The PartialNetwork package includes all functions for the replication of the results in Boucher and Houndetoungan (2020). The exact replication codes are located in the folder test. Below, we also provide detailed examples on how to use the estimators described in the paper.
- PartialNetwork package needs R version 3.0.0 or later which can be installed on Linux, Mac and Windows. See CRAN for installation details.
- devtools package should be installed on R. If not already done, install devtools using the code
install.packages("devtools"). - (Only for windows users) Windows users should install Rtools compatible with their R version.
PartialNetwork package can be installed from this GitHub repos using the install_github function of the devtools package. All the denpendencies will also be installed automatically.
library(devtools)
install_github("ahoundetoungan/PartialNetwork")Once the installation is done, PartialNetwork can be loaded as a common package in R.
library(PartialNetwork)We provide the function sim.IV(dnetwork, X, y, replication, power) where dnetwork is the network linking probabilities, X is a matrix of covariates, y (optional) is the vector of outcome, replication (optional, default = 1) is the number of replication, and power (optional, default = 1) is the number of powers of the interaction matrix used to generate the instruments. The function outputs a proxy for Gy and simulated instruments. See the help file (? sim.IV) of the function for details. The following code provides an example using a single group of network.
# initialize parameters
# number of groups
M <- 5
# size of the group
N <- 100
# value of lambda (precision parameter for the network formation model)
lambda <- 1
# individual effects
beta <- c(2, 1, 1.5)
# contextual effects
gamma <- c(5, -3)
# endogenous effect
alpha <- 0.4
# std-dev errors
se <- 1
# network probabilities
Prob <- list()
for (m in 1:M) {
# heterogeneity of the linking probabilities
c <- rnorm(N*N, 0, 1)
# generate linking probabilities
Pm <- matrix(exp(c / lambda) / (1 + exp(c / lambda)), N)
# no self-link
diag(Pm) <- 0
Prob[[m]] <- Pm
}
# generate data
X <- matrix(data = NA, nrow = 0, ncol = 2)
y <- c()
for (m in 1:M) {
# generate the 'observed network'
Gm <- sim.network(Prob[[m]])
rs <- rowSums(Gm)
rs[rs == 0] <- 1
# row-normalize
Wm <- Gm/rs
# covariates
Xm <- cbind(rnorm(N,0,5),rpois(N,6))
# endogenous variable, no contextual effect
ym <- solve(diag(N) - alpha * Wm) %*% (cbind(rep(1, N), Xm) %*% beta + rnorm(N,0,se))
y <- c(y, ym)
X <- rbind(X, Xm)
}
# generate instruments
instr <- sim.IV(Prob, X, y, replication = 1, power = 2)
GY1c1 <- instr[[1]]$G1y # proxy for Gy (draw 1)
GXc1 <- instr[[1]]$G1X[,,1] # proxy for GX (draw 1)
G2Xc1 <- instr[[1]]$G1X[,,2] # proxy for GGX (draw 1)
GXc2 <- instr[[1]]$G2X[,,1] # proxy for GX (draw 2)
G2Xc2 <- instr[[1]]$G2X[,,2] # proxy for GGX (draw 2)Once the instruments are generated, the estimation can be performed using standard tools, e.g. the function ivreg from the AER package. For example:
# build dataset
# keep only instrument constructed using a different draw than the one used to proxy Gy
dataset <- as.data.frame(cbind(y,X,GY1c1,GXc2,G2Xc2))
# rename variables
colnames(dataset) <- c("y","X1","X2","Gy1","Z1","Z2","ZZ1","ZZ2")
library(AER)
results <- ivreg(y ~ X1 + X2 + Gy1 | X1 + X2 + Z1 + Z2 + ZZ1 + ZZ2, data = dataset)
summary(results)The Bayesian estimator is neatly packed in the function mcmcSAR(formula, contextual, start, G0, hyperparms, iteration, ctrl.mcmc, data), where formula is the model equation, contextual indicates if the model has contextual effects, start (optional) is the parameter initialization, G0 (optional) is the starting of the network, hyperparam specify the prior distributions, including the network linking probabilities, iterations (optional, default = 2000) is the number of MCMC steps to be performed, ctrl.mcmc set some controls for the MCMC and data contains the data (if not specified R will search the variables in the global environment). See the help (? mcmcSAR) file for a complete description. Below, we provide a simple example using simulated data.
