Package 'QuantilePeer' reference manual

Title:	Quantile Peer Effect Models
Description:	Simulating and estimating peer effect models including the quantile-based specification (Houndetoungan, 2025 <doi:10.48550/arXiv.2506.12920>), and the models with Constant Elasticity of Substitution (CES)-based social norm (Boucher et al., 2024 <doi:10.3982/ECTA21048>).
Authors:	Aristide Houndetoungan [cre, aut] (ORCID: <https://orcid.org/0009-0003-2733-9362>)
Maintainer:	Aristide Houndetoungan <[email protected]>
License:	GPL-3
Version:	0.0.2
Built:	2026-06-01 09:45:36 UTC
Source:	https://github.com/ahoundetoungan/quantilepeer

The QuantilePeer package

Description

The QuantilePeer package simulates and estimates peer effect models including the quantile-based specification (Houndetoungan, 2025 doi:10.3982/ECTA21048), and the models with Constant Elasticity of Substitution (CES)-based social norm (Boucher et al., 2024 doi:10.3982/ECTA21048).

Author(s)

Maintainer: Aristide Houndetoungan [email protected] (ORCID)

References

Boucher, V., Rendall, M., Ushchev, P., & Zenou, Y. (2024). Toward a general theory of peer effects. Econometrica, 92(2), 543-565, doi:10.3982/ECTA21048.

Houndetoungan, A. (2025). Quantile peer effect models. arXiv preprint arXiv:2405.17290, doi:10.48550/arXiv.2506.12920.

Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361-365, doi:10.1080/00031305.1996.10473566.

Estimation of CES-Based Peer Effects Models

Description

cespeer estimates the CES-based peer effects model introduced by Boucher et al. (2024). See Details.

Usage

cespeer(
  formula,
  instrument,
  Glist,
  structural = FALSE,
  fixed.effects = FALSE,
  set.rho = NULL,
  grid.rho = seq(-400, 400, radius),
  radius = 5,
  tol = 1e-08,
  drop = NULL,
  compute.cov = TRUE,
  HAC = "iid",
  data
)
cespeer(
  formula,
  instrument,
  Glist,
  structural = FALSE,
  fixed.effects = FALSE,
  set.rho = NULL,
  grid.rho = seq(-400, 400, radius),
  radius = 5,
  tol = 1e-08,
  drop = NULL,
  compute.cov = TRUE,
  HAC = "iid",
  data
)

Arguments

formula

An object of class formula: a symbolic description of the model. formula should be specified as y ~ x1 + x2, where y is the outcome and x1 and x2 are control variables, which can include contextual variables such as averages or quantiles among peers.

instrument

An object of class formula indicating the excluded instrument. It should be specified as ~ z, where z is the excluded instrument for the outcome. Following Boucher et al. (2024), it can be an OLS exogenous prediction of y. This prediction is used to compute instruments for the CES function of peer outcomes.

Glist

The adjacency matrix. For networks consisting of multiple subnets (e.g., schools), Glist must be a list of subnets, with the m-th element being an $n_m \times n_m$ adjacency matrix, where $n_m$ is the number of nodes in the m-th subnet.

structural

A logical value indicating whether the reduced-form or structural specification should be estimated (see details).

fixed.effects

A logical value or string specifying whether the model includes subnet fixed effects. The fixed effects may differ between isolated and non-isolated nodes. Accepted values are "no" or "FALSE" (indicating no fixed effects), "join" or TRUE (indicating the same fixed effects for isolated and non-isolated nodes within each subnet), and "separate" (indicating different fixed effects for isolated and non-isolated nodes within each subnet). Note that "join" fixed effects are not applicable for structural models; "join" and TRUE are automatically converted to "separate".

set.rho

A fixed value for the CES substitution parameter to estimate a constrained model. Given this value, the other parameters can be estimated.

grid.rho

A finite grid of values for the CES substitution parameter $\rho$ (see Details). This grid is used to obtain the starting value and define the GMM weight. It is recommended to use a finely subdivided grid.

radius

The radius of the subset in which the estimate for $\rho$ is determined. The subset is a segment centered at the optimal $\rho$ found using grid.rho. For better numerical optimization performance, use a finely subdivided grid.rho and a small radius.

tol

A tolerance value used in the QR factorization to identify columns of explanatory variable and instrument matrices that ensure a full-rank matrix (see the qr function). The same tolerance is also used in the to minimize the concentrated GMM objective function (see optimise).

drop

A dummy vector of the same length as the sample, indicating whether an observation should be dropped. This can be used, for example, to remove false isolates or to estimate the model only on non-isolated agents. These observations cannot be directly removed from the network by the user because they may still be friends with other agents.

compute.cov

A logical value indicating whether the covariance matrix of the estimator should be computed.

HAC

A character string specifying the correlation structure among the idiosyncratic error terms for covariance computation. Options are "iid" for independent errors, "hetero" for heteroskedastic non-autocorrelated errors, and "cluster" for heteroskedastic errors with potential within-subnet correlation.

data

An optional data frame, list, or environment (or an object that can be coerced 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 cespeer is called.

Details

Let $\mathcal{N}$ be a set of $n$ agents indexed by the integer $i \in [1, n]$ . Agents are connected through a network characterized by an adjacency matrix $\mathbf{G} = [g_{ij}]$ of dimension $n \times n$ , where $g_{ij} = 1$ if agent $j$ is a friend of agent $i$ , and $g_{ij} = 0$ otherwise. In weighted networks, $g_{ij}$ can be a nonnegative variable (not necessarily binary) that measures the intensity of the outgoing link from $i$ to $j$ . The model can also accommodate such networks. Note that the network generally consists of multiple independent subnets (e.g., schools). The Glist argument is the list of subnets. In the case of a single subnet, Glist should be a list containing one matrix.

The reduced-form specification of the CES-based peer effects model is given by:

$y_i = \lambda\left(\sum_{j = 1}^n g_{ij}y_j^{\rho}\right)^{1/\rho} + \mathbf{x}_i^{\prime}\beta + \varepsilon_i,$

where $\varepsilon_i$ is an idiosyncratic error term, $\lambda$ captures the effect of the social norm $\left(\sum_{j = 1}^n g_{ij}y_j^{\rho}\right)^{1/\rho}$ , and $\beta$ captures the effect of $\mathbf{x}_i$ on $y_i$ . The parameter $\rho$ determines the form of the social norm in the model.

When $\rho > 1$ , individuals are more sensitive to peers with high outcomes.
When $\rho < 1$ , individuals are more sensitive to peers with low outcomes.
When $\rho = 1$ , peer effects are uniform across peer outcome values.

The structural specification of the model differs for isolated and non-isolated individuals. For an isolated $i$ , the specification is similar to a standard linear-in-means model without social interactions, given by:

$y_i = \mathbf{x}_i^{\prime}\beta + \varepsilon_i.$

If node $i$ is non-isolated, the specification is:

$y_i = \lambda\left(\sum_{j = 1}^n g_{ij}y_j^{\rho}\right)^{1/\rho} + (1 - \lambda_2)\mathbf{x}_i^{\prime}\beta + \varepsilon_i,$

where $\lambda_2$ determines whether preferences exhibit conformity or complementarity/substitution. Identification of $\beta$ and $\lambda_2$ requires the network to include a sufficient number of isolated individuals.

Value

A list containing:

model.info

A list with information about the model, including the number of subnets, the number of observations, and other key details.

gmm

A list of GMM estimation results, including parameter estimates, the covariance matrix, and related statistics.

first.search

A list containing initial estimations on the grid of values for $\rho$ .

References

Boucher, V., Rendall, M., Ushchev, P., & Zenou, Y. (2024). Toward a general theory of peer effects. Econometrica, 92(2), 543-565, doi:10.3982/ECTA21048.

