Wrong calculation of impacts in SLX

[nk027]

I was reproducing some results to compare spatialreg (version 1.2-1) and bsreg and stumbled upon an issue with the calculation of impacts for the SLX term (i.e. impacting SLX, SDM, and SDEM models). For the SLX, average indirect impacts are often assumed to be reflected by θ directly. However, this is not generally the case. Partial effects of a univariate SLX are given by

∂y/∂x = Iβ + Wθ

The average indirect effect (i.e. observation i on j != i) is given by sum(W) / N * theta. For row-stochastic W we have sum(W) == N, so the impact is given by θ, but not for other scaling types. I am currently wrapping up a working paper, where I go into some more detail, that I'll link here later.

The issue is that this is not accounted for in the impacts() methods for models with the LX term, as I demonstrate below. I think this can be fixed in a straightforward manner and I can start work on a PR, but will need some time. A code example of what I am talking about is below.

library("spatialreg")
set.seed(42)
N <- 100L

# Build connectivity ---
xy <- cbind(runif(N), runif(N)) # Simulate locations
dist <- as.matrix(dist(xy)) # Compute distance
diag(dist) <- Inf # We want zeros on the diagonal
W <- dist^-2 # Compute inverse-distance with decay parameter two
W_rs <- W / rowSums(W) # Use row-, eigenvalue-, and norm-scaling
W_ev <- W / max(eigen(W, symmetric = TRUE, only.values = TRUE)$values)
W_fn <- W / norm(W, "F")

# Build DGP ---
X <- matrix(rnorm(N), N)
beta <- 2
theta <- 1
y <- X * beta + W_ev %*% X * theta + rnorm(N, sd = 0.01) # No need for noise

# Estimate SLX (using lm and spatialreg) ---

# Use lm and compare theta
(m1 <- lm(y ~ 0 + X + W_ev %*% X)) # True model -- correct
(m2 <- lm(y ~ 0 + X + W_rs %*% X)) # Appears way off
(m3 <- lm(y ~ 0 + X + W_fn %*% X))
# Results mirror spatialreg
(s1 <- lmSLX(y ~ 0 + X, listw = spdep::mat2listw(W_ev)))
(s2 <- lmSLX(y ~ 0 + X, listw = spdep::mat2listw(W_rs)))
(s3 <- lmSLX(y ~ 0 + X, listw = spdep::mat2listw(W_fn)))
# The thetas are very different, depending on W's scale

# Compute average effects
c("Direct" = coef(m1)[[1]], "Indirect" = sum(W_ev) / N * coef(m1)[[2]])
c("Direct" = coef(m2)[[1]], "Indirect" = sum(W_rs) / N * coef(m2)[[2]])
c("Direct" = coef(m3)[[1]], "Indirect" = sum(W_fn) / N * coef(m3)[[2]])
# W_ev and W_fn are equivalent since they only differ by scaling -- W_rs is off
# since it's not symmetric, but effects are closer than theta would imply.

# Compute average effects using spatialreg
impacts(s1)
impacts(s2)
impacts(s3)
# The package just uses the coefficients and ignores the strucutre of W

# Check against true model ---
c("Direct" = beta, "Indirect" = sum(W_ev) / N * theta,
  "Direct (empirical)" = mean(((X + 1) * beta + W_ev %*% X %*% theta) - y),
  "Indirect (empirical)" = mean((X * beta + W_ev %*% (X + 1) %*% theta) - y))
# Partial effects computed manually seem valid