Is there a Causal Effect of High School Math on Labor Market Outcomes?


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1 Is there a Causal Effect of High School Math on Labor Market Outcomes? Juanna Schrøter Joensen Department of Economics, University of Aarhus Helena Skyt Nielsen Department of Economics, University of Aarhus May 5, Work in progress  Abstract In this paper we exploit a high school reform to identify the causal effect of advanced high school Math on labor market outcomes. Students who take advanced Math courses have more favourable labor market outcomes. We find some evidence of a causal relationship for the students who are induced to choose Math after being exposed to the reform which meant that they were able to combine advanced Math with a more flexible choice of additional advanced courses. The effect partly stems from the fact that those students end up with higher education and subjects of education leading to wellpaid jobs. JEL Classification: I20, J24. Keywords: Math, High School Reform, Instrumental Variable, Local Average Treatment Effect. Financial support from the Danish Research Agency is gratefully acknowledged. We thank participants at CEPR/IFAU/Uppsala University, DGPE conferences, and seminar participants at University of Aarhus for comments on earlier drafts. The usual disclaimer applies. 1
2 1 Introduction It is well established that high school graduates with advanced math qualifications perform better on a range of important economic performance measures. They have higher test scores, they attain a higher education and they earn a higher income than others, see e.g. Levine and Zimmerman (1995) or Zangenberg and Zeuthen (1997). The question is, however, to which extent these observations indicate a causal impact of Math on performance and to which extent it is a selection effect indicating that people with other favorable characteristics choose to acquire higher math qualifications. As a basis for policy discussion about changing high school curricula, we need hard evidence about the existence of a causal impact of Math courses on labor market success. Policy recommendations are distinctly different depending on the conclusion. If we fail to reject a causal impact of Math on outcomes, it implies that policy makers should consider enhanced weight on Math in high school curricula for all students. On the other hand, a rejection of the causal effect also rejects the need for an enhanced Math curriculum. The question of a causal effect of Math is closely related to the discussion of the human capital theory of education versus the signalling theory of education. The human capital theory implies that skills are acquired and added to the stock of human capital while attending school. The signalling theory of education says that acquired education is merely a signal which is cheaper to acquire for individuals with favorable skills. Drawing on human capital theory, the Math premium would indicate that more valuable skills are acquired during Math courses than during other coursework. Drawing on the signalling theory, advanced Math courses may work as a signal which is cheaper to acquire for individuals with favorable skills. The signalling theory is consistent with the selection story. Here a seemingly high return to Math courses is the result of selfselection of high ability individuals into advanced Math courses rather than causality, see e.g. Rose and Betts (2004). The policy implication would be that enhanced weight on Math would disturb the Math signal and lead to a potentially inefficient allocation of students and workers after high school. On the other hand, a failure to reject a causal impact of Math would serve to support the human capital explanation given above, whereas a rejection of the causal impact would confirm the selection story which is consistent with the signalling theory. A part of the literature studies the labor market returns to cognitive skills. In case there is an effect of Math courses on cognitive skills, this literature is informative for our study. Some of the skills learned in the high level Math course like clarity in expressions, logical reasoning and inference, as well as imagination and ingenuity would add to the general stock of cognitive skills which would prove powerful in any career in any field. Alexander and Pallas (1984) find that students who attended more Math courses in high school gained more knowledge and cognitive skills. Furthermore, consistent positive effects are found of general cognitive skills on employment, wages and earnings cf. Willis and Rosen (1979), Blackburn and Neumark (1993) and Cameron and Heckman 2
3 (1993). High school graduates with high cognitive skills tend to get more secure jobs, even if they do not obtain higher earnings in the early career, the earnings gain seem to show up later in the career. Murnane, Willet & Levy (1995) provide similar results for the effects of math skills on wages in addition to evidence of increasing importance of math skills in the labor market. A number of papers have studied the effect of high school curriculum on postsecondary schooling and earnings. Altonji (1995) pioneered this area of research. In a study based on the National Longitudinal Study of Youth (NLSY), he uses the variation in curricula across US high schools to identify the effects of coursework on wages and educational outcomes. He finds a negligible effect of specific course work including Math. Levine and Zimmerman (1995) find slightly stronger effects, while comparing the results of analyses based on NLSY with analysis of the 1980 cohort from High School and Beyond (HSB). However, still any potential effects are restricted to certain subgroups (low educated men and highly educated women), which raises doubt about the existence of a causal impact of coursework on labour market outcomes. 