DOES TRACKING INCREASE INEQUALITY BY REINFORCING RELATIVE AGE EFFECTS IN EDUCATION? 
 
ERASMUS UNIVERSITY ROTTERDAM 
Erasmus School of Economics (ESE) 
Department of Economics 
Supervisor ESE: Prof. Dr. H.D. Webbink 
Author: J.S. Rhuggenaath* 
Exam number: 328285 
 
Abstract 
There is empirical evidence that relative age differences in class leads to inequality between older students and younger students because relatively older students tend to have higher test scores and are less likely to be retained. This thesis uses data from TIMSS in a difference-in-difference approach in order to investigate whether early tracking increases the inequality between younger and older students that arises due to differences in relative age. The main findings suggest that early tracking tends to decrease the inequality between older and younger students that arises due to differences in relative age.  
Keywords: tracking, relative age, inequality, education, instrumental variable 

 
 
*Erasmus School of Economics, e-mail adress: jasonrhuggenaath@hotmail.com  
 
Acknowledgements 
First I would like to thank Eric Hanushek and Ludger Woessman for making their data on the ‘age of first selection’ that was used in their paper accessible to me.  
Also, I would like to thank Prof. Webbink. His seminar ‘Cases in Policy Evaluation’ provided me with the tools I needed in order to get started with this thesis. As my supervisor he was always very accessible and his comments were very useful.  

 

 

 

 

 

 
 
Table of Contents 
1 Introduction         4 
2 Previous Studies        7 
3 Empirical Strategy        15 
4 Data          20 
5 Economic Mechanisms and Hypothesis     25 
6 Main Results         28 
7 Interpretation of Main Results      39 
8 Sensitivity Analysis        43 
9 Conclusion and Discussion       58 
10 References         61 

 

 

 

 
 
1 Introduction 
A key institutional feature of education systems is the age at which students are tracked into different levels of education (the age of tracking). Countries such as Germany, Austria, Hungary and the Slovak Republic track students as early as age 10. While other countries, such as Norway, Sweden, Canada and the United Kingdom (UK) and Japan, have a comprehensive school system until the end of secondary education (Van Elk et al. (2011), Hanushek and Woessman (2006), PISA (2009)).  But what is ‘tracking’? Tracking refers to a characteristic of the education system of a particular country where students are grouped together based on their ability. Other terms for tracking are ability-based grouping and streaming. In this thesis I define tracking as the situation where students are separated into different type of schools which differ in educational programs or objectives. For example, some schools prepare students for a university education while other schools prepare students for more vocational education. There is no clear consensus on the effect of (early) tracking on the educational achievement of students. Some studies find that early tracking has no effect on mean test scores of students (Figlio and Page (2002), Jakubowski (2010)), while others find that early tracking may increase inequality among students (Hanushek and Woessman (2006)).  
Another key feature is the enrollment age of students in primary schools or kindergarten. All countries specify an appropriate age for primary school or kindergarten enrollment (UNESCO (2007), McEwan and Shapiro (2008)). Most countries also enforce these minimum ages by requiring that students’ birthdays precede a certain enrollment cutoff date. In the literature these rules are often referred to as cutoff dates or school-entry rules. For example, a child may be allowed to enter kindergarten as long as he is five years old by September 1 of the relevant year. Cutoff dates are important/ interesting because they cause some students to be older than others when they begin school: students that just meet the enrollment cutoff will be the youngest students and other students will be older. These cutoff dates thus cause some students in class to be relatively older than their classmates. Previous studies indicate that the effect of relative age differences on test scores and grade retention is significant1. These studies find that the oldest students in class (which are relatively older than their classmates) tend to score higher on tests and are less likely to be retained2. Thus there is empirical evidence that relative age differences in class leads to inequality between older students and younger students.  
1 These studies are discussed in section 2. 
2 This effect is called the relative age effect. 
This thesis investigates whether early tracking increases the inequality between younger and older students that arises due to differences in relative age. Why would early tracking increase the relative age effect? One possible explanation is that relatively older students might have a higher probability of being tracked into a school for high-ability students whereas relatively younger students have a higher probability of being tracked into a school for low-ability students, since older students tend to have higher test results.3 As a consequence, in an early-tracking system, older students tend to have (on average) a higher education level compared to younger students. Thus comparing older students with younger students in an early-tracking system would lead to a larger inequality between older and younger students compared to a comprehensive system.4  
3 Puhani and Weber (2007) provide empirical evidence that students who start school a year later, have a higher probability of attending the highest track in Germany.  
4 This is the main hypothesis of this thesis. This hypothesis is explained in more detail in section 5. 
5 In fact, according to my knowledge, there is no other research that explicitly studies relative age effects and tracking jointly. 
These topics are also relevant from a social point of view.  Some parents in the United States also voluntarily delay their children’s primary school enrollment (McEwan (2006), Stipek (2002), McEwan and Shapiro (2008)). Calculations by Cascio and Schanzenbach (2012) show that in 1968, only four percent of US six year olds were not in first grade and that by 2010 this figure had risen to about 20%. Only one-third of the recent trend is estimated to be explained by increases in the minimum kindergarten entry ages established by US states; the rest is thought to be due to teachers deeming more kindergartners unready to progress to first grade, and to more parents choosing to delay their child’s entrance into school (Deming and Dynarski (2008), Cascio and Schanzenbach (2012)). The decision of parents to delay enrollment into school is related to the concept of relative age because by delaying enrollment these students are more likely to be the oldest in their class and thus also more likely to score higher on tests.  
The approach used in this thesis to generate exogenous variation in relative age is based on Bedard and Dhuey (2006). Like Bedard and Dhuey (2006), I also use data from the Trends in International Mathematics and Science Study (TIMSS) which includes nationally representative mathematics and science achievement results for fourth and eighth graders for a large set of countries. However, I use TIMSS data from multiple TIMSS studies, while Bedard and Dhuey (2006) only consider data from the 1995 and 1999 TIMSS studies. This allows me to study whether the estimated effects are stable over time. Furthermore, Bedard and Dhuey (2006) do not study tracking and relative age effects jointly5. In addition to the approach by Bedard and Dhuey (2006), this thesis also uses a difference-in-difference strategy in order to investigate whether early tracking leads to an increase in the relative age effect. 
The findings6 indicate that relative age effect is positive at both the fourth and eighth grade level.  Furthermore, this relative age effect is stronger at the fourth grade level than at the eighth grade level. The implied gains in the test score distribution due to the relative age effect is about 7-10 percentiles in the fourth grade and 5-7 percentiles in the eighth grade. These results are consistent with previous research. The main findings for TIMSS 1995 and 2003 suggest that early tracking tends to decrease the inequality between older and younger students that arises due to differences in relative age. However, the point estimates for the sample of TIMSS 2007 where positive and not significantly different from zero. The implied loss in the test score distribution due to the negative effect of early tracking is about 3-4 percentiles which is quite large when compared with the main effect of relative age. Thus the evidence presented in this thesis does not support the hypothesis that early tracking would increase the relative age effect. A sensitivity analysis finds that the effect of early tracking on the relative age effect does not have a clear relationship with gender. The results of the sensitivity analysis however do provide suggestive evidence that the decrease in inequality between older and younger students is smaller for high-ability students compared to low-ability students. More precisely, I found that (i) when the effect of early tracking on the relative age effect is negative, students in the top half of the test score distribution experience a smaller decrease in the relative age effect compared with students in the bottom half of the test score distribution; (ii) when the effect of early tracking on the relative age effect is non-negative, students in the top half of the test score distribution experience a larger increase in the relative age effect compared with students in the bottom half of the test score distribution. Finally, when the sample of countries for different TIMSS studies are made as comparable to other a possible, the findings again indicate that the effect of early tracking on the relative age effect is negative.  
6 The results in this thesis are framed in terms of relative age differences between students (like in Bedard and Dhuey (2006)) and this is done for the convenience of the reader. Due to the identification strategy that is used in this thesis, the estimated effects should be interpreted as a combined effect of absolute age at the start of schooling, relative age differences in class and differences in age at the time of testing. For convenience I will not use the term ‘combined age effect’ in this thesis. This combined age effect will be explained in section 2. At first reading of (the main results of) this thesis the reader can disregard the distinction between the ‘combined age effect’ and relative age effect. 
The remainder of this thesis is organized as follows: Section 2 discusses previous studies about tracking and relative age effects; Section 3 presents the empirical strategy that has been used in this thesis; Section 4 describes the data that has been used; Section 5 discusses the main hypothesis of this thesis; Section 6 presents the main results of the analysis; Section 7 interprets the main results; Section 8 presents the results of a sensitivity analysis; and finally, in Section 9, the main conclusions of this thesis are summarized.  
2 Previous Studies 
In this section I will review the results of previous studies on relative age effects and educational tracking. But first the relationship between relative age differences, absolute age differences and differences in age at the time of testing will be explained. This discussion is important because it allows one to correctly interpret the results of this thesis and the results of previous studies.  
The combined age effect 
Figure 1 will be used in order to explain the combined age effect.  
Figure 1: Schematic overview of combined age effect and timing of schooling 
Differences in absolute starting age 
Differences in relative age and differences in ‘age at test’ 
 
Differences in relative age and differences in ‘age at test’ 

Location/phase in the education system 

 
Time period B: grade 8, TIMSS test 

 
Time period B: grade 4, TIMSS test 
Time period A: start of schooling 

Time period 

Figure 1 lists the different age effects and the timing of schooling at which they are relevant.  
At time period A students are starting with their schooling career. The enrollment cutoff dates determine which students are the old enough to enter the schooling system. Students whose birthdays just precede the enrollment cutoff should be the youngest students and the other students are older.7  Thus at the time that student start school some students are older than others and this is called the difference in absolute starting age.  
7 For simplicity I assume that there is perfect compliance with the cutoff rules. 
At time period B students have progressed through the schooling system and are now in grade 4. At time period A, some students were grouped together in the same class and assuming that none of them were retained or skipped a grade, they will end up together in class again but now at the grade 4 level. In grade 4 therefore, some students will again be older than others. This difference in age between students which are in the same class is referred to as the relative age difference. Furthermore, at time period B 
students also take the TIMSS test. Since all students take the TIMSS test at the same time, students which are older than their classmates are also older at the time they take the test. This leads to differences in the ‘age at test’.  
The analysis of time period C is similar to that of time period B, except that the students are in grade 8.  
Studies that use birth months and cutoff dates in order to generate exogenous variation in relative age essentially exploit variation in the absolute age at the start of schooling in order to explain variation in relative age (at later grades). The fundamental reason for doing this is due to the that fact that if (i) compliance with cutoff rules is perfect and (ii) grade retention or grade acceleration does not occur, then there is a (direct) one-to-one relationship between the absolute age at the start of schooling and the relative age (in later grades): being older (in a particular class) at the start of schooling implies being older in later grades.8 For this reason inference about relative age also implies inference about absolute starting age. Therefore, results should be interpreted as a ‘combined age effect’. 
8 In practice compliance with cutoff rules is not perfect and grade retention (and grade acceleration) does occur. However if the majority of individuals do comply with cutoff rules and progress to the next grade as scheduled, the relationship between absolute starting age and relative age will still be quite strong. And this is indeed the strategy that is used in previous studies (and in this thesis). See section 3 for more information about the empirical strategy.   
The ‘age at test’ effect is relevant when test scores are used as an outcome variable and (relative) age used as an explanatory variable. Since all students take the test at the same time, students which are older than their classmates are also older at the time they take the test such that the effect of (relative) age cannot be distinguished from the ‘age at test’ effect.  
Based on the discussion so far the following conclusion can be made. When (i) variation in absolute starting age is used in order to explain variation in relative age and (ii) students take a test at the same time (independent of age), then the effect of relative age on test scores reflects a combined age effect consisting of: (1) absolute age differences; (2) relative age differences and (3) differences in age at the time of testing. 
Now that the combined age effect has been discussed we can proceed to the discussion of previous studies. First I will discuss previous studies about relative effects and then I will discuss previous studies about educational tracking. 