Examples


set.seed(123)
ngr  <- 50  # Number of subnets
nvec <- rep(30, ngr)  # Size of subnets
n    <- sum(nvec)

### Simulating Data
## Network matrix
G <- lapply(1:ngr, function(z) {
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z], nvec[z])
  diag(Gz) <- 0
  # Adding isolated nodes (important for the structural model)
  niso <- sample(0:nvec[z], 1, prob = (nvec[z] + 1):1 / sum((nvec[z] + 1):1))
  if (niso > 0) {
    Gz[sample(1:nvec[z], niso), ] <- 0
  }
  # Row-normalization
  rs   <- rowSums(Gz); rs[rs == 0] <- 1
  Gz/rs
})

X   <- cbind(rnorm(n), rpois(n, 2))
l   <- 0.55
b   <- c(2, -0.5, 1)
rho <- -2
eps <- rnorm(n, 0, 0.4)

## Generating `y`
y <- cespeer.sim(formula = ~ X, Glist = G, rho = rho, lambda = l,
                 beta = b, epsilon = eps)$y

### Estimation
## Computing instruments
z <- fitted.values(lm(y ~ X))

## Reduced-form model
rest <- cespeer(formula = y ~ X, instrument = ~ z, Glist = G, fixed.effects = "yes",
                radius = 5, grid.rho = seq(-10, 10, 1))
summary(rest)

## Structural model
sest <- cespeer(formula = y ~ X, instrument = ~ z, Glist = G, fixed.effects = "yes",
                radius = 5, structural = TRUE, grid.rho = seq(-10, 10, 1))
summary(sest)

## Quantile model
z    <- qpeer.inst(formula = ~ X, Glist = G, tau = seq(0, 1, 0.1), max.distance = 2, 
                   checkrank = TRUE)$instruments
qest <- qpeer(formula = y ~ X, excluded.instruments  = ~ z, Glist = G, 
              fixed.effects = "yes", tau = seq(0, 1, 1/3), structural = TRUE)
summary(qest)

set.seed(123)
ngr  <- 50  # Number of subnets
nvec <- rep(30, ngr)  # Size of subnets
n    <- sum(nvec)

### Simulating Data
## Network matrix
G <- lapply(1:ngr, function(z) {
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z], nvec[z])
  diag(Gz) <- 0
  # Adding isolated nodes (important for the structural model)
  niso <- sample(0:nvec[z], 1, prob = (nvec[z] + 1):1 / sum((nvec[z] + 1):1))
  if (niso > 0) {
    Gz[sample(1:nvec[z], niso), ] <- 0
  }
  # Row-normalization
  rs   <- rowSums(Gz); rs[rs == 0] <- 1
  Gz/rs
})

X   <- cbind(rnorm(n), rpois(n, 2))
l   <- 0.55
b   <- c(2, -0.5, 1)
rho <- -2
eps <- rnorm(n, 0, 0.4)

## Generating `y`
y <- cespeer.sim(formula = ~ X, Glist = G, rho = rho, lambda = l,
                 beta = b, epsilon = eps)$y

### Estimation
## Computing instruments
z <- fitted.values(lm(y ~ X))

## Reduced-form model
rest <- cespeer(formula = y ~ X, instrument = ~ z, Glist = G, fixed.effects = "yes",
                radius = 5, grid.rho = seq(-10, 10, 1))
summary(rest)

## Structural model
sest <- cespeer(formula = y ~ X, instrument = ~ z, Glist = G, fixed.effects = "yes",
                radius = 5, structural = TRUE, grid.rho = seq(-10, 10, 1))
summary(sest)

## Quantile model
z    <- qpeer.inst(formula = ~ X, Glist = G, tau = seq(0, 1, 0.1), max.distance = 2, 
                   checkrank = TRUE)$instruments
qest <- qpeer(formula = y ~ X, excluded.instruments  = ~ z, Glist = G, 
              fixed.effects = "yes", tau = seq(0, 1, 1/3), structural = TRUE)
summary(qest)

Compute the CES Social Norm

Description

cespeer.data computes the CES social norm, along with the first and second derivatives of the CES social norm with respect to the substitution parameter $\rho$ .

Usage

cespeer.data(y, Glist, rho)
cespeer.data(y, Glist, rho)

Arguments

y

A vector of outcomes used to compute the social norm.

Glist

rho

The CES substitution parameter.

Value

A four-column matrix with the following columns:

y

The outcome;

ces(y, rho)

The CES social norm;

d[ces(y, rho)]

The first derivative of the social norm;

dd[ces(y, rho)]

The second derivative of the social norm.

Simulating Peer Effect Models with a CES Social Norm

Description

cespeer.sim simulates peer effect models with a Constant Elasticity of Substitution (CES) based social norm (Boucher et al., 2024).

Usage

cespeer.sim(
  formula,
  Glist,
  parms,
  rho,
  lambda,
  beta,
  epsilon,
  structural = FALSE,
  init,
  tol = 1e-10,
  maxit = 500,
  data
)
cespeer.sim(
  formula,
  Glist,
  parms,
  rho,
  lambda,
  beta,
  epsilon,
  structural = FALSE,
  init,
  tol = 1e-10,
  maxit = 500,
  data
)

Arguments

formula

A formula object (formula): a symbolic description of the model. formula should be specified as, for example, ~ x1 + x2, where x1 and x2 are control variables, which can include contextual variables such as averages or quantiles among peers.

Glist

parms

A vector defining the true values of $(\rho, \lambda', \beta')'$ , where $\rho$ is the substitution parameter of the CES function and $\lambda$ is either the peer effect parameter for the reduced-form specification or a 2-vector with the first component being conformity peer effects and the second component representing total peer effects. The parameters $\rho$ , $\lambda$ , and $\beta$ can also be specified separately using the arguments rho, lambda, and beta (see the Details section of cespeer).

rho

The true value of the substitution parameter of the CES function.

lambda

The true value of the peer effect parameter $\lambda$ . It must include conformity and total peer effects for the structural model.

beta

The true value of the vector $\beta$ .

epsilon

A vector of idiosyncratic error terms. If not specified, it will be simulated from a standard normal distribution (see the model specification in the Details section of cespeer).

structural

A logical value indicating whether simulations should be performed using the structural model. The default is the reduced-form model (see the Details section of cespeer).

init

An optional initial guess for the equilibrium.

tol

The tolerance value used in the Fixed Point Iteration Method to compute the outcome y. The process stops if the $\ell_1$ -distance between two consecutive values of y is less than tol.

maxit

The maximum number of iterations for the Fixed Point Iteration Method.

data

An optional data frame, list, or environment containing the model variables. If a variable is not found in data, it is retrieved from environment(formula), typically the environment from which cespeer.sim is called.

Value

A list containing:

y

The simulated variable.

epsilon

The idiosyncratic error.

init

The initial guess.

iteration

The number of iterations before convergence.

References

Boucher, V., Rendall, M., Ushchev, P., & Zenou, Y. (2024). Toward a general theory of peer effects. Econometrica, 92(2), 543-565, doi:10.3982/ECTA21048.

Examples

set.seed(123)
ngr  <- 50
nvec <- rep(30, ngr)
n    <- sum(nvec)
G    <- lapply(1:ngr, function(z){
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z])
  diag(Gz) <- 0
  Gz/rowSums(Gz) # Row-normalized network
})
tau  <- seq(0, 1, 0.25)
X    <- cbind(rnorm(n), rpois(n, 2))
l    <- 0.55
rho  <- 3
b    <- c(4, -0.5, 1)

out  <- cespeer.sim(formula = ~ X, Glist = G, rho = rho, lambda = l, beta = b)
summary(out$y)
out$iteration
set.seed(123)
ngr  <- 50
nvec <- rep(30, ngr)
n    <- sum(nvec)
G    <- lapply(1:ngr, function(z){
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z])
  diag(Gz) <- 0
  Gz/rowSums(Gz) # Row-normalized network
})
tau  <- seq(0, 1, 0.25)
X    <- cbind(rnorm(n), rpois(n, 2))
l    <- 0.55
rho  <- 3
b    <- c(4, -0.5, 1)

out  <- cespeer.sim(formula = ~ X, Glist = G, rho = rho, lambda = l, beta = b)
summary(out$y)
out$iteration

Demeaning Variables

Description

demean demeans variables by subtracting the within-subnetwork average. In each subnetwork, this transformation can be performed separately for isolated and non-isolated nodes.

Usage

demean(X, Glist, separate = FALSE, drop = NULL)
demean(X, Glist, separate = FALSE, drop = NULL)

Arguments

X

A matrix or vector to demean.

