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Homophily in preferences or meetings? Identifying and estimating an iterative network formation model
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Homophily, the observed tendency of agents with similar attributes to maintain relationships, is a salient feature in social and economic networks chandrasekhar2016econometrics,Jackson2010social. Inasmuch as it may drive network formation, homophily can produce relevant effects in outcomes as diverse as smoking behavior Badev2017, test scores Hsieh2016,Goldsmith2013, the adoption of health innovations Centola2011, and health outcomes Kadelka2021. Social connections are also an important driver of economic mobility Chetty2022socioeconomic, which suggests that a proper account of homophily may improve the design of policies that aim to reduce economic inequality Jackson2021,Chetty2022social. It is therefore unsurprising that the appropriate modelling of homophily has been a focus in the recent push for estimable econometric models of network formation Goldsmith2013, Mele2017, Graham2016, Graham2017,Jackson2016.
Homophily that is due to choice is distinct from homophily that is due to opportunity Jackson2010social. We shall label the former homophily in “preferences”; and the latter homophily in “meetings”. This distinction has an important role: public policy may be able to alter the meeting technology between agents (say, by desegregating environments), but it may be less successful in changing preferences. Thus, the effect of a public policy that aims at changing individual connections (e.g. a tracking policy or a policy that induces people to move to a different neighborhood) is expected to depend on the type of homophily that is prevalent in that network Chetty2022social. Theoretical models that distinguish between these mechanisms are studied by Currarini2009 and Bramoulle2012.\footnote{Currarini2010 provides estimates of a parametric version of the Currarini2009 model using AddHealth data.} These models have some relevant limitations, though: first, they focus on steady-state or “long-run” behaviour, which may not be appropriate in settings where transitional dynamics may matter (as in our empirical application in (ref)); second, they are either purely probabilistic\footnote{In contrast to strategic models of network formation. See Jackson2010social and dePaula2016 for examples.} Bramoulle2012 or, in the case of Currarini2009, allow for only a restrictive set of pay-offs from relationships.\footnote{Pay-offs of Currarini2009 depend only on the number of relationships with individuals of the same or different types. There is no role for indirect benefits.} These limitations render these models unfit for some empirical analyses.\footnote{In a static choice setting, Zeng2008 propose an ordered logistic model that accounts for both homophily in preferences and opportunities. In their model, however, the structure of homophily in opportunities must either be known a priori or depend on a disjoint set of traits than preferences, which limits its applicability even in settings where staticity may be a reasonable assumption.}
In this paper, we intend to fill the gap in the literature by analysing an estimable econometric model that accounts for both types of homophily. We study identification and estimation of a sequential network-formation algorithm originally developed by Mele2017 (see also Christakis2010 and Badev2017), where agents meet sequentially in pairs in order to revise their relationship status. The model is well-grounded in the theoretical literature of strategic network formation Jackson2002 and allows specifications that account for both “homophilies”. However, our approach differs from previous work in several aspects. Firstly, whereas Mele2017 discusses identification and estimation of utility parameters when either a single large or several networks drawn from the model's induced stationary distribution are observed, we consider identification and estimation of both preference- and meeting-related parameters in a setting where many networks are observed at two points of time.\footnote{Importantly, our results do not assume that networks are drawn from the model's stationary distribution, an assumption that can be rejected from the data when observation of a network in two periods of time is available (see Auerbach2022 for a statistical test). } Given observation of several networks at two instants in time, we are able to provide researchers with a menu of identification strategies, ranging from nonparametric exclusion restrictions to parametric assumptions on preferences and agents' matching technology, to recover both preference and meeting parameters. We then discuss how these parameters may be estimated through the expectation propagation approximate Bayesian computation (EP-ABC) algorithm Barthelme2014,Barthelme2018, a likelihood-free Bayesian approach that can deliver estimates relatively quickly.\footnote{Battaglini2021 use a variation of the standard ABC algorithm (see (ref)) in order to estimate a game of network formation. The authors use the structure of the game to speed up their implementation. In contrast, we consider a variation of the ABC algorithm that uses the structure of the data to reduce the computational toll of estimation.} Finally, we provide an empirical illustration on how to use these estimates to assess the counterfactual effects of changes in the meeting technology between agents -- something that previous has been unable to do.\footnote{In fact, as we show in Supplemental Appendix (ref), meeting parameters are unidentified in the setting of Mele2017.}
A second set of important differences with previous work is that, by identifying and estimating both meeting- and preference-related parameters, our methodology enables us to analyze the effects of policies along the transition to a new steady state, and not just at the model's stationary (long-run) distribution.\footnote{For example, the analyses in Mele2020 and gaonkar2023model, which are conducted using the identification and estimation approaches in Mele2017, are only able to assess the long-run effects of counterfactual policies that aim to, respectively, promote integration in schools, and restrict or incentivize firm entry on inter-firm collaboration.} Our results also cover a larger -- and arguably less restrictive -- class of preferences and meeting processes than those of Mele2017, who crucially assumes that meeting probabilities do not depend on the existence of a link between agents in the current network. By studying identification and estimation of general classes of preference- and meeting-related parameters in possibly off-stationary-equilibrium settings, we contribute to the model's applicability and empirical usefulness, especially in conducting counterfactual analyses.\footnote{In a recent paper, Chetty2022social use Facebook data to propose a decomposition of homophily in socioeconomic status between an “exposure bias”, the share of individuals with same socioeconomic status in the group (school, church) an individual participates vis-à-vis the share of individuals with same status in the overall population, and a friending bias, the share of same-status friendships within the group vis-à-vis the share of same-status individuals in the group. While they suggest these measures could be used to provide assessments of the effects of policies that aim to reduce segregation via changes in either exposure or friending bias, they do recognize that these measures need not be invariant to policy changes, e.g. a policy that is expected to reduce friending bias by $x$ p.p. may have a different overall effect than a calculation which treats exposure bias as fixed may suggest, inasmuch as it alters the incentives for group participation. In contrast, by working with a structural model, we are able to analyse more complex counterfactuals, where changes in exposure may interact with group participation dynamically. Our model also allows for welfare analyses and, by coupling a peer effects model to it, enables the analysis of the effects of network policies on outcomes (see our empirical application in (ref) and Supplemental Appendix (ref) for an example).}
\paragraph{Overview of identification results} In this paper, identification of meeting and preference parameters relies on different types of exclusion restrictions that limit some sources of variation to affect either the meeting process or agents’ preferences, but not both. We believe these conditions to be plausible in several empirical settings, as we seek to illustrate below.
In many environments, the opportunity to interact with other agents is shaped by institutional or logistical constraints that need not coincide with the determinants of agents’ preferences. In school settings, for example, meeting opportunities may be influenced by factors such as classroom seating arrangements, group assignments, schedule structures, or teacher-imposed activity groups. These features affect which students are more likely to interact during the school day, but need not directly enter the utility students derive from maintaining a friendship with a particular peer.
Conversely, some characteristics may affect agents’ preferences over relationships without substantially affecting the likelihood of interaction. For example, similarity in interests, academic ability, or personality traits may shape the utility students derive from friendships once interactions occur, even if these traits do not systematically influence the institutional structure determining who encounters whom.
In our identification analysis, we provide researchers with a menu of identifying assumptions that precisely exploit these ideas. As we show, there will be often a tension between the stringiness of the exclusion restriction and the degree of parametrization the researcher is willing to impose on preferences and meetings. In our model, fully nonparametric identification relies on the existence of several excluded “instruments” with a large support. In contrast, the adoption of parametric forms for preferences and meetings will allow us to considerably limit both the number of needed instruments and the support requirements, while still allowing for sufficient flexibility so as to capture known empirical regularities in network data, as well as “intuitive” patterns in institutional and individual behavior.
