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he impact of peer achievement and peer heterogeneity onwn achievement growth: Evidence from school transitions§

avid Kiss *

iversity of Erlangen-Nuremberg, Lange Gasse 20, D-90403 Nuremberg, Germany

Introduction

Peer characteristics are important determinants ofrental school choice decisions. Peer effects are also anue in debates on school vouchers, desegregation, abilitycking or anti-poverty programs. This paper studies thepact of average peer achievement and peer heterogene- on own achievement growth in reading and math.

To do so, I estimate value-added models for Berlin fifth-graders at the transition from primary to secondary school.Although school choice is endogenous, assignment ofstudents to classes within schools might be largelyexogenous in the data analyzed here for two reasons.First, the schools under investigation, called G5 schools inthe following, are highly selective as they enroll high-achieving primary school pupils only. Parents who seek toenroll their child in such a school have to apply for a slotabout six months in advance. Hence they cannot conditiontheir class choice on peer quality. Second, G5 schoolprincipals know little about incoming students at thebeginning of the fifth grade. For instance, course gradesfrom primary school are noisy measures of achievement ifstudents were taught by different teachers, who applydifferent grading standards.1 As will be shown, thecorrelation between peer quality at the beginning of aschool year and own predetermined characteristics is

R T I C L E I N F O

icle history:

ceived 3 December 2012

ceived in revised form 17 July 2013

cepted 5 August 2013

classification:

ywords:

ility peer effects among high-achievers

tural experiment

A B S T R A C T

This paper estimates ability peer effects on achievement growth in reading and math. It

exploits variation in peer characteristics generated at the transition from primary to

secondary school in a sample of Berlin fifth-graders. As will be discussed in detail, this

variation is exogenous in large parts. Results are similar for both achievement measures:

pupils benefit from abler peers, but high-achievers do so to a smaller extent. The variance

in peer skills has no impact on achievement growth – the corresponding estimates are

negative, but insignificant.

� 2013 Elsevier Ltd. All rights reserved.

An earlier version of this paper circulated as ‘‘The impact of peer

ility and heterogeneity on student achievement: Evidence from a

tural experiment’’. I wish to thank Regina T. Riphahn, Dinand Webbink,

ristoph Wunder, Michael Zibrowius and one anonymous referee for

luable comments and suggestions. Participants of the IZA Summer

ool, annual meeting of the CEA, International Workshop on Applied

onomics of Education, and the EALE conference provided numerous

lpful insights. Financial support by the Bavarian Graduate Program in

onomics (BGPE) is gratefully acknowledged. The data (ELEMENT) were

duced by the Humboldt University of Berlin and provided to me by the

earch data center (FDZ) at IQB Berlin, whose staff I thank for their help

d support. This paper is part of my Ph.D. dissertation, written at the

iversity of Erlangen-Nuremberg under the very helpful supervision of

gina T. Riphahn and Guido Heineck. Any errors or omissions are mine. Tel.: +49 911 5302 257.

E-mail address: David.Kiss@wiso.uni-erlangen.de.

1 In the data, the number of elementary schools (classes) exceeds the

number G5 schools (classes) by a factor of 13 (17). Teacher grading

standards are investigated by, among others, Figlio and Lucas (2004) and

Dardanoni, Modica, and Pennsi (2009).

Contents lists available at ScienceDirect

Economics of Education Review

jo u rn al h om epag e: ww w.els evier .c o m/lo c at e/eco n ed ur ev

72-7757/$ – see front matter � 2013 Elsevier Ltd. All rights reserved.

p://dx.doi.org/10.1016/j.econedurev.2013.08.002

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D. Kiss / Economics of Education Review 37 (2013) 58–65 59

significant for reading and weak, though not insignifi-ant, for math test scores.

The results suggest that students benefit from ablereers but high-achieving students do so to a smallerxtent. More precisely, students who lie in the topercentiles of the class-achievement distribution (at theeginning of the fifth grade) do not benefit from ablereers. The strongest relationship between peer quality andwn achievement growth is found for relatively low-chieving students. For a, compared to his/her classmates,edian-achieving student, a one-standard-deviation in-

rease in peer achievement raises own achievement by.07 standard deviations in reading and 0.10 standardeviations in math.2 The results further indicate thattudents in heterogeneous classes are not worse off thantudents in more homogenous classes: the correspondingstimates are negative, but insignificant.