Previous studies: Relative age effects 
Previous research has provided evidence that relative age effects can have significant effects on academic performance. For example, Bedard and Dhuey (2006) find that relative age effects can be quite large and that they can also be quite persistent9. The authors use nationally representative data for grade 4 and grade 8 from the Trends in International Mathematics and Science Study (TIMSS) of 1995 and 1999. Using an instrumental variables (IV) approach, they estimate relative age effects for each country separately and also give a pooled estimate. The main findings are that the youngest students in fourth and eighth grade classes have standardized test scores that are 2 -12 percentiles lower than the oldest students in the same classes. They also find that in the US and in British Columbia, relatively older students are about 10 – 12 percent more likely to take pre-university tests or to be enrolled in college. Note however that the reported results need to be interpreted carefully. The authors use the distance between the birth month and the cutoff date as an instrument for observed age (in months). Following the discussion in the introduction, relative age effects and absolute age effects cannot be separated in this design. This implies that the estimated relative age effects are not ‘pure’ relative age effects, but a combination of absolute and relative age effects. Bedard and Dhuey (2006) do attempt to correct for this by including the regulatory ‘age at entry’ as an additional control variable, but because there is non-compliance in the sample this correction is not sufficient.  
9 In this thesis the analysis of Bedard and Dhuey (2006) will be used as a starting point. See section 3 for more information about the study of Bedard and Dhuey (2006).  
10 More specifically, they tend to investigate the effect of school starting age on educational attainment and other long-term outcomes. 
There have also been a number of studies that investigate the effect of school entry laws on educational achievement10. Studies that focus on school entry laws are related to the concept of relative age effects in a natural way, because it is exactly these school entry laws that are the main cause of relative age differences in classes.  
Using Decennial Census Long Form Data for the states of California and Texas (which contain exact date of births), Dobkin and Ferreira (2010) apply a regression discontinuity (RD) design and find that school entry laws have a modest impact on educational attainment. They find that students born right before the cutoff date for school entry in Texas and California (the youngest students) are about a percentage point more likely to complete high school and they are also about half a percentage point more likely to complete 9th, 10th and 11th grades. Furthermore, the authors find no evidence that the school entry laws had a net impact on labor market outcomes such as employment rates and wages, or on other outcomes such as the probability of homeownership. While the RD estimates do give the causal effect of the school entry 
laws, the relationship with relative age is more subtle. Assuming that there is full compliance in the sample, students just above the cutoff date would correspond to the oldest students in class and students just below the cutoff date would correspond to the youngest students in class. However, due to the entry laws, the oldest students begin school a year later such that they effectively receive two treatments: (i) they will be the (relatively) oldest in their class and (ii) they start school at an older (absolute) age. Keeping the arguments just described in mind, Dobkin and Ferreira (2010) do control for age but they cannot test how high the compliance was in their sample11. However, using contemporary cohorts of students12, they also find evidence that the youngest students are more likely to be retained and this could indicate that the youngest students have lower academic performance (which is consistent with the findings regarding test scores of Bedard and Dhuey (2006)). 
11 This makes inferences about relative age effects unreliable. Inferences about the causal effect of schooling laws however are still valid. 
12 For these cohorts they do have information about the compliance. 
McEwan and Shapiro (2008) also use an RD design in order to determine the causal effect of school entry laws on grade retention (in grade 1) and test scores (in grade 4 and grade 8) in Chile. Similar to Dobkin and Ferreira (2010), the authors compare the outcomes of students just above the enrollment cutoff to the outcome of students just below the enrollment cutoff. The main findings are that, (1) delaying enrollment by a year lowers the probability of retention by about 2 percent (given a sample mean of 2.8 percent) and that, (2) delaying enrollment by a year increases test scores by about 0.3 to 0.4 standard deviations in fourth grade, with similar effects in eighth grade. These results are in line with those reported by Dobkin and Ferreira (2010) (in the sense that younger students are more likely to be retained) and Bedard and Dhuey (2006) (in the sense that older students tend to have higher test scores). However the study by McEwan and Shapiro (2008) also cannot separate relative age effects from absolute age effects.  
I now turn to a study that does manage to separate these two effects.  
A study by Cascio and Schanzenbach (2012) uses data from the Tennessee STAR experiment/ Project STAR and knowledge about cutoff dates in order to estimate the causal effects of absolute age and relative age on student outcomes, such as grade retention and test scores. The Project STAR was designed to estimate the effect of classroom size on student achievement. In fall 1995, approximately 6000 kindergartners and 300 teachers in 79 schools in the US state of Tennessee were assigned by lottery to one of three class types: small class type (with about 13-17 students), regular (with about 22-25 students) and regular with a full-time teachers aide (Cascio and 
Schanzenbach (2012)). Furthermore data is available on class assignment and on test scores through the eighth grade, as well as grade repetition and whether each participant took a college-entrance exam (Cascio and Schanzenbach (2012)). The design of Project STAR allowed Cascio and Schanzenbach (2012) to observe students with the same enrollment/starting age, but with kindergarten classmates with different ages on average. Also, the authors are able to instrument observed age by using information about the cutoff dates in order to determine the expected age at entry. The authors find that ‘pure’ relative age effects are not positive but negative for kindergarten students. To be more precise they find that students that are relatively older compared to their kindergarten classmates at the start of kindergarten (i) score lower on achievement tests (both at the end of kindergarten and in middle school), (ii) are more likely to be retained in grade by middle school and (iii) are less likely to take a college-entrance exam. These results are essentially the complete opposite of the results obtained by Bedard and Dhuey (2006).  
Cascio and Schanzenbach (2012) offer various explanations for the difference between their results and those obtained by Bedard and Dhuey (2006). One possible explanation for the difference in results is that, Bedard and Dhuey (2006) consider test results at the 4th and 8th grade whereas Cascio and Schanzenbach (2012) consider test results at the end of kindergarten: Cascio and Schanzenbach (2012) argue that the main objective of kindergarten is to let students socialize with each other and not on learning math and science. Also Cascio and Schanzenbach (2012) (only) consider data for the state of Tennessee, whereas Bedard and Dhuey (2006) use data from a nationally representative sample for various countries13. Finally, another possible explanation is related to the design of the studies and interpretation of results: Cascio and Schanzenbach (2012) estimate ‘pure’ relative age effects whereas Bedard and Dhuey (2006) estimate a combined effect. It could very well be that ‘pure’ relative age effects are negative, once relative age effects have been separated from absolute age effects.  
13 The authors suggest that the estimated negative effects of relative age could indicate the existence of positive peer effects. Another possible explanation for the difference in findings is that at the kindergarten level peer effects dominate the relative age effect (or maturity effect) and that the situation is reversed in the fourth and eighth grades.  
Another important issue is the fact that the combined effect of relative age and absolute age can be confounded with another effect when in-school test scores are used as the outcome variable of the analysis. More specifically, relative age effects can be confounded with the ‘age at test’ effect (Black et al. (2011), Datar (2006)). This is true because (under certain conditions) age at test can be computed as follows: age at test = age at entry + number of years of schooling. Assuming perfect compliance with 
cutoff rules, age at test and age at entry will be perfectly linearly related if every student takes the test at the same time14.  
14 Note that Bedard and Dhuey (2006) do not account for this. 
Black et al. (2011) study the effect of age at school entry on test scores and long-term outcomes such as earnings and receipt of social assistance. The authors are able to separate the effect of age at school entry and age at test because their outcome variable is the score on a military IQ test. Furthermore, there were specific rules such that the timing of the test and the month of birth of individuals was not perfectly linearly related. As a result Black et al. (2011) are able to apply an IV estimation procedure to instrument both age at test and age at entry. The main results indicate that the effect of age at test is strong and positive: being one year older when taking the test increases the score about one-fifth of a standard deviation. The effect of school starting age is negative and statistically significant but not very large: starting school a year later decreases test score by about one-twentieth of a standard deviation. Looking at the long-term results, the authors find that, a higher school starting age leads to lower earnings until about age 30. After about age 30, the estimates for both men and women become close to zero and are almost always insignificant from zero.  
Previous studies: Tracking 
Figlio and Page (2002) use US data from the National Education Longitudinal Study (NELS) to investigate the change in mathematics test score from 8th to the 10th grade for different ability groups. They separate students into different ability groups (top, middle and bottom third of the test score distribution) by using the 8th grade test score. For each ability group they run separate regressions where they exploit variation in tracking across schools. The main finding of Figlio and Page (2002) is that the effect of tracking is not significant across ability groups and they interpret this finding as evidence that tracking does not harm the low-ability students. Furthermore they also attempt to correct for possible endogeneity of the tracking programs of schools by using a two-stage least squares estimation procedure. The additional results of a two-stage-least-squares estimation suggest that low-ability students may even gain from tracking. 
Van Elk et al. (2011) exploit variation in the age of tracking within the Dutch education system to investigate the effect of the timing of educational tracking on completion of higher education. At the age of 12 Dutch students can enroll in tracked classes or in comprehensive classes. The comprehensive classes consist of a combination of school types and take one or two years. Basically these comprehensive classes delay enrolment into tracked classes until age 13 or 14. Van Elk et al. (2011) exploit this variation in the timing of tracking and compare higher education completion rates of 
students that are tracked at age 12 with the outcomes of students that are tracked at age 13 or 14. The main findings are that (i) students who were tracked early are less likely to complete higher education and that (ii) early tracking does not lead to significantly different completions rates for high-ability students. 
Duflo et al. (2011) provide evidence from a randomized experiment in Kenya. In this experiment 121 primary schools, who all had a single grade 1 class, received funding to hire an extra teacher and split this class into two sections. In 60 randomly chosen schools the pupils were randomly assigned to one of the two sections. In the other 61 schools the pupils were assigned based on a ranking of their previous educational performance. After 18 months pupils in the schools that selected based on previous educational performance scored on average 0.14 standard deviation higher than pupils in schools that selected randomly. In addition they find that pupils at all levels of the distribution benefited from tracking. Duflo et al. (2011) suggest that tracking was beneficial because it helped teachers to adjust their teaching to a level appropriate to most students in class.  
In their seminal paper Hanushek and Woessman (2006) study the effect of an early-tracking regime on average academic achievement and inequality by using a difference-in-difference approach in a cross-country analysis. They match international primary school tests to secondary school tests and compare differences in test scores across countries. They find that early tracking increases inequality, while it does not have a clear impact on average achievement15. 
15 Hanushek and Woessman (2006) measure inequality by the standard deviation in test scores. They claim that there is no clear gain in mean performance due to tracking. 
Jakubowski (2010) has criticized the results of Hanushek and Woessman (2006). Jakubowski (2010) uses data from TIMSS, PIRLS and PISA in a difference-in-difference approach in order to test the robustness of the findings of Hanushek and Woessman (2006). Jakubowski (2010) argues that there are crucial differences in the design of TIMSS, PIRLS (Progress in International Reading Literacy Study) and PISA (Programme for International Student Assesment) that could bias the results obtained the difference-in-difference approach. After adjusting the samples from TIMSS, PIRLS and PISA in order to make them more comparable to each other, he finds no evidence of a negative impact of tracking on mean performance and also no impact on inequality. Furthermore, when using another difference-in-difference approach (that is different from the approach of Hanushek and Woessman (2006)), he finds a negative impact of tracking on mean performance. But Jakubowski (2010) is cautious about this finding and argues that tracking was confounded with the effect of policies common in Eastern Europe, mainly Post-Communist, countries.  
Summary of Previous Studies 
In sum, we can conclude that the results of previous studies about relative age are mixed. Some studies find that relative age has a positive effect on test scores (Bedard and Dheuy (2006)) and that younger students are more likely to be retained (Dobkin and Ferreira (2010) and McEwan and Shapiro (2008)), but there are also studies that indicate that the youngest students have higher completion rates of education (Dobkin and Ferreira (2010)). The majority of empirical evidence tends to indicate that the short-term effect of relative age is positive. However, the long-term effects are a bit more mixed. In terms of earnings, there does not appear to be a significant long-term effect (Dobkin and Ferreira (2010), Black et al. (2011), Deming and Dynarski (2008)). 
Similarly, the results of previous studies about tracking are also mixed. Looking at cross-country analyses, one finds no clear impact of early tracking on mean performance and inequality. The study of Van Elk et al. (2011) suggests that early tracking may not be beneficial for (low-ability) students. On the other hand, the results of Figlio and Page (2002) and Duflo et al. (2011) indicate that tracking may be beneficial for all students.  

 

 

 

 
 
3 Empirical Strategy 
In this section I will explain the empirical method that will be used. The empirical method is based on the method used in Bedard and Dhuey (2006). I will briefly state the most important aspects in their approach here and when I introduce changes in their approach I will indicate that explicitly. I refer the reader to section III of their paper. Bedard and Dhuey (2006) start off with a simple model of the relationship between student outcomes and observed age: 
                                      (1) 
Where       is a test score on an internationally standardized test,      is observed age,       is a set of control variables and      is the error term. The index i denotes the student under consideration, the index c denotes the country under consideration and the index g denotes the grade under consideration. 
The causal interpretation of the parameter     rests on the assumption that unobservable variables do not confound the observed age effect, which is clearly untrue given nonrandom grade retention, for example. Bedard and Dhuey (2006) subsequently propose the following Instrumental Variables (IV) solution to this issue by using birth month relative to the school cutoff date, assigned relative age (R), as an exogenous determinant of observed age. More specifically, they estimate the parameters of equation (1) using two-stage least squares (TSLS) based on the following the first-stage equation for observed age: 
                                            (2) 
They use the cutoff date and month of birth to construct a linear measure of assigned relative age (R).16 More specifically, R = 0 for students born in the last eligible month and R = 11 for students born in the first eligible month. For example, if the cutoff date is January 1, December babies are the youngest (R = 0) and January babies are the oldest (R = 11). Actual age in months (A) is constructed using the test date and birth date, both of which are reported in months. 
16 Here the assumption that is made, is that birth month relative to the cutoff date is random. That is, it is assumed that parents do not time births around cutoff dates. Under this assumption, variation in R is indeed exogenous.  
By using equation 1 and 2 Bedard and Dhuey (2006) then arrive at the following main equation (the second stage equation): 
                                       (3) 
where    denotes the instrumented values of observed age  .  
So far, the models that have been considered are estimated separately for each country and each grade. The observations across countries can also be pooled in order to arrive at a pooled estimate of the relative age effect for each grade. The pooled model is similar to equation 3 and takes the following form 
                                             (4) 
where    is an additional set of control indicators.   includes a vector of month of birth dummy-variables (to capture season of birth effects that are common across countries) and also includes country dummy-variables. Unlike equation (3), this TSLS model is estimated using data pooled across countries, but separately by grade. 
All test score models include a basic set of socioeconomic controls (as used by Bedard and Dhuey (2006)). These include indicator (dummy) variables for sex, grade, native born mother, native born father, child living with both parents, child has a calculator, child has a computer, child has more than 100 books, and parental education (in eighth grade models only), and the number of people residing in the child’s household. Furthermore, all test scores are standardized to have a mean of 50 and a standard deviation of 10.  
Introducing tracking 
In order to estimate the effect of “educational tracking” on test results, we first need to construct a “proper variable”. To this end I will define a dummy variable T that takes the value of 1 if “educational tracking” takes place at age   or younger. The dummy variable T takes a value of zero otherwise.  
That is, 
                                                           (5) 
Where      denotes the age at which tracking occurs. Note that the event        is determined by the structure and organization of the educational system of a particular country: (    ) is a property of the educational system and does not represent the true age of a pupil. By defining the variable T in such a way, we can characterize educational systems as using either “early educational tracking” or “late educational tracking”.  
By modifying equation (1) and (2) and including an extra first-stage equation we arrive at the following model specification: 
                                                                 (6) 
                                                 (7) 
                                                        (8) 
                                                                            (9) 
Where   is a test score on a internationally standardized test,   is the dummy variable that indicates if a country uses “early educational tracking” or “late educational tracking” and   is a set of control variables. The index i denotes the student, the index c denotes the country under consideration and the index g denotes the grade under consideration. Variables with an upper bar denote instrumented values for the respective variable, that is,       denotes the instrumented values for     . 
Note that there is an extra first-stage equation (given by equation (8)) in order to instrument the variable given by           in equation (6).  
The parameter      gives the interaction effect of interest: it tells us whether the relative age effect differs depending on the degree of “educational tracking”. 
Introducing “grade effects”: difference-in-difference estimator 
Note that the parameter      does not necessarily represent the causal impact of tracking on the relative age effect. The parameter      only indicates whether the estimated relative age effect differs significantly across the tracking regime. In order to estimate the causal impact of tracking on the relative age effect, I use a difference-in-difference strategy. In order to illustrate this approach, first consider figure 2. 
Figure 2: Observed outcomes framework for the difference-in-difference strategy 

 	Tracking  = 0 	Tracking = 1 	
Grade 4 
(G = 0) 	                             
 
Scenario (A) 

Scenario (B) 	
Grade 8 
(G = 1) 	                             
 
Scenario (C) 	                             
 
Scenario (D) 

 
 
Figure 2 is essentially a schematic overview of the empirical strategy. The model defined by equation (4) can be estimated in 4 different scenarios: these are scenarios (A) through (D) that are indicated in figure 2. In order to determine whether the parameters      and      differ (due to the tracking regime) we can use the parameter      of equation (9). This procedure can be repeated for grade 4.  
Note that since students in grade 4 (regardless of the country under consideration) have not formally received education in a tracked environment yet, observations in grade 4 can be interpreted as ‘pre-treatment’ observations17. However, in grade 8, some students have formally received education in a tracked system, while other students have not such that observations in grade 8 can be interpreted as ‘post-treatment’ observations.   
17 The treatment here is ‘early tracking’, that is, Tracking = 1, or equivalently T = 1. 
18 Off course, the assumptions of the difference-in-difference strategy would also need to be satisfied in order for the causal interpretation to be valid. 
It should also be clear now that the parameter      of equation (9) estimated at the grade 8 level, does not have a causal interpretation because it represents a ‘naïve’ comparison of scenario (C) and scenario (D). By combining information from all four scenarios in a difference-in-difference strategy the causal effect of the tracking regime on the relative age effect could possibly be identified18. The causal effect that is identified by the difference-in-difference design is given by : 
                                       .        (10) 
The parameter              can be estimated by estimating the parameter    in the following main equation:  
                                                                                                                              (11) 
Where   = 0 if a student is in grade 4 and   = 1 if a student is in grade 8. Variables with an upper bar denote instrumented values for the respective variable, that is,                 denotes the instrumented values for the variable        . Note that the variable given by        is instrumented by       , the variable given by       is instrumented by      .  
Now the parameter     is of interest because it tells us what the causal effect of an early-tracking regime is on the relative age effect.  
This is not the first time that TIMSS data has been used in a difference-in-difference approach. Hanushek and Woessman (2006) and Jakubowski (2010) also use TIMSS data in a difference-in-difference strategy, but they match grade 4 observations from TIMSS with data of 15 year-olds from PISA. Another difference between the strategy presented in this section and the strategy of Hanushek and Woessman (2006) and Jakubowski (2010), is that they are interested in the effect of tracking on test results themselves and not interested in the relative age effects19.  
19 More specifically, they replace                               for               with       in figure 1.  

 

 

 

 

 
4 Data 
The data used in this thesis comes from various sources. First I will discuss the data from the TIMSS studies. Secondly, I will discuss the data about cutoff dates and finally I will discuss the data that was used for determining the tracking regime of various countries. 
TIMSS data 
The main data comes from the TIMSS studies of 1995, 2003 and 2007. The TIMSS data includes nationally representative mathematics and science achievement results for fourth and eighth graders for a large set of countries. The 1995 TIMSS includes test scores for two different grade groups. The first set of scores is for students enrolled in the two adjacent grades that contain the largest proportion of nine-year-olds— third and fourth graders in most countries. In order to keep things simple, we refer to these students as fourth graders. The second set of scores is for students enrolled in the two adjacent grades that contain the largest proportion of thirteen-year-olds—seventh and eighth graders in most countries. In order to keep things simple, we refer to these students as eighth graders20. In contrast, from 1999 onwards, TIMSS includes only one age group in a single grade. While the 2003 TIMSS21 uses the 1995 definition to target the two adjacent grades containing the most thirteen-year-olds, only students in the upper of the two grades were tested—eighth graders in most countries. The same reasoning also applies in order to target nine-year-olds from 2003 onwards.   
20 This is the definition that Bedard and Dhuey (2006) use.  
21 This also applies to later studies of TIMSS. 
The TIMSS data includes nationally representative mathematics and science achievement results for fourth and eighth graders. These achievement results are scores on tests about mathematics and science. Each student record in the TIMSS database contains imputed scores in mathematics and science, because students took tests from different booklets such that the mathematics and science questions are slightly different. Because each imputed score is a prediction based on limited information, it almost certainly includes some small amount of error. In order to judge the effect of this imputation on the analysis, the TIMSS database provides five separate imputed estimates (known as “plausible values”) for each test subject (TIMSS (2005)). That is, for each student that took a test, the TIMSS database provides five test scores (for both math and science). Accordingly, an analysis may be replicated as many as five times, using a different score each time, and the results compared to judge the impact of the imputation process (TIMSS (2005)).  In this thesis I use the mean of these five plausible 
values as the main outcome variable (test scores). Next, these test scores are standardized to have a mean of 50 and a standard deviation of 10.  
Besides personal information about the students, such as gender and birth dates, TIMSS data also contain information about the resources that students have at home, information about parents’ education level and information about the teachers.  
 
Data about cutoff dates 
Information about the cutoff dates comes from two sources. First, I used the cutoff dates that Bedard and Dhuey (2006) used in their analysis. Second, I also used cutoff dates from Gerritsen and Webbink (2013) in order to supplement the cutoff dates provided by Bedard and Dhuey (2006). 
Data about Tracking 
Information about the age of tracking comes from two sources. The main source is a document published by the PISA in 2009 that contains the age at which first selection takes place. This is used to define the variable tracking: if the age of first selection is 13 years or younger, then the dummy-variable of tracking   is set to 1, that is,      .  The second source for the tracking variable comes from Hanushek and Woessman (2006).22  
22 The authors have provided me with their information about the age of first selection, which they obtained from various sources. 
The information about cutoff dates and countries are summarized in the following two tables. Table 3 provides information about the number of early-tracking and late-tracking countries, and also on the number of observations available in these countries. 