1 Rose and Betts (2004) use data for the 1982cohort from HSB. This is an immense improvement upon the earlier studies for several reasons. First of all, the transcript data for the sampled individuals are more detailed than those used in the earlier papers, secondly, the individuals are observed in their thirties rather than in their twenties, and finally, the individuals are observed after the huge increase in the college premium around Rose and Betts (2004) find that Math matters. Consistent with this result, Ackerman (2000) finds a 6% earnings increase per additional year of Math classes. Twothirds of this effect is a direct effect, whereas onethird is an indirect effect running through the increased probability of college attendance. Perna and Titus (2004) investigate the impact of state public policies on the type of institutions that high school graduates attend after graduation. Due to state public policies only 27% of high school students in Alabama take upperlevel Math courses compared to 61% in Nebraska. After investigating the relationship between Math coursework and college attendance, they find that academic preparation, as measured by Mathematics coursework, is the strongest studentlevel predictor of college enrollment. Also crosscountry studies investigate the impact of Math on economic performance. Hanushek and Kimko (2000) were the first study to be concerned with the potential influence of the quality of education as measured by comparative Mathematics and Science test scores on growth. They found a consistent, strong and stable effect of quality measures on growth. In fact they found that one standard deviation increase leads to an increase in annual growth rates of 1 The fact that the returns cannot be attributed to specific courses is not necesarily a rejection of the human capital theory. Other skills than those related to specific coursework could be acquired. Bowles, Gintis and Osborne (2001) hypothesize that the most important human capital outcomes of high school are unrelated to the course work as such but rather to behavioural outcomes. They claim that high school teach students noncognitive skills as for instance punctuality, tactfulness and consistency, which are as important in the labour market as academic skills. 3
4 more than one percentage points, which only makes sense if there are strong externalities related to accumulating high quality human capital. Barro and Lee (2001) extend the same data set and investigate the opposite relationship, namely the effect of growth on test scores. They find a positive effect of growth on reading test scores but no effectonmathtestscores. Our study contributes to the litterature by getting closer to the causal impact of Math on earnings by use of better instruments for acquired Math qualifications. Previous studies have used the curriculum of the average student from the relevant high school as an instrument for acquired Math qualifications. However, as they point out, this experiment is not a clean natural experiment, because the curriculum of the average individual at a given high school may be correlated with the average family background, primary school preparation, ability of the student body, and the quality of the courses. However, the authors try to come across this problem by signing the potential bias. Altonji (1995) concludes that the potential bias is positive, which means that the small effect of specific course work that he finds may be interpreted as an upper bound. Furthermore, they try to control for the high school specific effects by observed variables and by inclusion of high school fixed effects. We suggest to use instruments based on a Danish high school reform implemented for the cohort starting high school in We use a brand new registerbased data set which covers four cohorts of high school students starting high school in the years We have information about complete detailed educational event histories and about the individual labor market histories, including actual labor market experience, unemployment degree and income. The data set is augmented with information about parents, courses taken in high school and high school GPA. Furthermore, information about the distance from the individual s place of residence to nearby high schools has been added. The dataset has been gathered for the particular purpose of this study. We use three different instruments that apply information about the high school reform of 1988 and the pilot versions of the reform implemented in terms of experimental curricula, both of which allowed for a more flexible combination of high school courses. After the reform, the number of high school students taking high level Math doubled, this change was one of the most dominant changes in course choices resulting from the reform. The reform was implemented for the cohort starting high school in 1988, and experimental curricula were implemented for the cohorts starting high school before the reform at some high schools. Our instrumental variable estimates may be interpreted as local average treatment effects, cf. Imbens and Angrist (1994). Common for the instrumental variables estimations are that the estimated coefficients identify the effect of Math on earnings for the group of students who were induced to choose high level Math because they were able to combine advanced Math courses with amoreflexible choice of additional advanced courses (e.g. Chemistry, Political Science, English, Biology or Music) which was not possible without the reform. Before the reform, advanced Math was only supplied in a package with advanced Physics, which scared away a lot of potentially Math interested students. Some 4
5 high schools got an exemption and were allowed to try out an experimental curriculum where advanced Math could be combined with Chemistry. The idea that restrictive coursepackages consisting of advanced Math and another advanced Science course scared away potentially Math interested students is consistent with the conclusion by Albæk (2003). He analyses the effect of restricted course packages on the choice of high school courses in a framework where the student maximizes the future entry probability at universities which is assumed to depend on GPA. This approach is consistent with the Danish postsecondary schooling system that screens students on GPA and high school course choices. Our empirical investigation of the effect of high school Math on labour market outcomes confirms the findings of Rose and Betts (2004) that Math matters. In accordance with previous studies, we find that students who choose advanced Math courses earn 2030% more than others. This earnings premium reflects a causal impact of Math on earnings for some subgroups of the population. Employing the actual high school reform as an instrument (#1), we identify the effect for those who are induced to choose Math because they are able to combine advanced Math with an advanced nonscience course. For this subgroup there is no (or even a negative) effect of Math on earnings. Employing our preferred instruments (#2 and #3) based on the pilot branches implemented prior to the reform of 1988, we find that the causal effect of high level Math is significant. This means that individuals who are induced to choose advanced Math because they at low travel costs may reach a high school with an experimental curriculum allowing them to combine advanced Math with advanced Chemistry instead of Physics, earn more than they would have earned had they not chosen advanced Math. The remainder of the paper is structured as follows: Section 2 presents the econometric methodology used to identify and estimate the causal impact of Math on labour market outcomes. Section 3 describes the data. Section 4 shows the results, whereas section 5 concludes the paper. 