Gender Peer Efects in School, a Birth Cohort Approach

by Antonio Ciccone Walter Garcia-Fontes^**

Documento de Trabajo 2014-07

June 2014

* University of Mannheim and Barcelona GSE.

Universitat Pompeu Fabra, Barcelona GSE and FEDEA.

Los Documentos de Trabajo se distribuyen gratuitamente a las Universidades e Instituciones de Investigación que lo solicitan. No obstante están disponibles en texto completo a través de Internet: http://www.fedea.es. These Working Paper are distributed free of charge to University Department and other Research Centres. They are also available through Internet: http://www.fedea.es. ISSN:1696-750

Gender Peer Efects in School, a Birth Cohort Approach

Antonio Ciccone

Walter Garcia-Fontes¹

June 2014

¹University of Mannheim and Barcelona GSE, and Universitat Pompeu Fabra and Barcelona GSE, respectively. We gratefully acknowledge the comments of Brindusa Anghel, Ghazala Azmat, Antonio Cabrales, Caterina Calsamiglia, Gabrielle Fack, Libertad Gonzalez, Stephan Litschig, Luis Ricardo Maertens, Marco Manacorda, Barbara Petrongolo, Joaquim Silvestre, Tetyana Surovtseva, Alessandro Tarozzi, Michele Tertilt, and seminar participants in Barcelona, Madrid, Mannheim, Munich, and London. The main data used have become available thanks to the eforts of Antonio Cabrales, Ismael Sanz Labrador, and Pablo Vazquez. We thank Luis Pires Jimenez for providing important additional data. We gratefully acknowledge financial support from CREI, fedea, Fundacio Catalunya - La Pedrera, Spanish research grants ECO2011-25272, ECO2011-30323-C03-02, and SEV-2011-0075 (Severo Ochoa Program for Centers of Excellence in R&D).

9 junio, 2014 - Evaluaciones Nacionales - Etiquetas: coeducación, cohorte, composición por sexos, Educación Primaria, familia,género, rendimiento,repetición - No hay comentarios

Fuente: FEDEAblog

¿Mejora el rendimiento escolar de las niñas cuando hay más niñas en las clases que niños? ¿Y cómo responde el desempeño escolar de los niños cuando hay más niñas en la clase? Los efectos de pares o iguales de las alumnas sobre las alumnas, o de las alumnas sobre los alumnos han sido objeto de debate desde que se generalizó la coeducación en todos los niveles educativos. Se trata además de un tema importante para el diseño eficiente del sistema escolar, ya que en algunos países se ha vivido un resurgimiento de escuelas separadas para niños y niñas.

Los primeros trabajos empíricos en el campo de la economía de la educación, que intentaron estudiar los efectos de pares relacionados con la distribución de niños y niñas, utilizaron una metodología basada en la explotación de la variación natural en la composición por sexos que existe entre escuelas. Sin embargo las familias juegan un papel importante en la selección de alumnos en distintas escuelas, por lo tanto la distribución de alumnos según sus aptitudes no es exógena, sino que es fruto de esa auto-selección. Dado que la aptitud está obviamente correlacionada con el rendimiento escolar, no es posible estimar de forma consistente el efecto de pares por sexo usando la variación en la distribución de niños y niñas entre escuelas cuando existe esta auto-selección por parte de las familias.

Algunos trabajos más recientes evitan este problema usando un enfoque basado en la variación en la composición por sexos dentro de las escuelas; usando las diferencias en esta composición dentro de las escuelas para un mismo curso escolar en diferentes años, analizan su efecto sobre el rendimiento escolar de niños y niñas. Con este nuevo enfoque, Hoxby (2000), con datos de Texas y Lavy y Schlosser (2011), con datos de Israel, han obtenido evidencia empírica de que la composición por sexos afecta el aprendizaje en la escuela. La idea detrás de este enfoque es que las diferencias en la composición por sexos dentro de las escuelas para un mismo curso escolar en diferentes años reflejan parcialmente la variación natural en los nacimientos de niños y niñas, una variación que es aleatoria y exógena, y por lo tanto no afecta la estimación econométrica con los problemas de auto-selección que mencionábamos anteriormente.

El tema de la coeducación y los efectos de pares relacionados con la distribución de alumnos y alumnas en las clases ha sido también objeto de debate en España, pero no se han realizado estudios dentro de la economía de la educación, ya que no se disponían de los datos necesarios para poder estudiar estos efectos empíricamente. Los primeros datos que permiten plantear este análisis empírico provienen de las pruebas de Conocimientos y Destrezas Indispensables de sexto de Primaria de la Comunidad de Madrid. Utilizando la muestra correspondiente a las pruebas realizadas los años 2009, 2010 y 2011 hemos estimado los efectos de pares por sexo en las escuelas de Primaria de esta comunidad. Nuestros principales resultados muestran un efecto de pares por sexo positivo y significativo de la proporción de niñas sobre los resultados académicos de los niños. Nuestra estimación muestra que un aumento alrededor de 10 puntos porcentuales en la proporción de niñas mejora los resultados generales y en matemáticas de los niños alrededor de 2,2 por ciento de una desviación estándar. Este efecto representa aproximadamente una quinta parte del efecto de tener padre o madre con estudios universitarios en lugar de estudios secundarios. En cuanto al efecto de la proporción de niñas sobre los resultados de las niñas, nuestras estimaciones muestran que es positivo pero no es estadísticamente significativo. El efecto que obtenemos es consistente con otros estudios que han obtenido evidencia a favor de que una proporción mayor de niñas en el aula implica un mejor ambiente que beneficia sobre todo a los niños, véase Lavy y Schlosser (2011) para la evidencia empírica. Mirando al efecto medio global, nuestros resultados implican que el rendimiento académico medio se maximiza cuando la proporción de niñas es igual a la de niños en las aulas.

En nuestro estudio “Gender Peer Effects in School,a Birth Cohort Approach” (Ciccone y García-Fontes, 2014) explicamos en detalle, mediante un modelo teórico, por qué, en presencia de una tasa positiva de repetición, el enfoque basado en cursos escolares usado en estos estudios previos, presenta ciertos problemas para la estimación. Las tasas de repetición en Primaria son relativamente altas en España, por esta razón, desarrollamos un nuevo enfoque sobre los efectos de pares por sexo dentro de la cohorte de nacimiento dentro del curso escolar. Usando este enfoque, los datos muestran el efecto contrario, es decir un efecto positivo y significativo de la proporción de niñas sobre el rendimiento de las niñas y no significativo sobre el rendimiento de los niños.

¿Cuál es la razón por la que el enfoque basado en cursos escolares genera una estimación sesgada en sistemas donde las tasas de repetición son positivas? La razón más importante detrás de este resultado se puede entender a través de una situación hipotética. Supongamos que las clases de primer curso de Primaria son todas del mismo tamaño, pero difieren en cuanto a la proporción de niños y niñas por razones naturales, es decir, la variación aleatoria de nacimientos en el área de influencia de la escuela en cuestión. Supongamos también que no hay efectos de pares por sexo reales, es decir, la aptitud individual de cada alumno no se ve afectada por la composición de niños y niñas. Por último, introduzcamos que el supuesto de que por razones aleatorias una clase comienza en primero de Primaria con una proporción de niñas más elevada que el promedio de los cursos de primero en la escuela. A medida que esta clase va avanzando hacia los cursos superiores, algunos alumnos repetirán curso y dejarán de estar junto a los alumnos de su cohorte de edad, mientras que la clase recibirá alumnos repetidores provenientes de una cohorte anterior. A pesar de estos cambios, la proporción de niñas entre los alumnos de la clase seguirá siendo más elevada que la de una clase que hubiera empezado primero de Primaria con una composición por sexos equilibrada.

Consideremos ahora la proporción de repetidoras dentro de todas las niñas de la clase. A medida que la clase progresa hacia cursos superiores, la proporción relativamente elevada de niñas en el curso inicial se traduce en una proporción relativamente baja de repetidoras entre todas las de la clase, siempre y cuando el sexo del alumnado repetidor sea independiente de la composición por sexos de la clase. En otras palabras, las fluctuaciones naturales en la composición por sexos de la cohorte de edad en el primer curso se traducen en una asociación negativa entre la proporción de niñas en la clase y la proporción de repetidoras entre todas las niñas en cursos superiores. Esta asociación permite entender los efectos espurios que se obtienen cuando hay una tasa de repetidores positiva y se usa el enfoque de estimación basado en el curso escolar. Si los alumnos que no han repetido curso tienen un rendimiento mejor que los repetidores, como sucede en nuestros datos, una proporción relativamente mayor de niñas en el primer curso se traducirá en un rendimiento medio de las niñas relativamente mejor en cursos superiores ya que la proporción de repetidoras, que tienen un rendimiento más bajo, es menor.

En este caso las fluctuaciones exógenas en la composición por sexos en las cohortes de edad se traduce en una asociación positiva entre la proporción de niñas y su rendimiento académico en cursos superiores, es decir, efectos de pares por sexo espurios positivos de niñas sobre el rendimiento de niñas, cuando se usa el enfoque de estimación basado en cursos escolares. Para los niños el argumento es simétrico e implica un efecto de pares por sexo espurio positivo de la proporción de niños en el curso sobre el rendimiento de niños, y por tanto, el efecto de pares por sexo espurio negativo de niñas sobre niños en el curso (ya que la proporción de niños y de niñas en un curso suman uno).

Otro tipo de fluctuaciones aleatorias iniciales, sobre las aptitudes de los alumnos o sobre los criterios que determinan la repetición de curso, también pueden llevar a estimaciones que lleven a conclusiones equivocadas cuando hay una tasa de repetición positiva y se usa el enfoque de estimación basado en repetición de curso. El enfoque de estimación que proponemos nosotros, basado en las cohortes de edad, no tiene estos inconvenientes, si la proporción de niñas en diferentes cohortes de edades dentro de las escuelas se debe a la variación natural y exógena de los nacimientos en el área de influencia de la escuela.

Si tenemos en cuenta las normas de matriculación en Primaria y la movilidad de alumnos entre escuelas en la Comunidad de Madrid, parece un supuesto razonable. Contamos además en los datos con una serie de variables que describen características individuales y familiares de los alumnos que nos permiten verificar que la proporción de niñas en los diferentes cohortes de edad no está correlacionada significativamente con estas variables.

Referencias

References

1. Ciccone, A y W. García-Fontes (2014). “Gender Peer Effects in School, a Birth Cohort Approach”. Documento de Trabajo, Universitat Pompeu Fabra.

References

1. Hoxby, C. (2000). “Peer Effects in the Classroom: Learning from Gender and Race” Variation. Working paper 7867, National Bureau of Economic Research.

References

1. Lavy, V. y Schlosser, A. (2011). “Mechanisms and Impacts of Gender Peer Effects at School”. American Economic Journal: Applied Economics, 3(2), páginas 1–33.

Abstract

We propose estimating gender peer efects in school by exploiting within-school variation in gender composition across birth cohorts. Our approach difers from the existing literature, which exploits variation in gender composition at a given grade level in diferent years. We argue that the birth cohort approach is a useful alternative as the grade level approach generally yields spurious gender peer efects when there is grade retention. The birth cohort approach applied to primary schools in Spain indicates statistically significant positive gender peer efects of girls on boys’ academic achievement and statistically insignificant efects of girls on girls’ achievement.

1 Introduction

Do girls learn more together with girls instead of boys? And what about boys? Possible gender peer efects in learning have been debated since the introduction of mixed-gender education and would have to be taken into account in the design of school systems as well as in the policy response to the recent revival of single-sex schools (e.g. Hoxby, 2000; Whitmore, 2005; Lavy and Schlosser, 2011).¹ Early empirical work looked for gender peer efects across schools but could not deal with the selection of students with diferent skills into diferent schools. The best evidence that gender composition afects learning in school comes from Hoxby’s (2000) and Lavy and Schlosser’s (2011) studies for Texas and Israel respectively. These studies bypass the selection of students with diferent skills into diferent schools by examining the response of academic achievement to within-school diferences in gender composition at a given grade level in diferent years.² Examining the consequences of such diferences in gender composition is appealing as they partly reflect natural variation in the births of girls and boys in school catchment areas (Lavy and Schlosser, 2011).

The within-school grade level approach to gender peer efects may not always be immune to selection issues however. Most school systems allow for grade retention of academically weak students and children in the same birth cohort may therefore end up in diferent grades depending on their academic skills. For example, only three of twenty-seven European Union countries rule out grade retention in primary school (European Commission, 2011). According to the program for international student assessment (PISA, 2009a), the share of students retained at least once in primary school averages to 8 percent across OECD countries and to 7 percent across EU countries.³ Retention rates in lower-secondary school are similar. Countries with comparatively high retention rates in primary school are Belgium and France (17 percent), Spain (11 percent), and the USA (10.6 percent).⁴ Retention rates in primary school are comparatively low in Canada and Israel (4.2 percent) and the UK (1.7

¹For an analysis of optimal (student) assignments in the presence of social spillovers see Graham (2011) and Graham, Imbens, and Ridder (2010, 2014).

²Whitmore (2005) studies the efect of gender composition at the classroom level on boys’ and girls’ academic achievement in kindergarten and school. She finds mostly statistically insignificant efects, maybe because of a relatively small sample size compared to Hoxby or Lavy and Schlosser.

³The length of primary schooling difers across countries. Children may also be retained in kindergarten or their entry into school may be delayed if parents and teachers consider the child not ready for school.

⁴There is no PISA data on the Texas school system studied by Hoxby (2000). The Texas education agency (1999, 2011) reports the share of students retained by grade and year between 1994 and 2009. If no student was retained more than once, the data would imply that 13-16 percent of students were retained once in primary school. As some students are retained more than once in primary school, 13-16 percent is an upper bound to the share of students retained at least once in primary school. The discrepancy should be small however as according to PISA (2009a) only around 0.5 percent of US students were retained more than once in primary school (there are no such data for Texas).

percent).

