Abstract: In this article, we propose the mutual influence regression (MIR) model to establish the relationship between the mutual influence matrix of actors and a set of similarity matrices induced by their associated attributes. This model is able to explain the heterogeneous structure of the mutual influence matrix by extending the commonly used spatial autoregressive model, while allowing it to change with time. To facilitate inferences using the MIR, we establish parameter estimation, weight matrices selection, and model testing. Specifically, we employ the quasi-maximum likelihood estimation method to estimate the unknown regression coefficients. Then, we demonstrate that the resulting estimator is asymptotically normal, without imposing the normality assumption and while allowing the number of similarity matrices to diverge. In addition, we introduce an extended BIC-type criterion for selecting relevant matrices from the divergent number of similarity matrices. To assess the adequacy of the proposed model, we propose an influence matrix test, and develop a novel approach to obtain the limiting distribution of the test. The results of our simulation studies support our theoretical findings, and a real example is presented to illustrate the usefulness of the proposed MIR model.

Key words and phrases: Extended Bayesian information criterion, mutual influence matrix, similarity matrices, spatial autoregressive model.