Penalized Likelihood for Dyadic Network Formation Models with Degree Heterogeneity

2026-05-01

AI Summary

This study addresses the non-existence of maximum likelihood estimators and incidental parameter bias arising from extreme nodes—such as isolates or fully connected vertices—in directed network formation models that incorporate degree heterogeneity and reciprocity. To overcome these issues, the paper proposes a penalized likelihood estimation approach that guarantees estimator existence in finite samples and improves accuracy through bias correction. Notably, the authors develop asymptotic theory without requiring compactness assumptions on fixed effects, instead allowing fixed effects to diverge at a logarithmic rate. This framework is well-suited for large-scale sparse networks and avoids selection bias induced by sample trimming. Empirical application to the global trade network demonstrates the method’s ability to yield robust parameter estimates, effectively circumventing the failures and biases inherent in conventional approaches.

Abstract

Estimating network formation models with degree heterogeneity raises two problems in empirical networks. First, agents that send no links, receive no links, or link to all remaining agents can make the fixed-effects MLE fail to exist. Trimming these agents changes the estimation sample and induces selection bias. Second, the incidental-parameter problem biases common parameters and average partial effects. We resolve both issues through a penalized likelihood approach. Our leading specification is a directed network model with reciprocity, nesting the standard undirected and non-reciprocal directed models. The penalty guarantees finite-sample existence and yields bias corrections for coefficients and partial effects. We establish asymptotic results without imposing compactness on the fixed-effects. Allowing the fixed effects to diverge at a logarithmic rate, our asymptotic framework accommodates the degree sparsity ubiquitous in large empirical networks. A global trade application demonstrates that our estimator avoids selection bias and recovers robust parameters where conventional methods fail.

Problem

• degree heterogeneity
• incidental-parameter problem
• network formation models
• selection bias
• fixed-effects MLE

Innovation

• penalized likelihood
• degree heterogeneity
• incidental-parameter problem
• network formation models
• bias correction