2 Identification and Estimation of the Causal Effect of Math We identify the causal effect of high level Math by exploiting the exogenous (cost) variation that is obtained from the high school reform in 1988 and from pilot versions of the reform introduced before In this section we briefly describe the applied selection model, the identification strategy and the IV estimation method. Consider the labor market outcome variable y i, which in our main analysis denotes log earnings of individual i 13 years after starting high school. The 2 See Appendix A for a detailed discussion of the Danish high school structure, contents, and the changes that the reform implied. 5
6 earnings equation is assumed linear and additive in the explanatory variables: y i,13 = β 0 + β 1 X i,13 + β 2 W i,0 + δmatha i + ε i, (1) where X i,13 is a vector of individual characteristics of individual i as measured 13 years after starting high school, including highest completed education and actual labor market experience; W i,0 is a vector of individual and family background variables of individual i, including the mother s as well as the father s highest completed education and income as measured in the year before the individual entered high school. It also includes time invariant individual background characteristics such as gender and high school GPA. The underlying latent variable for the selection equation is also assumed linear and additive in the explanatory variables. Individual i chooses the Math level, j = A, B, that provides the greatest value, V ij, where A denotes the advanced level. That is, individual i chooses high level Math if and only if V ia > V ib. By normalizing the value of choosing lthe less advanced Math courses, V ib, to zero, we obtain the selection rule: MathA i = 1 [α 0 + α 1 X i,13 + α 2 W i,0 + θz i + u i > 0], (2) where Z i is an instrumental variable which is either a reform indicator or a continuous distance measure. We assume that: µ µµ εi 0 σ2 ρ N 2,. u i 0 ρ 1 Since we assume that the error terms are independently and identically distributed across individuals, the Stable Unit Treatment Value Assumption (SUTVA) is implicitly satisfied. This assumption implies that potential earnings of each individual i is unrelated to the Math level of other individuals. Hence, we assume away general equilibrium effects due to for instance a general rise in the Math level in the labor force. In our basic case, we apply the two step IV estimator, where the probit estimates of the Math selection equation are obtained in the first stage: P (MathA i =1 X i,13,w i,0,z i )=Φ (α 0 + α 1 X i,13 + α 2 W i,0 + θz i ) From these estimates, the hazard for each individual is computed as: 3 γ 1i = φ( α 0+ α 1 X i,13 + α 2 W i,0 + θz i) Φ( α γ i = 0 + α 1 X i,13 + α 2 W i,0 +, if MathA θz i) i =1 γ 0i = φ( α0+ α1xi,13+ α2wi,0+ θz i) 1 Φ( α 0 + α 1 X i,13 + α 2 W i,0 +, if MathA θz i) i =0 (3) Then consistent estimates of the structural parameters β 0, β 1, β 2,andδ are obtained in the second stage by augmenting the outcome equation (1) with the hazard and estimating it by Ordinary Least Squares (OLS). 3 This is equivalent to the generalized residuals of the probit model describing the high level Math choice, cf. Gourieroux, Monfort, Renault and Trognon (1987). 6
7 The question immediately arises, which variables should be included among the explanatory variables? Post secondary schooling is most likely affected by Math qualifications from high school. As discussed by Altonji (1995), Math courses may induce individuals to take longer educations. If Math courses make longer education more profitable, for instance because they are cheaper to acquire because lower efforts are needed, this effect should be attributed to the Math course. However, on the other hand, the opportunity cost of higher education should be deducted, which means a number comparable to the real interest should be deducted (Altonji uses 4% as a maximum on the real interest). By choosing the high level Math course in high school, individuals may enhance the probability of finalizing educations leading to high paying occupations, which would increase δ. First of all, it might be easier to complete an education within the fields of Engineering, Natural Science or Economics, and secondly, the high school graduates simply extend their choice set when it comes to higher education. 4 We are able to distinguish between the direct effect and the total effect (direct plus indirect effect) of Math. 5 The direct effect of Math on earnings stem from Math affecting for instance logical reasoning and increasing cognitive skills which are useful in most occupations. The indirect effect would go through an enhanced probability of finalizing favorable university educations, and this effect would disappear when we include length and subjects of education in X i,13. In order to give high level Math full credit for all these effects, the length and type of higher education should be left out of the regression. In order to obtain an overview of the relationships, we estimate δ with and without controls for length and type of higher education. 6 A similar issue arises with respect to testscores (GPA), therefore we also estimate δ with and without GPA, because they may induce a positive or negative bias depending on the relationship between GPA, Math courses and unobserved ability, cf. Levine and Zimmerman (1995). The GPA is measured during high school, with highest weight on the grades obtained in the last year of high school. Hence the GPA may be affected by high school courses attended. However, Alexander and Pallas (1984) show that test scores at high school graduation, i.e. in 12th grade, are dominated by the effect of aptitude and prior achievements up until 9th grade, rather than learning, experience and achievements during high school. Thus we are confident that high school GPA is a reliable measure of aptitude and initial ability at high school entry, and not to any severe extent directly affected by high school courses. Parental background variables are measured in the year of high school entry, thus we do not have the same concern that they are influenced by student 4 By passing high level Math courses they can be admitted to university educations within Natural Science and Engineering without supplementary coursework. Up until 1990, even students with medium level math would be admitted without supplementary coursework, although they might have had a harder time following the courses. 5 This was done by Ackerman (2000), who found that one third of the total effect of Math on earnings is an indirect effect running through further education. 6 This approach was also used by Levine and Zimmerman (1995) and Rose and Betts (2004). 7
