Closed-form maximum likelihood estimates of nearest neighbor spatial dependence

Models of n(2) potential spatial dependencies among n observations spread i rregularly oz;er space seem unlikely to yield simple structure. However, th e use of the nearest neighbor leads to a very parsimonious eigenstructure o f the associated adjacency matrix which results in an extremely simple clos ed form for the log determinant. In turn, this leads to a closed-form solut ion for the maximum likelihood estimates of the spatially autoregressive an d mixed regressive spatially autoregressive models. With the closed-form so lution, one can find the neighbors and compute maximum likelihood estimates for 100,000 observations in under one minute. The model has theoretical, p edagogical, diagnostic, modeling, and methodological applications. For exam ple, the model could serve as a more enlightened null hypothesis for geogra phic data than spatial independence.