School Closure and Educational Inequality:
Parental Investment in the Pandemic*
Hyunjae KangTae Young Kang

Sunham Kim

November 8, 2022– Version 1.0
Preliminary
Abstract
Compulsory public education is a common policy tool to increase
participation in education, particularly from marginalized groups. The
COVID-19 pandemic brought unprecedented school closures aroundthe globe. Households seek their own mitigation strategy against education loss via private substitutes. We use the COVID-19 response ofSouth Korea as a laboratory for investigating the effects of pandemicinduced school closure on parental investment and educational outcomes. Specifically, we exploit the school-level variations in in-person
learning days driven by government and local authorities. Combining
administrative data and longitudinal survey, our fixed effects regression
identifies that losing 10 additional days implies a 9.5% increase in private
education expenditure. In addition, we build and estimate a structuralmodel tournament to quantify the impact of school closure on educational inequalities. A counterfactual experiment suggests that a $450
private tutoring voucher for low-income students may compensate forthe test score loss caused by a 20-day loss of in-person schooling. Our
results shed light on how households make schooling decisions and have
implications on the longer-run educational inequality induced by the
pandemic. Methodologically, we propose an estimation algorithm thattightly links the natural experiment estimates and the structural model.

JEL Classification: D30, I18, I21, I24, I28
Keywords: school closure, private education, learning disparity, COVID-19* H. Kang: Stony Brook University; T. Kang: KAIST; Kim: Purdue University. We are gratefulto Steven Stern, Juan Pantano, Miguel Sarzosa, Chong Xiang, and the seminar participants
at GEPS2021, SK-CSES2021, MEA2022, AMES2022 for comments and discussions. We also
thank Gyeonggi Institute of Education, Gyeonggi Province Office of Education, and Busan Metropolitan City Office of Education for sharing data. This research is supported by
Krannert Doctoral Research Funds (Purdue University, 2022). Any errors are our own.

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

1.

1

Introduction

Compulsory public education is a common policy tool to increase participation in education, particularly from marginalized groups. It operates
through exposing students to standardized educational environment for certain amount of time. Although the state-of-the-art educational environmentis provided, the education effect would not take place if students do notattend school. In general, absenteeism, either voluntary or involuntary, isreported to deteriorate the academic achievement of students. (Dobkin et al.
2010; Goodman 2014; Gottfried 2009, 2014; Aucejo and Romano 2016; Liu etal. 2021).
The COVID-19 pandemic brought unprecedented school closure aroundthe globe. (Figure 1) “The Great Equalizer shuts down” (Agostinelli et al. 2022),
households first seek their own mitigation strategy to offset education loss
via private substitutes, which was remarkably heterogeneous by socioeconomic status, and likely to incur a divergence in academic performance. (The
Economist 2021) While recent studies have used economic models to examine the impact of the COVID-19 school closure on educational outcomes andpotential mechanisms (e.g., Agostinelli et al. 2022; Fuchs-Schündeln et al.
2020; Grewenig et al. 2021), systematic evidence from the real world remains
to be provided. We first identify, empirically, that the impact of lost in-person
learning days on private education expenditure and academic achievement
using a novel panel dataset from South Korea. Second, we build and estimate
a tournament model of parental investment in which the number of schooldays is augmented as an input of education production function. We then use
the estimated model to quantify the impact of school closure on educational
inequality and conduct counterfactual analyses.
We start by documenting the empirical findings. In order to study the link
between in-person instructional days and educational inequality, we leverage

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H. KANG, T. KANG, AND KIM

Figure 1: Global School Opening Ratio During the Pandemic (2020-2021)

Data: World Banka novel administrative-survey linked data, the Gyeonggi Educational Panel
Survey (GEPS) from South Korea. We exploit the policy-driven school-levelvariation of in-person days to establish causality. In the data, we identify
that losing 10 additional in-person days implies a 3% increase in private
education expenditure, after controlling for individual fixed effects. The
estimate is higher for low-income households due to the base effect, whichsuggests the additional spending was more burdensome to them. Further,
the 10-day lose is associated with a 0.7%p decrease in a percentile score froma national standardized test.
We choose South Korea as a laboratory to facilitate casual interpretation.
The country is one of the countries with extensive school closure, allowing
only 6.2% of its entire school days to be fully opened during two years of thepandemic.1 The schools were fully closed during 24.7% of school days and1South Korea recorded the highest school closure ratio among Asia & Pacific countries
that are included in World Bank High Income Groups.

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

3

67.6% days were also under partial closure status. The exogenous nature ofschool closure dwarfs the caveat of reverse causality. The country has beensuccessful in “flattening the curve” with no blanket lockdown. It makes ourestimate less contaminated by the aggregate environment, compared to theones from other developed countries. Moreover, the fact that the country
is known for education fervor and has have no absenteeism alleviates theconcern of selection bias. Armed with the causal interpretation, we develop amodel that can feature the empirical facts.
Our framework takes the competition between students toward finite
seats for colleges as the main driver of parental investment in our setting,which motivates the use of the tournament model of parental investment.2(Lazear and Rosen 1981; Grau 2018; Tincani et al. 2021; Kang 2022) To capture
the substitution between public education and private parental investment,
our test score function is specified as the constant elasticity of substitution(CES) production function, following recent advances in the early childhood
development literature. (Cunha et al. 2010; Agostinelli and Wiswall 2016) The
maximum likelihood estimation of the structural model fits the data well.
Using the estimated model, we quantify the educational inequality induced by the Covid-19 school closure. Simulating the model with the preCovid in-person learning days increases the mean of test score percentile
by 8.9% and decreases the standard deviations of test score percentile by
23.1%, which suggests a negative impact of the Covid-19 school closure on
the academic performance, and the educational inequalities. Furthermore,
the estimated model suggests that the students from low-income households
undergo a more drastic decline in the test score. Taking a step further, motivated by the results, we conduct a counterfactual policy of private tutoring2

An alternative model with an assumption of individual caring about the absolute performance severely overpredicts the choice of parental investment. [TO BE ADDED IN THE
APPENDIX]