To understand the efects of grade retention on the grade level approach to gender peer efects we develop a theoretical model of a school system with grade retention that we can solve analytically. In our model, students with academic skills below a threshold are retained in a grade at some point during primary school. As a result, academically weak students end up in a lower grade than their academically stronger peers in the same birth cohort. Another important feature of the model is that the gender composition of birth cohorts and the skills of girls and boys in a birth cohort may be subject to exogenous shocks. The question we ask is if and why spurious gender peer efects in academic achievement may emerge when gender peer efects within schools are estimated at the grade level.

A main result of our theoretical model is that the grade level approach generally yields spurious gender peer efects in academic achievement even if grade level diferences in gender composition were solely driven by exogenous shocks to gender composition at the birth cohort level. Exogenous shocks to the skills of girls and boys at the birth cohort level also translate into spurious gender peer efects at the grade level. The direction of the spurious gender peer efects depends on the impact of grade repetition on students’ academic skills. If grade repetition improves retained students’ academic skills, exogenous shocks to the skills of girls and boys at the birth cohort level lead to a spurious positive gender peer efect of girls on girls’ academic achievement and a spurious negative gender peer efect of girls on boys’ achievement. If students who have been retained in the past perform on average worse academically than non-retained students in the same grade, exogenous shocks to the gender composition of birth cohorts also lead to a spurious positive gender peer efect of girls on girls’ academic achievement and a spurious negative gender peer efect of girls on boys’ achievement.

Because of the limitations of the grade level approach to gender peer efects in school systems with grade retention, we propose estimating gender peer efects in academic achievement by exploiting within-school diferences in gender composition across birth cohorts. Students are assigned to the same birth cohort if they should have started school in the same year according to the school system’s enrollment rule. The birth cohort approach examines whether girls or boys in a birth cohort with a greater share of girls do better academically than students of the same gender in other birth cohorts in the same school. Gender peer efects estimated using the birth cohort approach are intention-to-treat efects. Their magnitude depends on the strength of gender peer efects in the classroom but also, for example, on the rate of grade retention as this afects the link between the share of girls in the average student’s birth cohort and the share of girls in the average student’s classroom throughout primary school. A key feature of the birth cohort approach to gender peer efects is that it does not yield spurious gender peer efects when there is grade retention.⁵

We use the birth cohort approach to estimate gender peer efects in a sample of Spanish primary schools. Spain allows for grade retention in primary school and the rate of grade retention is above the OECD and EU average. An advantage of our primary school data compared to Hoxby’s (2000) data for Texas and Lavy and Schlosser’s (2011) data for Israel is that they allow estimating gender peer efects at the birth cohort level (as well as the grade level).⁶ Also, mobility across primary schools appears to be low compared to Texas and even to Israel, and compliance with the primary school enrollment rule is high.

When we estimate gender peer efects using the birth cohort approach, we find a statistically significant, positive gender peer efect of girls on the academic achievement of boys. A 10-percentage-point increase in the share of girls in a birth cohort improves boys’ overall academic achievement and their achievement in mathematics by around 3 to 4 percent of a standard deviation. (Girls do worse than boys on average and this efect can therefore not be explained by spillovers from high-skill to low-skill students.⁷) On the other hand, the efect of the share of girls in a birth cohort on the achievement of girls is statistically insignificant. The grade level approach tends to yield a diferent pattern of gender peer efects than the birth cohort approach. The diference between the birth cohort and the grade level approach is consistent with our theoretical model if (i) grade repetition improves retained students’ skills but students retained in the past still do worse on average than non-retained students in the same grade and (ii) grade level diferences in gender composition are mostly driven by shocks to gender composition or skills at the birth cohort level.

As rates of grade retention in Spain are comparatively high, it is natural to wonder whether the birth cohort and the grade level approach can yield diferent patterns of gender peer efects when retention rates are around or below the OECD or EU average. We examine this question in a series of counterfactual experiments based on our theoretical model. After calibrating the model to match our gender-peer-efect estimates for Spain, we lower the academic thresholds for grade retention – which leads to lower rates of grade retention – and simulate data. The simulated data allow us to estimate gender peer efects at successively lower rates of grade retention. Our findings indicate that the birth cohort and the grade level approach may yield diferent patterns of gender peer efects at rates of grade retention around or below the OECD and EU average. However, in interpreting these findings it should be kept in mind that they are based on counterfactual experiments calibrated to Spanish data and therefore cannot be generalized.

⁵It is natural to wonder whether spurious gender peer efects at the grade level could be avoided by instrumenting the share of girls at the grade level by the share of girls at the birth cohort level. We find that this approach would generally continue to yield spurious gender peer efects.

⁶Hoxby (2000) does not have the individual data necessary to assign students to birth cohorts (her data consist of grade level averages). Israeli primary schools participate only once every two years in the standardized tests used by Lavy and Schlosser (2011). As a result, test results are never available for both retained and non-retained students in a birth cohort.

⁷Hoxby (2000) and Lavy and Schlosser (2011) also find a positive efect of the share of girls on boys’ academic achievement in subjects where girls do worse than boys on average. See Sacerdote (2011) for a review of the literature on skill spillovers in school.

The community of Madrid data contain information on the primary school students attended when they took the test but do not specify where students went to school previously. This is also true for the Texas primary school data used by Hoxby (2000) and the Israeli primary school data used by Lavy and Schlosser (2011). Student mobility across primary schools in Israel is substantially lower than in Texas however. Lavy and Schlosser calculate that 7.9 percent of students in Israel left their primary school between first and second grade in 2002, while Hanushek, Kain, and Rivkin (2004) report an annual rate of student mobility in Texas of 24 percent. In our data, student mobility in primary school appears to be comparatively low. We estimate that the share of second grade students in 2013 who did not attend the same primary school a year earlier is around 2.8 percent.

The remainder of the paper is structured as follows. We discuss the related literature next and then preview our theoretical findings on spurious gender peer efects when using a grade level approach. Section 2 develops our theoretical model of a school system with grade retention and uses it to discuss the grade level approach and the birth cohort approach to gender peer efects. Section 3 presents the data and section 4 our empirical results. Section 5 contains our counterfactual experiments. Section 6 concludes.

1.1 Related Literature

Our study is closely related to the work of Hoxby (2000) and Lavy and Schlosser (2011), who estimate gender peer efects using within-school variation in gender composition at the grade level in Texas and Israel respectively. Lavy and Schlosser estimate gender peer efects in primary, middle, and high school. In primary school, they find a statistically significant efect of the share of girls on girls’ achievement in mathematics and an insignificant efect of girls on boys’ mathematics achievement. In science and technology, they find a statistically significant gender peer efect of girls on girls’ and on boys’ achievement, while gender peer efects are statistically insignificant in Hebrew and English. In middle school, Lavy and Schlosser find statistically significant gender peer efects of girls on girls’ achievement in mathematics and English but not in science and technology or Hebrew. Gender peer efects of girls on boys’ achievement are statistically insignificant in all subjects. In high school,

Lavy and Schlosser find gender peer efects of girls on girls’ as well as boys’ achievement for a range of academic outcomes. Lavy and Schlosser also provide evidence on mechanisms. They find that a greater share of girls in a grade lowers levels of classroom disruption and violence, improves inter-student and student-teacher relationships, and lessens teacher fatigue. These efects are not driven by changes in the behavior of individual girls or boys but by the share of (better behaved) girls. Hoxby’s study of gender peer efects in Texas primary schools finds statistically significant efects of girls on girls’ achievement in mathematics and reading in grades three to six. Gender peer efects of girls on boys’ achievement are also mostly significant in reading, with the exception of fourth grade. Gender peer efects of girls on boys’ achievement in mathematics are statistically insignificant for fourth and fifth grade but positive and significant in third and sixth grade.

As Israel, Spain, and Texas difer in their socioeconomics (e.g. the role of woman in society), their school systems, and their standardized tests, estimates of gender peer efects need not be similar. The 95 percent confidence intervals for our preferred estimates of gender peer efects in Spanish primary schools are broadly similar to Lavy and Schlosser’s (2011) for Israeli primary schools in mathematics, which is probably the most comparable subject. Hoxby’s (2000) estimates for Texas are somewhat diferent even at similar stages in primary school. Hoxby’s 95 percent confidence interval for the efect of a 10-percentagepoint increase in the share of girls on girls’ achievement in mathematics is from around 0 to around 12 percent of a standard deviation in fifth and sixth grade. The 95 percent confidence interval for Lavy and Schlosser’s estimates of gender peer efects of girls on girls’ achievement in mathematics is from around 0 to around 7 percent of a standard deviation, while it is from around −1 to around 4 percent of a standard deviation for our preferred estimates. The 95 percent confidence interval for Hoxby’s estimates of the efect of a 10-percentage-point increase in the share of girls on boys’ achievement in mathematics is from around −6 to around 5 percent of a standard deviation in fifth grade and from around 0 to around 16 percent of a standard deviation in sixth grade. The 95 percent confidence interval for Lavy and Schlosser’s estimates of gender peer efects of girls on boys’ achievement in mathematics is from around −1 percent to around 5 percent of a standard deviation, while it is from around 0 to around 5 percent of a standard deviation for our preferred estimates.

Our work is also related to Whitmore’s (2005) study of gender peer efects in kindergarten and primary school grades one to three using diferences in gender composition generated by the STAR project.⁸ Her approach exploits diferences in gender composition across classes within grades and schools. She finds that the efect of more girls in class on the average student’s academic achievement depends on the grade. The efect is significantly positive in kindergarten and grade two, insignificant in grade one, and significantly negative in grade three.⁹ When Whitmore allows for diferent efects on girls and boys, estimates become noisy and are mostly statistically insignificant (maybe because of the relatively small sample size compared to Hoxby or Lavy and Schlosser).¹⁰

⁸The project randomly assigned kindergarten children in participating schools and students entering participating schools to classes (of diferent types).

1.2 Preview of Main Theoretical Findings

A main finding from our theoretical model of a school system with grade retention is that the grade level approach generally yields spurious gender peer efects even if grade level diferences in gender composition were solely driven by natural fluctuations in the share of girls across birth cohorts. To understand this result consider the following scenario. Suppose that each incoming (first-grade) class in a primary school has the same size but that the share of girls is subject to exogenous shocks. Suppose also that individual students’ skills are unafected by the gender composition of their class (there are no true gender peer efects). Consider a class starting out with a larger share of girls than the average incoming class in the school. As this class proceeds from first grade to higher grades it will lose students who are retained and be joined by retained students. But the share of girls among all students in the class will tend to remain higher than in a class starting with the average gender composition.¹¹ Now consider the share of retained girls among all girls as the class proceeds from first grade to higher grades. As long as the gender of retained students is independent of the gender composition of the class they join, a larger-than-average share of girls in first grade leads to a smaller-than-average share of retained girls among all girls in the class in higher grades. As a result, exogenous shocks to the gender composition of birth cohorts translate into a negative association between the share of girls among all students and the share of retained girls among all girls in higher grades. This association is at the root of the spurious gender peer efects when using the grade level approach.

⁹Graham, Imbens, and Ridder (2010) develop an approach to quantify the gains from reallocating individuals across social groups in the presence of spillovers and illustrate their approach by studying the efects of gender segregation on mathematics achievements in kindergarten using STAR data. Their analysis difers from Whitmore in that it only looks at gender peer efects in math; only in kindergarten; and it allows for nonlinear efects.

¹⁰There is also a literature using intention-to-treat approaches based on primary school enrollment rules to examine whether diferences in initial maturity have long-lasting efects on academic achievement, see Bedard and Dhuey (2006).

¹¹This will be the case even if the likelihood of retention difers for girls and boys as long as the gender of retained students is independent of the gender composition of the class they join. A suficient condition for this to be the case is that gender composition is independently distributed across birth cohorts.

The direction of the spurious gender peer efects mainly depends on whether non-retained students do better or worse on average than students who were retained in the past. If (as in our data) non-retained students tend to do better, a class with a larger-than-average share of girls in first grade will tend to have girls who do better than average in higher grades because of a smaller-than-average share of worse-performing retained girls among all girls. As a result, exogenous shocks to the gender composition of birth cohorts translate into a positive association between the share of girls among all students and the academic achievement of girls in higher grades. This positive association between the share of girls and girls’ academic achievement in higher grades produces a spurious positive efect of the share of girls on girls’ achievement when using the grade level approach.¹² For boys, the argument is symmetric and implies a spurious positive gender peer efect of the share of boys in a grade on the academic achievement of boys. As the share of girls and boys in a grade sums to one, this translates into a spurious negative gender peer efect of the share of girls on the achievement of boys.

The grade level approach generally also yields spurious gender peer efects when there are exogenous shocks to the skills of girls and boys in a birth cohort. But such skill shocks do not – as a first analysis might suggest – necessarily translate into a spurious positive gender peer efect of girls on girls’ academic achievement and of boys on boys’ achievement. Instead, the direction of the spurious gender peer efects turns out to depend on whether grade repetition improves or worsens retained students’ skills. To understand the main forces shaping the direction of the spurious gender peer efects induced by skill shocks, consider an incoming (first-grade) class where the share of girls is equal to the average in incoming classes but girls have better-than-average skills. As this class proceeds from first grade to higher grades, a greater-than-average share of girls will be promoted (not retained). Moreover, the skills of non-retained girls will be better than average. Hence, this class will end up with a larger share of girls among all students in higher grades and with better-than-average girls. This would tend to translate into a spurious positive efect of the share of girls on the academic achievement of girls when using the grade level approach.

