Closed‐Form Maximum Likelihood Estimates of Nearest Neighbor Spatial Dependence

Models of n² potential spatial dependencies among n observations spread irregularly over space seem unlikely to yield simple structure. However, the use of the nearest neighbor leads to a very parsimonious eigenstructure of the associated adjacency matrix which results in an extremely simple closed form for the log determinant. In turn, this leads to a closed‐form solution for the maximum likelihood estimates of the spatially autoregressive and mixed regressive spatially autoregressive models. With the closed‐form solution, one can find the neighbors and compute maximum likelihood estimates for 100,000 observations in under one minute. The model has theoretical, pedagogical, diagnostic, modeling, and methodological applications. For example, the model could serve as a more enlightened null hypothesis for geographic data than spatial independence.