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H. KANG, T. KANG, AND KIM

vouchers targeting low-income households. A private tutoring voucher of$2,100 may let the households fully offset the decrease in the test score on average, which suggests the substantial effects of the school closure. Our results
shed light on how households make schooling decisions and has implicationon the longer-run educational inequality induced by the pandemic. These
results underscore the importance of public education as the balance wheelof the social machinery.
Related Literature This paper is connected to four different strands ofliterature. First, it is related to the literature of empirical studies of the effects
of school closures on education. Most previous papers focus on the impact ofschool closures on educational outcomes, particularly test scores. We bringnovel longitudinal data to the literature, which allows us to identify the effectof school closures on parental investment and educational outcomes controlling for unobserved individual heterogeneities. The granularity of the school
closure measure varies among the empirical studies. Our attendance data ismeasured school level, which helps control the factors that otherwise would
be unobservable. Secondly, this paper speaks to the literature of structuralmodels of school closure and parental inputs. we estimate a structural modelusing the longitudinal data from the real world. Thirdly, there are many micro
and macro papers that constructed structural models to tackle the questionof COVID-19 and education. Due to the data limitation, they are more in theflavor of calibration instead of estimation. We complement them by combining our panel estimation and the structural model. Last but not least,
this paper is linked to the papers trying to integrate the design-based andstructural approaches. (Todd and Wolpin forthcoming; Galiani and Pantano
2021)
The rest of the paper is organized as follows. Section 2 outlines the background and empirical framework. Section 3 describes the data we use. Section 4 presents the reduced-form results. Section 6 defines the tournament

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

5model framework. Section 6 elucidates how we implement the structural
estimation and reports the results. Section 7 conducts some counterfactual
policy experiments based on the structural model. Section 8 concludes.

2.

Institutions and Empirical Framework

2.1

Background on the Korean Education

South Korea has a highly centralized education system. All pre-college students follow a single track “6-3-3” system, which denotes six years of elementary school, three years of middle school, and three years of high school.3
Most schools follow the same official curriculum, the National Curriculum for
the Primary and Secondary Schools.4 The government requires all primaryand secondary schools to have 190 instructional days. Local school boards
exert limited power on local education policy. Parental involvement is nota major decision-making factor in school as well, compared to the UnitedStates. Almost all youths are enrolled in such official education enrollments.
Homeschooling, a form of decentralized education, is prohibited by law, so atiny fraction of student chooses to be a homeschooler.
The centralized system stems from a long-standing fervor for educationof Koreans. Education is regarded as the main driver of economic growth
and the key determinant of individual career and social mobility. The lowest
absence rate and highest literacy in OECD are the byproducts of the fervor. In
order to meet the demanding public’s expectations, especially to the fair anddependable education service, the government keeps control over education
3

Elementary school is often referred to as primary school. Secondary school consists ofmiddle and high school.
4
It applies to both public and private schools. A few selective high schools (autonomous
private schools and special purpose schools) may customize the curriculum to any meaningful extent.

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H. KANG, T. KANG, AND KIM

and puts top priority on it. Schools have never been closed before the COVID
pandemic under the various national crises after the Korean War, including
military coups (1961, 1979) and financial crises (1997, 2007). No exception
was taken to many epidemics before the COVID-19 such as MERS (2015),
H1N1 (2009), SARS (2004), and even to cholera (1969), which was a serious
threat to public health of the country before its economic takeoff.
While the official schooling system is centralized, education institutions
are not limited to schools. There are numerous private academic institutes,
called cram schools or “hagwon”, that offer a variety of tutoring services as acomplement of school education. The private education market is accessible
to anyone while the service quality may vary by cost. According to the Ministryof Education (2010), 45% of elementary, 50% of middle school, and 84% ofhigh school. Private education is often blamed being the “Great Unequalizer,”as the main incentive for hagwon enrollment is to ”win” the competition in
the early stages of the game. The government has tried to shackle the market,
based on concerns about its toll on households’ finances and students’ wellbeing. A law enacted in the late 2000s, which known as “curfew’, even set a
restriction on the business hours of hagwons. In sum, education is tauted as
the “Great Equalizer” in South Korea, as in the United States. The government
plays a major role as a service provider and wanted to control the private
education market, the “Great Unequalizer.”

2.2

The COVID Responses in Early 2020

The pandemic shock challenged the uninterrupted centralized system. Whenthe first confirmed case was reported in January 2020, the Ministry of Education stuck to the normal academic calendar. As the situation escalates
rapidly after the notorious super-spreader “Patient 31” and the first wave
arrived, however, the MoE declared the unprecedented universal school clo-

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

7

sure on February 23rd and postponed the school opening from March 2nd
to 9th. It took five more weeks to lift the closure, and the opening was fully
remote. All high school students and 9th graders went back to school on
April 9th. Younger groups, 5-8th and 1-4th graders did on April 16th and 20th,
respectively. The 2020 national academic calendar had to be overhauled.
As the country passes the peak of the first wave, the government tried
to open schools in person on May 13th. The schedule was postponed by aweek as the second wave arrived in early May. Eventually, the 12th graders
went back to school and all the other graders did in order by June 3rd. No
nationwide school closure is placed ever since this full reopening. Providing
some guidelines, the MoE left the open-or-remote decision to regional offices
and school principals. This decentralized school-level variation is central to
our identification strategy.
Importantly, private education institutions were open while schools wereclosed. The Korean government never placed a blanket lockdown, boasting
their effective ‘Trace, Test, and Treat (3T)’ strategy. All private businesses wereopen with certain degrees of social distancing, so were hagwons. The MoE
was not able to close hagwons as long as all the other private sectors were inbusiness. Limited by social distancing, hagwons could leverage the existing
online lecture programs. The online programs were out of control. Moreover,the no lockdown policy kept parents in workplaces so prevented them frommonitoring and educating their children. They had no other option but
private education. In other words, the virus shut down the “Great Equalizer”
and unleashed the “Great Unequalizer.”

2.3

Measurement Framework and Identification

We identify the effect of reduced in-person school days on parental investment. The treatment is the decentralized variations at the school level. In

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H. KANG, T. KANG, AND KIM

Figure 2: In-Person School Days in Gyeonggi Province

Data: SCHOOLINFO and administrative records.

order to evaluate the impact of the short in-person school days, we estimate
the following linear panel model:ccyikg=δ0 + δ1 schdaysckg + ΓZikg+ µi + αDg=12th + εcikg ,(1)cIn the above equation, yikg
is an outcome variable for individual i in schoolk, grade g, and cohort c. We focus on 11th and 12th graders in 2013 (control)and 2020 (treated) cohorts. y can be private education participation dummy,
log expenditure, and academic performance.5 schdaysckg is the in-person
5

We address the log of zero problem for the expenditure variable by applying the inverse
hyperbolic transformation.

