“Cramming” Before the Exam: Estimating the Causal Effect Of

“Cramming” Before the Exam: Estimating the Causal Eﬀect of Exam Preparatory Programs in a Non-randomized Study

Ming-sen Wang Department of Economics University of Arizona∗†

May 04, 2012

FIRST DRAFT: January 12, 2012

Abstract

In this empirical paper, I estimate the impact of attending exam preparatory programs, in particular “cram schools,” on students’ academic performance. I measure the outcome by admission to a public high school and an “elite” high school. Fo- cusing on the problem that students are not randomly assigned to “cram schools,” I approach the issue using propensity score matching and a Bayesian simultaneous- equations model. Using data from a survey of Taiwanese junior high school students in the Taiwan Youth Project, I ﬁnd evidence that there is an insigniﬁcantly negative

∗I am indebted for continuous guidance of Ronald Oaxaca and helpful comments and suggestions from Katherine Barnes, Price Fishback, Keisuke Hirano, and Tiemen Woutersen. I have beneﬁted from discussions with Mario Samano-Sanchez, Sandeep Shetty, and Ju-Chun Yen. All the remaining errors are of my own. E-mail: [email protected]; the latest version of the paper can be found at: http://www.u.arizona.edu/∼mswang. †Data analyzed in this paper were collected by the research project Taiwan Youth Project sponsored by the Academia Sinica ( AS-93-TP-C01). This research project was carried out by Institute of Sociology, Academia Sinica, and directed by Chin-Chun Yi. The Center for Survey Research of Academia Sinica is responsible for the data distribution. The authors appreciate the assistance in providing data by the institutes and individuals aforementioned. The views expressed herein are the authors’ own.

1 sorting into exam preparatory programs and attending an exam preparatory program improves a student’s possibility of being admitted to a public high school or an “elite” high school. Both approaches indicate similar positive treatment eﬀects.

1 Introduction

In many East Asian countries, such as Taiwan and Japan, attendance of the so-called “cram school” is prevalent. A “cram school” is a type of shadow education that is aimed at improving a student’s exam writing skills. Attending “cram school” imposes additional burdens on a student and her family. It puts additional stress on a student since it requires time and effort. It puts financial loads on parents because sending a child to a program for a month can cost more than tuition fees for a semester in a public school. Given the prevalence and important role of exam preparatory programs in the education system, it is surprising that there are few rigorous evaluations. One problem is that students often self-select into these prep-programs.(Jackson(2012)[33]) As shown in Figure (1), the number of “cram schools” in Taiwan grows steadily. However, there has never been a rigorous proof that attending exam prep-program indeed improves students placement of high school. In a seminal paper, Stevenson and Baker (1992)[47] point out possible factors that foster “cram schools”: (1) the use of a centrally administered examination, (2) the use of “con- test rules” instead of “sponsorship rules”, and (3) tight linkages between the outcomes of educational allocation in elementary and secondary schooling and future educational opportunities. Taiwanese society has all these factors. Graduates of an “elite” university in Taiwan have significant advantages in the labor market (Lin (1983) [36])1. A student’s performance in the Joint High-school Entrance Exam and the Joint College Entrance Exam is strongly linked to future opportunities. It causes a prevalence of “cram schools” in Taiwan and makes Taiwan an ideal candidate to study. The paper distinguishes itself from previous work in two ways (See Stevenson and Baker (1992)[47] and Lin et al.(2006)[37]). Firstly,while other literatures define exam performances as outcome, I focus on admission to public high school and “elite” high school as outcome of interest to avoid selection issue related to taking the Joint Entrance Exam. Since Taiwan has undergone a significant education reform lately as we will discuss in the next section, focusing on admission circumvents complication of modeling and necessity of exclusion restrictions.

1Notice this result can hardly be interpreted as causal since the research does not control for the selection that the graduates of an“elite” university in Taiwan is productive to begin with.

2 Besides, I estimate the effect of “cramming” using a dataset of junior high school students while previous work uses sample from high school students. The difference is meaningful in the sense that senior high school is an important stage of educational stratification in Taiwan. Whether attending prep-programs affects teenagers’ life trajectory to academic track or vocational track is an interesting question per se. I compare estimates from propensity score matching and a Bayesian simultaneous-equations model. Identification of the two approaches comes from different untestable assumptions: propensity score matching relies on conditional independence assumption (Rosenbaum and Rubin(1983)[43]) while the Bayesian model relies on exogeneity of the exclusion restrictions. Both approaches differ slightly in the interpretation of the estimate but indicate positive effects of attending “cram school” on admission to public high school or “elite” high school.

2500

2000

county Taipei City 1500 Taipei County Yilan County

1000 Number of Tutoring Schools Number of Tutoring

500

2002 2004 2006 2008 2010 year

Figure 1: Growth in Number of Tutoring Schools in Taiwan (2002 - 2010)

† Data of this bar chart comes from http://ap4.kh.edu.tw/. The database is maintained by the Education Bureau of Kaohsiung City Government. The database has county-level statistics for all cram schools and after-school tutoring in Taiwan. The ﬁgure shows the number of tutoring schools in the 3 countries under study increase over time from 2002 to 2010.

3 1.1 Institutional Backgrounds

In 1987, Taiwan ended the martial law that has been in effect since 1949. Along with the freer and more opener political atmosphere, many civil groups started to request reforms in the education system. One of the most significant changes was to replace the old Joint Exam System with the new Multi-Opportunities System. In the old system, every junior high graduate had to attend the Joint High-school Entrance Exam that took place in the summer after the graduation. Students were ranked based on their exam grades. The ranking determined their priorities to choose an academic high school or a vocational high school. Their performance on the Joint High-school Entrance Exam determined their high school. The Exam decided the educational stratification. In 2001, the Ministry of Education officially executed the new Multi-Opportunities Sys- tem. The main idea of the new system is to separate admissions from exams. Two joint exams, the Basic Scholastic Ability Test and the Joint High-school Entrance Exam, are held in a school year to provide students one more chance. Under the new system, students can be admitted to high schools through multiple channels, such as (1) the Joint Entrance Exam, (2) the Special Admission Quotas for Recommended Students, and (3) Other Chan- nels without Entrance Exam Grades. Even though using grades of the Joint Entrance Exam as outcome provides a universal measurement, it involves complication to handle selection to take the Exam. Defining admission as outcome very much simplifies the modeling.

1.2 Literature Review

Human capital investment has been a research focus ever since Becker(1962)[7]’s ﬁrst rigorous treatment on the topic. A large literature is dedicated to estimating the returns of the formal schooling.(See Ashenfelter and Krueger (1994)[6]; Card (1995)[11]; Card(2001)[12]; Belzil(2007)[10]) Regan et al.(2007)[41], on the other hand, focuses on the optimal level of stopping schooling instead of estimating the rate of returns. On the other hand, if a prep-program does not directly increase human capital and it only aﬀects a student’s exam performance, the program can be considered as a way to reduce high school costs. It is of particular interest to investigate whether “cram school” increases the likelihood of being admitted to public high school. Admission to an “elite” high school increases the likelihood of being admitted to a better public university2. Again, tuition fees

2Since Taiwanese government subsidizes higher education heavily, public universities in general are ranked as better universities.

