Rates of Return to University Education: An Application of Regression Discontinuity Design

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1 Rates of Return to University Education: An Application of Regression Discontinuity Design Elliott Fan Xin Meng y Zhichao Wei z Guochang Zhao x July 16, 2011 Abstract Estimating the rates of return to university education has always been di cult due to the problem of omitted variable bias. Bene ting from a special feature of the University Admission system in China, which has clear cuto s for university entry, combined with a unique data set with information on individual National College Entrance Examination (NCEE) scores, we estimate the Local Average Treatment E ects (LATE) of university education based on a Regression Discontinuity design. To the best of our knowledge, this is the rst study to use RD design to estimate the causal e ect of a university education on earnings. Our empirical results suggest that the return to a 4-year university education is a 36 to 59 percent increase of earnings, when compared to a control group consisting of individuals who attended NCEE but failed to gain admission to university (their highest education levels are either senior high school or three-year college). Further examinations suggest that the treatment e ects are likely to be heterogeneous for compliers and noncompliers and the LATEs are likely to be larger than the ATEs. Key word: Rate of return to education, Regression Discontinuity Design, China. JEL classi cation numbers: I21, I28, J24 Research School of Economics, CBE, Australian National University, Canberra ACT 0200, Australia. elliott.fan@anu.edu.au y Corresponding author: Research School of Economics, CBE, Australian National University, Canberra ACT 0200, Australia. Xin.Meng@anu.edu.au. z Department of Economics, Brown University, 64 Waterman Street, Providence, RI ( Zhichao_Wei@brown.edu) x Research School of Economics, CBE, Australian National University, Canberra ACT 0200, Australia. guochang.zhao@anu.edu.au

2 1 Introduction It is common knowledge that people who are more educated, on average, earn more than the less educated. A key question remains as to the extent to which higher education causes higher earnings. Perhaps higher earnings are caused by the more-educated being more able or having other unobserved advantages. Many studies have attempted to account for di erences in various unobserved endowments by using within-twin comparisons (see, for example, Ashenfelter and Krueger, 1994; Berhman, Rosenzweig, and Taubman, 1994; Miller, Mulvey, and Martin, 1995; Isacsson, 1999; and Bingley, Christinsen, and Jensen, 2009). Other studies have used natural experiments (see, for example, Angrist and Krueger, 1991; Card, 1995; Harmon and Walker, 1995; Acemoglu and Angrist, 2001; Lochner and Moretti, 2004; and Oreopoulos, 2006). Critics, however, have been skeptical about the returns to education estimated using these techniques (critics of within-twin variations include Bound and Solon, 1999; Neumark, 1999; Leigh and Ryan, 2008; and Lee and Lemieux, 2010; criticisms of natural experiments include Bound, Jaeger and Baker, 1995 and Oreopoulos, 2006). The best way to estimate the causal e ect of education on earnings is to use a randomized trial. However, education is a long run investment and measuring the resulting labor market outcomes demands a long window of observation, not to mention that such experiments on humans are not executable. This may be one of the reasons why we have not seen any studies on returns to education using a randomized trial. Recently, Lee and Lemieux (2010) have labeled Regression Discontinuity (RD) design as a closer cousin to randomized experiments. To date, however, Oreopoulos (2006) is the only study known to us which uses the Regression Discontinuity design to estimate the causality between one additional year of high school education and labor market earnings. This paper contributes to the literature by extending the application of RD design to assessing the returns to higher education. Utilizing an unique feature of the Chinese College Admission system (CAS), which uses test scores from a centralized examination the National College Entrance Examination (NCEE) as the benchmark to select students, we are able to nd well-de ned cuto s for university admission. This, together with a rich survey data set 1

3 with information on individual NCEE scores, provides a rare opportunity for us to apply fuzzy RD design with IV to estimate the local average treatment e ects (LATE) of the university education. To the best of our knowledge, our study is the rst to use RD design to estimate the causal e ect of university education on earnings. While our LATE estimate can be interpreted as the average treatment e ect for subpopulations whose treatment status is induced by the instrument, i.e., the compliers, it might not provide information on the average treatment e ect (ATE) for the population as a whole, unless the estimate can be extrapolated to all other subpopulations. However, knowing about the ATE is of special importance as it can be used to make predictions about the treatment e ect on a randomly selected individual. While it is impossible to consistently estimate the ATE for the subpopulations other than compliers in the presence of omitted variables and measurement error biases when there exist heterogeneous treatment e ects (Imbens and Wooldridge, 2009), in this paper we employ two strategies to gauge whether the ATE is likely to be larger or smaller than the LATE estimate. First, following a method proposed by Imbens and Wooldridge (2009), we calculate and compare the di erences in average earnings between eligible compliers and eligible always-takers, and between ineligible compliers and ineligible never-takers. The magnitudes of these di erences can assist in assessing whether the LATE estimate is similar to the treatment e ect for non-compliers. Second, we develop a method to explore whether the ATE is larger or smaller than the LATE through comparisons between estimates based on samples with and without observed non-compliers. Details of these strategies are presented in Section 6. The LATE estimates carry strong policy implications in China, where the annual enrolment at universities skyrocketed from just over 284,000 in 1979 to 5.99 million in 2008, a twenty-fold increase. 1 In particular, China expanded university enrolment by 47 percent in one year in 1999 and since then the enrolment gure has increased by almost three-fold. As a result, the proportion of the urban labor force with tertiary education increased from just over 10 percent in 1987 to 40 percent in 2007 (Meng, Shen, and Xue, 2009). Although studies have shown 1 China is not alone in expanding education investment. Despite the lack of consensus as to the size of the causal e ect of education on earnings, governments in many parts of the world are investing heavily in education. The 1993 World Bank Report The East Asian Miracle identi es the rapid growth of human capital as one of the two principal engines of economic growth in East Asian countries (World Bank, 1993). 2