 

 

Table 1: List of countries, cutoff dates and age of first selection 

Country ISO Code 	Country 	Cutoff date 	Age of first selection 	
40 
124 
196 
203 
208 
250 
276 
300 
344 
348 
352 
380 
392 
428 
440 
528 
554 
578 
620 
642 
643 
702 
703 
705 
724 
752 
756 
764 
926 
956 
957 
927 
158 
233 
364 	Austria* 
Canada 
Cyprus 
Czech Republic* 
Denmark 
France 
Germany* 
Greece 
Hong Kong 
Hungary* 
Iceland 
Italy 
Japan 
Latvia 
Lithuania 
Netherlands* 
New Zealand 
Norway 
Portugal 
Romania 
Russian Federation 
Singapore* 
Slovak Republic* 
Slovenia 
Spain 
Sweden 
Switzerland* 
Thailand 
England 
Belgium (Flemish)* 
Belgium (French)* 
Scotland 
Chinese – Taipei 
Estonia 
Iran 	September 1 
January 1 
March 1 
September 1 
January 1 
January 1 
July 1 
April 1 
January 1 
June 1 
January 1 
January 1 
April 1 
September 1 
September 1 
October 1 
May 1 
January 1 
January 1 
September 1 
October 1 
January 1 
September 1 
January 1 
January 1 
January 1 
January 1 
January 1 
September 1 
January 1 
January 1 
March 1 
September 1 
October 1 
October 1 	10 
16 
15 
11 
16 
15 
10 
15 
15 
11 
16 
14 
15 
16 
15 
12 
16 
16 
15 
14 
15 
12 
11 
14 
16 
16 
12 
15 
18 
12 
12 
18 
15 
15 
18 	

Notes: * indicates a country which can be classified as an ‘early-tracking’ country. Italic countries denote countries which are also in the sample of Bedard and Dhuey (2006) 

 
Table 2: List of countries available for each TIMSS study 

TIMSS 1995 	TIMMS 2003 	TIMMS 2007 	
 
Austria 
Canada 
Cyprus 
Czech Republic 
Denmark* 
France* 
Germany* 
Greece 
Hong Kong 
Hungary 
Iceland 
Italy 
Latvia 
Lithuania* 
Netherlands 
New Zealand 
Norway 
Portugal 
Romania* 
Russian Federation* 
Singapore 
Slovak Republic* 
Slovenia 
Spain* 
Sweden* 
Switzerland* 
Thailand 
Belgium (Flemish)* 
Belgium (French)* 
England 	 
Chinese - Taipei 
Cyprus 
Estonia* 
Hong Kong 
Hungary 
Italy 
Japan 
Latvia 
Lithuania 
Netherlands 
New Zealand 
Norway 
Romania* 
Russian Federation 
Singapore 
Slovak Republic* 
Slovenia 
Sweden* 
England 
Belgium (Flemish) 	 
Austria** 
Chinese – Taipei 
Cyprus* 
Czech Republic 
Denmark** 
Germany** 
Hong Kong 
Hungary 
Iran* 
Italy 
Japan 
Latvia** 
Lithuania 
Netherlands** 
New Zealand** 
Norway 
Romania* 
Russian Federation 
Singapore 
Slovak Republic** 
Slovenia 
Sweden 
England 

Notes: * indicates data available for eighth grade only, ** indicates data available for fourth grade only 

 

Table 3: Information on the number of observations and countries for different TIMSS studies. 

TIMSS study 	Grade 	Number of early tracking countries 
(T = 1) 	 Number of late tracking countries 
(T = 0) 	Number of observations for early tracking countries  	Number of observations for late tracking countries 	
1995 	4 	5 	12 	33221 	67350 	
 	8 	10 	20 	62679 	129062 	
2003 	4 	4 	12 	15683 	41910 	
 	8 	5 	15 	20346 	58069 	
2007 	4 	7 	13 	33189 	51101 	
 	8 	3 	13 	13216 	52637 	

 
From table 3 one can conclude that there are more late-tracking countries than early-tracking countries. Furthermore, the TIMSS study of 2007 contains the fewest early-tracking countries in grade 8 while the TIMSS 1995 study contains the most early-tracking countries in grade 8.  

 

 

 

 

 
5 Economic Mechanisms and Hypothesis 
Based on the previous studies discussed in section 2, the empirical strategy discussed in section 3 and structure of the available data discussed in section 4, I will formulate the main hypothesis of this thesis. I will also discuss the economic mechanism underlying this hypothesis, but I will not present formal economic models here.  
Economic Mechanisms and Hypothesis  
When formulating hypotheses it is important to take the structure of the TIMSS data into account: the TIMSS data is supposed to form a nationally representative sample. Therefore, statements about the effects of the tracking regime on the relative age effect should also take this information into account because they are not statements about the average student (see figure 3). 
Figure 3 gives an overview of the structure of the TIMSS sample. In particular it decomposes the TIMSS samples for each grade according to the school system (comprehensive and tracked system). The figure makes clear in which environment the students receive their education and which type of students are included in the TIMSS sample. In every class there is variation in the ability of students and in every class some students are relatively older than other students. In grade 4 both early-tracking and late-tracking countries have a comprehensive school system (students have not been tracked by that age). The main difference is in grade 8. In late-tracking countries, high-ability students and low-ability students are not separated (they are in the same class) and they receive the same type of education, as a consequence the grade 8 TIMSS sample for late-tracking countries is a combination of these mixed classes.  The situation is different in early-tracking countries. In those countries high-ability students receive their education separately from low-ability students (they are not in the same class). Furthermore, the content of the education receive is also different. High-ability students tend to receive more advanced education compared to low-ability students.   
Now that the structure of the TIMSS data has been discussed, the main hypothesis can be formulated23. 
23 Based on the results of Bedard and Dhuey (2006) and the similarity of the empirical strategy that was discussed in section 3, I assume that the main effect of relative age on test scores is positive.  

Figure 3: Structure of TIMSS data with respect to early and late tracking 
 
Grade 4: TIMSS Sample  
 
Grade 4: comprehensive system  

Combine the k classes to form a nationally representative sample  

H O  
L Y 

H O  
L Y 
 
H O  
L Y 
 
H O  
L Y 

  1                2 …….                 k 
There are k different classes. Among students there is variation in  ability and relative age. High-ability students in a comprehensive system are denoted by ‘H’ .Low ability students by ‘L’. Older students are denoted by ‘O’ and younger students by ‘Y’. 

 
Grade 8: comprehensive system (Late-tracking country) 

Grade 8: TIMSS Sample  

Combine the w classes to form a nationally representative sample  

HC O  
LC Y 

HC O  
LC Y 
 
HC O  
LC Y 
 
HC O  
LC Y 

  1                2 …….                 w 
There are w different classes. Among students there is variation in  ability and relative age. High-ability students in a comprehensive system are denoted by ‘HC’ .Low ability students by ‘LC’. Older students are denoted by ‘O’ and younger students by ‘Y’. 

 
 
Grade 8: tracked system (Early-tracking country) 

Grade 8: TIMSS Sample  

Combine the m high-track and n low-track classes and to form a nationally representative sample  

HT O  
 Y 
 
HT O  
 Y 
 
HT O  
 Y 

  1                2 …….                 m 

HT O  
LT Y 

 O  
LT Y 
 
 O  
LT Y 
 
 O  
LT Y 

  1                2 …….                 n 
There are m different high-track classes and n different low-track classes. Among students there is variation in  ability and relative age. High-ability students  (‘HT’ ) attend high-track classes and low-ability students (‘LC’)  attend low-track classes. Older students are denoted by ‘O’ and younger students by ‘Y’. 

Hypothesis 
Given a positive main effect of relative age in all grades and across all tracking regimes, one would expect that early tracking would be associated with increasing or at least non-decreasing relative age effects. That is, one would expect that early tracking would increase the inequality in test scores between older students and younger students. 
The reasoning behind this hypothesis is as follows. If one assumes that test scores can measure differences in ability of students and that admission into (high-track) schools is conditional on test scores, then relatively older students have a higher probability of being sent to a high-track school. In contrast, younger students tend to score lower on tests and might therefore have a higher probability of being tracked into a school for low-ability students. Thus (in terms of figure 3) in an early-tracking system, students who are older in grade 4 have a higher probability of being HT – students in grade 8 and younger student have a higher probability of being LT – students in grade 8. This implies that older students (in grade 8) in a tracked system receive (on average) more advanced education/ are better educated compared to younger students (in grade 8) in a tracked system.24 And under the assumption that students with a higher ability tend to score higher on tests, this would in turn tend to increase the inequality in test scores between younger and older students. Note the difference with a late-tracking country: students who are older in grade 4 tend to have higher test scores and have a higher probability of being (diagnosed as) a high-ability student, but (in grade 8) these high-ability students (HC – students) will not have received a substantially different education compared to low-ability students (LC – students) because they would still be in a comprehensive system. Therefore, in the grade 8 TIMSS sample of an early-tracking country variation in relative age is more likely to represent variation in education levels compared to the grade 8 TIMSS sample of a late-tracking country. Assuming that students with a higher level of education have a higher test score, early-tracking would be associated with an increase in the inequality between younger and older students.  
24 Recall that HT – students and LT – students receive fundamentally different education. That is, the curriculum of a high-ability school can be quite different from the curriculum of a low-ability school. 

 
 
6 Main Results 
In this section I will discuss the main results of the analysis. This section is split up in four parts. In part 1, I will discuss the validity of the instrument, in part 2 I will study the (main) effect of relative age, in part 3 I will introduce an interaction term in order to determine how early tracking is associated with the relative age effect. Finally, in part 4, I provide a difference-in-difference estimate of the impact of early tracking on the relative age effect. 
6.1 Validity instrument 
Note that assigned relative age (R) is a legitimate instrument if the variation in the instrument is exogenous. However the instrument may be correlated with unobservable characteristics and this would confound the estimated effect. While we cannot formally test whether the variation in R is unrelated to unobservable characteristics, we can test whether the variation of instrument can be explained by observable characteristics.  
In order to determine whether the variation of the instrument can be explained by observable characteristics, a regression can be performed. Table 1 gives estimates of a regression with the instrument as dependent variable and observable characteristics as independent variables. 
From table 1 we can conclude that, on average, most of the control variables do not have a significant effect on the instrument25. This gives us somewhat confidence that the conditional independence assumption holds, because on average, students have the same characteristics even if they have a different assigned relative age such that the variation in the instrument is arguably exogenous. 
25 Some variables are not included in the regression because otherwise some countries might be excluded from the analysis due to missing values and this could lead to a large loss in observations (this is more relevant for TIMMS 1995). Some variables are not available in for all TIMSS studies.  
26 From table 1, we can conclude that this is more relevant for the TIMMS 1995 study 
There are a few exceptions however. For example, it appears that students that do not possess a calculator are more likely to be born near the cutoff. This could indicate that students with a low SES are more likely to be relatively younger in class. To the extent that this would bias the estimates that will be presented later, this in turn could imply that the true effect of relative age on test score may be larger26.  

Table 4: Regression of the instrument on observable control variables 

 	TIMSS 2003 	TIMSS 2007 	TIMMS 1995 	
variable 	Grade 4 	Grade 8 	Grade 4 	Grade 8 	Grade 4 	Grade 8 	
Number of books 	0.023* 	0.007 	-0.006 	0.013 	0.006 	0.002 	
 	(0.013) 	(0.011) 	(0.012) 	(0.013) 	0.010 	(0.007) 	
Possess calculator † 	-0.054 	0.101 	-0.106** 	-0.021 	-0.134*** 	0.025 	
 	(0.058) 	(0.084) 	(0.049) 	(0.087) 	0.032 	(0.039) 	
Possess computer  † 	0.064 	0.050 	0.006 	-0.062 	-0.026 	 	
 	(0.049) 	(0.050) 	(0.043) 	(0.063) 	0.024 	 	
Possess Study desk † 	-0.004 	0.090* 	0.004 	-0.039 	-0.085*** 	0.022 	
 	(0.046) 	(0.049) 	(0.042) 	(0.056) 	(0.030) 	(0.029) 	
Possess dictionary † 	-0.129 	-0.126* 	-0.001 	-0.018 	-0.039 	 	
 	(0.051) 	(0.066) 	(0.043) 	(0.063) 	(0.034) 	 	
How often use  computer 	-0.011 	-0.014 	 	 	 	 	
 	(0.017) 	(0.018) 	 	 	 	 	
Possess internet connection at home † 	 	 	-0.055 	-0.052 	 	 	
 	 	 	(0.033) 	(0.045) 	 	 	
Mother born in country of test † 	0.064 	-0.042 	0.000 	-0.077 	-0.051 	 	
 	(0.050) 	(0.049) 	(0.046) 	(0.053) 	(0.036) 	 	
Father born in country of test  † 	-0.032 	-0.046 	0.000 	-0.080 	0.062* 	 	
 	(0.050) 	(0.047) 	(0.045) 	(0.054) 	(0.035) 	 	
How often speak language of test at home  	0.035* 	 	-0.008 	0.026 	 	0.039** 	
 	(0.021) 	 	(0.020) 	(0.023) 	 	(0.017) 	
Born in country of  test † 	-0.071 	0.059 	-0.041 	 	-0.053 	 	
 	(0.053) 	(0.055) 	(0.046) 	 	(0.045) 	 	
Gender 	-0.009 	-0.007 	0.002 	0.035 	 	0.020 	
 	(0.029) 	(0.025) 	(0.025) 	(0.029) 	 	(0.016) 	
# people living at home 	0.012 	0.008 	 	 	-0.010* 	-0.004 	
 	(0.011) 	(0.011) 	 	 	(0.006) 	(0.005) 	
Education mother 	 	-0.003 	 	-0.006 	 	 	
 	 	(0.007) 	 	(0.009) 	 	 	
Education father 	 	-0.009 	 	-0.014 	 	 	
 	 	(0.007) 	 	(0.008) 	 	 	
Lives with mother 	 	 	 	 	-0.038 	-0.012 	
 	 	 	 	 	(0.057) 	(0.039) 	
Lives with father 	 	 	 	 	0.037 	-0.003 	
 	 	 	 	 	(0.034) 	(0.022) 	

Notes: All estimates are from pooled regressions. All regressions are population-weighted and contain country fixed-effects and controls for month of birth. Robust standard errors in parenthesis. Variables with a ‘†’ have 2 values: 1=yes and 2=no.  *indicates a p-value <0.10 , **indicates a p-value <0.05 , ***indicates a p-value <0.01 

 
6.2 Main estimation results: main effects of relative age on test scores  
Table 5 provide the estimation results from equation (3) in the previous section. These are the estimates from a pooled regression (but separately by grade) of the effect of relative age on test results (the relative age effect). Table 4 also provides estimates of the relative age effect when the model defined by equation (3) is estimated separately by the tracking regime. 
Table 5: Grade 4 estimates of the effect of relative age on test scores 

 	 	Math 	 	Science 	 	
TIMSS study 	Sample 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	Whole sample 	0.665*** 	0.164*** 	-0.026*** 	0.247*** 	0.665*** 	0.174*** 	-0.016* 	0.260*** 	
 	 	(0.007) 	(0.011) 	(0.006) 	(0.017) 	(0.007) 	(0.011) 	(0.006) 	(0.016) 	
 	Early tracking (T=1) 	0.670*** 	0.155*** 	-0.086*** 	0.232*** 	0.670*** 	0.166*** 	-0.039*** 	0.248*** 	
 	 	(0.010) 	(0.016) 	(0.012) 	(0.025) 	(0.010) 	(0.016) 	(0.012) 	(0.024) 	
 	Late tracking (T=0) 	0.698*** 	0.180*** 	-0.008 	0.257*** 	0.698*** 	0.178*** 	-0.012 	0.254*** 	
 	 	(0.009) 	(0.015) 	(0.007) 	(0.022) 	(0.009) 	(0.015) 	(0.007) 	(0.022) 	
2003 	Whole sample 	0.552*** 	0.135*** 	-0.047*** 	0.247*** 	0.552*** 	0.143*** 	-0.01 	0.263*** 	
 	 	(0.007) 	(0.013) 	(0.01) 	(0.024) 	(0.007) 	(0.013) 	(0.009) 	(0.024) 	
 	Early tracking (T=1) 	0.576*** 	0.071*** 	-0.185*** 	0.123*** 	0.576*** 	0.116*** 	-0.119*** 	0.201*** 	
 	 	(0.017) 	(0.026) 	(0.017) 	(0.046) 	(0.017) 	(0.027) 	(0.016) 	(0.047) 	
 	Late tracking (T=0) 	0.548*** 	0.148*** 	0.009 	0.273*** 	0.548*** 	0.155*** 	0.037** 	0.288*** 	
 	 	(0.009) 	(0.015) 	(0.012) 	(0.028) 	(0.009) 	(0.015) 	(0.011) 	(0.029) 	
2007 	Whole sample 	0.578*** 	0.167*** 	-0.027* 	0.289*** 	0.578*** 	0.168*** 	-0.003 	0.291*** 	
 	 	(0.006) 	(0.011) 	(0.011) 	(0.020) 	(0.006) 	(0.011) 	(0.011) 	(0.02) 	
 	Early tracking (T=1) 	0.549*** 	0.147*** 	-0.13*** 	0.268*** 	0.549*** 	0.142*** 	-0.106*** 	0.259*** 	
 	 	(0.012) 	(0.019) 	(0.018) 	(0.035) 	(0.012) 	(0.018) 	(0.018) 	(0.034) 	
 	Late tracking (T=0) 	0.569*** 	0.167*** 	0.064*** 	0.294*** 	0.569*** 	0.170*** 	0.091*** 	0.300*** 	
 	 	(0.008) 	(0.015) 	(0.013) 	(0.027) 	(0.008) 	(0.015) 	(0.013) 	(0.027) 	