To date, it is still an open question whether high- orw-ability students benefit most from abler peers.

indings in Lavy, Paserman, and Schlosser (2011) andberman, Kugler, and Sacerdote (2012) are similar to this

tudy: in Lavy et al. (2011), low-achieving students sufferost from an increase in the share of low-ability peers.berman et al. (2012) also find that good peers have the

trongest (positive) impact on low-achievers. In contrast,urke and Sass (2013) and Ding and Lehrer (2007) showat high-achievers benefit most from an increase in peer

uality. Duflo, Dupas, and Kremer (2011) observe a U-haped relationship between peer quality and ownchievement growth: positive peer ability effects areund for both high- and low-achievers, but not foredian-achieving students.

Empirical evidence is further inconclusive regarding thepact of heterogeneity in peer skills on own achievement

rowth. Vigdor and Nechyba (2004) report a positiveelationship between peer heterogeneity and mathchievement growth whereas a negative impact is foundy Ding and Lehrer (2007) and Kang (2007). Similar to thistudy, related estimates in Hanushek, Kain, Markman andivkin (2003) and Duflo et al. (2011) are insignificant.

This paper complements the existing literature in twoays. First, this study adds to the scarce (quasi-xperimental evidence on ability peer effects. Most of

he related studies employ rich fixed-effects frameworkso overcome endogeneity issues. Sund (2009) andibbons and Telhaj (2012) belong to the small groupf peer effects papers that exploit variation in peerharacteristics generated at school transitions.3 Second,he analyzed sample comprises students that performbove-average in standardized tests. Therefore myesults may only apply to classrooms with large sharesf high-achievers. More importantly, the channelshrough which peer effects operate may largely dependn the track (in ability-tracked school systems), i.e. an

increase in peer quality in a lower-level track may affectachievement growth differently than the same increasein upper tracks.4 Differences in transmission channelsmight be a reason for the mixed empirical findings thathave been summarized in the previous paragraphs.

The remainder of the paper proceeds as follows. Thenext section briefly reviews the institutional backgroundand describes the data. Section 3 outlines the empiricalstrategy and discusses possible endogeneity problems.Results are presented in Section 4. Section 5 concludes.

2. Institutional background and data

2.1. Institutional background

In Germany, elementary school generally lasts until thefourth grade when children are 10 years old. Thereafter,students are segregated by ability into three types ofsecondary schools (called tracks): lower-secondary(Hauptschule), middle-secondary (Realschule), and upper-secondary school (Gymnasium). Upper-secondary school isthe most academic track and prepares students foruniversity study. The Berlin educational system is somewhatdifferent since primary education lasts six years. However,some Berlin upper-secondary schools allow transition tosecondary education already after four years of elementaryschool. These upper-secondary schools are referred to as G5schools. In the school year 2002/03, around 24,200 fourth-graders where enrolled in one of 402 Berlin elementaryschools. Only 7% of them attended one of 31 G5 schools in thefollowing school year. Transition to lower- or middle-secondary school is not possible after the fourth grade.

Admission to G5 schools is regulated by the educationact of the federal state Berlin.5 If the number of applicantsexceeds the number of available slots, selection basicallydepends on three criteria. These are (ranked in descendingorder): (i) the student’s track recommendation whichis issued by his/her primary school teacher. (ii) The G5school’s relative proximity to the student’s home (astudent is preferred if the second-nearest G5 school isfar away). (iii) If the number of applicants still exceeds a G5schools’ capacities, selection is made by lot (i.e. randomly).In addition, the first year in G5 schools is a probationaryperiod. Students who do not succeed have to switch to alower-level track in the following school year.

