Understanding Regression versus Variance Tests for Social Interactions

This article is designed to understand the relationship between two approaches to the empirical analysis of social interactions. The first approach is the use of linear regressions to uncover social interactions. This approach estimates social interaction effects through changes in the conditional (linear) expectation of individual outcomes when group-level variables are included in the conditioning set. In econometrics, this method was first analyzed by Manski (1993), who dubbed it the linearin-means model since the group-level variables are typically averages of individual variables within the group. This model is also studied by Brock and Durlauf (2001a, 2001b), CohenCole (2006) and Graham and Hahn (2005), among others. Durlauf (2004) surveyed empirical studies of social interactions based on such regressions. The second approach, originally proposed by Glaeser, Sacerdote, and Scheinkman (1996) and developed into a full inferential procedure by Graham (2005), who refers to this approach as variance contrasts, uses intragroup variation in behavior to infer social interactions. The idea in this approach is that social interactions will induce intragroup dependences in behavior that affect sample variances for intragroup outcomes. There has been relatively little work evaluating social interactions using variance contrasts. One reason is that with the exception of Graham’s work, there is little formal statistical theory for the approach. The purpose of this article was to clarify the relationship between regression and variance contrast tests.We clarify how these tests employ different informational assumptions to achieve identification of social interaction. Specifically, we argue that regression and variance contrast approaches correspond to identification via exclusion restrictions in the one case and covariance restrictions on errors in the other. As such, the two approaches replicate the classes of identification strategies that appear in the classical simultaneous equations literature (see Fisher (1966), for a classic treatment). The remainder of the article is organized as follows. In Section II, we define a general linear-in-means model and describe identification from the regression perspective. Section III provides identification analysis from the perspective of variance contrasts. Section IV concludes. As Graham’s analysis focuses on peer effects in classrooms, we use that example to motivate discussion.