4 in a public university are signiﬁcantly lower than in a private university. Lower tuition fees aﬀect a student’s decision of stopping schooling. As pointed out in Jackson(2010)[32],we can motivate the question in the context of the Becker–Willis-Rosen life cycle model of human capital investment (See Becker(1993)[9] and Willis and Rosen (1979)[52]). Suppose the log of earnings y is an increasing concave function of the years of schooling s: y = eg(s)

Individuals pay a cost c to attend school, and δ is the discount rate. Then in the Becker- Rosen framework, a student who considers two levels of schooling chooses T years over no schooling if:

Z ∞ Z T Z ∞ V (T ) ≥ V (0) = eg(T )e−δtdt − ce−δtdt ≥ eg(0)e−δtdt T 0 0

If c is lowered by the decision to attend a “cram school”, then a student’s utility when she acquires more education increases. A student will more likely acquire more education and postpone termination of schooling. If prep-programs have no effect or negative effects on placement of high school, then attending the programs is fundamentally a rent-seeking behavior.(See Krueger(1974)[34]) The motivation to send a teenager to “cram school” is affected by some behavioral factors, say unrealistic concerns that their children will be left behind if all other children go to “cram school.” Jackson(2010)[32] is the most similar study using a U.S. high school dataset. He looks at the short-term outcome of the Advanced Placement Incentive Program (APIP), which pays both teachers and students for passing grades of Advanced Placement (AP) examinations. Using propensity score matching methods, he finds that APIP adoption is associated with a 13 percent increase in the number of students scoring above 1100/24 on the SAT/ACT and 4.96 percent increase in the number of students matriculating in college. My study shows some similar patterns in Taiwan to his findings.

2 Data: Taiwan Youth Projects

The Taiwan Youth Project (TYP) was started in the spring of 2000, with junior high students from Taipei County, Taipei City, and Yilan County as the study population. In order to

5 examine the eﬀects of Taiwan’s educational reforms on the students, TYP takes two cohorts as the study subjects: the 1st year junior high students with an average age of 13 (those taking reformed high school entrance system) and the 3rd year junior high students with an average age of 15 (those taking old high school entrance system). TYP collects 1000 students in the junior high’s 1st and 3rd year from both Taipei City and Taipei County and 800 students in the junior high’s 1st and 3rd year from Yilan County. The total sampling size is 5600 students. I use the cohort of the ﬁrst year junior high students since I observe their program attendance history. After sample attrition, I am left with 2449 observations. In Table (1), I summarize the key variables in the dataset.

Table 1: Summary Statistics

Mean SD Mean SD “cram School” in Senior Year 0.48 0.50 Male 0.51 0.50 Sound Family 0.87 0.33 Number of Siblings 3.56 0.87 Ever Fail a Class 0.35 0.48 Admission to Public HS 0.30 0.46 Admission to Elite HS 0.10 0.31 Intent to Attend HS 0.68 0.47 Minutes to “cram School” 18.99 11.60 --- Cram School History for First 2 years 00 0.35 0.48 01 0.09 0.28 10 0.12 0.32 11 0.45 0.50 Counties Taipei City 0.39 0.49 Taipei County 0.39 0.49 Yilan County 0.22 0.41 --- Father’s Educ. Mother’s Educ. Elementary School 0.13 0.34 Elementary School 0.17 0.37 Junior High School 0.26 0.44 Junior High School 0.26 0.44 High School Graduate 0.25 0.43 High School Graduate 0.26 0.44 Vocational School 0.08 0.27 Vocational School 0.10 0.30 Vocational College 0.06 0.24 Vocational College 0.05 0.23 University 0.11 0.31 University 0.08 0.27 Grad School 0.04 0.18 Grad School 0.01 0.11 Not Applicable 0.00 0.05 Not Applicable 0.01 0.09 No Education 0.06 0.24 No Education 0.06 0.24 Family Income less than NTD 30,000 0.18 0.38 NTD 30,000 -NTD 49,999 0.22 0.41 NTD 50,000 -NTD 59,999 0.21 0.40 NTD 60,000 -NTD 69,999 0.07 0.26 NTD 70,000 -NTD 79,999 0.08 0.27 NTD 80,000 -NTD 89,999 0.05 0.22 NTD 90,000 -NTD 99,999 0.04 0.20 NTD 100,000 -NTD 109,999 0.04 0.20 NTD 110,000 -NTD 119,999 0.03 0.17 NTD 120,000 -NTD 129,999 0.02 0.14 NTD 130,000 -NTD 139,999 0.01 0.10 NTD 140,000 -NTD 149,999 0.01 0.10 more than NTD 150,000 0.04 0.20 ---

6 3 Propensity Score Matching

I approach the question firstly by propensity score matching. I define the treatment as attending an exam prep-program in the senior year because attending “cram school” in that year has the strongest linkage to placement of high school. Because in the data we only observe realized outcome of the treatment group, the propensity score matching approach is to construct a counterfactual outcome for each treated unit based on the propensity score. Identification of propensity score matching relies on conditional independence assumption (Rosenbaum and Rubin(1983)[43]):

Ti ⊥ Yi(1),Yi(0)|Xi

where Yi(1) and Yi(0) denote potential outcomes given treatment. Conditional on observable characteristics, potential outcomes are independent of treatment. In our context, I assume attending “cram school” is independent of the potential admission outcomes given attending “cram school” or not after controlling for the observed family background and students’ performance in school. It requires a strong but empirically untestable assumption on the mechanism that there is no unobserved characteristics that aﬀect both outcome and exam prep-program attendance. Hence, it is important to select covariates so that the conditional independence assumption is likely to hold. In addition to standard covariates in education literatures, I proxy for ability by whether a student ever fails a class and for motivation by whether she intends to attend high school. Given the richness of covariates I adopt propensity score approach. Rosenbaum and Rubin(1983)[43] shows that conditioning on the full covariates is equivalent to conditioning on the propensity score, which is the coarsest balancing score. I non-parametrically estimate the propensity score by series logit regression. By 10-fold cross-validation, the ﬁrst-order series yields the smallest predicted error. I present the estimates in the propensity score in Table(2).

3.1 Overlap Condition

An important issue that often hampers the propensity score matching approach is lack of overlap in the covariate distributions. Figure (2) shows the histogram of the estimated propensity scores of both treatment and control groups. Even though the treatment group is concentrated more to higher value of propensity score and the control group is concentrated

7 Table 2: Estimated Propensity Score

Estimate Std. Error z value Pr(>|z|) (Intercept) -4.1436 1.1167 -3.71 0.0002 Male -0.0747 0.1129 -0.66 0.5081 Num of Siblings -0.1443 0.0694 -2.08 0.0376 Sound Family 0.4229 0.1806 2.34 0.0192 Attendance Histories 11 3.3385 0.1450 23.02 0.0000 10 0.4082 0.1914 2.13 0.0329 01 2.8512 0.1986 14.36 0.0000 Fail a Class (Proxy for Ability) 0.5430 0.1269 4.28 0.0000 Intention to HS (Proxy for Motivation) 0.3859 0.1249 3.09 0.0020 Father’s Educ. Yes Mother’s Educ. Yes Father’s Occ. Yes Mother’s Occ. Yes School FE Yes Family Income Level Yes

more to the lower value, both share a common support. An implication of the ﬁgure is that we should use a small number of matches to avoid too much smoothing and extrapolation.

