{ "id": "epskjxhs9e", "slug": "spatial-autoregressive-model", "title": "Spatial autoregressive model", "updated": "2026-09-30", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference" }, { "id": "physical.math.statistics.stats_statistical_inference.regression-analysis", "label": "Regression analysis", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference.regression-analysis" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.regression-analysis", "label": "Non-geographic · 1946 to 2000: Regression analysis", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.regression-analysis", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.regression-analysis", "label": "Regression analysis", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.regression-analysis" } ] } ], "excerpt": "The spatial autoregressive (SAR) model is a regression model where each location's outcome depends partly on neighboring outcomes through a spatial weights matrix, used for property values.", "snippet": "The spatial autoregressive (SAR) model is a regression model where each location's outcome depends partly on neighboring outcomes through a spatial weights matrix, used for property values.", "node": "physical.math.statistics.stats_statistical_inference.regression-analysis", "markdown": "# Spatial autoregressive model\n\nA spatial autoregressive (SAR) model is a regression model in which the dependent variable at each location depends partly on the values of that variable at neighboring locations, entered through a spatial weights matrix and a scalar autoregressive parameter. It is widely used in spatial statistics and spatial econometrics for outcomes that exhibit spatial dependence, such as property values.[1](http://econweb.umd.edu/~kelejian/research/P071897.PDF)\n\nThe basic specification writes the outcome vector as \\( y = \\rho \\cdot W \\cdot y + X \\cdot \\beta + \\varepsilon \\), where \\( W \\) is a pre-specified \\( n \\times n \\) spatial weights matrix, \\( \\rho \\) measures the strength of spatial interaction, \\( X \\) collects exogenous covariates, and \\( \\varepsilon \\) is a disturbance vector.[1](http://econweb.umd.edu/~kelejian/research/P071897.PDF) In its simplest form the model considers spatial spillovers in the dependent variable only, expressing each unit's outcome as a weighted average of other units' outcomes plus a disturbance.[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC2888178/)\n\n| Key fact | Detail |\n|---|---|\n| Core equation | \\( y = \\rho \\cdot W \\cdot y + X \\cdot \\beta + \\varepsilon \\), with \\( |\\rho| < 1 \\)[1](http://econweb.umd.edu/~kelejian/research/P071897.PDF) |\n| Weights matrix | \\( W \\) is \\( n \\times n \\) with zero diagonal, usually row-standardized so rows sum to 1[3](https://www.statsref.com/HTML/sar_models.html) |\n| Autoregressive parameter | \\( \\rho \\) quantifies the strength of spatial association, usually between −1 and 1[4](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0327316) |\n| Why OLS fails | \\( W \\cdot y \\) is correlated with \\( \\varepsilon \\), so OLS is inconsistent[1](http://econweb.umd.edu/~kelejian/research/P071897.PDF) |\n| Standard estimators | Maximum likelihood and generalized spatial two-stage least squares (GS2SLS)[5](https://econ.umd.edu/sites/www.econ.umd.edu/files/pubs/SJ_SPREG%282013%29.pdf) |\n| Spatial multiplier | A uniform change in \\( x \\) has total effect \\( \\beta / (1 - \\rho) \\)[6](https://pysal.org/spreg/notebooks/13_ML_estimation_spatial_lag.html) |\n| Computational cost | Determinants and inverses grow in cost proportional to \\( n^3 \\); sparse-matrix methods reduce this sharply[7](https://onlinelibrary.wiley.com/doi/10.1111/j.1538-4632.1997.tb00959.x) |\n\n## How it works\n\nThe spatial lag term \\( W \\cdot y \\) is a weighted average of the outcomes at neighboring units. Row-standardizing \\( W \\) so that each row sums to 1 makes this term literally a weighted average of neighboring values.