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Estimating Social Effects with Randomized and Observational Network Data
\begin{bibunit}[jpe]
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\title{\bf Estimating Social Effects with Randomized and Observational Network Data}
\author{TszKin Julian Chan\\
Bates White Economic Consulting\\
Juan Estrada\\
Analysis Group Economic Consulting\\
Kim Huynh\thanks{We thank the Editor, Associate Editor, and two anonymous referees for very helpful corrections, comments, and suggestions that improved the overall readability and presentation of the article. The views expressed in this article are those of the authors. No responsibility for them should be attributed to the Bank of Canada. All remaining errors are the responsibility of the authors.}\hspace{.2cm}\\
Currency Department, Bank of Canada\\
David Jacho-Ch\'{a}vez\thanks{Corresponding Author}\\
Department of Economics, Emory University\\
Chungsang Tom Lam\\
Department of Finance, Florida State University\\
Leonardo S\'{a}nchez-Arag\'{o}n\\
Facultad de Ciencias Sociales y Human\'{i}sticas, ESPOL University}
\maketitle
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\begin{abstract}
This paper introduces an innovative approach to identifying and estimating the parameters of interest in the widely recognized linear-in-means regression model under conditions where the initial randomization of peers determines the observed network. We assert that peers who are initially randomized do not produce social effects. However, after randomization, agents can endogenously develop significant connections that potentially generate peer influences. We present a moment condition that compiles local heterogeneous identifying information for all agents within the population. Under the assumption of $\psi$-dependence in the endogenous network space, we propose a Generalized Method of Moments (GMM) estimator, which is proven to be consistent, asymptotically normally distributed, and straightforward to implement using commonly available statistical software due to its closed-form expression. Monte Carlo simulations demonstrate the GMM estimator's strong small-sample performance. An empirical analysis utilizing data from Hong Kong high school students reveals substantial positive spillover effects on math test scores among study partners in our sample, provided that their seatmates were exogenously assigned by their teachers.
\end{abstract}
\noindent
{\it Keywords:} Social Networks; Instrumental Variables; Causal Inference; $\psi$-dependence
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\section{Introduction}\label{introduction}
In many observational studies in economics and other social settings, a unit of observation's outcome (e.g. purchase of an item, health well-being of an individual, performance of a firm, or test scores of a student) depends not only on this unit's own observed characteristics (direct effect), but also on the outcome (peer effects) and characteristics (contextual effects) of other observations (peers) in the sample with which they have a link. The workhorse model in economics and other social sciences for this type of setting is the so-called \emph{linear-in-means} regression model (see \citeauthor{manski1993}, \citeyear{manski1993} and Section 3.1 in \citeauthor{Paula2017}, \citeyear{Paula2017}, pp. 275-289) where the outcome variable for observation $i$ (e.g., a person, a firm, or a country), $y_{n,i}$, is determined according to
\begin{equation}
\label{intro_lmm}
y_{n;i}=\beta_{0}\sum_{j \neq i}w_{n;i,j}y_{n;j}+\sum_{j \neq i}w_{n;i,j}\mathbf{x}_{n;j}^{\top}\boldsymbol{\delta}_{0}+ \widetilde{\mathbf{x}}_{n;i}^{\top}\boldsymbol{\gamma}_{0}+\varepsilon_{n;i}\text{,}
\end{equation}
\noindent where $i,j\in\{1,\dots,n\}$ are also known as \emph{nodes}, $\widetilde{\mathbf{x}}_{n;i}=[1,\mathbf{x}_{n;i}^\top]^\top$, and $\mathbf{x}_{n;i}$ is a vector of attributes that characterizes observations $i$, $w_{n;i,j} \in (0,1]$ if $j$ is connected to $i$ (a potentially weighted \emph{edge}), and 0 otherwise, $\varepsilon_{n;i}$ represents an unobserved latent error, and $n$ is the number of observations or nodes in the sample. The structure of the \emph{social} network is fully characterized by the square $n \times n$ matrix, $\mathbf{W}_{n}$, with the entry $(i,j)$ given by $w_{n;i,j}$, that is, the adjacency matrix. The structural parameters $\beta_0$ and $\boldsymbol{\delta}_0$ capture the peer and contextual effects, respectively, while $\boldsymbol{\gamma}_0$ captures the direct effects of the observation's own characteristics. They are jointly known as the social (or neighbor) effect parameters and the object of interest in empirical studies with network data; see, e.g., \cite{Sacerdote_QJE} in Economics, \cite{L_i_M_public_health} in Public Health, \cite{L_i_M_criminology} in Criminology, and \cite{L_i_M_sociology} in Sociology to mention just a few.
Sufficient conditions under which the social parameters in \eqref{intro_lmm} can be uniquely recovered from the estimating sample $\left\{y_{n;i},\mathbf{x}_{n;i}^{\top},\{w_{n;i,j}\}_{j=1,j\neq i}^n\right\}_{i=1}^n$ are well understood under the assumption that the adjacency matrix, $\mathbf{W}_{n}$, is exogenous; see, for example, \cite{Paula2017} and references therein. However, the endogenous case remains an active area of research among economists and social scientists due to simultaneity bias, measurement error in link information, or because there might be a natural correlation between covariates and the error term in \eqref{intro_lmm} (homophily); see, e.g., \cite{Johnsson2019} and references therein.
This article proposes the use of a type of multilayered (multidimensional) network data structure known as \emph{multiplex} networks \citep{Boccaletti2014,Kivela_multilayer_network_2014} to consistently estimate and perform correct inferences on the structural social parameters in \eqref{intro_lmm} with a potentially endogenous network structure, $\mathbf{W}_n$. In particular, we assume that the researcher observes another set of social ties among the original observations, $\{w_{n,0;i,j}\}_{j=1,j\neq i}^n$, in the form of $n\times n$ adjacency matrix $\mathbf{W}_{n,0}$ that are exogenous in the usual sense.
The assumption of network exogeneity in $\mathbf{W}_{n,0}$ is motivated from the literature on experimental settings with interactions; see, e.g., \cite{Athey2017}. Network randomization has been used in a wide range of applied settings, such as educational achievement \citep{Sacerdote_QJE}, entrepreneurship team performance \citep{Hasan2019}, and business performance \citep{Cai2018}, just to name a few. Furthermore, recent studies have emphasized the importance of endogenous choices of individuals to interact with or avoid their randomized peers as mediators of peer effects \citep{Hasan2019}. For example, \cite{Carrell2013} finds that after assigning cadets to squadrons (initial group assignment), i.e., $\mathbf{W}_{n,0}$, they endogenously sorted into homogeneous friendship groups based on their ability, i.e., $\mathbf{W}_n$. In the context of worker productivity, \cite{Mas2009}, \cite{Hjort2014}, and \cite{Kato2016} show that peer effects matter only when individuals interact frequently or share similar origins or ethnic divisions, even if they all belong to the same group.
Our method offers a benefit over earlier approaches for identifying peer and contextual effects in endogenous networks because it does not depend on a specific description of the network formation process, as seen in works such as \cite{Goldsmith-Pinkham2013}, \cite{Qu2015}, \cite{Johnsson2019}, \cite{Qu2021}, and \cite{Auerbach2022}. Apart from bypassing the estimation of the parameters of a potentially complex nonlinear outcome, the flexibility of our method permits us to address a broader spectrum of potential endogeneity sources. A notably significant benefit is that our method can manage endogeneity arising from the concurrent determination of the outcome and the network of interest. Current methods that require explicit modeling of network formation assume that networks depend on certain endogenous variables and that the outcomes are established once the endogenous network is formed. Extending those models to include a simultaneous system of equations that include both outcomes and dyadic linking choices requires one to fundamentally change the assumptions in the model, which can drastically affect the identification results. Nevertheless, since our method bypasses the direct modeling of link formation, it eliminates the need to explicitly account for the simultaneous determination of outcomes and the network, thereby making our straightforward approach highly effective in handling various sources of endogeneity.
The paper uses the exogenous network, $\mathbf{W}_{n,0}$, to construct a set of valid moment conditions that point-identify parameters in \eqref{intro_lmm}. The resulting linear Generalized Method of Moments (GMM) estimator is simple to implement in existing statistical software like \texttt{Python}, \texttt{R} or \texttt{Stata}; see, e.g., \cite{netivreg}. Furthermore, the estimator is shown to be asymptotically normal at the standard root--$n$ rate of convergence, and a consistent estimator of the efficient asymptotic variance-covariance is proposed to perform asymptotically valid inference.
Our proof method allows us to characterize the asymptotic variance-covariance that takes into account the presence of network dependence between individuals generated by the endogenous network $\mathbf{W}_{n}$. In particular, we assume that the triangular array for the random vector of observed and unobserved characteristics is $\psi$ dependent \citep{Doukhan1999}, and the levels of dependence between individuals decrease with their distance in the network space spanned by $\mathbf{W}_{n}$. Moreover, our asymptotic normality result requires that the levels of dependence decrease at a rate that is fast enough to compensate for any potential increase in the asymptotic levels of network density. Finally, our large sample variance-covariance characterization also connects the asymptotic levels of sparsity in the exogenous network, $\mathbf{W}_{n,0}$, with the precision of the network effects parameters. In particular, we show that the precision of the parameters increases with the number of nodes for which we can find indirect paths of distance of at least two in $\mathbf{W}_{n,0}$. Certainly, higher levels of network density correspond to a reduced probability of identifying indirect links between the individuals in that network, thereby affecting the accuracy of our suggested estimator.
The structure of this paper is as follows. Section \ref{prelim} provides various definitions and notation used throughout. Section \ref{sid} introduces the model and conditions for the parameters of the basic model to be uniquely recovered (point identification) from the estimation sample $\left\{y_i,\mathbf{x}_i^{\top},\{w_{i,j}\}_{j=1,j\neq i}^n,\{w_{0;i,j}\}_{j=1,j\neq i}^n\right\}_{i=1}^n$. Section \ref{estimation} describes the proposed linear GMM estimator, its asymptotic distribution, and how to calculate valid asymptotic standard errors. Section \ref{mc} provides the Monte Carlo exercises showing the small sample properties of the estimator, while an empirical illustration of the proposed methodology is discussed in Section \ref{emp}. Finally, Section \ref{discussion} concludes. The supplemental materials contain all mathematical proofs of the main results, as well as further details on the real data illustration.
\section{Preliminaries}\label{prelim}
We assume the full observability of two types of networks over the same set of nodes. One is the endogenous network of interest that can create social externalities; the other is the instrumental network induced by random assignment. For $N \in \mathbb{N}_{+} \equiv \{1, 2,\dots\}$, let $\mathcal{I}_{N}$ represent the set of agents in an arbitrarily large population, which we call \textit{nodes} in the network. We denote by $\mathcal{G}_{N}=(\mathcal{I}_{N}, E)$ the population network of interest and by $\mathcal{G}_{N,0}=(\mathcal{I}_{N}, E_{0})$ the instrumental population network, where $E$ and $E_{0}$ are the corresponding sets of links (Edges). Similarly, as in \cite{Graham2020}, the observed networks of size $n<N$, are denoted as $\mathcal{G}_{n}$ and $\mathcal{G}_{n,0}$, respectively, and are assumed to coincide with the subgraphs induced by $\mathcal{I}_{n}$ nodes sampled from their corresponding large population networks.
In the population, we represent the two networks $\mathcal{G}_{N}$ and $\mathcal{G}_{N,0}$ with their respective adjacency matrices; that is, $\mathbf{W}_{N}=[w_{N;i,j}]$ and $\mathbf{W}_{N,0}=[w_{N,0;i,j}]$, where $w_{N;i,j},w_{N,0;i,j} \in (0,1]$ are weights representing the importance of the $(i,j)$ connection in each of the networks and $w_{N;i,j}=0$ if $i$ and $j$ are not connected in $\mathcal{G}_{N}$. We do the same for $w_{N,0;i,j}$ in $\mathcal{G}_{N,0}$. Our framework permits both weighted as well as unweighted edges in networks. In the latter scenario, $w_{N;i,j}$ and $w_{N,0;i,j}$ equal to one whenever there exists a connection between individuals $i$ and $j$ and equal zero otherwise. The theory developed here works in both cases.