3.2 Results

The benchmark result of propensity score matching is presented in Table(3). I compares different matching approaches. In 1-nearest-neighbor matching, the counterfactual outcome is constructed based on the shortest distance in the control group to the treated. 10-nearest-neighbor-matching, instead, matches the closest 10 units. By using more com- parison units, the precision of the estimate increases at the cost of larger bias. The trade-off between 1-nearest-neighbor and 10-nearest-neighbor is well-known variance-bias trade-off in non-parametric literatures. On the other hand, caliper matching uses all the control units within the predefined caliper but drops the treated units that have no matches. The problem with caliper matching is that the choice of caliper is arbitrary to the researcher’s judgment and that dropping unmatched units alters the interpretation of the estimate. Instead of the average treatment effect on the treated (ATT), the estimate of caliper matching should be interpreted as conditional treatment effect on the treated given the matched subset (CATT). All the estimates for ATT are significantly positive, ranging from 15% to 18% improvement in chances of admission to public high school and from 3% to 5% improvement in chances of admission to “elite” high school. In words, the students who attended “cram school” would

8 0

1 density

0.0 0.2 0.4 0.6 0.8 1.0 propensity score

Figure 2: Histograms of Estimated Propensity Scores have lost 15% to 18% chances of being admitted to public high school and 3% to 5% chances of being admitted to “elite” high school if she had not attended “cram school.” Since I am interested in estimating ATT, I can apply the covariate balancing strategy proposed by Rubin (2006)[45] given overlap in covariate distributions is a concern. The idea is to select a more balanced subsample before estimating the ATT. The procedure works as follows:

1. Order the treated units by an estimated propensity score

2. Match without replacement by decreasing value of the estimated propensity score to select corresponding control units. This leads to a balanced sample with sample size

2 × N1.

3. Redo an analysis, say propensity score matching, on the balanced sample. Con

An advantage of the approach is that the interpretation of the estimate is not aﬀected by trimming control units as long as we are interested in ATT. I report the result in Table(4). Consistent with the previous results, attending an exam prep-program improves a student’s chance of being admitted to public high school by significantly 15% to 18% and to “elite” high school by 2% to 5%.

9 Table 3: Propensity Score Matching: Full Sample

Outcome Est. A-I S.E.† Num. Matched 1-Nearest-Neighbor Public High School 0.157∗∗∗ 0.035 1199 Elite High School 0.029 0.022 1199 10-Nearest-Neighbor Public High School 0.145∗∗∗ 0.030 1199 Elite High School 0.030 0.020 1199 Caliper δ = 0.001 Public High School 0.180∗∗∗ 0.011 457 Elite High School 0.045∗∗∗ 0.007 457

† The standard errors are calculated based on Abadie and Imbens(2006)[1].

Table 4: Propensity Score Matching: Rubin Subsample

Outcome Est. A-I S.E. Num. Matched 1-Nearest-Neighbor Public High School 0.190∗∗∗ 0.049 1199 Elite High School 0.050 0.036 1199 10-Nearest-Neighbor Public High School 0.187∗∗∗ 0.041 1199 Elite High School 0.053∗ 0.031 1199 Caliper δ = 0.001 Public High School 0.172∗∗∗ 0.013 376 Elite High School 0.024∗∗∗ 0.009 376

10 Table (5) shows the estimates of ATT using the subsample of students who intends to attend high school. The sample gets rid of observations that are interested in professional training or termination of schooling. This is the first attempt to deal with ability sorting issue. Students better at academics would like to attend high school; therefore, they are more likely to go to “cram school.” The estimated effect may be exaggerated. On the other hand, if students who go to “cram school” are those who would like to attend high school but do not have comparative advantage in academic, then we would expect the estimate to be downward biased. Again, the estimator relies on the assumption that the conditional independence assumption holds within the subsample even though some may doubt its validity on the full sample. Since the estimate only exploits a subsample, the interpretation of estimates is again changed from ATT to CATT: treatment effect on the treated given students who would like to go to high school. All estimates show slightly larger effects but still consistent with the previous estimates.

Table 5: Propensity Score Matching: Intention-to-HS Subsample

Outcome Est. A-I S.E. Num. Matched 1-Nearest-Neighbor Public High School 0.218∗∗ 0.061 931 Elite High School 0.096∗∗ 0.046 931 10-Nearest-Neighbor Public High School 0.236∗∗∗ 0.050 931 Elite High School 0.102∗∗ 0.040 931 Caliper δ = 0.001 Public High School 0.176∗∗∗ 0.013 224 Elite High School 0.068∗∗∗ 0.010 224

4 Bayesian Simultaneous Equations Model

As mentioned brieﬂy in the last section, some may be concerned about the validity of conditional independence assumption since students may select to attending “cram school” based on their motivation and ability. In this section, I set up a Bayesian simultaneous equations model that attempts to take possible selection into account. ∗ ∗ The model assumes latent potential outcomes Yi (0) and Yi (1) as a linear function of family characteristics, Xi, treatment (“cram school” attendance), Ti, and an unobserved random shock 1i.

11 In addition, I assume that the treatment eﬀect is constant over population

∗ ∗ Yi (1) − Yi (0) = τ, ∀i

and that the unobservable characteristics for each individual are the same whether she gets treatment or not. The constant treatment effect assumption is somehow unrealistic and restrictive. It may still be a good approximation. As noted in Angrist(2001)[2], in practice, more general estimation strategies allowing heterogeneous treatment effect often lead to similar average treatment effect. The assumption allows me to extrapolate the treatment effect on those whose decision is affected by the exclusion restriction to the whole population. I, in turn, express the latent potential outcomes as:

∗ Yi (1) = τ + Xiβ1 + 1i ∗ Yi (0) = Xiβ1 + 1i

The observed outcome becomes:

∗ Yi = Yi(0) + Ti[Yi(1) − Yi(0)] (1)

= τTi + Xiβ1 + 1i (2)

∗ I observe Yi = 1 if Yi > 0; Yi = 0 otherwise. In order to accommodate the selection problem, I follow the standard strategy of Heckman (1979)[30] to assume a household makes their optimal decision whether to send their children to an exam preparatory program. A household sends their children to a “cram school” if ∗ the utility is greater than a certain threshold. Therefore, I can interpret Ti as the latent normalized utility: ∗ Ti = γzi + Xiβ2 + 2i (3)

∗ I observe Ti = 1 if Ti > 0; Ti = 0 otherwise. ∗ ∗ The argument implies: given we know Yi and Ti , and I can solve for the simultaneous equations model, the estimate for τ is an estimate for the average treatment eﬀect. Identi- ﬁcation of the model boils down to whether I can solve the simultaneous equations. I will discuss the issue in Section (4.2).