\paragraph{Overview of empirical application} As an empirical application, we study how “homophilies” structure network formation in primary schools in Brazil Pinto2017. We consider 30 municipal elementary schools in Recife, Pernambuco, for which baseline (middle 2014) and follow-up (late 2014) data on 3rd- and 5th-grade intra-classroom friendship networks was collected. Using this information, we structurally estimate our model, using the distance between students in the alphabetically-ordered class-list as an instrument entering the meeting process but excluded from preferences. We then assess how changes in the meeting technology between classmates impact homophily in friendships. Our results suggest that removing biases in meeting opportunities (shutting down homophily in the meeting process) does not decrease observed homophily patterns in students' cognitive skills. By contrast, in a counterfactual scenario where the role of preferences is excluded from the network formation process, the probability that a student maintains a friendship with a classmate with a different level of cognitive skills increases. The results provide evidence that both types of homophily are important determinants of the edges in our networks; although homophily due to preferences does appear to be more important. In this context, a tracking policy that reallocates students between classrooms according to their cognitive skills leads to welfare improvements -- as students benefit from connecting with similar individuals -- though this benefit appears to diminish (vis-à-vis leaving the network process unchanged) in the long run. Given the opportunity to connect with similar people, students experience a positive jump in welfare in the short run. However, after they make their new friendships, they tend to keep these links, and the relative impact of the policy decreases in the long run -- inasmuch that, by the end of the school year, current welfare is roughly the same in both the base and counterfactual scenarios.
\paragraph{Structure of the paper} In the next sections, we introduce the network formation game under consideration ((ref)); explore identification when information on the network structure is available at two distinct points of time ((ref)); and discuss estimation ((ref)). (ref) presents the results of our application. (ref) concludes.
The setup builds upon the work of Mele2017. We consider a network game with a finite set of agents $\mathcal{I} \coloneqq \{1,2\ldots N\}$. Each agent $ i \in \mathcal{I}$ is endowed with a $k \times 1$ vector of exogenous characteristics $W_i$. These vectors are stacked on matrix $X \coloneqq
'$. Agents' characteristics are drawn according to law $\mathbb{P}_X$ before the game starts and remain \emph{fixed} throughout. We denote the support of $X$ by $\mathcal{X}$ and a realization of $X$ by an element $x \in \mathcal{X}$.
Time is discrete. At each round $t \in \mathbb{N}$ of the network formation process, agents' relations are described by a directed network. Information on the network is stored on an $N \times N$ adjacency matrix, with entry $g_{ij} = 1$ if $i$ lists $j$ as a friend and $0$ otherwise.\footnote{Our model and main results are readily extended to the case of an undirected network where friendships are forcibly symmetric.} By assumption, $g_{ii} = 0$ for all $i \in \mathcal{I}$. We denote the set of all $2^{N(N-1)}$ possible adjacency matrices by $\mathcal{G}$.
Agent $i$'s utility from a network $g$ when covariates are $X = x$ is described by a utility function $u_i: \mathcal{G} \times \mathcal{X} \mapsto \mathbb{R}$, $u_i(g,x)$. The utility may depend on the entire network and the entire set of agents' covariates.
Agents are myopic, i.e., they form, maintain, or sever relationships based on the current utility these bring. In each round, a matching process $m^t$ selects a pair of agents $(i,j)$.\footnote{We use the wording “matching process” or “meeting process” interchangeably throughout the paper.} The matching process $m^t$ is a stochastic process $\{m^t: t \in \mathbb{N}\}$ over $\mathcal{M} \coloneqq \{(i,j) \in \mathcal{I}\times \mathcal{I}: i \neq j\}$. If the pair $(i,j)$ is selected, agent $i$ will choose whether to form/maintain or not form/sever a relationship with $j$. After the matching process selects a pair of agents, a pair of choice-specific idiosyncratic shocks $(\epsilon_{ij,t}(0), \epsilon_{ij,t}(1))$ are drawn, where $\epsilon_{ij,t}(1)$ corresponds to the taste shock in forming/maintaining a relationship with $j$ at time $t$. These shocks are unobserved by the econometrician and enter additively in the utility of each choice.\footnote{Additive separability of unobserved shocks is a common assumption in the econometric literature on discrete choice and games Aguirregabiria2010, though it is not innocuous. In our setting, it precludes factors unobserved by the econometrician from affecting the marginal effect of covariates and network characteristics on utility (homophily in preferences), as taste shocks act as pure location shifts.} Given that choice is myopic, agent $i$ forms/maintain a relation with $j$ if, and only if
where $[a, g_{-ij}]$ denotes an adjacency matrix with all entries equal to matrix $g$ except for entry $ij$, which equals $a$.
The following two assumptions constrain the meeting process and the distribution of shocks.
Assumption (ref) constrains the matching function to assign positive probability to all possible meetings under all possible values of covariates and previous-round networks.
Conditional on $X=x$, we have that, under Assumptions (ref) and (ref) -- and given an initial distribution $\mu_0(x) \in \Delta(\mathcal{G})$\footnote{We denote by $\Delta(\mathcal{G})$ the set of all probability distributions on $\mathcal{G}$.} --, the network game just described induces a homogenous Markov chain $\{g^t : t \in \mathbb{N} \cup \{0\}\}$ on the set of netwotks $\mathcal{G}$. The $2^{N(N-1)} \times 2^{N(N-1)}$ transition matrix $\Pi(x)$ has entries $\Pi(x)_{gw}$, $g, w \in \mathcal{G}$, which specify the probability of transitioning to $w$ given the current period network $g$.
For each $g \in \mathcal{G}$, define $N(g) \coloneqq \{w \in \mathcal{G} \setminus \{g\}: \exists ! \ (i,j) \in \mathcal{M} , g_{ij} \neq w_{ij} \} $ as the set of networks that differ from $g$ in exactly one edge. Entries of $\Pi(x)$ take the form {
} where $F_{\epsilon(1) - \epsilon(0)}$ and $F_{\epsilon(0) - \epsilon(1)}$ denote the distribution function of the difference in shocks.
We next look for (conditional) stationary distributions. A stationary distribution is an element $\pi(x) \in \Delta (\mathcal{G}) $ satisfying $\pi(x) = \Pi(x)' \pi(x)$.
In this section, we study identification under many-network asymptotics. Specifically, we assume that we have access to a random sample (iid across $c$) of $C$ networks $\{G^{T_0}_c, G^{T_1}_c, X_c\}_{c=1}^C$ stemming from the network formation game described in (ref).\footnote{Under large-network asymptotics, we have access to a single (a few) network(s) with a large number of players. Identification in this setting consists of providing conditions under which any two sequences of elements of the structural parameter space (which is now indexed by the number of players) with pairwise distinct values lead to asymptotically distinguishable (in a suitable metric) network distributions. See Mele2017 for further discussion.} In this context, $G^{T_0}_c$ and $G^{T_1}_c$ are observations of network $c$ over two (possibly nonconsecutive) periods\footnote{In Supplemental Appendix (ref), we briefly discuss (non)identification when only one period of data stemming from the stationary distribution of the network formation game is available.} (labelled first and second); and $X_c$ is the set of covariates associated with network $c$. We recall the law of $X_c$ is $\mathbb{P}_{X}$, as $X_c$ is a copy of $X$ (i.e., a random variable with the same law as $X$). As in the previous section, realizations of $X$ are denoted by lower-case letters, i.e., elements $x \in \mathcal{X}$.
