Peer effects and educational achievement: Evidence of causal effectsusing age at school entry as exogenous variation for Peer quality

Raposo, Isabel Pessoa de Arruda; Gonçalves, Michela Barreto Camboim Article Peer effects and educational achievement: Evidence of causal effectsusing age at school entry as exogenous variation for Peer quality EconomiA Provided in Cooperation with: The Brazilian Association of Postgraduate Programs in Economics (ANPEC), Rio de Janeiro Suggested Citation: Raposo, Isabel Pessoa de Arruda; Gonçalves, Michela Barreto Camboim (2020) : Peer effects and educational achievement: Evidence of causal effectsusing age at school entry as exogenous variation for Peer quality, EconomiA, ISSN 1517-7580, Elsevier, Amsterdam, Vol. 21, Iss. 1, pp. 18-37, https://doi.org/10.1016/j.econ.2020.04.003 This Version is available at: https://hdl.handle.net/10419/266947 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ Available online at www.sciencedirect.com ScienceDirect HOSTED BY EconomiA 21 (2020) 18–37 Peer effects and educational achievement: evidence of causal effects using age at school entry as exogenous variation for Peer quality Isabel Pessoa de Arruda Raposo∗, Michela Barreto Camboim Gonc¸alves Researcher at the Funda¸cão Joaquim Nabuco – Ministry of Education, Rua Dois Irmãos, 92 - Ed. Anexo Anízio Teixeira - Apipucos – Recife/PE CEP: 52071-440, Brazil Received 12 March 2018; received in revised form 5 June 2019; accepted 23 April 2020 Available online 27 May 2020 Abstract This paper evaluates the diffusion of peer effects on academic achievement of 4th grade students in the Brazilian public school system. Using data from Prova Brasil 2013, the identification strategy builds on the use of an IV approach, in which the instruments for peers’ performance are the proportions of classmates born in the second semester of the year (and alternatively, in each quarter). The idea behind the instruments is that compulsory school enrolment laws generate variation in the child’s age at school entry, which, in turn, make the date of birth within the year an important determinant of educational achievement and, at the same time, plausibly exogenous to the quality of the student’s peers. The results demonstrate that classrooms with higher proportions of peers born in the 2nd semester (started school at a relatively older age) tend to perform better, on average, than those that concentrate children born in the 1st semester, even after the inclusion of a wide range of control variables. For the math and Portuguese language evaluations, a one standard deviation increase in the classmates’ test scores improves individual achievement by 30% of a SD. Keywords: Peer effects; Instrumental variables; School enrolment laws JEL Classification: I20; I21; C13 © 2020 The Authors. Production and hosting by Elsevier B.V. on behalf of National Association of Postgraduate Centers in Economics, ANPEC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/). 1. Introduction One of the prominent issues within the economics of education is devoted to understanding the role of peers in educational outcomes1. The behavioral influence received from friends in the social interaction process might affect educational results not only during the schooling period, but also latter in life, having an effect on standards of educational attainment to employment decisions. If the individual academic performance increases as the average class performance increases, then peer effects act as a social multiplier providing, as a result, implications for educational policies (Manski, 1993). These policies can guide group formation, and can be applied to school unit organization or to the whole educational system design. ∗Corresponding author. E-mail address: i [email protected] (I.P.d.A. Raposo). 1The sense of the term peers in this article refers to all members belonging to the same classroom. https://doi.org/10.1016/j.econ.2020.04.003 1517-7580 © 2020 The Authors. Production and hosting by Elsevier B.V. on behalf of National Association of Postgraduate Centers in Economics, ANPEC. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). I.P.d.A. Raposo, M.B.C. Gon¸calves / EconomiA 21 (2020) 18–37 19 Despite the fact that the role of peers in educational outcomes has already been vastly exploited in the international literature, in Brazil such literature is still scarce with few exceptions, such as the works of Koppensteiner (2018); Raposo (2015); Firpo et al. (2015); Oliveira (2015), and Pinto (2008). The use of instrumental variables is the most frequently reported method in the literature2, and the identification hypothesis is based on the use of an appropriate instrument, which determines peers’ educational performance, but which is nonetheless exogenous to any unobserved factor related to the potential individual outcome. For instance, Firpo et al. (2015) and Pinto (2008) build their empirical approach using the allocation rule of the students into classrooms and the teacher’s wage as instruments to the quality of peers. In France, Goux and Maurin (2007) use spatially represented data on households in France and attributes the birth month of neighboring adolescents as a tool for peer influence. Their idea is that the date of birth is an important determinant of children’s initial school performance and is exogenous to the quality of the neighborhood where they live. In the line of Goux and Maurin (2007) work, our study provides additional evidence for the Brazilian context and offers a new contribution for the literature since it develops the use of instruments not yet tested for this country. Specifically, our study seeks to answer the