine
majors. However, enrollment in these two majors are often strictly rationed regardless of
the admissions system. We mimic such rationing by adding one more constraint to Sys.S:
among all enrollees in college j, only those with law-specific (medicine-specific) ability that
meets a certain cutoff have the option to major in law (medicine). We conduct a series of
experiments with different cutoffs and report results from the one where the final number of
students in each law (medicine) program equals the number of available slots as proxied by
the enrollment size of the corresponding program under the baseline.
The last two columns of Table 3.15 show the equilibrium enrollment with rationing.
By construction, the fraction of students majoring in law (medicine) is cut down to its
capacity. It is not clear a priori how enrollment in unrationed majors may change because
two conflicting effects coexist. On the one hand, given total enrollment, enrollments in
unrationed majors should increase as rationed-out students reallocate themselves. On the
other hand, some students who would enroll without rationing may be discouraged from
enrolling at all as they are denied of the option to major in law and medicine. Indeed, as
shown in the last row of Table 3.15, 2.6% fewer students are enrolled in the first period when
rationing is imposed. Due to the dominance of this second effect, engineering, health and
science majors all become smaller compared to the case without rationing. The only major
where the first effect dominates is business, which becomes slightly larger.
3.7.2.3 Productivity
Table 3.16 shows the mean log starting wages (in 1,000 pesos) by major, which also reflects
the average productivity by major. With or without rationing, Sys.S improves the quality
of matches and hence productivity in all majors compared to the baseline. This is true even
though there is a 8.5% loss of human capital ceteris paribus for majors other than law and
medicine.
When enrollment in law and medicine is rationed, the average productivity increases
81
even further in both majors, which consist of only the very best students. As students who
are rationed out of law and medicine reallocate themselves, two conflicting effects occur for
the average productivity in other majors. On the one hand, some rationed-out students
have higher abilities in multiple majors over an average student, improving the average
productivity in the majors they flow into. On the other hand, some rationed-out students
are ill suited for other majors, dragging down the average productivity in the majors they
flow into. Comparing the last two columns of Table 3.16, we see that the resulting changes in
the productivity of unrationed majors are marginal. However, at least in one major we can
see the dominance of the second effect: the major of business gains not only in size (shown
in Table 3.15), but also in average productivity due to the inflow of high-ability students.
3.8 Conclusion
College-major-specific admissions system (Sys.J) and college-specific admissions system (Sys.S)
both have their advantages and disadvantages, whether or not the total welfare of students
under one system will improve under the alternative system becomes an empirical question,
one that has significant policy implications. However, answering this question is very diffi-
cult since one does not observe the same population of students under both regimes. In this
paper, we have taken a first step. We have developed and estimated an equilibrium college-
major choice model under Sys.J, allowing for uncertainty and peer effects. Our model has
been shown to match the data well.
We have modelled the counterfactual policy regime (Sys.S) as a Stackelberg game in which
a social planner chooses college-specific admissions policies and students make enrollment
decisions, learn about their fits to various majors and then choose their majors. We have
shown changes in the distribution of student educational outcomes and provided bounds on
potential welfare gains from adopting the new system. We have also explored a hybrid of
Sys.S and Sys.J that allows students to choose between early and postponed specialization,
82
which proves to be a very promising admissions policy regime.60
Although our empirical application is based on the case of Chile, our framework can be
easily adapted to cases in other countries with similar admissions systems. Due to data
limitations, we can only provide bounds on the welfare gains from adopting new admissions
policies. A natural and interesting extension is to model human capital production explicitly
as a cumulative process and to measure achievement at each stage of one’s college life. This
extension would allow for a more precise estimate of the loss of specific human capital due to
delayed specialization and hence a sharper prediction of the impacts on student welfare when
the admissions system changes. This extension requires information on student performance
in college and/or market returns to partial college training. With such data, it is also
feasible to relax our assumption about learning speed and model learning as a gradual process
where students update their beliefs about their major-specific suitability by observing college
performance overtime.61
Another extension is to introduce heterogeneity across colleges besides their student qual-
ity, which may also affect market returns. One modeling approach is to introduce exogenous
college fixed effect, however, as is true for student quality, college ”fixed effect” is likely
to change with admissions regimes, for example, via instructional investment. Therefore, a
more comprehensive model will allow the social planner to choose college investment together
with admissions policies. To implement this extension, information on college investment
becomes necessary.
Finally, one can also incorporate ex-ante unobserved heterogeneity in student abilities
into the framework. The planner will need to infer students’ ability from their observed test
scores in the making of admissions policies. This extension will be relatively straight forward
if the unobserved component of student ability is ”private” and does not affect peer quality.
In this case, the unobserved ability will play a role similar to student’s individual tastes
60Given the benefit of the hybrid system, one might wonder why a lot of countries choose inflexible systems.This is an important and interesting question that deserves future research.
61See, for example, Arcidiacono (2004).
83
except that the former will directly affect individual’s wages. If the unobserved component
of ability also contributes to peer quality, then the estimation strategy needs to deal with
the fact that the equilibrium objects are no longer observed from the data.
3.9 Tables and Figures
Table 3.1: Aggregation of Colleges
Tier No. Colleges Mean Scorea Total Enrollment Tuitionb
1 2 702 (4.2) 21440 (2171) 3609 (568.7)
2 10 616 (17.7) 10239 (4416) 2560 (337.2)
3 13 568 (7.2) 5276 (2043) 2219 (304.2)aThe average of
math+language2
across freshmen within a college.bThe average tuition (in 1,000 pesos) across majors within a college.cCross-college std. deviations are shown in parentheses.