Denote by $\Pi(X; \theta_0)$ the transition matrix under covariates $X$; where $\theta_0 \coloneqq ((u_i)_{i=1}^N, \rho)$ are the “true” parameters (functions). Further, write $\tau_0$ for the number of rounds of the network formation game that has taken place between the first and second periods. We can identify $\Pi(X;\theta_0)^{\tau_0}$, the transition matrix to the power of the number of rounds of the network formation game which took place between the first and second period ($\tau_0$), provided that the first-period conditional distribution, which we denote by $\pi_0(X)$, is such that $\pi_0(X) >> 0$ $\mathbb{P}_X$-a.s. To see this more clearly, suppose $X$ is empty. In this case, we could consistently estimate $\left(\Pi^{\tau_0}\right)_{gw}$, $g, w \in \mathcal{G}$, by $(\widehat{\Pi^{\tau_0}})_{gw} = \sum_{c=1}^C \mathbbm{1}\{G^{T_0}_c = g, G^{T_1}_c = w\}/\sum_{c=1}^C \mathbbm{1}\{G^{T_0}_c = g \}$,\footnote{In our setting, an agent is described by her set of exogenous characteristics $W_i$. When no such traits are included in the model, two adjacency matrices $g$ and $g'$ are deemed equal if they are equal up to relabeling of agents. When covariates are included, one compares adjacency matrices for a given labeling of agents -- and such labeling will simultaneously define both the distance between the covariate matrices X and X', and between $g$ and $g'$.} provided $\mathbb{P}[G^{T_0}_c = g] > 0$.\footnote{The case where $X$ has discrete support is similar to the case where no covariates are included: for each $x \in \mathcal{X}$, we consider observations $c$ such that $X_c = x$ for some labelling of agents in $c$. We then use these observations, with the adopted labelling of the agents, to compute the transition probability given $X=x$. When $X$ contains continuous covariates, a consistent estimator is given by a kernel estimator LiRacine2006. These estimators would behave poorly in most practical settings (even with few players). We do not suggest using them in practice, though; we rely on consistency only as an indirect argument for establishing identification of the conditional likelihood, i.e. for showing that a function of the distribution of observables recovers these objects.} The next assumption summarizes this requirement.
Since $\Pi(X;\theta_0)^{\tau_0}$ is identified from the data under (ref), the identification problem subsumes to (denoting by $\Theta$ the parameter space):\footnote{Abstracting from measurability concerns, this is the set of utilities and matching functions that satisfy the assumptions in (ref).}
where the inequality must hold with positive probability over the distribution of $X$. Requirement (ref) is equivalent to identification off from the model's conditional (on $G_0$ and $X$) likelihood, i.e. that the expected conditional log-likelihood is uniquely maximized at $(\theta_0,\tau_0)$ Newey1994.
In our statement of the identification problem, the number of rounds in the network formation game is assumed to be unknown. The researcher has no reason to expect $\tau_0$, the true number of rounds, to be known a priori unless the network formation algorithm has a clear empirical interpretation. Nonetheless, it is still possible to identify $\tau_0$ under some assumptions. For all $\theta \in \Theta$ and $x \in \mathcal{X}$, $\Pi(x;\theta)$ is irreducible and has a strictly positive main diagonal. It is clear, then, that the number of strictly positive entries in $\Pi(x;\theta)^\tau$ is nondecreasing in $\tau$. Moreover, this number is strictly increasing for $\tau \leq N(N-1)$\footnote{$N(N-1)$ is the minimum number of rounds required for the probability of transitioning from a “fully empty” network ($g_{ij}=0$ for all $ij$) to a “fully connected” network ($g_{ij}=1$ for all $ij$) to be strictly positive.} and does not depend on the choice of $(x,\theta)$. Thus, provided that $\tau_0 \leq N(N-1)$, we can identify $\tau_0$ by “counting”\footnote{We provide a consistent estimator for $\tau_0$ in (ref).} the number of positive entries in $\Pi(X;\theta_0)^{\tau_0}$.
A similar assumption is considered in Christakis2010, where the authors assume that $\tau_0 = N(N-1)/2$ (they work with an undirected network, so the set of available matches is divided by two) and that all meeting opportunities are played (though in an unknown order). In their setting, however, the assumption's primary purpose is to reduce the computational toll of evaluating the model likelihood (see (ref) for a similar discussion). In our case, we require it for identification. We also emphasize that the bound in (ref) is more or less restrictive, depending on the setting. Knowledge of the particular application in mind should help to assess its appropriability. {For example, one may know the number of time periods $\mu$ (e.g. schooldays) between $T_0$ and $T_1$, and expect that, in a particular setting, a maximum of $\ell$ meetings could have taken place at each period. In this case, one can check whether $N \cdot (N-1)$ is smaller than $\mu \cdot \ell $ to assess the plausibility of the bound. Finally, we note that our identification results on preferences and meeting parameters hold irrespective of the bound -- it suffices that $\tau_0$ is either known a priori or identified.}
In the following subsection, we discuss the identification of the vector of preference and meeting parameters $\theta_0$, taking the number of rounds as identified.
This section considers the power of exclusion restrictions in identifying preferences and meeting parameters. Our approach is motivated by the results in Supplemental Appendix (ref), which show that identification of preferences and meetings in the model defined by Assumptions (ref) and (ref) is generally impossible absent further restrictions.\footnote{Specifically, Supplemental Appendix (ref) shows that, if networks are drawn from the model's stationary distribution and observed at only a single period of time, then the model is generally unidentified. This result motivates our statement of the identification problem in Section (ref), which requires observation of the network at two periods of time. However, Supplemental Appendix (ref) shows that, even in our setting where networks are observed at two points of time, identification is generally impossible absent further restrictions on meetings and preferences.} We consider different types of exclusion restrictions that limit either the role of some covariate in shaping the meeting process, or that assume that some observed trait does not enter the decision of agents of, upon meeting, forming/mantaining a relationship status. We begin with the simplest cases, where $\tau_0 \in \{1,2\}$, or, alternatively, when the researcher imposes enough restrictions in the problem to ensure that $\Pi(X;\theta_0)$ or $\Pi(X;\theta_0)^2$ is identified from $\Pi(X;\theta_0)^{\tau_0}$. We will then provide sufficient conditions for the latter later in this section.
\paragraph{Notation} To make the dependence on covariates explicit, we denote by $X^m(g)$ the set of relevant transformations of $X$ entering the meeting probabilities of agents under network $g$. We thus write $\rho((i,j),g,X) = \rho_{ij}(g|X^m(g))$ for every $(i,j)\in \mathcal{M}$. Similarly, for every $g \in \mathcal{G}$ and $(i,j) \in \mathcal{M}$, we denote by $X_{ij}(g)$ the transformation of $X$ capturing the set of pay-off relevant covariates entering the decision of agent $i$ of altering her relationship status with $j$ under network $g$. We then write $u_i([1-g_{ij},g_{-ij}],X) - u_i(g,X) = \Delta u_{ij}(g|X_{i,j}(g))$.
Our first class of exclusion restrictions considers a situation where a covariate that is relevant for the pay-off of agent $i$ of moving from network $g$ to $w \in N(g)$, $w_{ij}\neq g_{ij}$, is excluded from the covariates relevant for the meeting process under $g$, i.e. $X^m(g)$.
Under the above Assumption, we obtain the following result
The above result establishes identification of preferences and meeting parameters pertaining to the decision of a pair $(i,j)$ moving to and from a network $g$. If Assumption (ref) holds for every pair $(i,j) \in \mathcal{M}$ and network $g \in \mathcal{G}$, then we are able to establish identification of all the relevant marginal utilities and meeting parameters. Under a location normalization of the level of utilities, e.g. that there exists some $g_0 \in \mathcal{G}$ such that $u_{i}(g_0,X)=0$ for every $i \in \mathcal{N}$, we may then establish identification of utilities in levels.
The previous result considered the case where $\tau_0 = 1$ or $\Pi(X;\theta_0)$ is identified. For the case where $\tau_0=2$, or $\Pi(X;\theta_0)^{2}$ is identified, we require a stronger exclusion restriction, which we state in the following. In what follows, we define $N^s(g)$ as the set of networks that differ from $g$ in exactly $s$ edges.
