15 November 2005
This entry follows up on earlier ones here and here on spatial statistics and spatial lag, and discusses another consequence of spatial dependence. Spatial error autocorrelation arises if error terms are correlated across observations, i.e., the error of an observation affects the errors of its neighbors. It is similar to serial correlation in time series analysis and leaves OLS coefficients unbiased but renders them inefficient. Because it's such a bothersome problem, spatial errors is also called "nuisance dependence in the error."
There are a number of instances in which spatial error can arise. For example, similar to what can happen in time series, a source of correlation may come from unmeasured variables that are related through space. Correlation can also arise from aggregation of spatially correlated variables and systematic measurement error.
So what to do if there is good reason to believe that there is spatial error? Maybe the most famous test is Moran's I which is based on the regression residuals and is also related to Moran's scatterplot of residuals which can be used to spot the problem graphically. There are other statistics like Lagrange multiplier and likelihood ratio tests, and each of them has different ways of getting at the same problem. If there is good reason to believe that spatial error is a problem, then the way forward is either model the error directly or to use autoregressive methods.
In any case it's probably a good idea to assess whether spatial error might apply to your research problem. Because of it's effect on OLS, there might be a better way to estimate the quantity you are interested in, and the results might improve quite a bit.