smmSAR: Simulated Method of Moments (SMM) Estimator of SAR model

Description

smmSAR implements the Simulated Method of Moments (SMM) estimator of the linear-in-mean SAR model when only the linking probabilities are available or can be estimated.

Usage

smmSAR(
  formula,
  contextual = FALSE,
  fixed.effects = FALSE,
  dnetwork,
  W = "identity",
  smm.ctr = list(R = 30L, iv.power = 2L, opt.tol = 1e-04, smoother = FALSE, print =
    FALSE),
  cond.var = TRUE,
  data
)

Value

A list consisting of:

n.group: number of groups.
N: vector of each group size.
time: elapsed time to run the SMM in second.
estimates: vector of estimated parameters.
formula: input value of formula.
contextual: input value of contextual.
fixed.effects: input value of fixed.effects.
smm.ctr: input value of smm.ctr.
details: other details of the model.

Arguments

formula: object of class formula: a symbolic description of the model. The formula should be as for example y ~ x1 + x2 | gy | gx1 + gx2 where y is the endogenous vector, the listed variables before the pipe, x1, x2 are the individual exogenous variables, gy is the average of y among friends, and gx1, gx2 are the contextual observed variables. If gy is observed and gx1, gx2 are not, the formula should be y ~ x1 + x2 | gy. If gy is not observed and gx1, gx2 are, the formula should be y ~ x1 + x2 || gx1 + gx2. If gy, gx1, and gx2 are not observed, the the formula should simply be y ~ x1 + x2.
contextual: logical; if true, this means that all individual variables will be set as contextual variables. In contrast mcmcSAR, formula as y ~ x1 + x2 and contextual as TRUE is not equivalent to set formula as y ~ x1 + x2 || gx1 + gx2. formula = y ~ x1 + x2 means that gy, gx1, and gx2 are not observed and contextual = TRUE means that the estimated model includes contextual effects.
fixed.effects: logical; if true, group heterogeneity is included as fixed effects.
dnetwork: a list, where the m-th elements is the matrix of link probability in the m-th sub-network.
W: is the weighted-matrix in the objective function of the SMM.
smm.ctr: is the list of some control parameters (see details).
cond.var: logical; if true the estimator variance conditional on dnetwork will be computed.
data: optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If missing, the variables are taken from environment(formula), typically the environment from which smmSAR is called.

Details

The parameter smm.ctr is the list of some control parameters such as:

R numbers of draws R (in the package, we assume S = 1 and T = 1);
iv.power number of powers of the network matrix G to be used to construct instruments;
opt.tol optimization tolerance that will be used in optimize;
smoother (logical) which indicates if draws should be performed using the smoother simulator;
h bandwith of the smoother (required if smoother = TRUE);
print (logical) indicates if the optimization process should be printed step by step.

Examples

Run this code

# Number of groups
M        <- 10
# size of each group
N        <- rep(20,M)
# covariates
X        <- cbind(rnorm(sum(N),0,5),rpois(sum(N),7))
# network formation model parameter
rho      <- c(-0.8, 0.2, -0.1)
# individual effects
beta     <- c(2, 1, 1.5, 5, -3)
# endogenous effects
alpha    <- 0.4
# std-dev errors
se       <- 1
# network
tmp      <- c(0, cumsum(N))
X1l      <- lapply(1:M, function(x) X[c(tmp[x] + 1):tmp[x+1],1])
X2l      <- lapply(1:M, function(x) X[c(tmp[x] + 1):tmp[x+1],2])
dist.net <- function(x, y) abs(x - y)
X1.mat   <- lapply(1:M, function(m) {
  matrix(kronecker(X1l[[m]], X1l[[m]], FUN = dist.net), N[m])})
X2.mat   <- lapply(1:M, function(m) {
  matrix(kronecker(X2l[[m]], X2l[[m]], FUN = dist.net), N[m])})
Xnet     <- as.matrix(cbind("Const" = 1,
                            "dX1"   = mat.to.vec(X1.mat),
                            "dX2"   = mat.to.vec(X2.mat)))
ynet     <- Xnet %*% rho
ynet     <- c(1*((ynet + rlogis(length(ynet))) > 0))
G0       <- vec.to.mat(ynet, N, normalise = FALSE)
# normalise
G0norm   <- norm.network(G0)
# Matrix GX
GX       <- peer.avg(G0norm, X)
# simulate the dependent variable use an external package
y        <- simSAR(~ X + GX, Glist = G0norm, parms = c(alpha, beta), 
                   epsilon = rnorm(sum(N), sd = se))
Gy       <- y$Gy
y        <- y$y
# build dataset
dataset           <- as.data.frame(cbind(y, X, Gy, GX))
colnames(dataset) <- c("y","X1","X2", "Gy", "GX1", "GX2")
nNet      <- nrow(Xnet) # network formation model sample size
Aobs      <- sample(1:nNet, round(0.3*nNet)) # We observed 30%
# We can estimate rho using the gml function from the stats package
logestim  <- glm(ynet[Aobs] ~ -1 + Xnet[Aobs,], family = binomial(link = "logit"))
slogestim <- summary(logestim)
rho.est   <- logestim$coefficients
rho.var   <- slogestim$cov.unscaled # we also need the covariance of the estimator

d.logit     <- lapply(1:M, function(x) {
  out       <- 1/(1 + exp(-rho.est[1] - rho.est[2]*X1.mat[[x]] -
                            rho.est[3]*X2.mat[[x]]))
  diag(out) <- 0
  out})
smm.logit   <- smmSAR(y ~ X1 + X2, dnetwork = d.logit, contextual = TRUE,
                      smm.ctr  = list(R = 100L, print = TRUE), data = dataset)
summary(smm.logit, dnetwork = d.logit, data = dataset)

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