8 achievements and course choices during high school. We employ a broad view of human capital investment and allow family background to influence both labor market ability and Math ability. Thus we include family background variables both among the observed abilities in the outcome equation, i.e. the factors that shift the marginal rate of return to investment in high level Math, and among the observed abilities affecting the selection into high level Math, i.e. the factors that shift the marginal supply of funds scedule. The explanatory variable of primary interest is the indicator for whether individual i passed the high level Math course in high school, MathA i. This variable is potentially endogenous, since there most likely exist unobserved variables affecting both future earnings and the choice of high level Math. Hence, endogeneity bias could arise due to individuals selfselecting into Math courses based on expected future earnings gains (selection on outcomes) or due to unobserved ability bias (selection on unobservables). Firstly, the choice of high level Math may be endogenous in the earnings equation if individuals who aspire to go into a highpaying occupation, e.g. as Engineers, choose the high level Math course in order to enhance the possibilities to succeed as an Engineer, cf. Levine and Zimmerman (1995). Secondly, unobserved ability bias arises if Math level depends on unobserved ability, e.g. if only the most talented individuals choose to attend the high level Math course and we fail to control for talent, then the estimate of δ will be upward biased. To the extent that GPA approximates the unobserved talent, including GPA as an explanatory variable corrects for the unobserved ability bias. However, this correction relies on the assumption that attending the high level Math course has no effect on GPA. 7 The instrumental variables approach that we use deals with both sources of endogeneity. 2.1 Instrumental Variables Estimation of LATE We use three alternative instrumental variables, Z j i,j =1, 2, 3, which are based on the policy reform of 1988 and the experimental curricula introduced prior to the reform. The reform and the experimental curricula work as exogenous cost shocks inducing more students to choose high level Math. Hence, the reform may be seen as a natural experiment from which we obtain a source of individual variation in students Math level, which does not influence the outcomes of interest. Before the reform, advanced Math was only supplied in a package with advanced Physics, which scared away a lot of potentially Math interested students. Some high schools got an exemption and were allowed to try out an experimental curriculum where advanced Math could be combined with Chemistry. We create three different instruments each of which exploits the exogenous variation in the exposure of students to the possibility of combining advanced Math courses with a various set of other advanced courses. Two of our instruments are binary reform indicators, whereas the third instrument is a continuous distance measure. In the following, we treat the instrumental variable, Z i, as a binary indicator knowing that the generalization to 7 See for example Hansen, Heckman and Mullen (2004) for a more comprehensive discussion of ability bias, and the effect of schooling on test scores. 8
9 the continuous case is straightforward. 8 For simplicity, we suppress superscript j, and let conditioning on the two matrices of explanatory variables, X i,13 and W i,0 be implicit. Associated with the two possible outcomes of Z i are counterfactual treatments, MathA i1 and MathA i0. These are the treatment statuses we would observe if i were exposed to the reform (Z i =1)ornot(Z i =0), respectively. Although, for each individual we only observe one of these. Likewise, there are counterfactual outcomes, y i1 and y i0, associated with the two possible treatment statuses. Where y i1 denotes the earnings of individual i if participating in the high level Math course (MathA i =1), and y i0 denotes the earnings if not (MathA i =0). Thus, the causal effect of the high level Math course on earnings is given by y i1 y i0. Since we observe only one of these (y i1 or y i0 )foreach individual we need to impose assumptions in order to identify the treatment effect of high level Math. To sum up, for each individual in a random sample of high school students, we observe (Z i,matha i,y i ), where the observed treatment indicator depends on whether the individual i was exposed to the reform or not, MathA i {MathA i1,matha i0 }, and the observed earnings depends on whether individual i has high level Math or not, y i {y i1,y i0 }. Instrumental variables estimation identifies the local average treatment effect (LATE), which is the causal effect of high level Math on earnings for those who are induced to choose high level Math because they were exposed to the reform. LATE has the advantage of being estimable using instrumental variables under weak conditions. But it also has two potential drawbacks: First of all, it measures the effect of treatment on a generally unidentifiable subpopulation, and secondly, the definition of LATE depends on the particular instrumental variable that we have available. In order to identify the LATE by use of a binary instrument, two conditions have to be satisfied: (a) Existence of instrument, and (b) Monotonicity. 9 In our framework, condition (a) is satisfied if the coefficient to Z i is strongly significant in the selection equation, and independent of ε i and u i. That is, the reform should influence earnings only through its effect on the probability of obtaining high level Math. This condition ensures that Z i is a proper instrument, and is very reasonable in our application, however, it is inherently untestable. Condition (b) guarantees identification of LATE by only allowing the instrument to affect the choice of Math level in a monotone way. More formaly, we assume that MathA i1 MathA i0. That is, since individuals on average are more likely to choose high level Math if they are exposed to the reform, E [MathA i Z i =1]>E[MathA i Z i =0], then anyone who would choose high level Math given that they were exposed to the reform would also have chosen high level Math had they not been exposed to it. In other words, when Z i switches from 0 to 1, we only have that students switch from not choosing the high level Math course to choosing the high level Math course (not the other way around). This seems reasonable because all prereform options 8 See e.g. Heckman, Lalonde and Smith (1999). 9 See e.g. Imbens & Angrist (1994) for technical details. 9
