Power Properties of Invariant Tests for Spatial Autocorrelation in Linear Regression

Abstract

This paper derives some exact power properties of tests for spatial autocorrelation in the context of a linear regression model. In particular, we characterize the circumstances in which the power vanishes as the autocorrelation increases, thus extending the work of Krämer (2005, Journal of Statistical Planning and Inference 128, 489-496). More generally, the analysis in the paper sheds new light on how the power of tests for spatial autocorrelation is affected by the matrix of regressors and by the spatial structure. We mainly focus on the problem of residual spatial autocorrelation, in which case it is appropriate to restrict attention to the class of invariant tests, but we also consider the case when the autocorrelation is due to the presence of a spatially lagged dependent variable among the regressors. A numerical study aimed at assessing the practical relevance of the theoretical results is included.

Testing for Spatial Autocorrelation: The Regressors that Make the Power Disappear

“…Indeed, if W has constant row-sums, the limiting power of any invariant test cannot vanish as long as an intercept is included in the regression; see Section 3.2.2 of Martellosio (2008). A discussion of the possible consequences of our results for the important case of a row-standardized W is deferred to Section 5.…”

“…In Krämer (2005) and Martellosio (2008) conditions are given for tests of spatial autocorrelation to have zero limiting power, where the limit is taken as the autocorrelation increases. The present paper has continued that work by addressing the question of whether it is always possible to run into regressors such that the limiting power vanishes.…”

“…That is, there are circumstances in which it may be very difficult to detect spatial autocorrelation when the autocorrelation is in fact large. Martellosio (2008) shows that Krämer's results can be extended to any test of spatial autocorrelation, and to other models, including a spatial lag model. Such extensions stem from the fact that any SAR(1) model tends, as the autocorrelation parameter goes to the right boundary of the parameter space, to a degenerate model.…”

“…How likely it is in a given application to run into the regression spaces causing the limiting power to vanish will depend to a very large extent on W , n − k, and α. Some simulation exercises analyzing this issue are reported in Krämer (2005) and Martellosio (2008). Here, we stress that Theorem 3.3 implies that in any simulation study of the power properties of the Cliff-Ord test for a fixed W / ∈ C and when X is drawn from a distribution supported on the whole R n×k , there must be repetitions such that the limiting power vanishes, provided only that the number of repetitions is large enough.…”

“…Both Krämer (2005) and Martellosio (2008) contain simulation results suggesting that, at least in some cases, a vanishing limiting power of tests for spatial autocorrelation is an empirically relevant phenomenon. In the present paper, we complement those simulation results with a theoretical investigation of whether, for any fixed weights matrix, there always are regression spaces (i.e., column spaces of the regressor matrix) such that the limiting power vanishes.…”