We also define the $1 \times N$ vectors $\mathbf{w}_{N,i}=[w_{N;i,1},\dots,w_{N;i,N}] \in [0,1]^{N}$ and $\mathbf{w}_{N,0;i}=[w_{N,0;i,1},\dots,w_{N,0;i,N}] \in [0,1]^{N}$ to be the $i$th row of the adjacency matrices $\mathbf{W}_{N}$ and $\mathbf{W}_{N,0}$, respectively. Similarly $\mathbf{w}_{N,0;i}^p$ represents the $i$th row of powers $p\ge 1$ of the adjacency matrix $\mathbf{W}_{N,0}^p$. The adjacency matrices of the observed sample, $\mathbf{W}_{n}$ and $\mathbf{W}_{n,0}$, are defined and formed accordingly.
Section \ref{estimation} uses the concept of $\psi-$dependence to bound the dependence among individuals as a function of their distance in the network space. Following the literature on graph theory, we use the shortest path length as our measure of distance; i.e., we denote $d_{n}(i,j)$ as the minimum path length connecting individuals $i$ and $j$ in the \textit{endogenous} network of interest $\mathcal{G}_{n}$ induced by the sample size $n$. Let $A$ and $B$ be any two sets of individuals of size $a, b \in \mathbb{N}_{+}$. We define the distance between sets as
$d_{n}(A,B)= \min_{i\in A}\min_{j\in B}d_{n}(i,j)$.
Based on the definition of set distance, we define the following group of node sets used in the asymptotic results presented in Section \ref{estimation}:
\refstepcounter{subassumption}
(\textit{\roman{subassumption}})~\ignorespaces \label{ds1} $\mathcal{P}^{+}_{n}(a, b , d)=\{(A, B): A, B \subset \mathcal{I}_{n},|A|=a,|B|=b\text{, and } d_{n}(A, B) \geq d\}$ containing groups of nodes at a distance of at least $d$ from each other;
\refstepcounter{subassumption}
(\textit{\roman{subassumption}})~\ignorespaces \label{ds2} $\mathcal{P}^{-}_{n}(a, b , d)=\{(A, B): A, B \subset \mathcal{I}_{n},|A|=a,|B|=b\text{, and } d_{n}(A, B) \leq d\}$ containing groups of nodes at a distance of at most $d$ from each other; and
\refstepcounter{subassumption}
(\textit{\roman{subassumption}})~\ignorespaces \label{ds3} $\mathcal{P}_{n}(a, b , d)=\{(A, B): A, B \subset \mathcal{I}_{n},|A|=a,|B|=b\text{, and } d_{n}(A, B) = d\}$ the set associated with groups of nodes at distance $d$ from each other, where $\mathcal{I}_{n}$ is the set of sampled individuals of size $n$. The associated set that contains all nodes at a certain distance from node $i$ is $\mathcal{P}_{n}^{+}(i,d)=\{j \in \mathcal{I}_{n}: d_{n}(i,j)\geq d\}$, $\mathcal{P}_{n}(i,d)=\{j \in \mathcal{I}_{n}: d_{n}=d\}$, and $\mathcal{P}_{n}^{-}(i,d)=\{j \in \mathcal{I}_{n}: d_{n}\leq d\}$.
For any random vector $\mathbf{r}_{N;i} \in \mathbb{R}^{L}$ for some $L \in \mathbb{N}_{+}$, we endow $\mathbb{R}^{L\times a}$ with the distance measure $\boldsymbol{d}_{a}(\mathbf{x}, \mathbf{y})=\sum_{l=1}^{a}\left\|x_{l}-y_{l}\right\|_{2}$, for $a\in \mathbb{N}_{+}$, and where $\|\cdot\|_{2}$ denotes the Euclidean norm and $(\mathbf{x}, \mathbf{y}) \in \mathbb{R}^{L\times a}$. We let $\mathscr{L}_{L, a}$ denote the collection of bounded Lipschitz real functions that map values from $\mathbb{R}^{L \times a}$ to $\mathbb{R}$. For any set of individuals $A$, let $\mathbf{r}_{N,A} = \left(\mathbf{r}_{N; i}\right)_{i \in A}$. In the following, we write triangular arrays simply as $\{\mathbf{r}_{N;i}\}$, and sequences such as $\{\lambda_n\}_{n\ge 1}$ as $\{\lambda_n\}$. As in \cite{Doukhan1999} and \cite{Kojevnikov2020}, we define $\psi-$dependence as follows.
\begin{definition}[$\psi$-dependence]
\label{depdef}
A triangular array $\{\mathbf{r}_{n;i}\}$, $n\geq 1$, $\mathbf{r}_{n;i} \in \mathbb{R}^{L}$ is $\psi$-dependent if for each ${n \in \mathbb{N}_{+}}$ there exist a sequence $\{\lambda_{n}\} \equiv\left\{\lambda_{n, d}\right\}_{d \geq 0}, \lambda_{n, 0}=1$ and a collection of non-random functions $\left(\psi_{a, b}\right)_{a, b \in \mathbb{N}}, \psi_{a, b}: \mathscr{L}_{v, a} \times \mathscr{L}_{v, b} \to [0, \infty)$, such that for all $A,B \in \mathcal{P}^{+}_{N}(a, b , d)$ for $d>0$ and all $f \in \mathscr{L}_{L, a}$ and $g \in \mathscr{L}_{L, b}$,
$$\left|\operatorname{cov}\left(f\left(\mathbf{r}_{n,A}\right), g\left(\mathbf{r}_{n,B}\right)\right)\right| \leq \psi_{a, b}(f, g) \lambda_{n, d}.$$
\end{definition}
The sequence $\{\lambda_{n}\}$ is called the \emph{dependence coefficients} of $\mathbf{r}_{n;i}$. The covariance of the nonlinear functions of the random vectors $\mathbf{r}_{n,A}$ and $\mathbf{r}_{n,B}$ are bounded by the dependence coefficients $\lambda_{n, d}$ and a functional $\psi_{a, b}(f, g)$, which depends on the size of the sets $A$ and $B$, and the aggregating nonlinear functions $f$ and $g$.
\section{Peer Effects Model and Identification}\label{sid}
There is an arbitrarily large population of agents in the set $\mathcal{I}_{N}$. Each agent $i\in\mathcal{I}_{N}$ is characterized by a set of $K$ observable characteristics $\mathbf{x}_{N;i}$, and an unobserved idiosyncratic shock (error) $\varepsilon_{N;i}$. The agents in the population are connected by two types of networks, that is, $\mathcal{G}_{N}$ and $\mathcal{G}_{N,0}$. The data generation process characterizing the two observed networks adheres to a framework where the network $\mathcal{G}_{N}$ contains connections formed by two agents making endogenous decisions to participate in social or professional relationships, while $\mathcal{G}_{N,0}$ represents the network formed through the random (or quasirandom) allocation of individuals into groups. The endogenous network formation determining $\mathcal{G}_{N}$ induces a potential correlation between the individuals' decisions to connect and their observed and unobserved characteristics \citep[see, e.g., ][]{Hasan2019}. Conversely, by the properties of randomization, the network $\mathcal{G}_{N,0}$ is strictly exogenous \citep[see, e.g., ][]{Athey2017}.
Let $\mathscr{G}$ be the discrete set of possible network configurations of size $N$, such that, for any possible realizations of the network of interest and the exogenous network, $\mathbf{g}$ and $\mathbf{g}_{0}$, it follows that $\left\{\mathbf{g},\mathbf{g}_{0}\right\} \in \mathscr{G}$. Let $\mathbf{X}_{N}=[\mathbf{x}_{N;1},\cdots,\mathbf{x}_{N;N}]^\top\in\mathscr{X}$. Here, $\mathscr{X}$ is the space that represents the support for the matrix of regressors, where we assume $\mathscr{X}\subseteq \mathbb{R}^{N\times K}$, without loss of generality. Finally, let $\boldsymbol{\varepsilon}_{N}=[\varepsilon_{N;1},\cdots,\varepsilon_{N;N}]^\top\in\mathbb{R}^N$. We denote the realizations of $\mathbf{X}_{N}$ and $\boldsymbol{\varepsilon}_{N}$ by $\mathbf{X}\in\mathscr{X}$ and $\boldsymbol{\varepsilon}\in\mathbb{R}^N$, respectively.
Formally, we assume that there is a population joint probability distribution function that determines the dependence patterns between the regressors, the networks, and the errors, that is, the joint distribution is defined as $f_{\mathbf{X}_{N}, \mathcal{G}_{N},\mathcal{G}_{N,0},\boldsymbol{\varepsilon}_{N}}(\mathbf{X},\mathbf{g},\mathbf{g}_{0},\boldsymbol{\varepsilon})= \Pr(\mathcal{G}_{N}=\mathbf{g},\mathcal{G}_{N,0}=\mathbf{g}_{0}\mid\mathbf{X}, \boldsymbol{\varepsilon})f_{\mathbf{X}_{N},\boldsymbol{\varepsilon}_{N}}(\mathbf{X}, \boldsymbol{\varepsilon})$, where $\Pr(\mathcal{G}_{N}=\mathbf{g},\mathcal{G}_{N,0}=\mathbf{g}_{0}\mid\mathbf{X}_{N}, \boldsymbol{\varepsilon}_{N})$ is the probability that $\mathcal{G}_{N}$ and $\mathcal{G}_{N,0}$ take on particular structures $\left\{\mathbf{g},\mathbf{g}_{0}\right\} \in \mathscr{G}$ conditional on the regressors and the errors, while $f_{\mathbf{X}_{N},\boldsymbol{\varepsilon}_{N}}$ represents the joint probability distribution function for $\mathbf{X}_{N}$ and $\boldsymbol{\varepsilon}_{N}$. Note that the joint distribution, $f_{\mathbf{X}_{N}, \mathcal{G}_{N},\mathcal{G}_{N,0},\boldsymbol{\varepsilon}_{N}}(\mathbf{X},\mathbf{g},\mathbf{g}_{0},\boldsymbol{\varepsilon})$, is characterized by two main features: First, \textit{validity}, the network $\mathcal{G}_{N,0}$ is independent of the agents' observed and unobserved characteristics because of randomization. Second, \textit{relevance}, agents randomly assigned to the same group in $\mathcal{G}_{N,0}$ are more likely to form connections in $\mathcal{G}_{N}$ \citep[see, e.g., ][]{Granovetter1973, Gargiulo2000, Kim2006, Goldsmith-Pinkham2013}.
