Title: Testing for Spatial Autocorrelation: The Regressors that Make the Power Disappear
Abstract: Abstract We show that for any sample size, any size of the test, and any weights matrix outside a small class of exceptions, there exists a positive measure set of regression spaces such that the power of the Cliff–Ord test vanishes as the autocorrelation increases in a spatial error model. This result extends to the tests that define the Gaussian power envelope of all invariant tests for residual spatial autocorrelation. In most cases, the regression spaces such that the problem occurs depend on the size of the test, but there also exist regression spaces such that the power vanishes regardless of the size. A characterization of such particularly hostile regression spaces is provided. Keywords: Cliff–Ord testPoint optimal testsPowerSpatial error modelSpatial lag modelSpatial unit rootJEL Classification: C12C21 ACKNOWLEDGMENTS I am grateful to two anonymous referees for their comments. I also wish to thank Ingmar Prucha for several valuable suggestions, Peter Burridge, Grant Hillier, Harry Kelejian, Patrick Marsh, Paolo Paruolo, Tony Smith, participants at the ESRC Econometrics Study Group, Bristol, 2008, and at the Spatial Econometrics Association World Conference, New York, 2008, for discussions and encouragement. Notes 1Lee and Yu (Citation2008) give several other references to the empirical economic literature. Regression models with strongly autocorrelated errors are also relevant in many applications outside economics; see, e.g., Basu and Reinsel (Citation1994) and Jones et al. (Citation2008). 2In particular, by the Perron–Frobenius theorem (e.g., Horn and Johnson, Citation1985, Ch. 8), our assumption is certainly satisfied if W is entrywise nonnegative and irreducible (see Section 3.3). Extensions of our setup to the cases when λmax is not defined (e.g., W is nilpotent) or has geometric multiplicity larger than one (e.g., W is block diagonal) are straightforward. 3More generally, in order for model (Equation2) to be invertible (so that u = ( I n − ρ W )−1 ϵ), ρ must be different from the reciprocal of the nonzero real eigenvalues of W . All such non-admissible values of ρ are outside H1. 4Note that y ′ M X y = 0 if and only if y belongs to the set {0} ∪ col( X ), which, since k < n, has zero measure. Hence, I is defined almost surely. 5The problem lies in inequality (12) of Krämer (Citation2005). In most cases, the critical value d 1 in that inequality can be positive or negative depending on α, and hence Krämer's proof holds only for sufficiently small α. In addition, there are weights matrices such that d 1 < 0 for any α; e.g., a W with zero diagonal entries and constant off-diagonal entries. For such matrices, inequality (12) is incorrect for all values of α. 6The two normalized eigenvectors are the same up to sign. Throughout the paper, it is irrelevant which of the two eigenvectors is chosen. Also, note that the normalization of f max is made here only for convenience, and will not be relevant until Section 4. 7Of course, X is assumed to be nonstochastic when constructing the Cliff–Ord test. We are now equipping G k, n with a probability measure only as a device to assess the practical relevance of the zero limiting power problem. One may think of an experiment where W is fixed, X is random, and the Cliff–Ord test is constructed for each realization of X . 8Note that 𝒞* is strictly smaller than 𝒞, because the condition m 1( W ′ W ) ≥ n − 1 is not necessarily satisfied by a matrix W in 𝒞. 9This part of Proposition 3.6 can be seen as a generalization of Proposition 5 in Smith (2009), which asserts that when W is an equal weights matrix and the regression contains an intercept, the Cliff–Ord statistic is degenerate. For a discussion of the associated identification problem, see Arnold (Citation1979), Kariya (Citation1980), and Martellosio (Citation2011). 10An example of a W ∈ 𝒞* ⊂ 𝒞 that does not satisfy Condition 3 is the 3 × 3 matrix with rows (0, 1, 2), (2, 0, 1), and (1, 2, 0). 11In general, m 1( W ) > 1 requires W to satisfy some symmetries; see Biggs (Citation1993, Ch. 15). An emblematic example is a block diagonal matrix W whose r blocks are all equal to an n × n equal weights matrix. For such a matrix, m 1( W ) = n − r. 12Here, for clarity and contrary to what is done in the rest of the paper, we use different notation for a random variable and its realizations. 13Of course, if E( z ) ≠ 0, one should demean the data before constructing an autocorrelation index. That is, in practice, one would not use I 0, but I ι : = v ′ M ι W M ι v / v ′ M ι v , where ι denotes the n × 1 vector of all ones. In general, v ∈ E 1( W ) (resp., v = f max) will be associated to a small (resp., large) value of I ι . 14In terms of Q , the coordinate system is that of the eigenvectors of the matrix Q + Q ′, which is a scalar multiple of (Equation5).