4 that the return to education increased during the 1990s, the rate of increase has slowed down signi cantly since the late 1990s (Zhang, Zhao, Park, and Song, 2005 and Meng et al. 2009). It is unfortunate that the drastic expansion of tertiary education in the late 1990s was not based on careful assessments of the returns to education, especially the rate of return for the group whose university attainment is more likely to be a ected by the expansion policy. Our LATE estimates, which measure the returns to university education for the group whose NCEE scores are around the cuto, and hence are more likely to be a ected by the university expansion policy, provide an important insight into the e ect of the education expansion. 2 In addition, our LATE estimates apply to 36 to 45 percent of our sample who participated in the NCEE (the compliers), and hence, should have a relatively general relevance. Using a fuzzy RD design with IV estimation, we nd that the return to a four-year university education is 36 to 59 percent relative to a control group of individuals who are either senior high school or three-year college graduates. Compared to the LATE, our analysis suggests that the ATE for the entire population is likely to be smaller a result consistent to the hypothesis of heterogeneous treatment e ects. The remainder of the paper is structured as follows. Section 2 describes the institutional background which a ects our research design. Section 3 introduces the methodologies. Data are presented in Section 4, which is followed by sections presenting our empirical results. Finally, conclusions are given in Section 7. 2 Background The Chinese schooling system is quite similar to that of the West. Figure 1 provides a sketch of the system. Students begin primary school at the age of 6-7 years. 3 The primary school normally requires six years to complete, and this is followed by three years at junior high school. Upon completion of junior high school, students have the option to continue studying 2 It is important to note that our sample does not include university graduates after Thus, our empirical results do not directly speak to the e ects of the 1999 expansion of the tertiary education. However, given that our LATE estimates are fairly large, it is di cult to imagine that the post-1999 e ect would be insigni cant, though a more careful examination us needed to make a precise assessment. 3 School starting age may di er across regions and over time. Currently it is six years in most of the urban areas. 3

5 for three years in an academic senior high school or entering a vocational secondary school for 2-4 years. Normally, those who complete the academic senior high school program participate in the NCEE to gain their undergraduate admission. Chinese higher education at the undergraduate level is divided into three-year college and four-year university programs. There are two tiers of four-year universities and the rst-tier is of higher quality and hence attracts greater central government funding. 4 For most universities, the undergraduate education is divided into two streams: the humanities/social sciences stream and the sciences stream. The National College Entrance Examination (NCEE) and College Admission system were established in During the Cultural Revolution ( ) the system stopped functioning for 11 years and resumed after the Cultural Revolution in The system operates on the basis of universal examination papers and marking standards across all regions in China. 5 The test subjects for the humanities/social sciences stream include Politics, Chinese, Math, Foreign Language, and History, while those for the sciences stream are Chinese, Math, Foreign Language, Physics and Chemistry. 6 Although in general the total score is uniform for di erent provinces within the same year, it varies over years because the number of subjects tested and the total score of each subject vary over years. 7 In addition, in a few provinces for several years 8 the original individual test scores were standardized based on their ranking in the distribution of the scores for all students in the province for that year. The full standardized scores are higher than the total original scores. We address these di erences by standardizing the test scores. 4 The two tier system was rst established by the central government in In that year six universities, including Peking University and Tsinghua University, were assigned to rst tier. Afterwards more universities gained the rst tier entitlement. By 1963, three years before the Cultural Revolution, there were 68 rst tier universities. In 1978, two years after the Cultural Revolution, the central government released a new list of 88 rst tier universities. In the 1990s this number reached 100 (China Ministry of Education, 2006 and Harbin Institute of Technology, 2008). 5 Since some provinces have gradually started to deviate from the system of universal exam papers. By 2006 the number of provinces not using National Examination papers had increased to 16. These, however, do not a ect our study as we only include individuals who participated the NCEE before For a detailed discussion of the NCEE and CAS in the earlier period, see Meng et al. (1989). 6 The exact subjects tested for di erent streams vary slightly across di erent years. 7 For example, in 1977, which marks the year the NCEE resumed after the Cultural Revolution, only four subjects were tested (Foreign Language was excluded) and the total score for each subject was set at 100. Thus, the full score for all subjects for that year was 400. In later years, however, the number of subjects tested increased to ve while the full score for each subject was set at 150 instead of 100. Consequently the total score increased to For detailed information on which provinces implemented the standardized scores, and in which years, please see Appendix A on cuto score data. 4

