# How to compute measures of dispersion and statistical significance of the impacts of a panel Spatial Durbin model (SDM)?

I ran a panel Spatial Durbin model (SDM) and computed the summary measures of impacts (direct, indirect and total). Now, I would like to get measures of dispersion for the impacts estimates as well as the statistical significance of the impacts.

How can I get such measures ?

Here is a reproducible example:

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set.seed(1234)
library(plm)
library(splm)
data(Produc, package = "plm")
data(usaww)

# Create spatially lagged independent variables WX using slag()
Produc2 <- pdata.frame(Produc, index = c("state", "year"))
Produc2$$pcap_SL <- slag(log(Produc2$$pcap), mat2listw(usaww))
Produc2$$pc_SL <- slag(log(Produc2$$pc), mat2listw(usaww))
Produc2$$emp_SL <- slag(log(Produc2$$emp), mat2listw(usaww))
Produc2$$unemp_SL <- slag(log(Produc2$$unemp), mat2listw(usaww))
# Define formula
fm <- log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp +
pcap_SL + pc_SL + emp_SL + unemp_SL
# Spatial Durbin Model : Y = ρ.WY + β.X + θ.WX + μ
summary(
sdm <- spml(fm, data = Produc2, index = c("dep", "period"),
model = "within", effect = "twoways", lag = TRUE,
listw = mat2listw(usaww), spatial.error = "none",
LeeYu = TRUE, Hess = FALSE)
)

## Compute impact measures of variable pcap
# Impacts are computed using equation (25) in LeSage (2008)
# LeSage (2008) is available at https://journals.openedition.org/rei/pdf/3887
# Note that I cannot use impacts.splm() since the current development of splm doesn’t allow impacts on SDM

Time <- length(unique(Produc$$year)) N <- length(unique(Produc$$state))
library(Matrix)
s.lws <- kronecker(Diagonal(Time) , listw2dgCMatrix(mat2listw(usaww)))
IrW <- Diagonal(N * Time) - sdm$$spat.coef * s.lws IrWi <- solve(IrW) S_pcap <- IrWi * (Diagonal(N * Time) * sdm$$coefficients +
s.lws * sdm\$coefficients)
# direct impact
dir_imp <- mean(diag(S_pcap))
# total impact
tot_imp <- mean(rowSums(S_pcap)) # or sum(S_pcap)/816
# indirect impact : total impact - direct impact
indir_imp <- tot_imp - dir_imp

# LeSage (2008) provides guidance for drawing inference regarding the significance
# of these impacts (section 4.2, page 35) but I'm struggling to implement it in R.

# Could you help me deal with this ?

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Thanks for your help.