[R] System GMM yields identical results for any weighting matrix

[Richard Hardy]

A copy of this question can be found on Cross Validated:
https://stats.stackexchange.com/questions/645362

I am estimating a system of seemingly unrelated regressions (SUR) in R.
Each of the equations has one unique regressor and one common regressor. I
am using `gmm::sysGmm` and am experimenting with different weighting
matrices. I get the same results (point estimates, standard errors and
anything else that I can see (**except** for the value of the $J$-test)
regardless of the weighting matrix. I do not think this is correct.
The phenomenon persists regardless of what type of covariance matrix
estimator I use: `MDS`, `CondHom` or `HAC`.
It also persists regardless of whether I use unrestricted estimation or
restrict the coefficients on one of the variables (the common regressor) to
be equal across equations.

**Question:** Why does system GMM via `gmm::sysGmm` yield identical results
for any weighting matrix? How can I make it yield proper results that vary
with the weighting matrix (if that makes sense and I am not mistaken, of
course)?

------------------------------ R code for a reproducible example

library(gmm)
library(systemfit)

# Generate and prepare the data
n <- 1000 # sample size
m <- 100  # length of the "second part" of the sample
N <- 3    # number of equations
set.seed(321); x <- matrix(rnorm(n*N),ncol=N); colnames(x) <-
paste0("x",1:N) # generate regressors
dummy <- c( rep(0,n-m), rep(1,m) ) # generate a common regressor
x <- cbind(x,dummy)                # include the common regressor with the
rest of the regressors
set.seed(123); y <- matrix(rnorm(n*N),ncol=N); colnames(y) <-
paste0("y",1:N) # a placeholder for dependent variables
for(i in 1:N){
 y[,i] <- i + sqrt(i)*x[,i] - i*dummy + y[,i]*15*sqrt(i)
 # y[,i] is a linear function of x[,i] and dummy,
 # plus an error term with equation-specific variance
}
data1 <- as.data.frame(cbind(y,x)) # create a data frame of all data (y and
x)

# Create the model equations and moment conditions
ES_g = ES_h <- list() # ES ~ equation system
for(i in 1:N){
 ES_g[[i]] <- as.formula(assign(paste0("eq",i), value=paste0("y",i," ~
x",i," + dummy"))) # define linear equations of SUR
 ES_h[[i]] <- as.formula(assign(paste0("eq",i), value=paste0(       "~
x",i," + dummy"))) # define the moment conditions for GMM
}

# Estimate a WLS-type weighting matrix to use as a user-specified weighting
matrix in GMM
m0 <- systemfit(formula=ES_g, method="OLS", data=data1)
OLSmat <- diag(diag(m0$residCov)); Wmat <- solve(OLSmat)

# Choose the type of covariance matrix in GMM
vc1 <- "MDS"
vc1 <- "CondHom"
vc1 <- "HAC"
#vc1 <- "TrueFixed"

# Choose between restricted and unrestricted estimation
cec1=NULL # unrestricted
cec1=3    # restrict the coefficient on the dummy to be equal across
equations

# Estimate the model with `sysGmm` using different weighting matrices:
identity, "optimal" and manually specified
m1a <- sysGmm(g=ES_g, h=ES_h, wmatrix="ident"  , weightsMatrix=NULL,
vcov=vc1, crossEquConst=cec1, data=data1); summary(m1a)
m1b <- sysGmm(g=ES_g, h=ES_h, wmatrix="optimal", weightsMatrix=NULL,
vcov=vc1, crossEquConst=cec1, data=data1); summary(m1b)
m1c <- sysGmm(g=ES_g, h=ES_h,                    weightsMatrix=Wmat,
vcov=vc1, crossEquConst=cec1, data=data1); summary(m1c)

------------------------------ R session info:

R version 4.3.3 (2024-02-29 ucrt)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 19045)

Matrix products: default

locale:
[1] LC_COLLATE=English_United States.utf8  LC_CTYPE=English_United
States.utf8
[3] LC_MONETARY=English_United States.utf8 LC_NUMERIC=C

[5] LC_TIME=English_United States.utf8

time zone: Europe/Berlin
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base

other attached packages:
[1] systemfit_1.1-30 lmtest_0.9-40    zoo_1.8-12       car_3.1-2
 carData_3.0-5    Matrix_1.6-1
[7] gmm_1.8          sandwich_3.0-2

loaded via a namespace (and not attached):
[1] MASS_7.3-60.0.1   compiler_4.3.3    tools_4.3.3       abind_1.4-5
rstudioapi_0.15.0 grid_4.3.3
[7] lattice_0.22-5

------------------------------

Kind regards,
Richard

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