sar: Estimating linear-in-mean models with social interactions

In CDatanet: Econometrics of Network Data

Estimating linear-in-mean models with social interactions

Description

sar computes quasi-maximum likelihood estimators for linear-in-mean models with social interactions (see Lee, 2004 and Lee et al., 2010).

Usage

sar(
  formula,
  Glist,
  lambda0 = NULL,
  fixed.effects = FALSE,
  optimizer = "optim",
  opt.ctr = list(),
  print = TRUE,
  cov = TRUE,
  cinfo = TRUE,
  data
)

Arguments

formula	a class object formula: a symbolic description of the model. formula must be as, for example, y ~ x1 + x2 + gx1 + gx2 where y is the endogenous vector and x1, x2, gx1 and gx2 are control variables, which can include contextual variables, i.e. averages among the peers. Peer averages can be computed using the function peer.avg.
Glist	The network matrix. For networks consisting of multiple subnets, Glist can be a list of subnets with the m-th element being an ns*ns adjacency matrix, where ns is the number of nodes in the m-th subnet.
lambda0	an optional starting value of \lambda.
fixed.effects	a Boolean indicating whether group heterogeneity must be included as fixed effects.
optimizer	is either nlm (referring to the function nlm) or optim (referring to the function optim). Arguments for these functions such as, control and method can be set via the argument opt.ctr.
opt.ctr	list of arguments of nlm or optim (the one set in optimizer) such as control, method, etc.
print	a Boolean indicating if the estimate should be printed at each step.
cov	a Boolean indicating if the covariance should be computed.
cinfo	a Boolean indicating whether information is complete (cinfo = TRUE) or incomplete (cinfo = FALSE). In the case of incomplete information, the model is defined under rational expectations.
data	an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which sar is called.

Details

For a complete information model, the outcome y_i is defined as:

y_i = \lambda \bar{y}_i + \mathbf{z}_i'\Gamma + \epsilon_i,

where \bar{y}_i is the average of y among peers, \mathbf{z}_i is a vector of control variables, and \epsilon_i \sim N(0, \sigma^2). In the case of incomplete information models with rational expectations, y_i is defined as:

y_i = \lambda E(\bar{y}_i) + \mathbf{z}_i'\Gamma + \epsilon_i.

Value

A list consisting of:

info	list of general information on the model.
estimate	Maximum Likelihood (ML) estimator.
cov	covariance matrix of the estimate.
details	outputs as returned by the optimizer.

References

Lee, L. F. (2004). Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models. Econometrica, 72(6), 1899-1925, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/j.1468-0262.2004.00558.x")}.

Lee, L. F., Liu, X., & Lin, X. (2010). Specification and estimation of social interaction models with network structures. The Econometrics Journal, 13(2), 145-176, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/j.1368-423X.2010.00310.x")}

See Also

sart, cdnet, simsar.

Examples

# Groups' size
set.seed(123)
M      <- 5 # Number of sub-groups
nvec   <- round(runif(M, 100, 1000))
n      <- sum(nvec)

# Parameters
lambda <- 0.4
Gamma  <- c(2, -1.9, 0.8, 1.5, -1.2)
sigma  <- 1.5
theta  <- c(lambda, Gamma, sigma)

# X
X      <- cbind(rnorm(n, 1, 1), rexp(n, 0.4))

# Network
G      <- list()

for (m in 1:M) {
  nm           <- nvec[m]
  Gm           <- matrix(0, nm, nm)
  max_d        <- 30
  for (i in 1:nm) {
    tmp        <- sample((1:nm)[-i], sample(0:max_d, 1))
    Gm[i, tmp] <- 1
  }
  rs           <- rowSums(Gm); rs[rs == 0] <- 1
  Gm           <- Gm/rs
  G[[m]]       <- Gm
}

# data
data   <- data.frame(X, peer.avg(G, cbind(x1 = X[,1], x2 =  X[,2])))
colnames(data) <- c("x1", "x2", "gx1", "gx2")

ytmp    <- simsar(formula = ~ x1 + x2 + gx1 + gx2, Glist = G, 
                  theta = theta, data = data) 
data$y  <- ytmp$y

out     <- sar(formula = y ~ x1 + x2 + + gx1 + gx2, Glist = G, 
               optimizer = "optim", data = data)
summary(out)

CDatanet documentation built on June 22, 2024, 11:14 a.m.