Triadic Network Formation

Abstract: We study estimation and inference for triadic link formation with dyad-level fixed effects in a nonlinear binary choice logit framework. Dyad-level effects provide a richer and more realistic representation of heterogeneity across pairs of dimensions (e.g. importer--exporter, importer--product, exporter--product), yet their sheer number creates a severe incidental parameter problem. We propose a novel ``hexad logit'' estimator and establish its consistency and asymptotic normality. Identification is achieved through a conditional likelihood approach that eliminates the fixed effects by conditioning on sufficient statistics, in the form of hexads -- wirings that involve two nodes from each part of the network. Our central finding is that dyad-level heterogeneity fundamentally changes how information accumulates. Unlike under node-level heterogeneity, where informative wirings automatically grow with link formation, under dyad-level heterogeneity the network may generate infinitely many links yet asymptotically zero informative wirings. We derive explicit sparsity thresholds that determine when consistency holds and when asymptotic normality is attainable. These results have important practical implications, as they reveal that there is a limit to how granular or disaggregate a dataset one can employ under dyad-level heterogeneity.