Anselin-Bera-Florax-Yoon (1996) Simple Diagnostic Tests for Spatial Dependence

Summary

Anselin, Bera, Florax and Yoon derive simple OLS-residual-based Lagrange multiplier (LM) tests for spatial dependence that remain valid when the other form of spatial dependence is locally present. Building on Bera and Yoon's (1993) adjustment for testing in the presence of a locally misspecified nuisance parameter, they construct robust LM tests — one for spatial error autocorrelation that is unaffected by a locally present spatial lag, and one for a spatial lag that is unaffected by locally present spatial error. A Monte Carlo study shows the adjusted tests have good finite-sample size and, unlike their unadjusted counterparts, correctly point to the source (lag vs error) of the dependence.

Key Claims

• Model. The mixed regressive–spatial-autoregressive model with spatial-AR disturbances, $y=\psi W_1 y + X\gamma + u,\; u=\phi W_2 u + \varepsilon,\; \varepsilon\sim N(0,\sigma^2 I)$, with row-standardized spatial-weights matrices $W_1,W_2$. Testing $H_0:\psi=0$ (no lag) or $H_0:\phi=0$ (no error) in the presence of the other parameter as a nuisance.
• The misspecification problem. If the nuisance parameter is locally non-zero, the ordinary one-directional test $LM_\psi$ acquires a non-centrality term $\lambda = \delta' J_{\psi\phi\cdot\gamma}J_{\phi\phi\cdot\gamma}^{-1}J_{\phi\psi\cdot\gamma}\delta$ and over-rejects — it has the wrong size even when its own parameter is truly zero. The crucial quantity is the conditional information $J_{\psi\phi\cdot\gamma}$; the ordinary test is only valid when it is zero.
• Robust (adjusted) tests. Applying Bera–Yoon (1993), the modified statistic recenters the score and adjusts its variance for the local presence of the nuisance parameter: $LM^*_\psi = \tfrac{1}{}\,[d_\psi-J_{\psi\phi\cdot\gamma}J_{\phi\phi\cdot\gamma}^{-1}d_\phi]'\,[J_{\psi\psi\cdot\gamma}-J_{\psi\phi\cdot\gamma}J_{\phi\phi\cdot\gamma}^{-1}J_{\phi\psi\cdot\gamma}]^{-1}[\,\cdot\,]$, asymptotically $\chi^2_1$ under the null and robust to a local nuisance value.
• Four tests. The unadjusted $LM_{err}$ (spatial error) and $LM_{lag}$ (spatial lag), plus the robust $RLM_{err}$ (error, robust to a local lag) and $RLM_{lag}$ (lag, robust to a local error) — all computed from OLS residuals and the weights matrix, no spatial estimation required.
• Decision rule (with Anselin–Rey 1991): compare the two one-directional tests; the robust versions discriminate which alternative — lag or error — actually generates the residual dependence, whereas the joint/one-directional unadjusted tests cannot.
• Monte Carlo. The adjusted LM tests have good finite-sample size and power across weights structures (rook/queen contiguity, $N=40,81,127$) and are more reliable than the unadjusted tests for identifying the source of dependence; power against the correctly-specified alternative remains high.

Concepts Introduced or Extended

• Spatial Autocorrelation
• Misspecification

Entities Mentioned

• Luc Anselin
• Anil K. Bera
• Raymond Florax; Mann J. Yoon (coauthors)

Quotes

"In this paper we propose simple diagnostic tests, based on ordinary least-squares (OLS) residuals, for spatial error autocorrelation in the presence of a spatially lagged dependent variable and for spatial lag dependence in the presence of spatial error autocorrelation."

"the adjusted LM tests have good finite sample properties … they prove to be more suitable for the identification of the source of dependence (lag or error) than their unadjusted counterparts."

My Take

This is the paper behind the now-standard "LM/RLM decision tree" every applied spatial-econometrics workflow uses: run OLS, compute $LM_{err}$, $LM_{lag}$ and their robust versions, and let the significant robust statistic select the spatial-lag vs spatial-error specification. Its elegance is that the whole apparatus is a special case of Bera–Yoon's general recentering for a locally-misspecified nuisance parameter — a genuinely reusable idea beyond the spatial setting. The main caveat is the usual one for this literature (flagged in Spatial Autocorrelation): everything is conditional on the analyst's weights matrix $W$, and the tests do not discriminate spatial dependence from spatial heterogeneity, which can masquerade as each other.