Notes: All estimates are from pooled regressions. The column marked with ‘FS’ provides estimates of the parameter    of equation (2). The column marked with ‘IV’ provides estimates of the parameter    of equation (4). The column marked with ‘RF’ provides estimates of the reduced form equation. The column marked with ‘OLS’ provides Ordinary Least Squares estimates.  All regressions are population-weighted and contain country fixed-effects and controls for birth months. Robust standard errors in parenthesis. *indicates a p-value <0.10 , ** indicates a p-value <0.05 , ***indicates a p-value <0.01 

 
Table 6: Grade 8 estimates of the effect of relative age on test scores 

 	 	Math 	 	Science 	 	
TIMSS study 	Sample 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	Whole sample 	0.590*** 	0.105*** 	-0.202*** 	0.178*** 	0.590*** 	0.105*** 	-0.176*** 	0.178*** 	
 	 	(0.006) 	(0.008) 	(0.005) 	(0.015) 	(0.006) 	(0.008) 	(0.004) 	(0.014) 	
 	Early tracking (T=1) 	0.606*** 	0.081*** 	-0.271*** 	0.133*** 	0.606*** 	0.095*** 	-0.216*** 	0.156*** 	
 	 	(0.010) 	(0.015) 	(0.008) 	(0.025) 	(0.010) 	(0.014) 	(0.008) 	(0.024) 	
 	Late tracking (T=0) 	0.568*** 	0.114*** 	-0.168*** 	0.201*** 	0.568*** 	0.107*** 	-0.156*** 	0.190*** 	
 	 	(0.007) 	(0.01) 	(0.005) 	(0.019) 	(0.007) 	(0.01) 	(0.005) 	(0.018) 	
2003 	Whole sample 	0.605*** 	0.097*** 	-0.161*** 	0.161*** 	0.605*** 	0.097*** 	-0.110*** 	0.160*** 	
 	 	(0.007) 	(0.011) 	(0.007) 	(0.019) 	(0.007) 	(0.011) 	(0.007) 	(0.019) 	
 	Early tracking (T=1) 	0.716*** 	0.06*** 	-0.285*** 	0.084*** 	0.716*** 	0.068*** 	-0.216*** 	0.095*** 	
 	 	(0.014) 	(0.023) 	(0.015) 	(0.033) 	(0.014) 	(0.023) 	(0.014) 	(0.032) 	
 	Late tracking (T=0) 	0.573*** 	0.109*** 	-0.119*** 	0.191*** 	0.573*** 	0.107*** 	-0.073*** 	0.186*** 	
 	 	(0.008) 	(0.013) 	(0.008) 	(0.024) 	(0.008) 	(0.013) 	(0.008) 	(0.023) 	
2007 	Whole sample 	0.545*** 	0.09*** 	-0.133*** 	0.165*** 	0.545*** 	0.095*** 	-0.113*** 	0.174*** 	
 	 	(0.007) 	(0.012) 	(0.008) 	(0.023) 	(0.007) 	(0.012) 	(0.008) 	(0.023) 	
 	Early tracking (T=1) 	0.573*** 	0.103*** 	-0.188*** 	0.180*** 	0.573*** 	0.098*** 	-0.199*** 	0.171*** 	
 	 	(0.016) 	(0.025) 	(0.016) 	(0.045) 	(0.016) 	(0.026) 	(0.016) 	(0.046) 	
 	Late tracking (T=0) 	0.531*** 	0.087*** 	-0.126*** 	0.164*** 	0.531*** 	0.095*** 	-0.095*** 	0.179*** 	
 	 	(0.009) 	(0.014) 	(0.009) 	(0.027) 	(0.009) 	(0.014) 	(0.009) 	(0.027) 	

Notes: All estimates are from pooled regressions. The column marked with ‘FS’ provides estimates of the parameter    of equation (2). The column marked with ‘IV’ provides estimates of the parameter    of equation (4). The column marked with ‘RF’ provides estimates of the reduced form equation. The column marked with ‘OLS’ provides Ordinary Least Squares estimates.  All regressions are population-weighted and contain country fixed-effects and controls for birth months. Robust standard errors in parenthesis. *indicates a p-value <0.10 , ** indicates a p-value <0.05 , ***indicates a p-value <0.01 
 
If we compare the OLS estimates with the reduced form (RF) and instrumental variable (IV) estimate we note that the OLS estimates are significant and negative in almost all cases. This is in line with the results from previous research by both Bedard and Dhuey (2006) and Cascio and Schanzenbach (2012). The fact that the OLS estimates are lower than their IV counterparts could be an indication that students that are older by actual age (and not predicted age) are negatively selected (Cascio and Schanzenbach (2012)).  
Looking at the reduced form (RF) estimates we conclude that relative age has a positive effect, with point estimates ranging from about 0.07-0.11 points. This implies that the youngest students in class (for which R = 0) score about 1-1.2 points lower on a standardized test with mean equal to 50 and standard deviation equal to 10 compared to the oldest students (for which R = 11). This implies a gain of about 3 – 7 percentiles in the distribution of test scores depending on the grade and subject of the test.  
The IV estimates are in general larger than the reduced form estimates and this is especially the case when the estimation sample is reduced to the late-tracking countries or when the whole sample is used. The implied gains in terms of the test score distribution range from 5 to 12 percentiles depending on the estimation sample.  
The IV estimates (for the whole sample of 1995) reported in table 5 and 6 are in general quite close to those reported in Bedard and Dhuey (2006). At the fourth grade level they find an IV estimate of 0.245 for mathematics and an estimate of 0.263 for science. At the eighth grade level they find an IV estimate of 0.163 for mathematics and an estimate of 0.159 for science. However, the reduced form and first-stage estimates in table 5 and 6 are smaller compared to those reported in Bedard and Dhuey (2006). Bedard and Dhuey (2006) report reduced form estimates that are closer to the IV estimates (but the reduced form estimates are still smaller than the IV estimates). More specifically, at the fourth (eighth) grade level they find a reduced form estimate of 0.213 (0.133) for mathematics and an estimate of 0.228 (0.129) for science. Bedard and Dhuey (2006) report first-stage estimates of about 0.869 at the fourth grade level and 0.814 at the eighth grade level.  
There are a couple of possible explanations for these observed differences. Firstly, Bedard and Dhuey (2006) impute missing data in their dataset, while I do not.  Secondly, they do not make a distinction between early-tracking and late-tracking countries such that they have more observations in their dataset. In this thesis however, countries for which information about tracking is not available are not included in the dataset. Thirdly, in Bedard and Dhuey (2006) the grade 8 data comes from both the 1995 TIMSS study and the 1999 TIMSS study, whereas I only use the 1995 TIMSS data for grade 8.  
The results of tables 5 and 6 provide preliminary evidence that early-tracking may be leads to a decrease in the inequality between older and younger students because students in grade 8 of early-tracking countries tend to have a lower relative age effect in 1995 and 2003. Note that this is not in line with the hypothesis formulated in section 5, instead it appears that the effect runs in the opposite direction. In grade 4 one would expect that the relative age effect would be roughly similar in early-tracking countries and late-tracking countries. Except for the IV estimates for math in 2003, it indeed 
appears to be the case that the relative age effect is roughly similar in early-tracking countries and late-tracking countries.  
In order to compare the implications of the estimated relative age effects, table 7 summarizes the implied gains in terms of percentiles of the test score distribution for the different samples and TIMSS studies.  
Table 7: Percentile gains in the test score distribution due to relative age effects 

 	 	Grade 4 	Grade 8 	
 	 	Math 	Science 	Math 	Science 	
TIMSS study 	Regression 	RF 	IV 	RF 	IV 	RF 	IV 	RF 	IV 	
1995 	Whole sample 	6 	10 	6 	10 	4 	6 	4 	7 	
 	Early tracking (T=1) 	6 	9 	7 	10 	3 	5 	4 	6 	
 	Late tracking (T=0) 	5 	10 	5 	10 	4 	7 	4 	7 	
2003 	Whole sample 	5 	10 	6 	10 	4 	6 	4 	7 	
 	Early tracking (T=1) 	5 	8 	6 	11 	3 	5 	3 	5 	
 	Late tracking (T=0) 	5 	10 	6 	11 	4 	7 	4 	9 	
2007 	Whole sample 	6 	11 	7 	12 	4 	7 	4 	7 	
 	Early tracking (T=1) 	6 	11 	6 	11 	4 	8 	4 	8 	
 	Late tracking (T=0) 	7 	11 	7 	12 	4 	6 	4 	7 	

Notes: In order to determine the gain in percentiles of the test score distribution, the median (50th percentile) was used as a base-case scenario. For example, for the 1995 TIMSS study in grade 4, the value of ‘9’ for math in the column denoted by IV, indicates that a student that initially is in the 50th percentile, gains 9 percentiles (due to the relative age effect) and thus ends up in the 59th percentile.  
 
The results from table 7 indicate that the relative age effect appears to be stronger at the fourth grade level compared with the eighth grade level. This result is comparable to the result found by Bedard and Dhuey (2006) for the 1995 and 1999 TIMSS studies. Furthermore, table 4 indicates that this result is also stable over time and also does not vary with the tracking regime.  
One possible explanation for this could be that relative age differences are more important at young ages because at young ages relatively older students will be more mature and hence score higher on achievement tests, independent of program placement (Bedard and Dhuey (2006)).  
Another way of formulating this explanation is that the relative age differences identified by the instrument decline with age: a difference of eleven months in age is not very large when students are thirteen years old compared to the situation when they are eight years old, for example. This argument can be summarized by the following relation:                                                                           
The same difference in relative age (denoted by               ) is more ‘valuable’ at young ages because it represents a greater difference in maturity when evaluated relative to actual age.  
Another interesting result is the fact that, in grade 8 of the 2003 TIMSS study, the gain in percentiles of the test score distribution appears to be lower for the early tracking regime compared with the late tracking regime. This result does not hold for the other TIMSS studies of 1995 and 2007.  This may be an indication that early tracking is associated with a decline in the relative age effect at higher grades but this association does not appear to be very stable.  
6.3 Main Results: Interaction effect between early tracking and relative age 
Next we examine whether the relative age effect that was just estimated differs significantly across the tracking regime. Table 8 provides the results from the model defined by equations (5) – (8).  
Table 8: estimates of the interaction effect between tracking and relative age 

 	 	Math 	 	Science 	 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4 	0.661*** 	-0.006 	-0.007 	-0.012 	0.661*** 	0.000 	0.022* 	-0.003 	
 	 	(0.009) 	(0.021) 	(0.009) 	(0.030) 	(0.009) 	(0.020) 	(0.008) 	(0.029) 	
 	8 	0.585*** 	-0.035** 	-0.060*** 	-0.062** 	0.585*** 	-0.012 	-0.041*** 	-0.026 	
 	 	(0.007) 	(0.017) 	(0.006) 	(0.029) 	(0.007) 	(0.016) 	(0.006) 	(0.028) 	
2003 	4 	0.543*** 	-0.028 	-0.167*** 	-0.057 	0.543*** 	-0.031 	-0.150*** 	-0.063 	
 	 	(0.008) 	(0.027) 	(0.019) 	(0.040) 	(0.008) 	(0.027) 	(0.018) 	(0.040) 	
 	8 	0.571*** 	-0.047* 	-0.178*** 	-0.098*** 	0.571*** 	-0.045* 	-0.164*** 	-0.094*** 	
 	 	(0.008) 	(0.024) 	(0.016) 	(0.035) 	(0.008) 	(0.024) 	(0.016) 	(0.034) 	
2007 	4 	0.574*** 	-0.018 	-0.200*** 	-0.037 	0.574*** 	-0.024 	-0.208*** 	-0.047 	
 	 	(0.008) 	(0.023) 	(0.022) 	(0.042) 	(0.008) 	(0.022) 	(0.022) 	(0.041) 	
 	8 	0.541*** 	0.017 	-0.026 	0.024 	0.541*** 	0.004 	-0.072*** 	0.000 	
 	 	(0.008) 	(0.027) 	(0.019) 	(0.049) 	(0.008) 	(0.028) 	(0.018) 	(0.049) 	

Notes: All estimates are from pooled regressions. The column marked with ‘FS’ provides estimates of the parameter    of equation (6). The column marked with ‘IV’ provides estimates of the parameter    of equation (8). The column marked with ‘RF’ provides estimates of the reduced form equation. The column marked with ‘OLS’ provides Ordinary Least Squares estimates.  All regressions are population-weighted and contain country fixed-effects and controls for birth months. Robust standard errors in parenthesis. *indicates a p-value <0.10 , ** indicates a p-value <0.05 , ***indicates a p-value <0.01 
 
The results from table 8 indicate that the OLS estimate of the interaction effect between tracking and relative age is almost always negative and significant, however the OLS estimate it not a reliable estimate. Most of the reduced form (RF) estimates appear to be negative. For the 1995 and 2003 study, the RF estimates are significantly negative for math. The 2003 TIMSS study is the only study for which the reduced form estimate is significant for science. Looking at the IV estimates it appears that most of the point estimates are negative and this result is also robust across test subjects (math or science) and the grade under consideration. Furthermore, most of the IV estimates are not significantly different from zero but there are a couple of exceptions: the TIMSS studies of 1995 and 2003. In grade 8 of the 2003 TIMSS study the results indicate that for math (science) there is a negative interaction effect of about -0.098 (-0.094).  This implies a loss of about 3-6 percentiles in the distribution of test scores, which is quite large given that the main effect of relative age leads to percentile gains of about 5 to 9 percentiles. For the 1995 TIMSS study only the IV estimate for math is significant and negative. 
The main message of table 8 is that early tracking is not associated with larger relative age effects, because most of the point estimates are negative (some of which are even significantly negative) and none are significantly larger than zero. At the fourth grade level we would expect an interaction effect that is not significantly different from zero, because formally tracking has not yet taken place. Looking at table 6, it appears that the estimation results provide evidence in favor of this expectation.  At the eighth grade level however, it is not clear which sign the interaction effect should have. The negative point estimate for grade 8 in TIMSS 1995 and 2003 could (again) be an indication that early tracking reduces the relative age effect.  
Note however that the estimated interaction effect does not have a causal interpretation but that it merely represents an association or correlation. In the next section I provide another estimate that could potentially approximate the causal effect of tracking on the relative age effect. 
6.4 Main Results: difference-in-difference estimate 
In this section I shall use a difference-in-difference strategy in order to estimate the causal effect of tracking on the relative age effect. Previous studies by Hanushek and Woessman (2006) and Jakubowski (2010) have also applied a difference-in-difference strategy and this section follows this previous literature (in particular Jakubowski (2010)).  
Before the impact of tracking on relative age can be estimated, the assumptions underlying the difference-in-difference methodology need to be tested. The main assumption that is made is the so called ‘common trend’ assumption. The common trend assumption states that the change in the outcome variable for the treatment group, would have been the same as the change in the outcome variable for the control group, had the treatment group not received the treatment. Essentially this implies that the control group serves as a counterfactual. This assumption is very difficult to test in general. This discussion can be summarized by using figure 2 from section 3. Below I reproduce a modified version of figure 2. 
Figure 4: Potential outcome framework for the difference-in-difference strategy 

 	Tracking  = 0 	Tracking = 1 	
Grade 4 
(G = 0) 	                             
 
Scenario (A) 

Scenario (B) 	
Grade 8 
(G = 1) 	                             
 
Scenario (C) 	                            
 
Scenario (D) 

 
The interpretation of all the parameters remain the same except for the parameter      . The parameter       represents the unobservable counterfactual of the parameter      : it is the effect of relative age on test results (in grade 8) in an early-tracking country, had it not applied early tracking (T = 1) but instead applied late tracking (T = 0).  Under the common trend assumption the following relation should hold: 
                                         (12) 
Usually one uses ‘pre-trend’ data in order to check whether the common trend assumption seems plausible. That is, one usually uses data that was collected before the estimation period of the current analysis. For example, in order to test the common trend assumption for the TIMSS 2003 study, one would use data that was collected before the 2003 TIMSS study was conducted. However, due to the fact that the TIMSS studies are conducted every 4 years, there is limited data available that can be used to test the common trend assumption.  The best alternative is to use data from previous TIMSS studies. For example, one could use data from the 1995 TIMSS study in order to check the common trend assumption for the 2003 TIMSS study. For the 2007 TIMSS study one can use data from the 1995 and 2003 TIMSS studies. Note that since the 1995 
TIMSS study was the very first study to be conducted, there isn’t any ‘pre-trend’ data available. 
There is one additional complication that arises in testing the common trend assumption. This has to do with the fact that the tracking policy of a particular country has been in place for a (relatively) long time. As a consequence there are in general no observations for eighth grade students of early tracking countries that have not applied tracking by the eighth grade27.  
27 If these observations do indeed exist, they are probably from a long time ago, such that they may no longer be representative for the current estimation sample.  
Because of the limited data to test the common trend assumption, I prefer to be cautious about the estimated effect of tracking on the relative age effect. Calling the estimated effect a causal effect would be a strong claim and therefore I prefer to think in terms of an ‘approximation of the causal effect’. Therefore, I shall refer to the estimate of this section as the ‘difference-in-difference estimate of the effect of tracking on the relative age effect’ and not as a causal effect.  
Another remark regarding the previous research of Hanushek and Woessman (2006) and  Jakubowski (2010) is in order. They match fourth grade data from TIMSS and PIRLS with data from 15 year-olds from PISA. As Jakubowski (2010) points out, this makes it harder to use the difference-in-difference strategy because of the difference in sampling design between PISA and TIMSS. Recall from section 2 that Jakubowski (2010) is quite critical about the results of Hanushek and Woessman (2006) because according to him Hanushek and Woessman (2006) do not pay sufficient attention this issue. This thesis, however, only uses data from TIMSS such that this issue is only of minor importance. Therefore, the empirical strategy in this section could even be seen as an improvement compared with the empirical strategy of previous studies.  
Now we can proceed to the difference-in-difference estimate of the effect of tracking on the relative age effect, which are provided in table 9. 