[3](https://www.statsref.com/HTML/sar_models.html) \\( W \\) can be built from an adjacency matrix, a distance matrix, or a standardized matrix following specific rules[4](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0327316); unlike the conditional autoregressive (CAR) model, \\( W \\) in the SAR model need not be symmetric.[3](https://www.statsref.com/HTML/sar_models.html)\n\nSolving the model gives the reduced form \\( y = (I - \\rho \\cdot W)^{-1} X \\cdot \\beta + (I - \\rho \\cdot W)^{-1} \\varepsilon \\).[5](https://econ.umd.edu/sites/www.econ.umd.edu/files/pubs/SJ_SPREG%282013%29.pdf) Writing \\( A = I - \\rho \\cdot W \\), the equilibrium solution \\( y = A^{-1}X\\beta + A^{-1}\\varepsilon \\) implies \\( \\mathrm{Var}(y) = \\sigma^2 A^{-1} \\cdot A^{-1\\prime} \\), so each observation is correlated with every other observation.[8](https://www.sciencedirect.com/science/article/abs/pii/S0304407606000509) • [24](https://pysal.org/spreg/notebooks/7_spatial_models.html) This is the mechanism behind global spillovers: a shock to one unit propagates through the network of neighbors. If the change in an explanatory variable is uniform across observations, the spatial multiplier is \\( 1/(1 - \\rho) \\), so the total effect of a change in \\( x_k \\) is \\( \\beta_k / (1 - \\rho) \\).[6](https://pysal.org/spreg/notebooks/13_ML_estimation_spatial_lag.html)\n\n## How it is done\n\nA practitioner first builds \\( W \\) from an adjacency matrix, a distance matrix, or a standardized matrix following specific rules, and typically row-standardizes it so that its rows sum to 1.[4](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0327316) • [3](https://www.statsref.com/HTML/sar_models.html) Model choice should match the substantive conceptualization: a SAR lag fits outcomes like neighboring prices affecting a price, spatially lagged explanatory variables (SLX) fit exposure such as surrounding property condition, and a spatial error model (SEM) fits unmeasured influences like neighborhood reputation.[9](https://gistbok-ltb.ucgis.org/current/print/concept/AM-03-032)\n\nSpecification is tested before or alongside estimation. Anselin's Lagrange Multiplier diagnostics formally test spatial residual autocorrelation in the presence of spatially lagged dependent variables and of heteroskedasticity.[10](https://doi.org/10.1111/j.1538-4632.1988.tb00159.x)\n\nBecause \\( \\rho \\neq 0 \\) makes regression coefficients misleading, reported effects should be direct and indirect impacts computed from \\( (I - \\rho \\cdot W)^{-1} \\): LeSage and Pace's average direct impact (ADI), average indirect impact (AII), and average total impact (ATI), where direct effects include feedback so they differ from \\( \\beta \\).[6](https://pysal.org/spreg/notebooks/13_ML_estimation_spatial_lag.html) Where the spatial coefficient is non-zero, these global spillover impacts should be reported rather than the regression coefficients.[11](https://r-spatial.org/book/17-Econometrics.html)\n\n## Origin\n\nThe Gaussian spatial autoregressive formulation requires \\( (I - B)^{-1} \\) to exist for the covariance to be defined.[12](https://repository.library.noaa.gov/view/noaa/54947/noaa_54947_DS1.pdf) Kelejian and Prucha describe the widely referenced version as a variant of Whittle's model[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC2888178/); Two one-parameter autoregressive forms were introduced, one for the dependent variable \\( (W \\cdot y) \\) and one for the error term \\( (W \\cdot e) \\).[13](https://brsa.org.br/wp-content/uploads/wpcf7-submissions/32125/SPATIAL-ECONOMETRIC-MODEL-SPECIFICATION-SEARCH.pdf) Maximum likelihood computation for these single-spatial-parameter models uses iterative methods.