Table 3.2: Summary Statistics By Tier (All Students)
Matha Language Log Wageb Dist. for All (%) Dist. for Female (%)
Tier 1 709 (80.9) 692 (58.5) 8.91 (0.59) 5.1 4.5
Tier 2 624 (69.0) 611 (68.9) 8.57 (0.66) 14.1 12.2
Tier 3 572 (58.8) 570 (62.4) 8.32 (0.69) 9.0 9.1
Outside 533 (67.5) 532 (67.4) - 71.8 74.2aThe maximum score for each subject is 850. Std. deviations across students are in parentheses.bLog of starting wage in 1000 pesos.
84
Table 3.3: Summary Statistics By Major (Enrollees)
Math Language Dist. for All (%) Dist. for Female (%)
Medicine 750 (66.0) 719 (55.5) 3.4 3.2
Law 607 (74.2) 671 (72.1) 4.6 4.8
Engineering 644 (79.7) 597 (75.4) 36.6 23.4
Business 620 (87.3) 605 (73.9) 9.9 10.5
Health 628 (58.3) 632 (64.3) 11.7 17.1
Science 631 (78.2) 606 (82.1) 8.5 8.3
Arts&Social 578 (70.7) 624 (72.4) 11.2 14.1
Education 569 (59.5) 593 (64.2) 14.0 18.6
Table 3.4: Human Capital Production
Peer Ability (γ1m) Own Ability (γ2m)
Medicine 0.01 (0.002) 0.18 (0.04)
Law 0.58 (0.04) 1.26 (0.02)
Engineering 0.70 (0.01) 1.53 (0.01)
Business 1.48 (0.01) 1.52 (0.01)
Health 0.53 (0.03) 0.48 (0.03)
Science 1.44 (0.01) 1.62 (0.01)
Arts&Social 0.91 (0.02) 1.03 (0.03)
Education 1.08 (0.01) 0.55 (0.02)
Efficiency Shock (ση) 0.60 (0.08)
Table 3.5: Consumption Value (Major-Specific Parameters)
Own Ability Peer Ability Female
Medicine 6.33 (0.73) -6.11 (0.62) -1982.2 (255.8)
Law 2.13 (0.41) -17.46 (1.30) -196.9 (66.8)
Engineering 2.22 (0.10) -0.01 (0.002) -1719.5 (59.7)
Business 0.004 (0.006) -2.52 (0.20) -196.6 (33.5)
Health 0.006 (0.004) -0.24 (0.02) 1668.6 (26.6)
Science 0.001 (0.003) -0.001 (0.001) -376.6 (30.6)
Arts&Social 1.50 (0.48) -4.14 (0.65) -393.3 (19.4)
Education 0.003 (0.008) -0.001 (0.02) 1302.5 (17.9)
85
Table 3.6: Enrollment by Tier
All Females
Data Model Data Model
Tier 1 5.1 4.5 4.5 3.4
Tier 2 14.1 14.7 12.2 12.1
Tier 3 9.0 9.9 9.1 8.8
Table 3.7: Enrollee Distribution Across Majors
All Females
Data Model Data Model
Medicine 3.4 5.0 3.2 2.9
Law 4.6 3.9 4.8 3.6
Engineering 36.6 36.5 23.4 24.2
Business 9.9 9.9 10.5 10.6
Health 11.7 10.7 17.1 17.9
Science 8.5 9.0 8.3 8.0
Arts&Social 11.2 11.0 14.1 10.8
Education 14.0 14.1 18.6 21.8
Table 3.8: Ability & Retention (by Tier)
Abilitya Retention (%)Tier Data Model Data Model
1 701 701 79.3 79.6
2 624 626 76.5 75.5
3 581 583 68.1 73.2aThe average of major-specific ability across majors in each tier.
Table 3.9: Ability & Retention (by Major)
Abilitya Retention (%)Data Model Data Model
Medicine 738 727 87.6 87.0
Law 658 649 81.3 80.8
Engineering 623 625 71.8 74.4
Business 619 619 74.6 73.4
Health 641 636 79.8 78.0
Science 622 614 63.7 72.0
Arts&Social 612 597 74.3 75.1
Education 590 592 77.1 73.8aAverage major-specific ability am in each major m.