 

 
Table 9: difference-in-difference estimate of the effect of tracking on the relative age effect. 

 	Math 	 	Science 	 	
TIMSS study 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	0.666*** 	-0.036** 	-0.081*** 	-0.065** 	0.666*** 	-0.017 	-0.059*** 	-0.035 	
 	(0.007) 	(0.016) 	(0.006) 	(0.028) 	(0.007) 	(0.016) 	(0.006) 	(0.027) 	
2003 	0.579*** 	-0.045* 	-0.171*** 	-0.094*** 	0.579*** 	-0.038* 	-0.160*** 	-0.083** 	
 	(0.007) 	(0.024) 	(0.015) 	(0.035) 	(0.007) 	(0.023) 	(0.014) 	(0.034) 	
2007 	0.561*** 	0.028 	0.052 	0.053 	0.561*** 	0.017 	0.023 	0.035 	
 	(0.007) 	(0.035) 	(0.027) 	(0.062) 	(0.007) 	(0.035) 	(0.026) 	(0.062) 	

Notes: All estimates are from pooled regressions. The column marked with ‘FS’ provides estimates of the coefficient of assigned relative age (R) in the first-stage equation for observed age (A).The column marked with ‘IV’ provides estimates of the parameter    of equation (9). The column marked with ‘RF’ provides estimates of the reduced form equation. The column marked with ‘OLS’ provides Ordinary Least Squares estimates.  All regressions are population-weighted and contain country fixed-effects and controls for birth months. Robust standard errors in parenthesis. *indicates a p-value <0.10 , ** indicates a p-value <0.05 , ***indicates a p-value <0.01.  
 
The results in table 9 indicate that the effect of early tracking on relative age is not constant. In 2003 we find, using the IV estimates, a significant negative effect for both math and science test scores, but in almost all other TIMSS studies the estimated effect is not significantly different from zero. One exception is the estimated effect for math test scores in the 1995 TIMSS study: here we again find a significant negative effect. The estimated change in percentiles of the test score distribution due to early tracking are about 3 – 4 percentiles when evaluated at the eighth grade level. This is again quite substantial given that the gain in percentiles due to the main effect is about 6 – 7 percentiles in grade 8.  
Another observation that can be made from table 9 is that the sign of the IV estimate is not stable: in 1995 and 2003 the IV point estimates are negative (although not always significant) while in 2007 the IV point estimate is positive (but also not significant).  Thus again the estimates do not give clear insight into the relationship between tracking and the relative age effect. The difference in point estimates may be due to the fact that the estimation sample consists of slightly different countries in 2007 compared to 1995 and 2003. In section 8 a sensitivity analysis will be performed in order to check how the results change if we make the sample of countries as comparable to each other as possible.  

 
7 Interpretation of Main Results 
In this section I will attempt to interpret the main results of the previous section but first I will discuss issues regarding the IV-framework and the identification strategy. 
Identification: timing of births 
In previous sections the assumption was made that variation in the assigned relative age (R) was arguably exogenous. Is this a reasonable assumption? In order to answer this question previous studies have generated histograms around cutoff dates and checked whether there were any clear discontinuities present in the data (for example, McEwan and Shapiro (2008)), and these studies find no clear discontinuities.  Others test for randomness by checking for smoothness in the average values of observable covariates such as parental education and race. Bedard and Dhuey (2006) for example, consider whether more highly educated women time births over the calendar year to ensure that their children are naturally older in class. The results of Bedard and Dhuey (2006) indicate that the variation in assigned relative age is arguably exogenous. Using U.S data from 1999 to 2004, Elder and Dickert-Conlin (2010), study whether parents manipulate the timing of births in the neighborhood of state-level school entrance cutoff dates and their analysis suggests that no such timing takes place. However, Gerritsen and Webbink (2013) point out that birth timing around cutoff dates does takes place in Japan. Based on the majority of the evidence from previous research it thus appears that the assumption of exogenous variation around cutoff dates is reasonable but one needs to be cautious because the estimates that use data from Japan may be biased.  
Interpretation of Reduced Form (RF) estimates  
As mentioned before, the estimates represent the combined effect of absolute age, relative age and age at testing on test scores and for convenience this was called the relative age effect. After I estimated the effect of relative age on test scores, I looked whether this relative age effect differs depending on whether countries apply early tracking or late tracking. How can one interpret the coefficients of the reduced form (RF) equations? The estimates of the reduced form effect represent the increase in test scores if the distance of birth month relative to the cutoff date (assigned relative age R) increases by 1 month. Note that this reduced form effect is a combination of different effects. This is true because assigned relative age also affects test scores through grade acceleration, grade retention and late entry28. The TIMSS database only contains records of students who are in a specific grade, but does not contain information on 
28 Note that late entry and grade retention cannot be distinguished from each other in the TIMSS database. 
whether some students skipped grades (grade acceleration) or have been retained. Also, the data does not contain information on the students that did not adhere to the cutoff rules and started school at a later age (late entry). While there is substantial variation in the amount of retention and acceleration across countries, there is a common pattern across assigned relative ages: younger students are more likely to be behind their assigned grade, and older students are more likely to be ahead of their assigned grade (Bedard and Dhuey (2006), McEwan and Shapiro (2008), Dobkin and Ferreira (2010)). Therefore, the reduced form effect also contains the effect of grade retention on test scores. 
Interpretation of Instrumental Variable (IV) estimates and LATE 
The instrumental variables (IV) estimates do not contain the effects of grade acceleration and grade retention/late entry. This is true because the age of students that are observed ahead or behind their assigned grade cannot be (systematically) predicted by assigned relative age (Bedard and Dhuey (2006)). Another important issue that is relevant for the IV estimates is the fact that the IV-framework identifies a local average treatment effect (LATE). The LATE is a local effect because it only applies to a certain subpopulation. This subpopulation is the population of individuals that comply with the treatment induced by the instrument. Therefore, it is important to identify the compliers29. However, within the IV-framework it is not possible to determine whether any individual is a complier. Suppose for example that we observe an individual in the TIMSS database with an assigned relative age (R) equal to a, that is, R = a for this individual. Under perfect compliance the observed age in grade 4 (denoted by A) would equal the predicted age based on R = a (denoted by ĀR = a). Suppose that we indeed observe that A = ĀR = a . Does this mean that this individual is a complier? The answer is no, because we do not know if A = ĀR = b  for a   b since we only observe the case R = a.  
29 Note that there are two stages where compliance is important. The first stage is when parents are making the enrollment decision (whether to comply or not with the cutoff date at the start of kindergarten) and the second stage of compliance refers to the potential grade retention and grade acceleration. 
30 According to calculations by Bedard and Dhuey (2006) the proportion of students who are too young for a particular grade is negligible. Therefore I will not pay a lot of attention to this issue.  
The LATE that is identified by the IV-framework is not representative for the subpopulation of non-compliers, so instead of looking at the compliers, one can also attempt to determine the subpopulation which corresponds to the non-compliers. Following the discussion above, non-compliance implies that A   ĀR = a .In practice this means that non-compliers are those individuals who are either too young for a particular grade or too old for a particular grade.30 The most common reasons for being 
too old for a particular grade are grade retention or late entry into school. Regarding late-entry into education, previous research suggests that non-compliers are students with upper-income, white and highly-educated parents (Deming and Dynarski (2008)). For the U.S states of California and Texas, it appears that white students and students with parents with more than a college degree are mostly likely to delay entry into school; Hispanics and students with parents with less than a college degree are the least likely; girls are also less likely to delay entry compared to boys (Dobkin and Ferreira (2010)). Using nationally representative data for the US, Bassok and Reardon (2013) find that delaying entry into kindergarten is about 3 times more likely for students with a high social economic status (SES) compared with students from a low SES. They also find that early-entry into kindergarten is more likely for students with a low SES compared with students with a high SES. An analysis by Frey (2005) finds that in particular, boys, African American and Hispanic students, and students living in poverty (low SES) are most likely to be retained. For other countries besides the US calculations made by Bedard and Dhuey (2006) show that, some countries such as Norway, England, Iceland and Japan, non-compliance is in general very small. If one generalizes the results of previous research, one may conclude that grade retention is more common for students with a low social economic status (SES) whereas late enrollment is more common among students with a high SES.  As such, the IV estimates would (probably) identify a LATE for students with a SES that lies somewhere in between.  
Sampling bias 
An important note regarding the data that has been used is related to the concept of sampling bias. Recall that TIMSS targets 13-year-olds and choose the highest grade with the most 13-year-olds (grade 8 in most cases). However, there are also 13-year-olds that are in grade 9 and in grade 7. The 13-year-olds that are in grade 9 could have for example started school early or they could have skipped grade. Similarly, 13-year-olds in grade 7 could have been retained. As a consequence of the sampling bias we do not observe all ‘types of 13-year-olds’. Therefore the estimated parameters are not representative for 13-year-olds in general but they should be interpreted conditional on being 13 years old and being in grade 8.31  
31 A similar reasoning also applies for students in grade 4. 
Interpretation and main hypothesis 
The results of the previous section suggest that the effect of tracking on the relative age effect is negative or at least not positive. This result is inconsistent with the hypothesis that was formulated in section 5. This result is quite surprising since one would expect 
that higher test scores would (ceterus paribus) increase the probability of attending a school for high-ability students.  

 

 

 

 

 

 

 

8 Sensitivity Analysis 
In this section a sensitivity analysis will be performed. The purpose of this section is to check how robust the results of section 6 are. The following robustness checks will be performed: 
1. I will check the sensitivity of the main results with respect to the choice of sampling weights 
2. I will check the sensitivity of the main results with respect to specific subsamples. First, I will check if the main results change if one restricts the estimation sample to countries which are also in the estimation sample of Bedard and Dhuey (2006).  Secondly, I will estimate the models of section 6 by gender (that is, separately for boys and girls). Thirdly, I will check how the main results are affected if one estimates the models separately for students which are in the top half of the test score distribution and for students which are in the bottom half of the test score distribution. Fourthly, I will check whether the results change if one holds the sample of countries roughly constant across TIMSS studies. Finally, I will check whether the estimated effect for 1995 changes if one makes a distinction between grade 7 and grade 8. 
8.1 Sensitivity analysis: sampling weights 
The models of section 6 were population weighted.  That is, when estimating the models, sampling weights were used. The main reason for doing so was to make the methodology as comparable to that of Bedard and Dhuey (2006), because they also used sampling weights. However, it is not clear from their paper which sampling weights were used. The TIMSS database provides various sampling weights and a description of some these sampling weights is as follows (TIMSS (2005)): 
“TOTWGT (Total Student Weight) 
TOTWGT is obtained by multiplying the variables WGTFAC1, WGTADJ1, WGTFAC2, WGTADJ2, WGTFAC3, and WGTADJ3 for each student. The sum of these weights within a sample provides an estimate of the size of the population.  
A key property of a sampling weight is that the same population estimates for means and proportions (although not the total or the number of units) will be obtained from any weighting variable that is proportional to the original weight (TOTWGT). For example, the sampling weights for a large country could be divided by a constant to make them smaller, and the weights of a smaller country could be multiplied by a constant to make them bigger. Regardless of 
which constant is used within a country, the weighted estimates of the means and proportions obtained from each of these proportional transformations of the weights will be exactly the same.  
SENWGT (Senate Weight) 
The SENWGT sampling weight is TOTWGT multiplied by 500 divided by the sum of the weights over all students in the target grade in each country. This results in a sample size of 500 in each country. SENWGT may be used in cross-country analyses in which each country should be treated equally. When SENWGT is used as the sampling weight for international estimates, the contribution of each country is the same, regardless of the size of the population. 
HOUWGT (House Weight) 
The HOUWGT sampling weight is TOTWGT multiplied by the ratio of the sample size (the number of students, n) in each country divided by the sum of the weights over all students in the target grade. HOUWGT may be used when the actual sample size is required for performing significance tests. Although some statistical computer software packages allow the sample size to be used as the divisor in the computation of standard errors, others will use the sum of the weights, which results in severely deflated standard errors for the statistics if TOTWGT is used as the weighting variable. HOUWGT is the preferred sampling weight for analyses using such software. Because of the clustering effect in most TIMSS samples, it may also be desirable to apply a correction factor such as a design effect to the HOUWGT variable.” 
Since this thesis is concerned with cross-country analysis, I did not use the sampling weight TOTWGT. Instead the sampling weight HOUWGT was used for estimation in section 6. Table 10 presents the estimation results when the sampling weight SENWGT is used.  
The results in table 10 suggests that the choice of sampling weights does not substantially alter the main results. The estimated main effects in table 10 all have the same sign and are of similar magnitude compared with the estimates presented in section 6. Looking at the interaction effect we do see some differences: now the interaction effect in grade 4 of the 2003 TIMSS Study is significant and negative. The reduced (RF) estimate for math equals -0.054 and the IV estimate equals  -0.091 and both are significant at the 10 % level. Looking at the difference-in-difference estimates there appears to be no substantial changes in the results. The negative interaction effect at the grade 4 level it rather hard to interpret because formally students in grade 4 have not been tracked yet. One possible explanation could be that the interaction effect is 
picking up differences between countries that are not captured by the country fixed-effects.  
Table 10: Sensitivity of results due to sampling weights 

PANEL A: sensitivity of main effects due to sampling weights (here senate weights are used) 	
 	 	Math 	 	Science 	 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4 	0.654*** 	0.164*** 	-0.047*** 	0.251*** 	0.654*** 	0.175*** 	-0.026*** 	0.267*** 	
 	 	(0.007) 	(0.011) 	(0.006) 	(0.017) 	(0.007) 	(0.011) 	(0.006) 	(0.017) 	
 	8 	0.614*** 	0.109*** 	-0.224*** 	0.178*** 	0.614*** 	0.109*** 	-0.185*** 	0.178*** 	
 	 	(0.006) 	(0.009) 	(0.005) 	(0.014) 	(0.006) 	(0.008) 	(0.004) 	(0.014) 	
2003 	4 	0.540*** 	0.129*** 	-0.066*** 	0.242*** 	0.540*** 	0.142*** 	-0.018 	0.266*** 	
 	 	(0.008) 	(0.013) 	(0.010) 	(0.025) 	(0.008) 	(0.013) 	(0.010) 	(0.025) 	
 	8 	0.601*** 	0.099*** 	-0.17*** 	0.164*** 	0.601*** 	0.097*** 	-0.116*** 	0.162*** 	
 	 	(0.007) 	(0.012) 	(0.007) 	(0.02) 	(0.007) 	(0.011) 	(0.007) 	(0.019) 	
2007 	4 	0.565*** 	0.166*** 	-0.029** 	0.294*** 	0.565*** 	0.166*** 	-0.003 	0.294*** 	
 	 	(0.006) 	(0.011) 	(0.010) 	(0.021) 	(0.006) 	(0.011) 	(0.010) 	(0.020) 	
 	8 	0.542*** 	0.091*** 	-0.133*** 	0.168*** 	0.542*** 	0.096*** 	-0.112*** 	0.176*** 	
 	 	(0.008) 	(0.012) 	(0.008) 	(0.023) 	(0.008) 	(0.012) 	(0.008) 	(0.023) 	
 	
PANEL B: sensitivity of the interaction effect due to sampling weights (here senate weights are used) 	
 	 	Math 	 	 	 	Science 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4 	0.679*** 	-0.029 	-0.016 	-0.019 	0.679*** 	-0.006 	0.011 	0.02 	
 	 	(0.008) 	(0.023) 	(0.010) 	(0.038) 	(0.008) 	(0.022) 	(0.009) 	(0.036) 	
 	8 	0.627*** 	-0.043** 	-0.069*** 	-0.061** 	0.627*** 	-0.016 	-0.051*** 	-0.018 	
 	 	(0.006) 	(0.017) 	(0.007) 	(0.029) 	(0.006) 	(0.017) 	(0.006) 	(0.028) 	
2003 	4 	0.539*** 	-0.054* 	-0.213*** 	-0.091** 	0.539*** 	-0.038 	-0.171*** 	-0.066 	
 	 	(0.008) 	(0.028) 	(0.019) 	(0.045) 	(0.008) 	(0.028) 	(0.019) 	(0.046) 	
 	8 	0.572*** 	-0.051** 	-0.196*** 	-0.105*** 	0.572*** 	-0.047** 	-0.169*** 	-0.098*** 	
 	 	(0.008) 	(0.025) 	(0.015) 	(0.037) 	(0.008) 	(0.024) 	(0.015) 	(0.036) 	
2007 	4 	0.562*** 	-0.012 	-0.203*** 	-0.026 	0.562*** 	-0.016 	-0.208*** 	-0.034 	
 	 	(0.008) 	(0.023) 	(0.021) 	(0.043) 	(0.008) 	(0.023) 	(0.021) 	(0.042) 	
 	8 	0.538*** 	0.014 	-0.031 	0.019 	0.538*** 	0.002 	-0.075*** 	-0.002 	
 	 	(0.009) 	(0.027) 	(0.019) 	(0.049) 	(0.009) 	(0.028) 	(0.018) 	(0.050) 	
 	
