Penalized Likelihood for Dyadic Network Formation Models with Degree Heterogeneity

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.

• The paper introduces a penalized likelihood approach that ensures estimator existence and eliminates first-order bias in network models with degree heterogeneity.
• It generalizes directed models with reciprocity, accommodating undirected networks via a unified estimation and inference framework that remains computationally tractable.
• Numerical experiments and a global trade application demonstrate that the method yields consistent, nearly unbiased estimates while retaining central nodes in sparse networks.

Penalized Likelihood Estimation for Dyadic Network Formation Models with Degree Heterogeneity

Overview and Motivation

This paper addresses the estimation challenges inherent in dyadic network formation models with agent-specific degree heterogeneity, particularly in the presence of sparse empirical networks. Conventional approaches, such as the fixed-effects maximum likelihood estimator (MLE), are known to face two key issues: (1) non-existence of the MLE when local degree boundary cases are present, and (2) first-order bias in parameter estimates induced by the incidental parameter problem. The authors propose a penalized likelihood (PL) method that resolves both concerns: it ensures estimator existence in all network samples (without node trimming) and corrects for bias in both coefficients and average partial effects (APEs). This approach generalizes directed network models with reciprocity—subsuming both undirected and non-reciprocal directed benchmarks—providing a unified estimation and inference framework.

Model Specification

The leading model is a directed dyadic formation model incorporating:

• Directed utility: $g_{ij} X_{ij}' \beta$ for a directed link from $i$ to $j$.
• Reciprocity (mutual utility): $g_{ij}g_{ji} Z_{ij}' \rho$.
• Degree heterogeneity: $g_{ij} (\alpha_i + \gamma_j)$, where $\alpha_i$ and $\gamma_j$ denote sender and receiver effects.

The model structurally nests the standard directed model (by setting $\rho = 0$) and the undirected network model (by restricting $\alpha_i = \gamma_i$ and $Z_{ij}$ symmetric).

Uniqueness of likelihood is secured through an ergodic stochastic best response dynamics, yielding a unique stationary distribution for dyads, with dyadic independence enabling feasible likelihood computation—unlike more complex, globally dependent network models.

Estimation Problems and the Penalized Likelihood Solution

Non-Existence of MLE

The fixed-effects MLE does not exist if nodes have degrees at the boundaries (zero or full degree). Existing ad hoc solutions involve iteratively trimming such nodes, resulting in selection bias and loss of central network nodes—a problem exacerbated in real-world networks such as trade, where local sparsity is ubiquitous.

Incidental Parameter Bias

Even when the MLE is defined, the dimensionality of node-level fixed effects (growing with network size) induces a first-order bias in the estimation of common parameters and APEs, as in other nonlinear panel frameworks.

Penalized Likelihood Construction

The proposed penalized likelihood adds to the log-likelihood a non-data-dependent penalty term derived from the block-diagonal part of the negative Hessian with respect to fixed effects. For the reciprocal model, this involves $i$0 blocks for each node. The penalty ensures:

• Existence: As the Hessian collapses toward the boundary (e.g., for an isolated node), the penalty term drives the objective to $i$1, precluding maximization at the boundary.
• Bias correction: The penalty introduces a likelihood-based correction analogous to higher-order analytical bias corrections, but more tractable. This extends naturally to both coefficient and APE estimation, avoiding difficult higher-order derivative calculations.

The same logic yields a diagonal penalty for non-reciprocal directed and scalar penalty for undirected models, providing a unified bias-correction framework across dyadic specifications.

Average Partial Effects (APE)

Structural functions such as APEs are directly estimable within this framework. The leading bias in plug-in APE estimators arises from fixed effects uncertainty and is corrected via a straightforward adjustment using the APE function’s second derivatives and the penalized Hessian approximation.

Asymptotic Theory

Sparse Network Asymptotics

Unlike most prior work, which assumes compactness of the fixed-effects parameter space (implying dense networks), the theory here allows fixed effects to diverge at a logarithmic rate. This admits degree sequences and link probabilities trending toward zero with growing network size, capturing realistic network sparsity. No compactness nor density assumptions are needed.

Explicit Hessian Inverse Approximation

Reciprocity introduces coupling between sender and receiver fixed effects, invalidating standard diagonal Hessian approximations. The authors derive a closed-form hybrid approximation using node-specific block-diagonal matrices and explicit low-rank corrections along aggregate weak directions. This innovation enables computational tractability for large, potentially sparse, reciprocal directed networks.

Consistency and Limit Distribution

Under the logarithmic divergence regime, the PL estimator enjoys:

• Uniform existence and consistency of both coefficient and fixed-effect estimates.
• Asymptotic normality for common parameters, centered at the true values—whereas in conventional MLE the center is shifted by first-order bias.
• The bias correction in APEs via penalized likelihood removes all leading-order bias, giving consistent estimation and valid inference without reliance on higher-order analytical corrections.

Numerical and Empirical Results

The Monte Carlo analysis demonstrates that:

• In sparse and heterogeneous networks, the conventional fixed-effects MLE is almost always undefined, while the PL estimator remains available and delivers accurate inference for both coefficients and APEs.
• The PL estimator maintains near-nominal empirical coverage across a wide range of simulated network designs, substantially outperforming standard approaches in settings of realistic local sparsity.
• Benchmarking against estimator-level corrections and conditional logit methods confirms accuracy and robust finite-sample performance.

In a global trade application using the 1986 CEPII-TradeProd textiles network, the PL estimator enables estimation on the full network (retaining all 133 countries), whereas the MLE trims away even core economies, leading to potentially severe selection bias. PL-corrected APEs on trade covariates show consistently smaller effects relative to MLE, indicating the practical magnitude of the bias correction for applied researchers.

Implications and Future Developments

From a theoretical perspective, penalizing the likelihood resolves both finite sample existence and large-sample bias in settings with high-dimensional fixed effects, reconciling issues that have previously required separate, often cumbersome, solutions. The framework is immediately applicable to empirical network research, as illustrated by the trade network case, and supports inference on both coefficients and structural effects without sample selection.

Practically, the method offers a generic and computationally convenient tool for researchers working with dyadic network data displaying local sparsity and reciprocal formation dynamics. The proposed Hessian approximation is well-suited for large-scale networks and could be extended to models with more complex forms of dyadic dependence.

Future directions include integrating likelihood-based inference directly into the penalized framework, extending to richer strategic network formation models beyond dyadic reciprocity, and further characterizing uniform inference procedures in non-compact, high-dimensional regimes.

Conclusion

By unifying the treatment of estimator existence and incidental parameter bias in dyadic network formation with degree heterogeneity, this paper provides a comprehensive and computationally feasible methodology applicable to a broad class of empirical networks. The penalized likelihood approach substantially improves finite-sample feasibility and large-sample accuracy for network models, particularly in the empirically pervasive case of sparse, heterogeneous, and reciprocal network linkages.