The following assumption imposes the primary validity condition that we call ex-ante exogeneity.
\begin{assumption}[Ex-Ante Exogeneity]
\label{id}
Let $f_{\mathbf{X}_{N}, \mathcal{G}_{N,0},\boldsymbol{\varepsilon}_{N}}(\mathbf{X}, \mathbf{g}_{0}, \boldsymbol{\varepsilon})\equiv\sum_{\mathbf{g}\in\mathscr{G}}f_{\mathbf{X}_{N}, \mathcal{G}_{N},\mathcal{G}_{N,0},\boldsymbol{\varepsilon}_{N}}\allowbreak(\mathbf{X},\mathbf{g},\mathbf{g}_{0},\boldsymbol{\varepsilon})$ be the resulting probability distribution from integrating $f_{\mathbf{X}_{N}, \mathcal{G}_{N},\mathcal{G}_{N,0},\boldsymbol{\varepsilon}_{N}}$ out with respect to the endogenous network $\mathcal{G}_{N}$. Thus, the probability distribution $f_{\mathbf{X}_{N}, \mathcal{G}_{N,0},\boldsymbol{\varepsilon}_{N}}$ is such that $\mathbb{E}[\mathbf{x}_{N;i}\varepsilon_{N;i}]=\boldsymbol{0}_K$ and $\mathbb{E}[(\mathbf{w}_{N,0;i}^{p}\mathbf{X}_{N})^{\top}\varepsilon_{N;i}]=\boldsymbol{0}_K,$ $\forall i\in\mathcal{I}_{N}$, and any $p\geq 1$, where $\boldsymbol{0}_K$ is a $K\times 1$ vector of zeros. Moreover, we assume $\mathbb{E}[\varepsilon_{N;i}]=0$, $\forall i\in\mathcal{I}_{N}$, where the expectation is taken with respect to the marginal distribution of $\boldsymbol{\varepsilon}_{N}$.
\end{assumption}
This assumption imposes the restriction of zero correlation between the observed and unobserved characteristics of any individual $i\in\mathcal{I}_{N}$. For simplicity, we impose the noncorrelation assumption across the entire vector of observed characteristics $\mathbf{x}_{N;i}$; however, our results also hold under the non-correlation premise for a minimum of one variable within $\mathbf{x}_{N;i}$.\footnote{The ex-ante exogeneity assumption can also accommodate correlation between the network $G_{N,0}$ and individuals' characteristics. If the network $\mathcal{G}_{N, 0}$ is determined by \emph{observed} characteristics, one can control for them in the outcome equation and the ex-ante exogeneity assumption becomes a \textit{conditional} ex-ante endogeneity assumption.} Furthermore, the constraint $\mathbb{E}[(\mathbf{w}_{N,0;i}^{p}\mathbf{X}_{N})^{\top}\varepsilon_{N;i}]=\boldsymbol{0}_K$ implies that the unobserved characteristics of the individual $i\in\mathcal{I}_{N}$ remain uncorrelated with the observed characteristics of any other $j\in\mathcal{I}_{N}$, selected exogenously from the population of individuals. Analogous formulations of this assumption, employing randomized networks, have previously been used to support the causal identification of peer effects \citep[see, e.g.,][]{Sacerdote_QJE, Carrell2009, Cai2018, Hasan2019}.
Note that the Assumption \ref{id} does not impose restrictions on the correlation between the vector $(\mathbf{w}_{N;i}\mathbf{X}_{N})$ and the unobserved characteristics $\varepsilon_{N;i}$. Thus, we allow for correlation between the observed and unobserved characteristics of any two individuals engaging in endogenous linking formation within the network $\mathcal{G}_{N}$. The assumption \ref{id} links the endogenous network formation process with the mechanism that induces correlation between the vectors of regressors and errors.
Under ex-ante exogeneity, a pair of agents $(i,j)$, exogenously allocated in an initial network $\mathcal{G}_{N,0}$, establish connections in the network $\mathcal{G}_{N}$ based on observed and unobserved characteristics. This behavior creates correlation between $\mathbf{x}_{N,i}$ and $\varepsilon_{N, j}$ through observed and unobserved homophily. For example, in the context of \cite{Carrell2013}, students with similar abilities may be more inclined to form connections, and observable characteristics such as race or gender could correlate with both link formation and individuals' abilities. Moreover, as previously mentioned, the source of endogeneity can arise not only from unobserved factors, but also because of an unspecified potential simultaneous determination of the outcome and the network of interest. In the same context of educational achievement, for example, students may care about the educational achievement of potential connections, which can generate an issue of simultaneous determination in the outcome equation.
In addition to ex-ante exogeneity, the following assumption imposes a linear model of peer effects, which has been shown to have a structural interpretation as the best response function of a Bayesian game of social interactions; see, i.e., \cite{Blume2015}.
\begin{assumption}[Linear Model]
\label{excrest}
The optimal choice (outcome), $y_{N;i}$, for agent $i$ is characterized by
\begin{equation}
\label{lmm}
y_{N;i}=\beta_{0}\sum_{j \neq i}w_{N;i,j}y_{N;j}+\sum_{j \neq i}w_{N;i,j}\mathbf{x}_{N;j}^{\top}\boldsymbol{\delta}_{0}+ \widetilde{\mathbf{x}}_{N;i}^{\top}\boldsymbol{\gamma}_{0}+\varepsilon_{N;i}\text{,}
\end{equation}
\noindent where $\widetilde{\mathbf{x}}_{N;i}=[1,\mathbf{x}_{N;i}^\top]^\top$, $w_{N;i,j}$ is the $ij$\emph{th} position in the adjacency matrix representing the endogenous network, $\mathbf{W}_{N}$, and $\boldsymbol{\theta}_0\equiv(\beta_{0}, \boldsymbol{\delta}_{0}^\top, \boldsymbol{\gamma}_{0}^\top)^\top$ belongs to the interior of the parameter space $\boldsymbol{\Theta}\subset\mathbb{R}^{2K+2}$, which is assumed to be compact.\footnote{Our Linear Model assumption includes both the case when the adjacency matrices are row normalized and when they are not. Our results hold for both the local aggregate and the local average models \citep{liu2014}.}
\end{assumption}
Assumption \ref{excrest} prescribes a linear model for social effects, effectively imposing an exclusion restriction on the network $\mathcal{G}_{N,0}$. The model posits that only optimally formed connections by agents can induce peer effects. Specifically, we contend that randomly grouped individuals are unlikely to generate peer effects, but, after randomization, agents can form endogenous connections that influence their behavior; see, for example, \cite{Hasan2019}. Similarly, \cite{Carrell2013} shows that groups designed to improve academic performance can produce negative effects due to the role of endogenous study partners and the formation of friendship bonds after the initial allocation of optimally designed to improve academic performance. Furthermore, \cite{Hjort2014} and \cite{Kato2016} find that the effects of peers on worker productivity manifest only between individuals of the same ethnic divisions and social origins, which could serve as proxies for closer social interactions. This evidence supports the plausibility of the exclusion restriction assumption of the initial exogenous network in a peer effects model.
Here, we implicitly impose a network \emph{exclusion} restriction in the sense that potential peer effects from the exogenous network $\mathcal{G}_{N,0}$ are precisely zero in \eqref{lmm}. Alternatively, it is possible to relax the exact exclusion restriction in the exogenous network by incorporating prior information where the effect of $\mathcal{G}_{N,0}$ on $y_{N;i}$ is proximate but not precisely zero, following the approach of \cite{Conley2012}.\footnote{We thank the Associate Editor and one of the referees for pointing this out.}
Finally, the assumption that the parameters $\boldsymbol{\theta}_0$ are in the interior of the parameter space is particularly relevant for the coefficient $\beta_{0}$, because \eqref{lmm} has a solution in terms of $w_{N;i}$, $\mathbf{X}_{N}$, and $\varepsilon_{N;i}$ only when $\beta_{0}<1/\lambda_{\max}$, where $\lambda_{\max}$ is the largest eigenvalue of $\mathbf{W}_{N}$. Assuming $K=1$ and that there is no constant for the sake of illustration, Assumption \ref{excrest} implies that the peer effects regressor can be written as
\begin{equation}
\mathbf{W}_{N} \mathbf{y}_{N}= \gamma_{0} \mathbf{W}_{N} \mathbf{x}_{N}+(\gamma_{0} \beta_{0}+\delta_{0}) \sum_{p=0}^{\infty} \beta_{0}^{p} \mathbf{W}_{N}^{p+2} \mathbf{x}_{N} +\sum_{p=0}^{\infty} \beta_{0}^{p}\mathbf{W}_{N}^{p+1}\varepsilon_{N},
\end{equation}
\noindent which under the condition that $\gamma_{0} \beta_{0}+\delta_{0}\neq0$ shows that, in principle, the powers of the adjacency matrix $\mathbf{W}_{N}$ could be used to instrument $\mathbf{W}_{N} \mathbf{y}_{N}$ \citep{Bramoulle2009, Degiorgi2010, manta2021}. This approach is not possible here because the network $\mathcal{G}_{N}$ is allowed to be endogenous. However, note that from Assumptions \ref{id}, the powers of the adjacency matrix $\mathbf{W}_{N,0}$ are natural candidates to replace $\mathbf{W}_{N}$ in this approach.
We propose to use the random assignment embodied in $\mathbf{W}_{N,0}$ to identify the parameters of a linear model defined on the network space spanned by $\mathbf{W}_{N}$ in Assumption \ref{excrest}. Formally, we define $\mathbf{D}_{N}=[\mathbf{W}_{N}\mathbf{y}_{N}, \mathbf{W}_{N}\mathbf{X}_{N}, \widetilde{\mathbf{X}}_{N}]$ as the matrix of regressors in the matrix notation counterpart of \eqref{lmm} and $\mathbf{Z}_{N}=[\mathbf{W}_{N,0}^{p}\mathbf{X}_{N},\mathbf{W}_{N,0}^{p-1}\mathbf{X}_{N},\dots,\mathbf{W}_{N,0}\mathbf{X}_{N}, \widetilde{\mathbf{X}}_{N}]$ as the matrix that produces the moment conditions formed based on Assumption \ref{id}, where $p>1$ is a constant parameter representing the powers of the adjacency matrix used as instruments. This framework allows for the option of using the characteristics of the so-called \textit{connections of connections}' as instruments by letting $p=2$. The flexibility of Assumption \ref{id} also allows the use of the characteristics of more indirect connections that are at distance $p>2$.
An important aspect differentiating the use of ex-ante exogeneity in Assumption \ref{id} and the standard validity of an instrumental variable is that ex-ante exogeneity allows us to form a large number of instruments. In particular, as long as $\mathbf{I}_{N}$, $\mathbf{W}_{N,0}$,$\mathbf{W}_{N,0}^{2}$,$\dots$, $\mathbf{W}_{N,0}^{p-1}$, and $\mathbf{W}_{N,0}^{p}$ are linearly independent, where $\mathbf{I}_{N}$ is the identity matrix of order $N$, we can form up to $K\cdot p$ different instruments using the $K$ ex-ante exogenous variables in $\mathbf{x}_{N,i}$. Note that as discussed in Section \ref{prelim}, the networks underlying the adjacency matrices $\mathbf{W}_{N}$ and $\mathbf{W}_{N,0}$ in $\mathbf{D}_{N}$ and $\mathbf{Z}_{N}$ can be weighted or unweighted. Considering that both adjacency matrices undergo post-multiplication by the matrix $\mathbf{X}$, we can think of the weighted and unweighted scenarios as computing averages and sums of the attributes of directly and indirectly connected individuals, respectively. As demonstrated by \cite{liu2014}, different weighting methodologies correspond to different behavioral mechanisms in social interactions, resulting in either local average or local aggregate effects. Although behavioral interpretations may differ, the identification argument here holds for both interpretations of peer effects. Another consideration when choosing between weighted or unweighted networks is that achieving the linear independence condition of the powers of adjacency matrices may be more feasible in the former case, see, i.e., \cite{Blume2015}.
The use of the two networks $\mathcal{G}_{N,0}$ and $\mathcal{G}_{N}$ for identification requires a \textit{relevance} condition that guarantees the two networks have enough overlap. The moment characterizing the correlation between the regressors and the instruments is the population average of the expected values of the random matrices $(\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top})$, which can be written as $\mathbb{E}[N^{-1}\sum_{i \in \mathcal{I}_{N}}\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}]$, where $\mathbf{d}_{N;i}$ and $\mathbf{z}_{N;i}$ contain the $i$th rows of the matrices $\mathbf{D}_N$ and $\mathbf{Z}_N$, respectively. Importantly, we do not assume that $\mathbb{E}[\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}]$ are equal for all $i$ and work directly with the population average of these expectations. The following assumption imposes a rank condition related to the strength of the correlation between $\mathbf{z}_{N;i}$ and $\mathbf{d}_{N;i}$.
\begin{assumption}[Relevance]
\label{relevance}
The matrix $\mathbb{E}[N^{-1}\sum_{i \in \mathcal{I}_{N}}\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}]<\infty$ has full column rank.