10 are also available postreform options. Hence, the IV estimator of the causal effect of high level Math on earnings corresponds to LAT E = E [y i1 y i0 MathA i0 =0,MathA i1 =1]= While the estimated parameter has predictive power for the subpopulation complying with the instrument, there is no reason to believe that the LATE corresponds to the average treatment effect in the population, AT E E [y i1 y i0 ], i.e. the expected effect of participating in the high level Math course for a randomly drawn high school student, or to the impact of treatment on the treated, TT E [y i1 y i0 MathA i1 =1], i.e. the expected effect of participation in the high level Math course for those who actually participated. 10 The instrumental variables method is very effective in estimating average treatment effects if a good instrument for treatment is available, and the reform based instruments are indeed strong: they predict the treatment of high level Math, they are also likely to be unrelated to unobserved heterogeneity (e.g. Math ability), and therefore redundant in the structural model E [y X, W, Z], i.e. any earnings differences between the individuals who are exposed to the reform and those who are not are assumed to be captured by the observed variables X and W. The validity of the instruments and the identifying assumptions are discussed in the next subsection, and Appendix B contain additional evidence about the validity of the instruments. According to Heckman (1997), economically meaningful IV estimates can be found using instruments measuring policy interventions that induce some people to switch participation status while leaving nonswitchers unaffected. A zero social cost would then allow us to interpret the LATE as the effect of the marginal policy change on per capita earnings. 2.2 Three reformbased instrumental variables Below we describe the three instruments in turn and discuss whether the assumptions are expected to be satisfied. The first instrumental variable, Zi 1,is equal to one if the individual started high school in 1988, and zero otherwise. 11 Students who start high school in 1988 are exposed to the reform which means that they are free to combine advanced Math with any other advanced course, e.g. Physics, Chemistry, English, Biology, Music, Political Science, etc. Students who start in high school in 1987 are allowed to take advanced Math if they also take the advanced Physics course, unless they are at a high school which has implemented an experimental curriculum in which case they can replace the advanced Physics course with advanced Chemistry. Hence, the IV approach using Zi 1 identifies the LATE for individuals who are induced to choose advanced 10 In a homogenous effects model or in a model with heterogenous effect which are not reacted upon by the individuals, the three parameters are identical. In the case of a continuous instrument, the LATE may recover the TT parameter if it is estimated for all possible values of the instrument, see Heckman, Lalonde and Smith (1999). 11 Using the terminology of Rosenzweig and Wolpin (2000), this instrument belongs to the group of instruments applying exogenous variation in birth dates. E[y i Z i =1] E[y i Z i =0] E[MathA i Z i =1] E[MathA i Z i =0]. 10
11 Math because they are able to combine with nonscience subjects. 12 Amajor concern is the validity condition. This instrument relies completely on cohort differences, and the condition fails if the earnings of individuals from the two cohorts differ for reasons other than the reform. Therefore, we carefully check whether results are robust across different cohorts and for different years of income, y i,13. The condition also fails if the reform influences earnings through other channels than through its effect on the probability of choosing advanced Math. This means that we need to rule out that choices of other courses which influence earnings also changed significantly. The validity condition also fails if high level Math is of lower quality after the reform. The monotonicity condition, requires that individuals who choose advanced Math in 1987 when they have to combine Math with an advanced Science course, would also choose high level Math had they been able to combine it with another advanced course. We are confident that the monotonicity assumption is reasonable in our application, although it cannot be tested. 13 The second and third instrumental variables are suitable for analyses focusing exclusively on cohort 1987 who were faced with experimental curricula before the implementation of the real reform. High schools with experimental curriculum allowed the students to combine advanced Math with either Physics or Chemistry, whereas high school with no experimental curriculum allowed only combination of advanced Math with advanced Physics. The two instrumental variables represent two polar cases with respect to the assumption about the selection into high schools: random distribution or selfselection based on potential preference for the experimental curriculum. In practice, students applied foracertainhighschool,andthenacentralcountybodydistributedstudents across institutions. Although the branch of studies was only chosen after the first year, the central body did in fact take account of students preferences for certain experimental branches of studies if such preferences existed before enrollment in high school. The second instrumental variable, Zi 2, is equal to one if the individual attended a high school which implemented an experimental curriculum allowing advanced Math to be combined with advanced Physics or Chemistry, and zero otherwise. This instrument builds on the assumption that individuals are randomly distributed across high schools with and without experimental curriculum. In order for this assumption to be violated, students should decide upon their branch of studies before enrolling, which may or may not be true. Hence, it rules out that forward looking high school applicants have the possibility to choose a high school which supplies the experimental curriculum in question. The third instrumental variable, Zi 3, equals the difference between the distance to the nearest high school with an experimental curriculum and the nearest high school. The instrument proxies the marginal costs of obtaining the option of the experimental curriculum. The assumption for this instrument to be valid 12 Before the reform, student could combine advanced Math with Physics or  if the travel toahighschooloffering the experimental curriculum  Chemistry. 13 We observe only 1.6% of students in the 1988 cohort who chose high level Physics without choosing high level Math. 11