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

9

school days in Spring 2020. The variable comes from the administrative
records and the unit is ten days.
We include the individual fixed effects µi to account for unobserved individual heterogeneity. All standard errors are clustered at the city level. We
estimate all models with the cohort-invariant grade dummy Dg=12th whichequals one if student i is a 12th grader. It captures an average trend in y bygrade. The grade dummy materializes the parallel trend assumption, whichcwill be discussed below. The vector Zikg
is a set of time-varying individual-

level controls.
We state our three key identifying assumptions below.
1. In-person Days as a Random Treatment The key variation for identification is the number of in-person days. It is determined by school principals
and the Gyeonggi Office of Education. No individual student could alter thedecision by the nature. Still, there could be a reverse causality if the reopening
is affected by households or correlated with any component of the (individual level) error term. It is unlikely to be true. First, although the decision isdelegated to local authorities, it was still in the hands of policymakers who
are familiar with the top-down approach, instead of parents and students.
Second, if the reverse causality existed, what would be the main channel,
and which schools might be affected by such bottom-up requests? The usual
suspect is the student and school quality. Along the same lines, selective
schools might or might not want to reopen. They might do because of theneed for intensive schooling. They might not do as the students can benefit
from private education. In sum, we can check if a school’s per capita spending, a proxy of school quality, predicts the length of in-school days. The data
exhibits no correlation. (Figure 3) Conditioning on a school characteristic(public v. private) does not change the result.

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H. KANG, T. KANG, AND KIM

Figure 3: In-person School Days (vertical) and log spending per capita (horizontal) by school

Data: GEPS and administrative records.

There could be a measurement error due to COVID infection. Say, if a
student and/or family members were infected, there would have been losses
in in-person days at the individual level. The error is nonclassical as it iscorrelated with the idiosyncratic error term: sick students cannot participate
in private education. It is also unlikely. The number of school-age cases waslimited: 80 out of 1,537 total cases in Gyeonggi as of July 30th 2020 (0.07%
of teenager population). We regressed the in-person days on the number ofcumulative cases of each city and found no association.
2. Parallel Trends in Expenditure Growth Fundamentally, The COVID
shock is an aggregate shock that absorbs into the time dimension. In order to
identify the COVID effect, we assume that the annual growth of expenditure is
the same across cohorts. That is, all seemingly time effects are time-invariant
grade effects. This parallel trend assumption enables us to use an earlier

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

11

cohort in the dataset as a control group and attributes all 2020-specific outcome variations to the changes in school days. Figure 4 visualizes the averagetutoring expenditure growth of the COVID and control cohorts between 10thand 12th grades.6
Figure 4: The Tutoring Expenditure Growth (10th -12th Graders) by Cohort

Data: GEPS and SCHOOLINFO.

3. Individual Fixed Effects We study changes from 2019 to 2020. Given the
short timeframe, there are not many relevant individual controls which varyover time. We assume that the individual fixed effects absorb all unobserved
heterogeneities, occasionally controlling for household incomes. We present
a set of subgroup analyses.
On top of the three identifying assumptions, we leverage the following
empirical facts to argue that South Korea is one of the best “laboratories” to6
It is unclear that the canonical pre-trend plot can support our version of parallel trend
assumption. It could have been better support if there were more data points. Unfortunately,
the control cohort data is available from the 10th grade.

12

H. KANG, T. KANG, AND KIM

tease out the school closure effect from the COVID impacts, and our estimates
are less prone to the aggregate confounder.
4. Massive Policy Responses and Limited Economic Damage All who travelin the rain get wet. It is hard to disentangle the school closure effect from theaggregate COVID effect. Parental job loss and school closure are intertwinedin the other developed economies, for example. Thanks to the successful“flattening the curve” without economywide lockdowns, the country’s macroeconomic response to the COVID shock was moderate and the labor market
disruption was minimal. While avoiding lockdowns, the policymakers combined social distancing, massive testing, financial stimulus, and other policies
to keep the economy on track. The first universal school closure in history
was part of the overall policy package aimed at fighting against COVID-19.
As a result, the monthly trend of the employment rate was not different
from usual, though the level was slightly lower, whereas the U.S labor market
response was wild. (Figure 5) This empirical fact suggests that the changes
in parental investment are likely to be driven by school closure, being lesscontaminated by indirect effects fromz‘z the aggregate shock.
5. “Education Fever” as a Full Compliance Condition Consider our environment as a language of the local average treatment effect framework. The
treatment is the school closure. The COVID sample is the treatment groupand the earlier sample is the control group. Alway-takers are the students
who are chronically absent. Never-takers are the ones who come to school
regardless of its operating status. Defiers are the ones who come to closed
school but never attend to opened school.
The nature of (centralized) school education eliminates the never-takerand defier groups. The low historical absence rate, which applies to ourcontrol group as well, suggests that the fraction of potential always-takers

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

13

is likely to be infinitesimal. As a result, we have nearly the whole student
population as a complier group.
Meanwhile, the private education market was open and thick. The market
offers a wide range of services from free online lectures to expensive one-onone tutoring. Students (and parents) could find many viable options. In otherwords, they had access to market alternatives. If there were no such market,
they would not be able to spend more money, at least in the short run.
Figure 5: Monthly Employments by Country, 2020(a) South Korea(b) United States

Data: Economically Active Population Survey (KR), CPS (US).

Under the three identifying assumptions and two supporting facts, δ̂1
captures the causal effect of in-person school days. As the regressor is thetotal length of in-person school days, δ̂1 is the change in y due to one unit
increase in schdays.

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H. KANG, T. KANG, AND KIM

3.

Data

3.1

The Gyeonggi Educational Panel Survey (GEPS)

We use the Gyeonggi Educational Panel Survey (GEPS).7 The GEPS is a Korean
longitudinal survey collected from Gyeonggi Province, the most populated
province in the country.8 The regionally representative survey started in 2012,sampling three cohorts of Gyeonggi students: 4th year elementary-school
students, 1st year middle-school students, 1st year high-school students. (4th, 7th , and 10th graders, respectively) They have been followed up through 9
years of primary and secondary education, being interviewed once a year. As
this is a critical period regarding adolescent development, the GEPS offers
decent opportunity to understand decisions and behavior of the surveyed
population.
Each year, information was collected in four separate questionnaires for:teenager, parents, teacher, and school principal. The survey is merged withthe administrative school information (the number of teachers, students,
classrooms, campus size, etc.) This design covers household information(household composition, parental education, occupation, income, etc.), covering information related to household composition and their parents education, occupation and income.
The GEPS pays special interest in the context of such decisions and collects data on learning behaviors. In particular, it asks detailed information
about enrollment in private academies, their type (e.g., group meetings, oneon-one, online), their topic (e.g., Korean, English, math), the hours spent onthem and their cost. Importantly, the GEPS offers information about academic achievement of adolescents, measured by a dedicated cognitive tests
7

The survey is based on Korean Education Longitudinal Study (KELS), which benchmarked the Education Longitudinal Study (ELS) of the United States.
8
A quarter of the Korean population lives in the province as of 2020. The country has 17
provinces.

