A recent line of research in analysing spatial data, mainly associated with Luc Anselin, has focused on how to establish the characteristics of the dependence between observations, whether dependence can be demonstrated and how it ought to be represented. (see for instance Anselin, 1988b, 1990, Anselin and Rey, 1991, Anselin and Florax, 1995, Anselin et al. 1996). Burridge (1980, 1981) made the first attempt to extend the tests for regression misspecification given by Cliff and Ord (1973), using Lagrange multiplier techniques to derive simpler procedures. These have been followed up by Anselin and collaborators, and are now at a stage at which their use in all cases in which geographical cross-sectional data are being analysed should be expected.
A problem solved in Anselin et al. (1996) is that of tests for spatial lag and spatial error specifications being mutually contaminated by each other, that is the original LM test for non-zero 
also responds to non-zero 
and vice-versa. The new tests take into account the possible non-zero value of the nuisance parameter, and appear to discriminate well between the two alternative forms. Results obtained by Bivand and Szymanski (1998) indicate that these refined LM tests are of considerable use in model specification, and that test results, drawn from OLS residuals from the initial model, are confirmed by likelihood ratio test results from maximum likelihood estimates of and for the spatial lag and spatial error models respectively.
Work on global tests for mis-specification is continuing, with Tiefelsdorf and Boots (1995) and Hepple (1998b) arriving independently at exact distributions of Moran's I as a test statistic for regression residuals, using results on ratios of quadratic forms in normal variables. Tiefelsdorf and Boots have also extended their results to the local Moran's 
statistic (1997).