Where W is the spatial weights matrix; X is the matrix of independent variables; β is the vector
of coefficients of independent variables; ρ is the autoregressive spatial coefficient and ε is the error
term. Adding Wy as an explanatory variable to model 4 means that the values of variable y in the
locality i are related to the values of this variable in neighboring localities. This model’s estimation
method must take into account the endogenous nature of variable Wy (Anselin, 1999). Its reduced form
gives a more precise interpretation of model 4:
Y =(I-ρW)-1Xβ +(I - ρW)-1ε
(5)
The expansion (I - ρW)-1 includes both the explanatory variables and the error terms. So the
economic interpretation of the causality relationship yj → yi may be considered as being the result of a
process involving global spatial correlation in the explanatory variables and error terms. This implies
that shocks in a given locality will affect all other localities through a global multiplier effect,
associated to both the explanatory variables and the ones that were not considered into it but are present
in the error terms.
In addition to the two models specified above, another model was used when so required by the
tests: SARSAR (or SARMA), which is a combination of the two previous models (error and spatial lag
models).8
The models were estimated using SpaceStat 1.80 (Anselin, 2001). The methods available for
estimating the spatial lag model are the maximum likelihood (ML) and instrumental variables - IV
(2SLS, Robust and Bootstrap). IV-Robust and IV-Bootstrap estimates are alternatives to 2SLS in the
case of nonnormality of residuals and heteroscedasticity. Both GMM estimation alternatives are robust
for nonnormality of errors.
Once the analysis of residuals for all models had produced strong evidence of nonnormality,
spatial error models were estimated using the 2-stage GM method, and spatial lag models using the VI-
Robust method. In regard to the SARSAR/SARMA model, Kelejian & Prucha IV-Generalized
procedure was used (1998).9
For this paper, the model estimation procedure comprised the following stages: (a) typical OLS
estimations; (b) use of specification tests to detect spatial patterns in OLS residuals; (c) model re-
estimation following more suitable specifications as shown by specification tests; (d) confirmatory test
for the final specification.
8 In practice, none of the specification tests based on MQO residuals is able to distinguish between an AR and a MA spatial
error, since these are considered to be locally equivalent alternatives (Anselin, 1999).
9 Jarque-Bera test results to be seen in all OLS-estimated equations.
16