An Empirical Model of Network Formation: Detecting Homophily When Agents Are Heterogenous

I formalize a widely-used empirical model of network formation. The model allows for assortative matching on observables (homophily) as well as unobserved agent level heterogeneity in link surplus (degree heterogeneity). The joint distribution of observed and unobserved agent-level characteristics is left unrestricted. Inferences about homophily do not depend upon untestable assumptions about this distribution. The model is non-standard since the dimension of the heterogeneity parameter grows with the number of agents, and hence network size. Nevertheless, under certain conditions, a joint maximum likelihood (ML) procedure, which simultaneously estimates the common and agent-level parameters governing link formation, is consistent. Although the asymptotic sampling distribution of the common parameter is Normal, it (i) contains a bias term and (ii) its variance does not coincide with the inverse of Fisher's information matrix. Standard ML asymptotic inference procedures are invalid. Forming confidence intervals with a bias-corrected maximum likelihood estimate, and appropriate standard error estimates, results in correct coverage. I assess the value of these results for understanding finite sample behavior via a set of Monte Carlo experiments and through an empirical analysis of risk-sharing links in a rural Tanzanian village (cf., De Weerdt, 2004).