bayespecon.diagnostics.lmtests.bayesian_robust_lm_wx_sem_test¶
- bayespecon.diagnostics.lmtests.bayesian_robust_lm_wx_sem_test(model)[source]¶
Bayesian robust LM-WX test in SEM context (H₀: γ = 0 | SEM).
Companion to
bayesian_robust_lm_lag_sem_test()with target and nuisance swapped: tests whether the SEM fit should be extended to SDEM by adding the WX block, robust to a locally-omitted spatial lag.Setup mirrors
bayesian_robust_lm_lag_sem_test(): per-draw whitened residuals \(\mathbf{u}^{(d)} = (I - \lambda^{(d)} W) (\mathbf{y} - X\beta^{(d)})\), filtered designs at \(\bar\lambda\), raw scores\[\mathbf{g}_\gamma^{(d)} = \tilde Z_\gamma^{\top} \mathbf{u}^{(d)}, \qquad g_\rho^{(d)} = \mathbf{u}^{(d)\,\top} \tilde z_\rho.\]The Neyman-orthogonal adjustment and Schur complement (with target \(\gamma\), nuisance \(\rho\)) give
\[\begin{split}\mathbf{g}_\gamma^{*\,(d)} &= \mathbf{g}_\gamma^{(d)} - V_{\gamma\rho} V_{\rho\rho}^{-1} g_\rho^{(d)},\\ V_{\gamma\,|\,\rho} &= V_{\gamma\gamma} - V_{\gamma\rho} V_{\rho\rho}^{-1} V_{\rho\gamma}.\end{split}\]The per-draw statistic is
\[\mathrm{LM}_R^{(d)} = \mathbf{g}_\gamma^{*\,(d)\,\top} V_{\gamma\,|\,\rho}^{-1} \mathbf{g}_\gamma^{*\,(d)} \;\xrightarrow{d}\; \chi^2_{k_{wx}} \quad \text{under } H_0,\]independent of local misspecification in \(\rho\).