following question: “what are the impacts of peer effects on 4th grade students’ academic outcomes from Brazilian public schools?” In order to address to the endogenous influence of classmates, we will instrument the quality of the student body with the proportions of peers born in the second semester of the year (and alternatively, with the proportion of births in each quarter), as proposed by the work of Goux and Maurin (2007)3. The intuition behind such instrument is based on the relative age literature4, which demonstrates that date of birth within the year is an important determinant of educational achievement and, at the same time, it is plausibly exogenous to the quality of the student’s peers. The idea is that educational achievement is related to age at school entry because of the age variation induced by compulsory school enrolment laws. In many countries there are enrolment laws that define the age at school entry based on a specified cut-off date, which induce some children to enter at a relatively older (or younger) age. For instance, in the US and other European countries enrolment laws require students to enter school in the fall of the year in which they turn six. In Chile and other Latin American countries this cut-off is the first semester of the year. In Brazil for the year 2009, when most of the students from our study sample entered the school, there was a Federal law establishing that a child should enter in the 1st year if he/ she turned six years old until the beginning of the school year, which usually occurred in March, those born after that should wait until the next year (BRASIL, 20085). From 27 Federal States, only six of them had different resolutions regarding the cut-off age for school entry, defined by their local boards of Education (see appendix Table A2). For the empirical estimations, we use data from the Prova Brasil6exams for the year of 2013 and find that classrooms with higher proportions of peers born in the 2nd semester tend to perform better than those that concentrate children 2Usually the empirical identification strategies rely either on the use of (i) exogenous instruments for peer’s educational outcome (Sund, 2009; Ding and Lehrer, 2007; Vigdor and Nechyba, 2007; Hanushek et al., 2003; Evans et al., 1992; Case and Katz, 1991); (ii) experimental or quasiexperimental data (Oosterbeek and Van Ewijk, 2014; Eisenkopf et al., 2011; Duflo et al., 2008; Zimmerman, 2003; Sacerdote, 2001) and (iii) mostly recently on the architecture of social networks (e.g. Patacchini and Venanzoni, 2014; Badev, 2013; Patacchini et al., 2011; Mele, 2010; Calvó-Armengol et al., 2009; Bramoullé et al., 2009; Ballester et al., 2006). 3Similar IV approach was firstly proposed by Angrist and Krueger (1991; 1992) and Angrist and Imbens (1995). Angrist and Krueger (1991; 1992) show that students’ quarter of birth interacts with compulsory enrolment laws in the US and age at school entry to generate variation in years of schooling. They show that educational attainment is related to age at school entry because children who enter at an older age (born in early quarters) are permitted to drop out after having completing less schooling than children who enter at a younger age. Angrist and Imbens (1995)based on such evidence also use student’s quarter of birth as an instrument for schooling in a wage equation. 4As pointed out by Goux and Maurin (2007), there is a literature on the influence of date of birth within the year (called the relative age effect) going back a few decades (Barnsley et al., 1985; Allen and Barnsley, 1993). It is shown that relative age has an impact on achievement in competitive activities (Hockey, Soccer), on achievement at school, on emotional development and even on the probability of committing suicide. 5According to legal guidance and standards set by the Brazilian National Board of Education / Ministry of Education in Resolution CNE/CEB 4/2008, to enter the first year of elementary education, the child must be six (6) years old by the beginning of the school year. Some States had different resolutions defined by their local boards of Education (see appendix Table A2). In the year of 2010, a new resolution explicitly defined the cut-off date on March, 31st at the Resolution CNE/CEB 6/2010, according to the following articles: Article III – To enter the first year of elementary education, the child must be six (6) years old by March 31st before they are enrolled. Article IV – Children that reach the age of six (6) years after the date set in Article III must be enrolled at pre-school. 6Prova Brasil is a census exam that evaluates students‘performances in Portuguese and mathematics on elementary education (4th and 8th grades) from all urban public schools in Brazil. It is the Brazilian National System to Evaluate Basic Education from the Brazilian Ministry of Education. 20 I.P.d.A. Raposo, M.B.C. Gon¸calves / EconomiA 21 (2020) 18–37 born in the 1st semester. For the math and Portuguese language evaluations, a one standard deviation increase in the peers’ test scores improves individual achievement by 30% of a standard deviation. The rest of the paper is organized into four additional sections. Section two describes Prova Brasil dataset and offers a descriptive analysis from the sample variables use. The third one presents the IV identification strategies using information on peer’s date of birth to estimate peer effects on academic performance. Section four provides the estimated results, and some robustness tests to verify the validity of the instruments. Finally, the last one brings together the main findings of the article. 