geostan (version 0.8.1)

spill: Spillover/diffusion effects for spatial lag models

Description

Spillover/diffusion effects for spatial lag models

Usage

spill(beta, gamma = 0, rho, W, approx = TRUE, K = 15)
impacts(object, approx = TRUE, K = 15)
# S3 method for impacts_slm
print(x, digits = 2, ...)

Arguments

beta

Coefficient for covariates (numeric vector)

gamma

Coefficient for spatial lag of covariates

rho

Spatial dependence parameter (single numeric value)

W

Spatial weights matrix

approx

For a computationally efficient approximation to the required matrix inverse (after LeSage and Pace 2009, pp. 114--115); if FALSE, then a proper matrix inverse will be computed using Matrix::solve.

K

Degree of polynomial in the expansion to use when 'approx = TRUE`.

object

A fitted spatial lag model (from stan_sar)

x

An object of class 'impacts_slm', as returned by geostan::impacts

digits

Round results to this many digits

...

Additional arguments will be passed to base::print

Details

These methods apply only to the spatial lag and spatial Durbin lag models (SLM and SDLM) as fit by geostan::stan_sar.

The equation for these SAR models specifies simultaneous feedback between all units, such that changing the outcome in one location has a spill-over effect that may extend to all other locations (a ripple or diffusion effect); the induced changes will also react back onto the first unit. (This presumably takes time, even if the observations are cross-sectional.)

These spill-overs have to be incorporated into the interpretation and reporting of the regression coefficients of SLM and SDLM models. A unit change in the value of \(X\) in one location will impact \(y\) in that same place ('direct' impact) and will also impact \(y\) elsewhere through the diffusion process ('indirect' impact). The 'total' expected impact of a unit change in X is the sum of the direct and indirect effects (LeSage and Pace 2009).

The spill function is for quickly calculating average spillover effects given point estimates of parameters.

The impacts function calculates the (average) direct, indirect, and total effects once for every MCMC sample to produce samples from the posterior distribution for the impacts; the samples are returned together with a summary of the posterior distribution (mean, median, and select quantiles).

Examples

##
## SDLM data
##

parts <- prep_sar_data2(row = 9, col = 9, quiet = TRUE)
W <- parts$W
x <- sim_sar(w=W, rho=.6)
Wx <- (W %*% x)[,1]
mu <- .5 * x + .25 * Wx
y <- sim_sar(w=W, rho=0.6, mu = mu, type = "SLM")
dat <- cbind(y, x)

# impacts per the above parameters
spill(0.5, 0.25, 0.6, W)

##
## impacts for SDLM
##

fit <- stan_sar(y ~ x, data = dat, sar = parts,
                type = "SDLM", iter = 500,
                slim = TRUE, quiet = TRUE) 

# impacts (posterior distribution)
impax <- impacts(fit)
print(impax)

# plot posterior distributions
og = par(mfrow = c(1, 3),
         mar = c(3, 3, 1, 1))
S <- impax$samples[[1]]
hist(S[,1], main = 'Direct')
hist(S[,2], main = 'Indirect')
hist(S[,3], main = 'Total')
par(og)

##
## The approximate method
##

# High rho value requires more K; rho^K must be near zero
Ks <- c(10, 15, 20, 30, 35, 40)
print(cbind(Ks, 0.9^Ks))

# understand sensitivity of results to K when rho is high
spill(0.5, -0.25, 0.9, W, approx = TRUE, K = 10)
spill(0.5, -0.25, 0.9, W, approx = TRUE, K = 20)
spill(0.5, -0.25, 0.9, W, approx = TRUE, K = 30)
spill(0.5, -0.25, 0.9, W, approx = TRUE, K = 50)

# the correct results
spill(0.5, -0.25, 0.9, W, approx = FALSE)

# moderate and low rho values are fine with smaller K
spill(0.5, -0.25, 0.7, W, approx = TRUE, K = 15)
spill(0.5, -0.25, 0.7, W, approx = FALSE)