Spatial Durbin Model (SDM): Insights & Applications

• Spatial Durbin Model (SDM) is a spatial econometric framework that captures both localized outcomes and neighboring covariate spillovers through its autoregressive structure.
• It employs a spatial weights matrix, often based on k-nearest neighbors, to accurately represent interdependencies among regional units.
• Maximum likelihood estimation in SDM allows for clear decomposition into direct, indirect, and total effects, enhancing insights into structural diffusion.

The Spatial Durbin Model (SDM) is a spatial econometric framework developed to analyze interdependent regional systems where both outcomes and explanatory variables may diffuse across space. In the context of "Modeling ICD-10 Morbidity and Multidimensional Poverty as a Spatial Network: Evidence from Thailand," the SDM is employed to elucidate how morbidity and deprivation measures spread through provincial networks, disentangling the localized and neighboring influences on health outcomes. This model’s explicit treatment of both spatial outcome autocorrelation and covariate spillovers enables precise inference on structural diffusion and spatially embedded phenomena.

1. Model Specification

The SDM is mathematically characterized for a cross-sectional spatial system as

$y = \rho W y + X \beta + W X \theta + \varepsilon$

where:

• $y$ is the $n \times 1$ vector of the dependent variable (e.g., province-level morbidity ratios);
• $W$ is the spatial weights matrix ($n \times n$), encoding neighbor relationships;
• $\rho$ is the spatial autoregressive coefficient, quantifying outcome spillover;
• $X$ is the matrix ($n \times p$) of exogenous predictors (e.g., multidimensional poverty indices);
• $\beta$ captures the direct (within-unit) effects;
• $\theta$ estimates spillover (neighbor-lagged) covariate effects;
• $y$0 is the disturbance.

For the ICD-10 chapters in Thailand, $y$1 quantifies the proportion of health visits accounted for by a given morbidity class per province (e.g., digestive diseases). $y$2 comprises seven TPMAP poverty indicators. The model structure simultaneously considers a province’s own deprivation and that of its neighbors, capturing both endogenous (outcome feedback) and exogenous (characteristics) spatial dependencies (Kukieattikool et al., 6 Jan 2026).

2. Construction of the Spatial Weights Matrix

Accurate representation of spatial structure is anchored in $y$3, the spatial weights matrix. In the referenced study, each province is positioned at its centroid, and its seven nearest neighbors in geodesic (Euclidean) distance are selected. The weights are specified as: $y$4 Each row of $y$5 is row-standardized to sum to one—this "KNN-7" (k-nearest neighbors, $y$6) approach ensures balanced neighborhood connectivity and statistical stability, mitigating heterogeneity intrinsic to adjacency-based systems. Every node has a fixed number of ties, facilitating robust inference regarding both local and spatial externalities.

3. Estimation via Maximum Likelihood

SDM parameters are typically estimated by maximizing the joint log-likelihood: $y$7 Estimation in the cited work utilized the lagsarlm(type="mixed") function from R’s spatialreg package, providing numerically stable maximum likelihood inference under the assumptions of:

• Exogenous covariates ($y$8 non-stochastic conditional on the model);
• Homoskedastic, uncorrelated error structure ($y$9);
• Invertibility ($n \times 1$0) for well-defined spatial processes.

These conditions ensure identification and consistent separation of local and network effects, accommodating both direct and indirect transmission mechanisms.

4. Decomposition of Effects

The SDM affords a decomposition of covariate impacts into direct, indirect, and total effects—essential for understanding spatial feedbacks. The reduced-form expectation is: $n \times 1$1 For each covariate $n \times 1$2, the $n \times 1$3 impact matrix is: $n \times 1$4 Definitions:

• Direct effect: average of diagonal elements of $n \times 1$5, measuring full-feedback local effect;
• Indirect effect: average row-sum of off-diagonal elements of $n \times 1$6, quantifying spatial spillover;
• Total effect: the sum of direct and indirect.

This decomposition clarifies the mechanism by which a marginal covariate change in one unit propagates spatially, distinguishing between local feedbacks and broader network influence.

5. Empirical Findings and Statistical Diagnostics

An application to ICD-10 chapter C2 (digestive system diseases) yielded:

• Spatial autoregressive coefficient: $n \times 1$7 ($n \times 1$8), indicating significant spatial clustering;
• Direct effects: only living-conditions deprivation ($n \times 1$9) was significant locally;
• Indirect effects: significant spillovers for neighbor-lagged health-service deprivation ($W$0), physical-accessibility deprivation ($W$1), and count of poor households ($W$2);
• Residual autocorrelation LM test ($W$3) indicated that SDM specification adequately absorbed spatial dependence.

These results exemplify how the SDM discerns the relative contribution of local and network-dependence in health outcomes, a crucial distinction for interpreting structural diffusion and regional risk.

6. Interpretation of Spillover Mechanisms

The estimated positive $W$4 for digestive diseases suggests an endogenous "contagion" process: elevated morbidity in neighboring provinces directly raises local incidence after conditioning on local deprivation. Neighbor-lagged covariates ($W$5) elucidate attribute spillovers—e.g., a higher count of poor households in adjacent provinces increases local morbidity, whereas higher neighbor deprivation in health-service access or physical accessibility correlates with lower local ratios, possibly reflecting cross-boundary healthcare utilization.

This pattern aligns with social network theory regarding contextual and contagion effects, supporting the conclusion that disease risk is embedded in a jointly determined spatial system, as opposed to being strictly a function of isolated provincial characteristics (Kukieattikool et al., 6 Jan 2026).

7. Advantages and Implementation Guidance

Relative to the simpler Spatial Autoregressive (SAR) or Spatial Error Model (SEM) frameworks, the SDM possesses two primary advantages:

• It simultaneously models spatial outcome-lag effects ($W$6) and neighbor-covariate spillovers ($W$7), mitigating omitted-variable bias when exogenous variable diffusion is plausible;
• SDM generalizes both SAR ($W$8) and, under restriction, SEM models, providing a robust, encompassing baseline.

Recommended implementation steps include:

1. Construct and row-standardize $W$9 (via KNN or adjacency);
2. Assemble $n \times n$0 and $n \times n$1;
3. Estimate parameters using appropriate software (e.g., lagsarlm(type="mixed"), or analogs in Stata/PySAL);
4. Conduct residual spatial-dependence tests (LM, Moran’s I);
5. Quantify and interpret direct, indirect, and total effects via the appropriate matrix decomposition.

The SDM is preferable when both endogenous and exogenous spatial processes are suspected or diagnostic tests on SAR/SEM indicate composite dependence modes. Application of this model in the context of ICD-10 morbidity and poverty in Thailand reveals the degree to which spatial networks drive health risks, highlighting the necessity for coordinated, network-aware policy interventions (Kukieattikool et al., 6 Jan 2026).