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Table 3.10: Enrollment, Retention & Welfare: Sys.S
Baseline 0 Extra Year 1 Extra Year 2 Extra Years
Enrollment (%) 29.1 39.1 27.5 19.2
Retention (%) 75.3 91.1 89.2 90.2
Mean Welfare (1,000 Peso) 93,931 98,574 95,185 93,093
Table 3.11: Enrollment, Retention & Welfare: Sys.S v.s. Hybrid
Baseline 0 Extra Year 1 Extra Year 2 Extra Years
Sys.S Hybrid Sys.S Hybrid Sys.S Hybrid
Enrollment (%) 29.1 39.1 40.8 27.5 36.2 19.2 34.7
Retention (%) 75.3 91.1 89.2 89.2 80.1 90.2 75.9
Welfare (1,000 Peso) 93,931 98,574 98,857 95,185 96,098 93,093 95,664
Table 3.12: Enrollment, Retention & Welfare: Hybrid
0 Extra Year 1 Extra Year 2 Extra Years
Specializer Diversifier Specializer Diversifier Specializer Diversifier
% of Enrollees 15.2 84.8 69.0 31.0 96.9 3.1
Retention (%) 79.8 90.9 76.3 88.6 75.5 88.8
Table 3.13: Enrollment, Retention & Welfare: Sys.S v.s. Rematch
Baseline 0 Extra Year 1 Extra Year 2 Extra Years
Sys.S Rematch Sys.S Rematch Sys.S Rematch
Enrollment (%) 29.1 39.1 32.4 27.5 29.5 19.2 29.4
Retention (%) 75.3 91.1 87.7 89.2 82.1 90.2 78.2
Welfare (1,000 Peso) 93,931 98,574 95,651 95,185 94,418 93,093 94,151
Table 3.14: Enrollment and RetentionBaseline New
Enrollment Retention Enrollment Retention
Tier 1 4.5 79.6 5.1 93.6
Tier 2 14.7 75.5 12.2 92.5
Tier 3 9.9 73.2 11.7 89.3
All 29.1 75.3 29.1 91.4
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Table 3.15: Distribution Across Majors
Baseline New Rationed New
1st Period 2nd Period 1st Period 2nd Period 1st Period 2nd Period
Medicine 1.5 1.3 - 3.3 - 1.5
Law 1.1 0.9 - 1.6 - 1.1
Engineering 10.6 7.9 - 7.3 - 7.2
Business 2.9 2.1 - 3.4 - 3.5
Health 3.1 2.4 - 2.8 - 2.6
Science 2.6 1.9 - 3.6 - 3.5
Arts&Social 3.2 2.4 - 2.1 - 2.1
Education 4.1 3.0 - 2.5 - 2.5
All 29.1 21.9 29.1 26.6 26.5 24.1
Table 3.16: Log Starting Wage
Baseline New Rationed New
Medicine 9.10 9.17 9.18
Law 9.20 9.59 9.63
Engineering 8.97 9.03 9.03
Business 8.51 8.74 8.76
Health 8.38 8.89 8.90
Science 8.36 9.07 9.08
Arts&Social 8.32 8.79 8.80
Education 8.06 8.35 8.35
88
Figure 3.1: Average Wage by Major and Experience
89
3.10 Appendix I
3.10.1 Detailed Functional Form and Distributional Assumptions
3.10.1.1 Cost of college for students:
Cjm(x,Ajm) = pjm + c1pjmI(y = low) + c2p2jmI(y = low) + c3mAjm + c4(Ajm − am)2,
where pjm is the tuition and fee for program (j,m). c1 and c2 allow for different tuition
impacts on low-family-income student. c3m and c4 measure the effect of peer quality on
effort costs.
3.10.1.2 Outside option and dropout value
The value of the outside option and that of dropout depend on one’s test scores (s) and
one’s family income (y). We assume that the intercepts of outside values differ across income
groups, and that the value of dropout is proportional to the value of the outside option:
V0 (x) =T∑
τ ′=1
βτ′−1
[L∑l=1
θlsl + θ01(I(y = high) + θ02I(y = low))
],
Vd (x) = ρV0 (x) .
3.10.1.3 Idiosyncratic tastes:
For major: each element in ε1 is independent and ε1m˜i.i.d.N(0, σ2major).
For programs: ε2jm = εj + εjm, where εj˜i.i.d.N(υj, σ2col) and εjm˜i.i.d.N(0, σ2
prog). υj is the
consumption value of college j for an average student.
90
3.10.1.4 Log wage function:
ln (wm (τ, x, ηm, Ajm, ζτ )) = α0m + α1mτ − α2mτ2 + α3mI(female) + ln(hm (am, ηm, Ajm)) + ζτ ,
hm (am, ηm, Ajm) = aγ1mm Aγ2mjm ηm.
ζτ˜N(−0.5σ2
ζ , σ2ζ
)is an i.i.d. transitory wage shock. Elements in the vector η are assumed
to be i.i.d. and ηm˜ lnN(−0.5σ2η, σ
2η).
62
3.10.2 Adjustment
3.10.2.1 Adjusted Value Functions
The first period in college lasts two years for all majors. Letting the total length of major
m be lm, the adjusted second-period value function is given by
ujm(x, ε, ηm|Ajm) =
max
∑lm
τ ′=3 βτ ′−3vjm(x, ε, Ajm)+∑T
τ ′=lm+1 βτ ′−3 [Eζ (wm(τ − lm − 1, x, ηm, Ajm, ζ)) + vm(x, ε)]
, Vd (x)
.
The adjusted first-period value function is given by
U(x, ε|a∗, A) = max
max(j,m)β2Eηm(ujm(x, ε, ηm|Ajm)) +
2∑τ ′=1
βτ′−1vjm(x, ε, Ajm), V0 (x)
s.t. Eηm(ujm(x, ε, ηm|Ajm)) = −∞ if am < a∗jm.
62Notice that student abilities across majors are correlated, but their efficiency levels are independentacross the aggregated majors. We make the independence assumption for identification concerns. Becausea student cannot observe η in making enrollment decisions, and because she can only stay in the majorof choice or drop out after the realization of η, the correlation between elements in η does not affect herdecisions and therefore cannot be identified. If efficiency shocks are positively correlated, we may over-statecollege retention rates in our counterfactual experiments.
91
3.10.2.2 Empirical Definitions of ω, a∗ and Retention Rates
1) Programs aggregated in major m have similar weights ωm. In case of discrepancy, we use
the enrollment-weighted average of ωmll across these programs.
2) For the cutoff a∗jm, we first calculate the adjusted cutoffs using weights defined in 1) and
then set a∗jm to be the lowest cutoff among all programs within the (j,m) group.
3) The retention rate in (j,m) is the ratio between the total number of students staying in
(j,m) and the total first-year enrollment in (j,m) .
3.10.3 Estimation and Equilibrium-Searching Algorithm
Without analytical solutions to the student problem, we integrate out their unobserved tastes
numerically: for every student x, draw R sets of taste vectors ε. The estimation involves
an outer loop searching over the parameter space and an inner loop searching for equilibria.
The algorithm for the inner loop is as follows:
0) For each parameter configuration, set the initial guess of o at the level we observe from
the data, which is the realized equilibrium.