\end{assumption}
Assumption \ref{relevance} guarantees that the population average of expectations $\mathbb{E}[\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}]$ is finite, and it provides the necessary conditions to ensure that a unique parameter value solves the identifying moment conditions. Unlike standard IV estimation, the relevance condition here imposes restrictions on the more primitive network architectures. \ref{Appendix_D} provides such a set of primitive conditions. They are empirically testable restrictions on the regressors and networks. Importantly, unlike previous results, identification in our model does not require linear independence between the matrices $\mathbf{I}_{N}$, $\mathbf{W}_{N}$, and $\mathbf{W}_{N}^{2}$. Instead, we show that for Assumption \ref{relevance} to hold, there should exist two \emph{ different} numbers $(r,s)\in\mathbb{N}_{+}\times\mathbb{N}_{+}$ such that $\mathbf{I}_{N}$, $\mathbf{W}_{N}^{r}$, and $\mathbf{W}_{N}^{s}$ are linearly independent. Thus, our primitive conditions for the network of interest are weaker in the sense that any two powers of the adjacency matrix have to be linearly independent, and not only the first and second powers.
Furthermore, we show that, as mentioned above, relevance requires that the matrices $\mathbf{I}_{N}$, $\mathbf{W}_{N,0}$,$\mathbf{W}_{N,0}^{2}$,$\dots$, $\mathbf{W}_{N,0}^{p-1}$, and $\mathbf{W}_{N,0}^{p}$ be linearly independent for some $p\in\mathbb{N}_{+}$ and $(\gamma_{0,k} \beta_{0}+\delta_{0, k})\neq 0$ for any $k\in \{1, \dots, K\}$. The first condition resembles the restrictions on the network structure imposed in previous literature, with the difference that in our model the linear independence condition is imposed on the excluded network. The second condition on the structural parameters of the linear model has also been used in the previous literature, see, i.e., \cite{Bramoulle2009}. It excludes the possibility that peer and contextual effects cancel each other out. The following Theorem formalizes the identification result.
\begin{theorem}[Identification]
\label{idtheorem}
Let Assumptions \ref{id}, \ref{excrest}, and \ref{relevance} hold, then $\mathbb{E}[\boldsymbol{m}_{N}(\boldsymbol{\theta})]=\boldsymbol{0}_{K}$ if and only if $\boldsymbol{\theta}=\boldsymbol{\theta}_{0}$, where $\boldsymbol{m}_{N}(\boldsymbol{\theta})\equiv N^{-1}\sum_{i\in\mathcal{I}_{N}}\mathbf{z}_{N;i}(y_{N;i}-\mathbf{d}_{N;i}^{\top}\boldsymbol{\theta})$.
\end{theorem}
Appendix \ref{Appendix_A} presents the proof of Theorem \ref{idtheorem}. This Theorem shows that identification is possible in a context where the network of interest $\mathcal{G}_{N}$ is formed endogenously by taking advantage of the randomization and exclusion restrictions on the exogenously imposed network $\mathcal{G}_{N,0}$. This approach allows us to attach a causal interpretation to the estimated parameters of a linear model of peer effects, which uses observational network data that emerge after an initial randomization. This method can be used to address research designs with randomized peers, as in \cite{Carrell2013}.
\section{Estimation}\label{estimation}
We propose a GMM estimator based on the identifying moment condition in Theorem \ref{idtheorem}. We assume that the analyst observes a sample of size $n<N$ from the population described in the previous section. In our sample scheme, $n$ agents are chosen at random without replacement and their observed characteristics, outcome, and connections in $\mathcal{G}_{N}$ and $\mathcal{G}_{N,0}$ are recorded.\footnote{In our empirical illustration we observe the complete graph for the schools' finite populations of students. However, as in \cite{Graham2020}, we consider the sampling process as a thought experiment that is useful in characterizing limiting distributions.} Therefore, the random sample consists of observations $\left\{y_i,\mathbf{x}_i^{\top},\{w_{i,j}\}_{j=1,j\neq i}^n,\{w_{0;i,j}\}_{j=1,j\neq i}^n\right\}_{i=1}^n$ from which it is possible to calculate the $n\times (2K+2)$ matrix of regressors $\mathbf{D}_{n}$ and the $n \times ((p-1)K+2K+1)$ matrix of \textit{instruments} $\mathbf{Z}_{n}$ (depending on the value of $K$, the system can be \emph{just}- or \emph{over}-identified). The population's GMM objective function is given by $J_{N}(\boldsymbol{\theta})=\mathbb{E}[\boldsymbol{m}_{N}(\boldsymbol{\theta})]^{\top}\mathbf{A}_{N}\mathbb{E}[\boldsymbol{m}_{N}(\boldsymbol{\theta})]$, where $\mathbf{A}_{N}$ is a constant full rank weighting matrix. The GMM estimator of $\boldsymbol{\theta}_0$ is defined as $\widehat{\boldsymbol{\theta}}_{\text{GMM}}=\arg\min_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}J_{n}(\boldsymbol{\theta})$, where $J_{n}(\boldsymbol{\theta})\equiv[n^{-1}\sum_{i\in \mathcal{I}_{n}}\mathbf{z}_{n;i}(y_{n;i}-\mathbf{d}_{n;i}^\top\boldsymbol{\theta})]^{\top}\mathbf{A}_{n}[n^{-1}\sum_{i\in \mathcal{I}_{n}}\mathbf{z}_{n;i}(y_{n;i}-\mathbf{d}_{n;i}^\top\boldsymbol{\theta})]$, the $((p-1)K+2K+1)\times ((p-1)K+2K+1)$ full rank weighting matrix $\mathbf{A}_{n}$ is assumed to converge in probability to $\mathbf{A}_{N}$. The linearity in \eqref{lmm} guarantees that the GMM estimator has a closed form solution given by
\begin{equation}
\widehat{\boldsymbol{\theta}}_{\text{GMM}} = [\mathbf{D}_{n}^{\top}\mathbf{Z}_{n}\mathbf{A}_{n} \mathbf{Z}_{n}^{\top}\mathbf{D}_{n}]^{-1} [\mathbf{D}_{n}^{\top}\mathbf{Z}_{n}\mathbf{A}_{n}\mathbf{Z}_{n}^{\top}\mathbf{y}_{n}]\text{.}\label{theta_GMM}
\end{equation}
To allow the possibility that the observed and unobserved characteristics of individuals are correlated in the joint distribution of the population $f_{\mathbf{X}_{N}, \mathcal{G}_{N},\mathcal{G}_{N,0},\boldsymbol{\varepsilon}_{N}}$, we use the concept of $\psi-$dependence in definition \ref{depdef} above. As mentioned there, we bound the correlation between nonlinear functions of random variables with the \textit{dependence coefficients}, which are decreasing functions of the network distance. We rule out a direct dependence structure based on the exogenous network $\mathcal{G}_{N,0}$, i.e., dependence is only generated through the endogenous network $\mathcal{G}_{N}$. Intuitively, this is justified here because individuals endogenously form connections in $\mathcal{G}_{N}$ based on observed and unobserved characteristics, and therefore we would expect relatively high levels of dependence between individuals close to each other in the network space spanned by $\mathcal{G}_{N}$. For example, in our empirical illustration, we expect the observed and unobserved characteristics of students who study together or are indirectly connected by study partners to be more correlated than those of students who are not study partners and are not indirectly connected. Given that the network $\mathcal{G}_{N}$ generates the dependence structure, when we talk about the distance in the network space, we refer to the distances in the network space $\mathcal{G}_{N}$ hereafter.
Let $\mathbf{r}_{N ; i} \equiv\left[\mathbf{x}_{N ; i}^{\top}, \varepsilon_{N ; i}\right]^{\top} \in \mathbb{R}^{K+1}$ be the vector that encompasses the observed and unobserved characteristics of the individual $i$. By choosing appropriate values for the functions $f$ and $g$ in Definition \ref{depdef}, the $\psi-$dependence framework allows us to bound the dependence between observed and unobserved characteristics among any set of individuals. Specifically, we impose the following assumption on the conditional population distribution $f_{\mathbf{X}_N, \varepsilon_N \mid \mathcal{G}_N}$.
\begin{assumption}[Weak Dependence]
\label{depas}
Consider the set $\mathscr{G}$ of all possible realizations of $\mathcal{G}_{N}$. $\forall\mathcal{G}_{N} \in \mathscr{G}$, and $\mathcal{N}$ denoting either $N\in \mathbb{N}_{+}$ or $n\in \mathbb{N}_{+}$, assume that the conditional distribution $f_{\mathbf{X}_{N}, \boldsymbol{\varepsilon}_{N}\mid \mathcal{G}_{N}}$ is such that:\medskip
\noindent
\refstepcounter{subassumption}
(\textit{\roman{subassumption}})~\ignorespaces \label{ea1} $\{\mathbf{r}_{\mathcal{N};i}\}$ is $\psi$-dependent with dependence coefficient $\lambda_{\mathcal{N}}$; \smallskip
\noindent
\refstepcounter{subassumption}
(\textit{\roman{subassumption}})~\ignorespaces \label{ea2} For a generic constant $C>0$, $\psi_{a, b}(f, g) \leq C \times a b\left(\|f\|_{\infty}+\operatorname{Lip}(f)\right)\left(\|g\|_{\infty}+\operatorname{Lip}(g)\right)$; \smallskip
\noindent
\refstepcounter{subassumption}
(\textit{\roman{subassumption}})~\ignorespaces \label{ea3} and for each ${\mathcal{N} \in \mathbb{N}_{+}}$, $\max_{d \geq 1} \lambda_{\mathcal{N}, d}<\infty$ and $\lim_{d\to\infty}\lambda_{\mathcal{N}, d}=0$.
\end{assumption}
We impose the Assumption \ref{depas} for both the population with conditional distribution $f_{\mathbf{X}_{N}, \boldsymbol{\varepsilon}_{N}\mid \mathcal{G}_{N}}$ and for the triangular array ${\mathbf{r}_{n;i}}$, where $n\geq 1$. This array is formed by randomly sampling networks $\mathcal{G}_{N}$ and the observed and unobserved characteristics that define ${\mathbf{r}_{n;i}}$. Condition \ref{ea2} bounds the functional $\psi_{a, b}(f, g)$ by an arbitrary constant $C$, the cardinality of the sets $A$ and $B$, and the sup-norm and Lipschitz constants of the aggregating functions $f$ and $g$. Intuitively, if the Lipschitz constants $\operatorname{Lip}(f)$ and $\operatorname{Lip}(g)$ increase, the values of the functions $f$ and $g$ can be higher for some values of $\mathbf{r}_{n,A}$ and $\mathbf{r}_{n,B}$, which requires larger constants to bound the covariance. This intuition is similar to the sup-norm. Finally, condition \ref{ea3} requires that the dependence coefficients be finite for any value of $d$ and that they dissipate to zero for a sufficiently large network distance between the random vectors $\mathbf{r}_{n,A}$ and $\mathbf{r}_{n,B}$.