12 is that the additional distance to those high schools is only related to earnings through its effect on the probability of choosing advanced Math. For instance, this assumption rules out that parents initially chose their location based on the distribution of high schools with experimental curriculum, and it also rules out that high schools implementing the experimental curriculum are systematically situated in areas of adolescents with high Math ability. Both of these assumptions seem reasonable. 3 Data For our empirical analysis we use a brand new rich panel data set comprising a complete sample of cohorts starting in high school in the years in Denmark. The data are administered by Statistics Denmark, who have gathered the data from different sources, mainly from administrative registers specifically for this particular purpose. For the main part of our analysis, we select a sample consisting of the cohorts of 1987 and 1988 in order to avoid having to deal with trends in educational choices over time. The cohort of 1987 is the last cohort entering high school before the high school reform and the cohort of 1988 is the first cohort entering high school after the reform. We use the cohorts of 1986 and 1989 for robustness checks only. For each individual we have data on complete detailed educational histories. These comprehend detailed codes for the type of education attended (level, subject, and educational institution), and the date for entering and exiting the education, along with an indication of whether the individual completed the education successfully, dropped out or is still enrolled as a student. Furthermore, we have information on the timeuse in high school in terms of optional courses chosen and high school GPA. 14 The GPA is a weighted average of the grades at the final exams of each course. Both the quality of the courses and the GPA are comparable across high schools, since the control of the high school is centralized at the Ministry of Education. Furthermore, allhighschoolstudentswithineach high school cohort are faced with identical written exams, and the oral exams and the major written assignments are evaluated both by the student s own teacher and an external examiner assigned by the Ministry of Education. Note that there are no tuition fees for education in Denmark, and all students above receive a study grant from the government that suffices to cover living expenses. Students living with their parents receive a reduced grant, but the grant is independent of parental income, educational effort and achievement as long as the student is less than one year behind scheduled study activity. We have yearly observations on labor income (earnings), gross income and net income for year All incomes are observed at yearend and deflated to real values measured in year 2000 DKK using the average wage index 14 In Denmark a numerical grading scale system is used. The possible grades are 00, 03, 5, 6, 7, 8, 9, 10, 11, 13, where 6 is the lowest passing grade, and 8 is given for the average performance. 15 Until 1996, this age limit was 19 years. 12
13 for the private sector. Other individual background variables that we use in our estimations are gender and actual labor market experience (including a squared term). Parental background variables that we use are: a set of mutually exclusive indicator variables for level of highest completed education of mother and father, respectively, whether the mother and the father took at least medium level Math in high school, and their income as observed in the year the individual starts high school. Among the gross population of high school entrants of 1987 and 1988, we select only high school graduates 16. Furthermore, we restrict the sample to individuals with nonmissing labor market income thirteen years after starting high school, hence we exclude individuals who have left the country, died, are fulltime all year unemployed or out of the labor force. 17 After these restrictions, the sample contains observations on 30,219 individuals from 139 high schools. Of these 14,970 start high school in 1987, and 15,249 start high school in Descriptive Statistics The descriptive statistics are shown in Tables 1 and 2. Table 1 shows summary statistics separately for the cohorts of 1987 and 1988 divided by gender, whereas Table 2 shows summary statistics divided by Math level. From Table 1 it is seen that the two cohorts have similar observable characteristics. Aside from the fact that a much higher proportion of the 1988 high school cohort has taken the advanced Math course, there are no striking differences. There are some small differences in choices of higher education. Another interesting point to note is that females in both high school cohorts have a disadvantaged parental background compared to males. Both parents of the female high school graduates have a lower Math level, and a lower level of highest completed education. First of all, some of this effect may be attributed to the fact that more females attend high school, as 57% of high school graduates are females. Secondly, empirical evidence also tends to be in line with this observation. Chevalier et. al. (2005) find that the effects of intergenerational transmission of education are stronger for sons than for daughters. It is also observed in Denmark that females have a higher probability (than males) to be upward mobile. 18 Hence, this observation should not be so surprising after all. In the present study, we only consider the traditional academic high school track, which is the only track affected by the reform. 19 There is no indication of any changes in the individuals choice of high school track after the high school reform, since the distribution of high schoolstudentsacrosstracksisalmost 16 We observe 18% who enter high school without completing in three years. The distribution of dropouts is fairly equal over cohorts and across mathematics and language based studies. Most of them drop out in the first year, hence before attending the advanced Math course. 17 We delete 3480 observations due to missing labor income. 18 These evidence stem from a publication from the Danish Ministry of Social Affairs (last item in list of references). 19 Four tracks are available: the traditional academic track, the business track, the technical track and a 2year equivalent of the traditional academic track. See Appendix A for further details. 13
