Identifying Social In teractions through Excess Variance ContrastsbyBry an S. Graham*(Initial Draft: June 2003)(This Draft: May 5th, 2005)Abst ractThis paper o utlines a new method f or detecting and assessing the strength of social in te ractions based oncontrastsinexcessvarianceacrosssocialgroupsofexogenouslydiffering sizes. An attractive feature of theapproach is its robustness to the presence of group-level heterogeneity and sorting. The proposed estimationstrategy is used to test for the presence of peer effects in learning using data from the Tennessee class sizereduction experimen t Project STAR. Size-induced contrasts of excess variance provide a powerful mechanismfor detecting peer group effects in this dataset. Switching from classroom where mean peer ability i s at the25th percen tile of the ability distribution to one where it is at the 75th percentile is associated with changesin math and reading achievemen t scores of 0.9 and 1.1 standard deviations respectively. These estimatessuggest that , at minimum, differences in peer composition are at least as important as those in teacherquality for explaining variation in academic achievement within Project STAR schools. While tests basedon excess variance contrasts provide strong evidence of peer group effects, conventi onal regression-basedexcess sensitivity tests do not. Calibrating asymptotic power functions for the two tests to the ProjectSTAR data suggests that across repeated samples the odds of detecting social inte ractions are roughly 20to 30 times greater with the proposed excess variance test. Generalized method of moments provides aunified framework for estimation and inference. The proposed approach is str aightforward to implementusing standard software.JEL Classification: C31, C39, I29, J24.Key Words: Social Interactions, Peer Group Effects, Reflection Problem , Linear-in-Means,Project STAR, Educational Production, Covariance Models.*I would like to thank Gary Chamb erlain, Larry K atz, Michael Kremer, and Caroline Hoxby for their encour-agement as well as many helpful comments, correct ions and suggestion s. Useful feedback from William Brock,David Card, Steven Durlauf, Bo H onoré, Guido Imbens, Richard Murnane, Mark Watson, John Willett, Jeff ZabelandparticipantsintheBanff Workshop on Social Interactions as well as seminars at Harvard, Harvard-MIT, UCSD,Dartmouth, Princeton, Stan ford, Stanford-GSB, UT-Austin, Berkeley, Chicago, Wisconsin, LSE, Yale, IFS, Toulouse-GREMAQ and Munich is also gratefully acknowl edged. Financial support provided by a National Science FoundationGraduate Fellowship, the Program on Justice, Welfare and Economics at Harvard University and by the Program ofFellowships for Junior Scholars, t he MacArthur Research Network o n Social Interactions a nd Economic Ineq ual ity.All the usu al di s claimers apply. Correspondence: Department o f Economics, University o f California - Berke ley,549 Evans Hall #3880, Berkeley, CA 94708. e-mail: [email protected] erkeley.edu.1Social Interactions and Excess Variance 21 IntroductionVariation in man y individual outcomes — such as earnings, academic ach ievemen t, substance abuse,crim inal behavior, and tec hnology adoption — includes a substantial between-group component .For example, a long economics of education literature documen ts that mean academic achievementvaries dramatically across different classrooms, even among those located within the same school(e.g., Hanushek 1971). Perh aps the most straightforward explanation for this finding is the presenceof classroom-level heterogeneity, such as differences in teacher quality.1An alternative explanation for excess variance is that it mirrors the relative salience of so cialinteractions or peer group effects. Social in t eractions are present if individual behavior is affectedb y reference or peer group behavior, c haracteristics or both. If students within the same classroomlearn from one another, then achievemen t levels will covary positively within a classroom and hencedisplay excess variation between classrooms. As with group-level heterogeneity, social interactionsare associated with a lack of independence in outcomes across members of the same social group.The two rival explanations for excess between-group variance, group-level heterogeneity andsocial interactions, are straigh tforwa rd to understand, but exceptionally difficult to discriminate be-tween empirically. Hoxb y (2002, p. 58) emphasizes the “formidable obstacles” faced by research erswhen attempting to detect peer effects in the learning process. In a recent and wide-ranging reviewDurlauf (2002, p. 20) concludes that “there is little reason why a skeptic should be persuadedto change his mind b y the statistical evidence [on social interactions] currently a vailable”. Oftenassociated with controversy, the empirical literature on so cial interactions is also characterized bywidely divergen t conclusions across different researchers.The indecisiveness of a vailable empirical evidence on social in teractions partly reflects the factthat it speaks to some of the most contentious contemporary social and political issues in society.For example, the merits of school choice, ability tracking, busing and other desegregation mea-sures, and different zoning l aws all relate to the ‘simple’ question of whether peer group effectsare important for the learning process.2A second reason for the diversity of conclusions found inthe literature is that no consensus exists on how to best identify and estim ate statistical models ofsocial interactions in the presence of group-level heterogeneity.This paper develops new methods for adducing the presence and magnitude of s ocial interac-tions based on excess variance contrasts. An attractive feature of the proposed methods is thatthey are able to identify social interactions in a way that is robust to the presence of group-levelheterogeneity. In the context of the economics of education example intro duced above, they providemechanisms by which excess between-classroom variation in student achievemen t can be decom-posed int o its teacher quality and p eer effect portions. Such a decomposition is useful for assessing1Another example is crime, which is endemic to some neig hborhoods a nd negligible in other seemingly similar ones(c.f., Glaeser, Sacerdote and Scheinkman 1996). Pi ketty (2000) and Becke r and M urphy (2000) survey theoreticalmodels generating excess