Spatial SAC Model
Spatial Autoregressive Combined (SAC) Model · Also known as: SARAR Model, Spatial Autoregressive Model with Autoregressive Disturbances, Cliff-Ord Combined Model, Uzamsal Otoregresif Birleşik Model
The Spatial Autoregressive Combined (SAC) model, also known as the SARAR model, simultaneously accounts for spatial dependence in both the dependent variable and the error term. Formalized by LeSage and Pace (2009), the SAC model combines the spatial lag model and the spatial error model into a single framework, estimating two distinct spatial autoregressive parameters — one capturing substantive spatial interaction among outcomes and another capturing residual spatial correlation among disturbances.
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When to use it
Use the SAC model when both substantive spatial interaction among outcomes and spatially structured residuals are theoretically plausible or detected by specification tests such as the Robust LM tests. It is appropriate for cross-sectional areal data where a meaningful spatial weights matrix can be constructed. Key assumptions include correct specification of W, homoskedastic errors conditional on spatial structure, and stationarity of the spatial processes. The SAC model is more demanding in data and computation than its nested alternatives; prefer the spatial lag or spatial error model when only one form of dependence is indicated by theory or diagnostics.
Strengths & limitations
- Simultaneously controls for two distinct sources of spatial dependence, reducing omitted-variable bias relative to simpler spatial models
- Nests both the spatial lag and spatial error models as special cases, enabling formal likelihood ratio tests of model restrictions
- Produces consistent and asymptotically efficient parameter estimates under correct specification via maximum likelihood
- Flexible enough to capture complex real-world spatial data-generating processes in economics, epidemiology, and regional science
- Requires specification of a spatial weights matrix W, and results can be sensitive to that choice
- Identification and precision of rho and lambda can be low when the two spatial processes are not well distinguished by the data
- Computationally more intensive than the spatial lag or error model alone due to two log-determinant evaluations
- Interpretation of direct, indirect, and total effects is more complex than in the spatial lag model alone
Frequently asked
How does the SAC model differ from the Spatial Durbin Model?
The Spatial Durbin Model (SDM) includes spatially lagged explanatory variables (WX) in addition to the spatial lag of y, whereas the SAC model combines a spatial lag of y with a spatial autoregressive error process. The SDM addresses spillovers in covariates; the SAC model addresses simultaneous spillovers in outcomes and residuals. Both are more general than the pure spatial lag or error model but through different generalizations.
Can rho and lambda both be statistically significant?
Yes. When the true data-generating process involves both substantive spatial interaction and spatially correlated disturbances, both parameters can be significantly different from zero. However, in finite samples with moderate spatial autocorrelation, the two parameters may be difficult to distinguish precisely, and it is advisable to rely on strong theoretical motivation and specification tests before interpreting both as meaningful.
What software can estimate the SAC model?
The SAC model is available in R through the spatialreg package (sacsarlm function) and the spdep package. Python users can use PySAL's spreg module (GM_Combo for GMM estimation or spreg.ML_Lag variants). Stata supports it via the spregress command with the disterror option. Matlab implementations are provided in LeSage and Pace's own spatial econometrics toolbox.
Sources
- LeSage, J., & Pace, R. K. (2009). Introduction to Spatial Econometrics. CRC Press. ISBN: 978-1-4200-6424-7
How to cite this page
ScholarGate. (2026, June 2). Spatial Autoregressive Combined (SAC) Model. ScholarGate. https://scholargate.app/en/spatial-analysis/spatial-sac-model
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Spatial Durbin ModelSpatial analysis↔ compare
- Spatial Error ModelSpatial analysis↔ compare
- Spatial Lag ModelSpatial analysis↔ compare