The most commonly used specification test for spatial autocorrelation is derived from a statistic developed by Moran (1948) as the two-dimensional analog of a test for univariate time series correlation (see also Cliff and Ord, 1973). In matrix notation, Moran’s I statistic is
I = [N/S0)(e’ We/e’e), (14.30)
with e as a vector of OLS residuals and S0 = X; XyWj, a standardization factor that corresponds to the sum of the weights for the nonzero cross-products. The statistic shows a striking similarity to the familiar Durbin-Watson test.29
Moran’s I test has been shown to be locally best invariant (King, 1981) and consistently outperforms other tests in terms of power in simulation experiments (for a recent review, see Anselin and Florax, 1995b). Its application has been extended to residuals in 2SLS regression in Anselin and Kelejian (1997), and to generalized residuals in probit models in Pinkse (2000). General formal conditions and proofs for the asymptotic normality of Moran’s I in a wide range of regression models are given in Pinkse (1998) and Kelejian and Prucha (1999b). The consideration of Moran’s I in conjunction with spatial heteroskedasticity is covered in Kelejian and Robinson (1998, 2000).
When spatial regression models are estimated by maximum likelihood, inference on the spatial autoregressive coefficients may be based on a Wald or asymptotic f-test (from the asymptotic variance matrix) or on a likelihood ratio test (see Anselin, 1988a, ch. 6; Anselin and Bera, 1998). Both approaches require that the alternative model (i. e. the spatial model) be estimated. In contrast, a series of test statistics based on the Lagrange Multiplier (LM) or Rao Score (RS) principle only require estimation of the model under the null. The LM/RS tests also allow for the distinction between a spatial error and a spatial lag alternative.30
An LM/RS test against a spatial error alternative was originally suggested by Burridge (1980) and takes the form
LMerr = [e’ We/(e’e/N )]2/[tr(W2 + W’W)]. (14.31)
This statistic has an asymptotic x2(1) distribution and, apart from a scaling factor, corresponds to the square of Moran’s I.31 From several simulation experiments (Anselin and Rey, 1991; Anselin and Florax, 1995b) it follows that Moran’s I has slightly better power than the LMerr test in small samples, but the performance of both tests becomes indistinguishable in medium and large size samples. The LM/RS test against a spatial lag alternative was outlined in Anselin (1988c) and takes the form
LMlag = [e’ Wy/(e’e/N)]2/D, (14.32)
where D = [(WXp)'(I – X(X’X)-1X’)(WXp)/o2] + tr(W2 + W’ W). This statistic also has an asymptotic x2(1) distribution.
Since both tests have power against the other alternative, it is important to take account of possible lag dependence when testing for error dependence and vice versa. This can be implemented by means of a joint test (Anselin, 1988c) or by constructing tests that are robust to the presence of local misspecification of the other form (Anselin et al., 1996).
The LM/RS principle can also be extended to more complex spatial alternatives, such as higher order processes, spatial error components and direct representation models (Anselin, 2000), to panel data settings (Anselin, 1988b), and to probit models (Pinkse, 1998, 2000; Pinkse and Slade, 1998). A common characteristic of the LM/RS tests against spatial alternatives is that they do not lend themselves readily to a formulation as an NR1 expression based on an auxiliary regression. However, as recently shown in Baltagi and Li (2000a), it is possible to obtain tests for spatial lag and spatial error dependence in a linear regression model by means of Davidson and MacKinnon’s (1988) double length artificial regression approach.