Space-Time Spatial Durbin Model (ST-SDM)
Space-Time Spatial Durbin Model · Also known as: ST-SDM, spatiotemporal Durbin model, spatial Durbin panel model, space-time SDM
The Space-Time Spatial Durbin Model extends the cross-sectional Spatial Durbin Model to panel data, simultaneously capturing spatial spillovers in both the dependent variable and the explanatory variables across space and over time. It is the most general and flexible specification in the spatial panel family, nesting the spatial lag and spatial error models as special cases.
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When to use it
Use the ST-SDM when you have a balanced or unbalanced panel of spatial units observed at multiple time points and you have a priori reason (or empirical evidence from LM tests) to believe both the dependent variable and the covariates exhibit spatial spillovers. It is especially appropriate in regional economics, epidemiology, environmental science, and political science where cross-unit contagion and covariate externalities coexist. Do not use it when spatial dependence is absent (verified by non-significant Moran's I and LM tests), when the panel is very short (few time periods make fixed-effect demeaning unreliable), or when the weights matrix is entirely arbitrary — misspecification of W propagates into biased spillover estimates.
Strengths & limitations
- Captures both outcome spillovers (spatial lag channel) and covariate spillovers (Durbin channel) simultaneously, making it the most complete spatial panel specification.
- Nests the spatial lag model (theta = 0) and the spatial error model (theta = -rho * beta) as testable restrictions, enabling formal model comparison.
- Separates direct effects from indirect (spillover) effects through the LeSage-Pace impact decomposition, providing substantively interpretable marginal effects.
- Controls for unobserved unit-level heterogeneity and common time trends via two-way fixed effects, reducing omitted-variable bias.
- Accommodates both cross-sectional and temporal dimensions, making full use of panel variation.
- Estimation requires the spatial weights matrix W to be correctly specified; results are sensitive to this choice, and there is no single accepted procedure for selecting W.
- Simultaneous spatial feedback makes computation more demanding than OLS; quasi-maximum likelihood is required, and convergence can be slow for large N.
- With short time series (T small relative to N) the Lee-Yu bias correction is needed; ignoring it produces biased fixed-effect estimates.
- Degrees of freedom are consumed by the large number of parameters (rho, beta, theta, unit and time effects), demanding reasonably large N and T.
Frequently asked
How does the ST-SDM differ from the spatial lag panel model?
The spatial lag panel model includes only the spatially lagged outcome W*y on the right-hand side. The ST-SDM also includes spatially lagged covariates W*X (the Durbin terms), capturing an additional channel through which neighbors' characteristics affect own outcomes. The SDM is strictly more general.
What is the LeSage-Pace impact decomposition and why is it necessary?
Because W*y creates a simultaneous spatial feedback loop, a unit change in one observation's covariate ripples through all connected units. LeSage and Pace (2009) showed that the total effect matrix is (I - rho*W)^-1(beta*I + theta*W). Averaging the diagonal gives direct effects; averaging the off-diagonal elements gives indirect (spillover) effects. Ignoring this and reading beta directly as the marginal effect is incorrect when rho is non-zero.
How do I choose the spatial weights matrix W?
There is no universally correct choice. Common options are queen/rook contiguity, k-nearest neighbors, or inverse-distance weights. The choice should be guided by the theoretical mechanism (e.g., contiguity for administrative spillovers, distance-decay for market interactions). Always perform a sensitivity analysis by re-estimating under alternative W specifications.
Can I include a time-lagged dependent variable as well?
Yes. When a temporal lag y_{i,t-1} is added alongside the spatial lag W*y_it and W*X_it, the model becomes a dynamic space-time SDM. This further captures inertia in the outcome, but requires an instrumental-variable or GMM estimator (e.g., Arellano-Bond style) to handle the endogeneity of the lagged outcome.
Which software packages implement the space-time SDM?
In R, the splm package supports spatial panel models including the SDM specification, and spatialreg provides cross-section SDM estimation. The sphet package handles heteroscedastic variants. In Stata, the xsmle command fits spatial panel models. Python users can use PySAL's spreg module, though full panel SDM support requires additional steps.
Sources
- LeSage, J. P., & Pace, R. K. (2009). Introduction to Spatial Econometrics. CRC Press / Taylor & Francis. ISBN: 978-1420064247
- Elhorst, J. P. (2014). Spatial Econometrics: From Cross-Sectional Data to Spatial Panels. Springer. ISBN: 978-3642403392
How to cite this page
ScholarGate. (2026, June 3). Space-Time Spatial Durbin Model. ScholarGate. https://scholargate.app/en/spatial-analysis/space-time-spatial-durbin-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.
- Panel Spatial Durbin ModelSpatial analysis↔ compare
- Space-Time Spatial Error ModelSpatial analysis↔ compare
- Space-Time Spatial Lag ModelSpatial analysis↔ compare
- Spatial Durbin ModelSpatial analysis↔ compare
- Spatial Error ModelSpatial analysis↔ compare
- Spatial Lag ModelSpatial analysis↔ compare