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In contrast, if we assume that the model is such that l = 0, Y = r ·WY +X · b +WX · q +e, known as the SDM (Spatial Durbin Model), then the estimators will be unbiased (and the test statistics valid) even if, in reality, we are in the presence of spatially auto-correlated errors (SEMs).
These two models - Kelejian-Prucha and SDM - include specific sub-models, i.e. the autore- gressive spatial model (SAR, AutoRegression spatial): Y = r ·WY +X · b +e) and the model with spatially auto-correlated errors (SEM, Spatial Error Model): Y = X · b +u and u = l ·Wu +e).
In this case, the SDM model is written: Y = X · b +r ·W (Y X · b)+e.
The SDM model is then estimated, and, using a likelihood ratio test (q = 0), the choice is made between the SAR model and the SDM model.
autocorrelation is present, i.e. r = 0 and l = 0 (right branch of the Figure 6.2), then the SDM model can be brought back (r = 0 and q = 0), followed by a test of the likelihood ratio on the common factor hypothesis (q = rb) to choose between SEM and SDM.
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