Spatial Durbin error, general nesting and CAR regression with tests

SdemML: spatial Durbin error model \(y = X\beta + WX_*\theta + u\), \(u = \lambda Wu + e\), fitted as the error model on the columns of \(X\) and \(WX_*\) (spatialreg::errorsarlm(Durbin = TRUE)), with covariance \(\sigma^2(Z_*^\top Z_*)^{-1}\) and impacts (direct \(\beta\), indirect \(\theta\), total) with standard errors. GnsML: general nesting model, the SAC model on the columns of \(X\) and \(WX_*\) (sacsarlm(Durbin = TRUE)), with SpatialImpacts. CarML: CAR regression (spautolm(family = "CAR")) by profile likelihood over \((1/e_{min}, 1/e_{max})\), with the LR test of \(\lambda = 0\). SpatialLrTest and SpatialWaldTest: chi-square tests. ResidualMoran: Moran's I of OLS residuals with exact moments (spdep::lm.morantest). LogJacobian: \(\log|I - \rho W| + \log|I - \lambda W|\). Identical to the Python arm morie.fn.spdurbin.

Usage

SdemML(y, X, W, interval = c(-0.999, 0.999))

GnsML(y, X, W, interval = c(-0.999, 0.999))

CarML(y, X, W, interval = NULL)

SpatialLrTest(ll_full, ll_restricted, df)

SpatialWaldTest(estimate, vcov)

ResidualMoran(y, X, W)

LogJacobian(W, rho, lam = 0)

Arguments

y: Response.
X: Design matrix including any intercept column.
W: Spatial weights matrix (symmetric for CarML).
interval: Search interval for the spatial parameter(s).
ll_full, ll_restricted: Log-likelihoods.
df: Degrees of freedom.
estimate: Estimates tested against zero.
vcov: Their covariance matrix.
rho, lam: Autoregressive parameters.

Value

List.

References

LeSage, J. and Pace, R. K. (2009). Introduction to Spatial Econometrics. CRC Press.

Elhorst, J. P. (2014). Spatial Econometrics: From Cross-Sectional Data to Spatial Panels. Springer.

Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society B 36, 192-236.

Cliff, A. D. and Ord, J. K. (1981). Spatial Processes: Models and Applications. Pion.

Examples

SpatialLrTest(-10, -12.5, 1)$pvalue
#> [1] 0.02534732
LogJacobian(matrix(c(0, 1, 1, 0), 2), 0.5)
#> [1] -0.2876821