Class formation, however, seems not to be regulated bylaw.6 As will be shown in Section 3.2, observable student

2 Ability peer effects of similar magnitude are found in many related

udies. These are well summarized in Sacerdote (2011) and Epple and

omano (2011).3

4 The large body of the tracking literature is reviewed by Betts (2010).5 ‘‘Schulgesetz fur das Land Berlin’’ from 2004, §56, see http://

gesetze.berlin.de (accessed 17.07.13).6 I carefully checked a large number of Berlin school laws for class

formation guidelines but did not find anything. According to an official

report I found online, schools are responsible for class formation. They

may apply some criteria to ensure that, for instance, the share of girls is

roughly constant across classes. At higher grades, students are allowed to

choose a specialization (e.g. additional foreign language vs. more science

classes), which may lead to student segregation. This issue should not be

relevant in my case as I investigate fifth-graders. Source: ‘‘Kleine Anfrage

des Abgeordneten Ozcan Mutlu (Bundnis 90/Die Grunen), Drucksache 16/

Duflo et al. (2011) (Kenyan schools) and Boozer and Cacciola (2001)

roject STAR) use experimental data to estimate peer effects.

14 030’’, http://www2.mutlu.de/uploads/ka16_14030%5B1%5D.pdf

(accessed 17.07.13).

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D. Kiss / Economics of Education Review 37 (2013) 58–6560

aracteristics cannot predict peer quality at the beginning the fifth grade in G5 schools.

. Data

The data set analyzed here (ELEMENT) is a longitudinalrvey on reading comprehension and math achievement

Berlin elementary and G5 pupils.7 It includes theiverse of fifth-graders who attended a G5 school in the

hool year 2003/04 (31 schools, 59 classes, 1700 pupils)d a random sample of 71 Berlin primary schools. Tocount for endogenous school choice (by adding schooled effects), G5 schools that run only a single class at theth grade are excluded. This reduces the sample to 22hools, 50 classes and 1467 pupils. Participation inndardized tests at the beginning and end of the fifth

ade is compulsory. Thus attrition in the data is solelyused by class repetitions, absence at the time of the test,

school switching. Test scores are comparable acrossades and were not made available to teachers or schoolincipals. Additional pupil information is collected from

questionnaires completed by students and parents on avoluntary basis.

Columns 1 and 2 in Table 1 contain summary statisticsfor the analyzed sample of G5 students. For each variable,the number (share) of missing values is reported in thethird (fourth) column. For comparisons, additional sum-mary statistics for primary school fifth-graders arereported in columns 5 and 6. Test scores in math andreading are normalized with mean 0 and standarddeviation 1 in G5 schools, i.e. for any student i in a G5or elementary school,

Ti;t �Ti;t � mG5

t

sG5t

Ti;t is the original math or reading test score of student i inthe data. t = 0 (t = 1) if Ti;t was measured at the beginning(end) of the fifth grade. mG5

t is the mean value of Tt in thesubsample of G5 students, and sG5

t is the standarddeviation of Tt among G5 students.

As indicated by the first four rows in Table 1, G5students have much better math and reading skills thanprimary school pupils. Elementary school pupils are alsomore likely to have a migration background and are 3months older on average.8 Most G5 students have a

ble 1

mmary statistics (Berlin fifth-graders).

chool type G5 Primary

Mean s.d. Missings Mean s.d.

(#) (%)

(1) (2) (3) (4) (5) (6)

wn characteristics

Reading achievement (beg.), Tread0 0.00 1.00 46 3.1 �1.40 1.36

Reading achievement (end), Tread1 0.00 1.00 47 3.2 �1.59 1.41

Math achievement (beginning), Tmath0 0.00 1.00 46 3.1 �1.27 1.01

Math achievement (end), Tmath1 0.00 1.00 50 3.4 �1.42 1.10

Girl 0.51 3 0.2 0.48

Migration background 0.28 222 15.0 0.41

Age (in years) 11.32 0.43 7 0.5 11.57 0.58

arental education

Lower-secondary 0.03 202 13.8 0.18

Middle-secondary 0.22 202 13.8 0.44

Upper-secondary 0.75 202 13.8 0.38

HISEI 2 [0,1] 0.64 0.21 225 15.3 0.44 0.23

eer characteristics

Peer reading achievement (beg.), Eread0 0.00 1.00 46 3.1 �3.74 1.89

Peer reading variance (beg.), V read0 0.00 1.00 46 3.1 0.93 1.50

Peer math achievement (beg.), Emath0 0.00 1.00 46 3.1 �3.50 1.39

Peer math variance (beginning), Vmath0 0.00 1.00 46 3.1 �0.65 1.10

(pupils) 1467 3169

(classes) 50 140

(schools) 22 71

ndard deviations are not reported for dummy variables. Columns (#) and (%) contain the number and share of missing values, respectively. ‘‘beginning/

d’’ refers to the beginning/end of the fifth grade (which is also indicated by subscript 0/1). Peer achievement E0 and peer variance V0 are the class-level

an and variance in test scores: for each pupil in the data, computation of E0 and V0 excludes that pupil’s own test score (see Section 2.2 for more details).