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

15(elementary and middle school) and national-level achievement test (highschool). It enables us to translate the student decisions to the changes inacademic achievement.
Our COVID sample is the elementary student cohort, who became 12th
graders in 2020, the first year of the pandemic. Students were initially sampledaccording to the proportion of the fourth year elementary-school students
present in each city. Two supplemental samples were added in the 4th and 7th
waves, respectively. Subjects were consistently interviewed for nine waves(six or three for each supplemental sample). The Control sample is the highschool student cohort, who was 12th graders in 2014. In sum, the working
sample consists of 4,500 youths and their parents or guardians (see descriptive statistics in Table 1).
Private education measures.

As with all other personal characteristics col-lected in the GEPS, private education experience is self-reported by the students and parents. Parents, the payers, report participation and monthly
expenditure (Korean won). Given that the GEPS collects its data during late
July and the first semester of Korean academic year runs from March to June,
one can interpret the information as asking for private education experienceduring the spring semester.
Test scores.

Dedicated achievement tests for earlier periods. Stannine struc-ture for the national tests in the later periods. [to be added]

3.2

SCHOOLINFO: The In-person School Days Data

The SCHOOLINFO is a rich administrative dataset about all primary andsecondary educational institutions in South Korea, which is managed by the
Ministry of Education. It has the number of instructional days by school

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H. KANG, T. KANG, AND KIM

Table 1: GEPS Descriptive Statistics
COVID cohort (2019-20)

Control cohort (2014-15)

N at 11th grade
N at 12th grade% Female% Ordinary school% Selective school

2,738
2,740
50.51
89.1
6.9

1,763
1,760
51.5
89.2
4.9

Mother’s Education
2-yr college or less4-yr college or more

60.5
39.5

66.6
33.4

5,062.5(2,471.3)

4,610.3(4,948.4)% Particpation
KoreanMathEnglish

54.7
81.7
85.8

42.7
78.0
83.8

$ spent on private educ.
KoreanMathEnglish

156.7
224.9
268.5

107.9
192.8
224.0

Mean HH income. ($)

Notes: Standard deviations in parentheses. All USD values are converted from
KRW. (1 USD = 1,200 KRW) “Selective school” is a sum of “autonomous” and“special purpose” schools.

and semester. The schooling mode, the number of in-person instructionaldays, is included as part of the data after the coronavirus outbreak. The
augmented information has not been disclosed to the public. We received theconfidential data for Gyeonggi schools and merged it with the GEPS. (Figure
6)

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

17

Figure 6: In-person School Days in Spring 2020 (GEPS Schools)

Data: GEPS and SCHOOLINFO.

4.

Empirical Results

We estimate Equation (1). Household income is the only time-varying control
included. “TWFE” is our preferred specification. “OLS” is a pooled crosssection implementation. “OWFE” exploits the panel structure but does nothave the grade dummy.
As a recap, Our δ̂1 is the marginal effect of one unit increase in school days.
As our focus is school closure, i.e. decrease in school days, we flip the sign forinterpretation.

4.1

Take-up and Expenditure

The private tutoring participation decreased by 1%p. A ten-day decrease ofin-person school days increased private education take-up by 2.6%p and
expenditure by 9.5%. (Table 2)

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H. KANG, T. KANG, AND KIM

Table 2: Impact of In-Person School Days on Tutoring Behaviors
Take-up(1)
OLS
schdays(2)
OWFE

log Expenditure
(3)
TWFE

(4)
OLS

(5)
OWFE

(6)
TWFE

-0.012*** -0.007*** -0.026*** -0.109*** -0.033*** -0.095***(0.002)(0.001)(0.002)(0.011)(0.004)(0.010)log Income 0.061***(0.011)0.016(0.013)0.007(0.014)0.642***(0.050)0.157***(0.054)0.128**(0.056)N
R-squared

8972
0.620

8972
0.643

4296
0.193

8990
0.760

8990
0.766

4295
0.083

Notes: OLS estimated using 2020 data only. Clustered Std Errors in parentheses (city level, N=30)

4.2

Subgroup Analyses

We explore socioeconomic variations in the impact of in-person learning days.
In these regressions, we estimate Equation (1) by gender, parental education,
lagged household income, and lagged academic performance. This allows us
to examine the potential heterogeneity in parental responses to the in-person
learning days shock. All reported results are TWFE.
4.2.1

Gender

The effect is similar across gender while female students are more responsive
by 0.7%p and 1.1%p for take-up and expenditure, respectively. (Table 3) It
is consistent with our prior as “son preference” has declined in South Korea
during last two decades. (Chung and Gupta 2007, Yoo et al. 2017, Choi and
Hwang 2020) We believe this pattern suggests that the estimates are more
likely to capture signals rather than noises.

19

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

Table 3: Impact of in-person school days on Tutoring Behaviors, by GenderTake-up
(1)
All

(2)
Femalelog Expenditure
(3)
Male

(4)
All

(5)
Female(6)Maleschdays-0.026*** -0.029*** -0.022*** -0.095*** -0.100*** -0.089***(0.002)(0.003)(0.002)(0.010)(0.014)(0.011)log Income

0.007(0.014)-0.005(0.018)0.017(0.015)0.128**(0.056)0.071(0.081)0.175***(0.062)N
R-squared

8972
0.643

4564
0.655

4408
0.629

8990
0.766

4574
0.786

4416
0.742

Notes: Clustered Std Errors in parentheses (city level, N=30)
4.2.2

Household Income and Parents’ Education

In Table 4, we estimate Equation (1) by one-year lagged income terciles. The
take-up and expenditure growth are higher for students from lower-income
families. We interpret it as a result of the base effect. Before the pandemic,
low-income students were less likely to participate in private education or tospend much money on it. Therefore they have a higher margin of adjustment
when the pandemic hit. Given the positive correlation between income andeducation, the same logic applies to the estimates by parental education.
(Tables 5-6) All results align with our prior.
4.2.3

Academic Performance

Lastly, we estimate the model by one-year lagged academic performance(test score terciles) and find that higher-performance groups better took
advantage of the school closure. (Table 7) The take-up and expenditure
growth are ordered by the lagged academic performance.

