Detecting Spatial Dependence in Panels with Common Factors through Permutations of Pesaran's CD(p) Test

In the spatial econometrics literature, spatial error dependence is characterized by spatial autoregressive processes, which relate every observation in the cross-section to any other with distance-decaying intensity: i.e., dependence obeys Tobler's First Law of Geography (''everything is related to everything else, but near things are more related than distant things''). In the literature on factor models, on the converse, the degree of correlation between cross-sectional units depends only on factor loadings. Standard spatial correlation tests have power against both types of dependence, while the economic meaning of the two can be much different; so it may be useful to devise a test for detecting ''distance-related'' dependence in the presence of a ''factor-type'' one. Pesaran's CD is a test for global cross-sectional dependence with good properties. The CD(p) variant only takes into account p-th order neighbouring units to test for local cross-sectional dependence. The pattern of CD(p) as p increases can be informative about the type of dependence in the errors, but the test power changes as new pairs of observations are taken into account. I propose a permutation test based on the values taken by the CD(p) test under permutations of the neighbourhood matrix, i.e. when ''reshuffling the neighbours''. I provide Montecarlo evidence of it being able to tell the presence of spatial-type dependence in the errors of a typical spatial panel irrespective of the presence of an unobserved factor structure.