Assumption (ref) considers network configurations $g \in \mathcal{G}$, and pairs of agents $(i,j)$ and $(k,l)$, such that, locally to $g$, the presence of a link between $k$ and $l$ ($i$ and $j$) does not affect the pay-off of $i$ and $j$ ($k$ and $l$) altering their relationship, i.e. we assume that this pay-off is the same under network $g$ and the network $[1-g_{kl},g_{-kl}] = s$ ($[1-g_{ij},g_{-ij}] = h$). This type of assumption is common in the network formation literature, where one usually assumes that the relationships “far away” from an agent's existing connections do not exert an effect over her preferences dePaula2018a,Jochmans2023.
For this subset of networks and pairs, parts (b) and (c) of Assumption (ref) assume the existence of a “large-support” covariate entering the relative pay-off of $(i,j)$ (the pay-off of $(k,l)$) changing their relationship status under network $g$, but excluded from the relative pay-off of $(k,l)$ changing their status under $g$ (the relative pay-off of $(i,j)$ changing their status under $g$), as well as meeting probabilities locally to $g$. These covariates should admit sufficiently “high” realizations that would ensure that, upon meeting, agents would almost certainly change their relationship status. The idea is then to leverage information from pairs that would amost certainly change their relationship status to infer about the meeting process, separating it from the role of preferences. This type of “identification-at-infinity” argument is commonly employed in the literature on the estimation of games and Industrial Organization Tamer2003, Bajari2010, Colas2020, as well as on recent research seeking to disentangle different sources of discrimination Hull2022,Baron2024.
Under Assumption (ref), we establish the following result.
Proposition (ref) establishes point identification of marginal utilities under the exclusion restriction in Assumption (ref). It also shows that, under the additional assumption that both pairs under consideration have the same probability of meeting under network configurations $g$ and $w \in N^2(g)$, meeting probabilities are partially identified, with one additional log-linear restriction being sufficient to establish point identification. For example, one can derive additional restrictions -- and thus identifying power -- by specifying parametric forms on preference and meeting parameters. These parametric forms can also be used to extrapolate identification results from pairs satisfying Assumption (ref) to other agents in the network. We discuss the role of parametric forms in extrapolating identified patterns in Section (ref).
To conclude, we consider an alternative type of exclusion restriction in the meeting process. Following Mele2017, we consider a setting where meeting probabilities do not depend on the existence of a link between agents, i.e. the meeting technology does not exhibit differential behavior on the basis of prior relationships. We show that, in our two-period setting, this assumption has identifying power, being able to recover preferences and meeting probabilities. This is in contrast to a setting with one period of data, wherein Supplemental Appendix (ref) shows that the assumption does not bring identifying power.
In this section, we consider a situation where large support covariates are included in the meeting process, but excluded from preferences. For the case where $\Pi(X;\theta_0)$ is identified, we consider the following assumption:
Assumption (ref) considers a situation where it is possible to find a covariate that affects the meeting probability of agents under network $g$, but is excluded from the relative pay-off of $(i,j)$ changing their relationship status under network $g$. Assumption (ref) requires this covariate to have a large support, insofar that it is possible to find a sequence of support points of $X^m(g)$ where the probability of agents $ij$ meeting under $g$ converges to one, while the relative pay-off of the pair changing status upon meeting remains unchanged.
For the case where $\Pi(X;\theta_0)^2$ is identified, we require a strengthening of Assumption (ref). First, we require the existence of a large support covariate entering $X^m(g)$ that is also be exluded from the covariates entering meeting probabilities in the “adjacent network” $w = [1-g_{ij},g_{-ij}]$. Second, we also require the existence of a limit where the relative probability of pair $ij$ meeting under networks $w$ and $g$ converges to one.
Using the above restrictions, we establish the following identification result
The previous identification results considered the case where $\Pi(X;\theta_0)$, or $\Pi(X;\theta_0)^2$, is identified. We now provide sufficient conditions for these objects to be identified from $\Pi(X;\theta_0)^{\tau_0}$. For that, we impose the following structure.
The assumption that taste shocks follow an extreme value type 1 distribution (Assumption (ref)) is a staple in structural models featuring discrete choices since the seminal work of McFadden1974. Assumption (ref) requires that preferences admit a potential function. It is satisfied by the class of preferences considered by Mele2017, which we also adopt in our empirical application in Section (ref).\footnote{More generally, if agents internalize the externality of their individual choices on other agents' pay-offs, then preferences will admit a potential function. Such representation is not restricted to these settings, though.} Finally, Assumption (ref) requires that the effect of a relationship on the meeting probability of a pair be representable by a “potential” meeting function. This assumption is a strict relaxation over Mele2017's setting, where meeting probabilities do not depend on the existence of a link between agents, i.e. $M$ is a constant function. We provide an example where the assumption holds (approximately) with a nontrivial $M$ in the next section, which we also adopt in our empirical application.
Under the above assumptions, we are able to state the following result.
Proposition (ref) leverages results on the spectral theory of Markov matrices to provide conditions under which it is possible to recover $\Pi(X;\theta_0)$ or $\Pi(X;\theta_0)^2$ from the identified transition matrix $\Pi(X;\theta_0)^{\tau_0}$. Part (a) shows that, if the number of rounds is odd, then $\Pi(x;\theta_0)$ is recoverable from $\Pi(x;\theta_0)^{\tau_0}$, for any $x$ in the support of $X$. Therefore, if $\tau_0$ is identified to be an odd number, one can leverage the identification results in previous sections that assumed $\Pi(X;\theta_0)$ to be identified in order to recover preferences and meeting probabilities. Similarly, part (b) shows that, if $\tau_0$ is even, then $\Pi(x;\theta_0)^2$ is recoverable from $\Pi(x;\theta_0)^{\tau_0}$. In this case, identification results that assumed $\Pi(X;\theta_0)^2$ to be known can be used to recover $\theta_0$. As an alternative to the latter, the second statement in item (b) shows that, with an even number of rounds, it is still possible to recover $\Pi(x;\theta_0)$ for the support points $x$ of $X$ where the transition matrix exhibits sufficient variability. If the identification conditions in previous subsections that assumed $\Pi(x;\theta_0)$ to be known can be shown to hold in this restricted region of the support, then it would be possible to identify preferences and meetings for this region of the support without relying on $\Pi(x;\theta_0)^2$; and then to extrapolate these patterns to the remainder of the support under additional assumptions. While potentially promising, we do not further exploit this alternative strategy in this paper. In the next subsection, we explore parametric forms and the role of extrapolation in aleviating the identification conditions when $\Pi(x;\theta_0)$ or $\Pi(x;\theta_0)^2$ are identified for every support point $x$ of $X$.
Finally, we consider the role of parametric restrictions in relaxing the exclusion restrictions previously considered. Following Mele2017, we consider the following parametric form for preferences.
where $W_{ij}$ is a vector of observed pair-level traits between agents $i$ and $j$, e.g. measures of pairwise distance in their observed characteristics. The first term of the specification allows agents $i \in \mathcal{I}$ to accrue a direct benefit from their connections $g_{ij}$, $j \neq i$. Moreover, this gain may vary according to the observed differences in agents' traits, i.e. one allows for homophily in direct connections. The second term of the specification further allows agents to accrue a differential gain from reciprocal relationships, i.e. to have an additional gain from establishing a connection $g_{ij}$ with a pair such that $g_{ji} = 1$. The third term allows agents to derive utility from their friends' connections, whereas the fourth term captures a “popularity” effect (individuals derive utility of serving as a “bridge” between agents).\footnote{The inclusion of preferences over indirect links and popularity may be seen as providing a rationale for observed clustering patterns that are common in friendship networks (see Badev2017; also Jackson2007). } Under the Assumption that $\beta_{\text{un}} = \beta_{\text{up}}$, Mele2017 shows that the utility function admits a potential function $Q$ as in Assumption (ref).