14 constant. In particular, 58% of the individuals who chose to enter high school in 1987 chose the traditional academic high school track, compared to 57% in However, the reform clearly affects the traditional high school students choice of high level Math. Only 27% of the students choose high level Math in the 1987 cohort as compared to 54% in the 1988 cohort. A higher proportion of male high school students choose high level Math. The proportion of males who start high school in 1987 and choose high level Math is 42%, as opposed to 73% of males in the 1988 high school cohort. For females these figures are 16% and 40%, respectively. Table 2 shows descriptive statistics categorized by the indicator of high level Math, and by high school cohort within each Math level. It reveals that high level Math students have very favorable labor market and educational outcomes. They have higher GPA from high school. More high level Math students attend and complete higher education at any level. Aside from having higher completion rates, high level Math students also complete a given educational level at a faster rate. Hence high level Math students seem to be more efficient in the higher educational system. In addition, they are also more successful after labor market entry where they are less unemployed and earn more. High level Math students log earnings are 0.28 higher than earnings of other high school students. As a point of reference, we note that the Math log earnings gap is as large as the gender log earnings gap for high school graduates. The Math earnings gap is larger within the 1987 high school cohort than within the 1988 cohort. Hence we set out to find out whether this huge earnings gap is due to the fact that students become more productive on the labor market because of the high level Math course, or it is only due to selection of the students with most favorable unobservable characteristics into the high level Math course. 4 Results The key explanatory variable in the empirical analysis is the indicator for completing the high level Math course in high school, MathA i. Aside from presenting the results using IV estimation with our three reformbased instruments, we replicate previous studies to the maximum feasible extent. This means that we also estimate the earnings equation (1) by OLS using the total estimation sample, the 1987 high school cohort, and the 1988 high school cohort, respectively. Furthermore, we estimate (1) including high school Fixed Effects (FE), hence controlling for unobserved differences across high schools, e.g. in quality of the student body and teachers. At last, we also apply a variant of the instrument suggested by Altonji (1995) which is the mean participation rate in the high level Math course at the students high school. 14
15 4.1 Estimation of the causal effect of Math on Earnings The key outcome variable is (yearly) log earnings 13 years after starting high school. 20 The preferred income measure would be lifetime income. However, since the individuals in our sample are relatively young (in their thirties), it is not possible to construct a sensible measure of lifetime income. We believe that the chosen income measure is sensible because individuals have on average been on the labor market for less than five years, and hence they are likely to have settled into careers. A separate note on variables reflecting higher education is needed. As argued in Section 2 above: On the one hand, higher education is a confounder of the direct effect of high school Math on earnings. On the other hand, it is important to control for it, since otherwise we might give high school Math credit for income increases that result from investments of time (and money) in higher education. To take this into account we run all our estimations with and without controlling for further education. We sequentially include two sets of mutually exclusive indicator variables for highest completed education  one for the level of education categories and one for the levelsubject categories, since the cost of adegreeaswellastheeffect of high level Math is presumably not independent of subject of education. 21 Similarly, we also run all estimations with and without the GPA proxy for aptitude, cf. discussion in Section 2 above. To account for differences in earnings profiles, we control for (linear quadratic) actual labor market experience in all specifications with additional explanatory variables. In principle, actual labor market experience could also be considered a confounder of the direct effect of high level Math (to the extent that high level Math affects employment). High level Math students are indeed more employed, cf. Table 2, but in the estimations the treatment effect of the high level Math course is not significantly affected by including (or excluding) actual labor market experience Results of estimation Estimation results for the total estimation sample of high school graduates entering high school in 1987 and 1988 are shown in Table 3. Estimates of the effect of high level Math on log earnings for all specifications and all estimation strategies are presented along with standard errors (in parantheses) and indications of significance. In addition to OLS, FE and estimation by use of Altonji s instrument, Table 3 presents the results which applies instrument #1, which is discussed in the next subsection. The results from OLS reveal that students who complete the high level Math course in high school receive 0.28 log points higher earnings. Some of this effect 20 We have done the analysis using three different income measures: gross income, net income, and labor market income. Furthermore, we have looked at income 12, 13 and 14 years after starting high school, respectively. Our qualitative results are robust to the change of income measure. 21 We have also estimated a specification with years of education, these results were similar, hence not reported. 15
