Identifying relationship-level effects using covariance restrictions

We propose a novel decomposition to identify relationship-specific effects or shocks in a bipartite network under covariance restrictions. We show existing two-way fixed effects decompositions are ill-suited to correlations consistent with realistic heterogeneity. Our strategy yields a simple consistent estimator. We estimate relationship-level credit demand and supply shocks across nine euro-area countries and three episodes. We find evidence demand and supply each have both firm and bank components, and within-firm/bank shock variation is of comparable scale to between-firm/bank variation. Regressions using firm fixed effects as demand controls are at odds with economic theory, while our method uncovers significant deleterious effects of the post-2022 monetary contraction on exposed firms.