Next, we consider the following parametrization for the meeting process:
where $Z_{ij}$ is a vector of pair-level traits that satisfties a certain exclusion restriction with respect to $W_{ij}$. Specification (ref) allows homophiliy in meeting probabilities with respect to observed traits $W_{ij}$, as well as dependence on the meeting process with respect to prior relationships. For example, if $\delta > 0$, the meeting process is biased towards agents who are already friends, as these are more likely to meet than those pairs of individuals who do not have a relationship. Moreover, our specification allows this effect to be heterogeneous with respect to the traits $Z_{ij}$.
Under the parametric form (ref), if we assume that $\mathcal{M}$ is sufficiently large relatively to the value of the coefficients $\delta$ and $\psi$, insamuch that the percentage impact of changing the value of one of the edge indicators over the denominator of (ref) is negligible, then we have that, for every $g \in \mathcal{G}$ and $w \in N(g)$ with $g_{ij} \neq w_{ij}$:
$$\log(\rho_{ij}(g|X)) - \log(\rho_{ij}(w|X)) \approx \delta (w_{ij} - g_{ij}) + (w_{ij} -g_{ij}) \psi'Z_{ij} \, $$ which shows that, up to approximation error, meeting probabilities admit the following potential meeting function $M$:
The following proposition provides identification results for the parametric model defined by equations (ref) and (ref), under the assumption of extreme-value type 1 taste shocks and the approximation (ref).
Proposition (ref) provides identification results for the parametric model defined by equations (ref) and (ref). The first part of the proposition shows that every preference parameter except the “intercept” in the direct-links part of pay-offs, as well as the role of the covariates $Z_{ij}$ in the meeting process, are identified under a standard rank condition for a single pair. The parametric restriction then effectively allow us to “extrapolate” these quantities to other pairs in the network. Notice that the rank condition effectively acts as an exclusion condition, since we do not allow $Z_{ij}$ to be collinear with $W_{ij}$.\footnote{While it is possible to allow some of the entries of $Z_{ij}$ to vary colinearly with $W_{i,j}$, this would require additional rank conditions in the second and third parts of the proposition, in order to be able to separately identify $\beta_{\text{ud}}$ from $\psi$. We opt not pursue this extension, as in our empirical specification $Z_{i,j}$ only includes “excluded” covariates.} The second and third parts of the proposition provide sufficient conditions for identification of the remainder parameters. Part (ii) shows that, when the number of rounds is odd, a two-part rank condition identifies the remaining parameters. The first part of the rank condition is implied by assumptions typically used in conditional logit analyses McFadden1974. We supplement this condition with an additional assumption that essentially requires the difference in the direct pay-offs of pairs $(a,b)$ and $(c,d)$ in forming a relationship to exhibit sufficient nonlinear variation so as not to be approximated, in expectation, by a linear function of $\Delta \tilde{W}$.\footnote{For example, this condition excludes the possibility of $\beta_{ud} =0$ and one of the differences between $W_{ab}-W_{cd}$ being constant.} Under the two-part rank condition, we are able to identify $\gamma$, and, having done so, it is then possible to separate $\delta$, which captures the role of existing links in the meeting process, from the intercept in $\beta_{\text{ud}}$, which captures the “base” pay-off of a direct connection. When the number of rounds is even, additional variation is needed, though. Part (iii) states that, in this case, the two-part rank condition must be satisfied while allowing for marginal perturbations in the covariates $Z$ of a third pair. These perturbations should affect the odds of the pair meeting when a link is present (the relevance condition). Finally, we note that the first item in part (iii) implies further exclusion conditions on $Z_{e,f}$ than those implied by part (i), since we assume variation in some direction conditionally on $\{W_m: m \in \mathcal{M}\}$.
In this section, we will analyze estimation. We have access to a sample of $C$ networks, $\{G^{T_0}_c, G^{T_1}_c, X_c\}_{c=1}^C$, stemming from the network formation game previously described.
We first propose to estimate $\tau_0$ as follows:
where $\lVert \cdot \rVert_{F}$ is the Froebenius norm. This estimator is intuitive: it amounts to “counting” the number of different edges between periods in each network and then taking the maximum. It turns out that, under iid sampling and the bound in (ref), $\hat{\tau} \overset{a.s.}{\to} \tau_0$.
As we argue in (ref), the previous lemma can be extended to accommodate a setting where the sequence of observations is independently drawn from games with common $\tau_0$, but where preferences and the meeting process, the distribution of covariates and the number of players may vary with $c$, provided that the number of rounds, $N_c$, is such that $\limsup_{c \to \infty} \mathbb{P}[N_c (N_c - 1) \geq \tau_0] > 0$ holds and that preferences, the meeting process and the distribution of covariates do not asymptotically concentrate on regions where the probability of a network changing by strictly less than $\tau_0$ edges is arbitrarily close to one. This extension is important, as in most practical settings, one expects variation in group sizes: indeed, this is the case in our empirical application in (ref). Notice that, in these settings, if the sample of networks is assumed to be randomly drawn from a population heterogeneous in $N$, then $\limsup_{c \to \infty} \mathbb{P}[N_c (N_c - 1) \geq \tau_0] > 0$ is equivalent to $\mathbb{P}[N (N - 1) \geq \tau_0] > 0$, i.e. the condition requires there exist a positive mass of networks whose number of players $N$ is such that $N (N - 1) \geq \tau_0$. In other words, we do not need all networks in the population to satisfy (ref), just that a positive mass do.
{Notice that, even though $\hat{\tau}$ is consistent for $\tau_0$ under the Assumptions required by Lemma (ref), our proposed estimator approaches $\tau_0$ from below, thus being generally downward biased. We thus recommend researchers to assess the sensitivity of their results to the estimated value of $\tau_0$. This can be achieved by directly veryfing how estimates of preference and meeting-related parameters vary when one changes the estimated value of $\tau_0$ -- an approach we undertake in our empirical application in Section (ref). Alternatively, if one adopts a Bayesian approach to estimation -- this is our recommended strategy to estimating preferences and meeting parameters in Section (ref) --, one could choose a data-driven prior for $\tau_0$ that takes our estimator $\hat{\tau}$ as a lower endpoint to its support. See Remark (ref) in the next section for further discussion on the specification of priors for $\tau_0$ in a Bayesian estimation framework. }
Let vector $\beta_0 \in \mathbb{B} \subseteq \mathbb{R}^l$ encompass a parametrization of preferences and meetings, i.e. $u_{i}(g,X) = u_{i}(g,X;\beta_0)$ and $\rho_{ij}(g, X) = \rho_{ij}(g,X;\beta_0)$ for all $(i,j) \in \mathcal{M}$, $g \in \mathcal{G}$. If the adopted parametrization satisfies one of the identification conditions in Section (ref), we know that $\beta_0$ is the unique minimizer of the expectation of the conditional log-likelihood, i.e. $$\beta_0 = \operatorname{argmin}_{\beta \in \mathbb{B}} \mathbb{E}[\log (\Pi(X_c;\beta)^{\tau_0}_{G^{T_0}_c, G^{T_1}_c})|X_c, G^{T_0}_c]\, .$$
As a consequence, a natural estimator to be considered in this case is a conditional MLE that replaces the expectation operator with its sample analog and $\tau_0$ with a consistent estimator of the number of rounds. To compute this estimator, we note that the log-likelihood of a second-period network $G^{T_1}_c$, conditional on $X_c$ and $G^{T_0}_c$, may be written as:
$$ l_c(G^{T_1}_c|G^{T_0}_c, X_c; \tau_0, \beta) = \sum_{g \in \mathcal{G}} \mathbbm{1}\{G^{T_1}_c = g\} \ln \left((\Pi(X_c;\beta)^{\tau_0})_{G^{T_0}_c g}\right) \,, $$ and the sample log-likelihood, under an independent sequence of observations, is
The second-step MLE estimator will thus be
$$\hat{\beta}_{\text{MLE}} \in \text{argmax}_{\beta \in \mathbb{B}} \mathcal{L}(\{G^{T_1}_c\}_{c=1}^C|\{G^{T_0}_c\}_{c=1}^C; \hat{\tau}, \beta) \,,$$ where $\hat{\tau}$ is the estimator discussed in the previous section. This formulation can also be easily modified to accommodate observations of networks with different numbers of players.
