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Spatial Regression

Definition

Spatial regression models quantify relationships between variables while accounting for spatial dependence and heterogeneity. Common forms are spatial lag models (SLM), where neighboring outcomes influence each other; spatial error models (SEM), where autocorrelated errors capture omitted spatial effects; and geographically weighted regression (GWR), which allows coefficients to vary across space. Model choice depends on theory, diagnostics (Moran’s I of residuals), and the nature of the spatial process.

Application

Urban economics links prices to amenities, transport, and zoning; public health relates disease rates to environment and access; ecology ties species richness to climate and habitat. Practitioners use spatial regression to avoid biased estimates and to reveal place-specific sensitivities that inform tailored policies.

FAQ

How do you decide between SLM and SEM?

If theory suggests true interaction among units (spillovers), SLM fits; if missing variables create spatially structured errors, SEM is appropriate. Hausman-type tests and Lagrange multipliers guide the choice.

What pitfalls accompany GWR?

Overfitting and multiple testing risks; results can be unstable where data are sparse. Use cross-validation to choose bandwidths, map diagnostics, and compare with hierarchical models as a check.

How should the spatial weights matrix be constructed?

Reflect true interaction pathways (contiguity, k-nearest neighbors, distance decay). Sensitivity to weights should be reported; arbitrary choices can change conclusions.

How do you interpret coefficients when variables are collinear and spatially autocorrelated?

Assess variance inflation, consider dimension reduction, and prefer models grounded in theory; report uncertainty honestly and avoid causal claims beyond the design.