However, there is a countervailing force operating in the class one year below the class that starts with better-than-average girls. As this class proceeds to higher grades, it receives retained girls from the class that starts with better-than-average girls. As the skills of these retained girls are above average and their number below average, the class one year below the class that starts with better-than-average girls ends up with better-than-average girls in higher grades and a smaller-than-average share of girls among all students. The dominant force in our theoretical model depends on whether grade repetition improves or worsens retained students’ skills. If grade repetition improves retained students’ skills, shocks to the skills of girls at the birth cohort level translate into a spurious positive gender peer efect of the share of girls in a grade on the academic achievement of girls. The efect of shocks to the skills of boys is symmetric and implies a spurious positive gender peer efect of the share of boys in a grade on the academic achievement of boys in the grade (or equivalently, a spurious negative efect of girls on boys’ achievement).

¹²There turns out to be a countervailing force that could in principle dominate and reverse the sign of the spurious gender peer efect of girls on girls’ achievement. This countervailing force emerges in the classes that receive retained students from the classes starting with a greater-than-average share of girls. We find that for this countervailing force to dominate, the grade retention policy has to be so stringent that more than half of the students repeat a grade. This scenario seems of little empirical relevance as rates of grade retention are below 50 percent in almost all countries (PISA, 2009a).

Two important follow-up issues are whether spurious gender peer efects at the grade level can be avoided by controlling for grade retention at the individual level or by using an instrumental-variables approach, with the share of girls at the grade level instrumented by the share of girls at the birth cohort level. For the instrumental-variables approach to work, the share of girls at the birth cohort level should afect students’ academic achievement solely through the share of girls at the grade level. We have already seen however that when there is grade retention and the average student retained in the past does not perform exactly as well as the average non-retained student in the same grade, the share of girls at the birth cohort level afects academic achievement at the grade level through the share of retained girls and boys among students of the same gender. This composition efect could be dealt with by controlling for grade retention at the individual level if within-school grade level diferences in the share of girls were solely driven by natural fluctuations in the share of girls across birth cohorts. But this estimation strategy becomes invalid when grade level diferences in the share of girls may also be driven by other factors, such as shocks to the academic skills of girls and boys in a birth cohort. The reason is that in this case, diferences between the academic achievement of retained and non-retained students in a grade may reflect the same factors as diferences in the academic achievement of students across school years.

2 A Theoretical Framework

To understand the consequences of grade retention for the estimation of gender peer efects within schools we develop a theoretical model that we can deal with analytically. The key feature of the model is that students with academic skills below a threshold are subject to grade retention. The model also features exogenous shocks to the gender composition of birth cohorts, to the academic skills of girls and boys, and to the thresholds used for grade retention. We first use the model to examine if and why the grade level approach may give rise to gender peer efects although there are no true gender peer efects. We also use the model to illustrate the birth cohort approach to gender peer efects.

2.1 A School System with Grade Retention

Children in birth cohort t start primary school in year t. For the first L school years, children attend what we will call lower grades (LG) and all children in the same birth cohort are in the same classroom. At the end of the Lth year in primary school, some children move to what we will call high grade (HG) and some are retained for an extra year in LG. Children in HG take a standardized test at the end of the school year and then leave primary school.

Whether students are retained for an extra year in LG depends on how their individual academic skills a after L years in LG compare with their school’s academic threshold for grade retention p. Students i of gender g in school s and birth cohort t move from LG to HG after L years if their academic skills at that point in time satisfy

\[a _ {i g s} ^ {t} \geq p _ {g s} ^ {t}.\tag{1}\]

Students with skills below the academic threshold, , are retained for one extra year in LG and therefore move to HG after years in LG. We sometimes refer to students in birth cohort t who move to HG in year and as students who enter HG on age and late respectively. The academic threshold may be subject to school, birth cohort, and gender specific shocks. (An alternative interpretation of shocks to are shocks to skills that are relevant for grade retention but irrelevant for the performance in the test students take in HG.)

Each year a continuum of children of measure one starts in each school. A share of the children entering school s in year t are girls and a share boys. The distribution of skills in birth cohort t in school s after L years in LG is taken to be uniform with density and a gender, school, and birth cohort specific mean

Students’ performance in the standardized test administered in HG depends on their skills when they reach HG and the skills they acquire in HG. The test performance of nonretained students from birth cohort t attending HG in the school year starting in is taken to reflect their academic skills where refers to the skills that student i of gender g acquires by attending HG in the school year starting in τ. We assume that the of diferent students are obtained as independent draws from a distribution with a constant variance and a mean that may depend on gender, school, and the school year. The test performance of students reaching HG after being retained for an extra year in LG is taken to reflect their academic skills where captures a gender, school, and birth cohort specific change in skills associated with grade repetition. This change in skills may be positive or negative. If , students who are retained for an extra year in LG accumulate additional skills and therefore do better in the standardized test than they would have done had they not been retained. Figure 1 summarizes the model.

¹³In our simulated models in section 5 we can consider a wider range of distributions.

Gender peer efects could end up afecting the academic skills of HG students and hence their test results in three main ways in the model. First, the share of girls among a student’s peers during the first L years of LG could afect the student’s academic skills at the end of the Lth school year. This can be captured by allowing average academic skills to depend on the share of girls in LG. Second, the share of girls among a student’s peers could afect the threshold used to determine who is promoted to HG after L years in LG. Third, the academic skills a student accumulates in HG could depend on the share of girls among the student’s HG peers. This can be captured by allowing average academic skills accumulated in HG to depend on the share of girls in HG.¹⁴

The rule for grade retention in (1) combined with the distribution of academic skills after L years in LG implies that the share of students of gender g in birth cohort t who reach HG in year (without being retained in LG) is

\[\lambda_ {g s} ^ {t} = \frac {1}{2 \theta} (\alpha_ {g s} ^ {t} + \theta - p _ {g s} ^ {t})\tag{2}\]

We assume throughout that some but not all students are retained for an extra year in LG in each school, , which amounts to the parameter restriction

\[- \theta < \alpha_ {g s} ^ {t} - p _ {g s} ^ {t} < \theta .\tag{3}\]

The average test performance of HG students from birth cohort t who are not retained in LG and reach HG in year is

\[E (t e s t _ {i g s} ^ {t} | n o n - r e t a i n e d) = E (a _ {i g s} ^ {t} \left| a _ {i g s} ^ {t} \geq p _ {g s} ^ {t}\right) + \omega_ {g s, t + L} = \frac {\alpha_ {g s} ^ {t} + \theta + p _ {g s} ^ {t}}{2} + \omega_ {g s, t + L}\tag{4}\]

where denotes the average skills after L years in LG of students who are not retained and ω denotes the average skills these students accumulate in HG in year The average test performance of HG students from birth cohort t who are retained for an extra year in LG and reach HG in year is

¹⁴We could also allow for gender peer efects on the change in skills associated with grade repetition. This complicates the model without generating additional insights as far as we can see.

\[\begin{array}{r c l} {E (t e s t _ {i g s} ^ {t} | r e t a i n e d)} & = & {E (a _ {i g s} ^ {t} \left| a _ {i g s} ^ {t} \leq p _ {g s} ^ {t}\right) + \delta_ {g s} ^ {t} + \omega_ {g s, t + L + 1}} \\ & = & {\frac {1}{2} (\alpha_ {g s} ^ {t} - \theta + p _ {g s} ^ {t}) + \delta_ {g s} ^ {t} + \omega_ {g s, t + L + 1}} \end{array}\tag{5}\]

where denotes the average skills after L years in LG of students who are retained, δ the change in skills associated with grade repetition, and the average skills these students accumulate in HG in year

The average test performance of girls and boys in HG can be derived by combining (4)- (5) and the shares of non-retained girls and boys among HG students of the same gender. The share of non-retained girls among girls in HG in school s in the school year starting in τ depends on the share of girls in birth cohorts and as well as the share of non-retained girls in these birth cohorts

\[\mu_ {f s \tau} = \frac {\phi_ {s} ^ {\tau - L} \lambda_ {f s} ^ {\tau - L}}{\phi_ {s} ^ {\tau - L} \lambda_ {f s} ^ {\tau - L} + \phi_ {s} ^ {\tau - L - 1} (1 - \lambda_ {f s} ^ {\tau - L - 1})}\tag{6}\]

where f stands for female, is the share of non-retained girls among all girls in birth cohort t, and is the share of girls among all students in birth cohort t. The share of non-retained boys among HG boys can be obtained analogously

\[\mu_ {m s \tau} = \frac {(1 - \phi_ {s} ^ {\tau - L}) \lambda_ {m s} ^ {\tau - L}}{(1 - \phi_ {s} ^ {\tau - L}) \lambda_ {m s} ^ {\tau - L} + (1 - \phi_ {s} ^ {\tau - L - 1}) (1 - \lambda_ {m s} ^ {\tau - L - 1})}\tag{7}\]

where m stands for male. The average test performance of HG students of gender in school s and school year can now be obtained by combining (4)-(7)

\[t e s t _ {g s \tau} = \mu_ {g s \tau} E (t e s t _ {i g s} ^ {\tau - L} | n o n - r e t a i n e d) + (1 - \mu_ {g s \tau}) E (t e s t _ {i g s} ^ {\tau - L - 1} | r e t a i n e d).\tag{8}\]

2.2 The Grade Level Approach to Gender Peer Efects

Could grade retention of academically weak students lead us to conclude that there are gender peer efects within schools although there are none? And in what direction might spurious gender peer efects To address these questions, we assume that there are no true gender peer efects within schools and ask what we would conclude if we assessed the strength of gender peer efects using a grade level approach.¹⁵

Suppose we have data on the test performance of HG students and the share of girls in HG for a large number of schools in school years and . The grade level approach would assess the strength of gender peer efects by regressing the test performance of HG girls or boys on the share of girls in HG and school fixed efects. This is equivalent to assessing the strength of gender peer efects by regressing the change in the average test performance of HG girls or boys between the school years starting in and ， , on the corresponding changes in the share of HG girls, . We have already derived in (8). The share of HG girls can be obtained by combining the share of girls and boys in birth cohorts and who are not retained in LG with the share of girls in these birth cohorts

¹⁵That is, we assume that and do not depend on the share of girls in LG or HG.

\[\begin{array}{c} g i r l s h _ {s \tau} = \\ \phi_ {s} ^ {\tau - L} \lambda_ {f s} ^ {\tau - L} + \phi_ {s} ^ {\tau - L - 1} (1 - \lambda_ {f s} ^ {\tau - L - 1}) \\ \hline \phi_ {s} ^ {\tau - L} \lambda_ {f s} ^ {\tau - L} + \phi_ {s} ^ {\tau - L - 1} (1 - \lambda_ {f s} ^ {\tau - L - 1}) + (1 - \phi_ {s} ^ {\tau - L}) \lambda_ {m s} ^ {\tau - L} + (1 - \phi_ {s} ^ {\tau - L - 1}) (1 - \lambda_ {m s} ^ {\tau - L - 1}). \end{array}\tag{9}\]

We find that such a grade level approach generally yields spurious gender peer efects and that the direction of these efects depends on the impact of grade repetition on retained students’ skills and the sources of shocks to the share of girls at the grade level. The possible sources of shocks in our model are shocks to the share of girls at the birth cohort level shocks to the academic skills of girls and boys at the birth cohort level , and shocks to the academic thresholds for grade retention . We examine the consequences of these three types of shocks in turn. To simplify the analysis somewhat, we assume throughout this section that the change in skills associated with grade repetition and mean skills accumulated in HG are constant, and

2.2.1 Shocks to the Share of Girls at the Birth Cohort Level

Suppose that academic skills and the thresholds for grade retention are the same across schools, birth cohorts, and gender, and . The only shocks are i.i.d. shocks with mean zero to the share of girls in a birth cohort

\[\phi_ {s} ^ {t} = 1 / 2 + \eta_ {s} ^ {t}\tag{10}\]

where . What would we obtain when estimating gender peer efects in HG using the grade level approach? Linearizing the share of girls in HG in (9) yields that the grade level approach would yield a spurious positive gender peer efect of girls on girls’ academic achievement and a spurious negative gender peer efect of girls on boys’ achievement if

\[(2 \lambda - 1) (\theta - \delta) > 0\tag{11}\]

where is the diference between the average academic skills in HG of non-retained students and students retained in the past, see (4) and (5), and λ is the share of non-retained students evaluated at and , the grade level approach would yield a spurious negative gender peer efect of girls on academic achievement and a spurious positive gender peer efect of girls on boys’ achievement. Only if would the grade level approach not yield spurious gender peer efects.

Hence, the grade level approach would indicate a spurious positive gender peer efect of girls on girls’ academic achievement and a spurious negative gender peer efect of girls on boys’ achievement if the school system’s retention rate is not too high, 2 and non-retained students do better on average in HG than students who were retained in the past, . To get an intuitive understanding of these spurious gender peer efects, it is useful to think through the consequences of a small positive shock to the share of girls in birth cohort t for the share of girls and their academic skills in HG in the school years starting in and . Non-retained students from birth cohort t reach HG in and retained students in . Making use of (9), the shock to the share of girls among HG students in and can be calculated as

\[\widehat {g i r l s h} _ {s, t + L} = \lambda \eta \quad \mathrm{and} \quad \widehat {g i r l s h} _ {s, t + L + 1} = (1 - \lambda) \eta\tag{12}\]

where denotes deviations of x from its value in the absence of shocks. These expressions reflect that a share λ of the η extra girls in birth cohort t reach HG in year while a share are retained for an additional year in LG and therefore only reach HG in year

The extra girls in birth cohort t who reach HG in year were not retained in the past and the increase in the share of girls among HG students in the school year starting in therefore goes together with an increase in the share of non-retained girls among HG girls. Using (6), the shock to the share of non-retained girls among HG girls can be calculated as

\[\widehat {\mu} _ {f s, t + L} = 2 \lambda (1 - \lambda) \eta .\tag{13}\]

The greater share of non-retained girls among HG girls translates into better average academic skills (and test performance) of HG girls in if non-retained students do better on average than students who were retained in the past. The shock to HG girls’ average test performance in the school year starting in can be calculated using (8) and (13)

\[\widehat {t e s t} _ {g s, t + L} = \widehat {\mu} _ {f s, t + L} (\theta - \delta) \eta = 2 \lambda (1 - \lambda) (\theta - \delta) \eta .\tag{14}\]

Hence, if non-retained HG students have better academic skills on average than students retained in the past, , a positive shock to the share of girls in birth cohort t translates into a positive shock to the average test performance of HG girls in the school year starting in

¹⁶See the appendix for a proof of this result.