1) Given o, solve student problem backwards for every (x, ε), and obtain enrollment decisionδ1jm (x, ε|a∗, A)
jm.63
2) Integrate over (x, ε) to calculate the aggregate Ajmjm , yielding onew.
3) If ‖onew − o‖ < υ, a small number, end the inner loop. If not, o = onew and go to step 1).
This algorithm uses the fact that all equilibrium objects are observed to deal with poten-
tial multiple equilibria: we always start the initial guess of o at the realized equilibrium level
and the algorithm should converge to o at the true parameter values, moreover, the realized
equilibrium o also serves as part of the moments we target.
63Conditional on enrollment in (j,m) , the solution to a student’s continuation problem follows a cutoffrule on the level of efficiency shock ηm, which yields closed-form expressions for Eηm(ujm(x, ε, ηm|Ajm)).Details are available upon request.
92
3.10.4 Additional Tables
3.10.4.1 Data
Table 3.17: Score Weights
Weightsa (%) Length
Language Math GPA Social Sc Science max(Social Sc., Science)b (years)Medicine 22 30 25 0 23 0 7
Law 33 19 27 21 0 0 5
Engineering 18 40 27 0 15 0 6
Business 21 36 31 0 0 12 5
Health 23 29 28 0 20 0 5
Science 19 36 30 0 15 0 5
Arts&Social 31 23 28 18 0 0 5
Education 30 25 30 0 0 15 5aWeights used to form the index in admissions decisions, weights on the six components add to 100%.bBusiness and education majors allow student to use either social science or science scores to form
their indices, students use the higher score if they took both tests.
Table 3.18: College-Major-Specific Cutoff Index
Medicine Law Engineering Business Health Science Arts&Social Education
Tier 1 716 679 597 609 640 597 578 602
Tier 2 663 546 449 494 520 442 459 468
Tier 3 643 475 444 450 469 438 447 460
The lowest admissible major-specific index across all programs within each tier-major category.
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Table 3.19: College-Major-Specific Annual Tuition (1,000 Pesos)
Medicine Law Engineering Business Health Science Arts&Social Education
Tier 1 4,546 3,606 4,000 3,811 3,085 3,297 3,086 3,012
Tier 2 4,066 2,845 2,869 2,869 2,547 2,121 2,292 1,728
Tier 3 4,229 2,703 2,366 2,366 2,391 2,323 2,032 1,763
The average tuition and fee across all programs within each tier-major category.
3.10.4.2 Parameter Estimates
We fix the annual discount rate at 0.9.64 Table 3.20 shows how the value of one’s outside
option varies with one’s characteristics.65 The constant term of the outside value for a
student from a low income family is only 70% of that for one from a high income family.
Relative to a high school graduate, the outside value faced by a college dropout is about 3%
higher.
Table 3.20: Outside ValueConstant (θ01) 8919.8 (98.1)
Low Income (θ02) 0.70 (0.01)
Language (θ1) 131.2 (9.8)
Math (θ2) 133.3 (17.0)
Dropout (ρ) 1.03 (0.01)
Table 3.21 shows major-independent parameters that govern one’s consumption value:
the left panel for college programs and the right panel for majors. Relative to Tier 3 colleges,
Tier 2 colleges are more attractive to an average student, while top-tier colleges are less
attractive.66 The standard deviations of student tastes suggest substantial heterogeneity in
student educational preferences.
64Annual discount rates used in other Chilean studies range from 0.8 to 0.96.65We cannot reject the hypothesis that the outside value depends only on math and language scores,
therefore, we restrict θl for other test scores to be zero.66One possible explanation is that the two top tier colleges are both located in the city of Santiago, where
the living expenses are much higher than the rest of Chile.
94
Table 3.21: Consumption Value (Major-Independent Parameters)
College Value Major Value
Tier 1 (υ1) -3311.1 (248.8) a2m (λ2m) 0.011 (0.0003)
Tier 2 (υ2) 1126.7 (141.1)
σcol 3197.1 (386.0) σmajor 2344.3 (86.1)
σprog 1618.5 (242.8)
υ3 is normalized to 0.
Table 3.22 shows major-independent cost parameters. The impact of tuition is larger for
low-family-income students than their counterpart. A student’s costs increase significantly
if her ability is far from her peers.
Table 3.22: College Cost (Major-Independent Parameters)
I(Low Inc)*Tuition (c1) 3.68 (0.19)
I(Low Inc)*Tuition2 (c2) -0.001 (0.00004)
(am − Ajm)2 (c4) 6.74 (0.61)
Table 3.23 shows parameters in the wage function, other than the effects of own ability
and peer quality. It is worth noting that females earn less than their male counterparts
across all majors, which contributes to the lower college enrollment rate among females.
