Social Interactions in Endogenous Groups

Social Interactions in Endogenous Groups

Introduction

Social interactions models are useful tools for exploring the interdependence of individual outcomes in various contexts, such as education, earnings, health, and crime. One notable feature of social interactions is their high tendency to occur among individuals within the same social or economic group. For instance, students interact with fellow students within their school, and residents engage with other residents within their neighborhood.\footnote{There is a massive literature on peer effects in schools, classrooms, or dorms (e.g., Evans1992; sacerdote2001peer; Duflo2011tracking). See Epple2011 for a survey. Examples of neighborhood effects include katz2001moving and bayer2008place.} Because individuals select the schools they apply to or the neighborhoods they reside in, the peers they ultimately interact with are often determined endogenously. Therefore, when we observe that individuals with more advantageous peers achieve better outcomes, it is unclear whether these outcomes are driven by the influence of their peers or by their choices of peer groups Epple2011,Sacerdote2011. This paper develops new econometric methods for identifying and estimating causal peer effects in the presence of endogenous groups.

Selection into groups has been a central topic in the literature on social interactions and group treatment effects (e.g., school value-added). Early studies on social interactions account for group selection in the framework of correlated effects (Manski1993; moffitt2001policy). This approach addresses selection by including group fixed effects, the validity of which relies on the restrictive assumption that an individual's group membership becomes exogenous once group-level unobservables (e.g., school resources, neighborhood amenities) are controlled for lee2007identification,bramoulle2009identification,Sacerdote2011. In contrast, empirical research has leveraged natural experiments with random group assignment to circumvent the selection issue.\footnote{Examples include classroom assignment Duflo2011tracking and dormitory assignment sacerdote2001peer,zimmerman2003peer.} In school choice, the literature has exploited exogenous variation under centralized school assignment (e.g., random lotteries or discontinuities around non-random test score cutoffs) to overcome selection into schools kirkeboen2016field,abdulkadirouglu2017research,abdulkadirouglu2022breaking,angrist2024credible. These empirical approaches require observing information on individuals' rank-order lists, priorities, and admission cutoffs. In this paper, we propose an approach to account for selection into groups that does not rely on random assignment or detailed information on applications and admissions. Our approach is applicable to observational data where only group memberships are observed, making it particularly relevant for contexts such as decentralized school choice and college admissions.

Specifically, we develop a model of group formation to examine how individuals sort into groups and how to effectively account for the impact of this sorting. We characterize group formation using a two-sided many-to-one matching model with nonparametric unobservables, where individuals select among groups based on their preferences, while groups rank and accept individuals according to qualifications until reaching capacity constraints azevedo_supply_2016,HSS_twosided. This framework closely mirrors real-world admission processes, such as those for schools gazmuri2017school,HSS_twosided, colleges dale_estimating_2002, residency programs roth1984evolution,agarwal2015empirical, and nursing homes gandhi2022picking. Our framework also covers one-sided group formation as a special case, in which individuals unilaterally determine the groups they join, such as in neighborhood choice brock2001interactions,brock2002multinomial,brock2005multinomial,ioannides2008interactions. In contrast to one-sided group formation, where an individual's choice set is assumed to include all groups, two-sided group formation allows the choice set to be latent and determined endogenously Barseghyan2021,Agarwal2022. To the best of our knowledge, we are the first to apply two-sided matching to analyze selection into groups.

The endogenous formation of groups leads to selection bias in the estimation of peer effects in linear-in-means social interactions models Manski1993. For example, students with higher capabilities or more advantageous family backgrounds may be sorted into more selective colleges, leading to the overestimation of peer effects dale_estimating_2002. The formulation of our selection bias supports the approach proposed by dale_estimating_2002, who overcame selection bias by matching students who applied to and were accepted by comparable sets of colleges. In contrast, we correct for selection bias in the absence of such application and admission information.

The selection bias in our model is further complicated by equilibrium effects. Because equilibrium groups depend on the (observed and unobserved) characteristics of all $n$ individuals in a market, selection bias is a high-dimensional function that involves the observed characteristics of the $n$ individuals. We overcome this dimensionality problem using the limiting approximation of a market as $n$ approaches infinity azevedo_supply_2016.

Moreover, we impose an exchangeability assumption on the distribution of unobservables, which yields two crucial properties that aid in addressing selection bias. First, it allows selection bias to be represented as a group-invariant selection function of preference and qualification indices, enabling the identification of the effects of group-level variables (e.g., group averages, group dummies). Second, under exchangeability, selection bias is symmetric in the impacts of alternative groups. For example, selection bias for attending college A is symmetrically affected by the preference and qualification indices associated with colleges B and C. This symmetry enables us to aggregate the indices of alternative groups using their elementary symmetric functions altonji_matzkin, thereby reducing the number of nuisance parameters in the nonparametric estimation of selection bias.

