Space-Time Spatial Lag Model

The cross-sectional SAR models spatial dependence at a single point in time. The space-time version adds a time dimension, allowing estimation over multiple periods while controlling for unit fixed effects, time fixed effects, and temporal dynamics simultaneously.

W should reflect the theoretical channel of interaction — contiguity (shared border), inverse distance, k-nearest neighbors, or economic distance. The choice must be motivated by prior theory, not data-driven optimization. Sensitivity analysis across two or three defensible W matrices is good practice.

Use a Hausman-type test. Fixed effects are preferred when unit-specific unobservables are likely correlated with covariates (the usual case in regional panels). Random effects are more efficient if that assumption is met, and are needed if time-invariant regressors must be included.

The spatial lag model treats spatial dependence as a substantive spillover effect (neighbors' outcomes cause your outcome). The spatial error model treats spatial dependence as nuisance correlation in the errors (common omitted factors). Lagrange multiplier tests after OLS can guide the choice: if LM-lag is significant but LM-error is not, prefer the lag model, and vice versa.

Yes — that yields the spatial dynamic panel model (SDM with temporal lag). Including both Wy_it and y_{i,t-1} captures spatial and temporal persistence simultaneously, but requires bias-corrected ML or GMM estimators (e.g., Elhorst 2010) to handle the incidental-parameters problem.