Table 3.23: Other Parameters in Log Wage Functions
Constant Experience Experience2 female
Medicine 7.78 (0.02) 0.09 (0.003) -0.002 (0.0001) -0.37 (0.09)
Law -2.63 (0.03) 0.11 (0.004) -0.007 (0.0002) -0.08 (0.03)
Engineering -5.38 (0.01) 0.10 (0.001) -0.002 (0.0003) -0.19 (0.01)
Business -10.67 (0.02) 0.11 (0.001) -0.003 (0.0001) -0.19 (0.02)
Health 2.30 (0.02) 0.02 (0.002) -0.0003 (0.0001) -0.19 (0.02)
Science -10.94 (0.01) 0.05 (0.001) -0.0007 (0.0001) -0.29 (0.03)
Arts&Social -3.80 (0.01) 0.02 (0.001) -0.0005 (0.0001) -0.11 (0.02)
Education -2.23 (0.02) 0.07 (0.002) -0.001 (0.0001) -0.30 (0.04)
Wage Shock (σζ) 0.683 (0.04)
95
3.11 Appendix II: For Online Publication
3.11.1 Illustration: Gender Differences
To explore the importance of gender-specific preferences in explaining different enrollment
patterns across genders, we compare the baseline model prediction with a new equilibrium
where females have the same preferences as males.67 Table B1 shows the distribution of
enrollees within each gender in the baseline equilibrium and the new equilibrium. When
females share the same preferences as males, there no longer exists a major that is obviously
dominated by one gender. Some differences between male and female choices still exist. For
example, although college enrollment rate among females increases from 24.3% to 27.1% (not
shown in the Table), it is still lower than that among males (35.9%) . Moreover, compared
with males, females are still less likely to enroll in medicine and science and more likely
to enroll in social science. One reason is that, on average, males have higher test scores
than females; and they have comparative advantage in majors that uses math more than
language.68
Table 3.24: Female Enrollee Distribution(%) Baseline New
Male Female Male Female
Medicine 6.7 2.9 9.3 6.6
Law 4.0 3.6 3.9 3.6
Engineering 46.3 24.2 45.9 45.7
Business 9.2 10.6 9.2 9.7
Health 4.9 17.9 3.9 4.8
Science 9.8 8.0 9.1 8.4
Arts&Social 11.1 10.8 10.5 12.6
Education 8.0 21.8 8.1 8.5
67The purpose of this simulation is simply to understand the importance of preferences; the simulationignores potential changes in admission cutoffs.
68The average math score for males (females) is 572 (547), and the average language score for males(females) is 557 (553).
96
3.11.2 Counterfactual Model Details: Sys.S
3.11.2.1 Student Problem
Continuation Decision
After the first period, a student with ability vector a learn about her fits to majors in her
interest set Ma. Given(x, ε, ηmm∈Ma
)and Aj ≡ Ajmm , an enrollee in college j chooses
one major of interest or drops out:
uj(x, ε, ηmm∈Ma|Aj) =
max
maxm∈Ma
vjm(x, ε, Ajm) + E
T∑τ ′=3
βτ′−2 (wm(τ − 3, x, ηm, Ajm, ζ) + vm(x, ε))
, Vd (x)
.
Let δ2m|j(x, ε, ηmm∈Ma
|Aj)
= 1 if an enrollee in j with(x, ε, ηmm∈Ma
)chooses major m.
Enrollment Decision
We assume that in the first period of college, an enrollee pays the averaged cost for and
derives the averaged consumption value from majors within her general academic interest.69
A student chooses the best among colleges she is admitted to and the outside option:
U (x, ε|q (a) , A) =
max
maxjβEηuj(x, ε, ηmm∈Ma
|Aj) +1
|Ma|∑m∈Ma
vjm(x, ε, Ajm), V0 (x)
s.t. Eηuj(x, ε, ηmm∈Ma|Aj) = −∞ if ψj (q (a)) = 0,
where q (a) is the planner’s admissions rule for a student with ability a, and ψj (q (a)) = 1 if
the student is admitted to college j. Let δ1j (x, ε|q (a)) = 1 if the student chooses college j.
69Presumably, there will be greater welfare gains if students are allowed more flexibility in their choicesof first-period courses. Our results provide a lower benchmark for potential welfare gains from the switch ofthe admissions system. We leave the extension for future work.
97
3.11.2.2 Planner’s Problem
To formalize the constraint on the planner’s strategy space, we introduce the following
notation. Let Ξ ≡ χ1, χ2, χ3, χ4 = [1, 1, 1] , [0, 1, 1] , [0, 0, 1] , [0, 0, 0] , where the j-th
component of each χn represents the admissions to college j, i.e., χnj = 1 if a student
is admitted to college j. Denote the planner’s admissions policy for student with abil-
ity a as q (a) , we restrict the planner’s strategy space to be probabilities over Ξ. That
is, for all a, q (a) ∈ Q ≡ ∆ ([1, 1, 1] , [0, 1, 1] , [0, 0, 1] , [0, 0, 0]) , a convex and compact set.
The probability that a student is admitted to college j, denoted as ψj (q (a)), is given by
ψj (q (a)) =∑4
n=1 qn (a)χnj.
Consistent with the assumptions on student course taking, we assume that in the first
period in college, a student with interest set Ma will take 1|Ma| slot in each m ∈Ma, and that
in the second period in college, she will take one slot in her chosen major and zero slot in
other majors, where |Ma| is the number of majors within the set Ma. Let z = [y, g] be the
part of x that is not observable to the planner, the planner’s problem reads:
π = maxq(a)∈Q
∫a
U (a|q (a) , A) fa(a)da
where U (a|q (a) , A) =∫z
∫εU (x, ε|q (a) , A) dFε (ε) dFz (z|a) is the expected utility of student
with ability a, integrating out student characteristics that are unobservable to the planner.70
For each a, one can take the first order conditions with respect to qn(a)4n=1 , subject
to the constraint that q (a) ∈ Q. Given the nature of this model, the solution is generically
at a corner with one of the qn (a)’s being one. Thus, we use the following algorithm to solve
the planner’s problem. For each student a, calculate the net benefit of each of the four
pure strategies ([1, 1, 1] , [0, 1, 1] , [0, 0, 1] , [0, 0, 0]). The (generically unique) strategy that
generates the highest net benefit is the optimal admissions policy for this student. Let ”·”70Given that test scores are continuous variables, we nonparametrically approximate Fz|a (z) by discretizing
test scores and calculating the data distribution of z conditional on discretized scores. In particular, we dividemath and language test scores each into n narrowly defined ranges and hence generate n2 bins of test scores.All a′s in the same bin share the same Fz|a (z) .
