Fixed-effect regressions on network data

This paper studies inference on fixed effects in a linear regression model estimated from network data. An important special case of our setup is the two-way regression model, which is a workhorse method in the analysis of matched data sets. Networks are typically quite sparse and it is difficult to see how the data carry information about certain parameters. We derive bounds on the variance of the fixed-effect estimator that uncover the importance of the structure of the network. These bounds depend on the smallest non-zero eigenvalue of the (normalized) Laplacian of the network and on the degree structure of the network. The Laplacian is a matrix that describes the network and its smallest non-zero eigenvalue is a measure of connectivity, with smaller values indicating less-connected networks. These bounds yield conditions for consistent estimation and convergence rates, and allow to evaluate the accuracy of first-order approximations to the variance of the fixed-effect estimator. The bounds are also used to assess the bias and variance of estimators of moments of the fixed effects.