Numerically, computation of the likelihood is complicated by the fact that we need to sum over all walks between $G^{T_0}_c$ and $G^{T_1}_c$.\footnote{A walk between $g$ and $w$ in $\tau$ rounds is a sequence of networks $g_1, \ldots, g_{\tau}$ such that $g_1=g$, $g_\tau = w$ and $g_t \in N(g_{t-1})\cup \{g_{t-1}\}$ for all $t=2,\ldots,\tau$.} For a small estimate of $\tau_0$, this is feasible, but for higher estimated values of $\tau_0$, it becomes impractical. Indeed, for a given number of rounds $\tau \in \mathbb{N}$, there are $[N(N-1) + 1]^{\tau}$ walks starting from $G^{T_0}_c$ and ending in some network $g \in \mathcal{G}$. Evaluating the model likelihood requires summing over all walks ending in $G^{T_1}_c$. A “recursive” approach for evaluating the model likelihood would consist of, for each $c \in \{1,2\ldots C\}$, “writing down” the formula for each walk iteratively, i.e., starting from $G^{T_0}_c$, compute all possible $N(N-1)+1$ transitions in the first round; then, for each of these $N(N-1)+1$ possible transitions, compute the $N(N-1)+1$ transitions in the second round and multiply each of these probabilities by the probability of the associated transition in the first round, and so on; and then summing over all walks ending in $G^{T_1}_c$. Walks that “strand off” from $G^{T_1}_c$ in some round $r < \tau$ can be excluded from the next steps in the recursion,\footnote{By “strand off” we mean a path of realizations of the stochastic process $g^t_c$ up to round $r$ such that the probability of reaching $G^{T_1}_c$ in $ \tau-r$ rounds is 0.} which ameliorates the computational toll, but does not solve it.
Given the above difficulty, an interesting alternative is to work with simulation-based methods, which allows us to bypass direct evaluation of the model likelihood. We take a Bayesian perspective\footnote{From the frequentist point of view, a simulated method of moments estimator would be a possibility in our case, though the nonsmoothness of the objective function (which involves indicators of simulated network observations) as well as the poor properties of GMM estimators with many moment conditions (transition probabilities) in finite samples NeweySmith2004 are unappealing. An indirect inference approach is also unappealing, as low-dimensional sufficient statistics are unknown in our context.} and follow an approach known as likelihood-free estimation or approximate Bayesian computation (ABC) Sisson2011.\footnote{From the Bayesian point of view, an alternative and well-known approach to simplify, but not bypass, a complicated model likelihood is data augmentation Hobert2011. However, this alternative is not useful in our setting due to the dimensionality of the support of the meeting process. Thus, an approach that conditions the likelihood on the (unobserved) matching process (reducing the number of walks starting at $G^{T_0}_c$ from $[N(N-1) + 1]^{\tau}$ to $2^\tau$) is unworkable here in our case, since we still have to draw from the matching process distribution conditional on the data and the model parameters.} This method bears a close correspondence to nonparametric (frequentist) estimation Blum2010 and indirect inference Frazier2018. The methodology requires the researcher to be able to draw a sample (or statistics thereof) from the model given the parameters. (ref) outlines the simplest accept-reject ABC algorithm in our setting, where $S$ is the maximum number of iterations and $p_0(\beta,\tau)$ is a prior distribution over $\mathbb{B} \times \mathbb{N}$:\footnote{In practice, a few improvements can be made upon (ref) Wentao2018a. First, we can use importance sampling: instead of drawing from the prior, we may draw from a proposal distribution $q_0(\beta,\tau)$ such that $\text{supp} \ p_0 \subseteq \text{supp} \ q_0$. Accepted draws should then be associated with weights $w_s \coloneqq p_0(\beta_s, \tau_s)/q_0(\beta_s, \tau_s)$. Wentao2018a provides a data-driven method to select the proposal. Second, we can use a “smooth” rejection rule, i.e., we accept a draw with probability $K(\lVert T_s - T_{\text{obs}}\rVert/\epsilon)$, where $K(\cdot)$ is a rescaled univariate kernel such that $K(0) = 1$. See Supplemental Appendix (ref) for details of this augmented algorithm.}
In this methodology, the researcher must make two crucial choices. One is the tolerance parameter. Here, we can use the recommendations of Wentao2018a: we may choose $\epsilon$ so the algorithm produces a “reasonable” acceptance rate. The second important choice is the vector of statistics. This is closely related to identification: for the proper working of the ABC algorithm, the chosen vector of statistics should be informative of the model's parameters Wentao2018a, Frazier2018.\footnote{Both authors study the frequentist properties of ABC algorithms. In their setting, where a finite-dimensional vector of summary statistics is considered, it is crucial that the binding function $b(\beta_s,\tau_s) = \operatorname{plim}_{C \to \infty} T_s$ identifies model parameters.} As an example, Battaglini2021 use a version of the ABC algorithm to estimate a network formation game. In their setting, the authors use the characterization of the model equilibrium to construct the summary statistics used to assess the quality of a parameter draw.
