Ordinary regression can explain a great deal—until the residuals draw a map.
Suppose a model predicts housing prices from floor area, age and transport access. If the remaining errors cluster geographically, something spatial is still unresolved. Nearby properties may share schools, neighbourhood reputation, planning rules or market expectations. The observations are not behaving like independent dots in a spreadsheet.
Spatial regression is a family of models designed for situations where geographic dependence, neighbourhood effects or spatially structured errors matter.
Spatial regression begins when “where” remains in the model after the obvious variables have already spoken.
Quick Read: The Mechanism
OUTCOME + EXPLANATORY VARIABLES + LOCATION → ORDINARY MODEL → SPATIAL DIAGNOSTICS → DEPENDENCE / ERROR / LOCAL VARIATION → SPATIAL MODEL → RECHECK
1. Start With the Ordinary Model
Spatial regression is not a licence to add geography automatically. Begin with a defensible substantive model, inspect residuals and ask whether spatial structure remains.
2. Spatial Dependence Violates Simple Independence
Nearby observations can share processes. If residuals are spatially autocorrelated, standard errors and inference from an ordinary model may be unreliable.
3. Spatial Lag Models
A spatial lag model can include neighbouring outcomes through the spatial lag. This is useful when interaction, diffusion or equilibrium dependence is theoretically plausible.
4. Spatial Error Models
Sometimes geography sits mainly in omitted or unobserved influences. Spatial error models represent correlation among model errors rather than treating neighbouring outcomes themselves as the central mechanism.
5. Spatial Durbin Models
Some models include both local explanatory variables and spatially lagged explanatory variables, allowing surrounding conditions to matter. Interpretation then includes direct and indirect effects.
6. Spatial Weights Are Part of the Model
Spatial Weights define the neighbourhood structure. A model using shared borders asks a different geographic question from one using travel time or distance decay.
7. Spatial Regression Is Not GWR
Spatial regression is a broad family concerned with dependence and spatial structure. Geographically weighted regression allows coefficients to vary locally and belongs more directly under Spatial Nonstationarity, which already owns that reader job.
8. Spatial Dependence Is Not Causation
A good-fitting spatial model does not prove neighbours cause one another. Shared omitted variables, sorting and common exposure can generate spatial dependence too.
9. Residual Maps Matter
After fitting the model, map the residuals again. If strong spatial clustering remains, the model may still be missing geography.
10. Primary Geography: The Leftovers Have Places
Children can predict playground temperature from shade, then map where predictions were too high or low. If the mistakes cluster, there may be another local condition they missed.
11. Secondary Geography: Model, Map, Diagnose
Students can learn the discipline even without advanced mathematics: explain a pattern, inspect what remains, ask whether the remaining errors cluster, then revise the explanation.
12. Advanced Geography: Direct and Indirect Effects
In spatial lag systems, changing one location can propagate through neighbours and feed back through the network. Coefficients therefore cannot always be interpreted like ordinary regression coefficients; impacts may include direct, indirect and total effects.
13. Singapore Example: Housing
Property prices share neighbourhood context, transport, schools and market expectations. Spatial regression can help distinguish measured attributes from remaining geographic dependence, while causal claims still require stronger research design.
14. Singapore Example: Heat
Urban temperature depends on local vegetation, materials and built form, but nearby cells also share atmospheric and morphological context. Residual spatial dependence can reveal where a simple heat model is incomplete.
15. Public Health Example
Area health outcomes can cluster because of shared demographics, environment, access or transmission. Spatial models can account for dependence, but they do not identify the medical mechanism by themselves.
16. Hostile Test: “The Spatial Model Fits Better, So It Is True”
Better fit is evidence of usefulness, not truth. Compare models, inspect residuals, test alternative weight matrices and ask whether the assumed spatial mechanism is substantively defensible.
17. Where Spatial-Regression Reasoning Breaks
- Spatial-by-default: adding spatial terms without evidence of spatial structure.
- W blindness: hiding the neighbourhood definition.
- Fit-equals-cause: treating better fit as causal proof.
- Coefficient literalism: interpreting spatial-lag coefficients like ordinary independent effects.
- Residual neglect: failing to check whether spatial structure remains.
- Model shopping: choosing the specification that produces preferred significance.
18. Ten Questions for Spatial Regression
- Why should geography matter here?
- What does the ordinary model leave unexplained?
- Are residuals spatially autocorrelated?
- How are neighbours defined?
- Is dependence in outcomes, errors or explanatory context?
- Are indirect effects plausible?
- Does the model survive alternative W choices?
- What causal claim, if any, is justified?
- Do residuals improve?
- Can the model predict new places?
19. Where This Fits
How Regression Works owns regression generally. Spatial Autocorrelation owns spatial dependence diagnostics. Spatial Lag owns the weighted-neighbour variable. This article owns what changes when regression must explicitly model geographic dependence.
The Idea to Keep
If the model’s mistakes still know where they are, the model has not finished learning the geography.