sim.IV: Instrument Variables for SAR model

Description

sim.IV generates Instrument Variables (IV) for linear-in-mean SAR models using only the distribution of the network. See Propositions 1 and 2 of Boucher and Houndetoungan (2020).

Usage

sim.IV(
  dnetwork,
  X,
  y = NULL,
  replication = 1L,
  power = 1L,
  exp.network = FALSE
)

Value

list of replication components. Each component is a list containing G1y (if the argument y was provided), G1 (if exp.network = TRUE), G2 (if exp.network = TRUE) , G1X, and G2X where G1 and G2 are independent draws of network from the distribution (see details).

G1y: is an approximation of \(Gy\).
G1X: is an approximation of \(G^pX\) with the same network draw as that used in G1y. G1X is an array of dimension \(N \times K \times power\), where \(K\) is the number of column in X. For any \(p \in \{1, 2, ..., power\}\), the approximation of \(G^pX\) is given by G1X[,,p].
G2X: is an approximation of \(G^pX\) with a different different network. G2X is an array of dimension \(N \times K \times power\). For any \(p \in \{1, 2, ..., power\}\), the approximation of \(G^pX\) is given by G2X[,,p].

Arguments

dnetwork: network matrix of list of sub-network matrices, where the (i, j)-th position is the probability that i be connected to j.
X: matrix of the individual observable characteristics.
y: (optional) the endogenous variable as a vector.
replication: (optional, default = 1) is the number of repetitions (see details).
power: (optional, default = 1) is the number of powers of the interaction matrix used to generate the instruments (see details).
exp.network: (optional, default = FALSE) indicates if simulated network should be exported.

Details

Bramoulle et al. (2009) show that one can use \(GX\), \(G^2X\), ..., \(G^P X\) as instruments for \(Gy\), where \(P\) is the maximal power desired. sim.IV generate approximation of those instruments, based on Propositions 1 and 2 in Boucher and Houndetoungan (2020) (see also below). The argument power is the maximal power desired.
When \(Gy\) and the instruments \(GX\), \(G^2X\), ..., \(G^P X\) are not observed, Boucher and Houndetoungan (2022) show that we can use one drawn from the distribution of the network in order to approximate \(Gy\), but that the same draw should not be used to approximate the instruments. Thus, each component in the function's output gives G1y and G1X computed with the same network and G2X computed with another network, which can be used in order to approximate the instruments. This process can be replicated several times and the argument replication can be used to set the number of replications desired.

Examples

Run this code

if (requireNamespace("ivreg", quietly = TRUE)) {
  library(ivreg)
  # Number of groups
  M             <- 30
  # size of each group
  N             <- rep(50,M)
  # individual effects
  beta          <- c(2,1,1.5)
  # endogenous effects
  alpha         <- 0.4
  # std-dev errors
  se            <- 2
  # prior distribution
  prior         <- runif(sum(N*(N-1)))
  prior         <- vec.to.mat(prior, N, normalise = FALSE)
  # covariates
  X             <- cbind(rnorm(sum(N),0,5),rpois(sum(N),7))
  # true network
  G0            <- sim.network(prior)
  # normalise
  G0norm        <- norm.network(G0)
  # simulate dependent variable use an external package
  y             <- simSAR(~ X, Glist = G0norm, parms = c(alpha, beta),
                          epsilon = rnorm(sum(N), sd = se))
  y             <- y$y
  # generate instruments
  instr         <- sim.IV(prior, X, y, replication = 1, power = 1)
  
  GY1c1         <- instr[[1]]$G1y       # proxy for Gy (draw 1)
  GXc1          <- instr[[1]]$G1X[,,1]  # proxy for GX (draw 1)
  GXc2          <- instr[[1]]$G2X[,,1]  # proxy for GX (draw 2)
  # 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, GXc1, GXc2))
  colnames(dataset) <- c("y","X1","X2","G1y", "G1X1", "G1X2", "G2X1", "G2X2")
  
  # Same draws
  out.iv1           <- ivreg(y ~ X1 + X2 + G1y | X1 + X2 + G1X1 + G1X2, data = dataset)
  summary(out.iv1)
  
  # Different draws
  out.iv2           <- ivreg(y ~ X1 + X2 + G1y | X1 + X2 + G2X1 + G2X2, data = dataset)
  summary(out.iv2)
} else {
  message("Suggested package 'ivreg' not available, skipping example.")
}