rental education is the highest secondary school degree of the parents. ISEI is the International Socio-Economic Index of (parental) occupational status,

Ganzeboom, De Graaf, and Treiman (1992). The higher the occupational status, the higher the ISEI-value. HISEI is the highest ISEI-value among the

dent’s parents.

‘‘Erhebung zum Lese- und Mathematikverstandnis: Entwicklungen in

n Jahrgangsstufen 4 bis 6 in Berlin’’, English translation: ‘‘Survey on

ding comprehension and math achievement in Berlin schools, grades 4

ough 6’’. Detailed data descriptions and a codebook (both in German)

available on the homepages of the Berlin senate department for8

ucation, science, and research (Berliner Senatsverwaltung fur Bildung,

issenschaft und Forschung).

Here, a student has a migration background if either he/she was born

abroad or if at least one parent was born abroad.

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D. Kiss / Economics of Education Review 37 (2013) 58–65 61

vorable socioeconomic background: in 75% of cases, atast one parent finished upper-secondary school and

verage HISEI values are high compared to the elementarychool sample.9

The last rows in Table 1 compare peer characteristics in5 and elementary schools. For any student i, peerchievement at the beginning of the fifth grade, denotedy E�i,0, is simply the mean test score of i’s classmates:

�i;0�1

nc � 1

Xj 2 Icnfig

T j;0

j,0 is the test score of classmate j at the beginning of thechool year. c identifies i’s class which is composed of nc

tudents whose IDs are collected in the set Ic. Similarly,eer variance is defined as

�i;0�1

nc � 1

Xj 2 Icnfig

ðT j;0 � mc0Þ

2

c0 is the class test score mean, which is constant for any

tudent i 2 Ic.Similar to own test scores, peer characteristics in

able 1 are normalized with mean 0 and standardeviation 1 in G5 classes. A comparison of the standardeviations in peer achievement reveals that the between-lass variation in peer achievement E0 is larger inlementary school (the standard deviations are 1 < 1.89

reading and 1 < 1.39 in math).10 The summary statisticsr peer variance V0 further indicate that within-class

ariation in reading skills (math skills) is smaller (larger) in5 schools, the corresponding mean values are 0 < 0.93

> �0.65). Hence, G5 classes are on average less (more)eterogeneous in reading (math) than elementary schoollasses.

. Empirical strategy

.1. Estimated models

The impact of peer characteristics on achievementrowth is estimated with two value-added models:

i;1 ¼ a þ b1E�i;0 þ g1V�i;0 þ d1Ti;0 þ d2Ri;0 þ p03Xi þ ls

þ ei

the baseline model. The interacted model is

i;1 ¼ a þ b1E�i;0 þ b2ðE�i;0 � Ri;0Þ þ g1V�i;0 þ g2ðV�i;0

� Ri;0Þ þ d1Ti;0 þ d2Ri;0 þ p03Xi þ ls þ ei

Ti,1 is pupil i’s math or reading test score at time 1, the endof a school year. All explanatory variables are measured attime 0, the beginning of the school year. This frameworkrules out reverse causation because realizations of thedependent variable are measured about 10 months laterthan realizations of the explanatory variables. The vari-ables of interest are peer achievement E0 and peer varianceV0, both measured at the class level at time 0. As discussedin the previous section, E�i,0 is the average test score of i’sclassmates and V�i,0 is the dispersion in i’s peers’ skills.Computation of both variables excludes i’s own test scorewhich is emphasized by the subscript �i.

Ti,0, i’s test score at the beginning of the fifth grade, isassumed to capture i’s past educational inputs. Ri,0 2 [0, 1]is pupil i’s class percentile rank in reading or math testscores. Within classes, the highest-achieving pupil hasrank one, the median-achiever has rank 0.5, and thelowest-achiever in the class has rank zero.11 In theinteracted model, peer effects are allowed to vary withthe rank. Xi is a column-vector of additional controlvariables (age, a girl dummy, indicator variables formigration and socioeconomic background) for student i.Missings in X are replaced with imputed values.12 ls is afixed effect for school s, and standard errors are clustered atthe class level.