2. The dataset and descriptive statistics of the variables This paper uses data from the Brazilian National System to Evaluate Basic Education also known as Prova Brasil, for the year 2013. Created in 2005 by the National Institute of Educational Research Anísio Teixeira from the Brazilian Ministry of Education (MEC/ Inep), the Prova Brasil evaluation consists of a census assessment involving students from 4th and 8th grades of elementary schools in Brazil, and offers a nationwide measurement of the quality of education. Schools with at least twenty students enrolled in the assessed grades are eligible to participate in the assessment and their pupils are submitted to two types of exams: mathematics and Portuguese language. At the end of the tests, the student answers a questionnaire containing questions about date of birth, age and sex, as well as, scholar history, studying habits and family socioeconomic background. Interviewers also administer two other questionnaires for the school Principals and teachers, in which they collect a large set of information on internal and external aspects of the school environment. For the present study we consider only the sample of 4th grade students from public schools that had at least ten students being evaluated in the Prova Brasil exams of 2013. The analysis could be conducted only after this year, in order to create a sample composed of children who entered school when the Federal law of 2008 was valid. As a consequence, each student in our dataset who was 10 years old in the beginning of the school year in 2013, entered in the first grade when he or she was 6 year old, as required by the Federal Law of 2008, which started to be valid in 2009. On average, children born after March start school relatively older. For our sample study, the average age at school entry in the 1st semester is 6.3 years old, while in the 2nd semester, it is 6.8. This difference is statistically significant at less than 1% level. We excluded all the schools located in the six states that did not follow the Resolution CNE/CEB 4/2008 (BRASIL, 2008) in the year of 2009 (the school entry year of the sample students), since for those units the students born in the second semester are not necessarily older than their peers, which breaks down the intensity status of the instrument. After the exclusion of individuals with inadequate or missing information, the final sample consists of about 1 million students from 4th grade and 47,902 schools distributed over the Brazilian states and the Federal District. Table 1 presents the descriptive statistics of the variables selected for the estimated models of this paper. The description of each variable is provided on the appendix section, Table A1. The scale of the Prova Brasil evaluations ranges from 0 to 500 and indicates in which position the student of a given school unit is located. Such position reveals the skills the students have already acquired in the learning process, or those that are still being developed. For the 4th grade, the minimum desired level for mathematics should be 225, while in Portuguese language this inferior limit should be 200. In the case of our sample, as presented in Table 1, the 4th graders achieve in average below the minimum level for both the math and Portuguese language evaluations, with average test score of 213 in mathematics, and 197 in Portuguese language. 3. Identification using information on peer’s date of birth The central question of this paper is whether student’s educational achievement is affected by the performance of other peers from the same classroom. The identification strategy builds on the use of an instrumental variable which determines peers’ performance at school, but which is nonetheless exogenous to any unobserved factor related to the potential individual outcome. Specifically, the instruments being considered here are the proportions of peers in the student’s classroom who were born in the second semester of the year, as proposed by the work of Goux and Maurin (2007). These authors use the distribution of neighbors’ dates of birth as instrumental variable to identify the effect of neighbors’ educational advancement on an adolescent’s performance at school. Their IV estimates suggest that a one standard deviation increase in the proportion of neighbors who have already been held back a grade at age 15 increases I.P.d.A. Raposo, M.B.C. Gon¸calves / EconomiA 21 (2020) 18–37 21 Table 1 Descriptive statistics of the variables. N Average Standard Deviation Dependent Variables (Y) Math grade at Prova Brasil exam (4th grade) 1,053,506 207.87 50.00 Portuguese Language grade at Prova Brasil exam (4th grade) 1,053,506 192.52 48.57 Instruments (Z) Proportion of peers born in the 2nd semester 1,506,897 0.48 0.12 Peers’ Educational Achievement (Y) Peers math grade (4th grade) 1,053,506 206.25 30.05 Peers Portuguese language grade (4th grade) 1,053,506 190.94 27.54 Students’ Characteristics (X) Semester II 1,506,897 0.49 0.50 Age 1,036,496 4.02 1.12 Male sex 1,020,405 0.50 0.50 White 884,898 0.31 0.46 Black 884,898 0.12 0.33 Student lives with the mother 1,040,255 0.89 0.31 Educational level (mother) 675,279 3.79 1.50 Literacy (mother) 1,048,732 0.94 0.23 Student lives with the father 1,033,909 0.63 0.48 Educational level (father) 563,820 3.70 1.59 Literacy (father) 1,031,954 0.91 0.29 Preschool 1,021,837 1.90 0.90 Grade retention 1,506,897 1.39 0.64 Student has already abandoned school 1,506,897 1.10 0.36 Schools’ Characteristics (W) Number of enrolled students 1,506,897 699.48 432.52 Day-shift classes 1,506,897 0.55 0.50 Night-shift classes 1,506,897 0.00 0.01 Federal schools 1,506,897 0.00 0.03 Municipal schools 1,506,897 0.75 0.44 Urbana 1,506,897 