Spatial Regression (Spatial Lag and Spatial Error Models)
Also known as: spatial econometrics, spatial lag model, spatial error model, SAR / SEM, Uzamsal Regresyon (Spatial Lag / Spatial Error)
Spatial regression is a family of regression models that build geographic neighbourhood relationships directly into the model, introduced by Luc Anselin in his 1988 treatment of spatial econometrics. It splits into a spatial lag model, where spatial dependence sits in the dependent variable, and a spatial error model, where the dependence sits in the error term.
Key highlights
- Explicitly models geographic neighbourhood dependence that ordinary regression assumes away.
- Offers a clear modelling choice: spatial lag for spillovers in the outcome versus spatial error for spatially correlated disturbances.
- Grounded in the established spatial econometrics framework of Anselin and of LeSage & Pace.
Intuition
This section is available to Pro members. Upgrade to Pro
How it works
This section is available to Pro members. Upgrade to Pro
When to use it
Use spatial regression when observations carry geographic coordinates and Moran's I shows significant spatial autocorrelation that ordinary regression would ignore. A spatial weight matrix W must be defined first, the residuals tested for spatial dependence with Moran's I, and a reasonable cross-sectional sample (at least about 50 units) is recommended. Choose the lag model when the outcome itself spills across neighbours and the error model when unobserved spatial factors contaminate the residuals.
Strengths & limitations
- Explicitly models geographic neighbourhood dependence that ordinary regression assumes away.
- Offers a clear modelling choice: spatial lag for spillovers in the outcome versus spatial error for spatially correlated disturbances.
- Grounded in the established spatial econometrics framework of Anselin and of LeSage & Pace.
- Requires a spatial weight matrix W defined in advance, and results can be sensitive to how neighbours are specified.
- Observations must have geographic coordinates and the framework here targets cross-sectional data.
- Needs a reasonable sample (at least about 50 units) and the maximum-likelihood estimation is more demanding than ordinary least squares.
Common pitfalls
This section is available to Pro members. Upgrade to Pro
Applications
This section is available to Pro members. Upgrade to Pro
Frequently asked
What is the difference between the spatial lag and spatial error models?
The spatial lag model places spatial dependence in the dependent variable — an observation's outcome depends on its neighbours' outcomes through the term ρWy. The spatial error model places the dependence in the disturbances, so spatially correlated unobserved factors leak through the errors via λWu. Lag captures genuine spillovers in the outcome; error captures spatial structure in what the model leaves out.
What is the spatial weight matrix W?
W encodes the neighbourhood structure: which observations are considered neighbours of which, and how strongly. It must be defined before estimation, typically from geographic coordinates (for example contiguity or distance), and the model's results depend on this choice.
Why test with Moran's I first?
Moran's I checks whether there is significant spatial autocorrelation in the data or residuals. If there is none, ordinary regression is adequate; if there is, ignoring it makes ordinary regression misleading and a spatial model is warranted.
How large a sample do I need?
A reasonable cross-sectional sample of at least about 50 georeferenced units is recommended so the maximum-likelihood spatial parameters can be estimated reliably.
Sources
- 1.Anselin, L. (1988). Spatial Econometrics: Methods and Models. Kluwer Academic Publishers.
- 2.LeSage, J. & Pace, R. K. (2009). Introduction to Spatial Econometrics. Chapman & Hall/CRC.
You have read it. What now?
Cite this page
ScholarGate. (2026, June 1). Spatial Regression. ScholarGate. https://scholargate.app/econometrics/spatial-regression