3.2. Are peer characteristics exogenous in new G5 classes?

Estimates of b and g are biased if some determinants ofthe class formation process that also affect achievementgrowth are unobserved. In this context, choices made byparents and school principals are considered as the mostrelevant sources of endogeneity. The impact of parentalschool choice on achievement growth is accounted for bythe school fixed effect l. Class choice made by parents isassumed to be exogenous because parents have to applyfor a slot in a G5 school about 6 months in advance. Thusparents cannot condition their class choice on peercharacteristics.

The second potential source of endogeneity are schoolprincipals. For instance, school principals may segregatestudents by ability or assign ‘‘good’’ teachers to ‘‘good’’students. However, there are two reasons why ability-grouping should not be relevant in G5 schools: first, G5schools are attended by students with above-average skillsin math and reading thus one would expect that ability-grouping is not the main objective of G5 schools. Second,even if G5 schools aim at grouping students by ability orassigning good teachers to better students, G5 schools havelittle information to do so: the most relevant informationon the students’ skills is summarized in their schoolreports from elementary school. These reports containcourse grades and written teacher assessments on thestudents’ educational progress. Compared to standardized

9 ISEI is the International Socio-Economic Index of (parental) occupa-

onal status, see Ganzeboom, De Graaf, and Treiman (1992). The higher

e occupational status, the higher the value of the ISEI. HISEI is the

ighest ISEI value among the student’s parents. Here, HISEI is normalized

lie between 0 and 1.0 One should not get confused by the large negative mean values of E0

primary school. In Table 1, test scores and peer characteristics are

andardized using the mean and standard deviation in G5 schools (thus

eer characteristics have mean 0 and standard deviation 1 in G5 schools).

11 R0 and T0 are conceptually different: two pupils who attend different

G5 classes may have similar ranks but large differences in T0 at the same

time.12 Imputed values were computed by the data provider. Results in

ompared to own test scores, there is much less variation in peer

chievement which inflates the magnitude of E0 in primary school.

Section 4 are insensitive to their in- or exclusion (estimates available on

request).

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D. Kiss / Economics of Education Review 37 (2013) 58–6562

hievement tests, course grades are subjective to sometent as they are assigned by teachers.13 As alreadyentioned, however, the number of elementary schools is

times larger than the number of G5 schools. As class sizesmaller in elementary school, the number of elementaryhool classes exceeds the number of G5 classes by a factor

17. Therefore it is reasonable to assume that most fifth-aders in G5 schools were taught by different elementaryhool teachers, i.e. students may be heterogeneous inills even if their course grades from elementary schoole similar. Still, G5 schools may have collected historicalformation on primary schools that would allow them toedict (with some noise) an incoming student’s skill. Inat case, ability-grouping becomes feasible to sometent. As will be shown below, own test scores T0 ate beginning of the fifth grade cannot predict peerhievement E0, i.e. empirical evidence suggests that G5hools do not group students by ability.

Estimates of b and g may still be contaminated if largeshares of pupils in newly created G5 classes previouslyattended the same elementary school class. To illustratethis potential source of bias, let cp

i denote a permanentshock on i’s achievement growth that has been deter-mined in primary school p. cp

i may encompass learningtechniques or problem-solving skills that have beenacquired through i’s elementary school teacher(s) andformer peers. Let j be a classmate of i in a new G5 class.c p

i ¼ cpj if i and j were already classmates in primary

school. To some extent, j’s test score at the beginning ofthe school year in a G5 school, Tj,0, is determined by his/her learning techniques cp

j from elementary school. Thusignoring cp

j violates the exogeneity assumption E[ei|E�i,0,V�i,0] = 0, because ei contains the omitted variablec p

i ¼ cpj , and Tj,0 is used to compute E�i,0 and V�i,0.