Glist

separate

A logical value specifying whether variables should be demeaned separately for isolated and non-isolated individuals. This is similar to setting fixed.effects = "separate" in qpeer.

drop

A logical vector of the same length as the sample, indicating whether an observation should be dropped. This can be used, for example, to remove false isolates or to estimate the model only on non-isolated agents. These observations cannot be directly removed from the network by the user, as they may still be connected to other agents.

Value

A matrix or vector with the same dimensions as X, containing the demeaned values.

Estimating Peer Effects Models

Description

qpeer estimates the quantile peer effect models introduced by Houndetoungan (2025). In the linpeer function, quantile peer variables are replaced with the average peer variable, and they can be replaced with other peer variables in the genpeer function.

Usage

genpeer(
  formula,
  excluded.instruments,
  endogenous.variables,
  Glist,
  data,
  estimator = "IV",
  structural = FALSE,
  drop = NULL,
  fixed.effects = FALSE,
  HAC = "iid",
  checkrank = FALSE,
  compute.cov = TRUE,
  boot = 500,
  nthreads = 1,
  tol = 1e-10,
  print = TRUE
)

linpeer(
  formula,
  excluded.instruments,
  Glist,
  data,
  estimator = "IV",
  structural = FALSE,
  drop = NULL,
  fixed.effects = FALSE,
  HAC = "iid",
  checkrank = FALSE,
  compute.cov = TRUE,
  boot = 500,
  nthreads = 1,
  tol = 1e-10,
  print = TRUE
)

qpeer(
  formula,
  excluded.instruments,
  Glist,
  tau,
  type = 7,
  data,
  estimator = "IV",
  structural = FALSE,
  fixed.effects = FALSE,
  HAC = "iid",
  checkrank = FALSE,
  drop = NULL,
  compute.cov = TRUE,
  boot = 500,
  nthreads = 1,
  tol = 1e-10,
  print = TRUE
)
genpeer(
  formula,
  excluded.instruments,
  endogenous.variables,
  Glist,
  data,
  estimator = "IV",
  structural = FALSE,
  drop = NULL,
  fixed.effects = FALSE,
  HAC = "iid",
  checkrank = FALSE,
  compute.cov = TRUE,
  boot = 500,
  nthreads = 1,
  tol = 1e-10,
  print = TRUE
)

linpeer(
  formula,
  excluded.instruments,
  Glist,
  data,
  estimator = "IV",
  structural = FALSE,
  drop = NULL,
  fixed.effects = FALSE,
  HAC = "iid",
  checkrank = FALSE,
  compute.cov = TRUE,
  boot = 500,
  nthreads = 1,
  tol = 1e-10,
  print = TRUE
)

qpeer(
  formula,
  excluded.instruments,
  Glist,
  tau,
  type = 7,
  data,
  estimator = "IV",
  structural = FALSE,
  fixed.effects = FALSE,
  HAC = "iid",
  checkrank = FALSE,
  drop = NULL,
  compute.cov = TRUE,
  boot = 500,
  nthreads = 1,
  tol = 1e-10,
  print = TRUE
)

Arguments

formula

excluded.instruments

An object of class formula to indicate excluded instruments. It should be specified as ~ z1 + z2, where z1 and z2 are excluded instruments for the quantile peer outcomes.

endogenous.variables

An object of class formula that allows specifying endogenous variables. It is used to indicate the peer variables whose effects will be estimated. These can include average peer variables, quantile peer variables, or a combination of multiple variables. It should be specified as ~ y1 + y2, where y1 and y2 are the endogenous peer variables.

Glist

data

estimator

A character string specifying the estimator to be used. The available options are: "IV" for the standard instrumental variable estimator, "gmm.identity" for the GMM estimator with the identity matrix as the weight, "gmm.optimal" for the GMM estimator with the optimal weight matrix, "JIVE" for the Jackknife instrumental variable estimator, and "JIVE2" for the Type 2 Jackknife instrumental variable estimator.

structural

A logical value indicating whether the reduced-form or structural specification should be estimated (see Details).

drop

fixed.effects

HAC

A character string specifying the correlation structure of the idiosyncratic error terms used for covariance computation. Options are "iid" for independent errors, "hetero" for heteroskedastic but non-autocorrelated errors, "cluster" for clustered errors at the subnet level, and "cluster-bootstrap" for clustered errors at the subnet level with bootstrap-based standard error estimation.

checkrank

A logical value indicating whether the instrument matrix should be checked for full rank. If the matrix is not of full rank, unimportant columns will be removed to obtain a full-rank matrix.

compute.cov

A logical value indicating whether the covariance matrix of the estimator should be computed.

boot

An integer indicating the number of bootstrap replications.

nthreads

A strictly positive integer indicating the number of threads to use when computing quantiles of peer variables or when performing the bootstrap.

tol

A tolerance value used in the QR factorization to identify columns of explanatory variable and instrument matrices that ensure a full-rank matrix (see the qr function).

print

A Boolean indicating whether the progression bar should be printed for bootstrap methods.

tau

A numeric vector specifying the quantile levels.

type

An integer between 1 and 9 selecting one of the nine quantile algorithms used to compute peer quantiles (see the quantile function).

Details

Let $\mathcal{N}$ be a set of $n$ agents indexed by the integer $i \in [1, n]$ . Agents are connected through a network that is characterized by an adjacency matrix $\mathbf{G} = [g_{ij}]$ of dimension $n \times n$ , where $g_{ij} = 1$ if agent $j$ is a friend of agent $i$ , and $g_{ij} = 0$ otherwise. In weighted networks, $g_{ij}$ can be a nonnegative variable (not necessarily binary) that measures the intensity of the outgoing link from $i$ to $j$ . The model can also accommodate such networks. Note that the network is generally constituted in many independent subnets (eg: schools). The Glist argument is the list of subnets. In the case of a single subnet, Glist will be a list containing one matrix.

Let $\mathcal{T}$ be a set of quantile levels. The reduced-form specification of quantile peer effect models is given by:

$y_i = \sum_{\tau \in \mathcal{T}} \lambda_{\tau} q_{\tau,i}(\mathbf{y}_{-i}) + \mathbf{x}_i^{\prime}\beta + \varepsilon_i,$

where $\mathbf{y}_{-i} = (y_1, \ldots, y_{i-1}, y_{i+1}, \ldots, y_n)^{\prime}$ is the vector of outcomes for other units, and $q_{\tau,i}(\mathbf{y}_{-i})$ is the sample $\tau$ -quantile of peer outcomes. The term $\varepsilon_i$ is an idiosyncratic error term, $\lambda_{\tau}$ captures the effect of the $\tau$ -quantile of peer outcomes on $y_i$ , and $\beta$ captures the effect of $\mathbf{x}_i$ on $y_i$ . For the definition of the sample $\tau$ -quantile, see Hyndman and Fan (1996). If the network matrix is weighted, the sample weighted quantile can be used, where the outcome for friend $j$ of $i$ is weighted by $g_{ij}$ . It can be shown that the sample $\tau$ -quantile is a weighted average of two peer outcomes. For more details, see the quantile and qpeer.instruments functions.

The quantile $q_{\tau,i}(\mathbf{y}_{-i})$ can be replaced with the average peer variable in linpeer or with other measures in genpeer through the endogenous.variables argument. In genpeer, it is possible to specify multiple peer variables, such as male peer averages and female peer averages. Additionally, both quantiles and averages can be included (genpeer is general and encompasses qpeer and linpeer). See examples.

One issue in linear peer effect models is that individual preferences with conformity and complementarity/substitution lead to the same reduced form. However, it is possible to disentangle both types of preferences using isolated individuals (individuals without friends). The structural specification of the model differs between isolated and nonisolated individuals. For isolated $i$ , the specification is similar to a standard linear-in-means model without social interactions, given by:

$y_i = \mathbf{x}_i^{\prime}\beta + \varepsilon_i.$

If node $i$ is non-isolated, the specification is given by:

$y_i = \sum_{\tau \in \mathcal{T}} \lambda_{\tau} q_{\tau,i}(\mathbf{y}_{-i}) + (1 - \lambda_2)(\mathbf{x}_i^{\prime}\beta + \varepsilon_i),$

where $\lambda_2$ determines whether preferences exhibit conformity or complementarity/substitution. In general, $\lambda_2 > 0$ and this means that that preferences are conformist (anti-conformity may be possible in some models when $\lambda_2 < 0$ ). In contrast, when $\lambda_2 = 0$ , there is complementarity/substitution between individuals depending on the signs of the $\lambda_{\tau}$ parameters. It is obvious that $\beta$ and $\lambda_2$ can be identified only if the network includes enough isolated individuals.