0.90 0.31 Source: MEC/ Inep (2013). an adolescent’s probability of grade repetition between the age of 15 and 16 by about 10–15 percentage points (i.e. about 20% of a SD). Following this approach, one simple way to identify the influence of scholar peer effects, in our study, is to test whether student’s performance at school is affected by the distribution of dates of birth within the year of other peers from the same class. As already mentioned, the intuition behind such instrument is based on the compulsory school enrolment laws that generate variation in the child’s age at school entry, which, in turn, make the date of birth within the year an important determinant of educational achievement and, at the same time, plausibly exogenous to the quality of the student’s peers7. Based on this argument, we propose to estimate the following system of equations based on a Two-Step Generalized Method of Moments (GMM-IV) procedure: 7In the related literature, there are studies that focuses on the potential linkage between birth events and socio-cultural aspects (Pinedo and Bermudez, 2015), which is particularly problematic for the purpose of our identification strategy, since birth seasonality could rather be reflecting family background. However, this has not been empirically observed. For instance, Lam and Miron (1987) find that the seasonal pattern of children’s births is unrelated to the wealth and marital status of their parents for several countries’ populations. In Brazil, Moreira (2013) does not find statistically significant differences of birth patterns among populations from different geographic regions, neither among socio-demographics characteristics of the mother, such as, age, marital status, education and occupation. According to this author, these evidences suggest that Brazilian conceptions are rather related to daily luminosity (photoperiod). As a result, the period of birth, in itself, should not be correlated with educational outcome, suggesting that this variable is plausibly exogenous for the purposes of this paper. 22 I.P.d.A. Raposo, M.B.C. Gon¸calves / EconomiA 21 (2020) 18–37 ¯ Y−icj = α1+ β1Zcj +K k=1θk 1Xk icj +S s=1δs 1Ws j+ νicj (1) Yicj = α2+ τ2ˆ ¯ Y−icj +K k=1θk 2Xk icj +S s=1δs 2Ws j+ (εicj + τ2ˆ νicj) (2) In the first step, Eq. (1), we regress the average peer’s grade ¯ Y−icj =1 ncj ncj −i=1y−icj, excluding student i, at the classroom c and school j8, on the excluded instrument Zcj composed by the proportion of 4th graders born in the second semester of the year at the classroom level, and on other predetermined variables of student i (X) and school j (W), as already described in Table 1. In the second step, Eq. (2), the instrumented average grade (ˆ ¯ Y−icj) is included to capture the effect of peer’s performance on the individual outcome of interest Yicj. There are two target outcomes being considered in this study: the individual achievement on math and on Portuguese language at the Prova Brasil exams of 2013. The estimation procedure is based on the GMM-IV technique and it is robust to a variety of violations of the assumption of independent and identically distributed errors (iid). In the IV approach there are two indispensable requirements necessary for the instruments to be considered valid, the independence and monotonicity assumptions, which cannot be directly checked because it involves unobservable variables9. Nevertheless, there are some testable implications that can be used as robustness tests and will be further discussed in the section of results. 4. Results The first results presented in Table 2 focus on the reduced form effect and analyse the student’s performance on math and Portuguese language as a function of own date of birth, and peer’s date of birth. This reduced form analysis does not distinguish endogenous peer’s behavior from exogenous social effects. Two models are estimated, Model 1 considers the semester as the period of birth and in Model 2 the periods of birth are divided in quarters. In both cases the estimates reveal that the student’s grades, either on math or Portuguese language, are higher if their classmates were born in the second semester rather than the first (Model 1) or if they were born in the later quarters rather than the first (Model 2). For both models, the maturity of peers (proportions) presents a stronger positive correlation with the student’s grade than own maturity (dummies), such results were also found by Goux and Maurin (2007). Table 3 presents the first stage results. Again two types of models are estimated, considering periods of birth as semester (Model 1) or quarters (Model 2). Columns (1) and (3) provide specifications without student and school characteristics, for peers’ average grades on math and Portuguese language, respectively, while columns (2) and (4), include all the control variables. We decided to include the states dummies even in the simplest specification, in order to control for regional specificities that might be related to student performance regardless of the instrument. For all the specifications and models, the estimates show a strong and significant correlation of the instruments with the potentially endogenous variable (peers’ achievement). In Model 1, classes with high proportions of peers born in the 2nd semester (or later quarters, Model 2) tend to perform better, in average, than those that concentrate children born in the 1st semester (or 1st quarter, Model 1), even after the inclusion of the student period of birth and a wide range of individual socio-economic and school features. Estimates do not change much between the models with and without the control variables. The first stage results