Testing endogeneity of spatial and social networks

The spatial autoregressive framework (SAR) has been widely used in studying spatial interdependence and social interaction. Previous research have assumed the spatial weight matrix (also known as sociomatrix or adjacency matrix) to be exogenously given. However, ignoring the possibility that adjacency matrices can be endogenous may lead to inconsistent estimates. Therefore, it is desirable to test whether spatial weights are endogenous or not. This paper constructs a Hausman specification test and one simple equivalent test under a general setting. We also provide LM test statistics for two representative SAR models with endogenous spatial/network relationships in the literature. We summarize their testing features and establish their asymptotic properties under the null and the alternative hypotheses. Monte Carlo simulations are conducted to examine the finite-sample performance of those tests.