The larger the share of former classmates in new G5classes, the more relevant this source of bias. TheELEMENT data contain no information on the previouslyattended elementary school of G5 students whichprecludes the inclusion of primary school class fixedeffects. As already mentioned, however, the number ofelementary school classes is 17 times larger than thenumber of G5 classes. Thus cp

i 6¼ cpj for most classmates i

and j in G5 classes.

ble 2

st for exogeneity in peer characteristics at the fifth grade in G5 schools.

ependent variable Eread0 V read

0 Emath0 Vmath

0

(1) (2) (3) (4)

A) No additional controls

Own reading/math achievement

(beginning of the school year)

�0.015

(0.01)

0.033

(0.02)

�0.029**

(0.01)

�0.056***

(0.02)

School fixed effects Yes Yes Yes Yes

2 (adj.) 0.7436 0.5611 0.7684 0.4275

B) With additional controls

Own reading/math achievement

(beginning of the school year)

�0.014

(0.01)

0.029

(0.02)

�0.021

(0.01)

�0.060***

(0.02)

Girl 0.018

(0.03)

0.021

(0.04)

0.065**

(0.03)

�0.057

(0.04)

Migration background �0.015

(0.03)

�0.002

(0.05)

0.011

(0.03)

0.051

(0.05)

Age 0.013

(0.03)

0.055

(0.04)

�0.044

(0.03)

�0.045

(0.05)

Parental education: middle-secondary

(Ref. category: lower-secondary)

�0.027

(0.10)

0.211

(0.14)

0.038

(0.08)

�0.143

(0.14)

Parental education: upper-secondary �0.081

(0.10)

0.169

(0.14)

0.066

(0.08)

�0.092

(0.14)

HISEI 0.019

(0.07)

0.041

(0.10)

�0.079

(0.07)

�0.057

(0.12)

School fixed effects Yes Yes Yes Yes

2 (adj.) 0.7432 0.5611 0.7691 0.4276

-Test for own ach. + add. controls (p-value) 0.6765 0.4801 0.0363 0.0912

-Test for additional controls only (p-value) 0.6842 0.6253 0.1273 0.4648

(pupils) 1421 1421 1421 1421

ndard errors (heteroskedasticity-robust, not clustered at the class or school level) in parentheses. Used data: ELMENT fifth-graders in G5 schools. Peer

ievement E0, peer variance V0 and own achievement have mean 0 and standard deviation 1, and are measured at the beginning of the fifth grade. The

mber of observations is reduced by missing values in test scores. Missing values in additional control variables are imputed, see Table 1 for details. The

t F-test is computed for the null hypothesis that, once school fixed effects are controlled for, own achievement and additional controls are not correlated

th the dependent variable. The second F-test is computed for the null hypothesis that, once school fixed effects and own achievement are controlled for,

additional controls are not correlated with the dependent variable.

Significance level: 10%.

* Significance level: 5%.

** Significance level: 1%.

For example, Dardanoni et al. (2009) find for 14 of 16 OECD countries

t schools with high shares of underperforming students tend to set

er grading standards. The relationship between educational standards

d norms or the degree of centralization is theoretically investigated by

strell (1994).

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D. Kiss / Economics of Education Review 37 (2013) 58–65 63

To summarize, there are good reasons why b and ghould not be severely biased in new G5 classes. Similar toarrell, Fullerton, and West (2009), one can test thexogeneity assumption by regressing E0 and V0 on T0 andther student characteristics (measured at time 0).tuitively, random mixing of pupils into classes impliesat individual pretreatment characteristics cannot predict

0 and V0.Estimates are reported in Table 2. In panel A, own

eading achievement (columns 1 and 2) or mathchievement (columns 3 and 4) is the only explanatoryariable along with school fixed effects. Throughout,dividual and aggregated measures of test scores haveean 0 and standard deviation 1. Panel A, column 1

hows a negative, but insignificant relationship betweenwn reading skills and peer reading skills at theeginning of the fifth grade. Similarly, there is no linearelationship between own reading skills and the disper-ion in peer reading skills once school fixed effects areccounted for (column 2). Column 3 suggests thattudents with good math skills have somewhat worseeers on average: an increase in own math achievementy one standard deviation is associated with a decrease

peer math achievement by 0.029 standard deviations.his association is of very small magnitude but signifi-ant at the 5% level. The relationship between own mathchievement and peer math variance is also small,hough significant.

