Spatial Lag Model (SAR / Spatial Autoregressive)
Also known as: SAR model, spatial autoregressive model, spatial lag, Uzamsal Gecikme Modeli (SAR / Spatial Lag)
The Spatial Lag Model is an autoregressive regression that assumes spatial dependence in the dependent variable itself: the outcome values of neighbouring units enter the model as an explanatory term (ρWy). It was formalised in Anselin's Spatial Econometrics (1988) and developed further by LeSage and Pace (2009), and it decomposes spillover effects into direct, indirect, and total impacts.
Key highlights
- Captures genuine spatial spillovers by modelling dependence directly in the dependent variable (ρWy).
- Decomposes effects into direct, indirect, and total impacts, separating a unit's own response from feedback through its neighbours.
- Maximum-likelihood (and IV/GMM) estimation gives consistent coefficients where OLS would be biased.
Intuition
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How it works
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When to use it
Use the spatial lag model with continuous, cross-sectional data that carry geographic coordinates, when there is reason to believe the outcome in one location is shaped by the outcomes of its neighbours (a substantive spillover). A reasonable sample is needed (at least about 50 units). It is appropriate when the spatial weights matrix W is correctly specified and an LM-Lag test is significant and favoured over the LM-Error test, indicating dependence in the dependent variable rather than in the errors.
Strengths & limitations
- Captures genuine spatial spillovers by modelling dependence directly in the dependent variable (ρWy).
- Decomposes effects into direct, indirect, and total impacts, separating a unit's own response from feedback through its neighbours.
- Maximum-likelihood (and IV/GMM) estimation gives consistent coefficients where OLS would be biased.
- Results hinge on a correctly specified spatial weights matrix W; a poorly chosen W misstates who the neighbours are.
- The spatially lagged outcome is endogenous, so OLS is inconsistent and ML or IV/GMM is required.
- Requires geographic coordinates and a reasonable sample size (at least about 50 units), and cannot be applied without spatial information.
Common pitfalls
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Applications
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Frequently asked
How is the spatial lag model different from the spatial error model?
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.
Why can't I just use OLS?
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.
What is the spatial weights matrix W?
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.
Why report direct, indirect, and total impacts?
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.
Sources
- 1.Anselin, L. (1988). Spatial Econometrics: Methods and Models. Kluwer Academic.
- 2.LeSage, J. & Pace, R. K. (2009). Introduction to Spatial Econometrics. CRC Press.
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Cite this page
ScholarGate. (2026, June 1). Spatial Lag Model. ScholarGate. https://scholargate.app/spatial-analysis/spatial-lag-model