Direct, indirect and total effects of a spatial lag model

In a model with a spatial lag of the outcome, y = rho W y + X beta + W X theta + e, a change in covariate k at every unit moves the outcomes by S_k = (I - rho W)^-1 (beta_k I + theta_k W): the mean of its diagonal is the direct effect, the mean row sum the total, and their difference the indirect effect, the spillover to other units (LeSage and Pace 2009). An analysis that reads beta_k alone can report "no effect" where the effect is all spillover. With row-standardised weights the total is (beta_k + theta_k) / (1 - rho), proved in Lean: see proof("spill"). Standard errors are by the delta method in rho, beta_k and theta_k, with the exact derivatives d (I - rho W)^-1 / d rho = (I - rho W)^-1 W (I - rho W)^-1. The effects are those of spatialreg::impacts(), which gives simulated rather than delta-method standard errors.

Usage

spill(fit = NULL, w = NULL, rho = NULL, b = NULL, th = NULL)

Arguments

fit

A sarml() fit of a lag or combined model ("lag" or "sac", with or without durbin); or NULL, with w, rho, b (and th) given.

w

The weights: spwt()'s list, an spdep listw object, or an n x n matrix of weights from 0 (zero diagonal).

rho

The spatial lag coefficient.

b

The coefficients of the covariates (named, or x1, x2, ...).

th

NULL, or the coefficients of their lags, as many as b.

Value

A data frame with one row per covariate: term, direct, indirect, total and their standard errors sed, sei, set (NaN without a fit).

References

LeSage, J. and Pace, R. K. (2009). Introduction to Spatial Econometrics. Chapman and Hall/CRC, Boca Raton. doi:10.1201/9781420064254

Examples

set.seed(2)
p <- cbind(x = runif(40), y = runif(40))
w <- spwt(p, k = 4)
spill(w = w, rho = 0.4, b = c(gent = 0.1), th = 0.3)
#>   term    direct  indirect     total sed sei set
#> 1 gent 0.1373729 0.5292938 0.6666667 NaN NaN NaN