homophily.fe {CDatanet} | R Documentation |

## Estimating network formation models with degree heterogeneity: the fixed effect approach

### Description

`homophily.fe`

implements a Logit estimator for 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,
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 |

`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 |

`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 |

`opt.ctr` |
(optional) is a list of |

`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 |

`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

### 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)
```

*CDatanet*version 2.2.0 Index]