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H. KANG, T. KANG, AND KIM

Table 4: Impact of in-person school days on expenditure, by previous year
IncomeTake-up
(1)
All

(2)
Low

(3)
Mid

log Expenditure
(4)
High

(5)
All

(6)
Low

(7)
Mid

(8)
High

schdays-0.026*** -0.037*** -0.021*** -0.015*** -0.095*** -0.119*** -0.088*** -0.073***(0.002)(0.003)(0.003)(0.003)(0.010)(0.017)(0.011)(0.015)log Income

0.008(0.013)N
R-squared

8994
0.643

0.132**(0.054)
3242
0.661

2990
0.633

2762
0.592

9012
0.767

3256
0.763

2990
0.736

2766
0.734

Notes: Clustered Std Errors in parentheses (city level, N=30)

Table 5: Impact of in-person school days on expenditure, by Mother’s Educ
Take-up(1)
All

log Expenditure(2)(3)
2yrCollorLess 4yrCollorMore

(4)
All

(5)
(6)
2yrCollorLess 4yrCollorMore

schdays-0.026***(0.002)-0.027***(0.002)-0.021***(0.003)-0.095***(0.010)-0.099***(0.011)-0.076***(0.014)log Income

0.007(0.014)0.003(0.020)0.025(0.024)0.128**(0.056)

0.124*(0.072)0.210*(0.108)N
R-squared

8972
0.643

5338
0.666

3124
0.588

8990
0.766

5350
0.765

3126
0.764

Notes: Clustered Std Errors in parentheses (city level, N=30)

5.

Theoretical Framework

5.1

From Quasi-Experimental Evidence to a StructuralModel

We utilize the structural model allowing for both parental investment and thenumber of school days affecting the academic achievement. The first purposeof employing a structural model is to evaluate the counterfactual policies that

21

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

Table 6: Impact of in-person school days on expenditure, by Fathers’ Educ
Take-up(1)
All

log Expenditure(2)(3)
2yrCollorLess 4yrCollorMore

(4)
All

(5)
(6)
2yrCollorLess 4yrCollorMore

schdays-0.026***(0.002)-0.030***(0.003)-0.021***(0.002)-0.095***(0.010)-0.112***(0.015)-0.074***(0.011)log Income

0.007(0.014)0.001(0.020)0.018(0.015)0.128**(0.056)0.079(0.064)0.208***(0.065)N
R-squared

8972
0.643

3950
0.681

4370
0.570

8990
0.766

3964
0.771

4374
0.752

Notes: Clustered Std Errors in parentheses (city level, N=30)

Table 7: Impact of in-person school days on expenditure, by lagged PerformanceTake-up
(1)
All

(2)
Low

(3)
Mid

log Expenditure
(4)
High

(5)
All

(6)
Low

(7)
Mid

(8)
High

schdays-0.026*** -0.014*** -0.027*** -0.033** -0.095*** -0.065*** -0.120*** -0.194***(0.002)(0.003)(0.002)(0.004)(0.010)(0.012)(0.015)(0.042)log Income

0.008(0.013)0.153(0.092)0.034(0.099)0.015(0.083)0.132**(0.054)0.139(0.087)0.049(0.093)0.409***(0.106)N
R-squared

8994
0.643

5542
0.822

2276
0.788

694
0.818

9012
0.766

5542
0.737

2276
0.741

694
0.755

Notes: Clustered Std Errors in parentheses (city level, N=30)

can help compensate the negative impact caused by the pandemic-induced
school closure. Having a model enables to conduct ex-ante policy evaluation
which accounts for the behavior of the targeted households. Second, bringing an structural model makes it explicit what underlying assumptions areneeded in identifying the relevant parameters. Third, the estimated structuralmodel is used to quantify the impact of school closures on the educationalloss and the inequalities. We employ the tournament model of parental in-

22

H. KANG, T. KANG, AND KIM

vestment (Kang 2022), and adjust the model to incorporate the notion ofschool closure. The choice of model is motivated by the features of secondary
school education in South Korea where students do their best to get into thebest college tier they can.9 As seats for prestigious colleges are finite, themotivation of relative performance governs the decision of the household.10
An alternative model would assume that household care about the absoluteperformance such as the level of human capital, rather than their relative
perforamnce. We find that such a model overpredicts the parental investmentof top-performing students and underpredicts the parental investment of
low-performing students compared to the tournament model.

5.2

Environment
Figure 7: Model Timeline
Consumption
shock εci,11realized

11th grade score

Consumption

12th grade score

realized with εqi,11

shock εci,12realized with εqi,12

11th grade

12th grade

End of secondary school

Decision

Decisionof tutoringof tutoring

expenditure ei,11expenditure ei,12Consider an economy composed of N households. We assume household
makes a unitary decision. Figure 7 presents the timeline of the model. The
model begins when the student becomes 11th grade. The initial conditions
are the complete income stream and the test score in 10th grade. We assume
9

See Section 2 of Kang (2022) for the relevant institutional features of secondary school
education in South Korea.
10
See Section 3 of Kang (2022) for the evidence of household using parental investment asa means to college admission competition.

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

23

households are myopic: they receive utility from the current return to academic outcome Rg , which is a function of the test score qi,g+1 . The first period
starts with the realization of the consumption shock εcig . Each household
makes a decision of private tutoring investment based on the realized consumption shock and the expectation over the test score shock εqit . After thedecision is made, test score qi,g+1 is produced with the realization of the timespecific test score shock εqig . The generated test score qi,12 is used as the input
of producing the subsequent tests score in the 12th grade. At the end of 12th
grade period, students compete against each other to get into better collegetier based on the final test score. The return function in the final period isformulated based on the tournament structure, which is explained in Section
5.3. As a result of the college admission competition, students are assigned to
one of the college tiers based on their ranking of the final test score.

5.3

HouseholdIn each period the household gets utility from consumption and the testscore, which is produced in period g. In particular, denoting g as school gradeof the student of household i, the utility function of household i is specified
asεcig ln(cig ) + Rg (qi,g+1 ),(2)

where cig is household consumption, Rg is a return from test score qi,g+1 ,and εcig is a shock to the marginal utility of consumption. Each household
decides how much it spends on private tutoring expenditure eig to maximize(2) subject to the budget constraint
cig + eig ≤ wig ,

(3)

24

H. KANG, T. KANG, AND KIM

where wig is household income. For the final period when th grade, the testscore tournament is realized through the return function Rg (qi,g+1 ), which is
specified as

R(qi,g+1 ) =αgJXln(vj ) × P rob(ln Q̃j−1 ≥ ln qi,g+1 ≥ ln Q̃j Γig ) ,

(4)

j=1

where Q̃j is the cutoff for college tier j. The cutoff Q̃j is where the competition
between household occurs. Denoting nj as the seats for college tier j, theP
cutoff Q̃j is the test score of Njth student, where Nj = j1 nj . That is, Q̃j is
the lowest test score of students who made it for college tier j. Note that weassume the return function for both 11th and 12th graders. The differential ofcollege structure is important in capturing the investment behavior of thehouseholds. We find that an alternative model, where a household cares
about the absolute performance of the student, fails to capture the parental
investment of the household.