PANEL C: sensitivity of the difference-in-difference estimate due to sampling weights (here senate weights are used) 	
 	 	Math 	 	 	 	Science 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4-8  	0.652*** 	-0.045*** 	-0.095*** 	-0.064** 	0.652*** 	-0.021 	-0.074*** 	-0.027 	
 	 	(0.007) 	(0.016) 	(0.006) 	(0.028) 	(0.007) 	(0.016) 	(0.006) 	(0.028) 	
2003 	4-8 	0.565*** 	-0.050** 	-0.189*** 	-0.105*** 	0.565*** 	-0.042* 	-0.168*** 	-0.092** 	
 	 	(0.007) 	(0.024) 	(0.014) 	(0.037) 	(0.007) 	(0.023) 	(0.014) 	(0.036) 	
2007 	4-8 	0.550*** 	0.018 	0.055* 	0.035 	0.550*** 	0.009 	0.025 	0.02 	
 	 	(0.008) 	(0.035) 	(0.026) 	(0.063) 	(0.008) 	(0.035) 	(0.026) 	(0.063) 	

Notes: All estimates are from pooled regressions. The interpretation of the estimates in Panel A is similar to the interpretation of Table 5 and Table 6 in section 6, except that another sampling weight (senate weights) have been used. Similarly, the interpretation of the estimates in Panel B is similar to the interpretation of Table 8 in section 6; the interpretation of the estimates in Panel C is similar to the interpretation of Table 9 in section 6. The column marked with ‘FS’ provides estimates of the coefficient of assigned relative age (R) in the first-stage equation for observed age (A). The column marked with ‘IV’ provides Instrumental Variables estimates. The column marked with ‘RF’ provides estimates of the reduced form equation. The column marked with ‘OLS’ provides Ordinary Least Squares estimates. All regressions are population-weighted and contain country fixed-effects and controls for birth months. Robust standard errors in parenthesis. *indicates a p-value <0.10 , ** indicates a p-value <0.05 , ***indicates a p-value <0.01 
8.2 Sensitivity analysis: Sample of Bedard and Dhuey (2006) 
Table 11 shows the estimation results if one restricts the estimation sample to countries that are also in the sample of Bedard and Dhuey (2006). There are a couple of reasons for considering this sample. First, it makes it easier to compare the results obtained here with the results obtained by Bedard and Dhuey (2006).  Secondly, according to Bedard and Dhuey (2006), the countries that they consider are stricter in enforcing the cutoff rules and this is an additional advantage when using the IV framework because it leads to an instrument that has a higher correlation with the endogenous variable.  
From table 11 one can indeed conclude that the first-stage coefficients are higher when the sample of Bedard and Dhuey (2006) is used: the first-stage coefficients now range from about 0.680 to about 0.960 which is higher compared with the ranges reported in section 6. Looking at Panel A of table 11 one can conclude that the main results remain unchanged. The RF and IV estimates all have the same sign and are of comparable magnitude compared with the results reported in section 6. An interesting observation is that the IV and RF effects tend to increase over time at the grade 4 level (but not at the grade 8 level). This was also true for the estimates in section 6 and section 8.1. The results in Panel B do indicate some differences with the main results. Now the interaction effect for math in grade 4 of the 1995 and 2007 TIMSS studies are negative and significant at the 1% level and for the 2007 study, the RF interaction effect at grade 4 is also significant. For grade 8 the results indicate that both the RF and IV the interaction effects for math in 1995 are no longer significant and that the RF and IV interaction effects for science in 2003 is also not significant anymore. The only significant interaction effect at the eighth grade level is for math in 2003 and the IV estimate is now only significant at the 10% level, whereas the RF is now larger and significant at the 5% level. In Panel C, the difference-in-difference estimate for math in 1995 is still negative and significant but now only at the 10% level. For the TIMSS study of 2003, the results indicate that the IV difference-in-difference estimate for science is no longer significant (but the RF still is). The biggest difference with the main results is the difference-in-difference estimate for 2007: both the RF and IV estimates are positive and significant at the 5% level. The IV estimate is very high compared with the estimates for other years, but the standard error is also considerably higher. Another thing to keep in mind is that the Bedard and Dhuey (2006) sample only contains 1 early-tracking country in grade 8 and 3 early-tracking countries in grade 4 of the 2007 study. Therefore, one needs to be cautious when interpreting the results because the difference-in-difference estimate may be picking up something that is country-specific rather than an effect related to early tracking. 
 
Table 11: Estimation results for Bedard and Dhuey(2006) sample 

PANEL A: Main effects for Bedard and Dhuey(2006) sample 	
 	 	Math 	 	Science 	 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4 	0.702*** 	0.149*** 	-0.033*** 	0.209*** 	0.702*** 	0.162*** 	-0.029*** 	0.226*** 	
 	 	(0.009) 	(0.016) 	(0.009) 	(0.023) 	(0.009) 	(0.016) 	(0.009) 	(0.023) 	
 	8 	0.683*** 	0.129*** 	-0.239*** 	0.189*** 	0.683*** 	0.131*** 	-0.21*** 	0.193*** 	
 	 	(0.007) 	(0.012) 	(0.006) 	(0.018) 	(0.007) 	(0.012) 	(0.006) 	(0.017) 	
2003 	4 	0.855*** 	0.237*** 	-0.049** 	0.282*** 	0.855*** 	0.223*** 	0.005 	0.267*** 	
 	 	(0.008) 	(0.022) 	(0.018) 	(0.026) 	(0.008) 	(0.022) 	(0.018) 	(0.026) 	
 	8 	0.819*** 	0.16*** 	-0.199*** 	0.195*** 	0.819*** 	0.143*** 	-0.15*** 	0.174*** 	
 	 	(0.008) 	(0.019) 	(0.015) 	(0.024) 	(0.008) 	(0.019) 	(0.014) 	(0.023) 	
2007 	4 	0.714*** 	0.216*** 	0.001 	0.303*** 	0.714*** 	0.212*** 	0.011 	0.297*** 	
 	 	(0.007) 	(0.015) 	(0.017) 	(0.022) 	(0.007) 	(0.015) 	(0.017) 	(0.022) 	
 	8 	0.841*** 	0.131*** 	-0.094*** 	0.156*** 	0.841*** 	0.127*** 	-0.076*** 	0.151*** 	
 	 	(0.007) 	(0.021) 	(0.016) 	(0.025) 	(0.007) 	(0.020) 	(0.016) 	(0.024) 	
 	
PANEL B: Interaction effect for Bedard and Dhuey(2006) sample 	
 	 	Math 	 	 	 	Science 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4 	0.790*** 	-0.14*** 	-0.059*** 	-0.12 	0.790*** 	-0.055 	-0.019 	0.029 	
 	 	(0.010) 	(0.045) 	(0.017) 	(0.079) 	(0.010) 	(0.040) 	(0.015) 	(0.073) 	
 	8 	0.679*** 	-0.046 	-0.146*** 	-0.068 	0.679*** 	-0.025 	-0.07*** 	-0.038 	
 	 	(0.009) 	(0.028) 	(0.010) 	(0.041) 	(0.009) 	(0.027) 	(0.010) 	(0.040) 	
2003 	4 	0.870*** 	-0.022 	-0.496*** 	-0.001 	0.870*** 	-0.026 	-0.35*** 	-0.009 	
 	 	(0.008) 	(0.054) 	(0.034) 	(0.071) 	(0.008) 	(0.054) 	(0.035) 	(0.070) 	
 	8 	0.846*** 	-0.088** 	-0.369*** 	-0.09* 	0.846*** 	-0.077** 	-0.270*** 	-0.077 	
 	 	(0.008) 	(0.039) 	(0.027) 	(0.051) 	(0.008) 	(0.037) 	(0.027) 	(0.049) 	
2007 	4 	0.814*** 	-0.177*** 	-0.332*** 	-0.142** 	0.814*** 	-0.129*** 	-0.323*** 	-0.068 	
 	 	(0.007) 	(0.038) 	(0.035) 	(0.064) 	(0.007) 	(0.037) 	(0.035) 	(0.062) 	
 	8 	0.961*** 	-0.041 	-0.223*** 	0.106 	0.961*** 	-0.048 	-0.246*** 	0.085 	
 	 	(0.006) 	(0.053) 	(0.035) 	(0.126) 	(0.006) 	(0.053) 	(0.035) 	(0.125) 	
 	
PANEL C: Difference-in-difference estimate for Bedard and Dhuey(2006) sample 	
 	 	Math 	 	 	 	Science 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	 	0.702*** 	-0.050* 	-0.154*** 	-0.075* 	0.702*** 	-0.027 	-0.083*** 	-0.042 	
 	 	(0.009) 	(0.026) 	(0.009) 	(0.039) 	(0.009) 	(0.025) 	(0.010) 	(0.037) 	
2003 	 	0.856*** 	-0.091** 	-0.384*** 	-0.09* 	0.856*** 	-0.076** 	-0.299*** 	-0.073 	
 	 	(0.008) 	(0.038) 	(0.025) 	(0.05) 	(0.008) 	(0.036) 	(0.025) 	(0.048) 	
2007 	 	0.810*** 	0.119** 	-0.101* 	0.267** 	0.810*** 	0.072 	-0.114** 	0.187 	
 	 	(0.007) 	(0.056) 	(0.044) 	(0.123) 	(0.007) 	(0.055) 	(0.043) 	(0.122) 	

Notes: All estimates are from pooled regressions. The interpretation of the estimates in Panel A is similar to the interpretation of Table 5 and Table 6 in section 6, except that the estimation sample is reduced to the countries which are in the sample of Bedard and Dhuey (2006). Similarly, the interpretation of the estimates in Panel B is similar to the interpretation of Table 8 in section 6; the interpretation of the estimates in Panel C is similar to the interpretation of Table 9 in section 6. The column marked with ‘FS’ provides estimates of the coefficient of assigned relative age (R) in the first-stage equation for observed age (A). The column marked with ‘IV’ provides Instrumental Variables estimates. The column marked with ‘RF’ provides estimates of the reduced form equation. The column marked with ‘OLS’ provides Ordinary Least Squares estimates. All regressions are population-weighted and contain country fixed-effects and controls for birth months. Robust standard errors in parenthesis. *indicates a p-value <0.10 , ** indicates a p-value <0.05 , ***indicates a p-value <0.01 

8.3 Sensitivity Analysis: Results by gender 
Some psychologists argue that older children acquire greater “readiness” for learning, and can acquire skills more quickly (Stipek (2002)).  Among young children, even a few months’ difference in age can lead to substantial differences in cognitive and emotional development (Deming and Dynarski (2008). Furthermore, there are a few studies that indicate that boys tend to perform worse than girls on many noncognitive and behavioral dimensions (Bertrand and Pan (2013)). Boys have well documented attention and behavioral difficulties (Beamen, Wheldall, and Kemp 2006; Gilliam 2005; Ready et al. 2005), have lower levels of inhibitory control and perceptual sensitivity (Else-Quest et al. 2006), are less able to delay gratification (Silverman 2003), and are more likely to be diagnosed with attention deficit hyperactivity disorder (see, for example, Szatmari, Offord, and Boyle 1989). To the extent that differences in maturity (induced by differences in relative age) lead students to perform better on tests and to the extent these maturity differences are related to the noncognitive and behavioral dimensions described above, it may be interesting to repeat the analysis of section 6 separately by gender.  
Table 12 provides the relevant estimates for girls and table 13 provides estimates for boys. According to the IV estimates in Panel A of table 12 and table 13, the relative age effect is stronger for boys in 2003 and 2007, whereas the relative age effect is stronger for girls in 1995. The RF estimates however, are pretty close to each other.  Looking at Panel B of table 12, the interaction effect between early tracking and the relative age effect is significantly negative for both grade 4 and grade 8 and also for both math and science (for the 2003 TIMSS study only). For boys, the interaction effect is significantly negative only at the grade 8 level in 1995 and only for math. Thus for boys early tracking is associated with a lower relative age effect in 1995 and for girls early tracking is associated with a lower relative age effect in 2003. This pattern does not change if one looks at the difference-in-difference estimates: for girls the difference-in-difference estimate is significant and negative for the 2003 study (for both RF and IV estimates and for both test subjects) and for boys the same result holds for the 1995 study. In 2007, it appears that the IV difference-in-difference point estimate is larger for boys than for girls, but these estimates are not significantly different from zero. The main message from the analysis is that there does not appear to be a clear relationship between gender and the effect of tracking on the relative age effect: in some cases the effect is negative for girls but not significantly differently from zero for boys and in other cases the opposite is true. Another conclusion is that the effect of tracking on the relative age effect does not appear to be significantly positive for either gender and this is consistent with the results from section 6. 
Table 12: Estimation results when sample is restricted to females/ girls. 

PANEL A: Main effects by gender (girls) 	
 	 	Math 	 	Science 	 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4 	0.678*** 	0.178*** 	-0.017 	0.263*** 	0.678*** 	0.176*** 	-0.012 	0.262*** 	
 	 	(0.009) 	(0.014) 	(0.009) 	(0.022) 	(0.009) 	(0.014) 	(0.009) 	(0.022) 	
 	8 	0.605*** 	0.11*** 	-0.191*** 	0.182*** 	0.605*** 	0.105*** 	-0.17*** 	0.177*** 	
 	 	(0.007) 	(0.012) 	(0.007) 	(0.020) 	(0.007) 	(0.011) 	(0.006) 	(0.019) 	
2003 	4 	0.560 	0.129*** 	-0.024 	0.231*** 	0.560 	0.143*** 	0.011 	0.256*** 	
 	 	(0.010) 	(0.018) 	(0.014) 	(0.032) 	(0.010) 	(0.018) 	(0.013) 	(0.032) 	
 	8 	0.600*** 	0.102*** 	-0.159*** 	0.171*** 	0.600*** 	0.103*** 	-0.106*** 	0.172*** 	
 	 	(0.009) 	(0.015) 	(0.010) 	(0.026) 	(0.009) 	(0.015) 	(0.010) 	(0.026) 	
2007 	4 	0.592*** 	0.168*** 	-0.014) 	0.283*** 	0.592*** 	0.167*** 	0.008 	0.282*** 	
 	 	(0.008) 	(0.016) 	(0.016 	(0.027) 	(0.008) 	(0.016) 	(0.015) 	(0.027) 	
 	8 	0.550*** 	0.086*** 	-0.114*** 	0.157*** 	0.550*** 	0.087*** 	-0.106*** 	0.159*** 	
 	 	(0.010) 	(0.017) 	(0.011) 	(0.031) 	(0.010) 	(0.017) 	(0.011) 	(0.030) 	
 	
PANEL B: Interaction effects by gender (girls) 	
 	 	Math 	Science 	
TIMSS study 	 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4 	0.667*** 	-0.028 	-0.023 	-0.048 	0.667*** 	-0.019 	0.005 	-0.036 	
 	 	(0.011) 	(0.027) 	(0.0120 	(0.040) 	(0.011) 	(0.027) 	(0.012) 	(0.039) 	
 	8 	0.602*** 	-0.011 	-0.059*** 	-0.018 	0.602*** 	0.01 	-0.048*** 	0.011 	
 	 	(0.009) 	(0.023) 	(0.009) 	(0.039) 	(0.009) 	(0.022) 	(0.009) 	(0.038) 	
2003 	4 	0.549*** 	-0.076** 	-0.228*** 	-0.122** 	0.549*** 	-0.075** 	-0.188*** 	-0.124** 	
 	 	(0.011) 	(0.037) 	(0.028) 	(0.053) 	(0.011) 	(0.036) 	(0.025) 	(0.053) 	
 	8 	0.555*** 	-0.078** 	-0.205*** 	-0.152*** 	0.555*** 	-0.055* 	-0.187*** 	-0.12*** 	
 	 	(0.011) 	(0.033) 	(0.023) 	(0.047) 	(0.011) 	(0.032) 	(0.022) 	(0.046) 	
2007 	4 	0.566*** 	-0.019 	-0.224*** 	-0.066 	0.566*** 	-0.031 	-0.229*** 	-0.087 	
 	 	(0.010) 	(0.032) 	(0.033) 	(0.056) 	(0.010) 	(0.031) 	(0.032) 	(0.054) 	
 	8 	0.533*** 	-0.019 	-0.035 	-0.052 	0.533*** 	-0.026 	-0.083** 	-0.063 	
 	 	(0.012) 	(0.038) 	(0.027) 	(0.063) 	(0.012) 	(0.037) 	(0.026) 	(0.062) 	
 	
PANEL C: Difference-in-difference estimates by gender (girls) 	
 	 	Math 	Science 	
TIMSS study 	 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	 	0.684*** 	-0.012 	-0.078*** 	-0.02 	0.684*** 	0.010 	-0.063*** 	0.011 	
 	 	(0.009) 	(0.022) 	(0.009) 	(0.038) 	(0.009) 	0.021 	(0.008) 	(0.036) 	
2003 	 	0.584*** 	-0.088*** 	-0.185*** 	-0.166*** 	0.584*** 	-0.066** 	-0.168*** 	-0.134*** 	
 	 	(0.010) 	(0.032) 	(0.021) 	(0.047) 	(0.010) 	0.031 	(0.020) 	(0.045) 	
2007 	 	0.555*** 	-0.001 	0.072 	0.014 	0.555*** 	0.005 	0.036 	0.024 	
 	 	(0.010) 	(0.049) 	(0.040) 	(0.081) 	(0.010) 	0.048 	(0.038) 	(0.080) 	

Notes: All estimates are from pooled regressions. The interpretation of the estimates in Panel A is similar to the interpretation of Table 5 and Table 6 in section 6, except that the estimation sample is reduced to females. Similarly, the interpretation of the estimates in Panel B is similar to the interpretation of Table 8 in section 6; the interpretation of the estimates in Panel C is similar to the interpretation of Table 9 in section 6. The column marked with ‘FS’ provides estimates of the coefficient of assigned relative age (R) in the first-stage equation for observed age (A). The column marked with ‘IV’ provides Instrumental Variables estimates. The column marked with ‘RF’ provides estimates of the reduced form equation. The column marked with ‘OLS’ provides Ordinary Least Squares estimates. All regressions are population-weighted and contain country fixed-effects and controls for birth months. Robust standard errors in parenthesis. *indicates a p-value <0.10 , ** indicates a p-value <0.05 , ***indicates a p-value <0.01 

Table 13: Estimation results when sample is restricted to males/ boys. 