6 Details of this standardization are discussed in the Data Section. Admissions into either a four-year university or a three-year college are based on the NCEE scores. From the point of view of students, gaining admission normally requires the following steps: (1) Students need to choose either a humanities-social sciences or a science stream for their undergraduate study before taking the NCEE. (2) They then need to submit an application form that usually has three panels, each corresponding to a type of university or college (the rst and second-tier universities, and three-year colleges) and the students are allowed to apply to up to four schools in each of the groups. This application form may be submitted before or after the students nish the NCEE. In some cases, submissions may be made after students are informed of their nal scores. But, in any case, the application occurs before the publication of the cuto scores. (3) They need to take the exams. And (4) after the admission letter from a university arrives, they need to decide whether to take the position or not. From the perspective of the administration, the admission process may be described as follows: (1) The central government allocates a quota to each university. (2) The universities then allocate their quota to di erent provinces. (3) After knowing the given quotas from all the universities and the distribution of students NCEE results, each province will then determine their own cuto s for the di erent tier universities and three-year-colleges separately. 9 These cuto scores will then be made publicly available through schools, local education bureaus, local newspapers, the internet and television channels. An important point to note here is that the cuto s are normally set at a point which is 10 to 20 percent higher than that implied by the quota. In other words, 10 to 20 percent more students may have NCEE scores above the cuto s than it is indicated by the actual number of students who can be admitted. (4) Based on the rule of better school, earlier admission, the rst-tier universities will start their admission process rst and continue until all their quotas are lled, followed by the second-tier universities, and then the three-year colleges. Within each university, the admissions are processed in priority order. That is, students who exceeded the cuto score and listed a particular university as their rst preference will be considered rst, based on the ranking of their NCEE scores among all 9 To be more speci c, the provinces de ne three cuto s for each of the science and the humanity-social science streams: a cuto for the rst-tire universities, a cuto for the second-tire universities, and a cuto for the three-year colleges. 5

7 students who applied to that school. If the university s quota for the province is less than the number of students whose NCEE score exceeded the cuto in the province, those whose NCEE scores ranked lower may not be admitted. If the quota is greater than the number of students with NCEE scores exceeding the cuto, the school will process students who listed the school as their second preference and so on. In this case, the process will stop at the point where all the quotas are lled. Inevitably, due to lack of demand, some schools may end up admitting students whose NCEE scores are below the cuto score. Two features of this admission system are worth emphasizing. First, for any individual student, the cuto s are exogenously determined. The design of the system ensures that, before participating in the NCEE, a student has no knowledge about exactly at where the cuto s would be. It is therefore implausible for any student to exercise complete control over his/her test score around any cuto point. This feature satis es the primary requirement for a valid RD design. The second feature is about non-compliance. The discussion above indicates that the cuto s will be fuzzy due to the following reasons: (i) Some universities may admit students with scores lower than their cuto due to lack of interest in the university; (ii) Normally the cuto s are set at the point where there are 10 to 20 percent more students with scores exceeding the cuto s than the available slots. Hence, students with an above-cuto score are not guaranteed admission; (iii) Students submit their applications before they are informed of the cuto s. Thus, some with an above-cuto score may miss out on admission because they aim too high the ranking of their score is too low to gain an admission to the schools they applied for; (iv) There may exist corruption, which may grant admission to individuals with a below-cuto score. The existence of these non-compliance cases will have signi cant implications on our research design and we will discuss the issue in detail in the methodology section. 6

8 3 Methodology fuzzy RD design In this paper, we examine the causal e ect of completing a four-year university degree on earnings. Consider the following equation: lnw i = + ED i + X i + i ; (1) where lnw i refers to the logarithm of annual earnings for individual i; ED i is a dummy variable indicating whether the individual possesses a four-year university degree; X i is a vector of control variables; and i is the error term. The OLS estimation of Equation (1) may provide a biased estimate of because may include components, such as ability and drive, which are correlated with ED. In this paper, we try to mitigate this problem using a Regression Discontinuity (RD) design. The basic idea of the RD design is to utilize the fact that a treatment is given to a group of people for whom a measurable characteristic (forcing variable) is equal to, or greater than, an exogenously set threshold value. This generates a sharp discontinuity in the treatment. If individuals are unable to precisely manipulate the forcing variable it is reasonable to attribute the discontinuous jump in the outcome to the causal e ect of the treatment (Lee and Lemieux, 2010). Under the continuity assumption, 10 the average treatment e ect can be written as: = lim c#c E(lnW j C = c) lime(lnw j C = c); (2) c"c where C refers to the forcing variable (in our case, the normalized NCEE test score); c is a certain value of C; and c is the cuto score. Imbens and Lemieux (2008) propose estimating the treatment e ect by tting a local linear regression to the observations within a distance h on either side of the discontinuity point, or equivalently estimating: 11 lnw i = + ED i + k(c i c) + med i (C i c) + X i + u 0 i: (3) 10 For the RD design to be legitimate for estimating the treatment e ect, it is necessary to assume that all factors other than the treatment and outcome variables are continuous with respect to the forcing variable. 11 Equation (3) can be estimated with various weights over the employed observations. In this paper, all estimations of the local linear regressions are conducted with rectangular kernel weights. 7

