Spatial Model Specification Tests

Moran's I Test

Based on the estimated linear regression Y = Xb + e and a row-standardized spatial weights matrix W, Moran's I test statistic is defined by

I =
e'We
e'e

Moran's I is approximately normally distributed with mean E(I) and Variance V(I) as follows:

E(I) = trace(MW)/(N-K), where M = I-X(X'X)-1X'
V(I) = {trace(MWMW')+trace[(MWMW)]+[trace(MW)]2}/((N-K)(N-K+2)) - E(I)2

Therefore, the standardized Moran's I is defined by

ZI =
I-E(I)
 ~ normal(0,1)
V(I)½

It is used to test for the first-order spatial correlation similar to Durbin-Watson test for the first-order time series correlation. For diagnostic checking of model specification, it can be applied to more general estimated residual vector e obtained from generalized least squares or maximum likelihood estimation method (that is, the data matrix X may be transformed according to the estimated model).

LM/RS Test for Spatial Error Model

Based on the restricted model (that is, assuming no spatial correlation), the Lagrange Multiplier (LM) or Rao Score (RS) test against a spatial error alternative takes the form

LM1 =
[e'We/(e'e/N)]2
T1

where T1 = trace(WW+W'W). LM1 test statistic follows an asymptotic c2(1) distribution and, apart from a scaling factor, corresponds to the square of Moran's I.

LM/RS Test for Spatial Lag Model

From the estimated regression model Y = Xb + e, the LM/RS test against a spatial lag alternative takes the form

LM2 =
[e'WY/(e'e/N)]2
T2

where T2 = [(WXb)'M(WXb)/(e'e/N)]+T1. LM2 also has an asymptotic c2(1) distribution. Since both LM1 and LM2 tests have power against the other alternatives, it is important to take account of possible lag dependence when testing for error dependence and vice versa.

The following robust tests are designed to work well under local model misspecification.

LM/RS Test for Spatial Error Model (Robust to Spatial Lag)

LM1* =
{[e'We/(e'e/N)]-(T1/T2)[e'WY/(e'e/N)]}2
 ~ c2(1)
T1(1-T1/T2)

LM/RS Test for Spatial Lag Model (Robust to Spatial Error)

LM2* =
{[e'WY/(e'e/N)]-[e'We/(e'e/N)]}2
 ~ c2(1)
T2-T1

The robust test statistics are similar to LM1* and LM2* extended with a correction factor to account for the local misspecification. LM1* is used to test for a spatial error process when the model specification contains a spatially lagged dependent variable. LM2* is used to test for a spatially lagged dependent variable in the presence of a spatial error process.

Wald Test and Likelihood Ratio Test

If the spatial model is estimated, the traditional form of Wald test or asymptotic t-test can be used to test for model specification. If both (restricted and unrestricted) models are estimated, likelihood ratio test can be used:

LR = -2 ln(L0/L1) ~ c2(q)

where L0 and L1 are the maximum likelihood function values for the restricted (null) and unrestricted (alternative) model, respectively. The degree of freedom q corresponds to the number of restrictions.