sart {CDatanet}

Estimate sart model

Description

sart is used to estimate peer effects on censored data (see details). The model is presented in Xu and Lee(2015).

Usage

sart(
  formula,
  contextual,
  Glist,
  theta0 = NULL,
  yb0 = NULL,
  optimizer = "fastlbfgs",
  npl.ctr = list(),
  opt.ctr = list(),
  print = TRUE,
  cov = TRUE,
  RE = FALSE,
  data
)

Arguments

formula

an object of class formula: a symbolic description of the model. The formula should be as for example y ~ x1 + x2 | x1 + x2 where y is the endogenous vector, the listed variables before the pipe, x1, x2 are the individual exogenous variables and the listed variables after the pipe, x1, x2 are the contextual observable variables. Other formulas may be y ~ x1 + x2 for the model without contextual effects, y ~ -1 + x1 + x2 | x1 + x2 for the model without intercept or y ~ x1 + x2 | x2 + x3 to allow the contextual variable to be different from the individual variables.

contextual

(optional) logical; if true, this means that all individual variables will be set as contextual variables. Set the formula as y ~ x1 + x2 and contextual as TRUE is equivalent to set the formula as y ~ x1 + x2 | x1 + x2.

Glist

the adjacency matrix or list sub-adjacency matrix.

theta0

(optional) starting value of \theta = (\lambda, \beta, \gamma, \sigma). The parameter \gamma should be removed if the model does not contain contextual effects (see details).

yb0

(optional) expectation of 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). Other arguments of these functions such as, control and method can be defined through the argument opt.ctr.

npl.ctr

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

opt.ctr

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, ...

print

a Boolean indicating if the estimate should be printed at each step.

cov

a Boolean indicating if the covariance should be computed.

RE

a Boolean which indicates if the model if under rational expectation of not.

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

Model

The left-censored variable \mathbf{y} is generated from a latent variable \mathbf{y}^*. The latent variable is given for all i as

y_i^* = \lambda \mathbf{g}_i y + \mathbf{x}_i'\beta + \mathbf{g}_i\mathbf{X}\gamma + \epsilon_i,

where \epsilon_i \sim N(0, \sigma^2).
The count variable y_i is then define that is y_i = 0 if y_i^* \leq 0 and y_i = y_i^* otherwise.

Value

A list consisting of:

info

list of general information on the model.

estimate

Maximum Likelihood (ML) estimator.

yb

ybar (see details), expectation of y.

Gyb

average of the expectation of y among friends.

cov

List of covariances.

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
M      <- 5 # Number of sub-groups
nvec   <- round(runif(M, 100, 1000))
n      <- sum(nvec)

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

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

# Network
Glist  <- 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
  Glist[[m]]   <- Gm
}


# data
data    <- data.frame(x1 = X[,1], x2 =  X[,2])

rm(list = ls()[!(ls() %in% c("Glist", "data", "theta"))])

ytmp    <- simsart(formula = ~ x1 + x2 | x1 + x2, Glist = Glist,
                   theta = theta, data = data)

y       <- ytmp$y

# plot histogram
hist(y)

opt.ctr <- list(method  = "Nelder-Mead", 
                control = list(abstol = 1e-16, abstol = 1e-11, maxit = 5e3))
data    <- data.frame(yt = y, x1 = data$x1, x2 = data$x2)
rm(list = ls()[!(ls() %in% c("Glist", "data"))])

out     <- sart(formula = yt ~ x1 + x2, optimizer = "nlm",
                  contextual = TRUE, Glist = Glist, data = data)
summary(out)