9 In the case where the forcing variable does not relate to the treatment receipt in a deterministic way, we have a fuzzy RD design, and the OLS estimate of in Equation (3) would be biased and inconsistent. However, an IV approach can provide an consistent estimate with the instrument being the eligibility rule, providing some important assumptions are satis ed. 12 The IV estimate can be interpreted as the weighted average treatment e ect (ATE) for the entire population only if the treatment e ect is homogeneous across subpopulations of various compliance types. Otherwise it can be interpreted as a weighted local average treatment e ect (LATE) for the compliers. As discussed in the background section, the NCEE, by design, is fuzzy because there are situations where individuals with scores lower than the cuto are admitted and those with scores above the cuto are denied admission. In this paper, therefore, we use the IV strategy to estimate the causal e ect of completing a four-year university education on earnings for a group of individuals whose treatment participation is induced by their eligibility status. As indicated earlier, the range and the distribution of the NCEE scores vary signi cantly across years, and in some years, even vary across provinces. In addition, the cuto scores are also di erent for di erent provinces over di erent years. It is therefore important to normalize the NCEE scores so that our forcing variable can be a comparable variable across di erent years and di erent provinces. To do so, we divide (C i c) by c. 13 Under the de nition of this normalized score, point zero always refers to the cuto point. Thus, Equation (3) with complete notations should read: lnw i = + ED i + k(s i ) + m 1fS i > 0g (S i ) + X i + u i (4) where S i = (C ipt c pt )=c pt with C ipt referring to the score of individual i who attended NCEE in province p at year t, and c pt referring to the cuto score of province p at year t: 12 As pointed out by Hahn et al. (2001), it requires two assumptions monotonicity and excludability for the LATE to be interpreted as a causal e ect. The monotonicity assumption states that the forcing variable crossing the cuto point cannot cause some individuals to accept and others to reject the treatment at the same time. The excludability assumption demands that the forcing variable crossing the cuto point can only a ect the outcome variable through its impact on the treatment. 13 An alternative way to normalize the NCEE score is to divide (C i c) by the standard deviation of the scores within the corresponding province-year cell. However, in many cases our cells are rather small and the standard deviations are di cult to derive. 8

10 Theoretically, it is unnecessary to incorporate any control variables X i in Equation (4) if the discontinuity in the treatment variable at the cuto point is exogenously determined. In practice, however, covariates can be used to eliminate small sample biases (Imbens and Lemieux, 2008). In the case of our sample, we impose the full set of province dummies as the only control variables. This is mainly because there is a substantial regional/provincial wage variation in China, and the observations in some provinces are fairly small, especially when the bandwidth used for the RD estimation is narrow. This raises a concern that observations in each province are unlikely to be evenly distributed on both side of the cuto point, so the provincial di erences in wage might lead to some sample biases. Later in the result section, we show that incorporating the province dummies makes a substantial di erence in the estimated LATE only when the bandwidth h is small. 4 Data The main data used in this paper are from the Urban Residents Education and Employment Survey (UREES) conducted in 2005 by the National Bureau of Statistics (NBS) of China. The survey covers 10,000 urban households from 12 provinces. 14 It uses the same NBS Urban Household Income and Expenditure Survey (UHIES) sampling frame, which is based on Probability Proportional to Size (PPS) sampling with strati cations at the province, city, county, town, and neighborhood community levels. Households are randomly selected within each chosen neighborhood community (see Gibson, Huang, and Rozelle, 2003; and Meng, Gregory and Wang, 2005 for detailed discussion of the sampling). The survey is somewhat non-conventional in that it records information for household heads, their spouses, parents of both heads and spouses including non-coresiding parents, up to three children, up to two grandchildren, and one other household member. In addition to individual demographic characteristics, income and wages in 2004, the UREES focuses mainly on the education and employment status of co-residing household members, including heads, spouses, and children. In particular, the survey asks a set of retrospective 14 The Provinces where the survey was conducted are: Beijing, Shanxi, Liaoning, Heilongjiang, Zhejiang, Anhui, Hubei, Guangdong, Sichuan, Guizhou, Shaanxi, and Gansu. 9