The use of the $\psi-$dependence framework in modeling network dependence has the advantage that it does not impose functional form restrictions on the errors and it allows for correlation between indirectly connected nodes. However, transformations of $\psi$-dependent random variables are not necessarily $\psi$-dependent. Therefore, in order to analyze the asymptotic behavior of $\widehat{\boldsymbol{\theta}}_{\text{GMM}}$, we now impose bounds to covariances of the form $\operatorname{cov}(r_{n; i, q} r_{n; j, \ell}, r_{n; h, q^{\prime}} r_{n; s, \ell^{\prime}})$, where $(i, j, h, s) \in \mathcal{I}_{n}$, $q, q^{\prime}, \ell$, and $\ell^{\prime}$ are components of the vector $\mathbf{r}_{n; i}$. These include covariances such as $\operatorname{cov}(\varepsilon_{n; i} \varepsilon_{n; j}, \varepsilon_{n; h} \varepsilon_{n; s})$ or $\operatorname{cov}(x_{n; i, q}x_{n; j, \ell}, \varepsilon_{n; h} \varepsilon_{n; s})$, for example.
\begin{assumption}[Bound Covariances]
\label{moments}
Define the functions $f_{q,\ell}$ and $g_{q^{\prime},\ell^{\prime}}$ mapping $\mathbb{R}^{(K+1)\times 2}$ to $\mathbb{R}$ to be such that $f_{q,\ell}(\mathbf{r}_{n;\{i,j\}}) = r_{n;i,q}r_{n;j,\ell}$ and $g_{q^{\prime},\ell^{\prime}}(\mathbf{r}_{n;\{h,s\}}) = r_{n;h,{q^{\prime}}}r_{n;s,{\ell^{\prime}}}$ for $(i,j,h,s) \in \mathcal{I}_{n}$, $i\neq j$, $h\neq s$, $q\neq \ell$ and $q^{\prime}\neq \ell^{\prime}$. The norms $\|f_{q,\ell}(\mathbf{r}_{n;\{i,j\}})\|_{p_{f}^{\ast}}+\|g_{q^{\prime},\ell^{\prime}}(\mathbf{r}_{n;\{h,s\}})\|_{p_{g}^{\ast}}<\infty$ for all $q,\ell$ where $p_{f}^{\ast}=\max\{p_{f,i},p_{f,j}\}$ (analogous for $p_{g}^{\ast}$) and $1/p_{f,i}+1/p_{f,j}+1/p_{g,h}+1/p_{g,s}<1$.
\end{assumption}
Assumption \ref{moments} provides sufficient conditions for the functions $f_{q,\ell}$ and $g_{q^{\prime},\ell^{\prime}}$ of $\psi$-dependent random variables to have bounded covariances. The weak dependence in Assumption \ref{depas} guarantees that the dependence coefficients vanish to zero when the network distance increases. However, the network distance $d_{n}(i,j)$ between any two individuals $i$ and $j$ is also a function of the sample size. Therefore, the asymptotic behavior of the dependence coefficients $\lambda_{n, d}$ depends on the asymptotic behavior of the network features determining the distance between nodes. In particular, the density of the network is explicitly related to the geodesic distance. When the density of the network is arbitrarily large, the geodesic distance is always one for any pair of nodes. Therefore, as noted by \cite{Kojevnikov2020}, there is a trade-off between network density and the rate of convergence of the dependence coefficients. Networks with higher density would require the dependence to decrease faster (and vice versa). The following assumption provides a necessary condition on the dependence coefficients for a Law of Large Numbers to apply.
\begin{assumption}[Dependence Rate of Decay]
\label{decay}
Let $\bar{D}_{n}(d)\equiv n^{-1}\sum_{i\in \mathcal{I}_{n}}|\mathcal{P}_{n}(i,d)|$ be the average number of distance-$d$ connections in the network $\mathcal{G}_{n}$. We assume that, for any realizations of the networks $\mathcal{G}_{n}$, for all $n$, it follows that $n^{-1}\sum_{d\geq 1}\bar{D}_{n}(d)\allowbreak \lambda_{n,d}{\longrightarrow}0$ as $n\longrightarrow\infty$.
\end{assumption}
Assumption \ref{decay} is similar to Assumption 3.2 in \cite{Kojevnikov2020}, using the notation in our paper. The key distinction lies in our use of the unconditional version of $\psi$-dependence, which results in the dependence coefficients in the sequence ${\lambda_{n,d}}$ not being random variables. Furthermore, we apply Assumption \ref{decay} conditionally to any realizations of the networks $\mathcal{G}_{n}$, ensuring that $n^{-1}\sum_{d\geq 1}\bar{D}_{n}(d)\allowbreak \lambda_{n,d}$ is nonrandom. As emphasized by \cite{Kojevnikov2020}, this assumption is implied by restrictions on dependence coefficients ($\lambda_{n, d} \leq \theta_{n, d}^{1-4 / p}$, for $p>4$) and the number of distance-$d$ connections on $\mathcal{G}_{n}$ (${D}_{n}(d)\leq 4c_{n}(d, m , k)$, where $c_{n}(d, m , k)$ is defined in \eqref{dense}). The following assumption imposes the existence of moments for products of $\psi$-dependent random variables.
\begin{assumption}[Existence of Moments]
\label{epmoments}
$\exists\epsilon>0$ such that $\sup _{n \geq 1} \max _{i \in \mathcal{I}_{n}}\|R_{n; i, j}\|_{1+\epsilon}<\infty$, where $R_{n;i,j}\equiv r_{n;i,q}r_{n;j,\ell}$, and $\left\|R_{n; i,j}\right\|_{p}\equiv (\mathbb{E}[|R_{n;i,j}|^{p} ])^{1 / p}$.
\end{assumption}
The previous assumptions are sufficient to guarantee that a Law of Large Numbers applies to products of $\psi$-dependent random variables. To show asymptotic normality, we again use the Central Limit Theorem result in \cite{Kojevnikov2020}. As mentioned above, for the asymptotic moments of network-dependent random variables to be well defined, we need to control the level of asymptotic density. In particular, following \cite{Kojevnikov2020}, we define a measure of the average neighborhood size as $\bar{D}_{n}(d , k)=n^{-1} \sum_{i \in \mathcal{I}_{n}}\left|\mathcal{P}_{n}(i , d)\right|^{k}$ and a measure of the average neighborhood shell size as $\bar{D}_{n}(d, m , k)^{-}=n^{-1} \sum_{i \in \mathcal{I}_{n}} \max _{j \in \mathcal{P}_{n}(i,d)}\left|\mathcal{P}_{n}^{-}(i,m) \setminus \mathcal{P}_{n}^{-}(j , d-1)\right|^{k}$, where $\mathcal{P}_{n}^{-}(j , d-1)=\left\{\emptyset\right\}$ when $d=0$. With these two measures of average density, construct the combined quantity,
\begin{equation}
\label{dense}
c_{n}(d, m , k)=\inf _{\alpha>1}\left[\bar{D}_{n}(d, m , k \alpha)^{-}\right]^{\frac{1}{\alpha}}\left[\bar{D}_{n}\left(d, \frac{\alpha}{\alpha-1}\right)\right]^{1-\frac{1}{\alpha}}.
\end{equation}
For some arbitrary position $q$ in the matrix $\mathbf{Z}_{n;i}$, let $S_{n}=\sum_{i\in\mathcal{I}_{n}}z_{n;i,q}\varepsilon_{n;i}$. Defining $\sigma_{n,q}^{2}\equiv\text{var}(S_{n})$, the following assumption guarantees the existence of higher-order moments, imposes asymptotic sparsity, and bounds the long-run variance.
\begin{assumption}[Average Sparsity]
\label{av_sparsity}
For all network realizations $\mathcal{G}_{n} \in \mathscr{G}$,
\refstepcounter{subassumption}
(\textit{\roman{subassumption}})~\ignorespaces \label{clt1} for some $p>4$, $\sup _{n \geq 1} \max _{i \in \mathcal{I}_{n}}\left\|z_{n;i,q}\varepsilon_{n;i}\right\|_{p}<\infty$. There exists a sequence $m_{n}\to\infty$, such that for $k=1,2$,
\refstepcounter{subassumption}
(\textit{\roman{subassumption}})~\ignorespaces \label{clt2} , $\frac{n}{\sigma_{n,q}^{2+k}}\sum_{d \geq 0} c_{n}\left(d, m_{n} , k\right) \lambda_{n, d}^{1-\frac{2+k}{p}} {\longrightarrow} 0$ as $n\to \infty$,
\refstepcounter{subassumption}
(\textit{\roman{subassumption}})~\ignorespaces \label{clt3} $\frac{n^{2} \lambda_{n, m_{n}}^{1-(1 / p)}}{\sigma_{n,q}} {\longrightarrow} 0$ as $n\to \infty$.
\end{assumption}
These conditions impose a convergence rate of the dependence coefficients $\lambda_{n, d}$ that is related to the density of the network. There is a trade-off in which a higher density requires a higher speed in dependence-decreasing patterns. The previous assumptions are sufficient to show that our GMM estimator is consistent and asymptotically normal. Formally, let $\mathbf{\Omega}_{n}=\text{var}(\mathbf{Z}_{n}^{\top}\boldsymbol{\varepsilon}_{n})$ be a variance defined over the set of sampled individuals. It converges (see Lemma \ref{finitevar} in the supplemental material) to the finite population variance,
\begin{equation}
\label{omega}
\mathbf{\Omega}_{N} = \lim_{n \to\infty} n^{-1}\left[\sum_{i=1}^{n} \text{var}(\mathbf{z}_{n;i}\varepsilon_{n;i}) + \sum_{i \neq j}\text{cov}(\mathbf{z}_{n;i}\varepsilon_{n;i}, \mathbf{z}_{n;j}\varepsilon_{n;j})\right]\equiv N^{-1}\sum_{d\geq 0} \mathbf{\Gamma}_{N}(d)<\infty,
\end{equation}
\noindent where $\mathbf{\Gamma}_{N}(d)=\sum_{i \in \mathcal{I}_{N}}\sum_{j \in \mathcal{P}_{N}(i,d)} \mathbb{E}[\mathbf{z}_{N;i}\varepsilon_{N;i}\varepsilon_{N;j}\mathbf{z}_{N;j}^{\top}]$ are the covariances between random variables of individuals at distance $d$. Therefore, the variance-covariance matrix $\mathbf{\Omega}_{N}$ can be calculated by summing the covariances for all possible distances $d\geq 0$. After characterizing $\mathbf{\Omega}_{N}$, Theorem \ref{t2} provides the asymptotic behavior of \eqref{theta_GMM}.
\begin{theorem}
\label{t2}
Let Assumptions \ref{id}--\ref{av_sparsity} hold, then as $n\to\infty$, $\widehat{\boldsymbol{\theta}}_{\text{\emph{GMM}}} = \boldsymbol{\theta}+o_{p}(1)$ and $\sqrt{n}(\widehat{\boldsymbol{\theta}}_{\text{\emph{GMM}}} - \boldsymbol{\theta}) \overset{d}{\to} \mathcal{N}(\boldsymbol{0}, \mathbf{\Sigma}_{N})$, where $\mathbf{\Sigma}_N\equiv(\mathbb{E}[N^{-1}\sum_{i \in \mathcal{I}_{N}}\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}]^{\top} \mathbf{A}_{N}\mathbb{E}[N^{-1}\sum_{i \in \mathcal{I}_{N}}\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}])^{-1}\times(\mathbb{E}[N^{-1}\sum_{i \in \mathcal{I}_{N}}\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}]^{\top}\allowbreak\mathbf{A}_{N}\mathbf{\Omega}_N\mathbf{A}_{N}\times\mathbb{E}[N^{-1}\sum_{i \in \mathcal{I}_{N}}\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}])(\mathbb{E}[N^{-1}\sum_{i \in \mathcal{I}_{N}}\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}]^{\top}
\mathbf{A}_{N}$
\noindent $\mathbb{E}[N^{-1}\sum_{i \in \mathcal{I}_{N}}\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}])^{-1}$, and when $\mathbf{A}_{N}=\mathbf{\Omega}_N^{-1}$, then
\begin{equation}
\mathbf{\Sigma}_N=(\mathbb{E}[N^{-1}\Sigma_{i \in \mathcal{I}_{N}}\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}]^{\top}\mathbf{\Omega}_{N}^{-1}\mathbb{E}[N^{-1}\Sigma_{i \in \mathcal{I}_{N}}\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}])^{-1}\text{.}\label{eff_Sigma}
\end{equation}
\end{theorem}
We present the proof for Theorem \ref{t2} in Appendix \ref{Appendix_A}. The following subsections discuss the relationship between the precision of the estimator in \eqref{theta_GMM} and the levels of sparsity of the endogenous population network of interest, $\mathcal{G}_{N}$.