98
stand for (q (a) , A) , it can be shown that the net benefit of applying some q (a) to student
with ability a is:
fa(a)
∫z
∫ε
U (x, ε|·) dFε(ε)dFz|a (z) (3.1)
+ fa(a)∑j
ψj(·)δ1j (a|·)
∑m∈Ma
(am − Ajm)
|Ma|bmγ2mA
γ2m−1jm Kjm
− fa(a)∑j
ψj(·)δ1j (a|·)
∑m∈Ma
(am − Ajm)
|Ma|
c3m(1 +∑2
τ ′=1 βτ ′−1 µ
2jm
µ1jm)
+2c4
∑2τ ′=1 β
τ ′−1 µ2jm
µ1jm(Ajm − A′jm)
.
Elements in (3.1) will be defined in the next paragraph. The first line of (3.1) is the expected
individual net benefit for student a. An individual student has effect on her peer’s net benefits
because of her effect on peer quality: the second line calculates her effect on her peers’ market
return; the third line calculates her effect on her peers’ effort costs. Peers of student a are
those who study in the programs she takes courses in. Student a′s effect on her peers is
weighted by her course-taking intensity 1|Ma| .
To be more specific, δ1j (a|·) =
∫z
∫εδ1j (x, ε|·)dFε(ε)dFz|a (z) is the probability that a stu-
dent with ability a matriculates in college j. ψj(·)δ1j (a|·) is the probability that student a is
enrolled in college j. µ1jm is the size of program (j,m) in the first period, where each student
a takes 1|Ma| seat in major m ∈Ma. Ajm is the average ability among these students.
µ1jm =
∫a
δ1j (a|·)ψj(·)I(m ∈Ma)
1
|Ma|fa(a)da,
Ajm =
∫aψj(·)δ1
j (a|·)I(m ∈Ma)1|Ma|amfa(a)da
µ1jm
.
The second line of (3.1) relates to market return. bm is the part of expected lifetime income
that is common to all graduates from major m.71 Kjm is the average individual contribution
71bm = E(eζ)∑T
τ ′=3 βτ ′−1e(α0m+α1m(τ ′−3)−α2m(τ ′−3)2), so that the expected major-m market value of
99
to the total market return among students who take courses in (j,m) :
Kjm ≡∫aψj(·)I (m ∈Ma) kjm (a) fa(a)da
µ1jm
,
where kjm (a) =
∫z
eα3mI(female)
∫ε
δ1j (x, ε|·)
∫η
δ2m|j (x, ε, η|Aj) aγ1mm ηmdFη (η) dFε(ε)dFz|a (z) .
Students with higher am contribute more to the total market return of their peers. The third
line of (3.1) relates to effort cost. µjm2 is the size of program (j,m) in the second period.
Ajmprime is the average ability among students enrolled in (j,m) in the second period.
Formally,
µ2jm =
∫a
δ1j (a|·)ψj(·)δ2
m|j (a|·) fa(a)da,
A′jm =
∫aψj(·)δ1
j (a|·)δ2m|j (a|·) amfa(a)da
µ2jm
,
where δm|j2 (a|·) =∫z
∫ε δ
1j (x,ε|·)
∫η δm|j
2(x,ε,η)dFη(η)dFε(ε)dFz |a(z)
δ1j (a|·) is the probability that student a
will take a full slot in (j,m) in the second period conditional on enrollment in j.
3.11.2.3 Equilibrium
An equilibrium in this new system consists of a set of student enrollment and continua-
tion strategiesδ1j (x, ε|q (a) , A),
δ2m|j(x, ε, ηmm∈Ma
|Aj)m
j, a set of admissions policies
q∗ (a) , and a set of program-specific vectors Ωjmjm ≡µ1jm, µ
2jm, Kjm, Ajm, A
′jm
jm,
such that
student with ability a can be written as
bm
∫z
eα3mI(female)
∫ε
δ1j (x, ε|·)∫η
δ2m|j (x, ε, η|Aj)h (am, Ajm, η) dFη (η) dFε(ε)dFz|a (z)
= bm
∫z
eα3mI(female)
∫ε
δ1j (x, ε|·)∫η
δ2m|j (x, ε, η|Aj) aγ1mm Aγ2mjm ηmdFη (η) dFε(ε)dFz|a (z)
= bmAγ2mjm
∫z
eα3mI(female)
∫ε
δ1j (x, ε|·)∫η
δ2m|j (x, ε, η|Aj) aγ1mm ηmdFη (η) dFε(ε)dFz|a (z) .
100
(a)δ2m|j(x, ε, ηmm∈Ma
|Aj)m
is an optimal choice of major for every (x, ε, ηmm∈Ma) and
Aj;
(b)δ1j (x, ε|q (a) , A)
j
is an optimal enrollment decision for every (x, ε) , for all q (a) and A;
(c) q∗ (a) is an optimal admissions policy for every a;
(d) Ωjm is consistent with q∗ (a) and student decisions.
Equilibrium-Searching Algorithm:
We use the same random taste vectors ε for each student as we did for the estimation. In
the new model, student continuation problem does not have analytical solutions, so we also
draw K sets of random efficiency vectors η. Finding a local equilibrium can be viewed as a
classical fixed-point problem, Γ : O ⇒ O, whereO =([0, 1]× [0, 1]×
[0, A
]×[0, A
]×[0, K
])JM,
o = Ωjm ∈ O. Such a mapping exists, based on this mapping, we design the following algo-
rithm to compute equilibria numerically.
0) Guess o = Ωjmjm ≡µ1jm, µ
2jm, Kjm, Ajm, A
′jm
jm.