11 questions regarding the respondent s participation in the National College Entrance Examination. The questions include whether the individual participated in the NCEE, and if so, the year and province of the participation, the total test score, whether he/she was admitted, the type of the education they completed (three-year college or four-year university), the name of the university/college, and the subject major. In addition to the information on tertiary education, the survey also asks about the quality of the senior high school the individual attended. With regard to employment, the information relevant to this study also includes individual earnings, income, the year the individual entered the labor market, the rst month s pay, and details of their employment situation. The purpose of this study is to use the RD design based on the NCEE scores to evaluate the return to completion of a four-year university degree. The control group comprises individuals who participated in NCEE but were not admitted to a four-year university. 15 Thus, only those who participated in the NCEE, reported NCEE scores, and were working with a positive earnings at the time of the survey are included in our sample. Of the 28,499 individuals in the total sample, 7,388 participated at some time in NCEE. Among these, 1,920 individuals are excluded because they obtained their nal degree through continuing education, rather than through direct admission to their nal degree based on their NCEE scores. 16 Another 1,623 individuals are left out because they either participated in NCEE before 1977 (the year the cuto scores became available) or after 2000 (the year after which individuals who participated in NCEE may not have graduated at the time of survey in 2004). In addition, individuals with missing values on the test score (754), on the province where they participated the NCEE (32), or with a nal education level above a four-year university degree (212) are excluded. Finally, we also restrict our sample to individuals who were working, with a positive earnings at the time 15 Those who were admitted but did not have a nal education level of four-year university are regarded as dropouts, but these are rare cases. We test the sensitivity of including or excluding them. Individuals whose nal degree is higher than four-year university are also excluded because their treatment is beyond the four-year university education. 16 Continuing education is a system that allows individuals to obtain four-year university education not through direct admission based on their NCEE scores. Among the 1,920 cases, 1,255 are individuals who were originally admitted to a three-year college program at a university and later on were allowed to sit a special examination given by the university to continue a fourth year of study to obtain a four-year university degree. As it is di cult to de ne the treatment e ect for this group, we decided to exclude them from the sample. The remaining 665 individuals are dropped because their admissions were based on cuto scores given by the continuing education system, which are di erent from the regular cuto s applied to general NCEE examinees. 10

12 of the survey, and exclude outliers on NCEE scores, earnings, and those with obvious reporting errors. 17 Our nal sample consists of 2,176 observations. 18 Table 1 reports the summary statistics for the treatment and the control groups. The main dependent variable used in this study is the logarithm of the 2004 annual earnings. 19 On average, the annual earnings for the treatment group is approximately 51 percent higher than that for the control group. Individuals in the control group are only slightly older than those in the treatment group. The ratio of females is, not surprisingly, lower in the treatment group than in the control group as the proportion of students completing university education is lower for females than for males. Father s years of schooling is higher for the treatment group than for the control group. Furthermore, more individuals in the treatment group are from richer families than are their counterparts in the control group. This is indicated by the proportion of individuals who reported that, at the time of their senior high school graduation, their family s consumption level was high compared to other households in the city they resided. In addition, almost half of the sample in the treatment group attended the best local senior high schools, compared to only 26% for the control group. Finally, as expected, the average NCEE test score is higher for the treatment group than for the control group. Another important data set we use in this paper is the cuto scores for four-year universities over the period 1977 to 2000 across di erent provinces for both the humanities-social sciences stream and the sciences stream. 20 We collected these data ourselves from various sources, including published books (for example, Meng, Yi, Xue, Qi, Xu, Liu, and Xia, 1988), local newspapers, and some o cial internet sites. Despite our widespread search e ort, there are 17 There are 28 individuals with NCEE scores lower than 10; 47 cases with annual earnings less than 2000 yuan or above 100,000 yuan; 8 individuals participated in NCEE in Ningxia or Guangxi province where the cuto score system is complicated and hence it is hard to decide whether they passed the cuto or not; 39 individuals with obvious reporting errors; and 549 individuals who were not working, did not work for the full year in 2004, or did not have positive earnings at the time of the survey. 18 To test whether our sample exclusion rules result in a biased sample we present comparisons of summary statistics between NCEE participants who are included in, and those who are excluded from our sample. These comparisons are reported in Appendix B. It shows that the two samples are quite similar in all the observable characteristics. 19 We also have a sub-sample of individuals with information on their hourly earnings and we test the sensitivity of our estimation to hourly earnings in Section We have some information on the cuto scores for the rst tier four-year universities and for the three-year colleges, but a large proportion are missing. We therefore decided not to examine the di erential returns to rst tier four-year universities or to three-year colleges. 11