5.4

Test Score Production Function

The channel in which compensation for school closure using parental investment occurs is the test score production function. We assume test score qi,g+1
is a function of previous test score qig , the number of school days schdayskg forschool k, private tutoring expenditure eig . The test score production functionis specified asqi,g+1 = f (θg , qig , eig , schdayskg , εqig ),where θg is set of the relavant parameters that may change over school grade.
To identify the subsitution between parental investment and the number ofschool days, we adopt a constant elasticity of subsitution (CES) production

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

25

function. (Cunha et al. 2010) In particular,1

φqi,g+1 =Ag (δqg qig+ δsg schdaysφig + δeg (1 + eig )φ ) φ εqig ,where Ag is the total factor productivity, φg is the substitution parameter, δqg
is the marignal effects of the previous test score, δsg is the marginal effectsof the number of school days, δeg is the marginal effects of private tutoring
expenditure, and εqig is the time-varying shock of test score.

6.

Structural Estimation

6.1

Likelihood Function

The individual likelihood contribution is composed of the contributions regarding private tutoring expenditure eig and test score qi,g+1 . For the testscores, the likelihood contribution is form of probability density function(PDF) for all individuals, as it is a continuous variable. For private tutoring
expenditure, the likelihood contribution differs depending upon the whether
the household participates in private tutoring activities or not. For the participants, likelihood contribution is form of PDF as tutoring expenditure isa continuous variable. On the other hand, the non-participants’ likelihood
contribution is a cumulative distribution function. Denoting Θcg as the set ofparameters, each household i’s likelihood contribution is1212
Li (Θcg |{qig }12
g=11 , {wig }g=11 ) =Πg=11

Z
εqig

Lig (θ|qig , wig )f (εqig )dεqig ,

(5)

26

H. KANG, T. KANG, AND KIMand

Lig (Θcg |qig , wig ) =deig
feig (eig ) · fqi,g+1 (qi,g+1 |eig )(1−deig )× Pr(eig = 0) · fqi,g+1 (qi,g+1 |eig );where deig is an indicator function of tutoring participation decision (i.e. eig >0 if dig = 1). The final form of the likelihood function isc1212
L(Θcg ) = ΠN
i=1 Li (θg |{qig }g=11 , {wig }g=11 ).

The likelihood contribution uses the equilibrium conditions of the theoretical framework. I denote ε̃cit and ε̃qit as the particular points of the shocks
where the utility of the household is maximized. They are assumed to be
jointly normal, and the likelihood function is based on ηitz = ln εzit for z = c, q.
In particular,Lig (Θcg |qig , wig ) =

dig

q
c
c
1
c (η̃ ) · f q (η̃ |η )|J |fηig
ηig ig ig
ig
i

 Z∞
×

(1−deig )

q
c
2
c
c
c (η ) · f q (η̃ |η )dη |J |fηig
ηig ig ig
ig i
ig

c
η̃ig

where |Jij |j=1,2 is the corresponding Jacobian transformation term. For the
non-participants of the tutoring activities, they do not spend on tutoringcexpenditure if they are above the threshold of the consumption shock η̃ig,

which is the minimum amount of the shock that makes household stop
spending on the tutoring expenditure. Note that θgc depends on the student’s
grade g and the cohort the student belongs to. In the subsequent section, I
explain the implications of the differentiation of the parameters.

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

6.2

27

Parameterization for Identification

We set the substitution parameter φ and the standard deviation of the unobserved heterogeneity parameters, σc and σq to be common across cohorts
and grades. To identify the effects of school days, we allow the parameter
governing the effects of number of schoool days δsg change by grade g, but
assume that the relative size of the effects of school days to the effects ofparental investment does not change over time. (i.e. δsg /δeg = κ, where κ isa constant.) As the only source of significant variation in school days comefrom 2020, it is impossible to identify δeg for different gs. Alternatively, we
identify the relative effects of school days to parental investment, namely κ.
The differentiation of a subset of parameters are necessary to implement theparallel trend assumption. Note that our parallel trend assumption is thegrowth of parental investment in 12th grade compared to 11th grade. Thus, theaverage cohort difference between the treatment group and control groupshould be captured in the structural estimation. To capture the mean difference, we differentiate the altruism parameter αg by cohorts. The parameter
governing the effects of tutoring expenditure, δeg is allowed to change for in
2020, the treatedment period. In addition, as the tests are different by cohortand grade, we differentiate the constant Ag by cohorts.

6.3

Implementation of the Parallel Trend Assumption

We propose an estimation algorithm that tightly links the natural experimentestimates with the structural model. The key assumption underlying theTWFE estimator is the parallel trend assumption. Note that the timing of
the treatment (Covid-19 school closure) happens when the students are in
12th grade. The trearment is realized as the increasing amount of tutoring
expenditure compared to 11th grade. Therefore, the parallel trend assumption implies the treated and the control groups would have had the same

28

H. KANG, T. KANG, AND KIMchange of tutoring expenditure, if there was no school closure. denotingẽAi,g as the counterfactual private tutoring expenditure of household i who
belongs to group A if schools were fully open and ebAi,group the predicted private
tutoring expenditure with the actual number of school days of household i
who belongs to group A the parallel trend assumption implies||

1
Ncontrolcontrol
NX)−− ebcontrol(becontrol
i11
i,12

i=1

1
NT reated

control
NX)|| =0,− ẽtreated(ẽtreated
i,11
i,12

i=1

which plays as a constraint of the estimation routine.
Finally, the objective function is a likelihood function with the counterfactual constraint. The set of structural parameters θb is estimated byb
θ = arg max log L + λ||control
NX

1
Ncontrol(b
econtrol− ebcontrol
)−
i,12
i11

i=1

1
NT reated

controlNX

i=1

where λ < 0 andL=NY

Lig (θ|qig , wig ).

i=1

6.4

Estimation Results

Table 8 presents the structural estimates. The estimates of the altruism parameter are similar across the cohorts and the grades. The overall differenceis within the range of 0.1. The relative size of the effects of the number ofschool days to the effects of tutoring expenditure is 1.722. The estimatedsubstitution parameter φ suggests that tutoring expenditure and school days
are gross complements. In Table 9, the effects of tutoring expenditure slightly
decreases in the 12th grade. This pattern is consistent with the early studies
in the literature of parental investment. (Del Boca et al. 2019)(ẽtreated− ẽtreated
)||
i,12
i,11

,

29

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

Table 8: Structural Estimation: Shock and Preference Parameters
A. Shock parametersσq

Test score shock std

σcConsumpt shock std

κrelative share of schdays

B. Preference Parameters
αtreated

altruism

αcontrolaltruism

0.463***(0.008)0.725***(0.006)1.722***(0.013)11th
0.596***(0.014)0.609***(0.015)12th

0.713***(0.019)0.657***(0.012)

Based on the estimates, we can compute the price of 10 days school
closure in terms of private tutoring expenditure. In terms of marginal contribution to the test score percentiles, 10 days of school closure is equivalent
to the $172.2 in private tutoring expenditure on average. However, this ismechanical estimates using the CES production function independent of
the household behavior. In the subsequent section, we introduce the simulation using structural model, which takes into account for the householdbehaviors.
The simulated results suggest that the parallel trend assumption is well
implemented. Figure 8 presents the simulated tutoring expenditure withthe parallel trend assumption. The vertical difference between solid line
and dotted line for treated group is the effects of school closure on parental
investment.