\subsection{Precision and Sparsity}
Some individuals in the population may not have any connections at distance $p$, and may affect the identifying moment condition in Theorem \ref{idtheorem}. To consider the effect of changes in identifying information on the asymptotic variance-covariance matrix in \eqref{eff_Sigma}, define $\eta_{N,0;i}^{p}$ to be a random variable equal to one if individual $i$ has at least one connection at distance $p$ and zero otherwise. Let $\kappa_{N,0;i}^{p}=\mathbb{E}[\eta_{N,0;i}^{p}]$ be the unconditional probability that the individual $i$ has at least one connection at distance $p$. Define $\boldsymbol{H}_{N,0;i}\equiv\text{diag}(\eta_{N,0;i}^{p}, \dots, \eta_{N,0;i}, 1, \dots, 1)$ to be a $[(p+1)K+1]\times[(p+1)K+1]$ matrix where the first $K$ elements contain the random variables that determine whether the individual $i$ has at least one connection at distance $p$ and the second $K$ elements are the random variables that determine whether or not the individual $i$ has at least one connection at distance $p-1$, etc., until the last $K$ elements associated with $\mathbf{W}_{N,0}\mathbf{X}_{N}$, where $\eta_{N;i}$ is the random variable that determines whether $i$ is isolated in the network $\mathcal{G}_{N,0}$. Finally, the last $K+1$ elements in the lower right submatrix, which coincide with the non-network regressors $\widetilde{\mathbf{x}}_{N;i}$, are ones. Define the $[(p+1)K+1]\times[(p+1)K+1]$-matrix $\boldsymbol{K}_{N,0;i}\equiv\text{diag}(\kappa_{N,0;i}^{p}, \dots, \kappa_{N,0;i}, 1, \dots, 1)$ to be $\mathbb{E}[\boldsymbol{H}_{N,0;i}]$.
Note that when $\eta_{N,0;i}^{p}=0$, the first $K$ elements of $\mathbf{z}_{N;i}$ equal zero. Similarly, when $\eta_{N,0;i}^{p-1}=0$, the second $K$ elements of $\mathbf{z}_{N;i}$ equal zero. The same argument repeats until the $K$ elements associated with $\mathbf{W}_{N,0}\mathbf{X}_{N}$, where if $\eta_{N;i}=0$ (individual $i$ is isolated), the elements of $\mathbf{z}_{N;i}$ associated with the component $(\mathbf{w}_{N,0;i}\mathbf{X}_{N})^{\top}$ are equal to zero. Therefore, by the law of total expectation, $\mathbb{E}[\sum_{i \in \mathcal{I}_{N}}\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}]$ can be written as $\boldsymbol{K}_{N,0;i}\mathbb{E}[\mathbf{z}_{N;i}\mathbf{d}_{N;i}^{\top}\mid \boldsymbol{H}_{N,0;i}^{\ast}\neq \boldsymbol{O}_{pK}]$, where $\boldsymbol{H}_{N,0;i}^{\ast}$ contains the left top $(pK\times pK)$-upper matrix of $\boldsymbol{H}_{N,0;i}$ and $\boldsymbol{O}_{pK}$ is the $pK \times pK$ zero matrix. Therefore, \eqref{eff_Sigma} depends on the population probabilities that an individual provides identification information. For low values of these probabilities, the upper right submatrix of $\boldsymbol{K}_{N;0,i}$ approaches the zero matrix, and the variance-covariance matrix could grow arbitrarily large. In the extreme case of nonidentification, \eqref{eff_Sigma} diverges to infinity. Theorem \ref{t2} exposes a relationship between the precision of network parameters and the sparsity of the network.
\subsection{Efficient Weight Matrix Estimation}
To construct an efficient version of the proposed GMM estimator, we need a consistent estimator of $\mathbf{\Omega}_{N}$. Here we use \citeauthor{Kojevnikov2020}'s \citeyearpar{Kojevnikov2020} network heteroskedasticity and autocorrelation-consistent (HAC) variance estimator. Let $D_{n}$ represent a bandwidth after which the dependence between individuals vanishes. For example, \cite{Kojevnikov2020} proposes $D_{n}=C \times [\log (\text{average degree} \vee(1+0.05))]^{-1} \times \log n$, and this rule of thumb is used in our Monte Carlo simulations and in our empirical study with $C=1.8$ according to their suggestion. The proposed variance-covariance matrix estimator is then given by
\begin{equation}
\widetilde{\mathbf{\Omega}}_{n} =\sum_{d\geq 0}K(d/D_{n}) \frac{1}{n}\sum_{i=1}^{n}\sum_{j \in \mathcal{P}_{n}(i,d)}\mathbf{z}_{n;i}\widetilde{\varepsilon}_{n;i}\widetilde{\varepsilon}_{n;j}\mathbf{z}_{n;j}^{\top},\label{Omega_tilde}
\end{equation}
\noindent where $\widetilde{\varepsilon}_{n;i} = y_{n;i} - \mathbf{d}_{n;i}^{\top}\widetilde{\boldsymbol{\theta}}_{\text{GMM}}$; $K(\cdot)$ is a kernel (weighting) function such that $K(0)=1$ and $K(u)=0$ for $u>1$; and $\widetilde{\boldsymbol{\theta}}_{\text{GMM}}$ is a preliminary consistent estimator. In \eqref{theta_GMM} with $\mathbf{A}_n$ equal to the identity matrix or $n^{-1}\mathbf{Z}_{n}^{\top}\mathbf{Z}_{n}$, the latter was chosen in Monte Carlo exercises and the empirical study. In the second step, the feasible efficient GMM estimator is defined with $\mathbf{A}_n=\widetilde{\mathbf{\Omega}}_{n}^{-1}$ in \eqref{theta_GMM}, which we call $\widehat{\boldsymbol{\theta}}_{\text{GMM}}^{\star}$.
\subsection{Standard Error Calculation}
It follows that the efficient variance-covariance matrix \eqref{eff_Sigma} can be estimated by
\begin{equation}
\left[n^{-1}\mathbf{D}_{n}^{\top}\mathbf{Z}_{n}\widehat{\mathbf{\Omega}}_{n}^{\star-1}n^{-1}\mathbf{Z}_{n}^{\top}\mathbf{D}_{n}\right]^{-1}\text{,}\label{feasible_var_cov}
\end{equation}
\noindent where $\widehat{\mathbf{\Omega}}_{n}^{\star}$ is calculated as in \eqref{Omega_tilde}, but using $\widehat{\boldsymbol{\theta}}_{\text{GMM}}^\ast$ instead. Standard errors can then be calculated taking the square root of the main diagonal elements of \eqref{feasible_var_cov} after dividing them by $n$.
Note that akin to any other IV procedure, potentially weak instruments can be a concern here. For example, within our specific framework, if both observed networks $\mathcal{G}_{n,0}$ and $\mathcal{G}_{n}$ exhibit a high sparsity with minimal overlap between connections, the identification power of the IV generated by the exogenous network would be inherently weak. Consequently, it is empirically recommended to validate the substantial overlap between the two networks and to confirm that the characteristics in $\mathbf{Z}_{n}$ significantly predict the components of $\mathbf{D}_{n}$ in an empirical application (see, e.g., Section \ref{validate_test} in Appendix \ref{Appendix_C}). The development of a theory addressing weak IVs in the network setting is beyond the scope of our paper and therefore left for future research.
\section{Monte Carlo Experiments}\label{mc}
To showcase the versatility of the proposed estimator in this paper, this section documents its performance in two different data-generating processes (hereafter DGPs) where the endogeneity is generated by a simultaneous determination of network formation and outcomes--also known as unobserved homophily--(Design 1) and measurement error in the connections (Design 2). A total of 1,500 data sets $\left\{y_{n;i},x_{n;i},\{w_{n;i,j}\}_{j=1,j\neq i}^n,\{w_{n,0;i,j}\}_{j=1,j\neq i}^n\right\}_{i=1}^n$; with $n\in\left\{50,100,200\right\}$, are generated from \eqref{lmm} by setting $k=1$ and drawing $\left\{x_{n,i}\right\}_{i=1}^n$ as a random sample from a normal distribution with a mean of zero and variance of 3. We set the true vector of the parameters at $\boldsymbol{\theta}_{0}=[\beta_{0}, \delta_{0}, \gamma_{0}]^\top=[0.7,1,1]^\top$. The other data components are constructed using the following rules.
\subsubsection*{Design 1: Unobserved Characteristics with Homophily\label{d3}}
In this design, the outcome variable of the individual $i$, $y_{n,i}$, and the connections $\left\{w_{n;i,j}\right\}_{j=1,j\neq i}^n$ are jointly determined by a common idiosyncratic homophily-related unobserved variable $\varepsilon_{n,1;i}^{\ast}$. First, an exogenous adjacency matrix $\mathbf{W}_{n,0}=[w_{n,0;i,j}]$ from an \citeauthor{Erdos1959}'s \citeyearpar{Erdos1959} random graph with a density of 0.01 is generated along with a $n\times 1$ vector $\boldsymbol{\varepsilon}_{n,1}^{\ast}=[\varepsilon_{n,1;1}^{\ast},\ldots,\varepsilon_{n,1;n}^{\ast}]^\top$ from a multivariate standard normal distribution.\footnote{The density of a network is the ratio between the total numbers of actual ties and of potential ties.} The elements of the endogenous adjacency matrix $\mathbf{W}_{n}=[w_{n;i,j}]$ are then calculated as
\begin{equation*}
w_{n;i,j}=
\begin{cases}
\mathds{1}(|\varepsilon_{n,1;i}^{\ast}-\varepsilon_{n,1;j}^{\ast}|<\widehat{F}_{\varepsilon_{n,1}^\ast}^{-1}(0.95))\times (1-w_{n,0;i,j}) + w_{n,0;i,j} & \text{, if $\varepsilon_{n,1;i}^{\ast}>\Phi^{-1}(0.95)$;} \\
\mathds{1}(|\varepsilon_{n,1;i}^{\ast}-\varepsilon_{n,1;j}^{\ast}|<\widehat{F}_{\varepsilon_{n,1}^\ast}^{-1}(0.95)) \times w_{n,0;i,j} & \text{ if $\varepsilon_{n,1;i}^{\ast}<\Phi^{-1}(0.05)$;} \\
w_{n,0;i,j} & \text{, otherwise};
\end{cases}
\end{equation*}
\noindent where $\widehat{F}_{\varepsilon_{n,1}^{\ast}}^{-1}(0.95)$, which represents the 95\% empirical quantile of the elements of $\boldsymbol{\varepsilon}_{n,1}^{\ast}$, $\varepsilon_{n,1;k}^{\ast}$ represents its $k$th element, and $\Phi^{-1}(\cdot)$ represents the inverse of the cumulative distribution function of a standard normal random variable. The $n\times 1$ vector of outcomes, $\mathbf{y}_{n}$, is then constructed from \eqref{lmm} by setting $\boldsymbol{\varepsilon}_{n}=m \times \boldsymbol{\varepsilon}_{n,1}+\boldsymbol{\varepsilon}_{n,2}$, where $m\in\left\{1,3\right\}$, $\boldsymbol{\varepsilon_{n,2}}$ is drawn from a multivariate standard normal distribution. The elements of $\boldsymbol{\varepsilon}_{n,1}$ are defined as
\begin{equation*}
\varepsilon_{n,1;i}=
\begin{cases}
\varepsilon_{n,1;i}^{\ast} & \text{, if $\varepsilon_{n,1;i}^{\ast}<\Phi^{-1}(0.05)$ or $\varepsilon_{n,1;i}^{\ast}>\Phi^{-1}(0.95)$;} \\
0 & \text{, otherwise}.