capture the relative age effect on the quality of peers, induced here by the exogenous variation of the age at school entry defined by enrolment laws. The 2nd stage IV and OLS results are exhibited in Table 4, again for Model 1 (semester) and Model 2 (quarters). Basically, OLS and IV estimates are similar in magnitude, although the IV standards errors are much higher. Focusing first on the IV estimates for Model 1 and math evaluation, the positive and significant 2nd stage IV coefficient (ˆ β= 0.79) and its corresponding standardized coefficient (ˆ b= 0.30)10, indicates that a one standard deviation increase in the peers’ grade improves individual achievement by 30% of a SD. For the Portuguese language exam, this effect represents an 8Where ncj is the total number of 4th graders students at classroom c and school j minus 1, the excluded student i. 9Suppose a specific classroom has a high proportion of students born in the second semester, and denote its average performance as ¯ Ycj 1. The non-observed counterfactual ¯ Ycj 0for that same unit would provide its average performance had that class concentrated more students born in the first semester. However, only one of ¯ Ycj 1(or ¯ Ycj 0) is observed. 10 The standardized coefficient is calculated as follows: ˆ b=ˆ β∗ [sd(Y)/sd (Y)]. I.P.d.A. Raposo, M.B.C. Gon¸calves / EconomiA 21 (2020) 18–37 23 Table 2 Reduced form effect of classmates’ date of birth on the student’s performance at the 4th grade. Model 1 Math Portuguese Language Proportion of peers born in the 2nd semester 4.94*** 4.35*** (0.60) (0.59) Student born in the 2nd semester (dummy) 2.91*** 2.87*** (0.15) (0.14) Additional controlsaYes Yes State dummies Yes Yes R20.25 0.26 Noof observations 380,712 Model 2 Math Portuguese Language Proportion of peers born the 2nd quarter 8.09*** 5.08*** (0.87) (0.84) Proportion of peers born the 3rd quarter 9.89*** 7.89*** (0.86) (0.84) Proportion of peers born the 4th quarter 8.00*** 5.86*** (0.87) (0.85) Student born in the 2nd quarter (dummy) 2.48*** 2.39*** (0.20) (0.19) Student born in the 3rd quarter (dummy) 5.20*** 5.05*** (0.21) (0.20) Student born in the 4th quarter (dummy) 3.23*** 3.20*** (0.20) (0.20) Additional controlsaYes Yes State dummies Yes Yes R20.25 0.26 Noof observations 380,712 Source: Original compilation based on MEC/ Inep (2013). aAdditional controls include student i (Xi,c,j) and school j (Wj) characteristics described in Table 1. Robust standard errors in between parentheses. ***, **, * represent the statistical significance at the 1%, 5% and 10% levels, respectively. increase of 29% SD of the student grade. These results are robust when the instruments consider different peers’ period of birth, semester or quarters. For Model 2, the corresponding standardized coefficient produces an impact of 32% SD on individual math grade, very similar to Model 1, while for Portuguese language this impact is a bit smaller, 25% SD of the student grade. The results also remain robust to a different sample of Prova Brasil. Estimations for the exams of 2015 are presented in appendix Tables A5 and A6 and even though some results are a little sensitive regarding their magnitudes, the direction of the effect of each variable is robust to the estimation with a different year of the exam (as the signs of the coefficients don’t change). Note that we find a stronger peer effects than the study of Goux and Maurin (2007), whose work found an impact of 20% SD on the probability of being held back a grade. Possibly the endogenous peer effects operate more sharply on the student performance in isolated disciplines, such as Portuguese language or mathematics, than on other outcomes, such as grade retention that capture the student’s overall performance. Also other studies that use student test scores as dependent variables, find a stronger peer influence than Goux & Maurin. For instance, for the Brazilian investigations, Koppensteiner (2018) find a peer impact of 40% SD of student grade, Firpo et al. (2015) also find 40%, Raposo (2015) encounters a smaller standardized effect of 6% and Oliveira (2015) provides estimates ranging from 15% to 18% SD. More mixed evidences are found for the international literature, for example Vardardottir (2013) find a peer impact of 32% SD of the student grade, Patacchini et al. (2011) find 10% SD, Calvó-Armengol et al. (2009) find 7%, Duflo et al. (2008) 14% SD, Hanushek et al. (2003) 35% SD and Sacerdote (2001) 5% SD11 . Returning to Table 4, when we compare IV with OLS estimates, we find that for both types of exams, the OLS coefficients seem to overestimate the influence of peers on individual achievement. In the OLS estimations, the corresponding standardized coefficient produces an impact of 42% SD on individual math grade (for both Model 1 and 2), 11 Comparisons to previous studies should be established with caution because of different methods used, different outcomes and mainly because of heterogeneities in the peer reference groups. 