Value

A list containing:

model.info

A list with information about the model, such as the number of subnets, number of observations, and other key details.

gmm

A list of GMM estimation results, including parameter estimates, the covariance matrix, and related statistics.

data

A list containing the outcome, outcome quantiles among peers, control variables, and excluded instruments used in the model.

References

Houndetoungan, A. (2025). Quantile peer effect models. arXiv preprint arXiv:2405.17290, doi:10.48550/arXiv.2506.12920.

Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361-365, doi:10.1080/00031305.1996.10473566.

Examples


set.seed(123)
ngr  <- 50  # Number of subnets
nvec <- rep(30, ngr)  # Size of subnets
n    <- sum(nvec)

### Simulating Data
## Network matrix
G <- lapply(1:ngr, function(z) {
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z], nvec[z])
  diag(Gz) <- 0
  # Adding isolated nodes (important for the structural model)
  niso <- sample(0:nvec[z], 1, prob = (nvec[z] + 1):1 / sum((nvec[z] + 1):1))
  if (niso > 0) {
    Gz[sample(1:nvec[z], niso), ] <- 0
  }
  Gz
})

tau <- seq(0, 1, 1/3)
X   <- cbind(rnorm(n), rpois(n, 2))
l   <- c(0.2, 0.15, 0.1, 0.2)
b   <- c(2, -0.5, 1)
eps <- rnorm(n, 0, 0.4)

## Generating `y`
y <- qpeer.sim(formula = ~ X, Glist = G, tau = tau, lambda = l, 
               beta = b, epsilon = eps)$y

### Estimation
## Computing instruments
Z <- qpeer.inst(formula = ~ X, Glist = G, tau = seq(0, 1, 0.1), 
                max.distance = 2, checkrank = TRUE)
Z <- Z$instruments

## Reduced-form model 
rest <- qpeer(formula = y ~ X, excluded.instruments = ~ Z, Glist = G, tau = tau)
summary(rest)
summary(rest, diagnostic = TRUE)  # Summary with diagnostics

## Structural model
sest <- qpeer(formula = y ~ X, excluded.instruments = ~ Z, Glist = G, tau = tau,
              structural = TRUE)
summary(sest, diagnostic = TRUE)
# The lambda^* parameter is y_q (conformity) in the outputs.
# There is no conformity in the data, so the estimate will be approximately 0.

## Structural model with double fixed effects per subnet using optimal GMM 
## and controlling for heteroskedasticity
sesto <- qpeer(formula = y ~ X, excluded.instruments = ~ Z, Glist = G, tau = tau,
               structural = TRUE, fixed.effects = "separate", HAC = "hetero", 
               estimator = "gmm.optimal")
summary(sesto, diagnostic = TRUE)

## Average peer effect model
# Row-normalized network to compute instruments
Gnorm <- lapply(G, function(g) {
  d <- rowSums(g)
  d[d == 0] <- 1
  g / d
})

# GX and GGX
Gall <- Matrix::bdiag(Gnorm)
GX   <- as.matrix(Gall %*% X)
GGX  <- as.matrix(Gall %*% GX)

# Standard linear model
lpeer <- linpeer(formula = y ~ X + GX, excluded.instruments = ~ GGX, Glist = Gnorm)
summary(lpeer, diagnostic = TRUE)
# Note: The normalized network is used here by definition of the model.
# Contextual effects are also included (this is also possible for the quantile model).

# The standard model can also be structural
lpeers <- linpeer(formula = y ~ X + GX, excluded.instruments = ~ GGX, Glist = Gnorm,
                  structural = TRUE, fixed.effects = "separate")
summary(lpeers, diagnostic = TRUE)

## Estimation using `genpeer`
# Average peer variable computed manually and included as an endogenous variable
Gy     <- as.vector(Gall %*% y)
gpeer1 <- genpeer(formula = y ~ X + GX, excluded.instruments = ~ GGX, 
                  endogenous.variables = ~ Gy, Glist = Gnorm, structural = TRUE, 
                  fixed.effects = "separate")
summary(gpeer1, diagnostic = TRUE)

# Using both average peer variables and quantile peer variables as endogenous,
# or only the quantile peer variable
# Quantile peer `y`
qy <- qpeer.inst(formula = y ~ 1, Glist = G, tau = tau)
qy <- qy$qy

# Model estimation
gpeer2 <- genpeer(formula = y ~ X + GX, excluded.instruments = ~ GGX + Z, 
                  endogenous.variables = ~ Gy + qy, Glist = Gnorm, structural = TRUE, 
                  fixed.effects = "separate")
summary(gpeer2, diagnostic = TRUE)
set.seed(123)
ngr  <- 50  # Number of subnets
nvec <- rep(30, ngr)  # Size of subnets
n    <- sum(nvec)

### Simulating Data
## Network matrix
G <- lapply(1:ngr, function(z) {
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z], nvec[z])
  diag(Gz) <- 0
  # Adding isolated nodes (important for the structural model)
  niso <- sample(0:nvec[z], 1, prob = (nvec[z] + 1):1 / sum((nvec[z] + 1):1))
  if (niso > 0) {
    Gz[sample(1:nvec[z], niso), ] <- 0
  }
  Gz
})

tau <- seq(0, 1, 1/3)
X   <- cbind(rnorm(n), rpois(n, 2))
l   <- c(0.2, 0.15, 0.1, 0.2)
b   <- c(2, -0.5, 1)
eps <- rnorm(n, 0, 0.4)

## Generating `y`
y <- qpeer.sim(formula = ~ X, Glist = G, tau = tau, lambda = l, 
               beta = b, epsilon = eps)$y

### Estimation
## Computing instruments
Z <- qpeer.inst(formula = ~ X, Glist = G, tau = seq(0, 1, 0.1), 
                max.distance = 2, checkrank = TRUE)
Z <- Z$instruments

## Reduced-form model 
rest <- qpeer(formula = y ~ X, excluded.instruments = ~ Z, Glist = G, tau = tau)
summary(rest)
summary(rest, diagnostic = TRUE)  # Summary with diagnostics

## Structural model
sest <- qpeer(formula = y ~ X, excluded.instruments = ~ Z, Glist = G, tau = tau,
              structural = TRUE)
summary(sest, diagnostic = TRUE)
# The lambda^* parameter is y_q (conformity) in the outputs.
# There is no conformity in the data, so the estimate will be approximately 0.

## Structural model with double fixed effects per subnet using optimal GMM 
## and controlling for heteroskedasticity
sesto <- qpeer(formula = y ~ X, excluded.instruments = ~ Z, Glist = G, tau = tau,
               structural = TRUE, fixed.effects = "separate", HAC = "hetero", 
               estimator = "gmm.optimal")
summary(sesto, diagnostic = TRUE)

## Average peer effect model
# Row-normalized network to compute instruments
Gnorm <- lapply(G, function(g) {
  d <- rowSums(g)
  d[d == 0] <- 1
  g / d
})

# GX and GGX
Gall <- Matrix::bdiag(Gnorm)
GX   <- as.matrix(Gall %*% X)
GGX  <- as.matrix(Gall %*% GX)

# Standard linear model
lpeer <- linpeer(formula = y ~ X + GX, excluded.instruments = ~ GGX, Glist = Gnorm)
summary(lpeer, diagnostic = TRUE)
# Note: The normalized network is used here by definition of the model.
# Contextual effects are also included (this is also possible for the quantile model).