Additional controls are accounted for in panel B ofable 2. Again, observable student characteristics cannotredict the mean or dispersion in peer reading skills. Inddition, the relationship between peer math achievementnd own math achievement becomes insignificant. To

summarize, Table 2 suggests that (i) peer characteristicsare as good as exogenous in reading. (ii) G5 schools do notgroup students by ability (once additional controls aretaken into account). (iii) As indicated by the F-tests,endogeneity cannot be ruled out for math but is likely toplay a minor role.

4. Results

Table 3a (Table 3b) contains estimates for the impact ofpeer characteristics on reading (math) achievement growthin G5 schools. For both outcomes, columns 1 and 2 reportestimates for the baseline model, the interacted model isestimated in columns 3 and 4 (see Section 3.1 for moredetails). Additional controls are omitted in columns 1 and 3,and included in columns 2 and 4.

As expected, own achievement at the beginning andend of the fifth grade are highly correlated. Regardless oftheir skill level, highly ranked pupils learn more thanotherwise comparable students with lower ranks. Thisresult is consistent with Cullen, Jacob, and Levitt (2006),who show that a student’s relative position among his/herpeers is a determinant of his/her school success. Studentswho are superior to their peers may feel more motivatedor follow the lessons with more ease (given that teachersadjust their curricular pace to the skill distribution in theirclasses).

Estimates of the interacted models (columns 3 and 4)reveal that the relationship between achievementgrowth and average peer achievement depends on astudent’s relative position among his/her classmates. b1,the peer achievement coefficient, becomes highlysignificant once peer achievement is interacted with

able 3a

pact of peer characteristics on reading achievement growth.

Dependent variable Reading achievement (end of fifth grade)

Estimated model Baseline Interacted

(1) (2) (3) (4)

Own reading achievement (beginning) 0.351***

(0.08)

0.340***

(0.07)

0.345***

(0.07)

0.329***

(0.07)

Rank (1 = best, 0 = worst) 0.518**

(0.23)

0.505**

(0.23)

0.535**

(0.22)

0.538**

(0.22)

Peer reading achievement 0.054

(0.06)

0.061

(0.06)

0.246***

(0.07)

0.257***

(0.07)

Peer reading variance �0.011

(0.05)

�0.012

(0.05)

�0.006

(0.06)

�0.017

(0.07)

Rank*peer reading achievement �0.370***

(0.08)

�0.377***

(0.08)

Rank*peer reading variance �0.031

(0.06)

�0.009

(0.06)

Additional controls No Yes No Yes

School fixed effects Yes Yes Yes Yes

R2 (adj.) 0.2953 0.3053 0.3063 0.3167

N (pupils) 1376 1376 1376 1376

tandard errors (in parentheses) are clustered at the class level. Used data: ELEMENT fifth-graders in G5 schools. All explanatory variables are measured at

e beginning of the fifth grade. ‘‘Rank’’ is a pupil’s class percentile rank in reading. By definition, the rank lies between zero and one. Additional controls are:

ummy variables for sex and migration background, age, indicator variables for parental education, and HISEI. The number of observations is reduced by

issing values in test scores. Missing values in additional control variables are imputed, see Table 1 for details.

* Significance level: 10%.

** Significance level: 5%.

*** Significance level: 1%.

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D. Kiss / Economics of Education Review 37 (2013) 58–6564

e student’s rank.14 For both achievement measures,udents with low ranks benefit to a greater extent fromtter peers. As discussed by Arcidiacono, Foster,odpaster and Kinsler (2012) and Carrell et al.

009), students with low ranks may have moreportunities to ask others for help.Estimates of the interacted model further imply that

signing an average pupil to a weak (in terms of test scores)ss is not necessarily harmful. To see why, consider the

tal differential of T1 ¼ EðT1jE0; V0; T0; R0; X; lÞ, the esti-ated conditional expectation function of end-of-the-yearst scores. Evaluation of the total differential dT1 at dR0 6¼ 0d dE0 6¼ 0 (i.e. other explanatory variables remainchanged) yields

1jdR0 6¼ 0;dE0 6¼ 0 ¼ ð0:612 � 0:206E0Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}> 0

dR0

þ ð0:206 � 0:206R0Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}� 0

dE0

this example, estimates were taken from column 4 inble 3b.15 Assignment of an average student with averageers (i.e. R0� 0.5 and E0� 0) to a class of underperformingers is equivalent to dR0> 0 and dE0< 0. That student istter off (in terms of achievement growth) if the increase

in the rank overcompensates the loss in peer quality, i.e. if(0.612 � 0.206E0) dR0> |(0.206 � 0.206R0) dE0|.