24 I.P.d.A. Raposo, M.B.C. Gon¸calves / EconomiA 21 (2020) 18–37 Table 3 1st stage IV estimates: correlations between peers’ educational achievement and the proportion of classmates by date of birth as the excluded instruments. Math Portuguese Language Model 1 (1) (2) (3) (4) Proportion of peers born in the 2nd semester 6.73*** (0.30) 6.50*** (0.39) 5.95*** (0.28) 5.66*** (0.36) Student born in the 2nd semester (dummy) – 0.57*** (0.07) – 0.53*** (0.07) Additional controlsaNo Yes No Yes State dummies Yes Yes Yes Yes R20.40 0.48 0.38 0.45 H0: “instruments are jointly valid” Sargan-Hansen J test Equation exactly identified Equation exactly identified H0: “equation is weakly identified” Kleibergen-Paap Wald rk F statistic 1150.75++ 418.69++ 1056.61++ 376.40++ Noof observations 1,053,506 378,034 1,053,506 378,034 Model 2(1) (2) (3) (4) Proportion of peers born the 2nd quarter 11.91*** (0.43) 9.87*** (0.56) 9.90*** (0.40) 7.85*** (0.52) Proportion of peers born the 3rd quarter 15.14*** (0.43) 13.54*** (0.56) 13.53*** (0.40) 11.89*** (0.51) Proportion of peers born the 4th quarter 10.12*** (0.44) 9.24*** (0.57) 8.17*** (0.41) 7.17*** (0.52) Student born in the 2nd quarter (dummy) – 0.64*** (0.11) – 0.55*** (0.09) Student born in the 3rd quarter (dummy) – 1.03*** (0.11) – 1.00*** (0.10) Student born in the 4th quarter (dummy) – 0.79*** (0.11) – 0.67*** (0.10) Additional controlsaNo Yes No Yes State dummies Yes Yes Yes Yes R20.28 0.35 0.24 0.33 H0: “instruments are jointly valid” Sargan-Hansen J test 3.02 (p = 0.22) 2.32 (p = 0.31) 2.75 (p = 0.25) 3.02 (p = 0.22) H0: “equation is weakly identified” Kleibergen-Paap Wald rk F statistic 1078.57+++ 326.82+++ 968.67+++ 284.17+++ Noof observations 1,053,506 378,034 1,053,506 378,034 Source: Original compilation based on MEC/ Inep (2013). aAdditional controls include student i (Xi,c,j) and school j (Wj) characteristics described in Table 1. Standard errors in between parentheses. Estimates efficient for arbitrary heteroskedasticity and autocorrelation. Statistics robust to heteroskedasticity and autocorrelation. ***, **, * represent the statistical significance at the 1%, 5% and 10% levels, respectively. ++ Test statistics to be compared with Stock-Yogo (2005) weak ID test critical values: 16.38, 8.96, 6.66 and 5.53 corresponding to 10%, 15%, 20% and 25% maximal IV size, respectively. H0is not accepted in all the cases. +++ Test statistics to be compared with Stock-Yogo (2005) weak ID test critical values: 22.30, 12.83, 9.54 and 7.80 corresponding to 10%, 15%, 20% and 25% maximal IV size, respectively, as well as, 13.91, 9.08, 6.46, and 5.39 corresponding to 5%, 10%, 20% and 30% maximal IV relative bias, respectively. H0is not accepted in all the cases. The estimated coefficients of all variables are presented in Tables A3–A6 from the appendix section. while for the Portuguese language exams this effect represents an increase of 37% SD of the student test score (again for both Models). This finding is somehow expected once endogenous group selection is likely to lead to upward bias in the OLS coefficient, given that members of the same group tend to share similar unobserved attributes. 4.1. Overidentification, weak identification and other robustness tests According to Nichols (2006) the standard errors on IV estimates are likely to be larger than OLS estimates, and much larger if the excluded instrumental variables are only weakly correlated with the endogenous regressors. Young I.P.d.A. Raposo, M.B.C. Gon¸calves / EconomiA 21 (2020) 18–37 25 Table 4 OLS and 2nd stage IV estimates: the effect of peers’ educational achievement on student’s performance at the 4th grade. Math Portuguese Language Model 1 OLS 2nd stage OLS 2nd stage Peers average math grade 0.70*** (0.003) 0.79*** (0.09) – – Peers average Port. Lang. grade – – 0.66*** (0.003) 0.77*** (0.10) Student born in the 2nd semester (dummy) 2.49*** (0.14) 2.47*** (0.15) 2.51*** (0.14) 2.45*** (0.15) Additional controlsaYes Yes Yes Yes State dummies Yes Yes Yes Yes R20.35 0.35 0.33 0.33 Noof observations 378,034 378,034 378,034 378,034 Model 2OLS 2nd stage OLS 2nd stage Peers average math grade 0.70*** (0.003) 0.78*** (0.06) – – Peers average Port. Lang. grade – – 0.66*** (0.003) 0.66*** (0.07) Student born in the 2nd quarter (dummy) 2.03*** (0.19) 2.00*** (0.19) 2.03*** (0.19) 2.02*** (0.19) Student born in the 3rd quarter (dummy) 4.46*** (0.19) 4.41*** (0.20) 4.04*** (0.19) 4.40*** (0.20) Student born in the 4th quarter (dummy) 2.66*** (0.19) 2.62*** (0.20) 2.75*** (0.19) 2.75*** (0.20) Additional controlsaYes Yes Yes Yes State dummies Yes Yes Yes Yes R20.35 0.35 0.33 0.33 Noof observations 378,034 378,034 378,034 378,034 Source: Original compilation based on MEC/ Inep (2013). aAdditional controls include student i (Xi,c,j) and school j (Wj) characteristics described in Table 1. The 1st stage estimates also include these additional controls. Standard errors in between parentheses. Estimates efficient for arbitrary heteroskedasticity and autocorrelation. Statistics robust to heteroskedasticity and autocorrelation. ***, **, * represent the statistical significance at the 1%, 5% and 10% levels, respectively. The estimated coefficients of all variables are presented in Tables A3 to A6 from the appendix section. (2018) in a review of 31 papers published in the journals of the American Economic Association, shows how non-iid errors weaken 1st stage relations, raising the relative bias of 2SLS and generating mean squared error that is larger than biased OLS in almost all published papers. Non-iid errors also increase the probability of spuriously large test statistics when the instruments are irrelevant, particularly in highly leveraged regressions and particularly in joint tests of coefficients, i.e. 1st stage F tests. As a consequence, weak instrument pre-tests based upon F-statistics are found to be largely uninformative of both size and bias. In order to avoid wrong inference, in our study all the estimated models are based on the IV-GMM technique, robust to heteroskedasticity and autocorrelation in the data process. Thus, all the tests implemented are consistent with violations on the iid error assumption. Referring again to Table 3, we first present the overidentification tests to check the validity of our instruments for peers’ academic achievement. The Sargent-Hansen J statistics test the null hypothesis that “instruments are jointly valid”, i.e., uncorrelated with the error term, and that the excluded instruments are correctly excluded from the estimated equation. The J statistic is consistent in the presence of heteroskedasticity and autocorrelation and it could only be calculated for Model 2, since it has more than one excluded instruments. As can be verified from Table 3 (Model 2 with control variables), the null hypothesis of the validity of our instruments cannot be rejected at 1% level, for both evaluations: math (p-value = 0.31 > 0.01) and Portuguese language (pvalue = 0.22 > 0.01). 