[11](https://r-spatial.org/book/17-Econometrics.html)\n\n[Luc Anselin](https://www.edgepedia.org/luc-anselin)'s 1988 book *Spatial Econometrics: Methods and Models*, published in the Studies in operational regional science series, consolidated the field with a typology of spatial autoregressive models and the spatial lag/error distinction[14](https://doi.org/10.1007/978-94-015-7799-1), and his 1988 *Geographical Analysis* paper derived the LM test diagnostics for spatial dependence and heterogeneity.[10](https://doi.org/10.1111/j.1538-4632.1988.tb00159.x) On the estimation side, Harry H. Kelejian and Ingmar R. Prucha proposed the generalized spatial two-stage least squares procedure for the model with autoregressive disturbances in 1998 in *The Journal of Real Estate Finance and Economics*[15](https://doi.org/10.1023/a:1007707430416), and R. Kelley Pace and Ronald Barry published quick sparse-matrix computation of spatial autoregressive estimators in 1997 in *Geographical Analysis*.[7](https://onlinelibrary.wiley.com/doi/10.1111/j.1538-4632.1997.tb00959.x)\n\n## Variants\n\nThe Kelejian–Prucha or SAC model keeps both a spatial lag and a spatially autoregressive disturbance; the spatial Durbin model (SDM) adds spatial lags of \\( X \\) as well as of \\( y \\).[9](https://gistbok-ltb.ucgis.org/current/print/concept/AM-03-032) The general nesting spatial (GNS) specification resembles the Manski neighborhood effects model but is rarely used in empirical work.[9](https://gistbok-ltb.ucgis.org/current/print/concept/AM-03-032) LeSage (2014) argues the SDM and SDEM best lend themselves to meaningful interpretation, recommending SDEM as a starting point for local spillovers and SDM for global spillovers.[9](https://gistbok-ltb.ucgis.org/current/print/concept/AM-03-032)\n\n## Applications\n\nApplied uses center on cross-sectional outcomes with plausible neighbor interactions; a canonical example is a spatial model explaining property values, where the value at each location relates to the property values of neighboring locations.[1](http://econweb.umd.edu/~kelejian/research/P071897.PDF) Cross-unit interaction of this kind is of interest in social science, biostatistical, and geographic science models.[16](https://econweb.umd.edu/~prucha/Papers/SJ_SPIVREG%282013%29.pdf) Software implementations include the spreg Python package (PySAL), which offers ML estimation of the spatial lag model and higher-order GMM estimators[6](https://pysal.org/spreg/notebooks/13_ML_estimation_spatial_lag.html) • [17](https://pysal.org/spreg/notebooks/17_GMM_higher_order.html), R's spatialreg, [GeoDa](https://www.edgepedia.org/geoda), and Stata's spregress, spivregress, and spxtregress.[18](https://www.stata.com/manuals/sp.pdf)\n\n## Limitations and alternatives\n\n**Estimation.** OLS is inconsistent because \\( W \\cdot y \\) is correlated with the disturbance.[1](http://econweb.umd.edu/~kelejian/research/P071897.PDF) • [8](https://www.sciencedirect.com/science/article/abs/pii/S0304407606000509) Formal asymptotics for the ML estimator include \\( n \\)-consistency, normality, and asymptotic efficiency.[8](https://www.sciencedirect.com/science/article/abs/pii/S0304407606000509) ML requires the log-determinant Jacobian \\( \\ln|I - \\rho \\cdot W| \\), which runs into numerical difficulties for large data sets; spreg implements three computation methods (full, ord, LU) that differ only in how this term is computed.[6](https://pysal.org/spreg/notebooks/13_ML_estimation_spatial_lag.html) Naive spatial estimators manipulate \\( n^2 \\) relations with determinants, eigenvalues, and inverses whose cost grows proportional to \\( n^3 \\), but exploiting the sparsity of \\( W \\) cuts this dramatically.[7](https://onlinelibrary.wiley.com/doi/10.1111/j.1538-4632.1997.tb00959.x) GeoDa implements an optimal ML approach based on an approximation, preferred for larger data sets, though it does not support the spatial Durbin specification.