# Number of groups
M <- 100
# size of each group
N <- rep(50,M)
# precision parameter for the network formation process
lambda <- 1
G <- list()
# individual effects
beta <- c(2,1,1.5)
# contextual effects
gamma <- c(5,-3)
# endogenous effect
alpha <- 0.4
# std-dev errors
se <- 1
prior <-list()
## generate network probabilities
for (m in 1:M) {
Nm <- N[m]
c <- rnorm(Nm*Nm,0,1)
# linking probabilities
Prob <- matrix(exp(c/lambda)/(1+exp(c/lambda)),Nm)
# no self-link
diag(Prob) <- 0
prior[[m]] <-Prob
}
## generate data
# covariates
X <- cbind(rnorm(sum(N),0,5),rpois(sum(N),7))
# dependent variable
y <- c()
for (m in 1:M) {
Nm <- N[m]
# true network
Gm <- matrix(runif(Nm^2),Nm,Nm) < prior[[m]]
# no self-link
diag(Gm) <- rep(0,Nm)
G[[m]] <- Gm
rsm <- rowSums(Gm)
rsm[rsm==0] <- 1
# normalize
Gm <- Gm/rsm
# rows index of group m
r2 <- sum(N[1:m])
r1 <- r2 - Nm + 1
# contextual effect
Xm <- X[r1:r2,]
GXm <- Gm %*% Xm
y[r1:r2] <- solve(diag(Nm)-alpha*Gm) %*% (cbind(rep(1,Nm),Xm) %*% beta + GXm %*% gamma + rnorm(Nm,0,se))
}# set the hyperparameter
# the hyperparameter is a list
hyperparms <- list("dnetwork" = prior)
# launch the MCMC
ctrl <- list(print.level = 2)
out <- mcmcSAR(y ~ X | X, hyperparms = hyperparms, ctrl.mcmc = ctrl)
# the packahe has summary and plot methods of class S3
# summary
summary(out)
# plot simulations
plot(out)
# plot density functions
plot(out, plot.type = "dens")The data is simulated following a procedure similar to the one in Breza et al. (2020). One notable exception is how we attribute traits to individuals. The code uses the functions rvMF (random variable generator von Mises-Fisher (vMF)), dvMF (density vMF), CpvMF (Normalizing constant, vMF), sim.dnetwork (linking probabilities) and sim.network (drawn network from probabilities). See the associated help files for details. Below, we provide an example using only one group.
# Sample size
N <- 500
# ARD parameters
genzeta <- 1
mu <- -1.35
sigma <- 0.37
K <- 12 # number of traits
P <- 3 # Sphere dimension
# Generate z (spherical coordinates)
genz <- rvMF(N,rep(0,P))
# Generate nu from a Normal distribution with parameters mu and sigma (The gregariousness)
gennu <- rnorm(N,mu,sigma)
# compute degrees
gend <- N*exp(gennu)*exp(mu+0.5*sigma^2)*exp(logCpvMF(P,0) - logCpvMF(P,genzeta))
# Link probabilities
Probabilities <- sim.dnetwork(gennu,gend,genzeta,genz)
# Adjacency matrix
G <- sim.network(Probabilities)
# Generate vk, the trait location
genv <- rvMF(K,rep(0,P))
# set fixed some vk distant
genv[1,] <- c(1,0,0)
genv[2,] <- c(0,1,0)
genv[3,] <- c(0,0,1)
# eta, the intensity parameter
geneta <-abs(rnorm(K,2,1))
# Build traits matrix
densityatz <- matrix(0,N,K)
for(k in 1:K){
densityatz[,k] <- dvMF(genz,genv[k,]*geneta[k])
}
trait <- matrix(0,N,K)
for(k in 1:K){
trait[,k] <- densityatz[,k]>sort(densityatz[,k],decreasing = T)[runif(1,0.05*N,0.25*N)]
}
# print a percentage of people having a trait
colSums(trait)*100/N
# Build ADR
ARD <- G %*% trait
# generate b
genb <- numeric(K)
for(k in 1:K){
genb[k] <- sum(G[,trait[,k]==1])/sum(G)
}
We present a simple function wrapping, mcmcARD, for the estimation procedure proposed by Breza et al. (2020). For specific information on the function, see the help file.
# initialization
d0 <- exp(rnorm(N)); b0 <- exp(rnorm(K)); eta0 <- rep(1,K);
zeta0 <- 05; z0 <- matrix(rvMF(N,rep(0,P)),N); v0 <- matrix(rvMF(K,rep(0,P)),K)
# We should fix some vk and bk
vfixcolumn <- 1:5
bfixcolumn <- c(3, 5)
b0[bfixcolumn] <- genb[bfixcolumn]
v0[vfixcolumn,] <- genv[vfixcolumn,]
start <- list("z" = z0, "v" = v0, "d" = d0, "b" = b0, "eta" = eta0, "zeta" = zeta0)
# MCMC
out <- mcmcARD(Y = ARD, traitARD = trait, start = start, fixv = vfixcolumn, consb = bfixcolumn, iteration = 5000)
# fit network distribution using the estimates from mcmcARD
dist <- fit.dnetwork(out, traitARD = trait)