homophily.fe: Estimating Network Formation Models with Degree Heterogeneity: the Fixed Effect Approach

Description

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

Usage

homophily.fe(
  network,
  formula,
  data,
  symmetry = FALSE,
  fe.way = 1,
  init = NULL,
  method = c("L-BFGS", "Block-NRaphson", "Mix"),
  ctr = list(maxit.opt = 10000, maxit.nr = 50, eps_f = 1e-09, eps_g = 1e-09, tol = 1e-04),
  print = TRUE
)

Value

A list consisting of:

model.info: A list of model information, such as the type of fixed effects, whether the model is symmetric, the number of observations, etc.
estimate: The maximizer of the log-likelihood.
loglike: The maximized log-likelihood.
optim: The returned value from the optimization solver, which contains details of the optimization. The solver used is optim_lbfgs from the RcppNumerical package.
init: The returned list of starting values.
loglike.init: The log-likelihood at the starting values.

Arguments

network: A matrix or list of sub-matrices 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, for example, ~ x1 + x2, where x1 and x2 are explanatory variables for link 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, a K-dimensional vector of the explanatory variables' parameters, 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 values for c(beta, mu, nu).
method: A character string specifying the optimization method. Expected values are "L-BFGS", "Block-NRaphson", or "Mix". "Block-NRaphson" refers to the Newton-Raphson method applied to each subnetwork, and "Mix" combines the Newton-Raphson method for beta with the L-BFGS method for the fixed effects.
ctr: (optional) A list containing control parameters for the solver. For the optim_lbfgs method from the RcppNumerical package, the list should include maxit.opt (corresponding to maxit for the L-BFGS method), eps_f, and eps_g. For the Block-NRaphson method, the list should include maxit.nr (corresponding to maxit for the Newton-Raphson method) and tol.
print: A boolean indicating if the estimation progression should be printed.

Details

Let $p_{ij}$ be the probability for a link to go from individual $i$ to individual $j$. This probability is specified for two-way effect models (fe.way = 2) as $$p_{ij} = F(\mathbf{x}_{ij}'\beta + \mu_i + \nu_j),$$ where $F$ is the cumulative distribution function of the standard logistic distribution. Unobserved degree heterogeneity is captured by $\mu_i$ and $\nu_j$. These 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 necessary but not implemented in this version. However, the estimators 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.

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, tools:::Rd_expr_doi("https://doi.org/10.1080/01621459.2018.1448829").

Examples

Run this code

# \donttest{
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)
# }

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