But it turns out that also implies that the positive shock to the share of girls in birth cohort t translates into a negative shock to the average test performance of HG girls in the school year starting in . To see this recall that the extra girls in birth cohort t who reach HG in year were all retained in the past. Hence, the share of retained girls among HG girls increases and this worsens the average academic skills (and test performance) of HG girls if non-retained students do better on average than students who were retained in the past. The shock to the average test performance of HG girls can be calculated using (8)

\[\widehat {t e s t} _ {g s, t + L + 1} = - \widehat {\mu} _ {f s, t + L + 1} (\theta - \delta) \eta = - 2 \lambda (1 - \lambda) (\theta - \delta) \eta .\tag{15}\]

Combining (12) and (14)-(15) yields that if a positive shock to the share of girls in birth cohort t leads to a positive grade level association between the average test performance of girls and the share of girls in HG in the school year starting in but a negative grade level association in the school year starting in . The grade level approach to gender peer efects ends up averaging these two associations and the sign of the spurious gender peer efect is determined by whether the positive or the negative association dominates.

To see this note that students in HG in any school year come either from birth cohort (non-repeaters) or birth cohort (repeaters). Let be the shock to gender composition in birth cohort and the independent shock to gender composition in birth cohort . It follows from (12) that the implied shock to the share of girls in HG in the school year starting in is . The implied shock to the average test performance of HG girls is and can be obtained from (14)-(15). As the grade level approach assesses gender peer efects in academic achievement by regressing within-school changes in the test performance of HG girls on the corresponding changes in the share of girls across schools, the sign of the gender peer efect is determined by the sign of

\[E (\widehat {g i r l s h} _ {s \tau} \widehat {t e s t} _ {f s \tau} \Big | \tau) = 2 \lambda (1 - \lambda) (2 \lambda - 1) (\theta - \delta) V a r (\eta)\tag{16}\]

where we used . This implies that a grade level regression of changes in HG girls’ test performance on changes in the share of HG girls yields a strictly positive efect if and only if (11) holds (we maintain throughout that . Intuitively, in this case, the grade level association between changes in the share of girls and their academic achievement in HG induced by non-retained girls is positive and dominates the negative

association induced by retained girls.

An analogous argument for the academic achievement of boys yields that if 1) , a regression of within-school changes in HG boys’ test performance on the corresponding changes in the share of HG boys yields a strictly positive efect. As the share of girls and boys in a grade sums to one, it follows that a regression of changes in HG boys’ test performance on changes in the share of HG girls yields a strictly negative efect in this case.

2.2.2 Shocks to the Skills of Girls and Boys at the Birth Cohort Level

A second source of shocks to the share of girls at the grade level are shocks to the skills of girls and boys in a birth cohort. To understand the efect of these shocks suppose that the average skills of girls and boys are independent across schools and birth cohorts but possibly correlated within schools and birth cohorts

\[\alpha_ {f s} ^ {t} = \alpha + \varepsilon_ {f s} ^ {t} \quad \mathrm{and} \quad \alpha_ {m s} ^ {t} = \alpha + \varepsilon_ {m s} ^ {t}\tag{17}\]

where are shocks with mean zero with and . Linearizing the share of girls in HG in (9) yields that shocks to academic skills do not translate into shocks to the share of girls in HG if . But if the correlation between the shocks to the average skills of boys and girls is less than perfect, , skill shocks afect the share of girls in HG. In this case the grade level approach to gender peer efects would lead to a spurious positive gender peer efect of girls on girls’ academic achievement and a spurious negative gender peer efect of girls on boys’ achievement if

\[(2 \lambda - 1) \delta > 0. ^ {1 7}\tag{18}\]

If , the grade level approach would yield a spurious negative gender peer efect of girls on girls’ academic achievement and a spurious positive gender peer efect of girls on boys’ achievement. Only if would the grade level approach not yield spurious gender peer efects.

Hence, the grade level approach would indicate a spurious positive gender peer efect of girls on girls’ academic achievement and a spurious negative gender peer efect of girls on boys’ achievement if the retention rate 1 − λ is not too high, , and if grade repetition improves retained students’ skills, . To get an intuitive understanding of these spurious gender peer efects, consider the implications of a small positive shock ε to the average skills of girls in birth cohort t for the share of HG girls and their academic skills in the school years starting in and . As the positive shock to the skills of girls implies that fewer girls in birth cohort t are retained, it translates into a positive shock to the share of girls in HG in school year and a negative shock of the same size to the share of girls in HG in ，

¹⁷See the appendix for a proof of this result.

\[\widehat {g i r l s h} _ {s, t + L} = \frac {1}{8 \theta} \varepsilon \quad \text { and } \quad \widehat {g i r l s h} _ {s, t + L + 1} = - \frac {1}{8 \theta} \varepsilon\tag{19}\]

where we made use of (2) and (9). The positive shock to the skills of girls in birth cohort t also afects the average academic skills (and test performance) of HG girls. Making use of (8) yields that the average test performance of HG girls in the school years starting in and are shocked by

\[\widehat {t e s t} _ {f s, t + L} = \lambda \frac {1}{2} \varepsilon + (1 - \lambda) \frac {1}{2} \left(1 - \frac {\delta}{\theta}\right) \varepsilon\tag{20}\]

and

\[\widehat {t e s t} _ {f s, t + L + 1} = (1 - \lambda) \frac {1}{2} \varepsilon + \lambda \frac {1}{2} \left(1 - \frac {\delta}{\theta}\right) \varepsilon .\tag{21}\]

These expressions capture two types of efects. An efect on test performance holding the share of non-retained girls constant and a composition efect that arises through a change in the share of non-retained girls. The efect holding the share of non-retained girls constant is captured by the first terms on the right-hand side of (20)-(21). These terms are obtained as the improvement in the average test performance of girls from birth cohort t who reach HG in the school years and multiplied by the weight of students from birth cohort t in these two school years. It is interesting to note that this efect on test performance is stronger in the school year where most girls in birth cohort t reach HG, which is school year as long as . The second terms in (20)-(21) capture a composition efect as the positive skill shock increases the share of non-retained girls among HG girls in and . If non-retained HG girls do better on average than girls retained in the past, this composition efect increase test performance in HG in both school years. This efect is weaker in the school year where most girls in birth cohort t reach HG.

Combining (19) and (20)-(21) yields that a positive shock ε to the average skills of girls in birth cohort t leads to a positive grade level association between the average test performance of girls and the share of girls in HG in the school year starting in but a negative grade level association in the school year starting in as long as . When the positive association is stronger than the negative association but this changes as falls. When , the two associations exactly ofset each other and the positive association is weaker than the negative association when . The grade level approach to gender peer efects ends up averaging these two associations and the sign of spurious gender peer efects is determined by whether the positive or the negative association dominates.

To see this recall that students in HG in any school year τ come either from birth cohort or birth cohort . Let be the shock to the academic skills of girls in birth cohort and the independent shock to the academic skills of girls in birth cohort . It follows from (19) that the implied shock to the share of girls in HG in the school year starting in . The implied shock to the average academic skills (and test performance) of HG girls can be calculated using (20)-(21) as . As the grade level approach assesses gender peer efects of girls on girls’ academic achievement by regressing within-school changes in the test performance of HG girls on the corresponding changes in the share of girls across schools, the sign of gender peer efects is determined by the sign of

\[E (\widehat {g i r l s h} _ {s \tau} \widehat {t e s t} _ {f s \tau} \Big | \tau) = \frac {(2 \lambda - 1) \delta}{1 6 \theta^ {2}} V a r (\varepsilon)\tag{22}\]

where we used that . This implies that the grade level approach yields a strictly positive efect of girls on girls’ academic achievement if the school system’s retention rate is not too high, , and . In this case the positive grade level association between changes in the share of girls in HG and their academic achievement induced by non-retained girls dominates the negative association induced by retained girls. If 2 the positive association induced by non-retained girls is either exactly ofset by the negative association induced by retained girls or is dominated by the negative association induced by retained girls

An analogous argument for the academic achievement of boys yields that if a regression of within-school changes in HG boys’ test performance on the corresponding changes in the share of HG boys yields a strictly positive efect. As the share of girls and boys in a grade sums to one, it follows that a regression of changes in HG boys’ test performance on changes in the share of HG girls yields a strictly negative efect in this case.

2.2.3 Shocks to the Academic Thresholds for Grade Retention

A third source of shocks to the share of girls at the grade level are shocks to the academic thresholds for grade retention.¹⁸ These shocks can be analyzed in a similar way as skill shocks. Suppose that the academic thresholds applied to girls and boys are subject to shocks that are independent across schools and birth cohorts but possibly correlated within

¹⁸As already mentioned an alternative interpretation of shocks to these academic thresholds are shocks to skills that are relevant for grade retention but irrelevant for the performance in the standardized test.

schools and birth cohorts

\[p _ {f s} ^ {t} = p + \nu_ {f s} ^ {t} \quad \mathrm{and} \quad p _ {m s} ^ {t} = p + \nu_ {m s} ^ {t}\tag{23}\]

where are shocks with mean zero with and Correl . Suppose also that the average skills of boys and girls are identical and the same across schools, birth cohorts, and gender, , and that there is the same share of boys and girls in each birth cohort,

Linearizing the share of girls in HG in (9) yields that shocks to the academic thresholds for grade retention do not translate into shocks to the share of girls in HG if the shocks to the academic thresholds applied to girls and boys are perfectly correlated. If the correlation is less than perfect, , the grade level approach to gender peer efects would lead to a spurious negative gender peer efect of girls on girls’ academic achievement and a spurious positive gender peer efect of girls on boys’ achievement if

\[(2 \lambda - 1) \left(1 - \frac {\delta}{2 \theta}\right) > 0. ^ {1 9}\tag{24}\]

If , the grade level approach would yield a spurious positive gender peer efect of girls on girls’ academic achievement and a spurious negative gender peer efect of girls on boys’ achievement. Only if would there be no spurious gender peer efects.

The condition in (24) can be derived analogously to (18). Intuitively, a positive shock to the academic threshold used for girls in birth cohort t leads to a negative grade level association between the average academic skills and the share of HG girls in the school year starting in but a positive grade level association in the school year starting in . When (24) holds, the negative association dominates and produces a spurious negative gender peer efect of girls on girls’ academic achievement when using the grade level approach. The argument for the efect on boys’ academic achievement is analogous.

2.2.4 A Grade Level Instrumental-Variables Approach?

It is natural to wonder whether spurious gender peer efects at the grade level could be avoided by using an instrumental-variables approach, with the share of girls in HG in the school year starting in τ instrumented by the share of girls in birth cohort . This approach eliminates the spurious gender peer efects due to shocks to academic skills at the birth cohort level and academic thresholds used for grade retention described in sections 2.2.2 and 2.2.3. But the approach would continue to yield spurious gender peer efects. When there is grade retention, birth cohorts with relatively more girls end up having a lower share of retained girls among all girls in HG and a greater share of retained boys among all boys in HG as described in section 2.2.1. This composition efect continues to translate into spurious gender peer efects when using an instrumental-variables approach as long as the average student retained in the past does not perform exactly as well as the average non-retained student in HG. The composition efect could be dealt with by controlling for grade retention at the individual level if grade level diferences in the share of girls were solely driven by exogenous shocks to the share of girls across birth cohorts. But this estimation strategy becomes invalid when grade level diferences in the share of girls are also driven by shocks to academic skills or retention thresholds, as diferences between the academic performance of retained and non-retained students in HG reflect some of the same factors as diferences in the academic performance of students across school years in this case.

¹⁹See the appendix for a proof of this result.

2.3 The Birth Cohort Approach to Gender Peer Efects

The birth cohort approach to gender peer efects examines whether girls or boys in a birth cohort with a greater share of girls do better academically than students of the same gender in other birth cohorts in the same school. We now illustrate the approach in our model.

As already mentioned when we set up the model, gender peer efects could afect academic achievement in three main ways. First, there could be gender peer efects in the accumulation of skills in lower grades (LG). To capture this efect we allow mean skills after L years in LG to depend on the share of girls in birth cohort t

\[\alpha_ {g s} ^ {t} = \widetilde {\alpha} _ {g s} ^ {t} + \pi_ {g} ^ {\alpha} \phi_ {s} ^ {t}\tag{25}\]

where is the strength of the gender peer efect on the academic skills of students of gender and captures exogenous shocks to skills.²⁰ Gender peer efects could also afect the academic thresholds used to determine who is retained for an extra year in LG. We therefore also allow the thresholds for grade retention to depend on the share of girls in birth cohort t

\[p _ {g s} ^ {t} = \widetilde {p} _ {g s} ^ {t} + \pi_ {g} ^ {p} \phi_ {s} ^ {t}\tag{26}\]

where is the strength of the gender peer efect on the threshold for grade retention applied to students of gender g and captures exogenous shocks. Moreover, there could be gender peer efects in high grade (HG) skill accumulation. To capture this efect we allow mean academic skills accumulated in HG in school year τ to depend on the share of girls in HG defined in (9)

²⁰In the model, the share of girls in a student’s birth cohort is identical to the share of girls in the student’s classroom and grade during the first L − 1 years of LG (we assumed one classroom per grade). In year L of LG, the share of girls in a student’s classroom and grade starts difering from the share of girls in the student’s birth cohort because of retained students from the birth cohort that is one year older. As a result, the formulation in (25) is an approximation to classroom or grade level gender peer efects in LG. It would be straightforward to extend the model to capture classroom or grade level gender peer efects in LG precisel but this would not generate additional insights as far as we can see.

\[\omega_ {g s \tau} = \widetilde {\omega} _ {g s \tau} + \pi_ {g} ^ {\omega} g i r l s h _ {s \tau}\tag{27}\]

where is the strength of the gender peer efect and captures exogenous shocks.

