Higher-Order Models

SACModel

A fitted SAC / SARAR model, returned by fit_sac: y = rho*W*y + X*beta + u with u = lambda*W*u + e.

Dependence in both y and the errors, so it nests SAR (lambda = 0) and the spatial error model (rho = 0).

SDEMModel

A fitted spatial Durbin error model, returned by fit_sdem: y = X*beta + W*X*theta + u with u = lambda*W*u + e.

Lagged covariates plus a spatial error, but no lag of y. Like SLXModel, the absence of feedback through y means theta reads as the spillover directly; unlike it, the errors are modelled too, so inference accounts for residual spatial correlation.

SLXModel

A fitted SLX model, y = X*beta + W*X*theta + e, returned by fit_slx.

The simplest way to admit spillovers: covariates of neighbours enter, but y does not depend on W*y. That makes it OLS on [X W*X], so there is no log-determinant and no endogeneity, and theta is the spillover directly — no (I - rho*W)^{-1} decomposition needed.

Often the right starting point before reaching for a lag model.

fit_sac(W_lag, W_error, X, y; varnames=nothing, method=ChebyshevApprox())

Fit the SAC (Spatial Autoregressive Combined) model via concentrated ML: y = ρW₁y + Xβ + u, u = λW₂u + ε

Also known as SARAR when W₁ = W₂. Kelejian & Prucha (1998).

fit_sdem(W, X, y; varnames=nothing, method=ChebyshevApprox())

Fit the SDEM (Spatial Durbin Error Model) via concentrated ML: y = Xβ + WXθ + u, u = λWu + ε

Combines the SDM covariate structure with SEM error structure. LeSage & Pace (2009).

fit_slx(W, X, y; varnames=nothing)

Fit the SLX (Spatial Lag of X) model by OLS: y = Xβ + WXθ + ε

No spatial lag of y and no spatial error structure. Recommended as a starting point before SAR/SDM (Halleck Vega & Elhorst, 2015).