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The LM decision rule: choosing between spatial models

Spatial methods05 / NOTE

Moran's I tells you spatial dependence exists but not which kind. The Lagrange-multiplier tests — and especially their robust variants — are how you decide between a spatial error and a spatial lag.

A residual Moran’s I that rejects tells you spatial dependence is present — but it is non-specific. It will not tell you whether to fit a spatial error or a spatial lag. The Lagrange-multiplier (LM) tests are built for exactly that decision, and the useful part is that they run on OLS residuals alone — you don’t have to estimate the spatial model to test whether you need one.

Two tests, each aimed at one alternative:

  • LM-error — H₁: spatially autocorrelated errors → the spatial error model (parameter λ).
  • LM-lag — H₁: a spatially lagged dependent variable → the spatial lag model (parameter ρ).

Each is χ²₁ under the null. The complication: LM-error is not invariant to a lag that is truly present, and vice versa, so both standard tests can reject even when only one form is real. The fix is the robust LM variants, which correct each test for the possible presence of the other form — they isolate one alternative net of the other, and are the decisive evidence when the plain tests disagree.

The standard procedure (Anselin et al., 1996; Florax et al., 2003):

1. Fit OLS. Residual Moran's I not significant?  → keep OLS.
2. Otherwise look at LM-lag and LM-error:
     only LM-lag significant    → spatial lag  (SLM)
     only LM-error significant  → spatial error (SEM)
     both significant           → step 3
3. Compare the ROBUST statistics:
     robust LM-lag sig, robust LM-error not   → SLM
     robust LM-error sig, robust LM-lag not   → SEM
     both robust significant → estimate both; consider a combined
                               (SARAR) or Durbin model, and report

Two cautions. First, every one of these statistics is computed under a fixed weight matrix W; change the neighbour definition and the verdict can move, so re-run the rule over alternative W. Second — the step people skip — a spatial model is not a substitute for a correct mean specification. If the “spatial dependence” is really an omitted variable that happens to cluster in space, the honest fix is the missing variable, not a spatial parameter that launders the misspecification. Use the decision rule to choose how to model dependence, only after you’re satisfied the model itself is right.