12 4.1 Model Assumptions

In order to estimate the behavioral model speciﬁed above, I adopt a parametric approach for the eﬃciency concern and simplicity.

Normality Assumption " # 1i Xi,Zi ∼ N (0, Σ) 2i

∗ The assumption speciﬁes how the unobserved characteristics aﬀect the outcome Yi and ∗ the selection rule Ti . Under normality assumption, the data augmentation approach ∗ ∗ comes into play. From an initial guess of the latent variables Yi and Ti , we can sequentially estimate the parameters and update the latent variables based on the estimates and the normality assumption.

Re-parametrization Assumption

2 var(2i) = 1 and 1i = δ2i + ηi, where ηi ∼ N 0, σ (4)

The assumption says the disturbance term of one equation is linear in the disturbance term of the other with an additive error term. The assumption implies " # σ2 + δ2 δ Σ = δ 1

Following the assumption, I can naturally re-parametrize the variance-covariance matrix. It has 3 advantages. First, it allows the researcher to explicitly estimate the components in Σ. Second, it normalizes a diagonal term in Σ to 1. Third, numeri- cally, the re-parametrization speeds up the convergence of Gibbs sampling described in details in Appendix (A).

4.2 Identiﬁcation

The model is fundamentally a special case of the simultaneous equations models presented in Heckman (1978) [29]. I would briefly summarize his identification arguments. Follow- ing Heckman’s argument, this class of simultaneous equations model is non-parametrically identified if 3 conditions are satisfied.

1. Principle Assumption

13 The principle assumption requires that the endogenous variable xi does not enter both equations. It is a sufficient and necessary condition for the class of simultaneous equations models to be well-defined. It guarantees we can uniquely solve each parameter from the equations. My model trivially satisfies the assumption.

2. Normalization of Variance Given the selection equation has an interpretation as utility, the utility is invariant to different scaling. The coefficients in the equations are identified up to a constant. I can normalize a diagonal term of the variance-covariance matrix. I adopt a re- parametrization approach presented in Section (4.1).

3. Exclusion Restrictions3 Even though I can purely rely on nonlinearity of normality assumption for identification, lacking in exclusion restrictions in simultaneous equations models usually hampers robustness of the estimates.(Manski(1989) [39]) On the other hand, a natural exclusion restriction is often difficult to find. Since selection to “cram school” can be interpreted as demand for “cram school,” it is natural to look for a cost shifter. I follow the insight of Card(1995)[11] to exploit the geographic variation. The idea is to use the distance between one’s school to “cram school” to be the exogenous variation. The cost of attending a “cram school” is composed of the time cost of transportation, the tuition fees, and the time cost the teenagers spend in the class. As the traveling time to “cram school” increases, the cost of attending “cram school” rises. Meanwhile, the traveling time does not affect students’ performance in the admission procedure. It satisfies the exogeneity condition for an ideal exclusion restriction. Since I only observe how long it takes the attendants to go to a “cram school” in the dataset, I estimate a censored regression of commuting time to “cram school” of the attendants against their family characteristics. I impute the missing commuting time of the non-attendants using the linear fitted values from the censored regression estimates. The imputation is valid because students go to school in their school district in junior high school. Without the rights to driver’s license, junior high students rely on public transportation, walking, biking, or their parents for mobility. The exam prep-programs are localized in the school districts. If teenagers with similar family

3I am indebted to Sandeep Shetty for the idea of the exclusion restriction.

14 backgrounds live in a similar neighborhood within the school district, the imputation based on the ﬁtted value of the censored regression will provide a good approximation for the missing commuting time for the non-attendants.

4.3 Reference Prior

To complete the Bayesian models, standard normal-gamma conjugate priors are imposed on the parameters ([24]).

0 2 β1 ∼ MVN (β1 , σβ1 I) 2 τ ∼ N (τ0, στ ) 0 2 β2 ∼ MVN (β2 , σβ2 I) 2 δ ∼ N (δ0, σδ ) σ2 ∼ IG(a, b)

This is a commonly used proper reference prior, which is an approximation to standard improper reference prior.(See Christensen(2011)[18]) The parameters on the means of the 0 0 normal distributions, β1 , β2 , τ, are set to 0. The choice of prior parameters is philosophically consistent with Zellner (2007) [5] in the sense that all variation is considered random or 2 2 2 6 nonsystematic unless shown otherwise. I set σβ, σα, and σγ to 10 and set the shape and scale parameters of the inverse-gamma distribution to 10−3. It gives the inverse gamma distribution an - form, which has concentrated density at 0+ and has a long tail. The choice of theses parameter values are standard. The reference prior corresponding to the standard frequentist MLE or least squared methods are well-known in the Bayesian literatures, such as Chib(1992)[15] and Christensen(2011)[18].

4.4 The Results

In Table (6), I present the empirical results estimated by the Bayesian simultaneous equations model. The first panel shows the results when the outcome is defined as admission to a public high school. In order to compare the result with the propensity score matching result, I calculate the average partial effect given treated, P (Y = 1|X,T = 1). Consistent with the matching estimates, the estimated effect is about 14% increase in the probability of being admitted to public high school. It also indicates an insignificant negative selection into “cram school.”

15 The second panel shows the results when the outcome is defined as admission to an “elite” high school. Conditional on participating in a prep-program, the partial effect of attending “cram school” increases the chances of being admitted to an “elite” high school by around 6.6%. The estimate is also consistent with matching results. Sorting to “cram school” in this case is also insignificantly negative.

Table 6: Empirical Results of the Key Variables

Post. Mean Post. SD APE† Post. Mean Post. SD APE Regressor Public HS Elite HS Cram School 1.128∗∗ 0.523 0.107 0.820∗∗∗ 0.208 0.084 Minutes to Cram School -0.050∗∗∗ 0.004 - -0.046∗∗∗ 0.004 - σ2 8.230 1.776 - 2.377 0.470 - δ 0.289 0.327 - -0.153 0.284 -

† The average partial eﬀect of “cram school” is deﬁned conditional on “cram school” attendance when “cram school” attendance switches from 0 to 1. 1 X ∆Pˆ(Y = 1|X,T = 1) ≈ φ(Xβˆ)βˆ ∆x N j j i

Figure (3) shows the Markov chains and the histograms of the posterior distributions of treatment eﬀect parameters. The posterior distributions are of standard shape for the normal-inverse-gamma model. Since both of them are unimodal and symmetric, the 90% conﬁdence set are simply represented by the 5% and 95% posterior quantiles. I present the full empirical results in the Appendix.