30

H. KANG, T. KANG, AND KIM

Table 9: Structural Estimation: Production Function
Grades
11th
Production Functionφ

CES Substitution

δe(ptexp)δs

Schdays*δecovidδscovid12th-0.238***(0.02)
0.287*** 0.272***(0.01)(0.01)
0.494*** 0.468***0.301***(0.01)
0.519***-

Figure 8: Parallel Trend Assumption in the Structural Estimation

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

7.

Counterfactual Experiment

7.1

School Open Counterfactual

31

To quantify the consequences of the Covid-19 pandemic educational outcomes, we simulate the structural model by imposing number of school daysof pre-pandemic. Specifically, we investigate how increasing the school days
to 60 and 95 (the pre-pandemic legal school days for one semester) changes
the educational outcomes. Table 10 presents the result of school open counterfactual. Compared to the estimated model, Full Open simulation increases
the test score by 0.61 SD, and the standard deviation decreases by 23.1%.

Figure 9: Experiment: School Open Simulation

32

H. KANG, T. KANG, AND KIM

Table 10: Experiment: School Open Counterfactual

Estimated Model
Partial Open (60 days)
Full Open (95 days)

7.2

Mean

Std. Deviation

4.27374.47584.6946

0.16250.13980.1249

Private Tutoring Voucher Simulation

Figure 10: Compensation of schooling loss for low income household

The result of voucher simulation suggests that tutoring voucher of about$2600 may fully compensate the average loss of the test scores for low income households. As private tutoring expenditure is measured monthly, theamount of expenditure needed for compensating the school closure is substantial. Although it is fairly expensive to fully compensate the average test

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

33

score loss from the reduced number of school days, tutoring expenditure
could be an effective way to partially mitigate the test score loss. Since there
exists diminishing marginal effects from private tutoring expenditure, themarginal returns from tutoring is greater in the low amount of tutoring expenditure. We find that the tutoring voucher of $450 can compensate thetest score loss about 15 days. Figure 11 shows the comparison of the 60-days
school open simulation and the $450 tutoring voucher simulation. The average test score of the two simulation is almost identical, which suggests thattutoring voucher of $450 has comparable effects of increasing the number ofschool days to 60 days.

Figure 11: Compensation of schooling loss for low income household

34

8.

H. KANG, T. KANG, AND KIM

Conclusion

This paper examines the effect of the COVID-induced school closure on
parental investment and the educational outcomes of students. We identified
the causal estimates using panel regression with the novel dataset. Motivatedby the empirical finding, we build and estimate a simple tournament model.
The model lets us conduct consistent policy experiments.
Our findings show that the short in-person instructional days increased
private education participation and expenditure. The responses were heterogeneous by income, parental education, and academic performance. It
implies that the school closure would have widened educational inequality.
All in all, our results elucidate the importance of public education andcompulsory schooling. The cost of school closure is huge and distributed
unevenly. We hope it sheds light on the policy debate of opening schools. It is
worth mentioning that our results do not deliver a clear-cut prescription alone
as our framework abstracted away from the tradeoff between in-classroomtransmission and learning loss. Still, it could be an informative building block
for a full cost-benefit analysis of school closure.

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

35

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, Matthias Doepke, Giuseppe Sorrenti, and Fabrizio Zilibotti, “When the great
equalizer shuts down: Schools, peers, and parents in pandemic times,” Journal of
public economics, February 2022, 206, 104574.
Aucejo, Esteban M and Teresa Foy Romano, “Assessing the effect of school days andabsences on test score performance,” Economics of education review, December
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Boca, Daniela Del, Christopher J Flinn, Ewout Verriest, and Matthew J Wiswall,“Actors in the Child Development Process,” February 2019.
Choi, Eleanor Jawon and Jisoo Hwang, “Transition of Son Preference: Evidence From
South Korea,” Demography, April 2020, 57 (2), 627–652.
Chung, Woojin and Monica Das Gupta, “The Decline of Son Preference in South
Korea: The Roles of Development and Public Policy,” Population and development
review, 2007, 33 (4), 757–783.
Cunha, Flavio, James Heckman, and Susanne Schennach, “Estimating the Technology of Cognitive and Noncognitive Skill Formation,” Econometrica: journal of the
Econometric Society, May 2010, 78 (3), 883–931.
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performance: Evidence from a regression discontinuity classroom experiment,”
Economics of education review, August 2010, 29 (4), 566–575.
Fuchs-Schündeln, Nicola, Dirk Krueger, Alexander Ludwig, and Irina Popova,“The Long-Term Distributional and Welfare Effects of Covid-19 School Closures,”
September 2020.

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H. KANG, T. KANG, AND KIM

Galiani, Sebastian and Juan Pantano, “Structural Models: Inception and Frontier,”
April 2021.
Goodman, Joshua, “Flaking Out: Student Absences and Snow Days as Disruptions ofInstructional Time,” June 2014.
Gottfried, Michael A, “Excused Versus Unexcused: How Student Absences in Elementary School Affect Academic Achievement,” Educational evaluation and policy
analysis, December 2009, 31 (4), 392–415.
, “Chronic absenteeism and its effects on students’ academic and socioemotional
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Grewenig, Elisabeth, Philipp Lergetporer, Katharina Werner, Ludger Woessmann,and Larissa Zierow, “COVID-19 and educational inequality: How school closures
affect low- and high-achieving students,” European economic review, November
2021, 140, 103920.
Kang, Hyunjae, “Parental investment, child’s efforts, and intergenerational mobility,”2022.
Lazear, Edward P and Sherwin Rosen, “Rank-Order Tournaments as Optimum Labor
Contracts,” The journal of political economy, 1981, 89 (5), 841–864.
Liu, Jing, Monica Lee, and Seth Gershenson, “The short- and long-run impacts ofsecondary school absences,” Journal of public economics, July 2021, 199, 104441.
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SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

37

Yoo, Sam Hyun, Sarah R Hayford, and Victor Agadjanian, “Old Habits Die Hard?
Lingering Son Preference in an Era of Normalizing Sex Ratios at Birth in South
Korea,” Population research and policy review, 2017, 36 (1), 25–54.

38

A.

H. KANG, T. KANG, AND KIM

More Discussion on Background

TO BE ADDED
Figure A.1: Monthly Employment by Province, 2020

Data: EAPS.

