Linear Regression with Centrality Measures

Abstract

This paper studies the properties of linear regression on centrality measures when network data is sparse and observed with error. We make three contributions in this setting. First, we show that OLS estimators can become inconsistent under sparsity and characterize the threshold at which this occurs, finding that regression on eigenvector centrality is less robust to sparsity than on degree and diffusion. Second, we derive the asymptotic distributions of the OLS estimators in regimes where they remain consistent. We show that when the target coefficients are non-zero, the estimators exhibit asymptotic bias that can be large relative to their variance, rendering conventional confidence intervals and t-tests invalid. Third, we propose bias correction and inference procedures for OLS with sparse, noisy networks. Simulations confirm that our methods perform well in such settings. We demonstrate the empirical relevance of our results in a stylized study of the relationship between consumption smoothing and informal insurance in Nyakatoke, Tanzania.