Using Matrix Exponentials to Explore Spatial Structure in Regression Relationships

Spatial regression relationships typically specify the structure of spatial relationships among observations using a fixed n by n weight matrix, where n denotes the number of observations. For computational convenience, this weight matrix is usually assumed to represent fixed sample information, making inferences conditional on the particular spatial structure embodied in the weight matrix used in the model. In an economic context, where the spatial structure can arise from externalities or spillovers, the magnitude and extent of influence from one observation or spatial location on other observations at nearby locations may be a subject of interest and inference. To address the dual goals of improving inference and understanding spatial structure, we set forth a flexible specification for the spatial weights that allows hyperparameters to control the number of neighboring entities as well as decay of influence over space. This specification is combined with the matrix exponential spatial specification (MESS) introduced in Pace and LeSage (2000) leading to a Bayesian model that allows posterior inferences regarding the magnitude and extent of spatial influence. We illustrate the method in an application that examines the role of geographical proximity in the economic growth of 219 urban zip-code areas in northeast Ohio. Inferences regarding the magnitude and extent of spatial influence in this setting may be of interest because theories of economic growth have increasingly stressed technological spillovers as a source of growth (see Romer, 1994; Grossman and Helpman, 1994).