PANEL A: Main effects by gender (boys) 	
 	 	Math 	 	Science 	 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4 	0.652*** 	0.150*** 	-0.033 	0.230*** 	0.652*** 	0.169*** 	-0.02* 	0.256*** 	
 	 	(0.010) 	(0.016) 	(0.009) 	(0.026) 	(0.010) 	(0.016) 	(0.009) 	(0.025) 	
 	8 	0.575*** 	0.099*** 	-0.214*** 	0.172*** 	0.575*** 	0.105*** 	-0.185*** 	0.182*** 	
 	 	(0.008) 	(0.012) 	(0.006) 	(0.022) 	(0.008) 	(0.012) 	(0.006) 	(0.021) 	
2003 	4 	0.544*** 	0.141*** 	-0.065*** 	0.264*** 	0.544*** 	0.146*** 	-0.027* 	0.274*** 	
 	 	(0.011) 	(0.019) 	(0.013) 	(0.035) 	(0.011) 	(0.019) 	(0.013) 	(0.035) 	
 	8 	0.612*** 	0.093*** 	-0.169*** 	0.152*** 	0.612*** 	0.093*** 	-0.122*** 	0.151*** 	
 	 	(0.010) 	(0.017) 	(0.010) 	(0.028) 	(0.010) 	(0.017) 	(0.010) 	(0.027) 	
2007 	4 	0.564*** 	0.166*** 	-0.037* 	0.295*** 	0.564*** 	0.169*** 	-0.013 	0.300*** 	
 	 	(0.009) 	(0.016) 	(0.015) 	(0.029) 	(0.009) 	(0.016) 	(0.015) 	(0.029) 	
 	8 	0.540*** 	0.094*** 	-0.151*** 	0.174*** 	0.540*** 	0.103*** 	-0.122*** 	0.190*** 	
 	 	(0.011) 	(0.018) 	(0.011) 	(0.033) 	(0.011) 	(0.018) 	(0.011) 	(0.033) 	
 	
PANEL B: Interaction effects by gender (boys) 	
 	 	Math 	Science 	
TIMSS study 	 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4 	0.654*** 	0.016 	0.005 	0.025 	0.654*** 	0.019 	0.032** 	0.029 	
 	 	(0.013) 	(0.031) 	(0.013) 	(0.045) 	(0.013) 	(0.030) 	(0.012) 	(0.043) 	
 	8 	0.568*** 	-0.056** 	-0.063*** 	-0.103** 	0.568*** 	-0.033 	-0.037*** 	-0.063 	
 	 	(0.010) 	(0.024) 	(0.009) 	(0.042) 	(0.010) 	(0.024) 	(0.009) 	(0.041) 	
2003 	4 	0.535*** 	0.014 	-0.116*** 	0.003 	0.535*** 	0.012 	-0.123*** 	-0.002 	
 	 	(0.012) 	(0.039) 	(0.026) 	(0.059) 	(0.012) 	(0.039) 	(0.026) 	(0.059) 	
 	8 	0.590*** 	-0.01 	-0.156*** 	-0.033 	0.590*** 	-0.026 	-0.148*** 	-0.056 	
 	 	(0.011) 	(0.035) 	(0.022) 	(0.052) 	(0.011) 	(0.034) 	(0.022) 	(0.050) 	
2007 	4 	0.583*** 	-0.017 	-0.19*** 	-0.006 	0.583*** 	-0.015 	-0.204*** 	-0.001 	
 	 	(0.011) 	(0.033) 	(0.030) 	(0.062) 	(0.011) 	(0.032) 	(0.030) 	(0.061) 	
 	8 	0.549*** 	0.051 	-0.023 	0.107 	0.549*** 	0.028 	-0.083*** 	0.065 	
 	 	(0.012) 	(0.040) 	(0.026) 	(0.076) 	(0.012) 	(0.040) 	(0.025) 	(0.077) 	
 	
PANEL C: Difference-in-difference estimates by gender (boys) 	
 	 	Math 	Science 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4-8 	0.646*** 	-0.060** 	-0.085*** 	-0.109*** 	0.646*** 	-0.044* 	-0.058*** 	-0.081** 	
 	 	(0.010) 	(0.023) 	(0.008) 	(0.041) 	(0.010) 	(0.023) 	(0.008) 	(0.040) 	
2003 	4-8 	0.574*** 	0.003 	-0.163*** 	-0.011 	0.574*** 	-0.002 	-0.163*** 	-0.017 	
 	 	(0.011) 	(0.034) 	(0.021) 	(0.053) 	(0.011) 	(0.033) 	(0.020) 	(0.051) 	
2007 	4-8 	0.568*** 	0.053 	0.048 	0.095 	0.568*** 	0.025 	0.009 	0.046 	
 	 	(0.011) 	(0.051) 	(0.036) 	(0.096) 	(0.011) 	(0.051) 	(0.036) 	(0.095) 	

Notes: All estimates are from pooled regressions. The interpretation of the estimates in Panel A is similar to the interpretation of Table 5 and Table 6 in section 6, except that the estimation sample is reduced to males. Similarly, the interpretation of the estimates in Panel B is similar to the interpretation of Table 8 in section 6; the interpretation of the estimates in Panel C is similar to the interpretation of Table 9 in section 6. The column marked with ‘FS’ provides estimates of the coefficient of assigned relative age (R) in the first-stage equation for observed age (A). The column marked with ‘IV’ provides Instrumental Variables estimates. The column marked with ‘RF’ provides estimates of the reduced form equation. The column marked with ‘OLS’ provides Ordinary Least Squares estimates. All regressions are population-weighted and contain country fixed-effects and controls for birth months. Robust standard errors in parenthesis. *indicates a p-value <0.10 , ** indicates a p-value <0.05 , ***indicates a p-value <0.01 
 
8.4 Sensitivity analysis: results for different parts of the test score distribution 
In the formulation of the main hypothesis of this thesis (in section 5) a distinction was made between high-ability and low-ability students. Therefore, it may be interesting to look how the results might change when they are estimated separately for different parts of the test score distribution. Students that have test scores above the median could be seen as a proxy for high-ability students and students that have test scores below the median could be seen as a proxy for low-ability students.32  
32 After determining which students were in the top half of the test score distribution, the test scores were re-standardized to a mean of 50 and a standard deviation of 10. The same was done for students in the bottom half of the test score distribution.  
Table 14 provides estimates for students that are in the 50th percentile or higher and table 15 provides the corresponding estimates for students that are in the 50th percentile or lower.  
From Panel A in table 14 and 15 it appears that the relative age effect is stronger for students in the top half of the test score distribution. Thus is appears that the inequality between older and younger students is larger for high-ability students. Again it appears that the relative age effect is stronger at the fourth grade level compared with the eighth grade level and this result holds independent of the position in the test score distribution.  
Looking at Panel B, the estimates indicate that for students in the top half of the test score distribution, early tracking is associated with a decline in the relative age effect at the fourth grade level in 2007 and an increase in the relative age effect at the eighth grade level in 2007. For students in the bottom half of the test score distribution it appears that early tracking is associated with a decline in the relative age effect at both the fourth grade and eighth grade level, but only for the 2003 TIMSS study.  
Finally, looking at Panel C of table 14, the RF and IV estimates for math show that for students in the top half of the test score distribution the relative age effect only increases but does not decline. However for science, it appears that the effect of early tracking on the relative age effect is negative for TIMSS 2003. For students in the bottom half of the test score distribution the estimates suggests that the inequality between older and younger students does not increase, because the estimates are either not significantly different from zero or negative. For the TIMSS study of 2003 early tracking has a significantly negative effect (both RF and IV estimates) for both math and science. Furthermore, while the RF and IV point estimates for 1995 are not significantly negative, they are smaller compared with the estimates for students in the 
top half of the test score distribution. Similarly for 2007, the increase in the relative age effect for students in the top half of the test score distribution is large, whereas the increase for students in the bottom half of the test score distribution is not significantly different from zero.  
The main message from the analysis is that: (i) when the effect of early tracking on the relative age effect is negative (such as in 1995 and 2003), it appears that students in the top half of the test score distribution experience a smaller decrease in the relative age effect compared with students in the bottom half of the test score distribution; (ii) when the effect of early tracking on the relative age effect is non-negative (such as in 2007), it appears that the students in the top half of the test score distribution experience a larger increase in the relative age effect compared with students in the bottom half of the test score distribution33. 
33 Note that the point estimates of the effect of tracking on the relative age effect were positive. Also remember that when restricted to the Bedard and Dhuey (2006) sample this effect was positive. Thus there is suggestive evidence that the effect in 2007 might be positive. 
Note however that the results presented here are only preliminary. By limiting the observations to students in the top or bottom half of the test score distribution one loses observations such that the estimates are less precise. Also the variation between early-tracking and late-tracking countries also decreases due to this loss of observations. Furthermore, the students were grouped based on the outcome variable of the analysis (test scores) such that the estimates may be biased.  

 

 

 
 
Table 14: Estimation results for students in the top half of the test score distribution 

PANEL A: Main effects by position in test score distribution (above the median) 	
 	 	Math 	 	Science 	 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4 	0.679*** 	0.128*** 	0.018 	0.186*** 	0.679*** 	0.126*** 	0.009 	0.185*** 	
 	 	(0.008) 	(0.017) 	(0.011) 	(0.026) 	(0.008) 	(0.017) 	(0.011) 	(0.026) 	
 	8 	0.613*** 	0.069*** 	-0.097*** 	0.112*** 	0.613*** 	0.069*** 	-0.082*** 	0.116*** 	
 	 	(0.007) 	(0.013) 	(0.007) 	(0.020) 	(0.007) 	(0.012) 	(0.007) 	(0.020) 	
2003 	4 	0.568*** 	0.125*** 	0.028* 	0.222*** 	0.568*** 	0.113*** 	0.048*** 	0.201*** 	
 	 	(0.009) 	(0.019) 	(0.013) 	(0.033) 	(0.009) 	(0.019) 	(0.014) 	(0.034) 	
 	8 	0.606*** 	0.089*** 	-0.056*** 	0.146*** 	0.606*** 	0.09*** 	-0.026* 	0.149*** 	
 	 	(0.008) 	(0.017) 	(0.011) 	(0.029) 	(0.008) 	(0.017) 	(0.011) 	(0.029) 	
2007 	4 	0.589*** 	0.134*** 	0.087*** 	0.223*** 	0.589*** 	0.142*** 	0.097*** 	0.242*** 	
 	 	(0.008) 	(0.017) 	(0.014) 	(0.028) 	(0.008) 	(0.017) 	(0.014) 	(0.028) 	
 	8 	0.559*** 	0.063*** 	-0.006 	0.112*** 	0.559*** 	0.046*** 	-0.005 	0.083*** 	
 	 	(0.009) 	(0.018) 	(0.013) 	(0.032) 	(0.009) 	(0.018) 	(0.012) 	(0.032) 	
 	
PANEL B: Interaction effects by position in test score distribution (above the median) 	
 	 	Math 	Science 	
TIMSS study 	 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4 	0.672*** 	-0.009 	0.000 	-0.028 	0.672*** 	-0.014 	0.004 	-0.024 	
 	 	(0.010) 	(0.033) 	(0.016) 	(0.050) 	(0.010) 	(0.032) 	(0.014) 	(0.048) 	
 	8 	0.604*** 	-0.009 	0.014 	-0.022 	0.604*** 	-0.001 	0.033*** 	-0.01 	
 	 	(0.008) 	(0.023) 	(0.009) 	(0.037) 	(0.008) 	(0.023) 	(0.009) 	(0.038) 	
2003 	4 	0.549*** 	0.01 	-0.115*** 	-0.007 	0.549*** 	0.022 	-0.114*** 	0.014 	
 	 	(0.010) 	(0.040) 	(0.026) 	(0.057) 	(0.010) 	(0.040) 	(0.028) 	(0.058) 	
 	8 	0.569*** 	-0.021 	-0.057* 	-0.06 	0.569*** 	-0.031 	-0.062** 	-0.074 	
 	 	(0.010) 	(0.033) 	(0.022) 	(0.047) 	(0.010) 	(0.034) 	(0.022) 	(0.048) 	
2007 	4 	0.595*** 	-0.06* 	-0.034 	-0.100* 	0.595*** 	-0.067** 	-0.067* 	-0.112* 	
 	 	(0.010) 	(0.034) 	(0.029) 	(0.059) 	(0.010) 	(0.033) 	(0.029) 	(0.059) 	
 	8 	0.547*** 	0.131** 	0.074** 	0.201*** 	0.547*** 	0.047 	-0.041 	0.071 	
 	 	(0.010) 	(0.041) 	(0.027) 	(0.067) 	(0.010) 	(0.041) 	(0.026) 	(0.067) 	
 	
PANEL C: Difference-in-difference estimates by position in test score distribution (above the median) 	
 	 	Math 	Science 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4-8 	0.677*** 	-0.002 	0.011* 	-0.012 	0.677*** 	-0.006 	0.019*** 	-0.02 	
 	 	(0.009) 	(0.013) 	(0.005) 	(0.036) 	(0.009) 	(0.013) 	(0.005) 	(0.037) 	
2003 	4-8 	0.583*** 	-0.01 	-0.013 	-0.059 	0.583*** 	-0.023 	-0.02 	-0.090* 	
 	 	(0.009) 	(0.018) 	(0.011) 	(0.049) 	(0.009) 	(0.018) 	(0.010) 	(0.048) 	
2007 	4-8 	0.577*** 	0.094*** 	0.038* 	0.279*** 	0.577*** 	0.054* 	-0.004 	0.168* 	
 	 	(0.010) 	(0.029) 	(0.019) 	(0.088) 	(0.010) 	(0.028) 	(0.018) 	(0.087) 	

Notes: All estimates are from pooled regressions. The interpretation of the estimates in Panel A is similar to the interpretation of Table 5 and Table 6 in section 6, except that the estimation sample is reduced to students with test scores in the 50th percentile or higher. Similarly, the interpretation of the estimates in Panel B is similar to the interpretation of Table 8 in section 6; the interpretation of the estimates in Panel C is similar to the interpretation of Table 9 in section 6. The column marked with ‘FS’ provides estimates of the coefficient of assigned relative age (R) in the first-stage equation for observed age (A). The column marked with ‘IV’ provides Instrumental Variables estimates. The column marked with ‘RF’ provides estimates of the reduced form equation. The column marked with ‘OLS’ provides Ordinary Least Squares estimates. All regressions are population-weighted and contain country fixed-effects and controls for birth months. Robust standard errors in parenthesis. *indicates a p-value <0.10 , ** indicates a p-value <0.05 , ***indicates a p-value <0.01 
 
Table 15: Estimation results for students in the bottom half of the test score distribution 

PANEL A: Main effects by position in test score distribution (below the median) 	
 	 	Math 	 	Science 	 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4 	0.652*** 	0.096*** 	-0.047*** 	0.153*** 	0.652*** 	0.090*** 	-0.044*** 	0.139*** 	
 	 	(0.011) 	(0.016) 	(0.008) 	(0.025) 	(0.011) 	(0.016) 	(0.009) 	(0.025) 	
 	8 	0.576*** 	0.035*** 	-0.167*** 	0.058** 	0.576*** 	0.053*** 	-0.159*** 	0.092*** 	
 	 	(0.009) 	(0.013) 	(0.006) 	(0.023) 	(0.009) 	(0.013) 	(0.006) 	(0.022) 	
2003 	4 	0.535*** 	0.073*** 	-0.108*** 	0.140*** 	0.535*** 	0.08*** 	-0.045*** 	0.153*** 	
 	 	(0.012) 	(0.019) 	(0.013) 	(0.037) 	(0.012) 	(0.019) 	(0.013) 	(0.037) 	
 	8 	0.604*** 	0.051*** 	-0.172*** 	0.083*** 	0.604*** 	0.04** 	-0.110*** 	0.067** 	
 	 	(0.011) 	(0.017) 	(0.011) 	(0.028) 	(0.011) 	(0.017) 	(0.011) 	(0.028) 	
2007 	4 	0.566*** 	0.097*** 	-0.114*** 	0.175*** 	0.566*** 	0.106*** 	-0.106*** 	0.187*** 	
 	 	(0.009) 	(0.017) 	(0.014) 	(0.031) 	(0.009) 	(0.016) 	(0.014) 	(0.030) 	
 	8 	0.535*** 	0.055*** 	-0.155*** 	0.104*** 	0.535*** 	0.074*** 	-0.138*** 	0.139*** 	
 	 	(0.012) 	(0.018) 	(0.011) 	(0.035) 	(0.012) 	(0.018) 	(0.011) 	(0.035) 	
 	
