sart {CDatanet}

Estimating Tobit models with social interactions

Description

sart estimates Tobit models with social interactions (Xu and Lee, 2015).

Usage

sart(
  formula,
  Glist,
  starting = NULL,
  Ey0 = NULL,
  optimizer = "fastlbfgs",
  npl.ctr = list(),
  opt.ctr = list(),
  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.

starting

(optional) a starting value for \theta = (\lambda, \Gamma, \sigma) (see the model specification in details).

Ey0

(optional) a starting value for E(y).

optimizer

is either fastlbfgs (L-BFGS optimization method of the package RcppNumerical), 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.

npl.ctr

a list of controls for the NPL method (see details of the function cdnet).

opt.ctr

a list of arguments to be passed in optim_lbfgs of the package RcppNumerical, nlm or optim (the solver set in optimizer), such as maxit, eps_f, eps_g, control, method, etc.

cov

a Boolean indicating if the covariance must 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 sart is called.

Details

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

\begin{cases}y_i^{\ast} = \lambda \bar{y}_i + \mathbf{z}_i'\Gamma + \epsilon_i, \\ y_i = \max(0, y_i^{\ast}),\end{cases}

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 modelswith rational expectations, y_i is defined as:

\begin{cases}y_i^{\ast} = \lambda E(\bar{y}_i) + \mathbf{z}_i'\Gamma + \epsilon_i, \\ y_i = \max(0, y_i^{\ast}).\end{cases}

Value

A list consisting of:

info

a list of general information on the model.

estimate

the Maximum Likelihood (ML) estimator.

Ey

E(y), the expectation of y.

GEy

the average of E(y) friends.

cov

a list including (if cov == TRUE) covariance matrices.

details

outputs as returned by the optimizer.

References

Xu, X., & Lee, L. F. (2015). Maximum likelihood estimation of a spatial autoregressive Tobit model. Journal of Econometrics, 188(1), 264-280, doi:10.1016/j.jeconom.2015.05.004.

See Also

sar, cdnet, simsart.

Examples

# Groups' size
set.seed(123)
M      <- 5 # Number of sub-groups
nvec   <- round(runif(M, 100, 200))
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")

## Complete information game
ytmp    <- simsart(formula = ~ x1 + x2 + gx1 + gx2, Glist = G, theta = theta, 
                   data = data, cinfo = TRUE)
data$yc <- ytmp$y

## Incomplete information game
ytmp    <- simsart(formula = ~ x1 + x2 + gx1 + gx2, Glist = G, theta = theta, 
                   data = data, cinfo = FALSE)
data$yi <- ytmp$y

# Complete information estimation for yc
outc1   <- sart(formula = yc ~ x1 + x2 + gx1 + gx2, optimizer = "nlm",
                Glist = G, data = data, cinfo = TRUE)
summary(outc1)

# Complete information estimation for yi
outc1   <- sart(formula = yi ~ x1 + x2 + gx1 + gx2, optimizer = "nlm",
                Glist = G, data = data, cinfo = TRUE)
summary(outc1)

# Incomplete information estimation for yc
outi1   <- sart(formula = yc ~ x1 + x2 + gx1 + gx2, optimizer = "nlm",
                Glist = G, data = data, cinfo = FALSE)
summary(outi1)

# Incomplete information estimation for yi
outi1   <- sart(formula = yi ~ x1 + x2 + gx1 + gx2, optimizer = "nlm",
                Glist = G, data = data, cinfo = FALSE)
summary(outi1)