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

39

Figure A.2: In-person School Days in South Korea (Province Level)

Data: SCHOOLINFO and administrative records.

B.

Comprehensive Description on All Data
Sources

TO BE ADDEDC.

Model Appendix

C.1

Construction of the prize term (vj )In this section I explain how the prize term vj is constructed. The only difference with the Section 3.1 of Kang (2022) is that the discount factor is changed
to 0.98, which reflects the change of interest rate. The prize term vj differs
by the college tiers. Conditional on the college-tier, the member colleges

40

H. KANG, T. KANG, AND KIM

are assumed to share the same vj . The tier-specific lifetime income vj isdefined as a discounted sum of the predicted income of the graduates, whichis specified as
∗

vj =

T
X

β t−T ŷjt ,t=T +1

where ŷjt is the income of the alumni of college tier j, and T ∗ is the retirementage. ŷjt is the tier-specific annual income at time t. The predicted lifetime
income ŷjt is predicted using the regression equation,ln yit =

J
X

T ier
(βj + δj · ageit )Di,j+ Zi γ + εit ,

(6)

j=1
T ier
where Di,jis the dummy variable indicating that person i graduated from atier j college, and Z is the set of explanatory variables including squared age,birth year, and gender of person i. See Kang (2022) for the estimates of thePooled-OLS estimates, which is used for the prediction. The resulted vector
V is

5×1

  

v1
357554.68  v  295126.48
 2 

  




V = v3  = 203498.74

5×1
  

v4  172310.24
  

v5
122007.46

SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

D.

Likelihood Function Details

D.1

Tournament model

41

First Order Conditions The first order condition with respect to eig is
J
εcig
∂ X
ej−1 ≥ ln qi,g+1 ≥ ln Q
ej ΩiT , eig ) = 0.
−
+ αgln(vj ) × P rob(ln Q
wig − eig
∂eig j=1(7)Equation 7 cannot be analytically solved in terms of eig , but the likelihood contribution is transformed from the analytical form of εcig . Denoting
the argument of the production function as B for simplicity, the first ordercondition g is specified as



J
εcig
∂ X
ej−1 ≥ ln qi,g+1 ≥ ln Q
ej )
g =−
+ αg
ln(vj ) × P rob(ln Q
wig − eig
∂eig j=1
XJ
ej−1 − ln\ej−1 − ln\
εcig
ln Q
qi,g+1∂ln Q
qi,g+1+ αg) − Φ()=−
ln(vj ) × Φ(
wig − eig
∂eig j=1
σq
σq
" J


X
ej−1 − ln\ej − ln\
εcig
1 ln Qqi,g+1
1 ln Qqi,g+1νg δeg (1 + e=−+ αgln(vj ) ×φ(
) − φ() (−wig − eig
σq
σq
σq
σq
B
j=1
"

J−1 
e0 − lnej − ln\[
εcig
1 ln Q
qig
1 X
1
ln Q
qig+1
=−
+ αg ln(v1 ) φ()−)−
ln(vj ) − ln(vj+1 ) φ(
wig − eigσqσq
σq j=1σqσ#eJ − ln\δ3 (1 + eig )φg −1
ln Q
qig+1
) ×(−
),
φ(
σq
B
φ
where B = δq qig+ δs Schdaysφig + δe (1 + eig )φ42

H. KANG, T. KANG, AND KIM

e0 = ∞ and ln(Qej ) = −∞, the first order condition H is specified
Since ln Q
as follows.
" J−1 #X
ej − ln\
εcig
1 ln Qqi,g+1
νg δg (1 + eig )φg −1
H =−+ αg −ln(vj ) − ln(vj+1 )
φ(
) (−
)
wig − eig
σ
σ
B
q
q
j=1
" J−1 #cX
ej − ln\
εig
1 ln Qqi,g+1
δg (1 + eig )φg −1+ αgln(vj ) − ln(vj+1 )
φ(
) (
)=0
=−
wig − eig
σq
σq
B
j=1
|
{z
}
ϕ

The likelihood contribution of eig is transformed from the specified shockln ε̃cig that makes H = 0. In particular,ln ε̃cig = ln αg + ln δeg + log ϕ + (φg − 1) ln(1 + eig ) + ln(wig − eig ) − ln B.
On the other hand,ln ε̃qig = ln qi,g+1 −



1
φg
φ
φ
δ0g +
ln(δqg qig + δsg schdaysig + δeg (1 + eig ) ) .
φg

Jacobian Transformation

I denote η̃itc = ln ε̃qig . For the participant of thetutoring expenditure, the Jacobian transformation is∂(η̃itc , η̃itq )|∂(eig , qi,g+1 )∂ η̃ c ∂ η̃itq∂ η̃ q ∂ η̃itc
= it
− it
∂eig ∂qi,g+1 ∂eig ∂qi,g+1∂ η̃itc ∂ η̃itq=∂eig ∂qi,g+1|J1 | = det |

x
x 1
x
∂ 1
Using ∂xσ φ( σ ) = − σ 2 ( σ φ( σ )),SCHOOL CLOSURE AND EDUCATIONAL INEQUALITY

43

c
∂ηig
1 ∂ϕ
φ−1
1
1
=
+
−
−
δe φ(1 + eig )(φ−1) ,φφφ∂eig ϕ ∂eig (1 + eig ) wig − eig δq qig + δs Schdaysig + δe (1 + eig )where ϕ =

PJ−1
j=1




e −ln q \(e )ln Qln(vj ) − ln(vj+1 ) σ1q φ( j σqi,g+1 ig ) and


J−1 
\
X
ej − ln q\
ej − ln q\
∂ϕ
ln Q
ln Q
i,g+1 (eig ) 1
i,g+1 (eig ) −∂ ln qi,g+1 (eig )=ln(vj ) − ln(vj+1 ) {−
φ(
)(
)
}
∂eig j=1
σq2
σq
σq
∂eig

J−1 
\
X
ej − ln q\
ej − ln q\
ln Q
ln Q
i,g+1 (eig ) 1
i,g+1 (eig ) ∂ ln qi,g+1 (eig )=ln(vj ) − ln(vj+1 ) {
}
φ(
)(
)
2
σ
σ
σ
∂e
q
q
ig
q
j=1


J−1
X 
ej − ln q\
ej − ln q\
ln Q
ln Q
δe φ(1 + eig )(φ−1)
i,g+1 (eig ) 1
i,g+1 (eit )=ln(vj ) − ln(vj+1 ) {
}
φ(
)
(
σq2
σq
σq
B
j=1
For the non-paritcipants, the Jacobian transformation term is∂ η̃itq|J2 | = det ||∂qi,g+1∂ η̃itq=∂qi,g+1