\end{cases}
\end{equation*}
This design captures the homophily idea; i.e., agents endowed with a large value of $\varepsilon_{n,1}$ will tend to create / maintain connections with those also endowed with large values of $\varepsilon_{n,1}$ and sever them with those with low values of this unusual unobserved characteristic.
\subsubsection*{Design 2: Misclassified Links\label{d2}}
This is a modified version of \citeauthor{Lewbel_Qu_Tang}'s \citeyearpar{Lewbel_Qu_Tang} Monte Carlo design. While the true DGP involves an unobserved adjacency matrix $\mathbf{W}_{n,0}^{\ast}=[w_{n,0;i,j}^{\ast}]$ generated from a standard \citeauthor{Erdos1959} \citeyearpar{Erdos1959} random network model with a density of 0.01 for size $n$, it is assumed that the empiricist only has access to an adjacency matrix $\mathbf{W}_{n}=[w_{n;i,j}]$, with randomly misclassified links; that is, $w_{n;i, j}=w_{n,0;i,j}^{\ast}e_{n,1;i,j}+(1-w_{n,0;i,j}^{\ast}) e_{n,2;i,j}$ for $i \neq j$ and an exogenous adjacency matrix $\mathbf{W}_{n,0}=[w_{n,0;i,j}]$, where $w_{n,0;i,j}=w_{n,0;i,j}^{\ast}b_{n,1;i,j}+(1-w_{n,0;i,j}^{\ast}) b_{n,2;i,j}$ for $i \neq j$. The $e_{n,1;i,j}$, $e_{n,2;i,j}$, $b_{n,1;i,j}$, and $b_{n,2;i,j}$ are Bernoulli random variables drawn independently from each other $\forall i\neq j$ with parameters $0.5$, $0$, $1-\tau$, and $0.002$, respectively. The design parameter $\tau\in\left\{0.01,0.05\right\}$ controls the probability of misclassification in $\mathbf{W}_{n,0}$. Notice that, as in \cite{Lewbel_Qu_Tang}, nonexisting links are never misclassified in $\mathbf{W}_n$, but misclassification of these nonexisting links is allowed in $\mathbf{W}_{n,0}$ with a very small probability of 0.2\%. However, this design makes the vector of individual outcomes an explicit function of the proportion of misclassification in $\mathbf{W}_n$ for each $i$; that is, the $n\times 1$ vector $\mathbf{y}$ is constructed following Equation \eqref{lmm}, where $\boldsymbol{\varepsilon}_{n}=\boldsymbol{\varepsilon}_{n,1}+\boldsymbol{\varepsilon}_{n,2}$, $\varepsilon_{n,1;i}=1/n\sum_{j=1}^{n}w_{n,0;i,j}^{\ast}e_{n,1;i,j}$, and $\boldsymbol{\varepsilon}_{n,2}$ is drawn from a multivariate standard normal distribution independently of everything else.
\subsubsection*{Results\label{MC_results}}
\noindent Figures \ref{boxplot} and \ref{qqplot} show the results in terms of box plots and Q-Q plots of the Monte Carlo replications. Apart from implementing the proposed efficient GMM estimator described in Theorem \ref{t2} for $p\in{2,3}$, the performance of the standard Ordinary Least Squares (OLS) estimator and the Generalized Two Stage Least Squares (G2SLS) estimator are also included. All adjacency matrices in all designs are row normalized prior to estimation \citep{liu2014}. The calculation of the efficient GMM requires an estimator of the variance-covariance matrix $\mathbf{\Omega}_{n}$. We use the standard two-stage GMM procedure to calculate the efficient weighting matrix. In the first step, we calculate the GMM estimator for $\boldsymbol{\theta}$ setting $\mathbf{A}_{n}=(\mathbf{Z}_{n}^{\top}\mathbf{Z}_{n})^{-1}$. We then use the estimated coefficients in the first step to calculate the network HAC variance estimator in \eqref{Omega_tilde}, where we choose $K(\cdot)$ to be the Parzen kernel, we set the bandwidth $D_{n}=1.8\times [\log (\text { average degree } \vee(1+0.05))]^{-1} \times \log n$ as in \cite{Kojevnikov2020}.
Each panel in Figure \ref{boxplot} displays the performance of the three estimators when the state of a design (Des.) changes by changing the relevant design parameter $m$ or $\tau$. The box plots are based on Monte Carlo replications of OLS (black), G2SLS (dark gray), and the two proposed GMM estimators for $p=2$ (gray) and $p=3$ (light gray) of the social effects. Peer effects ($\beta$), contextual effects ($\delta$), and direct effects ($\gamma$) in \eqref{lmm} are shown with whiskers that show the empirical Monte Carlo quantiles 5\% and 95\%. Across the board, for all parameters, designs, and sample sizes, the proposed GMM estimators perform better than the OLS and G2SLS estimators in terms of bias and sampling variability. As expected, the estimation variability decreases when going from $p=2$ to $p=3$. In contrast, these results also show that naive OLS and G2SLS could potentially lead to estimates with substantial biases in the presence of an endogenous network in a linear-in-means model. On average, the G2SLS underestimates the real value of the peer effects coefficient for the case of misclassified links (Design 2).
Similarly, Figure \ref{qqplot} displays the corresponding Q-Q plots for the GMM based on the standardized version of the Monte Carlo replications of the GMM estimator of the same social effects for sample sizes $n=50$ (light gray), $n=100$ (gray), and $n=200$ (black). The blue dashed line depicts the 45-degree line. This plot shows that the asymptotic normal approximation in Theorem \ref{t2} works well even with a sample as small as 50 observations. Furthermore, as the sample size increases, the approximation improves for all parameters and designs.
\begin{figure}
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\footnotesize
Note: Box plots depict the Monte Carlo performance of OLS (black), G2SLS (dark gray) and the proposed efficient GMM estimator for $p=2$ (gray) and $p=3$ (light gray). The box plots are based on 1,500 replications of Design 1 (Des.1 ) and Design 2 (Des. 2) for sample sizes $n\in\left\{50,100,200\right\}$. The whiskers display the 5\% and 95\% empirical quantiles. Parameters $m$ and $\tau$ control the level of endogeneity and the probability of misclassification in $\mathbf{W}_{n}$, respectively.
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\label{boxplot}
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\begin{figure}
\caption{Q-Q Plots for the GMM Estimator of Social Effects}
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\draw (axis cs:1.5,-1.2) node[
scale=0.5,
anchor=base west,
text=black,
rotate=0.0
]{Design 2};
\draw (axis cs:1.52,-2) node[
scale=0.5,
anchor=base west,
text=black,
rotate=0.0
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\end{groupplot}
\end{tikzpicture}
\vspace{0.4cm}
\begin{minipage}{1\textwidth}
\footnotesize
Note: Q-Q plots are based on the standardized sample of 1,500 Monte Carlo replications of the proposed GMM estimator of the parameters in \eqref{lmm} for Design 1 (Des.1 ) and Design 2 (Des. 2) for sample sizes $n=50$ (light gray), $n=100$ (gray), and $n=200$ (black); $p=3$. The blue dashed line shows the 45 degree line. Parameters $m$ and $\tau$ control the level of endogeneity and the probability of misclassification in $\mathbf{W}_{n}$, respectively.
\end{minipage}
\label{qqplot}
\end{figure}
\section{Empirical Illustration}\label{emp}
We provide an empirical illustration of the proposed methods in the framework of estimating peer effects on academic performance among high school students. The data set was collected between March and May 2011 as part of the Hong Kong Secondary Education Survey in Hong Kong (SESHK). The survey was carried out in the second semester before the final exams and involved three secondary schools with 868 students participating. The sample includes students in the seventh grade of all three schools and students in the eighth and ninth grades of one school (g $\in\{7,8,9\}$). Each grade within a school is made up of five different sections (cl $\in\{1,\ldots,5\}$). Table \ref{tables:table1} in the supplemental material shows the summary statistics for the variables we use. Additional details about these variables can be found in Section \ref{descrip} of the supplemental material.
In the survey, students were asked to report lists of up to ten peers within their grade with whom they discussed schoolwork issues and who were sitting nearby in class during the first semester. We used this information to build the \textit{study partner} and \textit{seatmate} networks, respectively. Classroom seat assignments undergo multiple changes throughout the semester, and the class teacher determines these adjustments. Unlike study partners, the seatmate network is based on proximity and is enforced by the school. Consequently, the seatmate network emerges as a compelling choice to serve as the instrumental network, $\mathbf{W}_{n,0}$, in our analysis. That is, we treat the seatmate network as ex-ante exogenous in this analysis. On the contrary, students have the autonomy to select their study partners, and this choice can be influenced by unobservable characteristics that also affect exam performance, or the study partner choices can depend on other exams results, making the study partner network $\mathbf{W}_{n}$, potentially endogenous. Table \ref{tables:table2} in the supplemental material reports the summary statistics of the network among all students by school.
We fit the Linear-in-Means model for social effects in \eqref{intro_lmm}. Specifically, for the individual $i$, we fit the following version of the model.
\begin{eqnarray} \label{emp_eq1}
\texttt{math}_{i,\text{s}\times\text{g}\times\text{cl}} &=& \alpha + \beta \sum_{j\neq i}^n w_{n;i,j}\texttt{math}_{j,\text{s}\times\text{g}\times\text{cl}} \\ \nonumber
& & {} + \sum_{j\neq i}^n w_{n;i,j}\texttt{characteristics}_{j,\text{s}\times\text{g}\times\text{cl}}^{\prime}\delta_{\texttt{characteristics}} \\ \nonumber
& & {} + \sum_{j\neq i}^n w_{n;i,j}\texttt{personality}_{j,\text{s}\times\text{g}\times\text{cl}}^{\prime}\delta_{\texttt{personality}} \\ \nonumber
& & {} + \texttt{characteristics}_{i,\text{s}\times\text{g}\times\text{cl}}^{\prime}\gamma_{\texttt{characteristics}}
+\texttt{personality}_{i,\text{s}\times\text{g}\times\text{cl}}^{\prime}\gamma_{\texttt{personality}} \\ \nonumber
& & {} + \sum_{\text{s}=1}^{3}\sum_{\text{g}=7}^{9}\sum_{\text{cl}=1}^{5}f_{\text{s}\times\text{g}\times\text{cl}}\times\mathbb{I}\{i\in{\text{s}\times\text{g}\times\text{cl}}\} + \epsilon_i\text{,}
\end{eqnarray}
\noindent where $\texttt{math}_{i,\text{s}\times\text{g}\times\text{cl}}$ is the natural logarithm of the first math test score of student $i$ in class cl, grade g, and school s; that is, $\mathbb{I}\{i\in{\text{s}\times\text{g}\times\text{cl}}\}=1$; $\texttt{personality}_{i,\text{s}\times\text{g}\times\text{cl}}$ includes the natural logarithm of cognitive ability, agreeableness, conscientiousness, extraversion, neuroticism, and openness test scores. Similarly, $\texttt{characteristics}_{i,\text{s}\times\text{g}\times\text{cl}}$ includes variables such as height, weight, indicator variables such as help from siblings, parents help, whether they commute to school by car or taxi; whether they play music; and whether the student is a man. We also include interaction terms between the male indicator and the test scores. Parameters $f_{\text{s}\times\text{g}\times\text{cl}}$ are jointly estimated with the social effects after setting $f_{3\times 9\times 5}=0$. The adjacency matrices are row-normalized before estimating the model as permitted by our theory. The effect of having more peers is captured by including the degree of the network (number of study partners). We also control for the fact that some students study alone (isolated). We set $\delta_{\texttt{personality}} = -\gamma_{\texttt{personality}} $ for all estimation routines to avoid potential collinearity problems. Behaviorally, this restriction implies that only deviations from the students' own personality characteristics from the average of their peers affect the students' tests scores; see, for example, \cite{liu2014}.
