## 1 INTRODUCTION

Evaluating peer effects in academic achievement is important for parents, teachers and schools. These effects also play a prominent role in policy debates concerning ability tracking, racial integration and school vouchers (for a recent survey, see Epple and Romano, 2011). However, despite a growing literature on the subject, the evidence regarding the magnitude of peer effects on student achievement is mixed (e.g. Sacerdote, 2001; Hanushek *et al*., 2003; Stinebrickner and Stinebrickner, 2006; Ammermueller and Pischke, 2009). This lack of consensus partly reflects various econometric issues that any empirical study on peer effects must address. Identifying and estimating peer effects raises three basic challenges. First, the relevant peer groups must be determined. Who interacts with whom? Second, peer effects must be identified from confounding factors. In particular, spurious correlation between students' outcomes may arise from self-selection into groups and from common unobserved shocks. Third, identifying the precise type of peer effect at work may be hard. Simultaneity, also called the *reflection problem* by Manski (1993), may prevent separating contextual effects, i.e. the influence of peers' characteristics, from the endogenous effect, i.e. the influence of peers' outcome. This issue is important since only the endogenous effect is the source of a *social multiplier*. Researchers have adopted various approaches to solve these three issues; we discuss the methods and results of previous studies in more detail in the next section. As will be clear, however, there is no simple methodological answer to these three challenges.

In this paper, we provide, to our knowledge, the first application of a novel approach developed by Lee (2007) for identifying and estimating peer effects. In principle, the approach is promising, as it allows to solve the problem of correlated effects and the reflection problem with standard observational (non-experimental) data. Moreover, the exclusion restrictions imposed by the model are explicitly derived from its structural specification and provide natural instruments. The econometric model does rely on a number of crucial assumptions, however, which makes its confrontation to real data particularly important. We empirically assess the approach using original administrative data on test scores at the end of secondary school in the Canadian Province of Québec. We investigate the presence of peer effects in student achievement in Mathematics, Science, French, and History. In the process, we also provide new economic insights regarding the sources of identification in the model. This matters in particular in assessing its robustness to alternative (nonlinear) approaches.

The econometric model relies on three key assumptions. First, individuals interact in groups known to the modeler. This means that the population of students is partitioned into groups (e.g. classes, grade levels) and that students are affected by all their peers in their groups but by none outside of it. This assumption is typical in studies of academic achievement but clearly arises from data constraints. Second, each individual's peer group is everyone in his group *excluding* himself. While this assumption seems innocuous and has been used in most empirical studies, it is a key source of identification in the model, as will become clear below. In fact, it is a main source of difference between Manski's (1993) and Lee's models. Manski's approach can be interpreted as one in which each individual's peer group *includes* himself.1 Third, individual outcome is determined by a linear-in-means model with group fixed effects. Thus the test score of a student is affected by his characteristics and by the average test score and characteristics in his peer group. In addition, it may be affected by any kind of correlated group-level unobservable.

Lee (2007) shows that peer effects are identified in such a framework when there are sufficient groups of different sizes. One important contribution of our paper is to clarify the economic intuition behind identification. Regarding the estimation of parameters, one potentially important limitation of the method, however, is that convergence in distribution of the peer effect estimates may occur at low rates when the average group size is large relative to the number of groups in the sample (Lee, 2007). This is also intuitive: excluding the individual or not from his peer group does not change much when its size is relatively large.

Here two remarks are in order. First, these results are to be distinguished from the idea that the group size is a factor in a school's production function (e.g. Krueger, 2003). In Lee's model, the effects of group sizes which are separable from the peer effects are controlled for by fixed effects in the structural model. Second, Lee's identification method differs from the *variance contrast* approach developed by Graham (2008). The basic idea in this approach is that peer effects will induce intra-group dependencies in behavior that introduce variance restrictions on the error terms. These restrictions are used to identify the composite (endogenous + contextual) social interaction effects under the assumption that the variance matrix parameters are independent of the reference group size.

We use administrative data on academic achievement for a large sample of secondary schools in the Province of Québec obtained from the Ministry of Education, Recreation and Sports (MERS). Our dependent variables are individual scores on four standardized tests taken in June 2005 (Mathematics, Science, French and History) by fourth- and fifth-grade secondary school students. All fourth- and fifth-grade students in the province must pass these tests to graduate. One advantage of these data is that all candidates in the province take the same exams, no matter what their school and location. This feature effectively allows us to consider test scores as draws from a common underlying distribution. Another advantage is that our sample is representative and quite large. We have the scores of all students for a 75% random sample of Québec schools which, over the four subjects, yields 194,553 test scores for 116,534 students. In terms of interaction patterns, the structure of the data leads us to make the following natural assumption. We assume that the peer group of a student contains all other students in the same school qualified to take the same test in June 2005. In practice, a small number of students postpone test-taking to August 2005. We extend Lee's methodology in the empirical modeling to address this issue. However, since the difference between observed group sizes and actual group sizes is small, the correction has little effect on the results. Following Lee (2007), we estimate the model in two ways: through generalized instrumental variables (IV) and, under stronger parametric conditions, through conditional maximum likelihood robust to non-normal disturbances (pseudo CML).

Our results are mixed though consistent with the model. We do provide evidence of some endogenous and contextual peer effects. Based on pseudo CML estimates, we find that the endogenous peer effect is positive, significant and quite high in Mathematics (0.83). Moreover it is within the range of previous estimates (see Sacerdote, 2011, for a recent survey). However, the effect is smaller and non-significant in History (0.64), French (0.30) and Science (− 0.23).2 Endogenous peer effects estimates obtained from IV methods are highly imprecise with our data, even in Mathematics. The higher precision of our pseudo CML estimates is consistent with results in Lee (2007) showing that CML estimators are asymptotically more efficient than IV estimators. As regards contextual peer effects, we find evidence that some of them matter, based on both pseudo CML and IV estimators. For instance, results from pseudo CML indicate that interacting with older students (a proxy for repeaters) has a negative effect on own test score in all subjects except Mathematics (not significant).

It is remarkable that even with large average group size relative to the number of groups we are able to identify some peer effects. However, there is also much dispersion in group sizes within our samples. We suspect that this helps identification. We study this issue systematically through Monte Carlo simulations. We find that indeed increasing group size dispersion has a positive impact on the precision of estimates.

The remainder of the paper is organized as follows. We discuss past research in Section 2 and present our econometric model and the estimation methods in Section 3. We describe our dataset in Section 4. We present our empirical results in Section 5 and run Monte Carlo experiments in Section 6. We conclude in Section 7.