Higher-Order Spatial Models

SLX — Spatial Lag of X

The SLX model includes spatially lagged covariates without a spatial lag of y:

\[y = X\beta + WX\theta + \varepsilon\]

Estimated by OLS. Recommended as a starting point before SAR/SDM (Halleck Vega & Elhorst, 2015).

m = fit_slx(W, X, y; varnames=["gdp", "pop"])

The SAC (also called SARAR) model combines spatial lag and spatial error:

\[y = \rho W_1 y + X\beta + u, \quad u = \lambda W_2 u + \varepsilon\]

Allows different weight matrices for the lag and error components (Kelejian & Prucha, 1998).

m = fit_sac(W_lag, W_error, X, y; varnames=["gdp", "pop"])

When W_lag == W_error, this is the classic SARAR model.

The SDEM combines the SDM covariate structure with the SEM error structure:

\[y = X\beta + WX\theta + u, \quad u = \lambda W u + \varepsilon\]

Estimated via concentrated maximum likelihood (LeSage & Pace, 2009).

m = fit_sdem(W, X, y; varnames=["gdp", "pop"])

See the full API Reference for fit_slx, fit_sac, and fit_sdem.