Division of Weight Matrix Based on Pair-Distances and Resulting Nonlinear Spatial Dependency in Spatial Autoregressive Models

In spatial autoregressive models, spatial autocorrelations in the dependent (or omitted) variable are modeled. Dependency is measured under known spatial structures, typically represented as a spatial weight matrix (W). For ordinal spatial autoregressive models, a unique W exists, and the strength of influence of the variable (or autocorrelation) is expressed through this matrix. Elements of W are obtained as a monotonically decreasing function of distance between points within the solution space. Since the coefficients of W are a measure of the autocorrelation, the model is a linear function of the dependency. In actual situations, for example, locational or tax competition, highly complex interdependency may exist. Simultaneously, the model may exhibit negative autocorrelation at short distances, caused by substitution effects, and positive autocorrelation at long distances, caused by complementary effects. In this paper, we discuss realistic models by dividing spatial terms. The resulting nonlinear dependency function is represented through the coefficients of a pair of spatial weight matrices. As an example, we formulate land price models for all prefectures in Japan. From comparisons between the conventional ordinal one-weight-matrix model and the proposed two-weight-matrices model, we established that the latter may produce lower information criterion values. Our results show that division of spatial terms is important for extending the variability of spatial autoregressive models