The birth cohort approach to gender peer efects within schools assesses the strength of gender peer efects by regressing the test performance of girls or boys in a birth cohort on the share of girls in the birth cohort and school fixed efects. This is equivalent to assessing the strength of gender peer efects by regressing the change in the average test performance of girls or boys between birth cohorts t and , on the change in the share of girls between these birth cohorts, . The average test performance of students in birth cohort t is the weighted average of the test performance of non-retained and retained students in the birth cohort

\[t e s t _ {g s} ^ {t} = \lambda_ {g s} ^ {t} E (t e s t _ {i g s} ^ {t} | n o n - r e t a i n e d) + (1 - \lambda_ {g s} ^ {t}) E (t e s t _ {i g s} ^ {t} | r e t a i n e d)\tag{28}\]

where is the share of non-retained students of gender g in birth cohort t and school s among all students of the same gender. Using (2), (4), and (5) this simplifies to

\[t e s t _ {g s} ^ {t} = \alpha_ {g s} ^ {t} + \left(\lambda_ {g s} ^ {t} \omega_ {g s, t + L} + (1 - \lambda_ {g s} ^ {t}) \omega_ {g s, t + L + 1}\right) + \delta_ {g s} ^ {t} (1 - \lambda_ {g s} ^ {t}).\tag{29}\]

The average test performance of students in a birth cohort is therefore the sum of three terms. The first term captures average academic skills of students in the birth cohort after L years in LG and the second term the academic skills these students accumulate in HG. The third term captures that retained students, who represent a share of the birth cohort, experience a change in academic skills during the extra year in LG.

Using (29) and (25)-(27), the strength of gender peer efects when using the birth cohort approach can be calculated as

\[\beta_ {g} ^ {B} = \pi_ {g} ^ {\alpha} + \pi_ {g} ^ {\omega} \left(\lambda^ {2} + (1 - \lambda) ^ {2} + \lambda \frac {\partial (\lambda_ {f s} - \lambda_ {m s})}{\partial \phi}\right) - \delta \frac {\partial \lambda_ {g s}}{\partial \phi}\tag{30}\]

where we have linearized (29) around as well as 2 and combines (2) and (25)-(26). The three terms in (30) correspond to the marginal efects of the share of girls in a birth cohort on the three terms in (29). The first term captures the strength of gender peer efects during the first L years in LG. The second term captures the expected strength of gender peer efects on the average girl or boy in HG. It is the product of the strength of gender peer efects in HG and a term (in parentheses) that captures the expected increase in the share of girls the average student in a birth cohort with more girls will be exposed to in HG.²¹

To understand the third term on the right-hand side of (30) it is useful to focus on the case where gender peer efects arise solely through the academic thresholds used for grade retention but in (25)-(27)). Suppose for example that more boys are retained in birth cohorts with a greater share of girls . As grade repetition changes skills by the average boy in a birth cohort with more girls would then do better in HG than the average boy in a birth cohort with a balanced gender composition if

A key feature of the birth cohort approach to gender peer efects that follows from (30) is that it only yields gender peer efects if gender composition afects the academic skills of students or the academic thresholds used for grade retention. To see this, note that combined with (2) and (25)-(27) implies

3 Background and Data

The community of Madrid, one of Spain’s largest and wealthiest regions, has been administering a standardized test to students in sixth grade (the last grade of primary school) since the school year starting in 2004.²² Since 2008, test results come accompanied by a range of student characteristics, that include, for example, birth year and month, education and occupation of students’ parents, and country of birth. These data plus the name of the school sixth graders attended have been made available to us for three school years (the school years starting in 2008, 2009, and 2010). The data cover around 50,000 students per school year in around 1150 primary schools. See appendix table 1 for summary statistics.

The Spanish primary school enrollment rule is that children start school in the year they turn six years old and more than 99 percent of children follow this rule in Spain and the community of Madrid (about 0.5 percent of children start a year later and 0.5 percent a year earlier).²³ Spanish primary schools permit grade retention and rates of grade retention in primary school in Spain and the community of Madrid are higher than the OECD or European Union average. The share of 15 year olds in 2009 who report having been retained at least once in primary school is 11 percent in Spain and 11.8 percent in the community of Madrid, while it averages to 8 percent across OECD countries and to 7 percent across EU countries (PISA, 2009a). The share of students in Spain and the community of Madrid who report having been retained twice or more during primary school is 0.5 percent, which is similar to the OECD and EU average. It is likely that many of these students were retained at least once outside of Spain, as the community of Madrid forbids that students are retained twice during primary school and the rest of Spain allows it only in very exceptional cases.²⁴

²¹To see this, note that λ and 1 − λ are the probabilities that a student is not retained and retained respectively. Moreover, if gender peer efects did not afect retention rates, (9) implies that a one-percentagepoint increase in the share of girls in a birth cohort increases the share of girls in HG retained students will be exposed to by 1 − λ percentage points and the share of girls in HG non-retained students will be exposed to by λ percentage points. Hence, the increase in the share of girls the “average” student in a birth cohort with a one-percentage-point greater share of girls will be exposed to in HG is . The third term in the parentheses in (30) captures that the share of girls in a birth cohort may also afect the share of girls in HG through a diferential gender peer efect of girls on the retention rates of girls and boys.

²²The community of Madrid’s population in 2013 was around 6.5 million and income per capita close to 30 000 euros (Eurostat Regional Yearbook, 2013).

²³This is higher than in countries like Iceland and Norway, which are known for almost all children complying with the enrollment rule (e.g. Bedard and Dhuey, 2006). We thank the community of Madrid’s department of education, youth, and sports for providing this information for the community of Madrid. For

Our data for sixth graders in the community of Madrid do not contain information on whether students have repeated a grade. We know students’ birth years however and can therefore check whether students born in year t − 6 took the test during the school year starting in year t + 5, as they should have if they started primary school according to the enrollment rule and were not retained. We find that 14.2 percent of students took the test one year late and 0.5 percent two or more years late. Hence, the share of students who repeated twice or more in primary school according to PISA coincides with the share of students who took the test two or more years late in our data. To see whether the PISA statistic on the share of students who repeated once in primary school in the community of Madrid is consistent with the share of students who took the test one year late in our data, we need to take into account that around 0.5 percent of students start primary school one year later than specified by the enrollment rule.²⁵ These students end up taking the test at least one year late even if they are not retained in primary school. Moreover, PISA allows participating countries and regions to exclude special needs students, who represent 2.3 percent of the students in our data, and students who have language dificulties because they arrived only recently from abroad (PISA, 2009b). As most of the students excluded from PISA are likely to have repeated a grade (95 percent of special needs students in our data take the test at least one year late) the PISA statistic and our statistic appear to be consistent.

The standardized test administered to sixth graders in the community of Madrid has four components: mathematics, reading, dictation, and general knowledge. Each component is scored between 0 and 10 (this is the usual range for grades in Spain). We transform these raw scores into standard scores , where is the raw score of test taker i in the school year starting in year and and are the mean and standard deviation of the raw scores. We report results on gender peer efects based on an average across all test components and on the mathematics component only. We also considered the reading component only and did not find evidence of gender peer efects.²⁶

Spain the data come from the national statistical institute’s INEbase (2013).

²⁴Throughout Spain it is not permitted to retain students more than once in the same grade. See Spanish laws 10/2002 and 2/2006 and decree 22/2007 of the community of Madrid.

²⁵These children can be thought of as being retained in kindergarten.

We implement the birth cohort approach to gender peer efects for students born in 1997 and 1998. According to the Spanish enrollment rule, these students should have started primary school in 2003 and 2004 respectively (we index birth cohorts by the year they should have started primary school). As our data cover sixth graders in the school years starting in 2008, 2009, and 2010, we observe students from the 2003 birth cohort if they started according to the enrollment rule and were not retained in any grade; if they took the test one year late; or if they took the test two years late. Students from the 2003 birth cohort are not in our data if they took the test one or more years early (during or before the school year starting in 2007) or three or more years late (during or after the school year starting in 2011). We observe students from the 2004 birth cohort if they started according to the enrollment rule and were not retained; if they took the test one year early; or if they took the test one year late. Students from the 2004 birth cohort are not in our data if they took the test two or more years early (during or before the school year starting in 2007) or two or more years late (during or after the school year starting in 2011). As only a small share of students enter primary school early or are retained more than once during primary school in the community of Madrid and Spain, we end up missing a small share of students from the 2003 and the 2004 birth cohorts due to data availability issues. We can assess how many students we miss from each birth cohort by calculating the share of sixth graders in the school years starting in 2008, 2009, and 2010 who took the test early or two years late. Around 0.5 percent of sixth graders took the test two or more years late in our data and 0.3 percent of sixth graders took the test one or more years early. As a result, we estimate that we miss around 0.4 percent of the students of the 2003 and 2004 birth cohorts because of data availability issues. We treat the 2003 and 2004 birth cohorts symmetrically and drop students from the 2004 birth cohort who took the test one year early (in the school year starting in 2008) because we cannot observe students from the 2003 birth cohort who took the test one year early (in the school year starting in 2007). We also drop students from the 2003 birth cohort who took the test two years late (in the school year starting in 2010) because we cannot observe students from the 2004 birth cohort who took the test two years late (in the school year starting in 2011). As we treat the 2003 and 2004 birth cohorts symmetrically, we end up missing around 0.9 percent of the students from these birth cohorts because of data availability.

²⁶Interestingly, Lavy and Schlosser (2011) find that gender peer efects are statistically insignificant in Hebrew, which is probably closest to reading in our data.

Students who were retained in sixth grade attend sixth grade twice. We want to focus on students’ academic achievement at the end of primary schooling (sixth grade is the last grade of primary school in Spain) and therefore only include these students in our empirical analysis in their last year of primary schooling. Our data do not specify whether students repeated sixth grade however. We therefore proceed as follows. Based on information on fifth and sixth graders in the community of Madrid in the school years starting in 2009 and 2010 from the Spanish national statistical institute’s INEbase (2013) database, we estimate that 2.5 percent of sixth graders were repeating sixth grade.²⁷ As there are approximately 50,000 students attending sixth grade in the school year starting in 2009 in our data, we estimate that around 1250 students are repeating sixth grade. These are students from the 2003 birth cohort who should have attended sixth grade for the first time one year earlier, in the school year starting in 2008. It is useful to denote the set of students from the 2003 birth cohort who were in sixth grade in the school year starting in 2008 by S(2003,2008) and the set of students from the 2003 birth cohort who were in sixth grade in the school year starting in 2009 by S(2003,2009). The first set contains students from the 2003 birth cohort who reached sixth grade on age while the second set contains students from the 2003 birth cohort who were attending sixth grade one year late. The approximately 1250 students from the 2003 birth cohort who were repeating sixth grade in the school year starting in 2009 should be in both sets. We therefore look for all pairs of observations (i, j) with i from S(2003,2009) and j from S(2003,2008) where i and j attend the same school; were born in the same month and country; arrived to Spain at the same age if they were born abroad; and also coincide in terms of the country of birth and the level of education of mothers.²⁸ This yields 1172 pairs, which is close to the 1250 pairs we expected based on the retention rates from INEbase (2013). We then proceed under the assumption that these are the students from the 2003 birth cohort who repeated sixth grade in the school year starting in 2009. We use the same approach to identify students from the 2004 birth cohort who were repeating sixth grade in the school year starting in 2010. In this case the approach yields 1246 matching pairs, again close to the 1250 pairs we expected.

²⁷INEbase contains the share of fifth and sixth graders in the community of Madrid who are repeating fifth or sixth grade in these school years. This share has been very stable at around 2.5 percent since there are data (the school year starting in 2001). INEbase does not have information for sixth grade only.

²⁸We focus on mothers because according to the Spanish national statistical institute’s INEbase (2013) database, 90 percent of children who live with one parent only, live with their mother. Around 9 percent of the observations have missing values for the education level of mothers. In these cases we look for observations i and j that coincide in that the education of mothers is missing.

Student mobility across primary schools in the community of Madrid appears to be low compared to Texas and Israel. Only 1.9 percent of the students in the second grade of public primary schools in the school year starting in 2012 did not attend the same school a year earlier and the analogous statistics for grades three through six are very similar.²⁹ For privately managed schools there are no oficial data on student mobility. We therefore surveyed 224 privately managed primary schools (50 percent of the privately managed schools in our sample). We obtained responses from 198 schools. In these schools, 3.9 percent of the students in second grade in the school year starting in 2013 did not attend the same school a year earlier. As around 55 percent of the students in our data attend public schools, we estimate average annual mobility between first and second grade of primary school in the community of Madrid to be 2.8 percent.³⁰ For comparison, Lavy and Schlosser calculate that 7.9 percent of students left their primary school between first and second grade in Israel in 2002, while Hanushek, Kain, and Rivkin (2004) report an annual rate of student mobility in Texas of 24 percent.

4 Empirical Results

We now employ both the birth cohort approach and the grade level approach to estimate gender peer efects in our sample of Spanish primary schools.

4.1 Results Using the Birth Cohort Approach

We start by checking the balancedness of student characteristics with respect to the share of girls in the birth cohort. We have data on a range of student characteristics: the education levels of mothers and fathers (4 categories); the profession/occupation of mothers and fathers (8 categories); with whom students live (6 categories); when students started preschool/school (4 categories); where students and their parents were born; at what age students arrived in Spain if they were born abroad; and whether students have some disability.³¹

²⁹They are between 1.7 and 2.2 percent. We are very grateful to the community of Madrid’s department of education, youth, and sports for providing this information.