32 I.P.d.A. Raposo, M.B.C. Gon¸calves / EconomiA 21 (2020) 18–37 Table A4 IV regressions for the effect of peers’ educational achievement on student’s performance: sample of Prova Brasil 2013 – Model 2. Math Portuguese language 1st stage: peers’ average grade (Dep. var.) 2nd stage: student grade (Dep. var.) 1st stage: peers’ average grade (Dep. var.) 2nd stage: student grade (Dep. var.) Model 2 Peers average math grade – 0.78*** (0.06) – – Peers average Port. Lang. grade – – – 0.66*** (0.07) Proportion of peers born in the 2nd quarter 9.87*** (0.56) – 7.85*** (0.52) – Proportion of peers born in the 3rd quarter 13.54*** (0.56) – 11.89*** (0.51) – Proportion of peers born in the 4th quarter 9.24*** (0.57) – 7.17*** (0.52) – Born in the 2nd quarter 0.64*** (0.11) 2.00*** (0.19) 0.55*** (0.09) 2.02*** (0.19) Born in the 3rd quarter 1.03*** (0.11) 4.41*** (0.20) 1.00*** (0.10) 4.40*** (0.20) Born in the 4th quarter 0.79*** (0.11) 2.62*** (0.20) 0.67*** (0.10) 2.75*** (0.20) Age −1.16*** (0.04) −2.61*** (0.10) −1.21*** (0.04) −2.48*** (0.11) Male sex −0.48*** (0.07) 3.83*** (0.14) −0.50*** (0.07) −10.23*** (0.14) White 1.33*** (0.08) −0.38** (0.17) 1.33*** (0.08) 0.60*** (0.18) Black −2.75*** (0.11) −8.13*** (0.26) −2.27*** (0.11) −6.77*** (0.25) Student lives with the mother 1.58*** (0.12) 2.17*** (0.25) 1.51*** (0.11) 2.05*** (0.25) Educational level (mother) 1.54*** (0.03) 2.77*** (0.10) 1.59*** (0.03) 2.99*** (0.12) Literacy (mother) 1.05*** (0.18) 0.73** (0.33) 0.87*** (0.16) −0.28 (0.31) Student lives with the father 1.54*** (0.08) 1.71*** (0.17) 1.36*** (0.07) 1.60*** (0.17) Educational level (father) 0.76*** (0.03) 0.81*** (0.07) 0.89*** (0.03) 1.13*** (0.08) Literacy (father) 1.54*** (0.15) 1.73*** (0.28) 1.48*** (0.13) 1.91*** (0.27) Preschool −0.50*** (0.04) −0.80*** (0.08) −0.48*** (0.04) −0.91*** (0.08) Grade retention −1.85*** (0.07) −10.46*** (0.17) −1.93*** (0.06) −11.00*** (0.18) Student has already abandoned school −3.45*** (0.11) −8.46*** (0.28) −3.09*** (0.10) −8.10*** (0.28) Day-shift classes 0.26*** (0.07) −0.34** (0.14) −0.05 (0.07) −0.38** (0.13) Night-shift classes 0.59 (2.36) 0.85 (4.47) −2.64 (1.95) −2.91 (4.07) Federal schools 37.36*** (0.92) 4.10 (2.78) 37.49*** (0.87) 7.11** (3.03) Municipal schools −3.36*** (0.10) 0.26 (0.26) −3.94*** (0.09) 0.12 (0.32) I.P.d.A. Raposo, M.B.C. Gon¸calves / EconomiA 21 (2020) 18–37 33 Table A4 (Continued) Math Portuguese language 1st stage: peers’ average grade (Dep. var.) 2nd stage: student grade (Dep. var.) 1st stage: peers’ average grade (Dep. var.) 2nd stage: student grade (Dep. var.) Urbana 11.40*** (0.13) −0.50 (0.69) 11.20*** (0.11) 0.20 (0.77) Number of enrolled students −0.00*** (0.00) −0.00*** (0.00) −0.00*** (0.00) −0.00*** (0.00) Constant 172.96*** (0.46) 70.22*** (10.39) 159.79*** (0.42) 85.83*** (11.02) State dummies Yes Yes Yes Yes R20.35 0.35 0.33 0.33 Noof observations 378,034 378,034 378,034 378,034 Source: Original compilation based on MEC/ Inep (2013). Standard errors in between parentheses. Estimates efficient for arbitrary heteroskedasticity and autocorrelation. Statistics robust to heteroskedasticity and autocorrelation. ***, **, * represent the statistical significance at the 1%, 5% and 10% levels, respectively. Table A5 IV regressions for the effect of peers’ educational achievement on student’s performance: sample of Prova Brasil 2015 – Model 1. Math Portuguese language 1st stage: peers’ average grade (Dep. var.) 2nd stage: student grade (Dep. var.) 1st stage: peers’ average grade (Dep. var.) 2nd stage: student grade (Dep. var.) Model 1 Peers average math grade – 0.89*** (0.22) – – Peers average Port. Lang. grade – – – 0.69*** (0.16) Proportion of peers born in the 2nd semester 3.57*** (0.51) – 5.22*** (0.52) – Born 2nd semester 0.25** (0.10) 1.66*** (0.20) 0.36*** (0.10) 1.94*** (0.21) Age −1.01*** (0.05) −1.33*** (0.25) −1.25*** (0.06) −1.42*** (0.23) Male sex −0.28*** (0.10) 4.53*** (0.20) −0.33*** (0.10) −8.83*** (0.21) White 1.25*** (0.11) −0.16 (0.35) 1.14*** (0.11) 0.10 (0.29) Black −3.01*** (0.16) −8.02*** (0.74) −2.64*** (0.16) −8.21*** (0.54) Student lives with the mother 1.31*** (0.16) 1.92*** (0.43) 1.26*** (0.16) 2.09*** (0.40) Educational level (mother) 1.25*** (0.04) 2.45*** (0.29) 1.39*** (0.04) 2.80*** (0.24) Literacy (mother) 0.17 (0.25) −0.24 (0.48) 0.74** (0.26) −0.34 (0.53) Student lives with the father 1.68*** (0.11) 1.37*** (0.43) 1.51*** (0.11) 1.37*** (0.33) Educational level (father) 0.66*** (0.04) 0.76*** (0.16) 0.87*** (0.04) 1.41*** (0.16) Literacy (father) 1.02*** (0.20) 1.75*** (0.44) 1.38*** (0.21) 2.55*** (0.46) Preschool −0.79*** (0.05) −1.06*** (0.21) −0.73*** (0.05) −0.94*** (0.16) Grade retention −1.71*** (0.10) −10.46*** (0.43) −1.75*** (0.10) −11.63*** (0.35) 34 I.P.d.A. Raposo, M.B.C. Gon¸calves / EconomiA 21 (2020) 18–37 Table A5 (Continued) Math Portuguese language 1st stage: peers’ average grade (Dep. var.) 2nd stage: student grade (Dep. var.) 1st stage: peers’ average grade (Dep. var.) 2nd stage: student grade (Dep. var.) Student has already abandoned school −2.02*** (0.14) −4.45*** (0.53) −2.19*** (0.15) −7.00*** (0.47) Day-shift