CDatanet (version 0.0.1)

netformation: Estimate Network Formation Model

Description

Estimate Network Formation Model

Usage

netformation(
  network,
  formula,
  data,
  fixed.effects = TRUE,
  init = list(),
  mcmc.ctr = list(),
  print = TRUE
)

Arguments

network

matrix or list of sub-matrix of social interactions containing 0 and 1, where links are represented by 1

formula

an object of class formula: a symbolic description of the model. The formula should be as for example ~ x1 + x2 where x1, x2 are explanatory variable of links formation

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 netformation is called.

fixed.effects

boolean indicating if sub-network heterogeneity as fixed effects should be included.

init

(optional) list of starting values containing beta, an K-dimensional vector of the explanatory variables parameter, mu an n-dimensional vector of unobserved parameters, sigmau2 the vector of the variances of mu in each sub-network (or single variance if fixed.effects = FALSE) and uu the vector of the of the means of mu in each sub-network (or single mean if fixed.effects = FALSE), where K is the number of explanatory variables and n is the number of individuals.

mcmc.ctr

(optional) list of MCMC control (see detail).

print

boolean indicating if the estimation progression should be printed.

Value

A list consisting of:

n

number of individuals in each network.

n.obs

number of observations.

n.links

number of links.

K

number of explanatory variables.

posterior

list of simulations from the posterior distribution and the posterior density.

acceptance.rate

acceptance rate of beta and mu.

mcmc.ctr

returned list of MCMC control.

init

returned list of starting values.

Details

The network formation model can be used to control the network endogeneity in the count data, Tobit, and SAR models (see the vignette for an example with the count data model). The MCMC control mcmc.ctr should be a list containing

• jscalebeta a K-dimensional vector of beta jumping scales.

• jscalemu an n-dimensional vector of mu jumping scales.

• burnin the number of iterations in the burn-in.

• iteration the number of simulations.

• tbeta target of beta.

• tmu target of mu.

The burn-in is replicated three times. The estimation is performed for each component of beta during the two firsts burn-in. The simulation from the second burn-in are used to compute covariance of beta. From the third burn-in and the remaining steps of the MCMC, all the components in beta are jointly simulated. As mu dimension is large, the simulation is performed for each component. The jumping scale are also updated during the MCMC following Atchad<U+00E9> and Rosenthal (2005).

Examples

# NOT RUN {
M            <- 5 # Number of sub-groups
nvec         <- round(runif(M, 100, 500))
beta         <- c(1, -1)
Glist        <- list()
dX           <- matrix(0, 0, 2)
mu           <- list()
uu           <- runif(M, -5, 5)
sigma2u      <- runif(M, 0.5, 16)
for (m in 1:M) {
  n          <- nvec[m]
  mum        <- rnorm(n, uu[m], sqrt(sigma2u[m]))
  X1         <- rnorm(n)
  X2         <- rbinom(n, 1, 0.2)
  Z1         <- matrix(0, n, n)  
  Z2         <- matrix(0, n, n)
  
  for (i in 1:n) {
    for (j in 1:n) {
      Z1[i, j] <- abs(X1[i] - X1[j])
      Z2[i, j] <- 1*(X2[i] == X2[j])
    }
  }
  
  Gm           <- 1*((Z1*beta[1] + Z2*beta[2] +
                        kronecker(mum, t(mum), "+") + rlogis(n^2)) > 0)
  diag(Gm)     <- 0
  
  diag(Z1)     <- NA
  diag(Z2)     <- NA
  Z1           <- Z1[!is.na(Z1)]
  Z2           <- Z2[!is.na(Z2)]
  
  dX           <- rbind(dX, cbind(Z1, Z2))
  Glist[[m]]   <- Gm
  mu[[m]]      <- mum
}

mu  <- unlist(mu)
out <- netformation(network =  Glist, formula = ~ dX, fixed.effects = T,
                    mcmc.ctr = list(burin = 1000, iteration = 5000))


# plot simulations
plot(out$posterior$beta[,1], type = "l")
abline(h = beta[1], col = "red")
plot(out$posterior$beta[,2], type = "l")
abline(h = beta[2], col = "red")

k <- 2
plot(out$posterior$sigmamu2[,2], type = "l")
abline(h = sigma2u[2], col = "red")


i <- 10
plot(out$posterior$mu[,i], type = "l", 
     ylim = c(min(out$posterior$mu[,i], mu[i]), max(out$posterior$mu[,i], mu[i])))
abline(h = mu[i], col = "red")

plot(out$posterior$uu[,k] , type = "l", 
     ylim = c(min(out$posterior$uu[,k], uu[k]), max(out$posterior$uu[,k], uu[k])))
abline(h = uu[k], col = "red")
# }