[6](https://pysal.org/spreg/notebooks/13_ML_estimation_spatial_lag.html)\n\nThe GS2SLS estimator remains consistent under disturbances with unknown-form heteroskedasticity, whereas ML is consistent in the IID case but generally not under heteroskedasticity.[5](https://econ.umd.edu/sites/www.econ.umd.edu/files/pubs/SJ_SPREG%282013%29.pdf) The best GMM estimator based on linear and quadratic moments is computationally simple, asymptotically as efficient as ML under normality, and asymptotically more efficient than Gaussian QML otherwise.[19](https://ideas.repec.org/a/eee/econom/v159y2010i2p303-319.html)\n\n**Failure modes.** Estimates are conditional on \\( W \\), and misspecifying \\( W \\) even by a constant causes inconsistent estimates; a regression-based K test can detect when \\( W \\) is inflated by a constant in expectation.[20](https://www.cambridge.org/core/journals/political-analysis/article/measurement-error-and-the-specification-of-the-weights-matrix-in-spatial-regression-models/1052D59CCDB8FA7BAA051B2739C966E4) Endogenous weights can bias conventional estimates and invalidate inference; an adjusted LM test of weight exogeneity has a central chi-square null distribution regardless of whether the true model contains a spatial lag.[21](https://www.sciencedirect.com/science/article/abs/pii/S0166046217301801) The GNS model's full parameter set is not identified, so ML cannot be applied, and ML estimation of SAR-SAR models suffers a \"banana\" convergence problem, switching between estimates of the spatial parameters; IV/GMM estimation is the practical alternative.[17](https://pysal.org/spreg/notebooks/17_GMM_higher_order.html)\n\n**Recent developments.** Huber IV and Huber GMM estimators extend robustness to heavy-tailed disturbances, requiring only first-order moment existence where traditional ML and GMM require moments above order 4.[22](https://www.cambridge.org/core/journals/econometric-theory/article/robust-estimation-for-the-spatial-autoregressive-model/6B6AA2026F503D8A39D8EB621768197D) A 2024 semiparametric Bayesian method for heterogeneous SAR models uses B-spline approximations of a nonparametric function with Gibbs and Metropolis–Hastings sampling.[23](https://www.mdpi.com/1099-4300/26/6/498) Published comparisons do not survey typical empirical values of \\( \\rho \\), nor do they compare SAR with [geographically weighted regression](https://www.edgepedia.org/geographically-weighted-regression) or machine-learning hybrids.\n\n## References\n\n1. [A Generalized Spatial Two Stage Least Squares Procedure for Estimating a Spatial Autoregressive Model with Autoregressive Disturbances (Kelejian & Prucha)](http://econweb.umd.edu/~kelejian/research/P071897.PDF)\n2. [Specification and Estimation of Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances (Kelejian & Prucha)](https://pmc.ncbi.nlm.nih.gov/articles/PMC2888178/)\n3. [StatsRef: SAR models](https://www.statsref.com/HTML/sar_models.html)\n4. [Characterization and estimation of heterogeneous spatial autocorrelation in spatial autoregressive models (PLOS One)](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0327316)\n5. [Maximum likelihood and generalized spatial two-stage least-squares estimators for a spatial-autoregressive model with spatial-autoregressive disturbances (Drukker, Egger, Prucha; Stata Journal 2013)](https://econ.umd.edu/sites/www.econ.umd.edu/files/pubs/SJ_SPREG%282013%29.pdf)\n6. [Maximum Likelihood Estimation - Spatial Lag Model, spreg Manual](https://pysal.org/spreg/notebooks/13_ML_estimation_spatial_lag.html)\n7. [Quick Computation of Spatial Autoregressive Estimators (Barry & Pace, Geographical Analysis, 1997)](https://onlinelibrary.wiley.com/doi/10.1111/j.1538-4632.1997.tb00959.x)\n8. [Finite sample properties of maximum likelihood estimator in spatial models (Journal of Econometrics)](https://www.sciencedirect.com/science/article/abs/pii/S0304407606000509)\n9. [UCGIS GIS&T