Finally, Tables 3a and 3b suggest that the dispersion inpeer skills does not harm achievement growth. Thecorresponding estimates are insignificant in all specifica-tions. All patterns reported in this section (magnitude ofpoint estimates, significance levels) are virtually the sameif observations with missing values in the additionalcontrol variables are excluded.16

5. Concluding remarks

This paper estimates the impact of average peerachievement and peer heterogeneity at the class level onachievement growth in reading and math. Making use of anatural experiment in a sample of students at the transitionfrom primary to upper-secondary school, the results suggestthat students benefit from abler peers, but pupils with highclass percentile ranks do so to a smaller extent. Holdingother things constant, a one-standard-deviation increase inpeer achievement raises achievement growth of a median-ranked student by 0.07 standard deviations in reading and0.10 standard deviations in math. Peer heterogeneity seemsnot to harm achievement growth. Even though estimatedachievement gains from better peers are of relatively smallmagnitude, Chetty et al. (2011) show that peer quality atearly stages matters for long-run outcomes like earnings orcollege attendance rates.

ble 3b

pact of peer characteristics on math achievement growth.

ependent variable Math achievement (end of fifth grade)

stimated model Baseline Interacted

(1) (2) (3) (4)

wn math achievement (beginning) 0.460***

(0.10)

0.455***

(0.09)

0.406***

(0.09)

0.403***

(0.09)

ank (1 = best, 0 = worst) 0.450

(0.29)

0.424

(0.29)

0.643**

(0.29)

0.612**

(0.28)

eer math achievement 0.092

(0.07)

0.088

(0.07)

0.214***

(0.07)

0.206***

(0.07)

eer math variance 0.004

(0.02)

0.002

(0.02)

�0.066

(0.05)

�0.068

(0.05)

ank*peer math achievement �0.212***

(0.07)

�0.206***

(0.07)

ank*peer math variance 0.143

(0.09)

0.141

(0.09)

dditional controls No Yes No Yes

chool fixed effects Yes Yes Yes Yes

2 (adj.) 0.3869 0.3914 0.3919 0.3961

(pupils) 1373 1373 1373 1373

ndard errors (in parentheses) are clustered at the class level. Used data: ELEMENT fifth-graders in G5 schools. All explanatory variables are measured at

beginning of the fifth grade. ‘‘Rank’’ is a pupil’s class percentile rank in math. By definition, the rank lies between zero and one. Additional controls are:

mmy variables for sex and migration background, age, indicator variables for parental education, and HISEI. The number of observations is reduced by

ssing values in test scores. Missing values in additional control variables are imputed, see Table 1 for details.

Significance level: 10%.

* Significance level: 5%.

** Significance level: 1%.

Burke and Sass (2013) also find small, but significant peer effects in

ear-in-means models. The magnitude of their peer effects coefficient

comes economically relevant once they interact peer achievement with

n achievement.

As long as R0 or E0 are not too large, the factors (0.612 � 0.206E0) and 16 Estimates are available on request. Missing values in additional

206 � 0.206R0) are positive. The same argument applies to the reading

efficients in Table 3a, columns 3 or 4.

control variables are imputed, see Table 1 for additional details. Test

scores or peer measures are never imputed.

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D. Kiss / Economics of Education Review 37 (2013) 58–65 65

The peer effects literature acknowledges that peerchievement or peer heterogeneity are proxies for unob-erved factors that ultimately affect achievement growth. Ifeer effects operate through many, mutually dependenthannels, reduced-from estimates of peer effects are ofmited use to inform policy about optimal grouping oftudents (see Carrell, Sacerdote, & West, 2011). As noted byanushek et al. (2003), ‘‘The role of peers can be complex.fluences may come from friends or role models, or peer

roup composition may alter the nature of instruction in thelassroom. . . The most common perspective is that peers,ke families, are sources of motivation, aspiration, andirect interactions in learning.’’ To arrive at reliable policyecommendations, further research needs to uncover the

ost relevant mechanisms that determine the relationshipetween peer quality and achievement growth.

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