classes −2.13*** (0.10) −0.03 (0.51) −2.46*** (0.10) −0.73 (0.45) Night-shift classes −1.19 (2.64) −5.14 (5.05) −5.00** (1.97) 3.55 (8.11) Federal schools 33.14*** (0.66) −2.37 (7.70) 31.28*** (0.86) 4.00 (5.62) Municipal schools −1.44*** (0.12) 0.63 (0.40) −2.60*** (0.12) 0.04 (0.49) Urbana 4.83*** (0.16) −1.54 (1.13) 7.08*** (0.17) 0.17 (1.20) Number of enrolled students 0.03*** (0.00) −0.01 (0.01) 0.03*** (0.00) −0.00 (0.01) Constant 190.53*** (0.53) 36.14 (42.99) 175.84*** (0.55) 79.06** (2910) State dummies Yes Yes Yes Yes R20.41 0.31 0.42 0.30 Noof observations 164,423 164,423 164,423 164,423 Source: Original compilation based on MEC/ Inep (2015). Standard errors in between parentheses. Estimates efficient for arbitrary heteroskedasticity and autocorrelation. Statistics robust to heteroskedasticity and autocorrelation. ***, **, * represent the statistical significance at the 1%, 5% and 10% levels, respectively. Table A6 IV regressions for the effect of peers’ educational achievement on student’s performance: sample of Prova Brasil 2015 – Model 2. Math Portuguese language 1st stage: peers’ average grade (Dep. var.) 2nd stage: student grade (Dep. var.) 1st stage: peers’ average grade (Dep. var.) 2nd stage: student grade (Dep. var.) Model 2 Peers average math grade – 0.68*** (0.15) – – Peers average Port. Lang. grade – – – 0.64*** (0.11) Proportion of peers born in the 2nd quarter 4.39*** (0.73) – 5.48*** (0.73) – Proportion of peers born in the 3rd quarter 7.44*** (0.72) – 10.47*** (0.73) – Proportion of peers born in the 4th quarter 3.73*** (0.73) – 4.96*** (0.75) – Born in the 2nd quarter 0.54*** (0.14) 2.89*** (0.28) 0.73*** (0.14) 2.91*** (0.29) Born in the 3rd quarter 0.77*** (0.14) 4.20*** (0.29) 1.08*** (0.14) 4.54*** (0.31) Born in the 4th quarter 0.28*** (0.14) 2.20*** (0.27) 0.38** (0.14) 2.34*** (0.28) Age −1.08*** (0.06) −1.89*** (0.19) −1.35*** (0.06) −1.83*** (0.19) I.P.d.A. Raposo, M.B.C. Gon¸calves / EconomiA 21 (2020) 18–37 35 Table A6 (Continued) Math Portuguese language 1st stage: peers’ average grade (Dep. var.) 2nd stage: student grade (Dep. var.) 1st stage: peers’ average grade (Dep. var.) 2nd stage: student grade (Dep. var.) Male sex −0.27** (0.10) 4.51*** (0.19) −0.31*** (0.10) −8.80*** (0.20) White 1.25*** (0.11) 0.11 (0.28) 1.14*** (0.11) 0.15 (0.26) Black −3.00*** (0.16) −8.66*** (0.54) −2.63*** (0.16) −8.32*** (0.44) Student lives with the mother 1.30*** (0.16) 2.18*** (0.37) 1.24*** (0.16) 2.12*** (0.37) Educational level (mother) 1.25*** (0.04) 2.71*** (0.20) 1.39*** (0.04) 2.85*** (0.18) Literacy (mother) 0.15 (0.25) −0.24 (0.48) 0.72** (0.26) −0.35 (0.52) Student lives with the father 1.67*** (0.11) 1.72*** (0.32) 1.51*** (0.11) 1.42*** (0.28) Educational level (father) 0.66*** (0.04) 0.90*** (0.12) 0.87*** (0.04) 1.45*** (0.13) Literacy (father) 1.00*** (0.20) 1.95*** (0.41) 1.36*** (0.21) 2.59*** (0.43) Preschool −0.79*** (0.05) −1.21*** (0.16) −0.73*** (0.05) −0.96*** (0.14) Grade retention −1.64*** (0.10) −10.49*** (0.31) −1.65*** (0.10) −11.37*** (0.28) Student has already abandoned school −2.00*** (0.14) −4.82*** (0.41) −2.17*** (0.15) −7.03*** (0.39) Day-shift classes 2.15*** (0.10) −0.51 (0.37) −2.48*** (0.10) −0.86** (0.34) Night-shift classes −0.56 (2.58) −4.89 (4.99) −4.20** (1.95) 3.81 (8.06) Federal schools 33.10*** (0.66) 4.83 (5.33) 31.24*** (0.86) 5.46 (4.24) Municipal schools −1.44*** (0.12) 0.33 (0.31) −2.59*** (0.12) −0.07 (0.38) Urbana 4.82*** (0.16) −0.47 (0.78) 7.08*** (0.17) 0.52 (0.87) Number of enrolled students 0.03*** (0.00) 0.00 (0.00) 0.03*** (0.00) 0.00 (0.00) Constant 188.37*** (0.60) 77.60** (28.47) 173.15*** (0.63) 87.08*** (20.21) State dummies Yes Yes Yes Yes R20.41 0.32 0.43 0.30 Noof observations 164,423 164,423 164,423 164,423 Source: Original compilation based on MEC/ Inep (2015). Standard errors in between parentheses. Estimates efficient for arbitrary heteroskedasticity and autocorrelation. Statistics robust to heteroskedasticity and autocorrelation. ***, **, * represent the statistical significance at the 1%, 5% and 10 % levels, respectively. 36 I.P.d.A. Raposo, M.B.C. Gon¸calves / EconomiA 21 (2020) 18–37 Table A7 Summary statistics of individual and school characteristics by schools with higher proportions of peers born by semester. Higher proportion of peers born in the 1st semester Higher proportion of peers born in the 2nd semestera Individual and school characteristics Mean Std. Dev. Mean Std. Dev. Age 3.97 1.11 4.07 1.11 Proportion of peers born in the 2nd semester in the classroom 0.40 0.07 0.58 0.07 Total number of enrolled students in the school 697.47 424.72 697.95 434.79 Male 0.53 0.50 0.53 0.50 White 0.32 0.47 0.32 0.47 Black 0.12 0.33 0.12 0.32 Student lives with the mother 0.91 0.29 0.91 0.29 Educational level (mother) 3.89 1.49 3.87 1.49 Literacy (mother) 0.95 0.22 0.95 0.22 Student lives with the father 0.70 0.46 0.70 0.46 Educational level (father) 3.81 1.56 3.76 1.55 Literacy (father) 0.92 0.27 0.92 0.27 Preschool 1.86 0.86 1.86 0.86 Grade retention 1.37 0.62 1.37 0.63 Student has already abandoned school 1.10 0.35 1.10 0.35 Day-shift classes 0.56 0.50 0.55 0.50 Night-shift classes 0.00 0.02 0.00 0.01 Federal school 0.00 0.03 0.00 0.04 Municipal school 0.74 0.44 0.73 0.45 Urban area 0.90 0.30 0.89 0.31 Noof observations 195,247 182,787 Source: Original compilation based on MEC/ Inep (2013). aIt is a dummy variable that takes the value 1 when the proportion of 4th graders born in the 2nd semester, in the student’s classroom, is at least as high as the proportion of 4th graders born in the 1st semester; and the value 0 otherwise. 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