Optimal Estimation of Two-Way Effects under Limited Mobility

We propose an empirical Bayes estimator for two-way effects in linked data sets based on a novel prior that leverages patterns of assortative matching observed in the data. To capture limited mobility we model the bipartite graph associated with the matched data in an asymptotic framework where its Laplacian matrix has small eigenvalues that converge to zero. The prior hyperparameters that control the shrinkage are determined by minimizing an unbiased risk estimate. We show the proposed empirical Bayes estimator is asymptotically optimal in compound loss, despite the weak connectivity of the bipartite graph and the potential misspecification of the prior. We estimate teacher values-added from a linked North Carolina Education Research Data Center student-teacher data set.