# The standard model can also be structural
lpeers <- linpeer(formula = y ~ X + GX, excluded.instruments = ~ GGX, Glist = Gnorm,
                  structural = TRUE, fixed.effects = "separate")
summary(lpeers, diagnostic = TRUE)

## Estimation using `genpeer`
# Average peer variable computed manually and included as an endogenous variable
Gy     <- as.vector(Gall %*% y)
gpeer1 <- genpeer(formula = y ~ X + GX, excluded.instruments = ~ GGX, 
                  endogenous.variables = ~ Gy, Glist = Gnorm, structural = TRUE, 
                  fixed.effects = "separate")
summary(gpeer1, diagnostic = TRUE)

# Using both average peer variables and quantile peer variables as endogenous,
# or only the quantile peer variable
# Quantile peer `y`
qy <- qpeer.inst(formula = y ~ 1, Glist = G, tau = tau)
qy <- qy$qy

# Model estimation
gpeer2 <- genpeer(formula = y ~ X + GX, excluded.instruments = ~ GGX + Z, 
                  endogenous.variables = ~ Gy + qy, Glist = Gnorm, structural = TRUE, 
                  fixed.effects = "separate")
summary(gpeer2, diagnostic = TRUE)

Simulating Linear Peer Effect Models

Description

linpeer.sim simulates linear peer effect models.

Usage

linpeer.sim(
  formula,
  Glist,
  parms,
  lambda,
  beta,
  epsilon,
  structural = FALSE,
  nthreads = 1,
  data
)
linpeer.sim(
  formula,
  Glist,
  parms,
  lambda,
  beta,
  epsilon,
  structural = FALSE,
  nthreads = 1,
  data
)

Arguments

formula

An object of class formula: a symbolic description of the model. formula should be specified as, for example, ~ x1 + x2, where x1 and x2 are control variables, which can include contextual variables such as averages or quantiles among peers.

Glist

parms

A vector defining the true values of $(\lambda', \beta')'$ , where $\lambda$ is either the peer effect parameter for the reduced-form specification or a 2-vector with the first component being conformity peer effects and the second component representing total peer effects. The parameters $\lambda$ and $\beta$ can also be specified separately using the arguments lambda, and beta.

lambda

The true value of the vector $\lambda$ .

beta

The true value of the vector $\beta$ .

epsilon

A vector of idiosyncratic error terms. If not specified, it will be simulated from a standard normal distribution.

structural

A logical value indicating whether simulations should be performed using the structural model. The default is the reduced-form model (see the Details section of qpeer).

nthreads

A strictly positive integer indicating the number of threads to use.

data

Value

A list containing:

y

The simulated variable.

Gy

the average of y among friends.

Examples

set.seed(123)
ngr  <- 50
nvec <- rep(30, ngr)
n    <- sum(nvec)
G    <- lapply(1:ngr, function(z){
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z])
  diag(Gz) <- 0
  Gz/rowSums(Gz) # Row-normalized network
})
X    <- cbind(rnorm(n), rpois(n, 2))
l    <- 0.5
b    <- c(2, -0.5, 1)

out  <- linpeer.sim(formula = ~ X, Glist = G, lambda = l, beta = b)
summary(out$y)
set.seed(123)
ngr  <- 50
nvec <- rep(30, ngr)
n    <- sum(nvec)
G    <- lapply(1:ngr, function(z){
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z])
  diag(Gz) <- 0
  Gz/rowSums(Gz) # Row-normalized network
})
X    <- cbind(rnorm(n), rpois(n, 2))
l    <- 0.5
b    <- c(2, -0.5, 1)

out  <- linpeer.sim(formula = ~ X, Glist = G, lambda = l, beta = b)
summary(out$y)

Printing Specification Tests for Peer Effects Models

Description

A print method for the class qpeer.test.

Usage

## S3 method for class 'qpeer.test'
print(x, ...)
## S3 method for class 'qpeer.test'
print(x, ...)

Arguments

x

an object of class qpeer.test

...

Further arguments passed to or from other methods.

Value

No return value, called for side effects

Computing Instruments for Linear Models with Quantile Peer Effects

Description

qpeer.instruments computes quantile peer variables.

Usage

qpeer.instruments(
  formula,
  Glist,
  tau,
  type = 7,
  data,
  max.distance = 1,
  checkrank = FALSE,
  nthreads = 1,
  tol = 1e-10
)

qpeer.instrument(
  formula,
  Glist,
  tau,
  type = 7,
  data,
  max.distance = 1,
  checkrank = FALSE,
  nthreads = 1,
  tol = 1e-10
)

qpeer.inst(
  formula,
  Glist,
  tau,
  type = 7,
  data,
  max.distance = 1,
  checkrank = FALSE,
  nthreads = 1,
  tol = 1e-10
)

qpeer.insts(
  formula,
  Glist,
  tau,
  type = 7,
  data,
  max.distance = 1,
  checkrank = FALSE,
  nthreads = 1,
  tol = 1e-10
)
qpeer.instruments(
  formula,
  Glist,
  tau,
  type = 7,
  data,
  max.distance = 1,
  checkrank = FALSE,
  nthreads = 1,
  tol = 1e-10
)

qpeer.instrument(
  formula,
  Glist,
  tau,
  type = 7,
  data,
  max.distance = 1,
  checkrank = FALSE,
  nthreads = 1,
  tol = 1e-10
)

qpeer.inst(
  formula,
  Glist,
  tau,
  type = 7,
  data,
  max.distance = 1,
  checkrank = FALSE,
  nthreads = 1,
  tol = 1e-10
)

qpeer.insts(
  formula,
  Glist,
  tau,
  type = 7,
  data,
  max.distance = 1,
  checkrank = FALSE,
  nthreads = 1,
  tol = 1e-10
)

Arguments

formula

An object of class formula: a symbolic description of the model. The formula should be specified as, for example, ~ x1 + x2 or y ~ x1 + x2, where x1 and x2 are variables for which the quantiles will be computed and y is the dependent variable. If y is specified, then the quantiles of x1 and x2 are computed by ranking observations according to the values of y (see details).

Glist

tau

The vector of quantile levels.

type

An integer between 1 and 9 selecting one of the nine quantile algorithms used to compute peer quantiles (see the quantile function).

data

max.distance

The maximum network distance of friends to consider in computing instruments.

checkrank

A logical value indicating whether the instrument matrix should be checked for full rank. If the matrix is not of full rank, unimportant columns will be removed to obtain a full-rank matrix.

nthreads

A strictly positive integer indicating the number of threads to use when computing the quantiles of peer variables.

tol

A tolerance value used in the QR factorization to identify columns that ensure a full-rank matrix (see the qr function).

Details

The sample quantile is computed as a weighted average of two peer outcomes (see Hyndman and Fan, 1996). Specifically:

$q_{\tau,i}(x_{-i}) = (1 - \omega_i)x_{i,(\pi_i)} + \omega_ix_{i,(\pi_i+1)},$

where $x_{i,(1)}, x_{i,(2)}, x_{i,(3)}, \ldots$ are the order statistics of the outcome within $i$ 's peers, and $q_{\tau,i}(x_{-i})$ represents the sample $\tau$ -quantile of the outcome within $i$ 's peer group. If y is specified, then the ranks $\pi_i$ and the weights $\omega_i$ for the variables in X are determined based on y. The network matrices in Glist can be weighted or unweighted. If weighted, the sample weighted quantile is computed, where the outcome for friend $j$ of $i$ is weighted by $g_{ij}$ , the $(i, j)$ entry of the network matrix.

Value

A matrix including quantile peer variables

A list containing:

qy

Quantiles of peer variable y.

instruments

Matrix of instruments.

index

The indices of the two peers whose weighted average gives the quantile.

weight

The weights of the two peers whose weighted average gives the quantile.

References

Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361-365, doi:10.1080/00031305.1996.10473566.

Examples

ngr  <- 50
nvec <- rep(30, ngr)
n    <- sum(nvec)
G    <- lapply(1:ngr, function(z){
  Gz <- matrix(rbinom(sum(nvec[z]*(nvec[z] - 1)), 1, 0.3), nvec[z])
  diag(Gz) <- 0
  Gz
}) 
tau  <- seq(0, 1, 0.25)
X    <- cbind(rnorm(n), rpois(n, 2))
l    <- c(0.2, 0.1, 0.05, 0.1, 0.2)
b    <- c(2, -0.5, 1)
y    <- qpeer.sim(formula = ~X, Glist = G, tau = tau, lambda = l, beta = b)$y
Inst <- qpeer.instruments(formula = ~ X, Glist = G, tau = tau, max.distance = 2)$instruments
summary(Inst)
ngr  <- 50
nvec <- rep(30, ngr)
n    <- sum(nvec)
G    <- lapply(1:ngr, function(z){
  Gz <- matrix(rbinom(sum(nvec[z]*(nvec[z] - 1)), 1, 0.3), nvec[z])
  diag(Gz) <- 0
  Gz
}) 
tau  <- seq(0, 1, 0.25)
X    <- cbind(rnorm(n), rpois(n, 2))
l    <- c(0.2, 0.1, 0.05, 0.1, 0.2)
b    <- c(2, -0.5, 1)
y    <- qpeer.sim(formula = ~X, Glist = G, tau = tau, lambda = l, beta = b)$y
Inst <- qpeer.instruments(formula = ~ X, Glist = G, tau = tau, max.distance = 2)$instruments
summary(Inst)

Simulating Optimal (or "Good") Instruments for Quantile Peer Effect Models

Description

qpeer.optimal.inst computes optimal (or "good") instruments for quantile peer effect models using a bootstrap approach.