The model \eqref{emp_eq1} is estimated using simple Ordinary Least Squares (OLS) with standard errors clustered at the $\text{s}\times\text{g}\times\text{cl}$ level, the Generalized Two-Stage Least Squares (G2SLS) of \cite{Kelejian1998,Kelejian_Prucha_1999_ER}, \cite{Lee2003}, and \cite{Bramoulle2009} with clustered standard errors as in the OLS estimator, and our proposed efficient GMM estimator with $p=5$, $C=1.8$ with the Tukey-Hanning kernel.
\begin{table}[H]
\vspace{-5em}
\setlength{\tabcolsep}{15pt}
\centering
\begin{threeparttable}
\caption{Estimations results}
\vspace{0.1em}
\footnotesize
\begin{tabular}{lccc}
\toprule
Variables & OLS & G2SLS & GMM \\
\midrule
\textbf{Peer effect} & & & \\
ln(Math Test) & 0.2455*** & 0.4534*** & 0.6065*** \\
& (0.0864) & (0.1027) & (0.1755) \\
\midrule
\textbf{Contextual effects} & & & \\
Male & -0.0232 & -0.0153 & -0.2455*** \\
& (0.0311) & (0.0276) & (0.0873) \\
ln(Height) & 0.0794 & 0.0651 & 2.1682*** \\
& (0.2229) & (0.1910) & (0.7400) \\
ln(Weight) & 0.1012 & 0.0399 & 0.0329 \\
& (0.0642) & (0.0600) & (0.1942) \\
Siblings Help & 0.0263 & 0.0305 & 0.0294 \\
& (0.0261) & (0.0217) & (0.0491) \\
Parents Help & 0.0345 & 0.0064 & -0.0106 \\
& (0.0287) & (0.0189) & (0.0517) \\
Commute by Car/Taxi & 0.0277 & 0.0250 & 0.1255** \\
& (0.0266) & (0.0224) & (0.0636) \\
Music & -0.0212 & -0.0090 & 0.1187** \\
& (0.0166) & (0.0160) & (0.0480) \\
\midrule
\textbf{$\dagger$} & & & \\
ln(Cognitive) & 0.0549 & 0.0734** & 0.1086*** \\
& (0.0357) & (0.0369) & (0.0286) \\
ln(Agreeableness) & -0.1145*** & -0.1275*** & -0.0712 \\
& (0.0300) & (0.0356) & (0.0457) \\
ln(Conscientiousness) & 0.0445 & 0.0537 & 0.0799** \\
& (0.0371) & (0.0396) & (0.0406) \\
ln(Extraversion) & -0.0998** & -0.0982** & -0.1164*** \\
& (0.0387) & (0.0384) & (0.0405) \\
ln(Neuroticism) & -0.0386 & -0.0409 & -0.0113 \\
& (0.0378) & (0.0373) & (0.0257) \\
ln(Openness) & 0.0461 & 0.0358 & 0.0272 \\
& (0.0485) & (0.0489) & (0.0414) \\
\midrule
\textbf{Direct effects} & & & \\
Male & -0.7922* & -0.8352** & -0.1648 \\
& (0.4596) & (0.4224) & (0.5417) \\
ln(Height) & -0.3366** & -0.2635** & -0.9904*** \\
& (0.1342) & (0.1210) & (0.2248) \\
ln(Weight) & -0.0088 & -0.0106 & -0.0494 \\
& (0.0293) & (0.0298) & (0.0453) \\
Siblings Help & -0.0099 & -0.0172 & -0.0395* \\
& (0.0176) & (0.0165) & (0.0204) \\
Parents Help & 0.0278* & 0.0225 & 0.0167 \\
& (0.0159) & (0.0142) & (0.0139) \\
Commute by Car/Taxi & -0.0079 & -0.0162 & -0.0251* \\
& (0.0117) & (0.0111) & (0.0136) \\
Degree & 0.0292*** & 0.0264*** & 0.0248*** \\
& (0.0041) & (0.0039) & (0.0034) \\
Isolate Students & 2.2787* & 0.4720* & 0.1425 \\
& (1.2954) & (0.2813) & (0.2935) \\
\midrule
$n$ & 868 & 868 & 868 \\
Adjusted $R^2$ & 0.3372 & 0.3936 & 0.2675 \\
RMSE & 0.1854 & 0.1716 & 0.2002 \\
\bottomrule
\end{tabular}
\label{tables:table3}
\begin{tablenotes}[para,flushleft]
\footnotesize
\raggedright
\item{Note:} (i) \text{*} \(p<0.10\), \text{**} \(p<0.05\), \text{***} \(p<0.01\); (ii) Standard errors are in parentheses. (iii) $\dagger$ These regressors are measured as the deviation of students' personality from their peers' average.
\end{tablenotes}
\end{threeparttable}
\end{table}
Tables \ref{tables:table3} show the results for a subset of all regressors included in \eqref{emp_eq1} (Table \ref{tables:table4} in the supplemental material contains the results for the remaining set of regressors). All estimators show positive spillover effects; however, both OLS and G2SLS are significantly smaller than what the proposed GMM estimator uncovers. Direct-effect coefficients are relatively consistent between different estimators, both in terms of size and direction. Importantly, the signs of most coefficients match previous findings in the literature. For example, cognitive ability and noncognitive characteristics, such as conscientiousness, have a positive effect on math grade achievement \citep{Heckman2001}. There is no gender gap between the grades of men and women. Interestingly, the estimated coefficient for the degree variable suggests that having a larger number of study partners has a positive and significant effect on achievement and that result is robust across different estimators.
Our results suggest that estimators that do not control network endogeneity tend to underestimate peer effects. The observation of a negative bias in estimators that do not account for endogeneity aligns with findings in the literature, which highlight a similar negative bias in standard maximum likelihood estimators of spatial autoregressive models that overlook the reflection issue \citep{Mizruchi2008, Neuman2010}. This trend of negative bias, as observed between OLS and G2SLS, also extends to our GMM estimator, which incorporates control for network endogeneity.
All results are qualitatively robust to different choices of $p$ and $D_n$, see, i.e. Section \ref{supp_estimates_output} in the supplemental material. We perform empirical tests to validate the exclusion restriction in Assumption \ref{excrest} and the relevance condition in Assumption \ref{relevance}. For the exclusion restriction, we perform a Least Squares (LS) regression that includes all variables in \eqref{emp_eq1}, but also includes our proposed instruments. The idea is to measure to what extent the instruments are good predictors of our variable of interest. We interpret the lack of predictability as evidence in favor of our exclusion restriction assumption. Table \ref{tables:table14} in the supplemental material. We find that all our instruments are not predictive of the outcome equation. For the relevance assumption, we perform a series of LS regressions in which the different outcomes are all different endogenous variables. We include all our instruments as regressors and calculate the $F$-statistic for each of the regressions. Consistent with the relevance condition, we reject the hypothesis that our instruments are jointly significant at the 1\% significance level for all our endogenous variables (see Section \ref{validate_test} in the supplemental material).
\section{Conclusion}\label{discussion}
This research adds to the literature on the identification and estimation of social effects with observational network data that often contain endogenous or mismeasured connections. Unlike current approaches, such as those in \cite{Johnsson2019} and \cite{Auerbach2022}, our method does not require the specification and estimation of a model characterizing how connections are created or misclassified. Our method circumvents the imposition of these modeling requirements (along with its potential misspecification issues) by showing how a fully observed set of exogenous connections can be used as an \emph{instrumental} network to uniquely identify and estimate parameters of interest in a widely used linear model of social interactions. Therefore, our approach is semiparametric in nature and hence avoids the usual drawbacks of strong modeling assumptions in this literature.
Another contribution of this research is technical in nature. A byproduct of acknowledging potential network mismeasurement or endogeneity is that it explicitly permits the observed and unobserved characteristics of individuals to be correlated; i.e., creating network dependence across observations in the sample. Our asymptotic results utilize the idea that dependence among observations decreases as a function of their distance in the network; that is, $\psi$-dependence. We show that the resulting estimator can be easily implemented utilizing standard linear GMM estimation routines in popular software like Python, R, or Stata, see, e.g., \cite{netivreg}. The estimator is consistent and asymptotically normal distributed at the standard parametric convergence rate. We characterize the form of the asymptotic variance-covariance matrix that accounts for the network dependence and illustrate how standard errors can be calculated in an empirical application.
Empirically, an important aspect of the proposed methodology is that it recognizes that exogenously imposed connections on individuals do not necessarily cause social effects. However, they can generate new types of freely formed connections that do so; i.e., resorting. The correlation between these two networks is at the heart of our identification and estimation strategy. In this sense, our approach provides an explicit solution to the resorting issue in random network identification strategies, such as in \cite{Moffitt2000}, by distinguishing what type of network creates peer effects (with whom you study, for example) and what other type simply influences these connections, but are otherwise exogenous to the model (for instance, to whom you are randomly assigned to share a physical space). Our empirical and Monte Carlo results show that ignoring the potential network endogeneity can severely bias the network effects estimators. We find significant positive network effects of math test scores among high schoolers from study partners in Hong Kong. These results are in line with previous literature in that they show the existence of strong positive network effects, but suggest that the magnitude of the effects can be larger. We postulate that this could be due to the fact that we focus directly on networks that endogenously emerge after an initial exogenous network assignment.
Finally, the idea of using the initial random assignment of network connections as an instrument differs from using other sources of exogenous variation in two critical aspects. First, under restrictions on exogenous network density, the shape of the network structure together with $K$ regressors can be used to form at least the $K+1$ instruments required to identify endogenous peer effects and contextual effects in the linear-in-means model fitted above. Without the exogenous variation of the randomized network, a researcher would have a difficult task finding $K+1$ different instrumental variables. Second, the standard relevance IV assumption imposes restrictions on the shape of the exogenous network and the process that determines the formation of the network of interest. In particular, we have shown that relevance requires that a number of $p$ powers of the adjacency matrix of the exogenous network to be linearly independent, and connections in the exogenous network need to have an effect on the decision of forming a connection on the endogenous network of interest. The need for linear independence imposes restrictions on the potential randomization of links, which researchers have to follow when designing their experiments. For example, researchers cannot randomly select individuals into groups of the same size if they want to estimate the effects of the network using our method. These are among the important empirical considerations left for future research.
\putbib[typ5.bib]
\end{bibunit}
\emptythanks
\clearpage
\setcounter{page}{1}
\title{\bf Estimating Social Effects with Randomized and Observational Network Data\\ -- Supplemental Materials --}
\author{TszKin Julian Chan\\
Bates White Economic Consulting\\
Juan Estrada\\
Analysis Group Economic Consulting\\
Kim Huynh\thanks{The views expressed in this article are those of the authors. No responsibility for them should be attributed to the Bank of Canada. All remaining errors are the responsibility of the authors.}\hspace{.2cm}\\
Currency Department, Bank of Canada\\
David Jacho-Ch\'{a}vez\\
Department of Economics, Emory University\\
Chungsang Tom Lam\\
Department of Finance, Florida State University\\
Leonardo S\'{a}nchez-Arag\'{o}n\\
Facultad de Ciencias Sociales y Human\'{i}sticas, ESPOL University}
\maketitle
\setcounter{section}{0}