Dyadic data with ordered outcome variables

TL;DR

This paper develops a novel estimation method for ordered logit models with fixed effects in directed network data, addressing the incidental parameter problem and enabling consistent inference even with sparse data or rare outcomes.

Contribution

It introduces the first tetrad-differencing conditional maximum likelihood approach for ordered outcomes with fixed effects, including two estimators with proven consistency properties.

Findings

The Pooled Tetrad Logit Estimator (PTLE) is more robust and consistent than alternative methods.

Simulations confirm the theoretical advantages of PTLE in various network sparsity scenarios.

Empirical application uncovers significant homophily effects in friendship networks.

Abstract

We consider ordered logit models for directed network data that allow for flexible sender and receiver fixed effects that can vary arbitrarily across outcome categories. This structure poses a significant incidental parameter problem, particularly challenging under network sparsity or when some outcome categories are rare. We develop the first estimation method for this setting by extending tetrad-differencing conditional maximum likelihood (CML) techniques from binary choice network models. This approach yields conditional probabilities free of the fixed effects, enabling consistent estimation even under sparsity. Applying the CML principle to ordered data yields multiple likelihood contributions corresponding to different outcome thresholds. We propose and analyze two distinct estimators based on aggregating these contributions: an Equally-Weighted Tetrad Logit Estimator (ETLE) and a Pooled Tetrad Logit Estimator (PTLE). We prove PTLE is consistent under weaker identification conditions, requiring only sufficient information when pooling across categories, rather than sufficient information in each category. Monte Carlo simulations confirm the theoretical preference for PTLE, and an empirical application to friendship networks among Dutch university students demonstrates the method's value. Our approach reveals significant positive homophily effects for gender, smoking behavior, and academic program similarities, while standard methods without fixed effects produce counterintuitive results.