Usage

qpeer.optimal.instruments(
  model,
  Glist,
  tau,
  boot = 100L,
  nthreads = 1L,
  seed,
  tol = 1e-10,
  maxit = 500
)

qpeer.optimal.instrument(
  model,
  Glist,
  tau,
  boot = 100L,
  nthreads = 1L,
  seed,
  tol = 1e-10,
  maxit = 500
)

qpeer.optimal.insts(
  model,
  Glist,
  tau,
  boot = 100L,
  nthreads = 1L,
  seed,
  tol = 1e-10,
  maxit = 500
)

qpeer.optimal.inst(
  model,
  Glist,
  tau,
  boot = 100L,
  nthreads = 1L,
  seed,
  tol = 1e-10,
  maxit = 500
)
qpeer.optimal.instruments(
  model,
  Glist,
  tau,
  boot = 100L,
  nthreads = 1L,
  seed,
  tol = 1e-10,
  maxit = 500
)

qpeer.optimal.instrument(
  model,
  Glist,
  tau,
  boot = 100L,
  nthreads = 1L,
  seed,
  tol = 1e-10,
  maxit = 500
)

qpeer.optimal.insts(
  model,
  Glist,
  tau,
  boot = 100L,
  nthreads = 1L,
  seed,
  tol = 1e-10,
  maxit = 500
)

qpeer.optimal.inst(
  model,
  Glist,
  tau,
  boot = 100L,
  nthreads = 1L,
  seed,
  tol = 1e-10,
  maxit = 500
)

Arguments

model

An object of class qpeer that contains an initial estimation of the model for which the "optimal" instruments will be computed.

Glist

tau

A numeric vector specifying the quantile levels for the instrument matrix. The default value is the vector of quantile levels used to estimate the model.

boot

A strictly positive integer indicating the number of bootstrap replications.

nthreads

A strictly positive integer indicating the number of threads to use when bootstrapping.

seed

The random number generator (RNG) state used for random number generation. This can also be set using the set.seed function.

tol

The tolerance value used in the fixed-point iteration method to compute the outcome y. The process stops if the $\ell_1$ -distance between two consecutive values of y is less than tol.

maxit

The maximum number of iterations for the fixed-point iteration method.

Value

A matrix of instruments, where the k-th column corresponds to the instrument for the k-th endogenous variable.

Examples


set.seed(123)
ngr  <- 30  # Number of subnets
nvec <- rep(30, ngr)  # Size of subnets
n    <- sum(nvec)

### Simulating Data
## Network matrix
G <- lapply(1:ngr, function(z) {
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z], nvec[z])
  diag(Gz) <- 0
  # Adding isolated nodes (important for the structural model)
  niso <- sample(0:nvec[z], 1, prob = (nvec[z] + 1):1 / sum((nvec[z] + 1):1))
  if (niso > 0) {
    Gz[sample(1:nvec[z], niso), ] <- 0
  }
  Gz
})

tau <- seq(0, 1, 1/3)
X   <- cbind(rnorm(n), rpois(n, 2))
l   <- c(0.2, 0.15, 0.1, 0.2)
b   <- c(2, -0.5, 1)
eps <- rnorm(n, 0, 0.4)

## Generating `y`
y <- qpeer.sim(formula = ~ X, Glist = G, tau = tau, lambda = l,
               beta = b, epsilon = eps)$y

### Estimation
## Computing instruments
Z <- qpeer.inst(formula = ~ X, Glist = G, tau = seq(0, 1, 0.1),
                max.distance = 2, checkrank = TRUE)
Z <- Z$instruments

## Reduced-form model
rest <- qpeer(formula = y ~ X, excluded.instruments = ~ Z, Glist = G, tau = tau)
summary(rest, diagnostic = TRUE)  

# Estimation using the optimal instrument
Ired <- qpeer.optimal.insts(rest, Glist = G, boot = 100, nthreads = 2)
summary(qpeer(formula = y ~ X, excluded.instruments = ~ Ired, Glist = G, tau = tau), 
        diagnostic = TRUE)  

## Structural model
sest <- qpeer(formula = y ~ X, excluded.instruments = ~ Z, Glist = G, tau = tau,
              structural = TRUE)
summary(sest, diagnostic = TRUE)

# Estimation using the optimal instrument
Istr <- qpeer.optimal.insts(sest, Glist = G, boot = 100, nthreads = 2)
summary(qpeer(formula = y ~ X, excluded.instruments = ~ Istr, Glist = G, tau = tau), 
        diagnostic = TRUE)

set.seed(123)
ngr  <- 30  # Number of subnets
nvec <- rep(30, ngr)  # Size of subnets
n    <- sum(nvec)

### Simulating Data
## Network matrix
G <- lapply(1:ngr, function(z) {
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z], nvec[z])
  diag(Gz) <- 0
  # Adding isolated nodes (important for the structural model)
  niso <- sample(0:nvec[z], 1, prob = (nvec[z] + 1):1 / sum((nvec[z] + 1):1))
  if (niso > 0) {
    Gz[sample(1:nvec[z], niso), ] <- 0
  }
  Gz
})

tau <- seq(0, 1, 1/3)
X   <- cbind(rnorm(n), rpois(n, 2))
l   <- c(0.2, 0.15, 0.1, 0.2)
b   <- c(2, -0.5, 1)
eps <- rnorm(n, 0, 0.4)

## Generating `y`
y <- qpeer.sim(formula = ~ X, Glist = G, tau = tau, lambda = l,
               beta = b, epsilon = eps)$y

### Estimation
## Computing instruments
Z <- qpeer.inst(formula = ~ X, Glist = G, tau = seq(0, 1, 0.1),
                max.distance = 2, checkrank = TRUE)
Z <- Z$instruments

## Reduced-form model
rest <- qpeer(formula = y ~ X, excluded.instruments = ~ Z, Glist = G, tau = tau)
summary(rest, diagnostic = TRUE)  

# Estimation using the optimal instrument
Ired <- qpeer.optimal.insts(rest, Glist = G, boot = 100, nthreads = 2)
summary(qpeer(formula = y ~ X, excluded.instruments = ~ Ired, Glist = G, tau = tau), 
        diagnostic = TRUE)  

## Structural model
sest <- qpeer(formula = y ~ X, excluded.instruments = ~ Z, Glist = G, tau = tau,
              structural = TRUE)
summary(sest, diagnostic = TRUE)

# Estimation using the optimal instrument
Istr <- qpeer.optimal.insts(sest, Glist = G, boot = 100, nthreads = 2)
summary(qpeer(formula = y ~ X, excluded.instruments = ~ Istr, Glist = G, tau = tau), 
        diagnostic = TRUE)

Simulating Linear Models with Quantile Peer Effects

Description

qpeer.sim simulates the quantile peer effect models developed by Houndetoungan (2025).