1) Given o, for every (x, ε) and every pure strategy q (a) , solve the student problem back-
wards, where the continuation decision involves numerical integration over efficiency shocks
η. Obtain δ2m|j (x, ε|q (a)) and δ1
j (x, ε|q (a)) .
2) Integrate over (ε, z) to obtain δ2m|j (a|q (a)) , δ1
j (a|q (a)) and U (a|q (a) , A) .
3) Compute the net benefit of each q (a) , and pick the best q (a) and the associated student
strategies. Do this for all students, yielding onew.
4) If ‖onew − o‖ < υ, where υ is a small number, stop. Otherwise, set o = onew and go to
step 1).
Global Optimality
After finding the local equilibrium, we verify ex post that the planner’s decisions satisfy
global optimality. Since it is infeasible to check all possible deviations, we use the following al-
gorithm to check global optimality.72 Given a local equilibrium o =µ1jm, µ
2jm, Kjm, Ajm, A
′jm
jm,
72Epple, Romano and Sieg (2006) use a similar method to verify global optimality ex post.
101
we perturb o by changing its components for a random program (j,m) and search for a new
equilibrium as described in B2.3.1. If the algorithm converges to a new equilibrium with
higher welfare, global optimality is violated. After a substantial random perturbations with
different magnitudes, we have not found such a case. This suggests that our local equilibrium
is a true equilibrium.
3.11.3 A Closer Look at Sys.S: Gainers and Losers
The two combinations that equalize student welfare between the Sys.J and Sys.S are either
1) law and medicine majors extend for 1 year, and φ = 23% for other majors; or 2) law and
medicine majors extend for 2 years, and φ = 19.5% for other majors.
Table 3.25: Different Treatments Across Majors
Baseline Combination 1 Combination 2
Extra Years in Law & Med - 1 2
φ: Loss in Other Majors (%) - 23.0 19.5
Enrollment (%) 29.1 22.7 22.3
Mean Welfare (1,000 Peso) 93,931 93,934 93,935
To illustrate who are more likely to gain/lose, we generate an indicator variable that
reflects whether the change in a student welfare is positive, zero or negative. Then, we run
an ordered logistic regression of this indicator on student observable characteristics, control-
ling for their idiosyncratic tastes, drawn from the distribution according to our estimates.
Table 3.26 shows the regression results for Combination 2, the results for Combination 1
are qualitatively similar. Males and students from low income families are more likely to be
gainers than their counterparts. Students with higher math scores are more likely to gain,
while neither language score nor high school GPA has significant effects. A welfare loss is
more likely for students with higher score in their track-specific subjects (science or social
science) and for those with a larger gap between language score and math score. In other
102
words, when a student has a clear comparative advantage, the cost of delayed specialization
is likely to outweigh its benefit.
Table 3.26: Welfare Gain and Student Characteristics
Female Low Income Language1000
Math1000
HSGPA1000
Subject1000
(language−math)2
1000
Coefficient -0.46∗∗ 0.28∗∗ -0.12 1.18∗ -0.17 -1.36∗ -0.018∗∗
Std. Dev. 0.08 0.07 0.56 0.53 0.44 0.54 0.0035
Ordered logistic regression, dependent variable in order: positive/zero/negative welfare change.
Control for student idiosyncratic tastes.∗ significant at 5% level, ∗∗ significant at 1% level.
3.11.4 Other Examples of Sys.J
73
3.11.4.1 China (Mainland)
1. High School Track: Students choose either science or social science track in the second
year of high school and receive more advanced training corresponding to the track of choice.
2. College Admissions: At the end of high school, college-bounding students take national
college entrance exams, including three mandatory exams in math, Chinese and English, and
track-specific exams. A weighted average of the national exam scores forms an index of the
student, used as the sole criterion for admissions. College admissions are college-major
specific: a student is admitted to a college-major pair if her index is above the program’s
cutoff.74
3. Transfer Policies: Transfers across majors are either near impossible (e.g., between a
social science major and a science major) or very rare (e.g., between similar majors).75
73Major Sources of Information: 1. ”Survey of Higher Education System” (2004), OECD Higher EducationProgramme, 2. OECD Reviews of Tertiary Education (by country), 3. Department of Education (bycountry), 4. Websites of major public colleges in each country.
74The cutoffs may be different based on the student’s home province.75In 2001, Peking University started a small and very selective experiment program which admits students
103
3.11.4.2 Japan
1. High School Track: similar to the case in 3.11.4.1.
2. College Admissions: Students applying to national or other public universities take
two entrance exams. The first is a nationally administered uniform achievement test, which
includes math, Japanese, English and specific subject exams. Different college programs
require students to take different subject exams. The second exam is administered by the
university that the student hopes to enter. A weighted average of scores in various subjects
from the national test forms the first component of the admissions index; a weighted average
of university-administered exam scores forms the other. The final index is a weighted average
of these two components. College admissions are college-major specific in most public uni-
versities, except for the University of Tokyo, which uses category-specific admissions (there
are six categories, each consists of a number of majors).
3. Transfer Policies:
1) University of Tokyo: Students choose one major within the broad category in their sopho-
more year. After that, a student can transfer to a different major within her current category
but only with special permission and she has to spend one extra year in college, besides meet-
ing the grade requirement of the intended major. Transfer across categories is rarely allowed.
2) Other public universities: Changing majors is normally possible only with special permis-
sion at the end of the sophomore year, and it may require much make-up or an extra year
in college.
3.11.4.3 Spain
1. High School Track: similar to the case in 3.11.4.1, but with three tracks to choose from:
arts, sciences and technology, and humanities and social sciences.
to two broad areas (social science or science) according to their high school track. Students are free to choosemajors within their areas in upper college years.
