Asymptotic Properties of Quasi-Maximum Likelihood Estimators for Heterogeneous Spatial Autoregressive Models

In this paper, we address a class of heterogeneous spatial autoregressive models with all n(n - 1) spatial coefficients taking m distinct true values, where m is independent of the sample size n, and we establish asymptotic properties of the maximum likelihood estimator and the quasi-maximum likelihood estimator for all parameters in the class of models, extending Lee's work (2004). The rates of convergence of those estimators depend on the features of values taken by elements of the spatial weights matrix in this model. Under the situations where, based on the values of the weights, each individual will not only influence a few neighbors but also be influenced by only a few neighbors, the estimator can enjoy an root n-rate of convergence and be asymptotically normal. However, when each individual can influence many neighbors or can be influenced by many neighbors and their number does not exceed o(n), singularity of the information matrix may occur, and various components of the estimators may have different (usually lower than root n) rates of convergence. An inconsistent estimator is provided if some important assumptions are violated. Finally, simulation studies demonstrate that the finite sample performances of maximum likelihood estimators are good.