Modeling and Estimating Two-Layer Network Interactions with Unknown Heteroskedasticity

This paper introduces a model featuring two hierarchically structured layers of spatial or social networks in a cross-sectional setting. Individuals interact within groups, while groups also interact with one another, generating network dependence at both the individual and group levels. The network structures can be flexibly specified using general measures of proximity. The model accommodates individual random effects with heteroskedasticity, as well as unobserved random group effects. Given the complex error structure, we consider a Generalized Method of Moments (GMM) approach for estimation. The linear moment conditions exploit exogenous variations in individual and group characteristics to identify the network parameters at both levels. To enhance identification when linear moments are weak, we also propose a new set of quadratic moments that are robust to heteroskedasticity. Building on the method of Lin and Lee (2010), we can consistently estimate the variance-covariance (VC) matrix of these heteroskedasticity-robust moments, enabling the construction of a GMM estimator with optimally weighted moments. The asymptotic properties of both a generic and the "optimal" GMM estimator are derived. Monte Carlo simulations demonstrate that the proposed estimators perform well in finite samples. The model is applicable to a variety of social and economic contexts where network effects at two distinct levels are of particular interest, with peer effects among students within the same class and spillovers between classes serving as a leading example.