Working Paper

An econometric model of link formation with degree heterogeneity


Bryan S. Graham

Published Date

4 August 2015


Working Paper (CWP43/15)

I formulate and study a model of undirected dyadic link formation which allows for assortative matching on observed agent characteristics (homophily) as well as unrestricted agent level heterogeneity in link surplus (degree heterogeneity). Similar to fixed effects panel data analyses, the joint distribution of observed and unobserved agent-level characteristics is left unrestricted. To motivate the introduction of degree heterogeneity, as well as its fixed effect treatment, I show how its presence can bias conventional homophily measures. Two estimators for the (common) homophily parameter, beta0, are developed and their properties studied under an asymptotic sequence involving a single network growing large. The first,tetrad logit (TL), estimator conditions on a sufficient statistic for the degree heterogeneity. The TL estimator is a fourth-order U-Process minimizer. Although the fourth-order summation in the TL criterion function is over the i = 1…N agents in the network, due to a degeneracy property, the leading variance term of hat-beta_TL is of order 1/n, where n = N*(N-1)/2 equals the number of observed dyads. Using martingale theory, I show that the limiting distribution of hat-beta_TL (appropriately scaled and normalized) is normal. The second, joint maximum likelihood (JML), estimator treats the degree heterogeneity as additional (incidental) parameters to be estimated. The properties of hat-beta_JML are also non-standard due to a parameter space which grows with the size of the network. Adapting and extending recent results from random graph theory and non-linear panel data analysis (e.g., Chatterjee, Diaconis and Sly, 2011; Hahn and Newey, 2004), I show that the limit distribution of hat-beta_JML is also normal, but contains a bias term. Accurate inference necessitates bias-correction. The TL estimate is consistent under sparse graph sequences, where the number of links per agent is small relative to the total number of agents, as well as dense graphs sequences, where the number of links per agent is proportional to the total number of agents in the limit. Consistency of the JML estimate, in contrast, is shown only under dense graph sequences. The finite sample properties of hat-beta_TL and hat-beta_JML are explored in a series of Monte Carlo experiments.