Using Convex Combinations of Spatial Weights in Spatial Autoregressive Models

Spatial regression models rely on simultaneous autoregressive processes that model spatial or cross-sectional dependence between cross-sectional observations using a weight matrix. A criticism of applied spatial regression methods is that reliance on geographic proximity of observations to form the weight matrix that specifies the structure of cross-sectional dependence might be unrealistic in some applied modeling situations. In cases where the structure of dependence or connectivity between (cross-sectional) observations arises from non-spatial relationships, spatial weights are theoretically unjustifiable. Some literature addresses the structure of dependence between observations by introducing geographic proximity as well as other types of non-spatial proximity, resulting in a model that utilizes multiple weight matrices. Each set of weights reflect a different type of dependence specified using linear combinations of observations defined by alternative characteristics. The multiple weight matrix approach results in a simultaneous autoregressive process that poses a number of challenges for parameter estimation and model interpretation. We focus on literature that relies on a single connectivity matrix constructed from a convex combination of multiple matrices, each of which reflects a different type of dependence or interaction structure. The advantage of this approach is that the resulting simultaneous autoregressive process is amenable to conventional spatial regression estimation algorithms as well as methods developed for interpretation of estimates from these models. Estimates of the scalar parameters used to form the convex combination of the weight matrices can be used to produce an inference regarding the relative importance of each type of dependence.