PANEL B: Interaction effects by position in test score distribution (below the median) 	
 	 	Math 	Science 	
TIMSS study 	 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4 	0.649*** 	0.001 	-0.027* 	0.014 	0.649*** 	0.011 	0.015 	0.016 	
 	 	(0.014) 	(0.030) 	(0.012) 	(0.045) 	(0.014) 	(0.030) 	(0.012) 	(0.044) 	
 	8 	0.575*** 	-0.041 	-0.091*** 	-0.067 	0.575*** 	-0.003 	-0.073*** 	-0.006 	
 	 	(0.011) 	(0.027) 	(0.010) 	(0.048) 	(0.011) 	(0.027) 	(0.010) 	(0.046) 	
2003 	4 	0.530*** 	-0.069* 	-0.094*** 	-0.11* 	0.530*** 	-0.023 	-0.084** 	-0.045 	
 	 	(0.013) 	(0.040) 	(0.026) 	(0.060) 	(0.013) 	(0.041) 	(0.027) 	(0.060) 	
 	8 	0.573*** 	-0.088** 	-0.200*** 	-0.141** 	0.573*** 	-0.06 	-0.181*** 	-0.098* 	
 	 	(0.012) 	0.039) 	(0.024) 	(0.056) 	(0.012) 	(0.038) 	(0.025) 	(0.054) 	
2007 	4 	0.548*** 	-0.038 	-0.206*** 	-0.085 	0.548*** 	-0.031 	-0.210*** 	-0.071 	
 	 	(0.011) 	(0.034) 	(0.027) 	(0.062) 	(0.011) 	(0.033) 	(0.027) 	(0.060) 	
 	8 	0.536*** 	-0.016 	-0.087*** 	-0.028 	0.536*** 	-0.077* 	-0.093*** 	-0.14* 	
 	 	(0.013) 	(0.042) 	(0.026) 	(0.079) 	(0.013) 	(0.041) 	(0.025) 	(0.076) 	
 	
PANEL C: Difference-in-difference estimates by position in test score distribution (below the median) 	
 	 	Math 	Science 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4-8 	0.656*** 	-0.027 	-0.068*** 	-0.068 	0.656*** 	-0.014 	-0.060*** 	-0.036 	
 	 	(0.011) 	(0.017) 	(0.006) 	(0.047) 	(0.011) 	(0.017) 	(0.006) 	(0.044) 	
2003 	4-8 	0.564*** 	-0.061** 	-0.140*** 	-0.143*** 	0.564*** 	-0.044* 	-0.122*** 	-0.089* 	
 	 	(0.011) 	(0.024) 	(0.014) 	(0.054) 	(0.011) 	(0.024) 	(0.015) 	(0.052) 	
2007 	4-8 	0.535*** 	-0.016 	0.022 	-0.03 	0.535*** 	-0.053 	0.031 	-0.125 	
 	 	(0.011) 	(0.034) 	(0.022) 	(0.095) 	(0.011) 	(0.035) 	(0.022) 	(0.092) 	

Notes: All estimates are from pooled regressions. The interpretation of the estimates in Panel A is similar to the interpretation of Table 5 and Table 6 in section 6, except that the estimation sample is reduced to students with test scores in the 50th percentile or lower. Similarly, the interpretation of the estimates in Panel B is similar to the interpretation of Table 8 in section 6; the interpretation of the estimates in Panel C is similar to the interpretation of Table 9 in section 6. The column marked with ‘FS’ provides estimates of the coefficient of assigned relative age (R) in the first-stage equation for observed age (A). The column marked with ‘IV’ provides Instrumental Variables estimates. The column marked with ‘RF’ provides estimates of the reduced form equation. The column marked with ‘OLS’ provides Ordinary Least Squares estimates. All regressions are population-weighted and contain country fixed-effects and controls for birth months. Robust standard errors in parenthesis. *indicates a p-value <0.10 , ** indicates a p-value <0.05 , ***indicates a p-value <0.01 
 
8.5 Sensitivity Analysis: Reducing differences between TIMSS 1995, 2003 and 2007 
The results for TIMSS 1995 and 2003 indicate that early-tracking tends to decrease the inequality between older and younger students. For TIMSS 2007 the point estimates were positive but not significantly different from zero. In this section I investigate whether this difference in results is due to a difference in the composition of countries included in the estimation sample. Table 16 presents difference-in-difference estimates (i) when the sample of TIMSS 1995 is made as comparable as possible to the sample of TIMSS 2007 and (ii) when the sample of TIMSS 2003 is made as comparable as possible to the sample of TIMSS 2007.  
Table 16: difference-in-difference estimates for different samples of countries 

PANEL A: Difference-in-difference estimates when sample of TIMMS 1995 is made similar to sample of TIMMS 2007 	
 	 	Math 	Science 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4-8 	0.646*** 	-0.084*** 	-0.047*** 	-0.147*** 	0.646*** 	-0.068*** 	-0.032*** 	-0.122*** 	
 	 	(0.007) 	(0.025) 	(0.010) 	(0.040) 	(0.007) 	(0.024) 	(0.009) 	(0.039) 	
2007 	4-8 	0.450*** 	-0.015 	-0.003 	-0.021 	0.450*** 	-0.004 	-0.023 	-0.002 	
 	 	(0.009) 	(0.039) 	(0.026) 	(0.071) 	(0.009) 	(0.039) 	(0.025) 	(0.070) 	
 	
PANEL B: Difference-in-difference estimates when sample of TIMMS 2003  is made similar to sample of TIMMS 2007 	
 	 	Math 	Science 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
2003 	4-8 	0.621*** 	-0.069** 	-0.01 	-0.110** 	0.621*** 	-0.052* 	-0.046* 	-0.087** 	
 	 	(0.008) 	(0.032) 	(0.021) 	(0.043) 	(0.008) 	(0.0320 	(0.0190 	(0.042) 	
2007 	4-8 	0.565*** 	-0.048 	-0.004 	-0.101 	0.565*** 	-0.025 	-0.022 	-0.064 	
 	 	(0.008) 	(0.044) 	(0.031) 	(0.071) 	(0.008) 	(0.044) 	(0.031) 	(0.070) 	

Notes: All estimates are from pooled regressions. The column marked with ‘FS’ provides estimates of the coefficient of assigned relative age (R) in the first-stage equation for observed age (A). The column marked with ‘IV’ provides estimates of the parameter    of equation (9). The column marked with ‘RF’ provides estimates of the reduced form equation. The column marked with ‘OLS’ provides Ordinary Least Squares estimates.  All regressions are population-weighted and contain country fixed-effects and controls for birth months. Robust standard errors in parenthesis. *indicates a p-value <0.10 , ** indicates a p-value <0.05 , ***indicates a p-value <0.01.  
According to table 16, the results for TIMSS 1995 and 2003 do not change substantially. For TIMSS 1995 and 2003, early tracking tends to decrease the inequality between older and younger students. For TIMSS 2007, the estimates are still not significantly different from zero but now the point estimates are negative (they were positive in section 6). The main message of table 16 is that early tracking does not increase the inequality between older and younger students. Note however, that in order to make the samples as comparable to each other as possible, the number of observations included in the regressions is reduced. Furthermore, the variation in early-tracking countries is also reduced. From table 16 it appears that TIMSS 2007 is probably most likely to be affected by this loss in observations, since the standard errors are quite large compared to TIMSS 1995 and 2003.  
8.6 Sensitivity Analysis: Splitting grade 8 of TIMSS 1995 
Note that there are no countries in the sample for which the ‘age of first selection’ is strictly equal to 13 years (see table 1 in section 4). As a consequence, observations from grade 7 of TIMSS 1995 also can be interpreted as ‘post-treatment’ observations, in the sense that some countries have applied early educational tracking by then and others have not. Up until now, I have used the definition of Bedard and Dhuey (2006) in the sense that their ‘grade 8’ sample consists of observations from both grade 7 and grade 8. If might be interesting to see how the results change if we treat grade 7 and grade 8 separately.  
Table 17 presents the results for the difference-in-difference estimate of the effect of early tracking on the relative age effect in 4 scenarios: (i) for grade 4 to grade 7 for the 1995 TIMSS sample of section 6; (ii) presented for grade 4 to grade 8 for the 1995 TIMSS sample of section 6, (iii) for grade 4 to grade 7 for the 1995 TIMSS sample of table 16 Panel A and (iv) for grade 4 to grade 8 for the 1995 TIMSS sample of table 16 Panel A.  
Table 17: difference-in-difference estimates for grade 7 and grade 8 of TIMSS 1995 

PANEL A: Difference-in-difference estimates for grade 4 to grade 7 	
 	 	Math 	Science 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4-7 	0.665*** 	-0.047** 	-0.114 	-0.083** 	0.665*** 	-0.024 	-0.079 	-0.045 	
 	 	(0.007) 	(0.021) 	(0.011) 	(0.036) 	(0.007) 	(0.020) 	(0.011) 	(0.035) 	
 	
PANEL B: Difference-in-difference estimates for grade 4 to grade 8 	
 	 	Math 	Science 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4-8 	0.667*** 	-0.023 	-0.191*** 	-0.045 	0.667*** 	-0.013 	-0.157*** 	-0.032 	
 	 	(0.007) 	(0.024) 	(0.012) 	(0.041) 	(0.007) 	(0.024) 	(0.012) 	(0.040) 	
 	
PANEL C: Difference-in-difference estimates for grade 4 to grade 7 (for TIMSS 1995 sample of table 16 PANEL A) 	
 	 	Math 	Science 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4-7 	0.645*** 	-0.087** 	-0.087 	-0.153*** 	0.645*** 	-0.049 	-0.074 	-0.091* 	
 	 	(0.007) 	(0.037) 	(0.021) 	(0.058) 	(0.007) 	(0.035) 	(0.020) 	(0.054) 	
 	
PANEL D: Difference-in-difference estimates for grade 4 to grade 8 (for TIMSS 1995 sample of table 16 PANEL A) 	
 	 	Math 	Science 	
TIMSS study 	Grade 	FS 	RF 	OLS 	IV 	FS 	RF 	OLS 	IV 	
1995 	4-8 	0.649*** 	-0.085** 	-0.188*** 	-0.145*** 	0.649*** 	-0.097*** 	-0.192*** 	-0.165*** 	
 	 	(0.007) 	(0.035) 	(0.018) 	(0.055) 	(0.007) 	(0.034) 	(0.017) 	(0.054) 	

Notes: All estimates are from pooled regressions. The column marked with ‘FS’ provides estimates of the coefficient of assigned relative age (R) in the first-stage equation for observed age (A). The column marked with ‘IV’ provides estimates of the parameter    of equation (9). The column marked with ‘RF’ provides estimates of the reduced form equation. The column marked with ‘OLS’ provides Ordinary Least Squares estimates.  All regressions are population-weighted and contain country fixed-effects and controls for birth months. Robust standard errors in parenthesis. *indicates a p-value <0.10 , ** indicates a p-value <0.05 , ***indicates a p-value <0.01.  
The results suggest that the negative effect of early tracking that was identified in section 6 is driven primarily by the change between grade 4 and grade 7. Both the RF and IV estimate for math are significantly negative at the 5% level. Notice however, that the point estimates for the difference-in-difference estimates for grade 4 to grade 8 are negative (although not significant). When the analysis is repeated for the subset of countries that is most similar to the sample of countries of TIMSS 2007 (see table 16, panel A) then the effect of early tracking on the relative age effect is significantly negative for grade 4 to grade 7 and for grade 4 to grade 8 (for both math and science). The results in this section thus indicate that early tracking tends to decrease the inequality between relatively older and relatively younger students for both (i) grade 4 to grade 7 and (ii) grade 4 to grade 8. An interesting question that arises is: What would the effect of early tracking be for grade 4 to grade 7 in TIMSS 2003 and 2007? Would one find the same results? Unfortunately, there is no data available in order to answer this question.  

 

 

 

 

 
9 Conclusion and Discussion 
This thesis has studied relative age effects in a cross-country setting. The findings indicate that relative age effect is positive at both the fourth and eighth grade level.  Furthermore, this relative age effect is stronger at the fourth grade level than at the eighth grade level. The implied gains in the test score distribution due to the relative age effect is about 7-10 percentiles in the fourth grade and 5-7 percentiles in the eighth grade. These results are consistent with previous research. The main findings for TIMSS 1995 and 2003 suggest that early tracking tends to decrease the inequality between older and younger students that arises due to differences in relative age. However, the point estimates for the sample of TIMSS 2007 where positive and not significantly different from zero. The implied loss in the test score distribution due to the negative effect of early tracking is about 3-4 percentiles which is quite large when compared with the main effect of relative age. Thus the evidence presented in this thesis does not support the hypothesis that early tracking would increase the relative age effect. A sensitivity analysis finds that the effect of early tracking on the relative age effect does not have a clear relationship with gender. The results of the sensitivity analysis however do provide suggestive evidence that the decrease in inequality between older and younger students is smaller for high-ability students compared to low-ability students. More precisely, I found that (i) when the effect of early tracking on the relative age effect is negative, students in the top half of the test score distribution experience a smaller decrease in the relative age effect compared with students in the bottom half of the test score distribution; (ii) when the effect of early tracking on the relative age effect is non-negative, students in the top half of the test score distribution experience a larger increase in the relative age effect compared with students in the bottom half of the test score distribution. However, these results may be biased since the main outcome variable was used in order to determine which students are in the top half of the test score distribution and which students are in the bottom half of the test score distribution. Finally, when the sample of countries for different TIMSS studies are made as comparable to each other a possible, the findings again indicate that the effect of early tracking on the relative age effect is negative.  
The results presented in this thesis may be difficult to interpret because it is difficult to pin down the subpopulation for which the LATE is identified. Note furthermore that the cross-country nature of this study makes it ever harder to pin down this subpopulation. In some countries non-compliance is very low but in other countries this is not the case. If one generalizes the results from previous research, one may tentatively conclude that grade retention is more common for students with a low social economic status (SES) whereas late enrollment is more common among students with a high SES. As such, the IV estimates would identify a LATE for students with a SES that lies 
somewhere in between. Also due to limited data it is very hard to test the common trend assumption underlying the difference-in-difference methodology. As a consequence, stating that the effect of early tracking on the combined age effect is a causal effect, would be a very strong claim.   
Note that the analysis in this thesis only considers test scores for mathematics and science. One could argue that the required skills for mathematics and science are different compared to the skills for other subjects such as for example reading comprehension. One possible implication is that the effect on tracking on the relative age effect may depend on the subject that is being considered. According to Pekkarinen (2008) several studies such as the PISA have shown that girls have an advantage in reading skills as compared to boys. Furthermore, Pekkarinen (2008) also states that mathematics and science are typically fields where boys tend to dominate girls. In this thesis the effect of early tracking on the relative age effect did not have a clear relationship with gender. But taking the comments of Pekkarinen (2008) into account, using another outcome variable such as reading test scores may lead to different results.  
Another remark regarding the estimated effects and the empirical framework is in order. The difference-in-difference design employed in this thesis compared the test scores of eighth-year-olds with those of thirteen-year-olds, as a consequence students (in grade 8) were only exposed to an early-tracking system for about 1 or 2 years.  One could argue that the effects of early tracking are not immediate but that it takes time for these effects to take effect. One solution to this issue is to for example use test scores of fifteen-year-olds taken from PISA. But this makes a difference-in-difference comparison more complicated because of the difference in sampling design between PISA and TIMSS (recall the discussion about the Jakubowski (2010) paper).   
What are the possible implications for society? According to the results presented, being relatively older increases test scores. Therefore, parents could have an incentive to hold their children back (‘red-shirt’ their children) and delay enrollment into kindergarten in order to give their children an advantage over other children. There is evidence that this is already occurring, despite the fact that the long-term benefits of red-shirting are very small or even zero (Deming and Dynarski (2008), Black et al. (2011)).  Furthermore, it appears that families with a high SES are more likely to red-shirt children and families with a low SES are more likely to comply with the cutoff rules. Therefore, students from a low SES will be disproportionally younger and hence benefit less from relative effects such that the (short-term) gap between high and low SES could increase. To the extent that an early tracking system reduces the relative age effect, the incentives for red-shirting would be weaker and this could prevent the gap between high and low SES from increasing further.  
A final remark regarding the analysis of this thesis is in order. The results presented here may depend on the countries that are included in the analysis. Information regarding cutoff dates and ‘age of tracking’ is not available for all countries, and by including more countries in the analysis the results might change. Based on the countries included in the analysis in this thesis, there does not appear to be evidence that early tracking increases the inequality between older and younger students. On the contrary, the results in this thesis indicate that early tracking decreases this inequality. It is however not clear why this would be the case. Previous research that jointly study tracking and relative age effects has been limited and more research needs to be done in order to fully understand the possible mechanisms that play a role.  

 

 

 

 

 

 
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