Usage

qpeer.sim(
  formula,
  Glist,
  tau,
  parms,
  lambda,
  beta,
  epsilon,
  structural = FALSE,
  init,
  type = 7,
  tol = 1e-10,
  maxit = 500,
  details = TRUE,
  nthreads = 1,
  data
)
qpeer.sim(
  formula,
  Glist,
  tau,
  parms,
  lambda,
  beta,
  epsilon,
  structural = FALSE,
  init,
  type = 7,
  tol = 1e-10,
  maxit = 500,
  details = TRUE,
  nthreads = 1,
  data
)

Arguments

formula

Glist

tau

The vector of quantile levels.

parms

A vector defining the true values of $(\lambda', \beta')'$ , where $\lambda$ is a vector of $\lambda_{\tau}$ for each quantile level $\tau$ . The parameters $\lambda$ and $\beta$ can also be specified separately using the arguments lambda and beta. For the structural model, $\lambda = (\lambda_2, \lambda_{\tau_1}, \lambda_{\tau_2}, \dots)^{\prime}$ (see the Details section of qpeer).

lambda

The true value of the vector $\lambda$ .

beta

The true value of the vector $\beta$ .

epsilon

A vector of idiosyncratic error terms. If not specified, it will be simulated from a standard normal distribution (see the model specification in the Details section of qpeer).

structural

A logical value indicating whether simulations should be performed using the structural model. The default is the reduced-form model (see the Details section of qpeer).

init

An optional initial guess for the equilibrium.

type

An integer between 1 and 9 selecting one of the nine quantile algorithms used to compute peer quantiles (see the quantile function).

tol

The tolerance value used in the Fixed Point Iteration Method to compute the outcome y. The process stops if the $\ell_1$ -distance between two consecutive values of y is less than tol.

maxit

The maximum number of iterations for the Fixed Point Iteration Method.

details

A logical value indicating whether to save the indices and weights of the two peers whose weighted average determines the quantile.

nthreads

A strictly positive integer indicating the number of threads to use.

data

Value

A list containing:

y

The simulated variable.

qy

Quantiles of the simulated variable among peers.

epsilon

The idiosyncratic error.

index

The indices of the two peers whose weighted average gives the quantile.

weight

The weights of the two peers whose weighted average gives the quantile.

iteration

The number of iterations before convergence.

References

Houndetoungan, A. (2025). Quantile peer effect models. arXiv preprint arXiv:2405.17290, doi:10.48550/arXiv.2506.12920.

Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361-365, doi:10.1080/00031305.1996.10473566.

Examples

set.seed(123)
ngr  <- 50
nvec <- rep(30, ngr)
n    <- sum(nvec)
G    <- lapply(1:ngr, function(z){
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z])
  diag(Gz) <- 0
  Gz
}) 
tau  <- seq(0, 1, 0.25)
X    <- cbind(rnorm(n), rpois(n, 2))
l    <- c(0.2, 0.1, 0.05, 0.1, 0.2)
b    <- c(2, -0.5, 1)

out  <- qpeer.sim(formula = ~ X, Glist = G, tau = tau, lambda = l, beta = b)
summary(out$y)
out$iteration
set.seed(123)
ngr  <- 50
nvec <- rep(30, ngr)
n    <- sum(nvec)
G    <- lapply(1:ngr, function(z){
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z])
  diag(Gz) <- 0
  Gz
}) 
tau  <- seq(0, 1, 0.25)
X    <- cbind(rnorm(n), rpois(n, 2))
l    <- c(0.2, 0.1, 0.05, 0.1, 0.2)
b    <- c(2, -0.5, 1)

out  <- qpeer.sim(formula = ~ X, Glist = G, tau = tau, lambda = l, beta = b)
summary(out$y)
out$iteration

Specification Tests for Peer Effects Models

Description

qpeer.test performs specification tests on peer effects models. These include monotonicity tests on quantile peer effects, as well as tests for instrument validity when an alternative set of instruments is available.

Usage

qpeer.test(
  model1,
  model2 = NULL,
  which,
  full = FALSE,
  boot = 500,
  nthreads = 1L,
  print = TRUE,
  maxit = 1e+06,
  eps_f = 1e-09,
  eps_g = 1e-09
)
qpeer.test(
  model1,
  model2 = NULL,
  which,
  full = FALSE,
  boot = 500,
  nthreads = 1L,
  print = TRUE,
  maxit = 1e+06,
  eps_f = 1e-09,
  eps_g = 1e-09
)

Arguments

model1, model2

Objects of class qpeer, linpeer, or genpeer.

which

A character string indicating the type of test to be implemented. The value must be one of "uniform", "increasing", "decreasing", "wald", "sargan", and "encompassing" (see Details).

full

A Boolean indicating whether the parameters associated with the exogenous variables should be compared. This is used for tests that compare two competing parameter sets.

boot

An integer indicating the number of bootstrap replications to use for the encompassing test and for computing p-values in the "increasing" and "decreasing" tests.

nthreads

Number of CPU cores (threads) used to run parts of the estimation in parallel.

print

A Boolean indicating whether the progression bar should be printed for bootstrap methods.

maxit, eps_f, eps_g

Control parameters for the optim_lbfgs solver used to optimize the objective function in the "increasing" and "decreasing" tests (see Kodde and Palm, 1986). The optim_lbfgs function is provided by the RcppNumerical package and is based on the L-BFGS method.

Details

The monotonicity tests evaluate whether the quantile peer effects $\lambda_{\tau}$ are constant, increasing, or decreasing. In this case, model1 must be an object of class qpeer, and the which argument specifies the null hypothesis: "uniform", "increasing", or "decreasing". For the "uniform" test, a standard Wald test is performed. For the "increasing" and "decreasing" tests, the procedure follows Kodde and Palm (1986).

The instrument validity tests assess whether a second set of instruments $Z_2$ is valid, assuming that a baseline set $Z_1$ is valid. In this case, both model1 and model2 must be objects of class qpeer, linpeer, or genpeer. The test compares the estimates obtained using each instrument set. If $Z_2$ nests $Z_1$ , it is recommended to compare the overidentification statistics from both estimations (see Hayashi, 2000, Proposition 3.7). If $Z_2$ does not nest $Z_1$ , the estimates themselves are compared. To compare the overidentification statistics, set the which argument to "sargan". To compare the estimates directly, set the which argument to "wald".

Given two competing models, it is possible to test whether one is significantly worse using an encompassing test by setting which to "encompassing". The null hypothesis is that model1 is not worse.

Value

A list containing:

test

A vector or matrix containing the test statistics, degrees of freedom, and p-values.

lambda

Peer effect estimates from tests based on a single model (monotonicity tests).

diff.theta

Differences in peer effect estimates from tests based on two models (endogeneity and encompassing tests).

delta

The estimate of $\delta$ for the encompassing test.

which

The value of which returned by the function.

boot

The value of boot returned by the function.

References

Hayashi, F. (2000). Econometrics. Princeton University Press.

Kodde, D. A., & Palm, F. C. (1986). Wald criteria for jointly testing equality and inequality restrictions. Econometrica, 54(5), 1243–1248.

Examples


set.seed(123)
ngr  <- 50  # Number of subnets
nvec <- rep(30, ngr)  # Size of subnets
n    <- sum(nvec)

### Simulating Data
## Network matrix
G <- lapply(1:ngr, function(z) {
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z], nvec[z])
  diag(Gz) <- 0
  # Adding isolated nodes (important for the structural model)
  niso <- sample(0:nvec[z], 1, prob = (nvec[z] + 1):1 / sum((nvec[z] + 1):1))
  if (niso > 0) {
    Gz[sample(1:nvec[z], niso), ] <- 0
  }
  Gz
})

tau <- seq(0, 1, 1/3)
X   <- cbind(rnorm(n), rpois(n, 2))
l   <- c(0.2, 0.15, 0.1, 0.2)
b   <- c(2, -0.5, 1)
eps <- rnorm(n, 0, 0.4)

## Generating `y`
y <- qpeer.sim(formula = ~ X, Glist = G, tau = tau, lambda = l, 
               beta = b, epsilon = eps)$y

### Estimation
## Computing instruments
Z1 <- qpeer.inst(formula = ~ X, Glist = G, tau = seq(0, 1, 0.1), 
                 max.distance = 2, checkrank = TRUE)$instruments
Z2 <- qpeer.inst(formula = y ~ X, Glist = G, tau = seq(0, 1, 0.1), 
                 max.distance = 2, checkrank = TRUE)$instruments

## Reduced-form model 
rest1 <- qpeer(formula = y ~ X, excluded.instruments = ~ Z1, Glist = G, tau = tau)
summary(rest1, diagnostic = TRUE)  
rest2 <- qpeer(formula = y ~ X, excluded.instruments = ~ Z1 + Z2, Glist = G, tau = tau)
summary(rest2, diagnostic = TRUE)  

qpeer.test(model1 = rest1, which = "increasing")
qpeer.test(model1 = rest1, which = "decreasing")
qpeer.test(model1 = rest1, model2 = rest2, which = "sargan")

