homophily.re: Estimating Network Formation Models with Degree...

In CDatanet: Econometrics of Network Data

Estimating Network Formation Models with Degree Heterogeneity: the Bayesian Random Effect Approach

Description

homophily.re implements a Bayesian Probit estimator for network formation model with homophily. The model includes degree heterogeneity using random effects (see details).

Usage

homophily.re(
  network,
  formula,
  data,
  symmetry = FALSE,
  group.fe = FALSE,
  re.way = 1,
  init = list(),
  iteration = 1000,
  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 variables for 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 homophily is called.
symmetry	indicates whether the network model is symmetric (see details).
group.fe	indicates whether the model includes group fixed effects.
re.way	indicates whether it is a one-way or two-way random effect model. The expected value is 1 or 2 (see details).
init	(optional) list of starting values containing beta, a K-dimensional vector of the explanatory variables parameter, mu, an n-dimensional vector, and nu, an n-dimensional vector, smu2 the variance of mu, and snu2 the variance of nu, where K is the number of explanatory variables and n is the number of individuals.
iteration	the number of iterations to be performed.
print	boolean indicating if the estimation progression should be printed.

Details

Let p_{ij} be a probability for a link to go from the individual i to the individual j. This probability is specified for two-way effect models (re.way = 2) as

p_{ij} = F(\mathbf{x}_{ij}'\beta + \mu_i + \nu_j),

where F is the cumulative of the standard normal distribution. Unobserved degree heterogeneity is captured by \mu_i and \nu_j. The latter are treated as random effects (see homophily.fe for fixed effect models).
For one-way random effect models (re.way = 1), \nu_j = \mu_j. For symmetric models, the network is not directed and the random effects need to be one way.

Value

A list consisting of:

model.info	list of model information, such as the type of random effects, whether the model is symmetric, number of observations, etc.
posterior	list of simulations from the posterior distribution.
init	returned list of starting values.

See Also

homophily.fe.

Examples

set.seed(1234)
library(MASS)
M            <- 4 # Number of sub-groups
nvec         <- round(runif(M, 100, 500))
beta         <- c(.1, -.1)
Glist        <- list()
dX           <- matrix(0, 0, 2)
mu           <- list()
nu           <- list()
cst          <- runif(M, -1.5, 0)
smu2         <- 0.2
snu2         <- 0.2
rho          <- 0.8
Smunu        <- matrix(c(smu2, rho*sqrt(smu2*snu2), rho*sqrt(smu2*snu2), snu2), 2)
for (m in 1:M) {
  n          <- nvec[m]
  tmp        <- mvrnorm(n, c(0, 0), Smunu)
  mum        <- tmp[,1] - mean(tmp[,1])
  num        <- tmp[,2] - mean(tmp[,2])
  X1         <- rnorm(n, 0, 1)
  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*((cst[m] + Z1*beta[1] + Z2*beta[2] +
                       kronecker(mum, t(num), "+") + rnorm(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
  nu[[m]]      <- num
}

mu  <- unlist(mu)
nu  <- unlist(nu)

out   <- homophily.re(network =  Glist, formula = ~ dX, group.fe = TRUE, 
                      re.way = 2, iteration = 1e3)

# plot simulations
plot(out$posterior$beta[,1], type = "l")
abline(h = cst[1], col = "red")
plot(out$posterior$beta[,2], type = "l")
abline(h = cst[2], col = "red")
plot(out$posterior$beta[,3], type = "l")
abline(h = cst[3], col = "red")
plot(out$posterior$beta[,4], type = "l")
abline(h = cst[4], col = "red")

plot(out$posterior$beta[,5], type = "l")
abline(h = beta[1], col = "red")
plot(out$posterior$beta[,6], type = "l")
abline(h = beta[2], col = "red")

plot(out$posterior$sigma2_mu, type = "l")
abline(h = smu2, col = "red")
plot(out$posterior$sigma2_nu, type = "l")
abline(h = snu2, col = "red")
plot(out$posterior$rho, type = "l")
abline(h = rho, col = "red")

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

CDatanet documentation built on April 3, 2025, 11:07 p.m.