Moment Restrictions for Dyadic Network Formation Models with Nontransferable Utility

This paper investigates the construction of moment restrictions in dyadic network formation models with unobserved individual heterogeneity under nontransferable utility. Using observed links and covariates from five-node pentads, we construct moment restrictions that do not depend on individual fixed effects. For a broad class of covariate specifications, the construction is minimal in the sense that it uses the least possible number of nodes and dyads. Based on these moment restrictions, we propose the pentad-GMM estimator. We establish asymptotic normality of the pentad-GMM estimator in dense, sparse, and ultra-sparse network regimes, with regime-specific convergence rates and asymptotic variances. These results provide a basis for inference across all three regimes. To make our method computationally efficient, we develop an algorithm that reduces the computational cost of the estimator from na\"ive \(O(N^5)\) to \(O(N^3)\). We apply the proposed method to an academic-discussion network, and find academic homophily and a positive association between link formation and a potential partner's openness to different perspectives.