Nontestability of Equal Weights Spatial Dependence

Abstract

We show that any invariant test for spatial autocorrelation in a spatial error or spatial lag model with equal weights matrix has power equal to size. This result holds under the assumption of an elliptical distribution. Under Gaussianity, we also show that any test whose power is larger than its size for at least one point in the parameter space must be biased.

On the Power of Invariant Tests for Hypotheses on a Covariance Matrix

“…We next verify that the spatial error model satisfies Assumptions 1, 3, and 4, and that it satisfies Assumption 2 under a mild condition on the distribution of ε. The first claim in Lemma 4.3 also appears in Martellosio (2011b), Lemma 3.3.…”

“…1 A further contribution of the present paper is a characterization of the situation where no invariant test can distinguish the null hypothesis of no correlation from the alternative. This characterization helps to explain, and provides a unifying framework for, phenomena observed in Kadiyala (1970), Arnold (1979), Kariya (1980), Martellosio (2010), and Martellosio (2011b).…”

“…We close our discussion of spatial regression models by applying the results on indistinguishability developed in Section 2.3 to these models. It turns out that a number of results in Martellosio (2010) (namely all parts of Proposition 3, 4, and 5 that are based on degeneracy of the test statistic) as well as the first part of the theorem in Martellosio (2011b) are consequences of an identification problem in the distribution of the maximal invariant statistic (more precisely, an identification problem in the "reduced" experiment). Theorem 2.30 and Corollary 2.31 thus provide a simple and systematic way to recognize when this identification problem occurs.…”

“…Compare Footnote 2 inMartellosio (2011b), where the author attempts to justify invariance in case of a spatial lag model. The argument given there to show that (I − ρW ) −1 for W an equal weights matrix maps span(X) into itself, however, does not make sense as it is based on an incorrect expression for E(y), which is incorrectly given as ∆ρX.…”

Spatial fixed effects and spatial dependence in a single cross‐section

“…Estimation with near unit spatial roots are considered using equal weights in Baltagi and Liu (), whereas an extension to a panel data setting is given in Baltagi (). Problems with standard tests for spatial autocorrelation, such as the Moran's I of Cliff and Ord and the Lagrange multiplier test are demonstrated in Baltagi and Liu () and Martellosio ().…”