Specification and Estimation of Network Formation and Network Interaction Models with the Exponential Probability Distribution

In this paper, we model network formation and network interactions under a unified framework. The key feature of our model is to allow individuals to respond to incentives that stem from interaction benefits of certain activities when they choose friends (network links), while capturing homophily in terms of unobserved characteristic variables in network formation and activities. There are two advantages of this modeling approach: first, one can evaluate whether incentives from certain interactions are important factors for friendship formation or not. Second, in addition to homophily effects in terms of unobserved characteristics, inclusion of incentive effects in the network formulation also corrects possible friendship selection bias on activity outcomes under network interactions. A theoretical foundation of this unified model is based on a sub-game perfect equilibrium of a two-stage game. A tractable Bayesian MCMC approach is proposed for the estimation of the model, and we demonstrate its finite sample performance in a simulation study. We apply the model to study empirically American high school students' friendship networks from the Add Health dataset. We consider two activity variables, GPA and smoking frequency, and find a significant incentive effect from GPA, but not from smoking, on friendship formation. These results suggest that the benefit of interactions in academic learning is an important factor for friendship formation, whereas the interaction benefit of smoking is not. On the other hand, from the perspective of network interactions, both GPA and smoking frequency are subject to significant positive interaction (peer) effects