³⁰The likely causes of low school mobility in Spain are low residential mobility and a school assignment system for public as well as publicly funded, privately managed schools that strongly incentivates parents to pick a neighborhood school (e.g. Caldera and Andrews, 2011; Calsamiglia and Guell, 2013).

³¹For around 13 percent of students, we do not have the education of fathers and for around 9 percent we do not have the education of mothers. For profession/occupation we do not have data for around 7 percent of fathers and 12 percent of mothers. For other characteristics, the data is almost complete (we miss data for at most 0.1 percent of students).

To check whether the characteristics of students are balanced with respect to the share of girls in their birth cohort, we use the following estimating equation

\[x _ {i g s} ^ {j t} = \gamma_ {g s} ^ {j} + \gamma_ {g} ^ {j t} + \gamma_ {g} ^ {j} B i r t h c o h o r t G i r l s h a r e _ {s} ^ {t} + u _ {i g s} ^ {j t}\tag{31}\]

where stands for student characteristic i refers to individuals, to gender, s to schools, t to the 2003 and 2004 birth cohorts; and is the residual. School level diferences and gender specific trends in characteristic for gender are captured by and respectively. The parameter of interest is which captures whether student characteristic varies with the share of girls in the birth cohort.

Table 1 contains our estimates of in (31) as well as robust standard errors clustered by school. We also report a statistic for the hypothesis that for girls as well as boys. The results in the table are for schools with at most two classes per grade as our main results are for these schools. The results in table 1 indicate that student characteristics are balanced with respect to the share of girls in a birth cohort.³² Of the 74 characteristics we examine for boys and girls, only 4 yield statistically significant at the or 10 percent level (the number one would expect based on type I error). Moreover, the hypothesis cannot be rejected at any conventional significance level (the p-values are 0.69 for girls and 0.64 for boys).

We also use a second approach to see whether students in a birth cohort with a greater share of girls may have characteristics that are associated with better academic performance. The starting point are separate regressions of girls’ and boys’ individual test scores on school and birth cohort fixed efects, student characteristics, and indicators for missing student characteristics.³³ We then use the results to obtain a predicted test score for each student. This predicted test score is our best estimate of how well a student does academically given his or her characteristics. We then use the predicted test score as the left-hand-side variable in (31) to check balancedness with respect to the share of girls in the birth cohort. The advantage of this approach is that it quantifies any non-balancedness of student characteristics in terms of test performance.

Table 2 summarizes our results on the link between the predicted test scores of students and the share of girls in their birth cohort. Panel A reports the results for schools with at most two classes per grade (the sample used in table 1). Predicted test scores of students appear to be balanced for girls as well as boys. For boys, we obtain a point estimate of −0.007 with a clustered standard error of 0.034 while for girls we obtain a point estimate of 0.007 with a standard error of 0.036. The point estimates imply that a 10-percentage-point increase in the share of girls in a birth cohort is associated with an improvement in the predicted test performance of less than one-tenth of one percent of a standard deviation for girls and a worsening in the predicted test performance of less than one-tenth of one percent of a standard deviation for boys. Panels B and C report the coeficient estimates for the sample of schools with only one class per grade and for all schools respectively. These results also indicate that there is no statistically significant link between the share of girls in a birth cohort and the predicted test performance of students in the birth cohort.

³²We also checked for evidence of unbalancedness in missing data. To do so we replaced the left-hand side of (31) by an indicator variable that takes the value of one if and only if we do not have information on characteristic for individual i. This did not yield evidence of unbalancedness in missing data.

³³The regressions are where refers to standard test scores, i to individuals, , 2004 to birth cohorts, g to gender, s to schools, is an indicator variable that takes the value of one if and only if we do not have information on the student’s characteristic and is the residual. Individual characteristics account for around 11 to 15 percent of the variance in test performance.

Table 3 contains our estimates of gender peer efects using the birth cohort approach. All results are based on the estimating equation

\[t e s t _ {i g s} ^ {t} = \alpha_ {g s} + \alpha_ {t g} + \alpha_ {g} X _ {i g s} ^ {t} + \beta_ {g} B i r t h c o h o r t G i r l s h a r e _ {s} ^ {t} + v _ {i g s} ^ {t}\tag{32}\]

where refers to individual standard test scores, i to individuals, to birth cohorts, g to gender, and s to schools; is a vector collecting the characteristics j of student i and indicators for missing characteristics (if any); and is the residual. The table reports the least squares estimates of with robust standard errors clustered by school. We report results for the average test score and the test score in mathematics for all schools, schools with at most two classes per grade, and schools with one class per grade.

Table 3, panel A examines the evidence for gender peer efects when we include all 1155 schools in our analysis. Around 20 percent of these schools have more than three classes per grade (the maximum number of classes per grade is five).³⁴ As the share of girls in a birth cohort and school is a noisy measure of the share of girls in any specific classroom, this could lead to noisy estimates of gender peer efects if these efects are at the classroom level. Our estimates in table 3, panel A indicate statistically insignificant gender peer efects when we use the average test score to measure academic achievement. This continues to be the case when we control for all the individual characteristics in table 1 plus indicators for missing characteristics and also when we control for peer group characteristics at the birth cohort level. The peer group characteristics included are the share of mothers as well as fathers with each of the four education levels in table 1, the share of immigrants, and the number of students. Only when we measure academic achievement with the mathematics test score is there some evidence that boys do better when there are more girls in their birth cohort.

³⁴The median number of students in schools with at most two classes per grade is 36 and the maximum number of students 58. The median number of students in schools with more than two classes per grade is 73 and the maximum number 152.

Table 3, panel B examines the evidence for gender peer efects when we only include the 908 schools with less than two classes per grade in our analysis. On the one hand, dropping schools with more than two classes per grade reduces the sample size, which may make estimates noisier. On the other hand, there should be a closer link between the share of girls at the birth cohort level and the classroom level in schools with fewer classes per grade and this should make it easier to pick up gender peer efects at the classroom level (if any). Our estimates in table 3, panel B indicate statistically significant gender peer efects of girls on the academic achievement of boys but not girls. This is the case whether we measure academic achievement using the average test score or the mathematics test score, and the estimates are very similar when we control for individual characteristics and peer group characteristics. The point estimates indicate that a 10-percentage-point increase in the share of girls in a birth cohort improves the test performance of boys by around 2.5 percent of a standard deviation.

Table panel C examines the evidence for gender peer efects when we only include the 331 schools with one class per grade in our analysis. This is a substantially smaller sample of schools, which could make estimates much noisier. On the other hand, the link between the share of girls at the birth cohort level and the classroom level is as close as can be. Our estimates continue to suggest statistically significant gender peer efects of girls on the academic achievement of boys but not girls for the average test score and the mathematics test score when we control for individual characteristics and peer group characteristics. The point estimates indicate that a 10-percentage-point increase in the share of girls in a birth cohort improves the average test score of boys by around 3 percent of a standard deviation and the mathematics test score by around 4 percent of a standard deviation.³⁵

4.2 Results Using the Grade Level Approach

Table 4 reports our estimates of gender peer efects using the grade level approach. Results are based on the grade level estimating equation

\[t e s t _ {i g s \tau} = \alpha_ {g s} + \alpha_ {g \tau} + \alpha X _ {i g s \tau} + \beta_ {g} G r a d e G i r l s h a r e _ {s \tau} + v _ {i g s \tau}\tag{33}\]

where all variables are defined analogously to (32) and refers to the school years starting in 2008 and 2009. Overall, the grade level estimates of gender peer efects point in a diferent direction than the birth cohort estimates in table 3. For the two larger samples in panels A and B, we obtain a statistically significant efect of the share of girls on the academic achievement of girls but not on the achievement of boys.³⁶

³⁵We do not find statistically significant efects of the share of girls in a birth cohort on the probability of grade retention of individual students. The grade retention efect captured by the last term on the right-hand side of (30) does therefore not appear to be at work in our data.

The diference in the pattern of gender peer efects using the birth cohort approach in table 3 and the grade level approach in table 4 is consistent with our theoretical model if (i) grade repetition improves retained students’ skills but students retained in the past still tend to do worse on average than non-retained students in the same grade and (ii) grade level diferences in gender composition are mostly driven by shocks to gender composition or skills at the birth cohort level. In this case, our model implies that even without any gender peer efects, the share of girls correlates positively with the academic skills of girls and negatively with the academic skills of boys at the grade level. As a result, the grade level approach overestimates the gender peer efect of girls on girls’ academic achievement and underestimates the gender peer efect of girls on boys’ academic achievement.

Table 5 replicates table 2 at the grade level to see whether there are signs of nonbalancedness in this case. The left-hand-side variable of the regressions underlying table 5 is the individual predicted test score also used in table 2. The right-hand-side variables are school fixed efects, school year fixed efects, and the share of girls at the grade level. For the two larger samples in panels A and C, there is now evidence of statistically significant non-balancedness. The grade level correlation between the predicted test score of girls and the share of girls is significantly positive in the sample of schools with at most two classes per grade and the grade level correlation between the predicted test score of boys and the share of girls is significantly negative in the sample of all schools.³⁷

³⁶We do not include the school year starting in 2010 to keep the empirical analysis using the grade level approach as close as possible to that using the birth cohort approach. When we include the 2010 school year, the results are very similar to those in table 4 except that estimates become more precise and that the (positive) efect of girls on girls’ academic achievement in schools with one class per grade is often statistically significant. It is interesting to note that the efect of the share of girls in a grade on the test performance of girls in table 4 is statistically significant in the sample with all schools, while the birth cohort approach yielded statistically insignificant efects for all schools. As already mentioned, if gender peer efects are at the classroom level, the results using the birth cohort approach can be explained by the share of girls at the birth cohort level being a noisy measure of the share of girls at the classroom level in schools with several classes per grade. On the other hand, the grade level approach will produce the spurious gender peer efects described in section 2.2 no matter how many classes there are per grade as long as grade level diferences in gender composition are mostly driven by shocks to gender composition or skills at the birth cohort level.

³⁷The signs of the efects in table 5 are consistent with our theoretical model’s explanation for the diference between grade level and birth cohort estimates of gender peer efects. When assessing the magnitude it should be kept in mind that much of students’ test performance cannot be predicted by the individual characteristics we observe (they account for around 11 to 15 percent of the variance in test performance). When we include the 2010 school year, the results in table 5 change in that a greater share of girls correlates significantly negatively with the predicted test scores of boys in all samples.

5 Counterfactual Analysis

As rates of grade retention in Spain are high compared to the OECD and EU average, it is natural to wonder whether the birth cohort approach could yield a diferent pattern of gender peer efects than the grade level approach when retention rates are around the OECD and EU average. We examine this question using counterfactual experiments based on a calibrated model. The basis is our theoretical model of schools with grade retention. We first calibrate the model to match our gender-peer-efect estimates for schools with at most two classes per grade in tables 3 and 4. Then we lower the academic thresholds for grade retention in the model – which implies that rates of grade retention fall – and simulate data. This allows us to estimate gender peer efects with the birth cohort and the grade level approach at successively lower rates of grade retention.

5.1 From the Theoretical to the Simulated Model

Our simulations are based on the model described in section 2 and summarized in figure 1. All students in birth cohort t start school in year t and spend the first L years of primary school together in lower grades (LG). After L years in LG, students in birth cohort t have academic skills drawn from a school, gender, and birth cohort specific distribution Students with skills above the academic threshold are promoted to high grade (HG). Students with skills below the academic threshold spend one extra year in LG before being admitted to HG.³⁸ After one year in HG, students leave primary school.

In the model in section 2 we took the distribution to be uniform. Now we can be more general and allow for three alternative skill distributions. On the other hand, we need to assume that the skills students accumulate in HG only depend on the school and on gender, This implies that the HG test scores of students in birth cohort t who have not been retained during primary school are with drawn from a distribution truncated below at and that the HG test scores of students in birth cohort t who have been retained during primary school are with drawn from truncated above at . This is equivalent to assuming that the HG test scores of non-retained students in birth cohort t are equal to drawn from shifted by and truncated below at and that the HG test scores of retained students are equal to with drawn from shifted by and truncated above at . We can therefore simplify the notation by absorbing HG skill accumulation into the mean of and the academic thresholds for grade retention and set . Gender peer efects in the model (if any) are assumed to afect the academic skills after years in LG or the thresholds used for grade retention as described in (25) and (26) respectively.

³⁸Hence, our simulated model continues to assume that all students who are retained are retained at the same stage in primary school. According to the information available from the Spanish national statistical institute’s (2013) INEbase database, roughly the same share of students is retained at diferent stages in primary school in the community of Madrid. But as our data do not specify in which grade students were retained, we cannot calibrate a model where students may be retained at any stage in primary school.

³⁹We need to make this assumption to be able to calibrate the model as our data do not allow us to calibrate the distribution of academic skills at diferent stages of primary school.

In the simulations there will be 1000 children per school and birth cohort with the number of girls determined by a binomial distribution with a probability that a student is female. The number of schools is set equal to the number of schools with at most two classes per grade in our data.

5.2 Baseline Calibration

We need to calibrate (a) the parameters of the skill distributions ; (b) the academic thresholds for grade retention the changes in skills that come with grade retention; and (d) the probability that a student is female . We choose these parameters so that when the calibrated model is used to simulate data and the simulated data is then used to estimate gender peer efects using the grade level and birth cohort approach, the results match what we obtained for schools with at most two classes per grade in tables 3 and 4. To do it is useful to note that the key estimation inputs of the grade level and birth cohort approach are (i) the school and gender specific average test scores of girls and boys at the birth cohort and the grade level, which are the left-hand-side variables of the gender-peer-efect regressions; and (ii) the school specific shares of girls at the birth cohort and the grade level, which are the right-hand-side variables. If the data simulated with the calibrated model reproduces these moments, it will reproduce our gender-peer-efect estimates at the birth cohort and the grade level. We therefore calibrate the model parameters to match these moments.