BoK [AM-03-032] Spatial Autoregressive Models](https://gistbok-ltb.ucgis.org/current/print/concept/AM-03-032)\n10. [Luc Anselin (1988). Lagrange Multiplier Test Diagnostics for Spatial Dependence and Spatial Heterogeneity. Geographical Analysis.](https://doi.org/10.1111/j.1538-4632.1988.tb00159.x)\n11. [Spatial Econometrics Models – Spatial Data Science (Bivand et al.)](https://r-spatial.org/book/17-Econometrics.html)\n12. [Spatial autoregressive models for statistical inference from ecological data](https://repository.library.noaa.gov/view/noaa/54947/noaa_54947_DS1.pdf)\n13. [Spatial Econometric Model Specification Search](https://brsa.org.br/wp-content/uploads/wpcf7-submissions/32125/SPATIAL-ECONOMETRIC-MODEL-SPECIFICATION-SEARCH.pdf)\n14. [Luc Anselin (1988). Spatial Econometrics: Methods and Models. Studies in operational regional science.](https://doi.org/10.1007/978-94-015-7799-1)\n15. [Harry H. Kelejian, Ingmar R. Prucha (1998). A Generalized Spatial Two-Stage Least Squares Procedure for Estimating a Spatial Autoregressive Model with Autoregressive Disturbances. The Journal of Real Estate Finance and Economics.](https://doi.org/10.1023/a:1007707430416)\n16. [SJ SPIVREG(2013) (econweb.umd.edu)](https://econweb.umd.edu/~prucha/Papers/SJ_SPIVREG%282013%29.pdf)\n17. [GMM Estimation - Higher Order Models, spreg Manual](https://pysal.org/spreg/notebooks/17_GMM_higher_order.html)\n18. [Stata Spatial Autoregressive Models Reference Manual](https://www.stata.com/manuals/sp.pdf)\n19. [An efficient GMM estimator of spatial autoregressive models (Journal of Econometrics, 2010)](https://ideas.repec.org/a/eee/econom/v159y2010i2p303-319.html)\n20. [Measurement Error and the Specification of the Weights Matrix in Spatial Regression Models (Political Analysis)](https://www.cambridge.org/core/journals/political-analysis/article/measurement-error-and-the-specification-of-the-weights-matrix-in-spatial-regression-models/1052D59CCDB8FA7BAA051B2739C966E4)\n21. [Simple tests for endogeneity of spatial weights matrices](https://www.sciencedirect.com/science/article/abs/pii/S0166046217301801)\n22. [Robust Estimation for the Spatial Autoregressive Model (Econometric Theory)](https://www.cambridge.org/core/journals/econometric-theory/article/robust-estimation-for-the-spatial-autoregressive-model/6B6AA2026F503D8A39D8EB621768197D)\n23. [A Semiparametric Bayesian Approach to Heterogeneous Spatial Autoregressive Models (Entropy, 2024)](https://www.mdpi.com/1099-4300/26/6/498)\n24. [Spatial Model Specifications, spreg v1.9.2.dev3+g4dbc80924 Manual](https://pysal.org/spreg/notebooks/7_spatial_models.html)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Statistical inference, estimation, sampling, and testing › Regression analysis*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "http://econweb.umd.edu/~kelejian/research/P071897.PDF", "https://econweb.umd.edu/~prucha/Papers/SJ_SPIVREG%282013%29.pdf" ], "url": "https://www.edgepedia.org/spatial-autoregressive-model", "markdown_url": "https://www.edgepedia.org/spatial-autoregressive-model.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Spatial autoregressive model\", Edgepedia (EdgeChat), https://www.edgepedia.org/spatial-autoregressive-model. Edgepedia Community License 1.0.", "credit_md": "\"[Spatial autoregressive model](https://www.edgepedia.org/spatial-autoregressive-model)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/spatial-autoregressive-model](https://www.edgepedia.org/spatial-autoregressive-model). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Spatial autoregressive model\", Edgepedia (EdgeChat), https://www.edgepedia.org/spatial-autoregressive-model. Edgepedia Community License 1.0.", "speakable": "The spatial autoregressive model is a regression model where each location's outcome depends partly on neighboring outcomes through a spatial weights matrix, used for property values." }