README.md

An R package for estimating Peer Effects Using Partial Network Data

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.

Installation

Requirements

How to install

PartialNetwork package can be installed from this GitHub repos using the install_github function of the devtools package. All the dependencies will also be installed automatically.

library(devtools) 
install_github("ahoundetoungan/PartialNetwork")

Load PartialNetwork

Once the installation is done, PartialNetwork can be loaded as a common package in R.

library(PartialNetwork)

Instrumental Variable procedure

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: ```R

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)


## Bayesian estimator

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.

### Simulate data
```R
# 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)) 
}

Estimate the model on simulated data

# 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")

ARD, Breza et al. (2020)

Simulation procedure

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)
}

Estimate the model on simulated data

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)


ahoundetoungan/PartialNetwork documentation built on Oct. 6, 2020, 1:51 a.m.