neighbayes.diagnostics.lmtests.bayesian_glm_lm_lag_test¶
- neighbayes.diagnostics.lmtests.bayesian_glm_lm_lag_test(model)[source]¶
Bayesian LM test for an omitted spatial lag of the linear predictor.
PG-augmented analog of
bayesian_lm_lag_test(). In the SAR-GLM family the linear predictor satisfies \(\psi = (I-\rho W)^{-1} X\beta\), so \(\rho\) enters only the mean of the augmented response \(\tilde z = \psi + \varepsilon\); the conditional covariance \(\mathrm{Cov}(\tilde z\mid\psi,\omega) = \Omega^{-1}\) is \(\rho\)-independent. This is the structural difference from Gaussian SAR (where \(\mathrm{Var}(y) = \sigma^2 (I-\rho W)^{-1} (I-\rho W)^{-\top}\) depends on \(\rho\)) and it drops the \(T_{WW}\) trace term that appears in Anselin (1996, eq. 13).The score reduces to the standard GLM score test for adding a single regressor \(z = WX\beta\) to the linear predictor:
\[S^{(d)} = (WX\beta^{(d)})^\top\, \Omega^{(d)}\, \tilde e^{(d)},\]with concentrated variance
\[V = (WX\bar\beta)^\top \bar\Omega\, M_X^{\bar\Omega}\, (WX\bar\beta), \qquad M_X^{\bar\Omega} = I - X(X^\top\bar\Omega X)^{-1} X^\top\bar\Omega.\]Asymptotically \(\chi^2_1\) under \(H_0:\rho=0\).
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