4.5 Robustness Check

A possible concern about the empirical results may be: Is the result robust in absence of the Bayesian model? In this section, I implement a standard bivariate Probit model and compare the results with the Bayesian approach. Table (7) shows the empirical results using standard bivariate Probit model. Notice that compared with the Bayesian model developed in the previous section, Probit model imposes an additional constraint of equal variances. The results indicate that “cram school” raises students’ chances of being admitted to public high school by 18.8% while it increases their chances of being admitted to “elite” high school by 3.3%. Both speciﬁcations also indicate slightly negative sorting into “cram school.”

16 Histogram: Public High School Markov Chain: Public High School 0.8 2.5

2.0 0.6 1.5

0.4 1.0 density

Cram School Cram 0.5 0.2 0.0

0.0 0 1 2 3 200 400 600 800 1000 Cram School Iterations

Histogram: Elite High School Markov Chain: Elite High School 0.8 2.0

0.6 1.5

0.4 1.0 density

Cram School Cram 0.5 0.2 0.0

0.0 0 1 2 200 400 600 800 1000 Cram School Iterations

Figure 3: Posterior Distributions

† The Markov chains plot every 5 draws of the simulated chains.

Table 7: Robustness Check: Bivariate Probit Model

Post. Mean Post. SD APE† Post. Mean Post. SD APE Regressor Public HS Elite HS Cram School 0.730∗ 0.366 0.220 1.035∗∗ 0.316 0.065 Minutes to Cram School -0.051∗∗ 0.005 - -0.047∗∗ 0.005 - ρ -0.083 0.211 - -0.368∗ 0.205 -

17 4.6 Individual Decision Problem

An advantage of applying Bayesian methods to program evaluation is that it allows the researchers to think of the problem as a decision problem. (Dehejia(2005)[21]) Imagine a student wonder whether she should attend “cram school” given her performance in school and family background. She may be concerned about the uncertainty of the model estimates. The researcher can help her out by exploiting the Bayesian model. The decision problem for a student to decide whether to enroll in an exam prep-program is associated with the outcome. It is important to embody the uncertainty of the outcomes from the model by allowing for parameter uncertainty. The predictive posterior distribution of the Bayesian model constructs a distribution of outcome based on the posterior distribution of the parameters. I simulate the predictive posterior distribution in the following way: for each individual i in the cohort of interest, say group of family income less than NTD30,000 per month, living ˜1 in Taipei City, and having failed a class. Given the covariates Xi as observed, I set Ti = 1 ˜0 to simulate for the treated and Ti = 0 to simulate for the control. Using the stored draws (j) (j) 2(j) 5000 ∗1 ˜ (j) from the posterior distributions {τ , α , σ }j=1 , I draw for Yi |{Ti,Xi} ∼ N (τ + (j) 2(j) ∗0 ˜ (j) 2(j) Xiα , σ ) and Yi |{Ti,Xi} ∼ N (Xiα , σ ). Finally, I obtain predicted outcome by 1 ∗1 0 ∗0 Yi = 1(Yi > 0) and Yi = 1(Yi > 0). In Table (8), I show the average predicted probability of being admitted to a public high school given different levels of family income and whether she has failed a course. I define low income as earning less than NTD30,000 per month, median income as NTD 50,000 - NTD 59,999 per month, and high school as earning more than 150,000 per month. The result shows that the likelihood of being admitted to a public high school is significantly higher for students from higher income families. Among students who have failed a class, or less able in academia, the predicted improvement in probability by going to “cram school” is larger for higher income students than lower income students. However, we do not observe the same pattern among students who have never failed a class. Comparing students who fail a class with those who have never failed one, the predicted effect is also significantly higher for the students who are more able. The effect of “cram school” for less motivated or less able children is smaller than for motivated and able students. This suggests that parents should think twice before sending their children who are not interested in studying to a “cram school” to ”force” them to academic track. The effect may not outweigh the costs of time, tuition fees, and unnecessary additional pressure.

18 Table 8: Mean and Variance of Predicted Probability of being Admitted to Public HS

Treated Control Treated - Control Cohorts Pred. Prob. S.D. Pred. Prob. S.D. Pred. Diﬀ. S.D. Num. Obs. Taipei City Low Income; Fail 0.170 0.376 0.095 0.293 0.075 0.455 111 Median Income; Fail 0.176 0.381 0.099 0.298 0.077 0.466 122 High Income; Fail 0.302 0.459 0.189 0.391 0.114 0.575 25 Low Income; Never Fail 0.594 0.491 0.448 0.497 0.146 0.663 36 Median Income; Never Fail 0.683 0.465 0.542 0.498 0.141 0.651 69 High Income; Never Fail 0.728 0.445 0.593 0.491 0.134 0.633 26 Taipei County Low Income; Fail 0.110 0.313 0.056 0.230 0.054 0.375 142 Median Income; Fail 0.136 0.342 0.070 0.256 0.065 0.415 127 High Income; Fail 0.119 0.324 0.059 0.236 0.060 0.392 20 Low Income; Never Fail 0.516 0.500 0.371 0.483 0.145 0.662 46 Median Income; Never Fail 0.534 0.499 0.389 0.487 0.145 0.664 73 High Income; Never Fail 0.599 0.490 0.455 0.498 0.145 0.669 9 Yilan County Low Income; Fail 0.148 0.355 0.081 0.273 0.067 0.430 90 Median Income; Fail 0.175 0.380 0.098 0.297 0.077 0.464 81 High Income; Fail 0.179 0.383 0.103 0.304 0.076 0.467 11 Low Income; Never Fail 0.579 0.494 0.433 0.495 0.146 0.656 18 Median Income; Never Fail 0.677 0.468 0.541 0.498 0.136 0.643 36 High Income; Never Fail 0.608 0.488 0.458 0.498 0.150 0.674 5

19 5 Discussions

Exam preparatory programs are prevalent in many East Asian countries because of the usage of a centrally administered exam system to allocate scarce educational resources. However, because attendants to these programs may be highly self-selected, there is a lack of rigorous study on evaluation of the programs. The research question of whether an exam prep-program increases likelihood of being admitted to a public high school or an “elite” high school can be motivated by Rosen-Willis life cycle model of human capital investment. Investment in exam preparatory programs can be considered as a current investment to decrease future educational costs. We expect “cram school” increases propensity of admission to a public high school or an “elite” high school. The paper provides two alternative empirical approaches to evaluate the effectiveness of “cram schools.” Identification of propensity score matching approach relies on unconfoundedness assumption, which states: given the observed characteristics, attending “cram school” is independent of potential placements. If unconfoundedness assumption holds, then the average treatment effect on the treated can be obtained by matching each treated unit with a control unit with the shortest distance in propensity score. The result suggests that “cram school” increases chances to be admitted to public high school by 16% to 20% and to “elite” high school by 4% to 7%. Alternatively, I set up a Bayesian simultaneous equations model that specifies the selection rule. Identification of the model relies on exogeneity of exclusion restriction. I assume commuting time from school to “cram school” is relevant to students’ attendance decision and exogenous to their high school placement. Imposing the constant treatment effect assumption, I can extrapolate the effect of the ”compliers,” who are discouraged from participating a program due to longer commuting time, to the population of interest. I find evidence that average partial effect given treated is around 11% in chances of being admitted to a public high school and 8.4% to an “elite” high school. The result also indicates the correlation of the unobservable characteristics that affect both selection and outcome is not significantly different from 0. The paper adds an important policy perspective to the ongoing debate of educational reform in Taiwan. The findings suggest that “cram schools” pass the test of the market. Attending “cram school,” indeed, improves students’ chances to be admitted to a public high school and an “elite” high school. Even though the Taiwanese policy makers consider “cram schools” as an unnecessary sources of pressure, these programs will continue to play an significant role in Taiwanese teenagers’ life without a fundamental change in the centrally

20 administered admission system and the belief of ”elitism.” Whenever there is a demand for “elite” high school, the market will persist.