In Supplemental Appendix (ref), we show that, if we take the vector of summary statistics as the whole second-period network data, then, as $\epsilon \to 0$ and $S \to \infty$, the mean of the accepted draws $h(\theta_s)$ converges in probability to the expectation of $h(\cdot)$ with respect to the posterior computed using the conditional likelihood yielded by (ref) in the Bayes rule, where $h(\cdot)$ is a function with finite moments (with respect to the prior distribution). Using the entire dataset as the vector statistics is important, as it ensures we do not discard any information in the likelihood that, under our assumptions, enables point identification of the model parameters.\footnote{As remarked in an earlier footnote, our model does not yield any obvious low-dimensional vector of sufficient statistics to be used in the analysis without compromising identifiability.} The result in Supplemental Appendix (ref) motivates the computation of approximations to the posterior mean and credible intervals using the accepted simulated draws. However, it should be noted that, if there are many networks in the sample, it will generally be very difficult to generate a good approximation to posterior quantities without running a very large (and often computationally unfeasible) number of simulations. Intuitively, with many networks, it will take many tries to get a draw that approximates the entire vector of observed data well. With a fixed number of simulations, this means that, to have a reasonably small Monte Carlo variance, we tolerate high levels of bias in the ABC approximation to the true posterior.\footnote{This “curse of dimensionality” in the dimension of the vector of statistics used in ABC methods bears resemblance to the notion of a “curse of dimensionality” in the rate of convergence in nonparametric kernel estimation methods, as the tolerance may be seen as a type of bandwidth. See Blum2010 for further exploration of this connection.}
To ameliorate the curse of dimensionality associated with taking the whole second-period data as the vector of summary statistics, we propose to use an alternative version of the ABC algorithm: the Expectation Propagation (EP) ABC Barthelme2014, Barthelme2018. In this method, we leverage the assumption of an independent sample of networks to factor the posterior distribution as:
where $\mathbf{G}_{0} =(G_c^{T_0})_{c=1}^C$; $\mathbf{G}_{1} =(G_c^{T_1})_{c=1}^C$; $\mathbf{X} = (X_c)_{c=1}^C$; and the prior $F$ only ranges over utility and meeting parameters because we assume that $\tau_0$ is either fixed or estimated in a first step via (ref) (see (ref) for further discussion). EP-ABC assumes a Gaussian prior for $\beta_0$, and further considers a Gaussian approximation with mean $\mu_c$ and covariance matrix $\Omega_c$ to each network likelihood function $\beta\mapsto \mathbb{P}[G^{T_1}_c|\beta,G^{T_0}_c, X_c]$. It then proceeds by sequentially updating the approximations as follows. Suppose we are currently at network $c$. The goal is to find $(\mu_{c}, \Omega_c)$ to minimize the Kullback-Leibler (KL) divergence between (i) a hybrid approximation that uses the true density at the $c$-th network and the Gaussian approximation at the remaining networks; and (ii) the “full” Gaussian approximation. Let us denote the prior distribution by $l_0(\beta)$ ($dF(\beta)\eqqcolon l_0(\beta)$), the "true" likelihood function at each network by $l_c(\beta)$ ($\mathbb{P}[G^{T_1}_c|\beta, G^{T_0}_c, X_c] \eqqcolon l_c(\beta)$), and let $f_c(\beta)$ be a Gaussian density with parameters $(\mu_c,\Omega_c)$. Barthelme2014 show that the choice of $(\mu_{c}, \Omega_c)$ that minimizes this KL divergence is
These quantities are approximated by running a variant of the ABC algorithm (ref) with prior $ l_0(\beta) \times \prod_{i \neq c}^C f_i(\beta)$, which is Gaussian and can be evaluated; and likelihood $l_c(\beta)= \mathbb{P}[G^{T_1}_c|\beta, G^{T_0}_c, X_c]$, which involves simulating the second-period adjacency matrices only in network $c$. This ameliorates the curse of dimensionality since, at any given step, a draw must be good at reproducing features of the current network. Starting from initial guesses for the $(\mu_c,\Omega_c)$ equal to the prior parameters, the algorithm proceeds by sequentially going through all networks $c=1,\ldots C$ and updating $(\mu_c,\Omega_c)$ until meeting a convergence criterion or a full number of passes through all networks.\footnote{In Supplemental Appendix (ref), we describe another method: “expectation propagation with `local' summary statistics”, as an alternative to the EP-ABC described above. To further reduce dimensionality, this method replaces the vector of network edge indicators in the EP-ABC approach with data-driven “local” summary statistics.}
Our application considers data on friendship networks from Pinto2017. The dataset comprises information on 3rd- and 5th-graders from 30 elementary schools in Recife, Brazil. Data on students' traits and intraclassroom friendship networks was collected at the middle (baseline) and end (follow-up) of the 2014 school year.\footnote{Specifically, students could nominate up to 8 classmates in each of the three categories: classmates with whom they would (i) study, (ii) talk, or (iii) play. We consider an individual a friend if she appears on at least one of the three lists. No student exceeds eight friends since, in most cases, the three criteria coincide. In our raw dataset, only 1.01% of students reported eight friends at the baseline, which falls to 0.3% at the follow-up. For a complete dataset description, see Pinto2017.} Once missing observations are removed, our working sample comprises 161 classrooms (networks), totaling 1,589 students.\footnote{ Figure (ref) in Supplemental Appendix (ref) plots one such network.} The median number of students in each classroom is $10$.
As a first step in our analysis, we attest that homophily is a salient feature of our data. For that, we run the following regression model:
where $g_{ij,c,1}$ equals $1$ if, in classroom $c$, individual $i$ nominates $j$ as a friend at the followup period. Vector $W_{ij,c,0}$ consists of pairwise distances in gender, age (in years), and the logarithm of measures of cognitive and non-cognitive skills between $i$ and $j$ at the baseline period.\footnote{Table (ref) in Supplemental Appendix (ref) presents summary statistics of our dyad-level covariates. See Pinto2017 for details on the construction of the measures of cognitive and noncognitive skills.} The specification controls for sender ($\alpha_i$) and receiver ($\gamma_j$) fixed effects. We cluster standard errors at the classroom level.
Column (1) in (ref) reports estimates obtained from running the above specification. Results indicate that homophily is pervasive, e.g., same-sex classmates are, on average, 19.8 pp more likely to be friends than boy-and-girl pairs.
In implementing our model of network formation, we consider the parametrisation of preferences given by (ref), with $\beta_{\text{un}} = \beta_{\text{up}}$, and $W_{i,j}$ taken to be the vector $W_{ij,c,0}$ of pair-level covariates at the baseline. The difference in preference shocks is drawn from a logistic distribution. For the meeting process, we consider a similar specification to (ref), where, for individuals $i$ and $j$ in classroom $c$, we assume that:
with $Z_{ij,c,0}$ a pair-level trait that, according to the discussion in Proposition (ref), should be crucially excluded from pair-level covariates $\{W_{m,c,0}\}$, and affect network formation ($\psi\neq0$).\footnote{Even though, in the specification discussed in Section (ref), $Z_{ij}$ appears interacted with $g_{ij}$, it is immediate to adapt Proposition (ref) to hold in the case where this interaction occurs with $(1-g_{ij})$, as is the case our empirical specification.} In our implementation, we take this covariate to be the distance between two students in the alphabetically-ordered class list. We do so because we expect the distance in the classlist to affect the odds of a pair meeting, e.g. through in-class activities conducted in alphabetically-arranged groups. Still, we do not expect this variable to impact the utility of individuals over social relations, especially since, in specifying preferences, we control for distances in age, gender, and cognitive and noncognitive skill measures. Finally, our proposed specification also embodies the assumption that the class list meeting mechanism is only important for pairs that are not currently friends.
Column (2) in (ref) provides reduced-form evidence of the relevance of our class list distance variable. Again, we run the specification in (ref), but include our class list distance variable and exclude sender and receiver fixed effects. The covariate is statistically significant at the 1% level, with the expected sign: all else equal, classmates “one more student away” in the class list is 0.2 pp less likely to be friends.
{We estimate our network formation model using a two-step approach. In the first step, we assume $\tau_0$ to be constant across classrooms and estimate it through (ref). Our first-step estimate of $\tau_0$ leads to 76 rounds. This estimate satisfies the inequality $\hat{\tau} < N_c (N_c - 1)$ for 81 out of 161 networks in our sample, indicating that, for about half of the classrooms, the estimated number of rounds is such that it would be impossible for the existing network in mid 2014 to be completely overturned by the end of the school year. We also find our estimate to be consistent with contextual information on the number of school days between the baseline and followup. Indeed, collection of baseline data started on July 21st, whereas the followup ended on December 12th, totalling around 100 school days. Our estimate would thus imply that three opportunities to revise friedship status -- our model notion of meetings -- occur every four days. As a sensitivity test, Supplemental Appendix (ref) reports estimates of preference and meeting parameters when we double our estimate of the number of rounds -- i.e. when we set $\hat \tau = 152$, or equivalently that three meeting opportunities occur every two days. Qualitatively, our main results are mostly unchanged.}
In the second step of our estimation approach, we use the EP-ABC method described in (ref) to approximate the posterior of preference- and matching-related parameters, where we take the first-step estimate of $\tau_0$ as given.\footnote{From a frequentist perspective, we can motivate our estimator by appealing to some Bernstein-von-Mises theorem Vaart1998 which ensures posterior asymptotic normality as $C \to \infty$ (see also the discussion of Frazier2018). From a Bayesian point of view, our estimator imposes a degenerate prior on $\tau_0$ at the frequentist estimator (ref).} We adopt independent zero-mean Gaussian distributions with a standard deviation equal to two as priors for both meeting and preference parameters.\footnote{The value of $2$ is chosen according to the reduced-form patterns in (ref), to cover with reasonable prior probability parameter values that enable either homophily in preferences by itself or homophily in meetings by itself to account for observed patterns.} We follow Barthelme2014's (Barthelme2014) method by making a single pass through the dataset, which appears to be sufficient for convergence. In the ABC step, we follow the suggestion of these authors and use a simple nonstochastic accept-reject rule for draws. We use the follow-up network data in the current classroom as the vector of statistics. Dimensionality, in this case, is smaller than using a standard ABC algorithm since we consider just a single classroom at each step. For each classroom, we simulate 100,000 draws and aim for an acceptance rate of 1% (1,000 accepted draws). We use Halton random numbers to generate draws from the prior, as suggested by Barthelme2014, to improve numerical stability.
(ref) presents the posterior mean of the coefficients of the utility function and matching obtained by the Expectation Propagation ABC method described above. We also report posterior quantiles and the posterior probability of a negative parameter.
The results indicate several mean estimates of utility parameters associated with covariates are negative. This suggests that homophily in preferences is pervasive and not restricted to direct links. We find high posterior probabilities of a negative effect for the role of gender in preferences, not only for the direct links but also for reciprocal links and popularity. Moreover, we find high posterior probabilities of a negative effect for the role of cognitive and noncognitive skills in different utility components. As for the matching parameters, the posterior probability of the coefficient associated with our instrument being negative is near 100%, which is reassuring. We also find evidence of heterophilia in the matching function with respect to age and gender and dependence of meeting opportunities on a previous link between agents. However, these results are not robust when we estimate our model using the alternative method discussed in Supplemental Appendix (ref).\footnote{See Supplemental Appendix (ref) for results under this alternative method. The main conclusions of our counterfactual exercise remain essentially unchanged.}
Next, we proceed to counterfactual exercises. We consider the evolution of networks, starting from their baseline value, under four different sequences of matching parameters: (i) when these are kept at their estimated value (base case); (ii) when random unbiased matching is imposed across networks (random meetings); (iii) when keeping grade and classroom size in schools fixed, we track students according to their cognitive skills (tracking case); and (iv) when, upon meeting, friendships are formed at random with probability 1/2 (random friendships). It is important to emphasize that we view counterfactuals (ii) and (iv) less as implementable policies, and more as a means of quantifying the role of preferences and meetings in network formation and welfare. Counterfactual (iii) illustrates how our model can be used in policy analysis.
(ref) reports posterior means and 95% credible intervals of the projection coefficients of edge indicators at the followup period ($g_{ij,c,1}$) on an intercept and our main controls at the baseline. Compared with the observed data, magnitudes in the base case are broadly in line with the frequentist reduced form (Column (2) in (ref), reproduced as Column “Data” in (ref)). Nonetheless, our model appears to overstate the role of homophily in age; and we also understate the average value of $g_{ij,c,1}$ in the data. Comparing the second and third columns, we see that imposing random unbiased matching increases observed homophily patterns in gender, cognitive skills, and conscientiousness. This indicates that shutting down biases in meeting opportunities does not lead to a decrease in observed homophily patterns. Moving on to the last column, when we close the channel of homophily due to preferences, the coefficients associated with homophily due to gender, cognitive skills, and conscientiousness decrease in magnitude compared to the base case. This is evidence that, in this example, homophily in preferences is more important than homophily in meetings. In the fourth column, tracking leads to a weak reduced-form estimate of homophily in cognitive skills. This is expected as students now interact in homogeneous groups. It also leads to a weaker pattern in the gender coefficient, which may be due to the correlation of this attribute with cognitive skills at the baseline.\footnote{Girls have, on average, $1.84 \%$ more cognitive skills at the baseline than boys, and this difference is statistically significant at the 1% level.}
Figures (ref), (ref), and (ref) compare the evolution of aggregate utility in the base case with each of our counterfactual scenarios. We plot posterior means and 95% credible intervals of the aggregate utility index, $\sum_{c=1}^C \sum_{i=1}^{N_c} u_i(g^t_c, X_c)$, in the counterfactual scenario minus the same index in the base case, for each round from the baseline to the follow-up period. We normalize the difference in indices by the number of students in our sample multiplied by minus the posterior mean estimate of the distance-in-gender direct utility parameter; so results can be interpreted as the number of direct same-sex links each student should receive in the base case so that they are indifferent between policies (without taking into account spillovers on the remaining components of utility).\footnote{Since we only report the difference in aggregate utility between alternative policy scenarios, our welfare analyses are invariant to the normalization adopted to recover the levels of utilities (observe that parametrization (ref) adopts the normalization that the utility of an empty network is zero). Computing welfare relatively to a baseline scenario thus helps us avoid the typical concern in counterfactual analyses in discrete choice models that require correctly identifying the levels of pay-offs (see kalouptsidi2021counterfactual for related discussion).}
Imposing random matching leads to a lower trajectory in aggregate utility over the school semester. As expected from the previous analysis, random friendship formation leads to even lower welfare than random matching. The decrease in welfare in both cases comes from a decrease in two components of the utility function: direct links and popularity. We also see that tracking leads to an improvement in welfare, though this benefit diminishes as time passes on. This indicates that a tracking policy in these schools has a positive effect on welfare in the short run, but this effect decreases over time, getting close to zero at the end of 76 rounds of iteration. The impact of tracking policies depends on the network structure, as pointed out by Jackson2021. Our example shows that homophily due to preferences is stronger than homophily due to opportunities. Then, when we change the meeting opportunities and place individuals in homogeneous classrooms, their welfare increases substantially in the first rounds of the game. However, as soon as they make their new connections, the relative effect of the tracking policy starts to decrease vis-à-vis the base case. At the end of the game, the relative impact of the policy is close to zero.
In Supplemental Appendix (ref), we show how our model can be used to assess the effects of counterfactual policies on measures of productivity and inequality in cognitive skills by coupling a peer effects model to our network formation algorithm. Our counterfactual exercises indicate that shutting down the preference channel in network formation may lead to higher average cognitive skills, though possibly at the expense of increased within-classroom inequality. All in all, these results suggest that policies aimed at changing the determinants of network formation (see Chetty2022social for examples) may have nonnegligible impacts on students' outcomes.
In this article, we studied the identification and estimation of a network formation model that distinguishes between homophily due to preferences and homophily due to meeting opportunities. The model builds upon the algorithm of Mele2017 by allowing for general classes of utilities and meeting processes. It is also well-grounded in the theoretical literature of network formation Jackson2002, Jackson2010social. We provided researchers with a menu of identification assumptions that leverage different types of exclusion restriction to point identify both preference and meeting parameters. We also discussed a Bayesian estimation procedure that bypasses direct evaluation of the model likelihood -- a task that can be computationally unfeasible even for a moderate number of rounds of the network algorithm. All in all, our approach enables users to estimate the counterfactual effects of changes in the meeting technology between agents across time, something previous work could not do.
In the applied section of our article, we studied network formation in elementary schools in Northeastern Brazil. Our results suggest that tracking students according to their cognitive skills improves welfare, though the benefits reduce over time. The effect of this policy can be associated with the structure of the networks. In these classroom networks, homophily due to preferences seems more salient than homophily due to meeting opportunities. This can explain the large positive short-run effect of the tracking policies but the almost zero long-run impact.
{The identification strategy developed in this paper relies on a structural feature common to many models of decentralized interaction: the separation between a meeting process, which determines which agents interact, and a choice process, which governs the actions taken conditional on a meeting. In the model studied here, the meeting process determines which pair of agents obtains the opportunity to revise their relationship status, while agents’ preferences determine whether links are formed or maintained.
This structure is not specific to network formation models. Similar sequential protocols arise in a wide range of economic environments, including search models of financial and product markets Lagos2017,Wright2021. In these settings, outcomes are generated by a combination of arrival processes (which determine who meets whom) and choice behavior conditional on interaction opportunities.
The identification arguments developed here suggest that, when observations are available on the evolution of outcomes across time and across many independent environments, it may be possible to recover both the meeting technology and the underlying preferences governing agents’ decisions. Exploring our identification strategies in these other contexts constitutes a promising direction for future research.}