To get there we have to take diferent approaches depending on the birth cohort. Our data allow us to observe students in the 2003 and 2004 birth cohorts (the two birth cohorts we used to estimate gender peer efects at the birth cohort level) whether they were retained or not. We can therefore calibrate the parameters corresponding to these birth cohorts so as to match the average test score of non-retained girls and boys in each school; the average test score of retained girls and boys in each school; the share of retained girls and boys in each school; and the share of girls in each school.

Our estimates of gender peer efects at the grade level are based on sixth graders in the school years starting in 2008 and 2009. Sixth graders in 2008 who have been retained once come from the 2002 birth cohort. We therefore also have to calibrate the parameters of the

2002 birth cohort (in addition to the parameters of the 2003 and 2004 birth cohorts). But as we lack data on non-retained students from the 2002 birth cohort we need to take a diferent approach. We set some parameters equal to the average of the calibrated parameters of the 2003 and 2004 birth cohorts and other parameters to match grade level statistics.

As already mentioned, our simulated model employs both the uniform distribution we assumed for the distribution of academic skills in section 2 and three alternative skill distributions. As a result, we end up with four calibrated models that we can use for counterfactual experiments. We now discuss the calibration of each model in more detail.

5.2.1 Uniform Skill Distribution

In the model in section 2 the distribution of skills was uniform The distribution parameters we need to calibrate are the school, birth cohort, and gender specific means and the parameter θ governing skill dispersion.

Calibrating the Parameters of the 2003 and 2004 Birth Cohorts We calibrate the school, birth cohort, and gender specific means and the parameter θ governing the dispersion of skills jointly with the academic thresholds for grade retention to match (i) the share of girls and boys among students of the same gender who were not retained in any grade at the school and birth cohort level, ; (ii) the average test score of non-retained students at the school, birth cohort, and gender level, ; and (iii) the variance of the test score of non-retained students, , averaged across schools, birth cohorts, and gender. Denoting the cumulative distribution of the uniform by , this yields the first three calibration equations

(34)

\[1 - F _ {g s} ^ {t} (p _ {g s} ^ {t}) = \overline {{\lambda}} _ {g s} ^ {t}\tag{35}\]

\[E _ {g s} ^ {t} (a \left| a \geq p _ {g s} ^ {t}\right) = \overline {{E}} _ {g s} ^ {t} (t e s t \mid n o n - r e t a i n e d)\tag{36}\]

\[\frac {1}{4 S} \sum_ {t, s, g} V a r _ {g s} ^ {t} (a | a \geq p _ {g s} ^ {t}) = \frac {1}{4 S} \sum_ {t, s, g} \overline {{V a r}} _ {g s} ^ {t} (t e s t | n o n - r e t a i n e d)\]

where S is the number of schools. The changes in skills that come with grade repetition are calibrated to get the average test score of retained students in the model to match the average test score of retained students (if any) at the school, birth cohort, and gender level in the data,

\[\delta_ {g s} ^ {t} + E _ {g s} ^ {t} (a \left| a < p _ {g s} ^ {t}\right) = \overline {{E}} _ {g s} ^ {t} (t e s t \left| r e t a i n e d\right). ^ {4 0}\tag{37}\]

⁴⁰Our calibration yields positive average and median changes across schools for girls as well as boys. There are some empirical estimates of the efect of grade retention on academic skills. The findings of Jacob

This yields a system of calibration equations with parameters, which can be solved in closed form. 41

The probability of a student in school s and birth cohort t being female is set equal to the share of girls in the school and birth cohort.

Calibrating the Parameters of the 2002 Birth Cohort Students who start sixth grade in 2008 and have been retained once come from the 2002 birth cohort. To do counterfactual experiments using the grade level approach to gender peer efects, we therefore need to calibrate the parameters of the 2002 birth cohort. We do not observe the full 2002 birth cohort however, as non-retained students from this birth cohort attended sixth grade outside of the period covered by our data. As a result, we have to take a diferent approach to that used for the 2003 and 2004 birth cohorts. We proceed in the following way. The dispersion parameter for the 2002 birth cohort is set equal to the calibrated value for the 2003 and 2004 birth cohorts. The probability of a student being female is set equal to the average for the same school that we calibrated for the 2003 and 2004 birth cohorts. The parameters of the skill distribution for girls and boys and are set equal to the averages that we calibrated for the same school and gender for the 2003 and 2004 birth cohorts. The skill changes associated with grade repetition for girls and boys and and the academic thresholds used for grade retention and are calibrated to match the average test score of retained girls and boys from the 2002 birth cohort as well as the share of (retained) girls and boys from the 2002 birth cohort among sixth graders in the school year starting in 2008 at the school level.

5.2.2 Normal Skill Distribution

We also calibrate a model where the distribution of skills is taken to be normal with mean and standard deviation θ, . The calibration follows the same steps as the calibration of the uniform distribution. As there is no closed-form solution for the calibration equations in (34)-(36), we solve the equations numerically.⁴²

and Lefgren (2004) for the US suggest a positive short-term efect on skills of grade retention in third grade but no efect of grade retention in sixth grade. Jacob and Lefgren (2009) find that retained sixth graders in the US are less likely to repeat eighth grade than students who were narrowly not retained in sixth grade. Both Jacob and Lefgren’s (2009) study and Manacorda’s (2012) analysis for Uruguay find that retained students are more likely to drop out in secondary school than narrowly not retained students.

⁴¹Using the formula for the variance of uniform distributions and (2), (34), and (36) yields that θ is the positive root of . Given θ, (2) and (4) imply that (34) and (35) are linear in and

⁴²While these and subsequent numerical solutions could in principle yield multiple solutions, the solutions found appeared to be unique in practice.

5.2.3 A First Skill Distribution Calibrated to Ensure Raw Test Scores in the [0, 10] Range

So far we have calibrated the skill distributions using standard test scores and ignored that standard test scores are obtained from raw test scores that lie between 0 and 10. We now present a first approach that accounts for these bounds on raw test scores.⁴³

To do so we assume that students’ raw skills are generated according to

\[z _ {i g s} ^ {t} = 1 0 \left(\frac {1}{1 + \exp (- v _ {i g s} ^ {t})}\right) \mathrm{with} v _ {i g s} ^ {t} \sim N (\alpha_ {g s} ^ {t}, \theta^ {2})\tag{38}\]

which ensures . We use (38) to obtain the density and cumulative distribution of students’ standard test scores as where and are the mean and standard deviation of raw scores in the year non-retained students from birth cohort t took the test. The parameters in (38) as well as and θ are then calibrated by solving (34)-(36) for the 2003 and 2004 birth cohorts. As there is no closed-form solution for the calibration equations, we solve the equations numerically. Once we have calibrated 2 and θ for the 2003 and 2004 birth cohorts, we obtain the values of for these birth cohorts using (37). For the parameters of the 2002 birth cohort we proceed as in the case of the uniform distribution.

5.2.4 A Second Skill Distribution Calibrated to Ensure Raw Test Scores in the [0, 10] Range

A second skill distribution that we calibrate to account for the bounds on raw test scores is the triangular distribution where l and u are the lower and upper bound and the mode. Raw test scores can be ensured to lie between 0 and 10 by assuming and . Standard test scores are then again obtained as where and are the mean and standard deviation of raw scores in the year non-retained students from birth cohort t took the test. The triangular distribution generally has three birth cohort, school, and gender specific parameters to be calibrated. The calibration has to therefore proceed somewhat diferently than in previous cases.

Our calibration of the triangular skill distributions and the academic thresholds for grade retention for the 2003 and 2004 birth cohorts matches the share of non-retained students and the standard test scores of non-retained students by solving (34)-(35). As there is no closedform solution, we again rely on numerical solutions. We first try to solve these equations for a given birth cohort, school, and gender assuming full support on [0, 10] , that is and . In this case the calibration equations (34)-(35) have only two unknown parameters for a given birth cohort, school, and gender (the mode of the triangular distribution and the academic threshold for grade retention . For birth cohort, school, and gender combinations where (34)-(35) have no solution with full support on [0, 10] we either (i) relax the assumption that the lower bound of the triangular distribution is 0 and solve for the mode as well as the lower bound of the distribution or (ii) we relax the assumption that the upper bound is 10 and solve for the mode as well as the upper bound of the distribution.⁴⁴ Once we have obtained a solution for the parameters of the triangular skill distribution and the academic thresholds, we calibrate the values of for the 2003 and 2004 birth cohorts using (37). For the parameters of the 2002 birth cohort we proceed as in the case of the uniform distribution.

⁴³This allows us to check that the results of our counterfactual experiments are not driven by raw test scores outside of the 0 to 10 range.

5.3 Counterfactual Simulations of Lower Grade Retention Thresholds

We now use the calibrated models to simulate data and estimate gender peer efects with the birth cohort and the grade level approach. All simulations are based on 1000 children per school and birth cohort. The gender of students is determined by a binomial distribution with a school and birth cohort specific probability of a student being a girl . Once the gender of students has been determined, we draw their skills from the school, birth cohort, and gender specific skill distributions. Students with skills above the threshold for grade retention are assumed to enter sixth grade in year and achieve a test score in the standardized test. Students with skills below the threshold for grade retention are assumed to enter sixth grade in year and achieve a test score . The student data simulated with the model are then used to estimate gender peer efects with the grade level approach and the birth cohort approach.

The grade level approach relates the average test score of girls and boys in sixth grade to the share of girls in sixth grade. The average test score of girls and boys in sixth grade in the school year starting in , 2009 is obtained as

\[t e s t _ {g s \tau} = \mu_ {g s \tau} E (t e s t _ {i g s} ^ {\tau - 5} | n o n - r e t a i n e d) + (1 - \mu_ {g s \tau}) E (t e s t _ {i g s} ^ {\tau - 6} | r e t a i n e d)\tag{39}\]

⁴⁴It is straightforward to determine which of the two assumptions has to be relaxed to find a solution. For each birth cohort, school, and gender we first obtain all pairs that solve (34) assuming and . Then we obtain the highest and the lowest among these combinations. If the highest is lower than , the average test score of non-retained students in the data is too high to be matched with a triangular distribution with full support and we relax the assumption that the lower bound is zero. If the lowest is higher than |non-repeater ) , the average test score of non-retained students in the data is too low to be matched with a triangular distribution with full support and we relax the assumption that the upper bound is ten.

where is the simulated share of non-retained girls or boys among sixth graders of the same gender, while and are the simulated average test scores of non-retained students from birth cohort and retained students from birth cohort respectively. The share of girls at the grade level, , in the simulations is obtained as the number of non-retained girls in birth cohort plus the number of retained girls in birth cohort divided by the number of sixth graders in the school year starting in τ. Gender peer efects at the grade level are then estimated by regressing within-school changes in the average test score of girls or boys, , on the corresponding changes in the share of girls in sixth grade,

The birth cohort approach relates the average test score of girls and boys in a birth cohort to the share of girls in the birth cohort. The average test score of girls and boys in birth cohorts , 2004 is obtained as

\[t e s t _ {g s} ^ {t} = \lambda_ {g s} ^ {t} E (t e s t _ {i g s} ^ {t} | n o n - r e t a i n e d) + (1 - \lambda_ {g s} ^ {t}) E (t e s t _ {i g s} ^ {t} | r e t a i n e d)\tag{40}\]

where is the simulated share of non-retained girls or boys among students of the same gender in birth cohort t, while and are the simulated average test scores of non-retained and retained students from birth cohort t respectively. Gender peer efects at the birth cohort level are then estimated by regressing within-school changes in the average test score of girls or boys, tes , on the corresponding changes in the share of girls in these birth cohorts.

5.3.1 Baseline Simulation and Estimation

In the simulation baseline, we simulate data assuming that all model parameters are at their calibrated values. These data are then used to estimate gender peer efects with the birth cohort and the grade level approach. The results are in the first rows of tables 6 and 7. We report point estimates and standard errors averaged across 100 simulations.⁴⁵ The point estimates are very similar to the empirical results we obtained for schools with at most two classes per grade in tables 3 and 4. This is not surprising of course. As our calibration matches the average test score at the birth cohort, school, and gender level for the 2003 and 2004 birth cohorts, as well as the share of girls in these birth cohorts, our gender-peer-efect estimates at the birth cohort level with the simulated data have to be very close to our empirical results. As our calibration also matches the average test score of sixth grade girls and boys as well as the share of girls in sixth grade in the school years starting in 2008 and 2009, the same has to be true for gender-peer-efect estimates at the grade level. The main reason why the gender-peer-efect estimates in the simulation baseline do not coincide exactly with our empirical results is that all birth cohorts are of the same size in our simulations, while the size of birth cohorts in our data difers somewhat across schools and years.

⁴⁵As we have 1000 students per school and birth cohort, results vary little across simulations.

5.3.2 Counterfactual Experiments

In our counterfactual experiments we successively lower the academic thresholds used for grade retention by the same amount for girls and boys in all schools and birth cohorts, starting from the calibrated thresholds. The amount by which we lower these thresholds is chosen to hit certain targets for the share of retained boys averaged across all schools (10, 7.5, 5, 2.5, and 0 percent).⁴⁶ As we change the academic thresholds used for grade retention by the same amount independently of the share of girls in the birth cohort, these changes can be thought of as afecting the intercept in (26) but not the marginal efect of the share of girls on the threshold (if any). All model parameters except the thresholds for grade retention are held at their baseline values.