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24 A Gibbs Sampling Algorithm

The posterior distributions of the parameters of the Bayesian selection models are obtained through Gibbs Sampling procedure. The Gibbs sampling is a Markov chain Monte Carlo simulation techniques. (Dehejia (2003)[20]) The algorithm allows me to simulate random variables from a distribution indirectly without having to calculate its density. A good introductory survey of this method is Casella and George (1992)[14]. The basic idea of Gibbs Sampling is that, by sequentially sampling from the conditional distribution of each parameter on the remaining parameters, the simulated draws would converge in distribution to a stationary distribution that is the joint distribution of interest under some regularity conditions. Given the conjugate priors, all conditional distributions have closed forms. It simpliﬁes the algorithm to sequential drawings from the following conditionals after I complete the data augmentation steps. The steps of the Gibbs sampler are as follows4:

Step 1

δ σ2 T ∗|T = 1 ∼ tN γz + X β + (Y ∗ − β − X β ), i i [0,∞) i i 2 σ2 + δ2 i i 1 σ2 + δ2 δ σ2 T ∗|T = 0 ∼ tN γz + X β + (Y ∗ − X β ), i i (−∞,0] i i 2 σ2 + δ2 i i 1 σ2 + δ2

where tN denotes truncated normal distribution.

Step 2

∗ ∗ 2 Yi |Yi = 1 ∼ tN(0,∞) τTi + Xiβ1 + δ(Ti − γzi − Xiβ2), σ ∗ ∗ 2 Yi |Yi = 0 ∼ tN(−∞,0) τTi + Xiβ1 + δ(Ti − γzi − Xiβ2), σ

Step 1 and Step 2 are often called ”data augmentation” steps by the seminal work in Tanner and

Wong (1987)[48]. Intuitively, given I observe Ti and the normality assumption on the disturbances, I ∗ can ”observe” the latent variables Ti . In addition, given the ﬁxed censored point assumption and the ∗ normality assumption, I can impute the missing values of Yi by drawing from the truncated normal distribution. The above argument leads to an algorithm of successive substitution to solve for a ﬁxed point. Nat- urally, the Gibbs sampling algorithm is ideally applicable.

Step 3 " N N N #! N 1 X X X σ2 ∼ IG a + , b + 2 − 2δ + δ2 2 2 2 1i 1i 2i 2i i=1 i=1 i=1 ∗ ∗ where I denote 1i = Yi − τ − Xiβ1 and 2i = Ti − γzi − Xiβ2 Step 4 ! δ σ2 + σ2 PN σ2σ2 δ ∼ N 0 δ i=1 1i 2i , δ 2 2 PN 2 2 2 PN 2 σ + σδ i=1 2i σ + σδ i=1 2i

4Notice that I suppress the conditionals for simplicity of notations.

25 Step 5 β, α, γ are simulated by Bayesian Regressions: Notice that I can rearrange Equation (2) and (3):

δ σ2 T ∗ − (Y ∗ − τT − X β ) = γz + X β + ξ , where ξ ∼ N (0, ) i σ2 + δ2 i i i 1 i i 2 2i 2i σ2 + δ2

and ∗ ∗ 2 Yi − δ(Ti − γzi − Xiβ2) = τTi + Xiβ1 + ξ1i, where ξ1i ∼ N (0, σ )

To simulate the conditional requires 2 steps. Firstly, notice that the joint normality assumption immediately leads to " # " # " #! Y ∗ τT + X β σ2 + δ2 δ i ∼ N i i 1 , ∗ Ti γzi + Xiβ2 δ 1 By the property of joint normal distribution, I have

∗ ∗ ∗ E[Yi |Ti ] = τTi + Xiβ1 + δ(Ti − γzi − Xiβ2) ∗ ∗ 2 var[Yi |Ti ] = σ

Now, we can estimate β and α using standard Bayesian regression, which is a special case of Lindley and Smith (1974)[38]:

∗ ∗ 2 Yi − δ(Ti − γzi − Xiβ2) = τTi + Xiβ + η1i, where η1i ∼ N (0, σ )

Secondly, we observe that

δ E[T ∗|Y ∗] = γz + X β + (y∗ − τT − X β ) i i i i 2 σ2 + δ2 i i i 1 σ2 var[T ∗|Y ∗] = i i σ2 + δ2

Again, I can estimate γ by standard Bayesian regression5:

δ σ2 T ∗ − (Y ∗ − τT − X β ) = γz + X β + η , where η ∼ N (0, ) i σ2 + δ2 i i i 1 i i 2 2i 2i σ2 + δ2

5In the previous literatures, such as Li (1998)[35], these 2 steps are usually completed by Zellner’s seemingly unrelated regressions in one step. Even though these 2 approaches are theoretically equivalent, SUR model requires computation of the inverse of a sparse matrix of high dimensionality. The approximation error in the computer routines is likely to slow down the convergence of the chains or even bias the estimates. Considering I do not intend to run a more complex model, such as a multilevel model, the 2-step method can be more suitable.

26 B Full Empirical Results

In this section, I report the full empirical results. I obtain all the results by simulating 55000 draws and discarding the first 5000 draws as the burn-in period. It is a standard procedure in Bayesian estimation to minimize the impact of the choice of initial points on the simulated posterior distribution. Since I use the standard normal-gamma model, the posterior distributions are all unimodal so that the 90% high-propensity confidence set can be easily obtained by looking at 5% and 95% quantiles. I also report the probability that a parameter is greater than 0 for one-sided significance test. In addition, shrinkage factors are computed using Gelman-Rubin convergence diagnostics (Cowles (1996)[19] and Gelman et al.(1992)[25]) by simulating 4 parallel chains with initial points disperse around the original initial points. All shrinkage factors in all specifications are stabilized around 1 implying the convergence of the Markov chains to the stationary distribution. I do not report the shrinkage factors in the table.

