smmSAR {PartialNetwork}

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
)

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.

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.

Examples

# Number of groups
M        <- 100
# size of each group
N        <- rep(30,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 dependent variable use an external package
y        <- CDatanet::simsar(~ X, contextual = TRUE, Glist = G0norm,
                             theta = c(alpha, beta, 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)