13 still 8 percent of the year-province cells with missing cuto data. To handle the problem of the missing cuto s, we use existing data to impute missing values. The basic idea is to use variations within a province over time and within one year across di erent provinces to extrapolate the missing cuto scores. The details of our imputation method, as well as the data sources are presented in the Appendix A. 21 Figure 2 presents the NCEE four-year university cuto s for both the humanities-social sciences and the sciences streams by year. The hollow triangles show the original scores, while the solid dots are the imputed scores. The gure shows a signi cant increase in the value of the cuto scores between 1977 and 1988, and since then these have not been changed much. As discussed in the background section, the early increase in cuto scores was mainly due to the change in the NCEE settings (variations in the number of subjects examined and the change in the full scores for each subject). Another important point revealed from Figure 2 is that since the late 1980s there are a few outlier provinces, where the cuto scores are much higher than those for other provinces. These outliers are the provinces which adopted the standardized scores (see Background Section for detailed discussion). Finally, the gure also shows that including or excluding imputed missing cuto scores does not change the range and the trend of the cuto scores. 5 Fuzzy RD results LATE 5.1 Validity of the RD design Before presenting our fuzzy RD results (LATE), it is important to conduct the validity tests for the RD design. The most important assumption underlying the validity of the RD design is that each individual cannot exercise precise control over the forcing variable around the cuto point. Although this assumption cannot be directly tested (Lee and Lemieux, 2010), it is di cult to imagine that individuals have precise control over the test scores around the cuto point, based on how the Chinese National College Entrance Examination and the Chinese College Admission system operate(as described in the background section). This is mainly because the cuto s are determined by provincial governments after the NCEE is nished each year, not to 21 In the empirical work conducted below, we test the sensitivity of including these imputed cuto scores. 12

14 mention that it is extremely di cult for a student to locate his/her total score relative to the cuto point even if it were revealed before the exam took place. However, because our data on NCEE scores are collected retrospectively through individual self reporting rather than through administrative records, it is possible that individuals do not remember the scores precisely and mistakenly reported them based on their knowledge of the cuto points. 22 If this is the case, our estimation may su er from the violation of this underlying assumption. Fortunately, there are two implicit features of the RD underlying assumption that may be testable. First, if individuals do not have precise control over the forcing variable around the cuto point, the density of the forcing variable should not exhibit any discontinuity around the cuto. A jump in the density at the cuto is direct evidence of some degree of sorting around the threshold, and should cast doubt about the appropriateness of the RD design (Lee and Lemieux, 2010). Second, the mean of the observable characteristics in relation to the outcome variable should be continuous through the cuto. Any signi cant discontinuity in these characteristics implies that individuals with particular characteristics are to some extent able to manipulate their NCEE scores around the cuto point so as to be eligible for university admission. 23 Below, we test these two implications. We adopt a test proposed by McCrary (2008) to examine whether the density of the forcing variable exhibits any discontinuity around the cuto. Panels A, B, and C of Figure 3 present the graphs that show the discrepancy at each side of the cuto point in the density distribution of the normalized score, using di erent bandwidths 0.35, 0.25, and 0.15, respectively, with binwidth of An eyeball examination suggests no signi cant discontinuity at the cuto point in any of the graphs. The result is corroborated by the reported estimates (shown on the top of each graph) suggesting that the log distance in the density distribution of the normalized score on the two sides of the cuto point is not signi cant. Thus, the hypothesis that the density distribution is continuous around the cuto cannot be rejected. 24 In addition, in Panel D of Figure 3 we also report the number of observations in each bin. The gure shows that although 22 This issue is similar to the mis-reporting problem raised in Lemieux and Milligan (2008). 23 See Lindo et al. (2010) and Carpenter and Dobkin (2009) for applications of this type of test. 24 We also conducted the test using a smaller binwidths. The only case the test failed is when the bandwidth is 0.15 and the binwidth is In that particular case the result marginally rejects the hypothesis at the 90% signi cance level. 13

15 the distribution appears to be skewed to the right, there is no large discrepancy in the number of observations in the bins just above and below the cuto point. This further con rms that individuals are unlikely to precisely control the test score around the cuto point. To examine whether the means of the observable characteristics are continuous at the cuto, we rst graphically present the mean values of the outcome and the treatment variables, as well as the observable characteristics presented in Table 1, against the normalized scores. Figure 4 presents the outcome and the treatment variables with binwidths of 0.06 and The gure shows a clear discontinuity in outcome and the treatment variables at the cuto point. For the outcome variable, we observe a di erence of around 20 log points in annual earnings at the cuto point, while for the probability of obtaining a four-year university degree the discontinuity is around 20 percentage points. In addition, the graph for the treatment variable indicates that the discontinuity is fuzzy as the treatment reception is not unequivocally determined by the eligibility. 25 Figure 5 plots the means of eight observable characteristics. It shows that none of the variables has an obvious discontinuity at the cuto point. These, however, are only eyeball tests. More formally, we follow Lee and Lemieux (2010) to estimate the local linear regressions using these observable characteristics as dependent variables to test whether their means are continuous at the cuto point. We use bandwidths of 0.35, 0.25, and Two sets of results are summarized in Table 2, one regressing these variables on the dummy variable indicating eligibility for university, and the other on the dummy variable for university and using eligibility as the instrument. In either case, we nd no signi cant discontinuities in any of these characteristics. In particular we would like to stress that there is no signi cant discontinuity in employment status at the cuto point, implying that it is unlikely that the treatment induces selection into the labor market. 25 These graphs use a binwidth of We also test for smaller binwidths and the results are consistent. The curve on either side of the cuto point is the kernel-weighted local polynomial regression line. 14

