Testing for Spatial Autocorrelation in Areal Data

Consider a linear model for a spatial system of n areal units, (1) , ~ (0,) Y X u u N V    where both  and V are unknown. The simplest procedure for specifying the covariance matrix, V , is to start by assuming that (2) 2 n V I   , so that  can be estimated by OLS. Given this estimate, one can then test to see whether there is sufficient spatial autocorrelation in the residuals to warrant more elaborate specifications of V. If for a given sample, y (i.e., observed realization of Y) we denote the OLS estimate of  by (3) 1 ˆ () X X Xy      and corresponding OLS residuals by (4) ˆ ˆ u y X   then the task is to develop a test for spatial autocorrelation using the residuals, ˆ u. To do so, we shall assume that the underlying spatial structure of these n areal units is representable by a given spatial weight matrix, ij W w i j n  . In terms of W, it is then hypothesized that all relevant spatial autocorrelation among the residuals, u , in (1) can be captured by the spatial autoregressive model, (5) 2 , ~ (0,) n u Wu N I       The key feature of this hypothesis is that testing for spatial autocorrelation then reduces to testing the null hypothesis: