Two-Step Estimation of a Strategic Network Formation Model with Clustering

Two-Step Estimation of a Strategic Network Formation Model with Clustering

Introduction

Network formation has attracted considerable interest from economists due to its many applications to phenomena such as job referrals Beaman2012, favor exchange Jackson_2012, interbank lending Elliott2014, and production networks Acemoglu2020. A challenge in modeling the formation of social and economic networks is that the formation of a link may be influenced by the presence of other links (Jackson_2008; Jackson_2017). For example, individuals get to know each other as friends of friends Jackson_Rogers_2007. Having a friend in common can be an incentive for establishing a relationship Jackson_2012. These externalities from indirect connections can create strategic interactions between links that complicate the empirical analysis of network formation.

To account for this link dependence, empirical models of strategic network formation typically exploit a game-theoretic framework in which the latent utility from forming a link depends on the other links in a network, and the network that individuals form is an equilibrium outcome (see Graham_2020 and Paula_2020 for surveys). Depending on how we specify the information individuals possess, and the strategies they take, significant challenges can arise in the identification, estimation, and computation of model parameters. Miyauchi_2016, Paula_2018, and Sheng_2020 assumed that individuals form links simultaneously under complete information. Because of the prevalence of multiple equilibria, the parameters in general are partially identified.\footnote{These papers considered undirected networks and used pairwise stability Jackson_Wolinsky_1996 as the equilibrium solution.} Mele_2017 and Christakis_2020 circumvented multiplicity by assuming that links in a network are formed in a random sequence. This evolutionary process of network formation provides a particular equilibrium selection mechanism that yields either a unique network or a unique stationary distribution over networks (Jackson_Watts_2002). Strategic interactions under complete information also generate a complex dependence structure, which makes it difficult to establish asymptotic results if one only observes a single large network. Leung_2019 and Menzel_2017 proved a Law of Large Numbers, and Leung_Moon_2021 proved a Central Limit Theorem with further restrictions on sparsity and preferences.

In this paper, we develop a model of strategic network formation under incomplete information. We assume that individuals know the unobserved (by the researcher) utility shocks for their own potential links, but not the unobserved utility shocks for the potential links of the other individuals. Individuals simultaneously choose the links they wish to form, and the directed network they form is a Bayesian Nash equilibrium. The Bayesian Nash equilibrium has been widely used in other network-related models.\footnote{Examples include Blume_2015, who developed an incomplete information game of social interactions where individuals do not observe the utility shocks of other individuals in a network, and Galeotti_2010 and Jackson_Yariv_2007, who explored more general games played on a network where individuals do not observe the private costs or degrees of other individuals.} For network formation, we provide evolutionary results resembling those under complete information (Jackson_Watts_2002; Mele_2017). We argue that a Bayesian Nash equilibrium can be regarded as a long-term equilibrium in a dynamic process of network formation (Myatt_Wallace_2003; Myatt_Wallace_2004; Jackson_Yariv_2007). Incomplete information can offer an advantage over complete information in the econometric analysis. The microfounded assumption of independent private information yields conditional independence between links formed by different individuals, thereby simplifying the asymptotic analysis in a single large network. Leung_2015 pioneered the study of strategic network formation under incomplete information. He assumed that the utility function is additively separable in one's own links. We extend his work to a more general utility function that is nonseparable in one's own links.

Our extension to Leung_2015 is motivated by the empirical regularity that social and economic networks typically present a high degree of clustering (Jackson_2008; Jackson_2017; Graham_2016). This phenomenon occurs in part because two individuals who have a mutual friend may have an increased chance of knowing each other or have stronger incentives to build a cooperative relationship to share risks or exchange favors Jackson_2012. To capture the preference for friends in common, we allow the utility function to depend on the interaction between an individual's two link choices.\footnote{The clustering considered here is different from that in the statistical literature on community detection, which typically assumes that there is a latent community structure in the data (Abbe_2018; Mele_2022).} In a network with $n$ individuals and nonseparable utility, an individual chooses between $2^{n-1}$ overlapping portfolios of links, a seemingly intractable discrete choice problem, as we assume that $n$ grows large. We propose a novel approach that applies the Legendre transform Rockafellar_1970 to the utility function so that the intractable discrete choice problem is transformed into an equivalent tractable sequence of correlated binary choice problems.\footnote{We are grateful to Terence Tao for suggesting this approach.} The dependence between an individual's link choices that results from the preference for friends in common is captured by an auxiliary variable introduced by the Legendre transform. After the transformation we can derive the optimal link choices of an individual explicitly.

We propose a two-step estimation procedure where we estimate the link choice probabilities in the first step, and estimate the model parameters in the second step. Two-step estimation has been widely used in dynamic discrete choice models and games of incomplete information.\footnote{Seminal papers on two-step estimation include Hotz_1993, BBL_2007, Aguirregabiria2007, and BHKN_2010.} We extend this approach to network formation using data from a single large network. Our framework requires that the network is dense, so that the probability of forming a link does not vanish as $n\rightarrow\infty$. The asymptotic analysis is complicated by the fact that the preference for friends in common leads to dependence between an individual's link choices. The auxiliary variable in the Legendre transform provides a useful tool for investigating how the link dependence affects the asymptotic properties of the estimator. We show that the two-step estimator is consistent and asymptotically normal. The link dependence does not affect the rate of convergence but increases the asymptotic variance of the estimator.

While the two-step estimation facilitates computation, accounting for link dependence when computing a link choice probability can be computationally costly in a large network. To make the estimation procedure practical, we show that a link choice probability in a finite-$n$ network converges to a limiting link probability as $n\rightarrow\infty$. The limiting link probability has a closed form that is simple to compute. We provide simulation evidence that using the limiting approximation in the second step yields estimates that are similar to those from the finite-$n$ model, provided that the networks are sufficiently large. In addition, we also discuss how to extend our approach to undirected networks.

We apply our approach to favor exchange networks in rural India. We extend Jackson_2012 by investigating the directed links in favor exchange, that is, who offers a favor to whom. We find that indirect links have significant effects on favor provision; ignoring these spillover effects will overestimate the homophily effects. More strikingly, we find that the effect of support from a mutual link depends critically on the direction of the support from the provider's perspective. Individual $i$ is more likely to offer a favor to individual $j$ if $i$ offers a favor to $j$'s favor-exchange companion $k$ instead of being offered a favor by $k$. These directional results complement the findings in Jackson_2012 and Leung_2015 and shed further light on the outcomes of policy interventions.

The remainder of the paper is organized as follows. Section (ref) introduces the model, including the utility function, the information structure, and the equilibrium. Section (ref) derives an explicit expression for the optimal link choices of an individual. Section (ref) presents the two-step estimator and its asymptotic properties. Section (ref) explores extensions to our approach, including undirected networks and the limiting approximation. Section (ref) discusses the empirical application. Section (ref) concludes the paper. Additional results are presented in the Online Appendix.

Model

Consider a set of $n$ individuals who choose to form a network. Each individual $i$ is endowed with a vector of observed characteristics $X_{i}$ with support $\mathcal{X}$, and a vector of unobserved link-specific utility shocks $\epsilon_{i}=(\epsilon_{i1},\ldots,\epsilon_{i,i-1},\epsilon_{i,i+1},\ldots,\epsilon_{in})'\in\mathbb{R}^{n-1}$, where $\epsilon_{ij}$ is the utility shock for link $ij$. Let $X=(X_{1}',\ldots,X_{n}')'\in\mathcal{X}^{n}$ denote the characteristic profile and $\epsilon=(\epsilon_{1}',\ldots,\epsilon_{n}')'\in\mathbb{R}^{n(n-1)}$ denote the utility shock profile.

The network formed is denoted by an $n\times n$ binary matrix $G\in\mathcal{G}$, where the $ij$th entry $G_{ij}=1$ if individual $i$ forms a link to individual $j$ and $G_{ij}=0$ otherwise. The diagonal elements $G_{ii}$ are set to $0$ for all $i$, so there are no self-links. In this paper, we focus on directed networks, that is, $G_{ij}$ and $G_{ji}$ can be different. While relationships such as friendships and collaborations are typically undirected, many economic networks are in fact formed as a result of directed individual decisions. Examples include a village resident lending money to another resident, a buyer purchasing a product from a seller, and an employee referring a candidate for a job. Following Bala_2000, Mele_2017, and Leung_2015, we consider a noncooperative framework where individual $i$ unilaterally decides to form the link $ij$.\footnote{Equivalently, we can characterize the formation of a link as a bilateral decision between the provider and the recipient in which the recipient always prefers the link. For example, a village resident always likes to receive a favor, a seller always wants to sell a product (given the price), and a job candidate always wishes to get a job referral.} In Section (ref), we extend our analysis to undirected networks.

\paragraph*{Utility.}

For a given characteristic profile $X$ and utility shock vector $\epsilon_{i}$, individual $i$'s utility in a network $G$ is given by

where $G_{-i}$ is the submatrix of $G$ with the $i$th row deleted, that is, the links formed by individuals other than $i$. We assume that the utility function is known up to the parameter $\theta_{u}=(\beta',\gamma')'$ in a compact set $\Theta_{u}\subset\mathbb{R}^{d_{u}}$.

In this specification, the term $u_{ij}(G_{-i},X;\beta)$ represents individual $i$'s incremental utility from linking to individual $j$ that does not depend on the other links that $i$ forms. A typical specification of $u_{ij}(G_{-i},X;\beta)$ is

where $d(X_{i},X_{j})$ represents a vector of known functions of $X_{i}$ and $X_{j}$ that measure the social proximity between $i$ and $j$; for example, whether they have the same gender, age, education, and caste. This term captures the homophily effect (Jackson_2008; Jackson_2017). The last three terms in equation ((ref)) capture the spillover effects from other links that involve $j$, including the reciprocity effect of link $ji$ ($\beta_{4}$) and the effects of $j$'s in-degree ($\beta_{5}$) and out-degree ($\beta_{6}$). Note that we normalize $j$'s in-degree and out-degree by $n-2$ to ensure that these terms remain bounded as $n\rightarrow\infty$, so that they do not dominate when $n$ is large. The specification in equation ((ref)) is similar to that in Leung_2015.

In addition to utility that is separable in individual $i$'s links, we also allow for utility that is nonseparable in individual $i$'s links. The term $v_{i,jk}(G_{-i},X;\gamma)$ represents the incremental utility that $i$ derives from linking to both individual $j$ and individual $k$, if $j$ and $k$ link to each other. An important example is

The two terms are motivated by the prevalence of triadic closure ($\gamma_{1}>0$) and cyclic closure ($\gamma_{2}>0$), which mean that individual $i$ is more likely to link to individual $j$ if $i$ links to a third individual $k$ who is connected to $j$ directly or indirectly (Kossinets_Watts_2006; Jackson_2008; Jackson_2017).\footnote{The terms $G_{jk}+G_{kj}$ and $G_{jl}G_{lk}+G_{kl}G_{lj}$ in equation ((ref)) can be replaced by other functions that are symmetric in $j$ and $k$. For example, we can replace $G_{jk}+G_{kj}$ by $G_{jk}G_{kj}$ (i.e., both $j$ links to $k$ and $k$ links to $j$) or $1\{G_{jk}+G_{kj}\geq1\}$ (i.e., either $j$ links to $k$ or $k$ links to $j$).} One possible reason for triadic/cyclic closure is that via a mutual friend $k$, individual $i$ may have an increased chance to know $j$. Another reason---one relevant particularly in favor exchange and risk sharing networks---is that linking to a third individual $k$ whom $i$ trusts and who trusts $j$ may give $i$ the basis to trust $j$ (Easley_2010; Karlan2009; Jackson_2012).\footnote{In the context of directed links, while certain variants of triadic and cyclic closure statistics can be specified through the separable component $u_{ij}$ (e.g., the supported trust in Leung_2015), we demonstrate in the empirical application in Section (ref) that the empirically relevant variants of these statistics may inevitably require the nonseparable component $v_{i,jk}$.} These mutual-friend effects can also depend on the social proximity between $j$ and $k$, as captured by $\gamma_{1}(X_{j},X_{k})$ and $\gamma_{2}(X_{j},X_{k})$, which consist of known nonnegative functions of $X_{j}$ and $X_{k}$, such as whether $j$ and $k$ share certain characteristics and a vector of parameters.\footnote{One example is $\gamma_{1}(X_{j},X_{k})=d_{1}(X_{j},X_{k})'\gamma_{1}$ and $\gamma_{2}(X_{j},X_{k})=d_{2}(X_{j},X_{k})'\gamma_{2}$, where $d_{1}(X_{j},X_{k})$ and $d_{2}(X_{j},X_{k})$ are vectors that measure the social distance between $j$ and $k$.} Note that $v_{i,jk}(G_{-i},X;\gamma)$ is symmetric in $j$ and $k$ so that the utility function does not depend on how we label the individuals.\footnote{This requires that $\gamma_{1}(X_{j},X_{k})$ and $\gamma_{2}(X_{j},X_{k})$ are symmetric in $j$ and $k$.} The second term in equation ((ref)) is also normalized to guarantee its boundedness for large $n$.

\paragraph*{Information.}

Most literature on network formation games assumes that individuals have complete information about the game Jackson_Wolinsky_1996,Bala_2000,Christakis_2020,Mele_2017,Miyauchi_2016,Paula_2018,Sheng_2020,Menzel_2017. While this is appropriate in small networks, in a large network an individual may not observe every aspect of the other individuals. In this paper, we follow Leung_2015 and assume that each individual only has partial information about the other individuals. In particular, we assume that the characteristic profile $X$ is observed by all the individuals, but the utility shock vector $\epsilon_{i}$ is observed by individual $i$ only.\footnote{This is a standard setup for games of incomplete information (e.g., BHKN_2010).} We also assume that the utility shocks are i.i.d. and are independent of the characteristics. Formally,

The independence of $\epsilon_{i}$ across $i$ is a crucial assumption. It enables us to break the link dependence across individuals and reduce the complexity of the model. The independence of $\epsilon_{ij}$ and $\epsilon_{ik}$ is imposed for simplicity.\footnote{Leung_2015 allows $\epsilon_{ij}$ and $\epsilon_{ik}$ to be arbitrarily correlated, which generates persistent correlation between $G_{ij}$ and $G_{ik}$, leading to a rate of convergence slower than ours.} Assumptions (ref)(ii)--(iii) are standard regularity assumptions.

\paragraph*{Equilibrium.}

We assume that individuals form links simultaneously. Let $G_{i}$ be the $i$th row of network $G$, that is, the links formed by individual $i$, and $\mathcal{G}_{i}=\{0,1\}^{n-1}$ the set of all possible $G_{i}$. A strategy of individual $i$ is a function $G_{i}(X,\epsilon_{i}):\mathcal{X}^{n}\times\mathbb{R}^{n-1}\rightarrow\mathcal{G}_{i}$ that maps $i$'s information $(X,\epsilon_{i})$ to a row vector of links $G_{i}$. Denote the strategy profile of all individuals by $G(X,\epsilon)=(G_{1}(X,\epsilon_{1})',\ldots,G_{n}(X,\epsilon_{n})')'$. A Bayesian Nash equilibrium (or an equilibrium for short) of the game is a strategy profile $G(X,\epsilon)$ such that each $G_{i}(X,\epsilon_{i})$ maximizes the expected utility $\mathbb{E}[U_{i}(G_{i},G_{-i},X,\epsilon_{i})|X,\epsilon_{i}]$, where the expectation is taken with respect to the strategies of individuals other than $i$, $G_{-i}$.

For the utility function in ((ref)), the expected utility of individual $i$ is

Under the specifications in ((ref))--((ref)), we have

and

where $\sigma_{ij}(X)=\mathbb{E}[G_{ij}|X]$. The expressions for $\mathbb{E}[u_{ij}(G_{-i},X)|X]$ and $\mathbb{E}[v_{i,jk}(G_{-i},X)|X]$ follow because the independence of $\epsilon_{i}$ across $i$ implies that $\epsilon_{i}$ is independent of the strategies of others $G_{-i}$ conditional on $X$ and hence $\mathbb{E}[u_{ij}(G_{-i},X)|X]$ and $\mathbb{E}[v_{i,jk}(G_{-i},X)|X]$ only depend on the public information $X$. Equation ((ref)) holds also because, conditional on $X,$ the strategies $G_{j}$ (or $G_{k}$) and $G_{l}$ are independent.

Following the literature on incomplete information games BHKN_2010, we can represent an equilibrium in the space of conditional choice probabilities. Given $X$, let $\sigma_{i}(g_{i}|X)$ denote the conditional probability that individual $i$ chooses link vector $g_{i}$

and $\sigma(X)=\{\sigma_{i}(g_{i}|X),g_{i}\in\mathcal{G}_{i},i=1,\ldots,n\}$ the conditional choice probability (CCP) profile. The right-hand side of equation ((ref)) defines a mapping from $\sigma_{-i}(X)=\{\sigma_{j}(g_{j}|X),g_{j}\in\mathcal{G}_{j},j\neq i\}$ to $\sigma_{i}(g_{i}|X)$ that we denote by $\mathcal{P}_{i}(g_{i}|X,\sigma_{-i}(X))$. An equilibrium CCP profile $\sigma^{\ast}(X)$ is a fixed point of the equation

for all $g_{i}\in\mathcal{G}_{i}$ and all $i=1,\ldots,n$. From an equilibrium CCP profile, we can derive the equilibrium strategy profile from the optimal decision of each individual $i$ for a given $X$ and $\epsilon_{i}$. Therefore, we can represent an equilibrium equivalently by the CCP profile.

The Bayesian Nash equilibrium, though less common in network formation, has been widely used in other network-related models. For example, Blume_2015 developed an incomplete information game of social interactions where individuals do not observe the private utility shocks of other individuals in a network. They show that Bayesian Nash equilibria of the game provide a microfoundation that can nest the standard social interaction models such as Manski_1993. Galeotti_2010 and Jackson_Yariv_2007 considered more general games played on a network, where an individual's payoff depends on the actions taken by neighbors. In such a game, individuals do not observe the private costs or degrees of other individuals. They form beliefs about the degrees of their neighbors based on the degree distribution in the network. Both Galeotti_2010 and Jackson_Yariv_2007 investigated Bayesian Nash equilibria of the game.

One concern regarding the application of the Bayesian Nash equilibrium to network formation is that links may not be formed simultaneously. In network formation games with complete information, equilibrium solutions that assume simultaneous move (e.g., the Nash equilibrium for directed networks and pairwise stability for undirected networks) are usually justified by an evolutionary process that converges to equilibria in the static game. For example, Jackson_Watts_2002 developed a dynamic process of network formation that converges to pairwise stable networks if cycles are ruled out. Mele_2017 considered a similar dynamic process for directed networks that converges to Nash equilibria in the static game. In Online Appendix (ref), we show that similar evolutionary results can be established for the Bayesian Nash equilibrium. Specifically, we construct two dynamic processes of network formation where links are formed over time. The first process assumes that one link is updated in each period as in Myatt_Wallace_2004, and the second process assumes that all links are updated in each period as in Myatt_Wallace_2003. Unlike the dynamic processes in Jackson_Watts_2002 and Mele_2017, where an active individual observes all the links formed in previous periods, we assume that an active individual only observes the distribution of the links formed previously. Following Myatt_Wallace_2004 and Myatt_Wallace_2003, we show that for a sufficiently large network, the first process generates a Markov chain of networks that has a unique limiting distribution with local modes coinciding with stable Bayesian Nash equilibria, and the second process converges to a Bayesian Nash equilibrium in probability. The result in our second process is in line with Jackson_Yariv_2007, who also showed that Bayesian Nash equilibria in a static game are equivalent to steady states of a dynamic process. These evolutionary results suggest that a Bayesian Nash equilibrium can be regarded as a long-term equilibrium in a dynamic process of network formation.

\paragraph*{Symmetric Equilibrium.}

In this paper, we focus on symmetric equilibria where observationally identical individuals have the same choice probabilities. In a symmetric equilibrium, the CCP profile $\sigma(X)$ satisfies that for any individuals $i$ and $j$ with $X_{i}=X_{j}$, we have $\sigma_{i}(g_{i}|X)=\sigma_{j}(g_{j}|X)$ for all $g_{i}\in\mathcal{G}_{i}$ and $g_{j}\in\mathcal{G}_{j}$, with $g_{j}$ obtained from $g_{i}$ by swapping its $i$th and $j$th components $g_{ii}$ and $g_{ij}$. Simply put, individuals with the same observed characteristics choose their links with the same probability.\footnote{This restriction does not rule out the possibility that two observationally equivalent individuals form different links in an observed network because they can have different unobserved utility shocks.} This restriction is motivated by the observation that the utility function is the same for all individuals, so if they hold symmetric beliefs about the decisions of others (as specified in a symmetric CCP profile), then individuals of any given $X_{i}$ and $\epsilon_{i}$ face the same decision problem. The optimal decision in it is unique with probability one, leading to symmetry in the choice probabilities. The symmetry of an equilibrium guarantees that the conditional choice probabilities of an individual do not depend on how we label the individuals, a desirable feature in most networks where the identities of individuals do not play any role and individuals are labeled arbitrarily.

In Proposition (ref), we establish the existence of a symmetric equilibrium. Our proof is similar to that in Leung_2015. We assume that in observed data, individuals coordinate on a symmetric equilibrium, independently of the utility shocks $\epsilon$.\footnote{The symmetry implies that the expected utility terms in ((ref)) and ((ref)) depend on $i$, $j$ and $k$ only through $X_{i}$, $X_{j}$, and $X_{k}$.} There may be multiple symmetric equilibria that satisfy condition ((ref)).

A game with multiple equilibria is considered incomplete unless restrictions are imposed on the equilibrium selection mechanism. In data scenarios with many markets, it is often assumed that the selection mechanism is degenerate, meaning that any market with observationally equivalent individuals must select the same equilibrium BHKN_2010. This assumption ensures that the CCPs can be estimated by pooling observations across markets Paula_2013. In contrast, we build on the insight of Leung_2015 and restrict the selection mechanism to select only symmetric equilibria. This approach allows us to estimate the CCPs by pooling pairs of individuals in a single large network.

The main challenge in analyzing the model involves characterizing the optimal decision of an individual. Because the expected utility depends on the interaction $G_{ij}G_{ik}$, an individual no longer chooses between separable links as in Leung_2015, but between portfolios of links. This is a multinomial discrete choice problem with $2^{n-1}$ overlapping alternatives. Note that links $G_{ij}$ and $G_{ik}$ are strategic complements (substitutes) if $\gamma_{1},\gamma_{2}>0$ ($<0$). The nonseparable decision over links naturally leads to the links being dependent on one another. In the subsequent section, we develop a novel method to derive the optimal decision of an individual and characterize the link dependence.

In addition, this challenge may arise in other applications where individuals select a set of binary choices that are complements or substitutes for one another. Formally, suppose individual $i$ has a set of binary choices $D_{ij},j\in\mathcal{C}=\{1,\dots,n_{c}\}$, where the number of choices $n_{c}$ is large. The utility of individual $i$ is nonseparable in $D_{ij}$ and includes the term $\frac{1}{n_{c}(n_{c}-1)}\sum_{j}\sum_{k\neq j}D_{ij}D_{ik}v_{i,jk}$, where $v_{i,jk}$ captures the complementarity or substitutability between $D_{ij}$ and $D_{ik}$.\footnote{$v_{i,jk}$ does not need to depend on an equilibrium, as it does in our model. Instead, it can be flexibly specified according to the context.} We illustrate this setting with several examples. Our method can be used to derive the optimal choices in these cases.

Optimal Link Choices

In this section, we develop an approach that yields an explicit expression for the optimal link choices of an individual. The idea is to find an auxiliary variable that captures the strategic interactions between an individual's link choices, so that after the inclusion of this auxiliary variable the link choices become correlated binary choices, with the correlation captured by the auxiliary variable.

Recall that the incremental utility $v_{i,jk}(G_{-i},X)$ is symmetric in $j$ and $k$. Moreover, in a symmetric equilibrium $\sigma$ the expected incremental utility $\mathbb{E}[v_{i,jk}(G_{-i},X)|X,\sigma]$ depends on $j$ and $k$ only through the values of $X_{j}$ and $X_{k}$.\footnote{The inclusion of $\sigma$ in the notation indicates that the expectation is taken according to $\sigma$. } These symmetry properties imply that $\mathbb{E}[v_{i,jk}(G_{-i},X)|X,\sigma]$ is a symmetric function of $X_{j}$ and $X_{k}$.

To facilitate the exposition, we focus on the case where $X_{i}$ is discrete. Assume that $X_{i}$ takes a finite number of values, which we refer to as the types of an individual.\footnote{It is more complicated to derive the optimal link choices when $X_{i}$ is continuous, as the matrix notation must be replaced with linear operators. Exploring the theoretical results for continuous $X_{i}$ is beyond the scope of this paper. In practice, our approach can be applied by discretizing continuous covariates, as demonstrated in the empirical application in Section (ref).}

Under Assumption (ref), we can represent the expected utility in ((ref)) in matrix form. For $1\leq s,t\leq T$, let $V_{i,st}(X,\sigma)$ denote the value of $\mathbb{E}[v_{i,jk}(G_{-i},X)|X,\sigma]$ if individuals $j$ and $k$ are of types $x_{s}$ and $x_{t}$ respectively; that is, $V_{i,st}(X,\sigma)=\mathbb{E}[v_{i,jk}(G_{-i},X)|X_{j}=x_{s},X_{k}=x_{t},X,\sigma]$. Arrange the $T^{2}$ type-specific expected incremental utilities $V_{i,st}(X,\sigma)$ in a $T\times T$ matrix $V_{i}(X,\sigma)=(V_{i,st}(X,\sigma))\in\mathbb{R}^{T\times T}$. Because $V_{i,st}(X,\sigma)$ is symmetric in $s$ and $t$, $V_{i}(X,\sigma)$ is a symmetric matrix. Using the matrix notation, we can represent the expected utility in ((ref)) as

where $Z_{j}=(1\{X_{j}=x_{1}\},\ldots,1\{X_{j}=x_{T}\})'$ is a $T\times1$ vector of binary variables that indicates the type of individual $j$, and $U_{ij}(X,\sigma)=\mathbb{E}[u_{ij}(G_{-i},X)|X,\sigma]-\frac{1}{2(n-2)}Z'_{j}V_{i}(X,\sigma)Z_{j}$. The term $Z'_{j}V_{i}(X,\sigma)Z_{k}$ represents the expected incremental utility that individual $i$ receives from linking to both $j$ and $k$.

To derive the optimal decision of individual $i$, we \textquotedbl linearize\textquotedbl the quadratic term in ((ref)) using the Legendre transform Rockafellar_1970. Observe that $V_{i}(X,\sigma)$ is real and symmetric and thus has a real spectral decomposition

where $\Lambda_{i}(X,\sigma)=\text{diag}(\lambda_{i1}(X,\sigma),\ldots,\lambda_{iT}(X,\sigma))$ denotes the $T\times T$ diagonal matrix of eigenvalues in $\mathbb{R}$ and $\Phi_{i}(X,\sigma)=(\phi_{i1}(X,\sigma),\ldots,\phi_{iT}(X,\sigma))$ denotes the $T\times T$ orthogonal matrix of eigenvectors in $\mathbb{R}^{T}$. Using the spectral decomposition, we can express the quadratic term in ((ref)) as a function of the squares of $\frac{1}{n-1}\sum_{j\neq i}G_{ij}Z'_{j}\phi_{it}(X,\sigma)$, $t=1,\ldots,T$, which are linear in link choices $G_{ij}$.

Next, we “linearize” these squares of linear functions using a special case of the Legendre transform. In particular, for any scalar $y\in\mathbb{R}$, we have

where $\omega\in\mathbb{R}$ is a scalar auxiliary variable. By choosing $y=\frac{1}{n-1}\sum_{j\neq i}G_{ij}Z'_{j}\phi_{it}(X,\sigma)$, we can replace its square by the maximization on the right-hand-side of ((ref)). This maximization has an objective function that is linear in $y$ and thus linear in the link choices $G_{ij}$. The transformation of the expected utility is presented in Lemma (ref).

The optimal decision of individual $i$ is a link vector $G_{i}\in\mathcal{G}_{i}$ that maximizes her expected utility. By Lemma (ref), the expected utility can be expressed as the optimal value obtained from optimizing over the components of the auxiliary variable $\omega=(\omega_{1},\dots,\omega_{T})'\in\mathbb{R}^{T}$. Note that the transformed expected utility in ((ref)) is separable in each maximization. Therefore, if we move $\lambda_{it}(X,\sigma)$ inside the maximization over $\omega_{t}$, the maximization remains unchanged if $\lambda_{it}(X,\sigma)>0$ and switches to a minimization if $\lambda_{it}(X,\sigma)<0$, leading to a maximin problem over $\omega$ in general. The separability also implies that the order of the maximizations and minimizations does not matter. If we can further interchange the maximization over $G_{i}$ and the maximin over $\omega$, we can solve for the optimal $G_{i}$ first from a simple maximization with an objective function linear in $G_{i}$. This optimal $G_{i}$ is evidently a function of $\omega$. By solving for the optimal $\omega$ next and evaluating the optimal $G_{i}$ at the optimal $\omega$, we can derive the optimal decision that maximizes the expected utility. The validity of this approach and the derivation of the optimal decision are demonstrated in Theorem (ref).

To gain some intuition about the auxiliary variable $\omega_{i}(\epsilon_{i},X,\sigma)$ in ((ref)), multiplying the first-order condition of problem ((ref)) (see Lemma (ref)) by $\frac{n-1}{n-2}Z'_{j}\Phi_{i}(X,\sigma)$, we derive that

almost surely, where $G_{ik}(\epsilon_{i},X,\sigma)$ is defined in ((ref)). The left-hand side of ((ref)) is the component added to the latent utility in ((ref)). The right-hand side of ((ref)) interprets this component as the expected incremental utility from friends in common. We interpret it as such because if individual $i$ contemplates a link to individual $j$, she anticipates that her friend $k$ can potentially become a mutual friend with $j$. If individual $j$ is of type $x_{s}$ and individual $i$'s friend $k$ is of type $x_{t}$, then $i$'s expected utility from this potential friend in common is $V_{i,st}(X,\sigma)$. Taking the average over all friends of individual $i$, we obtain the expected incremental utility from friends in common if individual $i$ links to individual $j$. By adding this component to the latent utility, we internalize the strategic interactions between the link choices due to the preference for friends in common, so that the optimal decision breaks down into a collection of binary choices.

The auxiliary variable $\omega_{i}(\epsilon_{i},X,\sigma)$ provides an explicit expression for the dependence of the links formed by individual $i$. Note that $\omega_{i}(\epsilon_{i},X,\sigma)$ is a function of $\epsilon_{i}$ because it is an optimal solution to problem ((ref)) whose objective function depends on $\epsilon_{i}$. The randomness in $\omega_{i}(\epsilon_{i},X,\sigma)$ leads to dependence between the link choices. From ((ref)) we can see that two link choices $G_{ij}$ and $G_{ik}$ are dependent either through the presence of $\omega_{i}(\epsilon_{i},X,\sigma)$ in both links, or through the dependence between the utility shock $\epsilon_{ij}$ and $\omega_{i}(\epsilon_{i},X,\sigma)$ in $G_{ik}$ or symmetrically between the utility shock $\epsilon_{ik}$ and $\omega_{i}(\epsilon_{i},X,\sigma)$ in $G_{ij}$. This explicit characterization of the link dependence is useful for studying the asymptotic properties of an estimator.

Estimation

We now turn our attention to estimating the parameter $\theta=(\theta'_{u},\theta'_{\epsilon})'$. We propose a two-step estimation procedure, where we estimate conditional link choice probabilities in the first step and estimate the parameter $\theta$ in the second step Leung_2015. The asymptotic analysis of the estimator is complicated by the fact that link choices of an individual are correlated due to the preference for friends in common. We exploit the optimal link choices in Theorem (ref) to investigate the link dependence and derive the asymptotic properties of the estimator.

We start with the data generating process. We consider the scenario in which a single large network is observed. In the asymptotic analysis, we assume that the number of individuals in the network $n$ goes to infinity. Because the network depends on $n$, we denote it by $G_{n}=(G_{n,ij})$ hereafter. We assume that links in a network are generated as follows. First, we draw a vector of characteristics $X=(X_{1}^{\prime},\ldots X_{n}^{\prime})^{\prime}$ from a joint discrete distribution, where $X_{i}$ represents the observed characteristics of individual $i$. Because $X$ is ancillary, we treat it as deterministic. Note that $X_{i}$ can be dependent across $i$. Next, we draw an $(n-1)\times1$ vector of unobserved preferences $\epsilon_{i}\in\mathbb{R}^{n-1}$ for each $i$, independently across $i$. After that, each individual chooses to form links, and an equilibrium (a fixed point of ((ref))) emerges. There can be multiple equilibria, and nature selects one equilibrium $\sigma_{n}$ among the equilibria. The network $G_{n}$ observed in the data is obtained from the optimal links chosen under $\sigma_{n}$.

\paragraph*{First step.}

The optimal link choices in ((ref)) depend on the equilibrium $\sigma_{n}$ only through the conditional probabilities of forming each link, denoted by $p_{n,ij}=\mathbb{E}[G_{n,ij}|X]$, $1\leq i\neq j\leq n$. Moreover, the symmetry of the equilibrium (Assumption (ref)) implies that each $p_{n,ij}$ depends on $i$ and $j$ only through their types $X_{i}$ and $X_{j}$. Under Assumption (ref), it is thus sufficient to consider the type-specific conditional link probabilities $p_{n,(st)}=\mathbb{E}[G_{n,ij}|X_{i}=x_{s},X_{j}=x_{t},X]$ for $1\leq s,t\leq T$. Denote $p_{n}=(p_{n,(st)},1\leq s,t\leq T)'$. This is the parameter we need to estimate in the first step.

Specifically, for each $1\leq s,t\leq T$, we estimate $p_{n,(st)}$ by the relative frequency of forming a link among the pairs of individuals that are of types $x_{s}$ and $x_{t}$

Let $\hat{p}_{n}=(\hat{p}_{n,(st)},1\leq s,t\leq T)'$ denote the first-step estimator.

\paragraph*{Second step.}

To estimate $\theta_{0}$, we define $P_{n,ij}(\theta,p)=\mathbb{E}[G_{n,ij}(\epsilon_{i},\theta,p)|X]$ as the model-implied probability that $i$ forms a link to $j$, where $G_{n,ij}(\epsilon_{i},\theta,p)$ represents the optimal link choice in ((ref)) given $\theta$ and $p$. The equilibrium condition in ((ref)) yields a set of conditional moment restrictions

where each value of $(X_{i},X_{j})$ gives one moment restriction. Based on ((ref)), we can construct a GMM estimator for $\theta_{0}$. Let $q_{n,ij}=q_{n}(X_{i},X_{j})$ denote a $d_{\theta}\times1$ vector of instruments that can depend on $X$ as well as $\theta_{0}$ and $p_{n}$, and $\hat{q}_{n,ij}$ denote an estimator of $q_{n,ij}$. Define

to be the sample moment, where $\hat{p}_{n}$ is the first-step estimator. The minimizer of the function $\hat{m}_{n}(\theta,\hat{p}_{n})'\hat{m}_{n}(\theta,\hat{p}_{n})$ gives a GMM estimator $\hat{\theta}_{n}$.\footnote{We formulate the instrument in a way so that the weighting matrix is absorbed into the instrument. See Newey_McFadden_1994 for justification of this general formulation.} Suppose that the estimator $\hat{\theta}_{n}$ satisfies $\hat{m}_{n}(\hat{\theta}_{n},\hat{p}_{n})=o_{p}(n^{-1})$.

\paragraph*{Asymptotic analysis.}

We now investigate the asymptotic properties of the estimator $\hat{\theta}_{n}$. Given $X$, define the population counterpart of $\hat{m}_{n}(\theta,p)$ by

Equation ((ref)) implies that $m_{n}(\theta_{0},p_{n})=0$.\footnote{The population moment $m_{n}(\theta,p)$ has the subscript $n$ because the probability of forming a link depends on the network size.} Because $p_{n}$ is uniquely determined in the first step,\footnote{The first step can be characterized by the moment restrictions $\mathbb{E}[G_{n,ij}-p_{n,ij}|X]=0$; or equivalently, $\mathbb{E}[G_{n,ij}-p_{n,(st)}|X_{i}=x_{s},X_{j}=x_{t},X]=0$ for all $1\leq s,t\leq T$, where $p_{n}$ is a unique solution.} we assume that $m_{n}(\theta,p_{n})=0$ has a unique solution at $\theta_{0}$. Stacking the moments in the first and second steps then uniquely identifies $\theta_{0}$ and $p_{n}$. With abuse of notation, we write $(\theta,p)$ for $(\theta',p')'$.

With the addition of Assumption (ref), we show that $(\hat{\theta}_{n},\hat{p}_{n})$ is consistent for $(\theta_{0},p_{n})$.

Assumption (ref)(i) is a standard regularity condition. Assumption (ref)(ii) is the identification condition previously discussed. This assumption requires that $\mathbb{E}[G_{n,ij}-P_{n,ij}(\theta,p_{n})|X]=0$ has a unique solution at $\theta_{0}$. Equivalently, we can consider the model-implied type-specific link choice probabilities $P_{n,(st)}(\theta,p)=\mathbb{E}[G_{n,ij}(\epsilon_{i},\theta,p)|X_{i}=x_{s},X_{j}=x_{t},X]$, $1\leq s,t\leq T$. The assumption requires that for any $\theta\neq\theta_{0}$, there exist $1\leq s,t\leq T$ such that $P_{n,(st)}(\theta,p_{n})\neq P_{n,(st)}(\theta_{0},p_{n})$. When there is no effect from friends in common ($\gamma_{1},\gamma_{2}=0$), this assumption reduces to a standard rank condition that the regressors in ((ref)) evaluated at $p_{n}$ are linearly independent Leung_2015. Moreover, note that the auxiliary term $\Phi_{ni}\Lambda_{ni}\omega_{ni}(\epsilon_{i})$ in ((ref)) can be viewed as a solution to the first-order condition in ((ref)) (multiplied by $\Phi_{ni}$). The assumption requires that the solution to this first-order condition under $\theta\neq\theta_{0}$ must differ from the solution under $\theta_{0}$, so that $\gamma_{1}$ and $\gamma_{2}$ can be identified.\footnote{A necessary condition is that the two network statistic terms in ((ref)) must be linearly independent, so that $V_{ni}$ does not remain the same for different values of $\gamma_{1}$ and $\gamma_{2}$.} Assumption (ref)(iii) is a standard assumption that the instrument is bounded and its estimator is consistent, both uniformly over $i$ and $j$. This assumption ensures that estimating the instrument has no impact on the asymptotic distribution of $\hat{\theta}_{n}$, as the terms involving the estimation error $\hat{q}_{n,ij}-q_{n,ij}$ are of a smaller order compared to those involving the true instrument $q_{n,ij}$. Assumption (ref)(iv) imposes a mild restriction on $X$. It requires that the fractions of pairs of each type remain positive as $n\rightarrow\infty$, so that the numbers of pairs of each type grow without bounds, and we can identify and estimate each $p_{n,(st)}$. If $X_{i}$ is i.i.d. or has limited dependence across $i$ such that $\frac{1}{n(n-1)}\sum_{i}\sum_{j\neq i}1\{X_{i}=x_{s},X_{j}=x_{t}\}$ converges to $\Pr(X_{i}=x_{s},X_{j}=x_{t})$ almost surely, and if $\Pr(X_{i}=x_{s},X_{j}=x_{t})>0$ for all $1\leq s,t\leq T$, then Assumption (ref)(iv) holds for almost every realization of $X$.

The consistency is established as a result of the fact that given $X$ links formed by different individuals are independent, although links formed by the same individual are correlated. The conditional independence allows us to establish a uniform LLN for the stacked sample moment, which together with the identification condition yields consistency.

Analyzing the asymptotic distribution of $\hat{\theta}_{n}$ is more complicated because links formed by an individual are correlated. Theorem (ref) shows that link choices $G_{n,ij}$ and $G_{n,ik}$ are correlated because they both depend on the auxiliary variable $\omega_{ni}(\epsilon_{i})$, which is a maximin solution of the function

In the expression, we add subscript $n$ to $\omega_{ni}$, $\Pi_{ni}$, $U_{n,ij}$ and $V_{ni}$ to indicate their dependence on $n$, and all of the terms are evaluated at $(\theta_{0},p_{n})$, abbreviated for simplicity. To investigate how the link dependence will affect the asymptotic distribution of $\hat{\theta}_{n}$, we represent $\omega_{ni}(\epsilon_{i})$ in an asymptotically linear form. Specifically, let $\Pi_{ni}^{\ast}(\omega)$ denote the population counterpart of $\Pi_{ni}(\omega,\epsilon_{i})$ given $X$

and $\omega_{ni}^{\ast}$ denote a maximin solution of $\Pi_{ni}^{\ast}(\omega)$. Under the regularity conditions in Assumption (ref), we show in Lemma (ref) that $\omega_{ni}(\epsilon_{i})$ has an asymptotically linear representation

where $\phi_{n,ij}^{\omega}(\omega_{ni}^{\ast},\epsilon_{ij})$ is an influence function defined in the lemma. Observe that $\omega_{ni}^{\ast}$ is deterministic, so link choices evaluated at $\omega_{ni}^{\ast}$ are independent. The representation indicates that the link dependence due to $\omega_{ni}(\epsilon_{i})$ vanishes at the rate of $n^{-1/2}$, which is crucial in determining the asymptotic distribution of $\hat{\theta}_{n}$.

With the addition of Assumptions (ref) and (ref), we show that $\hat{\theta}_{n}$ is asymptotically normal.

Assumption (ref)(i) imposes a smoothness restriction on $P_{n,ij}(\theta,p)$. We show in Lemma (ref) that $P_{n,ij}(\theta,p)$ is continuous in $\theta$ and $p$, by exploiting the fact that there is a one-to-one mapping between the optimal link choices of an individual and a partition of the $\epsilon_{i}$ space $\mathbb{R}^{n-1}$ (Online Appendix (ref)), where the function that defines the boundary of each set in the partition is continuous in $\theta$ and $p$. The proof suggests that $P_{n,ij}(\theta,p)$ can have kinks if the binding inequalities that define the partition vary with $\theta$ and $p$. This assumption requires that there is a neighborhood of $(\theta_{0},p_{n})$ that has no kinks. In fact, we show in Proposition (ref) that $P_{n,ij}(\theta,p)$ converges (pointwise in $\theta$ and $p$) to a limit as $n\rightarrow\infty$, which is continuously differentiable in $\theta$ and $p$. Therefore, the assumption is less of a concern for larger $n$. Assumption (ref)(ii) is a standard regularity condition for $\hat{\theta}_{n}$ to have a well-behaved asymptotic distribution. It also ensures that $(\theta_{0},p_{n})$ is locally identified in a small neighborhood of $(\theta_{0},p_{n})$. Assumption (ref) imposes additional regularity conditions on the auxiliary variable $\omega$ so that we can derive the asymptotically linear representation in ((ref)) as well as other needed asymptotic properties of $\omega_{ni}(\epsilon_{i})$.

We derive the asymptotic distribution by decomposing the sample moment into two leading terms, corresponding to the two components in the influence function $\phi_{n,ij}^{\theta}$. The first captures the sampling variation in link choices that does not account for the link dependence due to $\omega_{ni}(\epsilon_{i})$. The second captures the contribution of the link dependence to the asymptotic distribution. Note that $\hat{\theta}_{n}$ converges to $\theta_{0}$ at the rate of $n$ (the square root of the sample size). The link dependence vanishes sufficiently fast so that it does not slow down the rate at which $\hat{\theta}_{n}$ converges, but increases its asymptotic variance.

The asymptotic variance of $\hat{\theta}_{n}$ can be calculated as $\frac{1}{n(n-1)}J_{n}^{-1}\Sigma_{n}(J'_{n})^{-1}$. We can estimate the asymptotic variance consistently by a plug-in estimator, where we replace $\theta_{0}$ and $p_{n}$ by their estimators $\hat{\theta}_{n}$ and $\hat{p}_{n}$. In Online Appendix (ref), we provide detailed guidance on how to estimate $\theta$ and compute the standard errors in practice.

\paragraph*{Instrument.}

In practice, we need to choose an instrument. We suggest using the instrument derived from quasi-maximum likelihood estimation (QMLE).\footnote{This is also the optimal instrument given the conditional moment restrictions in ((ref)) Chamberlain_1987.} Let $\mathcal{L}_{n}(\theta,\hat{p}_{n})$ denote the log of the quasi-likelihood function evaluated at the first-step estimator $\hat{p}_{n}$\footnote{The quasi-likelihood function does not take into account the joint distribution of link choices $G_{n,ij}$ and $G_{n,ik}$, which can be informative about $\theta$.}

Taking the derivative with respect to $\theta$, we obtain the quasi-likelihood equation \[ \frac{1}{n(n-1)}\sum_{i}\sum_{j\neq i}\frac{\nabla_{\theta}P_{n,ij}(\theta,\hat{p}_{n})}{P_{n,ij}(\theta,\hat{p}_{n})(1-P_{n,ij}(\theta,\hat{p}_{n}))}(G_{n,ij}-P_{n,ij}(\theta,\hat{p}_{n}))=0. \] Comparing this equation with the sample moment in ((ref)) suggests the instrument

Note that the instrument depends on $\theta$. We can either construct a preliminary estimator of $\theta$ using an initial instrument\footnote{For example, we can use powers and interactions of $X_{i}$ and $X_{j}$ to construct an initial instrument.} or use continuous updating, as in Hansen_1996.\footnote{Using instrument ((ref)) with continuous mapping is equivalent to QMLE based on ((ref)). However, GMM provides more flexibility in choosing the instrument, especially when the link choice probability $P_{n,ij}(\theta,p)$ is not fully differentiable in $\theta$ and $p$. See Section (ref) for more discussions.}

Extensions

Undirected Networks

In some applications, the networks observed by researchers are undirected, such as friendship and coauthorship networks Jackson_2008. In this section, we demonstrate that the approach developed in Section (ref) can be extended to undirected networks. However, because the underlying decisions that induce undirected links are not observed, the identification and estimation of parameters become more challenging.

Let $G_{ij}$ denote an undirected link between individuals $i$ and $j$, and thus $G_{ij}=G_{ji}$. We assume that individuals propose the links they wish to form as in a link announcement game Myerson_1991. The undirected link $G_{ij}$ is formed if both $i$ and $j$ propose to form it. Specifically, let $D_{ij}$ indicate whether $i$ proposes a link to $j$. We have $G_{ij}=D_{ij}D_{ji}$.

We adapt the utility specifications in ((ref)) and ((ref)) to accommodate undirected links. By removing the reciprocity effect and replacing the spillover effects of directed links with their undirected counterparts, we specify \[ u_{ij}(G_{-i},X;\beta)=\beta_{1}+X_{i}^{\prime}\beta_{2}+d(X_{i},X_{j})'\beta_{3}+\frac{1}{n-2}\sum_{k\neq i,j}G_{jk}\beta_{4} \] and \[ v_{i,jk}(G_{-i},X;\gamma)=G_{jk}\gamma_{1}(X_{j},X_{k})+\frac{1}{n-3}\sum_{l\neq i,j,k}G_{jl}G_{kl}\gamma_{2}(X_{j},X_{k}). \] This utility specification is more general than that of Comola_Dekel_2023, who also extend Leung_2015's approach to undirected networks. Comola_Dekel_2023 maintain Leung_2015's assumption of separable utility, which is more restrictive in the undirected context because it excludes any triadic closure statistic.\footnote{For example, Leung_2015 allows for the supported trust $\frac{1}{n-2}\sum_{k\neq i,j}G_{ki}G_{kj}$ in $u_{ij}$. In an undirected setting, however, this statistic takes the form $\frac{1}{n-2}\sum_{k\neq i,j}D_{ik}D_{ki}D_{jk}G_{kj}$, which depends on $i$'s decision $D_{ik}$ and therefore violates the separability assumption. }

Because $G_{ij}=D_{ij}D_{ji}$, we write $G=G(D_{i},D_{-i})$, where $D_{i}=(D_{ij},j\neq i)\in\mathcal{D}_{i}=\{0,1\}^{n-1}$ represents the links proposed by individual $i$, and $D_{-i}=(D_{j},j\neq i)$ the links proposed by individuals other than $i$. By taking the expectation with respect to $D_{-i}$, we can calculate the expected utility of individual $i$ as follows:

where

and

In these expressions, we denote $\sigma_{ij}(X)=\mathbb{E}[D_{ij}|X]$ and $\sigma_{i,jk}(X)=\mathbb{E}[D_{ij}D_{ik}|X]$. Equations ((ref)) and ((ref)) hold because, conditional on $X$, the proposals $D_{j}$ and $D_{k}$ are independent. Note that ((ref)) and ((ref)) involve the probability of an individual proposing two links (e.g., $\sigma_{j,ik}(X)$).

The expected utility in ((ref)) is similar to that in ((ref)) when viewed as a function of proposals. Therefore, we can apply the approach in Section (ref) to derive the optimal proposals. For $1\leq s,t\leq T$, let $V_{i,st}^{u}(X,\sigma)$ denote the value of $\mathbb{E}[D_{ji}D_{ki}v_{i,jk}(G_{-i},X)|X]$ if individuals $j$ and $k$ are of types $s$ and $t$, respectively; that is, $V_{i,st}^{u}(X,\sigma)=\mathbb{E}[D_{ji}D_{ki}v_{i,jk}(G_{-i},X)|X_{j}=x_{s},X_{k}=x_{t},X,\sigma]$. The superscript $u$ indicates an undirected network. Arrange the $T^{2}$ type-specific expected incremental utilities $V_{i,st}^{u}(X,\sigma)$ in a $T\times T$ matrix $V_{i}^{u}(X,\sigma)=(V_{i,st}^{u}(X,\sigma))\in\mathbb{R}^{T\times T}$. Because $V_{i,st}^{u}(X,\sigma)$ is symmetric in $s$ and $t$, $V_{i}^{u}(X,\sigma)$ is a symmetric matrix. Hence, it has a real spectral decomposition \[ V_{i}^{u}(X,\sigma)=\Phi_{i}^{u}(X,\sigma)\Lambda_{i}^{u}(X,\sigma)\Phi_{i}^{u\prime}(X,\sigma), \] where $\Lambda_{i}^{u}(X,\sigma)=\text{diag}(\lambda_{i1}^{u}(X,\sigma),\ldots,\lambda_{iT}^{u}(X,\sigma))$ denotes the $T\times T$ diagonal matrix of eigenvalues in $\mathbb{R}$, and $\Phi_{i}^{u}(X,\sigma)=(\phi_{i1}^{u}(X,\sigma),\ldots,\phi_{iT}^{u}(X,\sigma))$ denotes the $T\times T$ orthogonal matrix of eigenvectors in $\mathbb{R}^{T}$. Define $U_{ij}^{u}(X,\sigma)=\mathbb{E}[D_{ji}u_{ij}(G_{-i},X)|X]-\frac{1}{2(n-2)}Z_{j}^{\prime}V_{i}^{u}(X,\sigma)Z_{j}$.

Following Theorem (ref), we derive the optimal proposals in Corollary (ref).

Corollary (ref) shows that the optimal proposals can be expressed as binary choices, with the addition of an auxillary variable $\omega_{i}^{u}(\epsilon_{i},X,\sigma)$, which serves the same role as $\omega_{i}(\varepsilon_{i},X,\sigma)$ in directed networks. We anticipate that proposals in an undirected network exhibit a dependence structure analogous to that of links in a directed network. However, since we observe links rather than proposals, the estimation method must be adapted. In Online Appendix (ref), we discuss how to extend the estimation procedure in Section (ref) to undirected networks. A complete econometric analysis for undirected networks is left for future research.

Limiting Approximation

In this section, we demonstrate that under certain conditions, a link choice probability in the finite-$n$ game converges to a limit as $n\rightarrow\infty$. In contrast to its finite-$n$ counterpart, the limiting link probability is continuously differentiable in the parameters and can be calculated analytically. It provides a useful approximation to facilitate the estimation and computation of the parameters.

Given a characteristic profile $X$ and equilibrium $p=(p_{(st)},1\leq s,t\leq T)'$, recall that the probability that individual $i$ forms a link to individual $j$ is given by

where the auxiliary variable $\omega_{ni}(\epsilon_{i},X,p)$ is a maximin solution of the objective function $\Pi_{ni}(\omega,\epsilon_{i},X,p)$ in problem ((ref)). Suppose that $U_{n,ij}(X,p)$ and $V_{ni}(X,p)$ converge to some limites $U^{*}(X_{i},X_{j},p)$ and $V^{*}(X_{i},p)$ as $n\rightarrow\infty$, and $V^{*}(X_{i},p)$ has a real spectral decomposition $V^{*}(X_{i},p)=\Phi^{*}(X_{i},p)\Lambda^{*}(X_{i},p)\Phi^{*\prime}(X_{i},p)$, where $\Lambda^{*}(X_{i},p)=\text{diag}(\lambda_{1}^{*}(X_{i},p),\dots,\lambda_{T}^{*}(X_{i},p))$. Let $\omega^{*}(X_{i},p)\in\mathbb{R}^{T}$ denote an optimal solution to the maximin problem

where $\mathcal{T}_{+}=\{1\leq t\le T:\lambda_{t}^{*}(X_{i},p)>0\}$ and $\mathcal{T}_{-}=\{1\leq t\leq T:\lambda_{t}^{*}(X_{i},p)>0\}$. We set $\omega_{t}^{*}(X_{i},p)=0$ if $\lambda_{t}^{*}(X_{i},p)=0$. Let $\Pi^{*}(\omega,X_{i},p)$ denote the objective function in ((ref)), where we condition on $X_{i}$ and take expectation with respect to $X_{j}$ and $\epsilon_{ij}$, $j\neq i$. We can regard problem ((ref)) as the limiting counterpart of problem ((ref)) and show that the finite-$n$ solution $\omega_{ni}(\epsilon_{i},X,p)$ converges to the the limiting solution $\omega^{*}(X_{i},p)$ as a result. From these results, we can derive that the finite-$n$ link probability $P_{n,ij}(X,p)$ converges to a limit defined by

We refer to $P^{*}(X_{i},X_{j},p)$ as the limiting link probability.

To formally establish the convergence result, we impose the following assumptions.

Because $\frac{\partial}{\partial c}\mathbb{E}[c-\epsilon]_{+}=\frac{\partial}{\partial c}\int_{-\infty}^{c}(c-\epsilon)f_{\epsilon}(\epsilon)d\epsilon=F_{\epsilon}(c)$, ((ref)) has the first-order condition

Any solution to this first-order condition must be bounded. Therefore, it is reasonable to assume that $\omega$ lies in a compact set $\Omega\subseteq\mathbb{R}^{T}$ as in Assumption (ref)(i).\footnote{This assumption resembles Assumption (ref)(i), which is imposed to derive the asymptotic properties of $\omega_{ni}(\epsilon_{i})$ conditional on $X$. } Assumption (ref)(ii) is an identification condition.\footnote{This assumption resembles Assumption (ref)(ii) for fixed $X$.} It is imposed to derive the consistency result for $\omega_{ni}(\epsilon_{i},X,p)$ in Lemma (ref).\footnote{We require $\Lambda^{*}(X_{i},p)\omega^{*}(X_{i},p)$, rather than $\omega^{*}(X_{i},p)$, to be unique because $V^{*}(X_{i},p)$ may be singular, but it is $\Lambda^{*}(X_{i},p)\omega^{*}(X_{i},p)$ that affects the link choices.} Our analysis in the previous sections avoided making assumptions about how $X_{i}$ are generated. In Assumption (ref)(iii), we assume that $X_{i}$ is i.i.d. in order to establish the limiting approximation. Assumption (ref)(iv) posits the convergence of the expected utilities. In Example (ref), we demonstrate that Assumption (ref)(iv) holds under Assumption (ref)(iii) for the expected utility specified in ((ref))--((ref)). Intuitively, the spillover effects in ((ref))--((ref)) take the form of sample averages over functions of $X_{k}$, $k\neq i,j$, (e.g., $p(X_{k},X_{j})$). These averages converge to population means when $X_{i}$ is i.i.d..

Proposition (ref) shows that the finite-$n$ link probabilities converge in probability to the limiting link probabilities as $n\rightarrow\infty$.

We establish the result in Proposition (ref) by first noting that the finite-$n$ first-order condition in ((ref)) takes the form of a sample average over functions of $X_{j}$, $j\neq i$. Under the assumption of i.i.d. $X_{i}$ and converging expected utilities, we can show that the finite-$n$ first-order condition converges to the limiting first-order condition in ((ref)). Consequently, the solution to the finite-$n$ first-order condition also converges to the solution to the limiting counterpart. While the finite-$n$ auxiliary variable $\omega_{ni}(\epsilon_{i},X,p)$ depends on both $\epsilon_{i}$ and the entire $X$, its limiting counterpart $\omega^{*}(X_{i},p)$ depends on $X_{i}$ only. Therefore, conditional on $X_{i}$, individual $i$'s link choices in the limit become independent.\footnote{In equation ((ref)), the latent utility of forming a link depends on the equilibrium $p$, indicating that strategic interactions among link choices do not vanish in the limit. The presence of $\omega^{*}(X_{i},p)$ further suggests that, even in the limit, strategic interactions due to the preference for friends in common persist. By incorporating $\omega^{*}(X_{i},p)$ in the latent utility, we internalize the limiting approximation of the spillover effects caused by this preference.} Our result aligns with the literature that employs large-market approximations as a simplification for finite-$n$ markets, which are often challenging to analyze due to complex equilibria.\footnote{For example, Menzel_2015 discovered the large-market approximation for a one-to-one matching model under non-transferable utility. Azevedo2016 established the convergence of equilibrium cutoffs for a many-to-one matching model under non-transferable utility as the market size grows large.} In our context, we derive the limiting approximation to simplify the link dependence arising from the preference for friends in common, thereby yielding simpler link choice probabilities.

\paragraph{Advantage of the limiting approximation.}

The two-step estimator proposed in Section (ref) requires an instrument in the second stage. We suggested using the instrument derived from quasi-maximum likelihood (equation ((ref))); however, this instrument involves the derivative of a link choice probability. Because the limiting auxiliary variable $\omega^{*}(X_{i},p)$ does not depend on $\epsilon_{i}$, the limiting link probability $P^{*}(X_{i},X_{j},p)$ is continuously differentiable in the parameters. Therefore, we can use the derivative of a limiting link probability to construct the instrument, addressing the concern that finite-$n$ link probabilities may have kinks. Given that finite-$n$ and limiting link probabilities are asymptotically close (Proposition (ref)), the instrument based on limiting link probabilities should achieve asymptotic efficiency similar to that of the instrument based on finite-$n$ link probabilities.\footnote{Because we only approximate the instrument, the consistency of the estimator remains unaffected.}

We can further simplify the moment condition by replacing the finite-$n$ link probabilities in the moment function with their limiting counterparts. This approximation improves computational efficiency, as limiting link probabilities can be computed without simulation. Although the approximated moment function yields a misspecified model, the misspecification vanishes asymptotically.\footnote{To analyze the asymptotic properties of such an estimator, we must examine the extent to which the limiting link probabilities evaluated at a finite-$n$ equilibrium differ from that equilibrium. In the presence of multiple equilibria, additional assumptions ensuring the convergence of a sequence of equilibrium selection mechanisms would be needed to achieve the consistency of the estimator. This issue is related to the convergence of equilibria explored in Menzel_2016.}

\paragraph{Simulation evidence.}

Given the scope of this paper, we do not investigate the theoretical properties of the limiting approximation. However, we provide simulation evidence on its performance. In Online Appendix (ref), we evaluate our approach in a simulation study, where limiting link probabilities are used to approximate the instrument and/or the moment function. The estimates that use limiting link probabilities for the instrument (Table (ref) Case (ii)) are similar to those that use the finite-$n$ counterparts (Table (ref) Case (i)), although they are biased and have larger root MSEs in small networks ($n\leq25$). The estimates that use limiting link probabilities for both the moment function and the instrument (Table (ref) Case (iii)) are the most biased and have the largest root MSEs in small networks, but once networks become moderately large ($n\geq100$), they perform similarly to the other estimates -- remaining unbiased with comparable root MSEs. These results suggest that limiting link probabilities provide a useful approximation in sufficiently large networks.

Empirical Application

\paragraph*{Data and setup.}

We apply our approach to investigate favor exchange networks in rural India. The dataset was collected from 75 rural villages in southern India as part of a study on a microfinance program (see Jackson_2012 and Banerjee_2013 for detailed descriptions of the data). Respondents in the survey were asked whether they provided monetary, in-kind (kerorice), advisory, or medical help to -- or received such favors from -- other individuals surveyed in the same village. Because providing and receiving favors represent distinct decisions, we keep the directed relationships and construct a directed network of favor exchange in each village.

In particular, we say that individual $i$ lends money or kerorice to individual $j$ if either $i$ reports lending money or kerorice to $j$ or $j$ reports borrowing money or kerorice from $i$. Similarly, we say that individual $i$ gives advice or medical help to individual $j$ if either $i$ reports providing such help to $j$ or $j$ reports receiving such help from $i$.\footnote{These directed relationships are constructed using the variables Borrow-money, Lend-money, Borrow-kerorice, Lend-kerorice, Advice-come, Advice-go and Medical-help in the data. For detailed descriptions of these variables, see Jackson_2012. } We say that individual $i$ does a favor for individual $j$ if $i$ lends money or kerorice, or gives advice or medical help to $j$. This creates a directed link from $i$ to $j$ in a favor exchange network.

Our empirical study is motivated by Jackson_2012, who found that the provision of a favor is supported by mutual relationships with other individuals. We aim to provide further evidence on self-support within a directed network of favor exchange. Given the intrinsic nature of a favor, we assume that whoever receives a favor accepts it, so that the presence of a favor is determined unilaterally by the provider. Inspired by the findings of Jackson_2012, we allow individual $i$'s marginal utility from providing a favor to individual $j$ to depend on the support from the connections that $i$ and $j$ have with another individual $k$. From individual $i$'s perspective, her incentive to provide a favor to $j$ may differ depending on whether she provides a favor to $k$ or receives a favor from $k$. Therefore, we distinguish the supporting connections based on the direction of the link $ik$. We define the inward support for the link $ij$ as $\frac{1}{n-2}\sum_{k\neq i,j}G_{ki}G_{kj}$, where $i$ receives a favor from a supporting individual $k$.\footnote{There are other possible variants of inward support, such as $\frac{1}{n-2}\sum_{k\neq i,j}G_{ki}(G_{jk}+G_{kj})$, which aligns with our definition of outward support. However, we choose $\frac{1}{n-2}\sum_{k\neq i,j}G_{ki}G_{kj}$, as it coincides with the supported trust defined in Leung_2015 and thus facilitates comparison.} In contrast, we define the outward support for the link $ij$ as $\frac{1}{n-2}\sum_{k\neq i,j}G_{ik}(G_{jk}+G_{kj})$, where $i$ provides a favor to a supporting individual $k$.

Specifically, we consider the utility function in ((ref)), where the unobservable $\epsilon_{ij}$ is assumed to follow a logistic distribution. Our specification of the separable utility $u_{ij}$ in ((ref)) includes the provider $i$'s characteristics (gender, age, education, caste), homophily measures (same gender, same age, same education, same caste), and spillover effects that are separable in $i$'s links: reciprocity ($G_{ji}$), recipient's in-degree ($\frac{1}{n-2}\sum_{k\neq i,j}G_{kj}$), recipient's out-degree ($\frac{1}{n-2}\sum_{k\neq i,j}G_{jk}$), and inward support ($\frac{1}{n-2}\sum_{k\neq i,j}G_{ki}G_{kj}$). Our specification of the nonseparable utility includes outward support ($\frac{1}{n-2}\sum_{k\neq i,j}G_{ik}(G_{jk}+G_{kj})$), with $v_{i,jk}=(G_{jk}+G_{kj})\gamma_{1}$, where $\gamma_{1}$ is constant.\footnote{We do no consider the second term in ((ref)) and set $\gamma_{2}=0$.} While the spillover effects in $u_{ij}$ can be estimated using the approach in Leung_2015, estimating the effect of outward support requires our approach.

Our approach requires discrete types. We discretize age into three categories (under 29, 30--49 and over 50) and education into two categories (below and above the median).\footnote{The median number of years of schooling in the dataset is 5.} Castes are classified into three categories: scheduled (including scheduled castes and scheduled tribes), other backward class (OBC), and general. This discretization and categorization result in a type space of $36$ types ($T=36$).

To align with the asymptotic framework in the paper, we use only one village from the dataset for our empirical analysis. Our sample consists of $n=395$ individuals in the village. Among these individuals, there are $n(n-1)=155,630$ potential directed links.

\paragraph*{Estimation and inference.}

We estimate the utility parameters in two steps. In the first step, we estimate the probability that individual $i$ provides a favor to individual $j$ given the characteristics of $i$ and $j$. In our sample, certain pair types are absent.\footnote{Our sample consists of 1,084 pair types. In fact, no village in the dataset contains all 1,296 pair types.} Therefore, rather than using a frequency estimator, as discussed in Section (ref), we use a series logit estimator Hirano_2003. Specifically, we run a logit regression of favor provision on a second-order polynomial series of provider and recipient characteristics. The predicted link choice probabilities for each pair type yield our first-step estimates.

In the second step, we estimate the utility parameters in $u_{ij}$ and $v_{i,jk}$ by GMM. We use the moment in ((ref)), with the instrument given by ((ref)).\footnote{In practice, we implement GMM by weighted nonlinear least squares (NLS), where we use the optimal weight $1/(P_{n,ij}(1-P_{n,ij}))$ for link $G_{ij}$. The first-order condition of the weighted NLS coincide with that of GMM, so the estimates should be equivalent. An advantage of weighted NLS is that we can use the built-in command in MATLAB $\textit{nlinfit}$ to calculate the estimates.} To reduce the computational burden, we approximate the finite-$n$ link choice probability $P_{n,ij}$ in ((ref)) using a variant of the limiting approximation developed in Section (ref). This approximation retains all terms from $P_{n,ij}$, except that the auxiliary variable $\omega_{ni}(\epsilon_{i})$, a maximin solution of ((ref)), is replaced by its population counterpart $\omega_{ni}^{\ast}$, a maximin solution of ((ref)). Unlike $\omega_{ni}(\epsilon_{i})$, which must be calculated for each individual by simulation (Online Appendix (ref)), $\omega_{ni}^{\ast}$ depends on $i$ only through her type and does not involve unobservables. Consequently, it can be calculated for each type without simulation.\footnote{On an 8-core CPU, a single evaluation of the approximated link probabilities for all the 1296 pair types in our sample takes 0.03 seconds.} While $\omega_{ni}^{\ast}$ is a maximin solution, we compute it by solving the first-order condition of ((ref)) using a standard fixed-point algorithm.

In practice, to get an educated guess for the initial values of the parameters, we first estimate the parameters approximately by logit, where we use the right-hand side of ((ref))---calculated based on the observed links---to approximate the auxiliary term on the left-hand side. Given that the logit approximation runs quite fast, it is also useful to researchers who want to experiment with specifications.\footnote{For our specification with all spillover effects---the specification that is most costly in computation---on an 8-core CPU, the logit approximation takes 0.4 seconds, while the GMM estimation, implemented by weighted NLS, takes 32.7 seconds.}

The standard errors are calculated using the asymptotic variance in Theorem (ref) with modifications. First, the use of the population $\omega_{ni}^{\ast}$ implies that the links are conditional independent. This property simplifies the asymptotic distribution in Theorem (ref), leaving only the first term in the influence function $\ensuremath{\phi_{n,ij}^{\theta}}$. Moreover, recall that the first step is estimated by series logit. Following Ackerberg2012, we can account for the contribution of a nonparametric first step to the asymptotic variance in the same way as in two-step estimation with a parametric first step.\footnote{Ackerberg2012 established the numerical equivalence result when the first step is estimated using sieves. This result can be extended to series logit by applying the approach in Chernozhukov2021.}

\paragraph*{Results.}

Table (ref) presents the second-step GMM estimates and their standard errors. We consider four specifications of spillover effects. Column 1 assumes no spillover effects. Column 2 allows for four separable spillover effects (reciprocity, recipient's in-degree, recipient's out-degree, and inward support).\footnote{To estimate the effect of inward support, we need a first-step estimate for $\mathbb{E}[G_{ki}G_{kj}|X]$, that is, the conditional probability that individual $k$ forms a link with both $i$ and $j$. Under the limiting approximation, the two links are conditional independent. Therefore, we estimate $\mathbb{E}[G_{ki}G_{kj}|X]$ approximately by the products of the estimated $\mathbb{E}[G_{ki}|X]$ and $\mathbb{E}[G_{kj}|X]$.} Column 3 allows for outward support only. Column 4 considers all five spillover effects. In all the four specifications, we control for homophily measures and the provider's characteristics. Across specifications, we find that individuals sharing the same gender, age, and caste are significantly more likely to exchange favors, with caste and gender similarity having the greatest impacts. Moreover, individuals are significantly more likely to provide a favor if they are male, over the age of 50, and belong to a higher caste. These findings are consistent with evidence documented in the literature Jackson_2012. Additionally, we observe that homophily effects tend to be smaller in the specifications with spillover (Column 1 vs. Columns 2-4). This suggests that ignoring spillover effects and estimating a dyadic model may overestimate homophily effects.

In addition to the dyadic factors, Table (ref) provides evidence of spillover effects. Columns 2 and 4 show a positive reciprocity effect, suggesting that individuals are more willing to do favors for those who also do favors for them, although this effect is insignificant. Furthermore, individuals are significantly more likely to provide favors to recipients with higher in-degrees and less likely to do so for those with higher out-degrees. This indicates that providers interpret these network metrics as signals of need: a high in-degree implies greater reliance on others (and thus more need), while a high out-degree reflects the capacity to help others (and thus less need). In short, favors tend to flow toward those perceived as needing help and away from those already helping others.

More importantly, Table (ref) highlights the distinct effects of inward and outward support on favor provision. Inward support shows a positive but insignificant effect (Columns 2 and 4), suggesting that $i$'s decision to help $j$ is unaffected by receiving favors from a third party $k$ connected to $j$. This finding is consistent with the results of Leung_2015. In contrast, outward support has a positive and significant effect (Columns 3 and 4), indicating that $i$ is more likely to help $j$ if $i$ provides favors to a third party $k$ connected to $j$. While mutual connections with a third party matter, as shown in Jackson_2012, the direction of these connections is crucial: receiving a favor from $k$ has no impoact on $i$'s decision to help $j$, whereas providing a favor to $k$ increases the likelihood of $i$ helping $j$. These findings suggest that policies prioritizing outward support over inward support are more effective in promoting favor exchange. For example, targeting active providers with high out-degrees can amplify support and strengthen favor provision throughout the network.

\paragraph*{Variance decomposition of predicted log odds ratio.}

In our framework, favor provision is influenced by three types of factors: (i) dyadic attributes (homophily measures and provider's characteristics), (ii) separable spillover (reciprocity, recipient's degrees, and inward support), and (iii) nonseparable spillover (outward support). Using the estimates in Column 4 of Table (ref), we calculate the predicted values of these components for each link. The sum of the three gives the predicted log odds ratio of the link ($\log(\hat{P}_{n,ij}/(1-\hat{P}_{n,ij}))$).\footnote{We calculate the impact of dyadic attributes as $\hat{\beta}_{1}+X_{i}^{\prime}\hat{\beta}_{2}+d(X_{i},X_{j})^{\prime}\hat{\beta}_{3}$, separable spillover as $\hat{p}_{ji}\hat{\beta}_{4}+\frac{1}{n-2}\sum_{k\neq i,j}\hat{p}_{kj}\hat{\beta}_{5}+\frac{1}{n-2}\sum_{k\neq i,j}\hat{p}_{jk}\hat{\beta}_{6}+\frac{1}{n-2}\sum_{k\neq i,j}\hat{p}_{ki}\hat{p}_{kj}\hat{\beta}_{7}-\frac{1}{2(n-2)}Z'_{j}\hat{V}_{ni}Z_{j}$, and nonseparable spillover as $\frac{n-1}{n-2}Z'_{j}\hat{\Phi}_{ni}\hat{\Lambda}_{ni}\hat{\omega}_{ni}^{\ast}$.}

Table (ref) decomposes the total variance of the predicted log odds ratio into the variances of dyadic attributes, separable spillover, and nonseparable spillover, along with their covariances. Dyadic attributes alone account for only 58% of the total variance in the predicted log odds ratio. Including separable spillover increases this proportion to 67%. However, 33% of the total variance remains unexplained without nonseparable spillover. These results highlight the importance of accounting for nonseparable spillover.

\paragraph*{Predicting support distribution.}

Next we investigate our model's performance in predicting support measures under different specifications of spillover effects. Using the estimates in Column 4 of Table (ref), we simulate three directed networks. In the first network, all spillover effects are set to zero. In the second network, only the effect of outward support is set to zero. In the third network, all spillover effects are included.\footnote{We fix the first-step estimates when predicting a network under alternative parameter values. The predicted links do not reflect the potential change in the equilibrium.} The support measure introduced by Jackson_2012 is defined at the network level for undirected networks. We adapt it to the individual level for directed networks. Specifically, we calculate the support measure of individual $i$ in network $G$ as \[ \text{Supp}_{i}(G)=\frac{\sum_{j\neq i}G_{ij}\max_{k\neq i,j}((G_{ik}\lor G_{ki})\land(G_{jk}\lor G_{kj}))}{\sum_{j\neq i}G_{ij}}, \] where $x\lor y=\max\{x,y\}$ and $x\land y=\min\{x,y\}$. A link $G_{ij}$ is supported in network $G$ if there exists a third party $k$ that is connected (in any direction) to both $i$ and $j$. The support measure of individual $i$ in network $G$ is calculated as the ratio of the number of supported links $i$ forms to the total number of links $i$ forms.

Figure (ref) plots the cumulative distribution function (CDF) of the support measure across individuals in the observed network and three predicted networks. Predictions from the specification with no spillover severely understate the support distribution in the data. Including separable spillover (reciprocity, recipient's degrees, and inward support) improves the predicted support distribution to some extent. However, the specification that yields predictions best matching the data is the one that includes both separable and nonseparable spillover. These findings underscore the importance of spillover effects, in particular nonseparable spillover effects (outward support), in predicting the support distribution.

Conclusion

In this paper, we develop an econometric methodology for strategic network formation under incomplete information using data from a single large network. The utility function can be nonseparable in an individual's link choices because of the spillover effects from friends in common. We develop a novel approach that applies the Legendre transform to the utility function so that the optimal decision of an individual can be represented equivalently as a sequence of correlated binary choices. We propose a two-step estimation procedure, where we estimate the link choice probabilities in the first step and estimate the model parameters in the second step. We show that the two-step estimator is consistent and asymptotically normal. The link dependence due to the preference for friends in common does not affect the rate of convergence, but increases the asymptotic variance of the estimator. We also explore a scenario of undirected networks and derive a limiting approximation of the game that simplifies the computation in large networks.

There are a few more extensions of our approach that might be of interest. We may relax the i.i.d. assumption on the utility shocks by adding an individual-invariant heterogeneity Graham_2017. Both the individual heterogeneity and the strategic interactions considered in this paper can generate link dependence. It would be valuable to investigate the extent to which each of them accounts for the link dependence in network data. A recent strand of literature explores social interactions in endogenous networks where the endogeneity of a network is characterized through a network formation model (Goldsmith_Imbens_2013; Hsieh_Lee_2016; Johnsson_Moon_2021; Auerbach2022). These studies typically model network formation by a dyadic regression or a sequential process. Our paper provides an alternative model of network formation that is simple to analyze and allows for strategic interactions.\footnote{Other related studies along this line include Badev_2021, who developed a joint model of network formation and individual outcomes, and Battaglini_2021, who developed a model of network formation to recover unobserved social networks using only observable outcomes.}

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Appendix

\paragraph{Notation}

We use $\|\cdot\|$ to denote the Euclidean norm. For an $n\times1$ vector $x\in\mathbb{R}^{n}$ and an $n\times n$ matrix $A\in\mathbb{R}^{n^{2}}$, we have $\|x\|=(\sum_{i=1}^{n}x_{i}^{2})^{1/2}$ and $\|A\|=(\text{tr}(AA'))^{1/2}=(\text{\ensuremath{\sum}}_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^{2})^{1/2}$. $I_{T}$ denotes the $T\times T$ identity matrix. The notation $o_{p}(1)$ and $O_{p}(1)$ are defined conditionally on $X$ or certain components of $X$, depending on the context. For example, the statement “$Y_{n}=o_{p}(1)$ conditional on $X$” means that for any $\delta>0$, $\lim_{n\rightarrow\infty}\Pr(\|Y_{n}\|>\delta|X)=0$.

Proofs in Section (ref)

Proofs in Section (ref)

Proofs in Section (ref)

Proofs in Section (ref)