27 Table 9: Bayesian Model: Public High School

Mean SD 5% 25% 50% 75% 95% P(x>0) Equation 1: Public HS Cram School 1.128 0.523 0.281 0.784 1.120 1.465 2.018 0.985 Cram before 0.449 0.322 -0.068 0.230 0.441 0.660 0.984 0.923 male -0.166 0.176 -0.465 -0.281 -0.159 -0.048 0.117 0.167 Num. Siblings -0.449 0.132 -0.676 -0.532 -0.442 -0.358 -0.246 0.000 Intention to HS 1.515 0.244 1.136 1.349 1.504 1.665 1.927 1.000 less than NTD 30,000 -1.411 0.337 -1.961 -1.637 -1.407 -1.183 -0.864 0.000 NTD 30,000 -NTD 49,999 -0.893 0.316 -1.405 -1.108 -0.891 -0.678 -0.376 0.002 NTD 50,000 -NTD 59,999 -1.155 0.327 -1.695 -1.372 -1.150 -0.934 -0.627 0.000 NTD 60,000 -NTD 69,999 -0.590 0.390 -1.234 -0.849 -0.585 -0.336 0.042 0.067 NTD 70,000 -NTD 79,999 -1.005 0.390 -1.653 -1.270 -0.999 -0.732 -0.371 0.004 NTD 80,000 -NTD 89,999 -0.080 0.448 -0.818 -0.380 -0.073 0.217 0.650 0.430 NTD 90,000 -NTD 99,999 -0.089 0.438 -0.792 -0.390 -0.097 0.198 0.642 0.411 NTD 100,000 -NTD 109,999 -0.516 0.472 -1.288 -0.836 -0.516 -0.189 0.257 0.141 NTD 110,000 -NTD 119,999 -0.748 0.502 -1.560 -1.088 -0.753 -0.409 0.083 0.068 NTD 120,000 -NTD 129,999 -1.009 0.580 -1.966 -1.395 -1.011 -0.622 -0.048 0.042 NTD 130,000 -NTD 139,999 -0.437 0.682 -1.567 -0.898 -0.440 0.025 0.674 0.262 NTD 140,000 -NTD 149,999 0.630 0.760 -0.628 0.112 0.639 1.147 1.886 0.799 more than NTD 150,000 -0.960 0.468 -1.729 -1.269 -0.960 -0.652 -0.199 0.021 Sound family -0.055 0.270 -0.506 -0.229 -0.057 0.125 0.388 0.419 Fail a subject 3.509 0.372 2.942 3.254 3.484 3.746 4.146 1.000 School FE Yes Parents’ Educ. Yes Parents’ Occ. Yes Equation 2: Cram School Commuting time -0.050 0.004 -0.057 -0.053 -0.050 -0.047 -0.044 0.000 Cram before 1.510 0.065 1.406 1.466 1.508 1.552 1.618 1.000 male -0.084 0.061 -0.183 -0.124 -0.084 -0.043 0.017 0.085 Num. Siblings -0.107 0.037 -0.168 -0.132 -0.107 -0.082 -0.046 0.003 Intention to HS 0.290 0.068 0.178 0.244 0.289 0.336 0.399 1.000 less than NTD 30,000 0.125 0.220 -0.240 -0.018 0.121 0.269 0.491 0.717 NTD 30,000 -NTD 49,999 0.119 0.218 -0.235 -0.027 0.118 0.261 0.482 0.711 NTD 50,000 -NTD 59,999 0.140 0.221 -0.220 -0.010 0.137 0.287 0.512 0.736 NTD 60,000 -NTD 69,999 0.097 0.237 -0.293 -0.065 0.100 0.253 0.495 0.657 NTD 70,000 -NTD 79,999 0.048 0.235 -0.331 -0.106 0.046 0.204 0.446 0.574 NTD 80,000 -NTD 89,999 -0.042 0.251 -0.459 -0.208 -0.045 0.123 0.378 0.425 NTD 90,000 -NTD 99,999 0.441 0.251 0.026 0.271 0.441 0.611 0.860 0.959 NTD 100,000 -NTD 109,999 -0.109 0.261 -0.534 -0.285 -0.113 0.067 0.321 0.335 NTD 110,000 -NTD 119,999 0.007 0.278 -0.447 -0.178 0.004 0.191 0.475 0.505 NTD 120,000 -NTD 129,999 0.229 0.319 -0.293 0.012 0.222 0.442 0.768 0.762 NTD 130,000 -NTD 139,999 0.005 0.363 -0.596 -0.242 0.005 0.251 0.597 0.506 NTD 140,000 -NTD 149,999 -0.235 0.412 -0.904 -0.512 -0.238 0.042 0.447 0.281 more than NTD 150,000 0.127 0.266 -0.309 -0.051 0.124 0.303 0.572 0.686 Sound family 0.080 0.100 -0.084 0.012 0.079 0.147 0.245 0.791 Fail a subject 0.299 0.071 0.181 0.252 0.300 0.346 0.417 1.000 School FE Yes Parents’ Educ. Yes Parents’ Occ. Yes σ2 8.230 1.776 5.794 6.938 8.006 9.264 11.527 1.000 δ 0.289 0.327 -0.261 0.089 0.290 0.506 0.818 0.819 28 Table 10: Bayesian Model: “elite” High School

