Power properties of invariant tests for spatial autocorrelation in linear regression

Many popular tests for residual spatial autocorrelation in the context of the linear regression model belong to the class of invariant tests. This paper derives a number of exact properties of the power function of such tests. In particular, we extend the work of Krämer (2005, Journal of Statistical Planning and Inference 128, 489-496) by characterizing the circumstances under which the limiting power, as the autocorrelation increases, vanishes. More generally, the analysis in the paper sheds new light on how the power of invariant tests for spatial autocorrelation is affected by the matrix of regressors and by the spatial structure. A numerical study aimed at assessing the practical relevance of the theoretical results is included.