bayespecon.diagnostics.bayesian_lmtests.bayesian_robust_lm_wx_test¶
- bayespecon.diagnostics.bayesian_lmtests.bayesian_robust_lm_wx_test(model)[source]¶
Bayesian robust LM-WX test (H₀: γ = 0, robust to ρ).
Tests the null hypothesis that the WX coefficients are zero, robust to the local presence of a spatial lag (ρ). Uses the Neyman orthogonal score adjustment from Doğan et al. [2021], Proposition 3, which is the Bayesian analogue of the robust LM-WX test in Koley and Bera [2024].
The alternative model is SAR (includes ρ but not γ). For each posterior draw from the SAR model, residuals are:
\[\mathbf{e} = \mathbf{y} - \rho W \mathbf{y} - X\beta\]The unadjusted scores are:
\[\begin{split}g_\rho &= \mathbf{e}^\top W \mathbf{y} \\ \boldsymbol{g}_\gamma &= (WX)^\top \mathbf{e}\end{split}\]The Neyman-adjusted score for γ is:
\[\boldsymbol{g}_\gamma^* = \boldsymbol{g}_\gamma - J_{\gamma\rho \cdot \sigma} J_{\rho\rho \cdot \sigma}^{-1} g_\rho\]The adjusted weight matrix is:
\[C_{\gamma\gamma}^* = P_{\gamma\gamma} J_{\gamma \cdot \rho}\]where \(P_{\gamma\gamma} = I - J_{\gamma\rho \cdot \sigma} J_{\rho\rho \cdot \sigma}^{-1} J_{\rho\gamma \cdot \sigma} J_{\gamma \cdot \rho}^{-1}\) and \(J_{\gamma \cdot \rho} = J_{\gamma\gamma \cdot \sigma} - J_{\gamma\rho \cdot \sigma} J_{\rho\rho \cdot \sigma}^{-1} J_{\rho\gamma \cdot \sigma}\).
The robust LM statistic for each draw is:
\[\mathrm{LM}_R = (\boldsymbol{g}_\gamma^*)^\top (C_{\gamma\gamma}^*)^{-1} \boldsymbol{g}_\gamma^*\]which is distributed as \(\chi^2_{k_{wx}}\) under H₀.