Spatial Error Model (SEM)
Also known as: SEM, spatial error regression, spatial autoregressive error model, Uzamsal Hata Modeli (SEM / Spatial Error)
The Spatial Error Model, developed within Anselin's spatial econometrics framework (1988), is a regression model that assumes spatial dependence enters through the error term: the disturbances of neighbouring units are correlated. It is used when unobserved shared factors make the errors of nearby observations move together, and it is estimated by maximum likelihood or GMM rather than ordinary least squares.
Key highlights
- Corrects the inefficiency and invalid standard errors that ordinary least squares suffers when errors are spatially correlated.
- Captures spillovers from unobserved shared factors (climate, policy, culture) without needing to measure them directly.
- Coefficients on the predictors keep their usual interpretation, since the spatial dependence is confined to the error term rather than the outcome.
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 SEM with continuous outcomes on cross-sectional data where each unit has geographic coordinates and a sample of at least about 50 observations. It is the right choice when there is no direct spatial spillover in the dependent variable itself but spatial dependence works through unobserved common factors in the errors — confirmed when a Lagrange Multiplier error test (LM-Error) is significant and favoured over the LM-Lag test. A correctly specified spatial weights matrix W is essential; ML estimation also assumes normal errors, while GMM relaxes that.
Strengths & limitations
- Corrects the inefficiency and invalid standard errors that ordinary least squares suffers when errors are spatially correlated.
- Captures spillovers from unobserved shared factors (climate, policy, culture) without needing to measure them directly.
- Coefficients on the predictors keep their usual interpretation, since the spatial dependence is confined to the error term rather than the outcome.
- Results hinge entirely on a correctly specified spatial weights matrix W; a poorly defined neighbourhood structure biases the estimates.
- Cannot be applied without geographic coordinates — with no spatial information, plain OLS regression is the appropriate fallback.
- Maximum likelihood estimation assumes normally distributed errors; when that fails, GMM is needed instead.
- Designed for cross-sectional data, requiring at least roughly 50 observations for reliable estimation.
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
How is SEM different from the spatial lag model?
In a spatial lag model the dependent variable of a unit depends directly on its neighbours' outcomes (substantive spillover). In SEM there is no direct spillover in the outcome; instead the unobserved errors are spatially correlated. The LM-Error and LM-Lag diagnostic tests help decide which specification fits the data.
Why not just use OLS?
When errors are spatially correlated, OLS remains consistent but is no longer efficient and its standard errors are invalid, so the t and F tests mislead. ML or GMM estimation of SEM restores efficiency and gives valid inference.
What is the spatial weights matrix W?
W encodes the neighbourhood structure — which units are considered neighbours and how strongly — for example by contiguity or distance. SEM results depend critically on W being correctly specified and exogenous.
When should I prefer GMM over maximum likelihood?
Maximum likelihood estimation of SEM assumes normally distributed errors. If that assumption is doubtful, GMM provides efficient estimation without requiring normality.
Sources
- 1.Anselin, L. (1988). Spatial Econometrics: Methods and Models. Kluwer Academic.
You have read it. What now?
Cite this page
ScholarGate. (2026, June 1). Spatial Error Model. ScholarGate. https://scholargate.app/spatial-analysis/spatial-error-model