Early contributions by Veblen (1899), Duesenberry (1949), Leibenstein (1950), Pollak (1976), and others show that economists have recognized the potential importance of social interactions for a long time. The slow progress of empirical research in this area is to a large extent related to a number of methodological problems. As described by Manski (1993, 2000) and others, a major difficulty is to disentangle endogenous social interactions (which imply a social multiplier effect) from other types of social interactions (which do not imply a multiplier effect). Another problem is the endogeneity of reference groups. Recent years have shown an increasing number of empirical studies searching for credible empirical evidence on social interactions, in part by using data that are quasi-experimental in nature; see Sacerdote (2001), Durlauf and Moffitt (2003), and Duflo and Saez (2003) for examples.
The present paper focuses on methodological problems related to a specific but frequently encoutered situation: social interactions in small groups when choice variables are discrete. In a discrete-choice model with endogenous social interactions, the choices of other individuals are explanatory variables in the equation describing the choice behavior of a given individual. For estimation and other purposes, the reduced form (or ‘social equilibrium’ or ‘solution’) of the model is required. While the reduced form is straightforwardly obtained in a linear model with continuous variables, its derivation is more complicated in the case of discrete variables. As already noted by authors analyzing the simultaneous probit model (see, for example, Heckman, 1978; Maddala, 1983), such models may not have a solution or may have multiple solutions. This in turn may yield problems regarding the statistical coherency of the model.
In Section 2 we present the model and characterize its equilibrium properties, in particular the correspondence between interaction strength, number of agents, and the set of equilibria. Section 3 proposes to estimate the model by means of simulation methods, assuming that observed choices represent an equilibrium of the static discrete game played by all interacting agents. Section 4 is devoted to an empirical application. We analyze a sample of 485 high school classes with detailed information on the individual behavior of the students within each class. As all students in a sampled class are interviewed in principle, the dataset has rich information on the behavior of potentially important peers of each respondent. We estimate the model for five types of discrete choices made by teenagers: smoking, truancy, moped ownership, cell phone ownership, and asking parents' permission for purchases. To control for sorting into schools and omitted variables that induce a positive correlation between peers, we also estimate versions that allow for within-class correlation of error terms and for school-specific fixed effects. We find strong social interaction effects for behavior closely related to school (truancy), somewhat weaker social interaction effects for behavior partly related to school (smoking, moped and cell phone ownership) and no social interaction effects for behavior far away from school (asking parents' permission for purchases). Intra-gender interactions are generally much stronger than cross-gender interactions. Once we control for school-specific fixed effects, social interaction effects become insignificant, with the exception of intra-gender interactions for truancy.
A number of recent papers have analyzed social interactions in a discrete-choice framework. Brock and Durlauf (2001a, 2006) use a random-fields approach to study aggregate behavioral outcomes in an economy in which social interactions are imbedded in individual decisions. Equilibrium properties of this model are derived by imposing a rational expectations condition on the subjective choice probabilities of the agents and by assuming that the number of agents is sufficiently large that each agent ignores the effect of his own choice on the average choice level. In contrast, the present paper describes behavior in relatively small groups of a given size in which choices of other individuals can be assumed to be fully observable. For this reason, it is more appropriate to model the interactions as a non-cooperative game, by making an individual's pay-off dependent on the actual choice of others in his group. In the analysis, we will focus on the one-shot pure Nash equilibria of this game. In a recent paper Tamer (2003) proposes a semi-parametric estimator which allows—under certain conditions—for consistent point estimation of the model in the N = 2 case without making assumptions regarding non-unique outcomes. Its extension and empricial implementation to have not been fully developed as yet. Gaviria and Raphael (2001) analyze school-based peer effects in the individual discrete-choice behavior of tenth-graders. However, their econometric model ignores multiplicity of equilibria.