homophily.fe {CDatanet}

Estimate Network Formation Model with Degree Heterogeneity as Fixed Effects

Description

homophily.fe implements a Logit estimator for network formation model with homophily. The model includes degree heterogeneity as fixed effects (see details).

Usage

homophily.fe(
  network,
  formula,
  data,
  symmetry = FALSE,
  fe.way = 1,
  init = NULL,
  opt.ctr = list(maxit = 10000, eps_f = 1e-09, eps_g = 1e-09),
  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. If missing, the model is estimated with fixed effects only.

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).

fe.way

indicates whether it is a one-way or two-way fixed effect model. The expected value is 1 or 2 (see details).

init

(optional) either a list of starting values containing beta, an K-dimensional vector of the explanatory variables parameter, mu an n-dimensional vector, and nu an n-dimensional vector, where K is the number of explanatory variables and n is the number of individuals; or a vector of starting value for c(beta, mu, nu).

opt.ctr

(optional) is a list of maxit, eps_f, and eps_g, which are control parameters used by the solver optim_lbfgs, of the package RcppNumerical.

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 (fe.way = 2) as

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

where F is the cumulative of the standard logistic distribution. Unobserved degree heterogeneity is captured by \mu_i and \nu_j. The latter are treated as fixed effects (see homophily.re for random effect models). As shown by Yan et al. (2019), the estimator of the parameter \beta is biased. A bias correction is then necessary and is not implemented in this version. However the estimator of \mu_i and \nu_j are consistent.
For one-way fixed effect models (fe.way = 1), \nu_j = \mu_j. For symmetric models, the network is not directed and the fixed effects need to be one way.

Value

A list consisting of:

model.info

list of model information, such as the type of fixed effects, whether the model is symmetric, number of observations, etc.

estimate

maximizer of the log-likelihood.

loglike

maximized log-likelihood.

optim

returned value of the optimization solver, which contains details of the optimization. The solver used is optim_lbfgs of the package RcppNumerical.

init

returned list of starting value.

loglike(init)

log-likelihood at the starting value.

References

Yan, T., Jiang, B., Fienberg, S. E., & Leng, C. (2019). Statistical inference in a directed network model with covariates. Journal of the American Statistical Association, 114(526), 857-868, doi:10.1080/01621459.2018.1448829.

See Also

homophily.re.

Examples

set.seed(1234)
M            <- 2 # Number of sub-groups
nvec         <- round(runif(M, 20, 50))
beta         <- c(.1, -.1)
Glist        <- list()
dX           <- matrix(0, 0, 2)
mu           <- list()
nu           <- list()
Emunu        <- runif(M, -1.5, 0) #expectation of mu + nu
smu2         <- 0.2
snu2         <- 0.2
for (m in 1:M) {
  n          <- nvec[m]
  mum        <- rnorm(n, 0.7*Emunu[m], smu2)
  num        <- rnorm(n, 0.3*Emunu[m], snu2)
  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*((Z1*beta[1] + Z2*beta[2] +
                       kronecker(mum, t(num), "+") + 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
  nu[[m]]      <- num
}

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

out   <- homophily.fe(network =  Glist, formula = ~ -1 + dX, fe.way = 2)
muhat <- out$estimate$mu
nuhat <- out$estimate$nu
plot(mu, muhat)
plot(nu, nuhat)