The LM decision rule: choosing between spatial models

Video lesson · 01:41 Move from residual diagnosis to model choice

Vera starts with OLS residuals and Moran's I, then reads the classical and robust LM-lag and LM-error tests as a sequence. The revised decision path comes first; the note then restores the constraints imposed by mechanism, weights, and residual diagnostics.

The lesson organises the comparative LM path; tests do not replace theoretical judgment, whose limits remain in the full note.

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.