#' A model with a mispecified tau
rest3 <- qpeer(formula = y ~ X, excluded.instruments = ~ Z1 + Z2, Glist = G, tau = c(0, 1))
summary(rest3)
#' Test is rest3 is worse than rest1
qpeer.test(model1 = rest3, model2 = rest1, which = "encompassing")

set.seed(123)
ngr  <- 50  # Number of subnets
nvec <- rep(30, ngr)  # Size of subnets
n    <- sum(nvec)

### Simulating Data
## Network matrix
G <- lapply(1:ngr, function(z) {
  Gz <- matrix(rbinom(nvec[z]^2, 1, 0.3), nvec[z], nvec[z])
  diag(Gz) <- 0
  # Adding isolated nodes (important for the structural model)
  niso <- sample(0:nvec[z], 1, prob = (nvec[z] + 1):1 / sum((nvec[z] + 1):1))
  if (niso > 0) {
    Gz[sample(1:nvec[z], niso), ] <- 0
  }
  Gz
})

tau <- seq(0, 1, 1/3)
X   <- cbind(rnorm(n), rpois(n, 2))
l   <- c(0.2, 0.15, 0.1, 0.2)
b   <- c(2, -0.5, 1)
eps <- rnorm(n, 0, 0.4)

## Generating `y`
y <- qpeer.sim(formula = ~ X, Glist = G, tau = tau, lambda = l, 
               beta = b, epsilon = eps)$y

### Estimation
## Computing instruments
Z1 <- qpeer.inst(formula = ~ X, Glist = G, tau = seq(0, 1, 0.1), 
                 max.distance = 2, checkrank = TRUE)$instruments
Z2 <- qpeer.inst(formula = y ~ X, Glist = G, tau = seq(0, 1, 0.1), 
                 max.distance = 2, checkrank = TRUE)$instruments

## Reduced-form model 
rest1 <- qpeer(formula = y ~ X, excluded.instruments = ~ Z1, Glist = G, tau = tau)
summary(rest1, diagnostic = TRUE)  
rest2 <- qpeer(formula = y ~ X, excluded.instruments = ~ Z1 + Z2, Glist = G, tau = tau)
summary(rest2, diagnostic = TRUE)  

qpeer.test(model1 = rest1, which = "increasing")
qpeer.test(model1 = rest1, which = "decreasing")
qpeer.test(model1 = rest1, model2 = rest2, which = "sargan")

#' A model with a mispecified tau
rest3 <- qpeer(formula = y ~ X, excluded.instruments = ~ Z1 + Z2, Glist = G, tau = c(0, 1))
summary(rest3)
#' Test is rest3 is worse than rest1
qpeer.test(model1 = rest3, model2 = rest1, which = "encompassing")

Summary for the Estimation of CES-based Peer Effects Models

Description

Summary and print methods for the class cespeer.

Usage

## S3 method for class 'cespeer'
summary(object, fullparameters = TRUE, ...)

## S3 method for class 'summary.cespeer'
print(x, ...)

## S3 method for class 'cespeer'
print(x, ...)
## S3 method for class 'cespeer'
summary(object, fullparameters = TRUE, ...)

## S3 method for class 'summary.cespeer'
print(x, ...)

## S3 method for class 'cespeer'
print(x, ...)

Arguments

object

An object of class cespeer.

fullparameters

A logical value indicating whether all parameters should be summarized (may be useful for the structural model).

...

Further arguments passed to or from other methods.

x

An object of class summary.cespeer or cespeer.

Value

A list containing:

model.info

A list with information about the model, such as the number of subnets, number of observations, and other key details.

coefficients

A summary of the estimates, standard errors, and p-values.

gmm

A list of GMM estimation results, including parameter estimates, the covariance matrix, and related statistics.

Summary for the Estimation of Quantile Peer Effects Models

Description

Summary and print methods for the class qpeer.

Usage

## S3 method for class 'genpeer'
summary(
  object,
  fullparameters = TRUE,
  diagnostic = FALSE,
  diagnostics = FALSE,
  boot = NULL,
  nthreads = 1L,
  print = TRUE,
  ...
)

## S3 method for class 'summary.genpeer'
print(x, ...)

## S3 method for class 'genpeer'
print(x, ...)

## S3 method for class 'linpeer'
summary(
  object,
  fullparameters = TRUE,
  diagnostic = FALSE,
  diagnostics = FALSE,
  boot = NULL,
  nthreads = 1L,
  print = TRUE,
  ...
)

## S3 method for class 'summary.linpeer'
print(x, ...)

## S3 method for class 'linpeer'
print(x, ...)

## S3 method for class 'qpeer'
summary(
  object,
  fullparameters = TRUE,
  diagnostic = FALSE,
  diagnostics = FALSE,
  boot = NULL,
  nthreads = 1L,
  print = TRUE,
  ...
)

## S3 method for class 'summary.qpeer'
print(x, ...)

## S3 method for class 'qpeer'
print(x, ...)
## S3 method for class 'genpeer'
summary(
  object,
  fullparameters = TRUE,
  diagnostic = FALSE,
  diagnostics = FALSE,
  boot = NULL,
  nthreads = 1L,
  print = TRUE,
  ...
)

## S3 method for class 'summary.genpeer'
print(x, ...)

## S3 method for class 'genpeer'
print(x, ...)

## S3 method for class 'linpeer'
summary(
  object,
  fullparameters = TRUE,
  diagnostic = FALSE,
  diagnostics = FALSE,
  boot = NULL,
  nthreads = 1L,
  print = TRUE,
  ...
)

## S3 method for class 'summary.linpeer'
print(x, ...)

## S3 method for class 'linpeer'
print(x, ...)

## S3 method for class 'qpeer'
summary(
  object,
  fullparameters = TRUE,
  diagnostic = FALSE,
  diagnostics = FALSE,
  boot = NULL,
  nthreads = 1L,
  print = TRUE,
  ...
)

## S3 method for class 'summary.qpeer'
print(x, ...)

## S3 method for class 'qpeer'
print(x, ...)

Arguments

object

An object of class qpeer.

fullparameters

A logical value indicating whether all parameters should be summarized (may be useful for the structural model).

diagnostics, diagnostic

Logical. Should diagnostic tests for the instrumental-variable regression be performed? These include an F-test of the first-stage regression for weak instruments, a Wu-Hausman test for endogeneity, and a Hansen's J-test for overidentifying restrictions (only if there are more instruments than regressors).

boot

An integer indicating the number of bootstrap replications for the encompassing test.

nthreads

A strictly positive integer indicating the number of threads used to perform the bootstrap.

print

A logical indicating whether a progress bar should be displayed for bootstrap methods.

...

Further arguments passed to or from other methods.

x

An object of class summary.qpeer or qpeer.

Value

A list containing:

model.info

A list with information about the model, such as the number of subnets, number of observations, and other key details.

coefficients

A summary of the estimates, standard errors, and p-values.

diagnostics

A summary of the diagnostic tests for the instrumental-variable regression if requested.

KP.cv

Critical values for the Kleibergen–Paap Wald test (5% level).

gmm

A list of GMM estimation results, including parameter estimates, the covariance matrix, and related statistics.

Package 'QuantilePeer'

Help Index

The QuantilePeer package

Description

Author(s)

References

See Also

Estimation of CES-Based Peer Effects Models

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Compute the CES Social Norm

Description

Usage

Arguments

Value

Simulating Peer Effect Models with a CES Social Norm

Description

Usage

Arguments

Value

References

See Also

Examples

Demeaning Variables

Description

Usage

Arguments

Value

Estimating Peer Effects Models

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Simulating Linear Peer Effect Models

Description

Usage

Arguments

Value

See Also

Examples

Printing Specification Tests for Peer Effects Models

Description

Usage

Arguments

Value

Computing Instruments for Linear Models with Quantile Peer Effects

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Simulating Optimal (or "Good") Instruments for Quantile Peer Effect Models

Description

Usage

Arguments

Value

Examples

Simulating Linear Models with Quantile Peer Effects

Description

Usage

Arguments

Value

References

See Also

Examples

Specification Tests for Peer Effects Models

Description