Siao Xu

We study the latent group structure in a two-way effect model for matched data represented by a bipartite network. A directed acyclic graph (DAG) encodes the assumed relationships among the observed outcomes, network links, latent effects, group memberships, and prior parameters, thereby clarifying the data-generating process and facilitating estimation. Unobserved heterogeneity is modeled using group-specific prior distributions, allowing individual effects to vary both across and within groups. Although the two-way effects can be estimated by ordinary least squares (OLS), this estimator may be imprecise when the underlying bipartite network is sparse or connected across latent groups. We therefore develop a machine-learning-assisted EB estimator that first learns the latent group structure from the network and then exploits this structure to shrink the estimated effects toward their corresponding group-specific distributions. We establish identification conditions, consistency of prior-parameter estimates, oracle equivalence, and the conditional MSE/bias–variance characterization. The Monte Carlo simulations show that it substantially reduces estimation error relative to OLS under the data-generating processes considered. Finally, in an empirical application to Amazon user–product ratings, we use the proposed method to estimate and characterize heterogeneity in the user- and product-specific components of ratings.