16 5.2 Estimation results As discussed earlier, due to the fuzziness of our treatment, we employ the IV approach to estimate the Local Average Treatment E ect (LATE). The instrument used is eligibility a dummy variable indicating whether an individual passed the cuto score in the year and province where he/she participated in the NCEE. Our estimations compare individuals who completed a four-year university education with those in the control group, i.e. individuals who attended the NCEE but were not admitted to a university. The IV results from estimating Equation (4) are presented in Panel 1 of Table 3. To examine the robustness of the IV estimates, we present the results based on three di erent bandwidths 0.35, 0.25, and 0.15 with the numbers of observations being 1,761, 1,520, and 1,121, respectively. The X vector included in the rst three columns of Table 3 is a group of dummy variables indicating individuals residential locality (province). Column (1) shows that when using a bandwidth of 0.35, the estimated LATE is 0.36 and signi cant at the 5% level. This estimate increases to 0.59 as the bandwidth reduces to 0.25 (column 2 of Table 3), and it changes to 0.53 with a bandwidth of 0.15 (column 3) and the estimate turns insigni cant. Panels 2 and 3 of Table 3 present the corresponding results from the rst stage and the reduced-form estimations. The results from the rst stage estimation show that the instrument is very strong for the estimation with bandwidths of 0.35 and 0.25, but it does not pass the rule of thumb test for a strong instrument when the bandwidth is narrowed to 0.15 (see the F-tests presented at the bottom row of Panel 2). This is not surprising given that with a narrower range of normalized test score around the cuto, the treatment compliance becomes fuzzier, and the outcome di erence becomes smaller. As a result, eligibility becomes a weaker instrument. This might be the reason why the IV estimate becomes insigni cant with 0.15 bandwidth. Fortunately, the magnitude of the coe cient does not change much when we reduce the bandwidth from 0.25 to The reduced-form results for the 0.35 and 0.25 bandwidths show correct signs and are statistically signi cant. But for the narrowest sample, the size of the coe cient reduced signi cantly and becomes statistically insigni cant. In columns (4) to (6) of Table 3 we add age and female dummy as two additional control 15

17 variables in Equation (4) to examine the robustness of these LATE estimates to di erent covariates. The results are very similar to those observed in columns (1) to (3). We then present the results with no control variable in columns (7) to (9). While the LATE estimate based on the 0.35 bandwidth remains signi cant and is fairly close to the corresponding estimate in column (1), the LATE estimates in columns (8) and (9) are substantially larger than the corresponding estimates in columns (2) and (3), respectively. As argued earlier in the methodology section, the marked di erences might re ect sample biases caused by the substantial wage variations across provinces in China, and the small numbers of observations in some provinces are unlikely to be evenly distributed on the two sides of the cuto point. If this is the case, the biases should be more severe when the bandwidth is small, which is what we found here. 5.3 Robustness checks We conduct ve robustness checks. First, we explore an alternative speci cation of the RD design featuring the following polynomial function: lnw i = + ED i + k(s i ) + X i + i ; (5) where S i is the normalized NCEE score for individual i, and k(s i ) is a exible function of S, which is a vector of high order polynomial terms. We estimate Equation (5) using the TSLS method with the same instrumental variable. Due to the small sample size, we use the full sample in the estimation, but we test whether using di erent functional forms of k(s) makes a di erence. We use from the second order to the fourth order of the polynomial terms of the forcing variable, although only the linear and quadratic terms are statistically signi cant. The rst row of Table 4 presents the results and they show that the estimated LATEs are all signi cantly positive, ranging from 0.50 to This is a range that approximately matches the magnitudes of the LATEs estimated from the local linear regression: 0.36 to 0.59, as displayed in columns (1) to (3) of Table 3. We also tried to make the polynomial function di erentially exible on each side of the cuto point by interacting eligibility with the polynomial terms and the results are reported 16