104
2. College Admissions: All public colleges use the same admissions procedure. College-
bounding students take the nation-wide Prueba de Acceso a la Universidad (PAU) exams,
which consist of both mandatory exams and track-specific exams. Admissions are college-
major specific, and the admissions criterion is a weighted average of student high school
GPA and the PAU exam scores.
3. Transfer Policies: Transfers across majors require that the student have accumulated
a minimum credit in the previous program that is recognized by the intended program,
where the recognition depends on the similarity of the contents taught in the two programs.
Transfers across similar majors can happen, although not common, in which cases, the
student usually has to spend one extra year in college. Transfers across very different majors
are rarely allowed.
3.11.4.4 Turkey
1. High School Track: Students in regular high schools choose, in their second year, one of four
tracks: Turkish language–Mathematics, Science, Social Sciences, and Foreign Languages. In
Science High Schools only the Science tracks are offered.
2. College Admissions: Within the Turkish education system, the only way to enter a
university is through the Higher Education Examination-Undergraduate Placement Exami-
nation (YGS-LYS). Students take the Transition to Higher Education Examination (YGS)
in April. Those who pass the YGS are then entitled to take the Undergraduate Placement
Examination (LYS) in June, in which students have to answer 160 questions(Turkish lan-
guage(40), math(40), philosophy(8), geography(12), history(15), religion culture and moral-
ity knowledge(5), biology(13), physics(14) and chemistry(13)) in 160 minutes. Only these
students are able to apply for degree programs. Admissions are college-major specific and
students are placed in courses according to their weighted scores in YGS-LYS.
3. Transfer Policies: Most universities require a student meet strict course and GPA
requirement and provide faculty reference in order to transfer majors. In a few universities,
105
the transfer policies are more flexible. However, transfers across very different majors are
near infeasible and transfers across similar majors are uncommon as well.
3.11.5 Proof of existence in a simplified (baseline) model.
Assume there are two programs m ∈ 1, 2 and a continuum of students with ability a ∈[0, A
]2that are eligible for both programs. Let the average ability in program j be Am.76
The utility of the outside option is normalized to 0. The utility of attending program 1 is
v1(a,A1) for all who have ability a, and v2(a,A2)− ε, where ε is i.i.d. idiosyncratic taste, a
continuous random variable.
A sorting equilibrium consists of a set of student enrollment strategies δm(a, ε|, ·)m ,
and the vector of peer quality A = [A1, A2] , such that
(a) δm (a, ε|A)m is an optimal enrollment decision for every (a, ε) ;
(c) A is consistent with individual decisions such that, for m ∈ 1, 2 ,
Am =
∫a
∫εδm(a, ε|A)amdFε (ε) dFx(x)∫
a
∫εδm(a, ε|A)dFε (ε) dFx(x)
. (3.2)
A sorting equilibrium exists.
Proof. The model can be viewed as a mapping
Γ : O ⇒ O,
where O =[0, A
]2, o = [A1, A2] .. The following shows that the conditions required by
Brouwer are satisfied and hence a fixed point exists.
1) The domain of the mapping O =[0, A
]2is compact and convex.
76It can be shown that conditional on enrollment in a program, the solution to a student’s continuationproblem follows a cutoff rule on the level of efficiency shock ηm, which yields closed-form expressions forEηm(u(a, ε, ηm|Am)). As such, vm (·) can be viewed as the net expected utility of enrollment, i.e., thedifference between Eηm(u(a, ε, ηm|Am)) and the cost Cm (am, Am) , both are continuous functions. Detailsare available upon request.
106
2) Generically, each student has a unique optimal enrollment decision. In particular, let
ε∗ (a,A) ≡ v2 (a,A2)−max 0, v1 (a,A1)
δ (a, ε|A) =
[0, 1] if ε < ε∗ (a,A)
[1, 0] if v1 (a,A1) > 0 and ε ≥ ε∗ (a,A)
[0, 0] if v1 (a,A1) ≤ 0 and ε ≥ ε∗ (a,A)
.
Given that both va (a,Aa) are continuous functions of (a,A) , so are max 0, v1 (a,A1) and
ε∗ (a,A) .
3) Given the result from 2), the population of students with different (a, ε) can be aggregated
continuously into the total enrollment in program m via∫a
∫εδm(a, ε|A)dFε (ε) dFx(x) and
the total ability in m via∫a
∫εδm(a, ε|A)amdFε (ε) dFx(x), hence the right hand side of (3.2) ,
being a ratio of two continuous functions, is continuous in A. That is, the mapping Γ is
continuous.
4) ”Every continuous function from a convex compact subset K of a Euclidean space to K
itself has a fixed point.” (Brouwer’s fixed-point theorem)
In the full model, where there are more than two programs and the taste shock is a
vector, there will be cutoff hyperplanes. It is cumbersome to show, but the logic of the proof
above applies.
107
3.11.6 Model Fit
Table 3.27: Enrollment (Low Income)
Data Model
Tier 1 2.3 2.6
Tier 2 12.6 12.4
Tier 3 9.7 9.7
Enrollment among students with low family income.
Table 3.28: Enrollee Distribution Across Majors (Low Income)
Data Model
Medicine 1.7 3.0
Law 3.4 3.2
Engineering 35.1 34.8
Business 10.0 9.9
Health 12.2 10.4
Science 8.2 9.6
Arts&Social 11.0 12.6
Education 18.5 16.4
Distribution across majors among enrollees with low family income.
Table 3.29: Mean Test Scores Among Outsiders
Data Model
Math 533 531
Language 532 532
HS GPA 542 541
Max(Science, Soc Science) 531 530
Mean test scores among students who chose the outside option.
108
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