16 is an indirect effect through parental background variables, labor market experience and gender. Controlling for these background variables, the Math earnings gap is reduced to Likewise, some of the total effect is due to aptitude. Controlling for GPA further reduces the income premium by another four percentage points. However, if further educational choices are also controlled for, theincomepremiumisnotaffected by controlling for GPA. This is what would have been expected given the consistent evidence of individuals selfselecting into educational levels based on ability, see e.g. Willis and Rosen (1979). Furthermore, GPA is used as an admission criteria to the universities and other institutions of higher education. Since GPA is used by educational institutions to screen individuals, it can directly affect further educational choices. The effect goes down from 0.17 to 0.10 after controlling for higher educational level choices, and further down to 0.07 when also controlling for educational subjects. Hence, more than half of the total effect of high level Math on earnings is an indirect effect running through higher education. This indicates that the potential gains from having high level Math are not independent of which length and subject of higher education individuals choose. The result is intuitively clear, since individuals with high level Math will probably be more successful in completing a Science education and other technical educations that traditionally lead to a better paid job. On the other hand, high level Math students are probably also less likely to choose postsecondary education within Humanities that traditionally lead to lower paid jobs. 22 However, it is not clear to which extent, high level Math students have learned skills that make them better able to successfully complete postsecondary education within the Humanities. Some of the skills learned in the high level Math course like clarity in expressions, logical reasoning and inference, as well as imagination and ingenuity, should prove to be powerful tools for completing a higher education in any field. 23 The separate OLS regressions for each high school cohort, reveal that the high level Math earnings premium is higher for the 1987 high school cohort. The high level Math students in the 1987 experience an earnings gain of 34 percent, which diminishes to 11 percent when controlling for all observable background variables. On the other hand high level Math students in the 1988 high school cohort have a significant earnings premium of 29 percent, which is reduced to 7 percent when controlling for all observable background variables. The estimated effects of the high level Math course from the FE are slightly lower than the OLS estimates. Nevertheless, the pattern is similar. The income premium from having high level Math is 27 percent in total, but diminishes to 7 percent when we control for all background variables, including levelsubject categories of higher education. This shows that selection of the most talented students into the best quality high schools is not that important. This was also to be expected, since the centralized control of the high schools means that the course quality should be similar across high schools, cf. Section 3 above. 22 Descriptive statistics confirm these two presumptions (not reported, but available upon request). 23 See the elaboration by Ackerman (2000), and Appendix A in this paper for further details on the objectives of the high level Math course. 16
17 Applying the proportion of high level Math students in the year group in the student s high school as an instrument for the student s own Math level we find smaller estimates of the effect of high level Math on earnings. The estimated IV coefficient measures the effect of those who are induced to take the advanced Math course because they move to another high school where marginally more students attend the high level Math course. The total effect is 0.11, but diminishes to 0.05 when controlling for parental background caracteristics, gender and experience, and becomes insignificant when controlling for either GPA or further education. Although the instrument has a high preditive power of attending the high level Math course, it is not considered a clean instrument for several reasons (cf. Section 1) Instrument #1 The last (lower right) panel in Table 3 presents the results which applies instrument #1. Instrument #1 is an indicator for belonging to cohort 1988 and hence being exposed to the actual reform of Table 4 shows a full set of estimation results for one of the specifications. The estimated coefficients of the selection equation show that the instrument is very strong with a tstatistic above 50. Therefore the precision on the estimate of the effect of the endogenous variable is accordingly high. The IV estimates of the causal effect of high level Math on earnings using our instrument, show no effect of high level Math on earnings. Thus the high school students who are induced to choose high level Math because of the wider set of nonscience options do not gain (in terms of earnings) from attending the extra Math course. Hence, the seemingly large effect of high level Math on earnings seems to be a selection effect of individuals with more favorable unobservable characteristics into the high level Math course, and not acausaleffect. 24 To explore the selection effect in more detail, and since part of the indirect effect of Math on earnings runs through gender, we estimate all specifications on the subsamples of males and females, respectively. The results for males are shown in Table 5a and for females in Table 5b. We find high estimated coefficients to the high level Math course indicators for males and much smaller coefficients for females. The OLS estimates for males range from 0.28 (with no explanatory variables) to 0.10 (controlling for all background variables), and similarly for the FE estimates. While the OLS and FE estimates are large and significant, the IV estimates are small, negative and most often insignificant. Hence yet again, the apparent effect of Math on earnings is not a causal effect. The gender difference in the estimated coefficients indicates that the selection of males with more favorable unobservable characteristics into the high level Math course seems to be much more prevalent than the similar selection process for females. 24 Sensitivity analysis of the IV estimates corroborates that these results are very robust to estimation strategy and model specification, cf. Appendix B. 17
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