The results of our counterfactual experiments for the four diferent distributions for students’ academic skills are summarized in tables 6 and 7.⁴⁷ The results using the birth cohort approach to gender peer efects indicate a statistically significant gender peer efect of girls on boys’ academic achievement and a statistically insignificant efect of girls on girls’ academic achievement in the simulation baseline and all counterfactual experiments. The size of these efects varies little across counterfactual experiments.⁴⁸

In the simulation baseline, the grade level approach yields a diferent pattern of gender peer efects than the birth cohort approach. The estimates indicate a statistically significant gender peer efect of girls on girls’ academic achievement and a statistically insignificant efect of girls on boys’ academic achievement. This is not surprising as the model calibration targets the gender-peer-efect estimates for schools with at most two classes per grade in tables 3 and 4. The main finding in tables 6 and 7 is that the grade level approach continues to yield a statistically significant gender peer efect of girls on girls’ academic achievement in the counterfactual experiments where the average retention rate for girls and boys is around or somewhat below the average retention rate in the EU or OECD (7 and 8 percent respectively). The grade level approach also continues to indicate a statistically insignificant efect of girls on boys’ academic achievement at average EU or OECD retention rates for all distributions except the uniform. Hence, the results in tables 6 and 7 suggest that the birth cohort and the grade level approach may yield diferent patterns of gender peer efects at rates of grade retention around or below the OECD and EU average. Grade level and birth cohort estimates of gender peer efects coincide when the academic thresholds for grade retention are so low that retention rates are zero (the two approaches become identical in this case).

⁴⁶In the case of the normal distribution we can never achieve a retention rate that is exactly zero and we therefore targeted a retention rate of 0.001 percent.

⁴⁷As the models used for the counterfactual experiments are calibrated to match our estimates of gender peer efects in schools with at most two classes per grade, the retention rate in the simulation baseline difers from the overall retention rate in our data.

⁴⁸To see why gender-peer-efect estimates using the birth cohort approach may vary little across counterfactual experiments it is useful to return to the expression for the birth cohort gender peer efect in (30). As the model underlying our simulations assumes , all the variation in the birth cohort gender peer efect across counterfactual experiments comes from the product between the change in skills associated with grade repetition and the marginal efect of the share of girls in a birth cohort on the retention rate. As the marginal efect of the share of girls in a birth cohort on retention rates turns out to vary little across counterfactual experiments, estimates of gender peer efects at the birth cohort level are similar at diferent retention rates. With , changes in the academic thresholds for grade retention would also afect the strength of gender peer efects through the second term on the right-hand side of (30). This term captures that changes in the retention policy/rate afect the strength of gender peer efects using the birth cohort approach because they afect the expected share of girls the average student in a birth cohort is exposed to in higher grades (when some students will have been retained). We need to assume as our data do not allow us to calibrate the distribution of academic skills at diferent stages of primary school.

6 Conclusions

Schools are the obvious place to look for gender peer efects that may arise when girls and boys learn together. The selection of students with diferent skills into diferent schools can be bypassed by exploiting within-school variation in gender composition as in Hoxby (2000) and Lavy and Schlosser (2011). Our main theoretical objective has been to understand within-school selection issues that may arise in school systems allowing for grade retention. We found that grade retention generally leads to spurious gender peer efects in academic achievement when gender peer efects are estimated by exploiting within-school diferences in gender composition at a given grade level in diferent years. The direction of spurious gender peer efects depended mainly on the source of shocks to grade level gender composition and the impact of grade repetition on student skills.

Because of the limitations of the grade level approach in school systems with grade retention, we have proposed estimating gender peer efects in academic achievement by exploiting within-school diferences in gender composition across birth cohorts. Students are assigned to the same birth cohort if they should have started school in the same year according to the school system’s enrollment rule. The birth cohort approach examines whether girls or boys in a birth cohort with a greater share of girls do better academically than students of the same gender in other birth cohorts in the same school. A key feature of the approach is that it does not yield spurious gender peer efects when there is grade retention.

Our main empirical objective has been to estimate gender peer efects in Spanish primary schools, where rates of grade retention are high by international comparison. The birth cohort approach yielded a statistically significant positive gender peer efect of girls on boys’ academic achievement by the end of primary school but a statistically insignificant efect of girls on girls’ achievement (or equivalently, a statistically significant negative gender peer efect of boys on boys’ academic achievement but a statistically insignificant efect of boys on girls’ achievement). The grade level approach to gender peer efects tended to yield a diferent pattern of gender peer efects. The diference between the birth cohort and grade level estimates of gender peer efects turned out to be consistent with our theoretical model if grade repetition improves retained students’ skills but students retained in the past still tend to do worse on average than non-retained students in the same grade and if grade level diferences in gender composition are mostly driven by shocks to gender composition or skills at the birth cohort level.

Appendix

To prove the results around the inequalities in (11), (18), and (24), we first linearize the within-school grade level change in the share of girls in HG, 2 around , and . Making use of (2) and (9), this yields

\[\begin{array}{r c l} \Delta g i r l s h _ {s \tau} & = & \left(\Delta \lambda_ {f s} ^ {\tau - L} - \Delta \lambda_ {m s} ^ {\tau - L} - \Delta \lambda_ {f s} ^ {\tau - L - 1} + \Delta \lambda_ {m s} ^ {\tau - L - 1}\right) / 4 \\ & & + \lambda \Delta \phi_ {s} ^ {\tau - L} + (1 - \lambda) \Delta \phi_ {s} ^ {\tau - L - 1} \end{array}\tag{A1}\]

where , and Linearizing the within-school grade level change in the average test score of girls and boys in HG, , using (4)-(9) yields

\[\begin{array}{r l} & {\Delta t e s t _ {g s \tau} = \lambda (\Delta \alpha_ {g s} ^ {\tau - L} + \Delta p _ {g s} ^ {\tau - L - 1}) / 2 + (1 - \lambda) (\Delta \alpha_ {g s} ^ {\tau - L - 1} + \Delta p _ {g s} ^ {\tau - L - 1}) / 2} \\ & {+ (\theta - \delta) [ 2 (1 - \lambda) \lambda (2 \phi_ {s} ^ {\tau - L - 1} - \phi_ {s} ^ {\tau - L} - \phi_ {s} ^ {\tau - L - 2}) + (1 - \lambda) \Delta \lambda_ {g s} ^ {\tau - L} + \lambda \Delta \lambda_ {g s} ^ {\tau - L - 1} ]} \end{array}\tag{A2}\]

where and

The standard formula for the least squares regression slope implies that the sign of the least squares slope when regressing (A2) on (A1) across schools is equal to the sign of the covariance between and across schools. We therefore proceed to calculate this covariance under the diferent assumptions underlying the results around inequalities in (11), (18), and (24). For example, the inequality in (11) was derived assuming 2 and hence . In this case (A1) simplifies to and (A2) to . Hence, the assumption of i.i.d. shocks to the share of girls in a birth cohort implies . As (3) implies it follows that a least squares regression of within-school changes of HG girls’ average test scores on within-school changes of the share of girls in HG yields a strictly positive least squares slope if , which proves (11).

The inequality in (18) was derived assuming and . Substituting in (A1) and (A2) yields , this implies a strictly positive least squares slope when regressing (A2) on (A1) across schools if , which proves (18). The result in (24) can be proven analogously.

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Table 1: Balancedness at the Birth Cohort Level (Schools with at Most Two Classes per Grade)

(Continued on next page)

(Continued from previous page)

Notes: Estimates of the efect of the share of girls in a birth cohort on listed characteristics, see (31) in the main text for the estimating equation. Robust standard errors clustered at school level in parentheses. The null hypothesis of the joint F-statistic is that the efect of the share of girls in the birth cohort on all listed characteristics is zero. Professions: (1) military, (2) manager or civil servant, (3) professional or technician, (4) secretarial, (5) police, fireman, salesperson, hotel or restaurant staf, (6) construction or maintenance, (7) manufacturing, (8) domestic or cleaning service, private security, concierge. Household composition: with whom the student lives. School starting age: at what age pre-school or school was started. Arrival age: at what age the student arrived to Spain (for students born in Spain the value is 0). Immigrant: student and at least one parent were born outside of Spain. There are 908 schools with at most two classes per grade. *, **, *** significant at the 10 percent, 5 percent, and 1 percent significance level respectively.

Table 2: Balancedness of Predicted Test Scores at the Birth Cohort Level

Notes: Estimates of the efect of the share of girls in a birth cohort on predicted individual test scores. All regressions include school and birth cohort fixed efects. Test scores are predicted separately for girls and boys based on regressions of test scores on all characteristics in table 1 and missing data dummies plus school and birth cohort fixed efects. The estimating equation is (31) in the main text. Robust standard errors clustered at the school level in parentheses. There are 1155 schools in total; 908 schools with at most two classes per grade; and 331 with one class per grade. *, **, *** significant at the 10 percent, 5 percent, and 1 percent significance level respectively.

Table 3: Gender Peer Efects Using the Birth Cohort Approach

Notes: Estimates of the efect of the share of girls in a birth cohort on the test scores of boys and girls. The estimating equation is (32) in the main text. Robust standard errors clustered at the school level in parentheses. Individual controls refer to the individual characteristics in table 1 and missing data dummies. Peer group controls are the shares of mothers and of fathers in the birth cohort with the four educations levels in table 1, the share of immigrants in the birth cohort, and the size of the birth cohort. There are 1155 schools in total; 908 schools with at most two classes per grade; and 331 with one class per grade. *, **, *** significant at the 10 percent, 5 percent, and 1 percent significance level respectively.

Table 4: Gender Peer Efects Using the Grade Level Approach

Notes: Estimates of the efect of the share of girls in sixth grade on the test scores of boys and girls. The estimating equation is (33) in the main text. Robust standard errors clustered at the school level in parentheses. Individual controls refer to the individual characteristics in table 1 and missing data dummies. Peer group controls are the shares of mothers and of fathers in the grade with the four educations levels in table 1, the share of immigrants in the grade, and the number of students in the grade. There are 1155 schools in total; 908 schools with at most two classes per grade; and 331 with one class per grade. *, **, *** significant at the 10 percent, 5 percent, and 1 percent significance level respectively.

Table 5: Balancedness of Predicted Test Scores at the Grade Level

Panel B: One-Class Schools

Panel C: All Schools

Notes: Estimates of the efect of the share of girls in sixth grade on predicted individual test scores. All regressions include school and school year fixed efects. Test scores are predicted separately for girls and boys based on regressions of test scores on all characteristics in table 1 and missing data dummies plus school and grade fixed efects. The estimating equation is as in (31) in the main text but at the grade level. Robust standard errors clustered at the school level in parentheses. There are 1155 schools in total; 908 schools wit at most two classes per grade; and 331 with one class per grade. *, **, *** significant at the 10 percent, 5 percent, and 1 percent significance level respectively.

Table 6 Counterfactual Experiments of Lower Thresholds for Grade Retention Based on the Uniform and Normal Distribution of S kills

Notes: The coeficients and standard errors in parentheses are averages across 1 00 simulations .

Table 7: Counterfactual Experiments of Lower Thresholds for Grade Retention Based on Distributions Ensuring Raw Skills in the 0- 1 0 Range

Notes: The coeficients and standard errors in parentheses are averages across 1 00 simulations .

App endix Table 1 Descriptive St atistics

Notes: Average score is an average of mathematics , reading , general knowledge , and dictation .

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16. 2013-17: “Rainfall Risk and Religious Membership in the Late Nineteenth-Century US”, Philipp Ager y Antonio Ciccone.
17. 2013-16: “Immigration in Europe: Trends, Policies and Empirical Evidence”, Sara de la Rica, Albrecht Glitz y Francesc Ortega.
18. 2013-15: “The impact of family-friendly policies on the labor market: Evidence from Spain and Austria”, Sara de la Rica y Lucía Gorjón García.
19. 2013-14: “Gender Gaps in Performance Pay: New Evidence from Spain”, Sara de la Rica, Juan J. Dolado y Raquel Vegas.
20. 2013-13: “On Gender Gaps and Self-Fulfilling Expectation: Alternative Implications of Paid-For Training”, Juan J. Dolado, Cecilia García-Peñalosa y Sara de la Rica.
21. 2013-12: “Financial incentives, health and retirement in Spain”, Pilar García‐Gómez, Sergi Jiménez‐Martín y Judit Vall Castelló.
22. 2013-11: “Gender quotas and the quality of politicians”, Audinga Baltrunaite, Piera Bello, Alessandra Casarico y Paola Profeta.
23. 2013-10: “Brechas de Género en los Resultados de PISA: El Impacto de las Normas Sociales y la Transmisión Intergeneracional de las Actitudes de Género”, Sara de la Rica y Ainara González de San Román.
24. 2013-09: “¿Cómo escogen los padres la escuela de sus hijos? Teoría y evidencia para España”, Caterina Calsamiglia, Maia Güell.
25. 2013-08: “Evaluación de un programa de educación bilingüe en España: El impacto más allá del aprendizaje del idioma extranjero”, Brindusa Anghel, Antonio Cabrales y Jesús M. Carro.
26. 2013-07: “Publicación de los resultados de las pruebas estandarizadas externas: ¿Tiene ello un efecto sobre los resultados escolares?”, Brindusa Anghel, Antonio Cabrales, Jorge Sainz e Ismael Sanz.
27. 2013-06: “DYPES: A Microsimulation model for the Spanish retirement pension system”, F. J. Fernández-Díaz, C. Patxot y G. Souto.
28. 2013-05: “Vertical differentiation, schedule delay and entry deterrence: Low cost vs. full service airlines”, Jorge Validoa, M. Pilar Socorroa y Francesca Medda.
29. 2013-04: “Dropout Trends and Educational Reforms: The Role of the LOGSE in Spain”, Florentino Felgueroso, María Gutiérrez‐Domènech y Sergi Jiménez‐Martín.
30. 2013-03: “The European Crisis and Migration to Germany: Expectations and the Diversion of Migration Flows”, Simone Bertoli, Herbert Brücker y Jesús Fernández-Huertas Moraga.
31. 2013-02: “Understanding Different Migrant Selection Patterns in Rural and Urban Mexico”, Jesús Fernández-Huertas Moraga.
32. 2013-01: “Publicizing the results of standardized external tests: Does it have an effect on school outcomes?, Brindusa Anghel, Antonio Cabrales, Jorge Sainz y Ismael Sanz.