Mean SD 5% 25% 50% 75% 95% P(x>0) Equation 1: Elite HS Cram School 0.820 0.208 0.485 0.677 0.819 0.955 1.172 1.000 Cram before -0.094 0.269 -0.525 -0.278 -0.100 0.093 0.349 0.366 male -0.056 0.142 -0.289 -0.149 -0.052 0.040 0.167 0.356 Num. Siblings -0.320 0.092 -0.480 -0.377 -0.315 -0.258 -0.180 0.000 Intention to HS 0.469 0.189 0.164 0.343 0.464 0.594 0.781 0.993 less than NTD 30,000 -1.320 0.334 -1.874 -1.545 -1.324 -1.093 -0.777 0.000 NTD 30,000 -NTD 49,999 -0.838 0.300 -1.333 -1.040 -0.837 -0.632 -0.349 0.002 NTD 50,000 -NTD 59,999 -0.824 0.297 -1.317 -1.020 -0.825 -0.628 -0.335 0.004 NTD 60,000 -NTD 69,999 -0.366 0.342 -0.930 -0.588 -0.370 -0.143 0.193 0.139 NTD 70,000 -NTD 79,999 -0.938 0.347 -1.513 -1.169 -0.934 -0.705 -0.358 0.003 NTD 80,000 -NTD 89,999 -0.783 0.391 -1.428 -1.048 -0.776 -0.522 -0.148 0.020 NTD 90,000 -NTD 99,999 -0.274 0.365 -0.871 -0.523 -0.278 -0.030 0.327 0.229 NTD 100,000 -NTD 109,999 -0.380 0.404 -1.049 -0.654 -0.367 -0.108 0.283 0.171 NTD 110,000 -NTD 119,999 -0.980 0.436 -1.704 -1.274 -0.978 -0.679 -0.264 0.012 NTD 120,000 -NTD 129,999 -0.848 0.493 -1.684 -1.170 -0.849 -0.514 -0.042 0.041 NTD 130,000 -NTD 139,999 0.197 0.553 -0.720 -0.171 0.193 0.562 1.118 0.643 NTD 140,000 -NTD 149,999 -0.703 0.640 -1.738 -1.121 -0.700 -0.277 0.326 0.140 more than NTD 150,000 -0.605 0.381 -1.236 -0.859 -0.600 -0.348 0.026 0.056 Sound family -0.336 0.227 -0.711 -0.491 -0.336 -0.182 0.037 0.070 Fail a subject 1.984 0.222 1.629 1.834 1.981 2.126 2.344 1.000 School FE Yes Parents’ Educ. Yes Parents’ Occ. Yes Equation 2: Cram School Commuting time -0.046 0.004 -0.053 -0.049 -0.046 -0.043 -0.038 0.000 Cram before 1.509 0.069 1.395 1.462 1.509 1.556 1.623 1.000 male -0.034 0.066 -0.142 -0.078 -0.034 0.010 0.075 0.304 Num. Siblings -0.079 0.040 -0.146 -0.106 -0.079 -0.051 -0.012 0.027 Intention to HS 0.319 0.074 0.199 0.270 0.320 0.369 0.438 1.000 less than NTD 30,000 0.192 0.228 -0.178 0.037 0.187 0.346 0.568 0.799 NTD 30,000 -NTD 49,999 0.133 0.228 -0.239 -0.022 0.129 0.287 0.513 0.724 NTD 50,000 -NTD 59,999 0.169 0.229 -0.208 0.012 0.167 0.322 0.553 0.769 NTD 60,000 -NTD 69,999 0.234 0.252 -0.170 0.068 0.231 0.398 0.659 0.826 NTD 70,000 -NTD 79,999 0.031 0.245 -0.367 -0.141 0.034 0.197 0.435 0.553 NTD 80,000 -NTD 89,999 -0.004 0.263 -0.436 -0.177 -0.007 0.177 0.426 0.487 NTD 90,000 -NTD 99,999 0.403 0.268 -0.037 0.218 0.405 0.579 0.848 0.935 NTD 100,000 -NTD 109,999 -0.144 0.270 -0.586 -0.329 -0.143 0.041 0.299 0.302 NTD 110,000 -NTD 119,999 0.245 0.294 -0.236 0.050 0.245 0.441 0.734 0.800 NTD 120,000 -NTD 129,999 0.413 0.343 -0.149 0.177 0.416 0.641 0.978 0.885 NTD 130,000 -NTD 139,999 0.105 0.375 -0.507 -0.151 0.100 0.359 0.717 0.606 NTD 140,000 -NTD 149,999 -0.081 0.424 -0.788 -0.368 -0.071 0.207 0.592 0.430 more than NTD 150,000 0.136 0.268 -0.304 -0.046 0.134 0.315 0.579 0.691 Sound family 0.090 0.107 -0.084 0.018 0.090 0.162 0.270 0.798 Fail a subject 0.327 0.078 0.199 0.277 0.327 0.378 0.453 1.000 School FE Yes Parents’ Educ. Yes Parents’ Occ. Yes σ2 2.377 0.470 1.676 2.073 2.331 2.630 3.219 1.000 δ -0.153 0.284 -0.628 -0.344 -0.156 0.037 0.323 0.292 29 Table 11: Bivariate Probit Model: Public High School

Variable Coeﬃcient Std. Err. Coeﬃcient Std. Err. Equation 1 : Public HS Equation 2 : Cram School Cram School 0.730∗ 0.366 - - Commuting time - - -0.051∗∗ 0.005 Cram before 0.021 0.196 1.494∗∗ 0.082 Male 0.016 0.060 -0.080 0.079 Num. Siblings -0.043 0.049 -0.103∗∗ 0.029 less than NTD 30,000 -0.761 0.524 -0.072 0.311 NTD 30,000 -NTD 49,999 -0.560 0.538 -0.076 0.289 NTD 50,000 -NTD 59,999 -0.663 0.539 -0.062 0.288 NTD 60,000 -NTD 69,999 -0.503 0.556 -0.106 0.307 NTD 70,000 -NTD 79,999 -0.641 0.535 -0.152 0.332 NTD 80,000 -NTD 89,999 -0.286 0.539 -0.252 0.335 NTD 90,000 -NTD 99,999 -0.313 0.575 0.249 0.307 NTD 100,000 -NTD 109,999 -0.475 0.511 -0.306 0.325 NTD 110,000 -NTD 119,999 -0.585 0.546 -0.174 0.327 NTD 120,000 -NTD 129,999 -0.793 0.572 0.034 0.327 NTD 130,000 -NTD 139,999 -0.514 0.600 -0.201 0.424 NTD 140,000 -NTD 149,999 0.397 0.655 -0.450 0.405 more than NTD 150,000 -0.639 0.549 -0.075 0.345 Sound family 0.128 0.094 0.062 0.094 Intention to HS 0.617∗∗ 0.079 0.295∗∗ 0.073 Fail a subject 1.288∗∗ 0.092 0.303∗∗ 0.090 School FE Yes Parents’ Educ. Yes Parents’ Occ. Yes ρ -0.083 0.211

30 Table 12: Bivariate Probit Model: “elite” High School

Variable Coeﬃcient Std. Err. Coeﬃcient Std. Err. Equation 1 : Elite HS Equation 2 : Cram School Cram School 1.035∗∗ 0.316 - - Commuting time - - -0.047∗∗ 0.005 Cram before -0.251 0.172 1.486∗∗ 0.087 Male 0.039 0.075 -0.043 0.084 Num. Siblings -0.077 0.065 -0.080∗∗ 0.030 less than NTD 30,000 -2.339∗∗ 0.474 0.466 0.402 NTD 30,000 -NTD 49,999 -2.076∗∗ 0.480 0.417 0.378 NTD 50,000 -NTD 59,999 -2.041∗∗ 0.473 0.448 0.371 NTD 60,000 -NTD 69,999 -1.774∗∗ 0.436 0.501 0.411 NTD 70,000 -NTD 79,999 -2.125∗∗ 0.443 0.304 0.413 NTD 80,000 -NTD 89,999 -2.074∗∗ 0.477 0.267 0.412 NTD 90,000 -NTD 99,999 -1.670∗∗ 0.481 0.677† 0.376 NTD 100,000 -NTD 109,999 -1.789∗∗ 0.439 0.122 0.395 NTD 110,000 -NTD 119,999 -2.298∗∗ 0.499 0.551 0.432 NTD 120,000 -NTD 129,999 -2.200∗∗ 0.535 0.710† 0.422 NTD 130,000 -NTD 139,999 -1.365∗∗ 0.506 0.402 0.513 NTD 140,000 -NTD 149,999 -2.253∗∗ 0.643 0.187 0.499 more than 150,000 -1.886∗∗ 0.490 0.398 0.392 Sound family -0.104 0.127 0.058 0.110 Intention to HS 0.426∗∗ 0.138 0.317∗∗ 0.074 Fail a subject 1.391∗∗ 0.154 0.319∗∗ 0.101 School FE Yes Parents’ Educ. Yes Parents’ Occ. Yes ρ -0.368† 0.205