The two workhorse spatial models — and the one that nests them

Once a diagnostic says spatial dependence is real, you face a modelling choice: where does the dependence live? Two canonical answers, plus a general model that contains both.

Spatial error model (SEM). Put the dependence in the disturbance:

y = Xβ + u,    u = λWu + ε

The systematic part Xβ is ordinary; the errors are spatially correlated, usually because of omitted, spatially structured factors — an unmeasured neighbourhood quality that bleeds across borders. The parameter λ measures how strongly neighbouring residuals move together. Crucially, the marginal effect of X on y is still just β — there are no spillovers. SEM is a nuisance correction: it repairs OLS’s wrong standard errors without changing the economic story. Do not read λ as “my neighbour’s outcome affects mine.”

Spatial lag / autoregressive model (SLM / SAR). Put the dependence in the outcome:

y = ρWy + Xβ + ε

Now each unit’s y depends on its neighbours’ y — genuine interaction or diffusion (reference pricing, imitation, congestion). Two consequences follow. First, Wy is correlated with ε by construction, so OLS is biased and inconsistent — estimate by maximum likelihood, or by IV/GMM using WX, W²X as instruments. Second, because a shock feeds back through the spatial multiplier (I − ρW)⁻¹, the coefficient β is no longer the marginal effect. You have to decompose impacts into direct (own location, including feedback), indirect (spillovers to others), and total.

The Durbin model (SDM) that nests them. Add the neighbours’ covariates too:

y = ρWy + WXθ + Xβ + ε

SDM lets both neighbours’ outcomes and neighbours’ attributes matter, which makes it robust to an omitted variable that is spatially correlated with an included one. It nests the others: θ = 0 gives SLM; the “common-factor” restriction θ = −ρβ gives SEM; ρ = 0 gives spatially-lagged-X only. Because it contains the workhorses, a common strategy is to start general (SDM) and test down to the simpler forms.

The through-line: name where the dependence sits before you estimate. In the error model the coefficient keeps its plain meaning and there are no spillovers. In the lag (or Durbin) model there are spillovers, Wy is endogenous, and “the coefficient” must be replaced by an impact decomposition. Getting this wrong doesn’t just cost efficiency — it changes what your numbers mean.