18 in the second row of Table 4. The result is quite consistent with those observed in row (1) when only the quadratic term and its interaction term with eligibility dummy are added. The estimated coe cient is 0.48 and is statistically signi cant. However, when the cubic and its interaction terms are included the results become statistically insigni cant but the magnitude remains in the range observed from the local linear regression estimations [0.36, 0.59]. It is important to note that when we add the interaction term in all the speci cations in row (2), none of the interaction term is statistically signi cant. Perhaps the polynomial function does not vary much on the two sides of the cuto point and that our sample is not large enough to provide precise estimates at the level of exibility allowed in these functional forms. Second, we test the robustness of our results with regard to whether using annually or hourly earnings as the dependent variable makes a di erence. In the survey the household heads and spouses are asked the questions on the number of days per week and number of hours per day they worked in The third row of Table 4 presents the results using log hourly earnings as the dependent variable for this subsample. To make the comparison sensible, the fourth row presents the results for the same subsample but uses the log annual earnings as the dependent variable. We nd that, with this restricted sample, the estimated returns are very similar no matter which dependent variable is used. Furthermore, they are also very similar to the results found in the full sample estimations as shown in columns (1) to (3) of Table 3. The third robustness test we conduct is related to the concern that our test score variable is based on individuals retrospective reporting rather than administrative data, and thus may su er from the measurement error problem. To this end we believe that if we assume the measurement error is random and given that we estimate an IV regression, the measurement error in the eligibility variable may a ect both the numerator and denominator in the Wald estimator proportionally and hence these may cancel out each other. 26 Therefore, the measurement error problem should not seriously a ect our results. Nevertheless, we test this empirically by adding a set of six cohort dummies classi ed by the years the individuals participated in the NCEE. The results are reported in the fth row of Table 4. These estimates are very similar to the 26 Intuitively, the attenuation bias caused by the measurement error reduces the RD gaps in the treatment and outcome proportionally, leaving the LATE estimation unchanged. For detailed discussion, see Appendix C. 17

19 original treatment e ects observed in columns (1) to (3) of Table 3, suggesting that the possible impact of elapsed memory is unlikely to be signi cant. Next, we examine whether including individuals who participated in the NCEE in the year and province where we used imputed cuto scores a ects our estimated results. To do so, we rst add a dummy variable indicating whether the cuto score used for an individual is imputed in our estimation. Second, we exclude individuals with imputed cuto scores from our estimation. Both results are similar to our original estimates (see rows (6) and (7) of Table 4). Finally, we check the robustness of our results against the de nition of the treatment. Throughout the paper we de ne the treatment group as those who completed a four-year university degree. Thus, everybody who is in the treatment group has a four-year university degree. However, as the data do not provide information on whether an individual is a university dropout or not, it is possible that some individuals who were accepted by a four-year university but did not complete would be in our control group. We identify these individuals and test if including them alters our results. In our nal sample of 2,176 individuals, the only possible dropouts are the 38 people reported going to a four-year university but with the nal education level of a three-year college The nal row (row (8)) of Table 4 presents the results from the regression which includes a dummy variable indicating these potential dropouts. The results are almost identical to those presented in columns (1) to (3) of Table 3. 6 Examining the direction of the di erence between the LATE and ATE The above analysis using the RD design provide a range of the Local Average Treatment E ect (LATE) of four-year university education relative to a control group of individuals who participated in the NCEE but were not admitted to the universities. The estimated returns range from 27 It is not clear to us whether these people are dropouts as it is possible that the inconsistency presented here comes from mis-reporting. 28 We could not nd the national dropout rate gure for China. However, the available information for Peking University suggests that the graduation rate for that university in 2004 is 94.2% (see rst.cn/cl_in.html?id=1972). Given that Peking University is one of the two best universities in the country and hence has the strictest rules for graduation, we assume that the national dropout rate should be lower. 18

20 0.36 to As indicated in Oreopoulos (2006), the Average Treatment E ect (ATE) for the population may be a more important policy parameter than the LATE, as it is a theoretically more stable parameter than the LATE when considering potential gains for anyone receiving university education. In this section, therefore, we try to assess whether the ATE and the LATE are di erent and, if so, the direction of the di erence. Previous studies provide no estimate on returns to university degree on a comparable sample of treatment and control groups for China. Zhang et. al. (2005) estimated the OLS regression using individuals with three-year college and above education as the treatment group and senior high school graduates as the control group. Their estimated average return for 2001 is around 37 percent. Applying OLS estimation to our own sample of the same treatment and control groups, as used in our RD design presented above, to estimate log annual earnings regression controlling for age, gender and province of residents, we obtain estimated coe cients of 0.48, 0.41, and 0.38 for the total sample and the sample with bandwidths of 0.35 and 0.25, respectively. It seems that, for the narrow bandwidth of 0.25, the RD-IV estimate is much larger than the OLS estimate. We also employ the National Bureau of Statistics (NBS) annual survey data the Urban Household Income and Expenditure Survey 2004 to estimate the OLS regression using the same model speci cation. The estimated coe cient on a four-year university education relative to a senior high school or a 3-year college degree is 0.45, which is quite close to our own total sample OLS estimate of These OLS estimates, however, may not represent the average treatment e ect (ATE) due to the possible omitted variable bias. At the same time, although our RD estimates are clean of the omitted variable bias, they may be very di erent from the ATE if there exists an heterogeneous treatment e ect. We therefore assess whether the ATE and the LATE are di erent, and the direction of the di erence, by rst examining if there are heterogeneous treatment e ects. To understand this issue, we follow Imbens and Wooldridge (2009) to calculate the proportions for the following groups: (i) those who received treatment but were not eligible (observed always-takers) out of total ineligibles ( oa ); (ii) those who did not receive treatment but were eligible (observed never-takers) out of total eligibles ( on ); (iii) those who received treatment and were eligible (including compliers and unobserved always-takers) out of total eligibles ( ce + noa ); and (iv) 19