Spatial Lag Model (SAR / Spatial Autoregressive)

The lag model puts spatial dependence in the dependent variable (ρWy), representing a substantive spillover where neighbours' outcomes affect one another. The spatial error model instead puts the dependence in the error term (λWu), representing unobserved common factors that correlate neighbours' errors. An LM-Lag versus LM-Error test helps decide which specification fits.

Because the spatially lagged outcome Wy is correlated with the error term, the lagged term is endogenous and OLS becomes inconsistent. The model is estimated by maximum likelihood, or alternatively by instrumental variables or GMM.

W is a matrix that defines the neighbourhood structure — which units are treated as neighbours and how strongly. Every result depends on it, so it must be specified deliberately (for example by contiguity or distance) and is assumed to be correctly defined and exogenous.

Because of the feedback through ρWy, a change in a predictor affects a unit's own outcome (direct), the outcomes of its neighbours (indirect), and the sum of both (total). The raw β coefficients do not reflect these spillovers, so LeSage and Pace recommend interpreting the impact decomposition instead.