Our results on selection bias make three important contributions to the literature. First, our model yields selection bias that depends on individual characteristics involved in group formation. This finding highlights that simply including group fixed effects is insufficient to correct for selection bias. The result also complements the work of Altonji2018, who showed that selection bias can be captured by group averages of covariates under specific linear restrictions. In contrast, our framework imposes no such linear restrictions. We show that effectively correcting for selection bias requires exploiting individual-specific information in group formation. Second, our results emphasize the importance of accounting for the impacts of alternative groups.\footnote{For instance, in college admissions, whether a student attends a particular college depends not only on the attributes of that college, but also on the attributes of the colleges they choose to forgo. This feature suggests that the attributes of alternative groups play a crucial role in selection correction.} Unlike existing studies that directly restrict the impacts of alternative groups Dahl2002, our approach aggregates these impacts by leveraging the symmetry of selection bias under exchangeability. This makes the selection correction tractable, even when the number of groups is moderately large. Related to our work, abdulkadirouglu2020parents proposed a similar selection correction method in centralized school choice, where selection occurs only through unobserved preferences. Their selection correction aggregates the impacts of alternative schools constructed using rank-order lists. We provide a microfoundation for the aggregation, but in the context of decentralized markets with selection on both unobserved preferences and qualifications, where rank-order lists are unavailable. Third, we separately identify group treatment effects (e.g., school fixed effects that measure school effectiveness) and self-selection, thereby allowing for the disentanglement of correlated effects due to group-level unobservables from those due to self-selection into groups Manski1993,moffitt2001policy.\footnote{Manski1993 was the first to discuss correlated effects, defining them as situations in which “individuals in the same group tend to behave similarly because they have similar individual characteristics or face similar institutional environments.” moffitt2001policy further categorized correlated effects into two types: those arising from shared environmental influences (e.g., unobserved school characteristics), and those arising from sorting, where individuals with similar traits tend to group together.}

To identify the social interaction parameters, we partial out selection bias as in a partially linear model Robinson1988. A key challenge in identifying linear-in-means social interactions is that peer outcomes and peer characteristics may be linearly dependent, leading to the well-known reflection problem Manski1993. Some studies have overcome this problem by leveraging group size variation or intransitive triads (lee2007identification; graham2008identifying; Davezies2009; bramoulle2009identification). By comparison, brock2001interactions demonstrated that self-selection can aid in the identification of social interactions. The intuition is straightforward: self-selection enters the outcome equation as an individual-specific term that is excluded from group-level averages, thereby providing an exclusion restriction to disentangle endogenous social effects from exogenous social effects. brock2001interactions considered one-sided group formation with parametric unobservables. We extend their insight to a more general setting with two-sided group formation under nonparametric unobservables. Similarly, gu2024peer leveraged self-selection to achieve the identification of peer effects in a binary selection setting, where individuals decide whether to participate in a program, under parametric unobservables.

We propose a sieve OLS estimator for the social interaction parameters, where we control for selection bias using a sieve approximation. While using OLS instead of IV might appear counter-intuitive---given that peer outcomes could be endogenous due to simultaneity---we demonstrate that this endogeneity vanishes asymptotically in large groups where within-group connections are dense (e.g., group averages). This finding is consistent with the results in Lee2002, who demonstrated the consistency and efficiency of OLS estimators for peer effects under a deterministic adjacency matrix. The asymptotic properties of our estimator cannot be established using existing methods because our adjacency matrix is stochastic due to randomness in group memberships and within-group connections. By recognizing that the leading terms in our estimator take the form of weighted $U$-statistics with random weights, we develop an innovative approach that generalizes the asymptotic methods for standard weighted $\mathit{U}$-statistics Lee1990 to accommodate randomness in weights. Furthermore, we impose additional conditions to ensure that the network dependence introduced by random weights diminishes sufficiently fast. Under these conditions, we prove that our estimator is $\sqrt{n}$-consistent and asymptotically normal. Finally, we provide simulation evidence showing that our estimator performs well.

We apply our approach to analyze social interactions among high school students in Chile. We use data from the SIMCE along with administrative records from the Ministry of Education. Our sample consists of 6,872 tenth-grade students enrolled in 53 high schools in the Biob�o Region in 2006. The Chilean education system is highly segregated, with substantial sorting across high schools. While our dataset provides comprehensive student information both before high school admissions (eighth grade) and after (from tenth grade through college graduation), it lacks detailed data on the high school admission process (e.g., rank-order lists, priorities, and admission thresholds), which are essential for conventional identification strategies such as regression discontinuity or lottery-based designs. Our approach is particularly useful in this decentralized market, as it accounts for selection bias using only school enrollment information, without requiring data on the admission process. We define a student's peers as their classmates and measure student outcomes using academic performance both in the short term (tenth-grade math and language scores and high school graduation) and in the long term (post-secondary and college enrollment and graduation). Our objective is to estimate the causal effects of peer outcomes and peer characteristics on these academic outcomes, in the presence of endogenous selection into high schools.

We find that both peer outcomes and peer characteristics have significant effects on tenth graders. Including school fixed effects only partially corrects for selection bias compared to simple OLS. Additionally controlling for our selection correction further adjusts the estimates of peer effects. For example, the selection-corrected coefficients of peer outcomes are 10-45% lower than those obtained using school fixed effects alone. A variance decomposition of predicted outcomes into peer influence, school effectiveness, and self-selection reveals that self-selection accounts for the largest share of explained variance in most outcomes, highlighting the importance of properly correcting for selection bias in school evaluation. Notably, peer influence is overestimated when selection is not adequately accounted for, particularly in highly selective schools. Moreover, failing to account for selection distorts the estimation of school effectiveness, primarily by underestimating the value-added of less selective schools. The selection-corrected estimates suggest that more selective schools may actually provide lower value-added. These findings underscore the critical need for proper selection correction to evaluate school performance and ensure equitable resource allocation.

Our work is related to the econometric literature that addresses the endogeneity of peer relationships in social interactions models (goldsmith2013social,qu2015estimating,hsieh2016social,hsieh2018smoking,hsieh2020specification,johnsson2021estimation,Auerbach2022,griffith2024random). These studies typically assume the presence of unobserved individual heterogeneity that affects both the formation of links and individual outcomes, thereby generating endogeneity in peer relationships. This type of endogeneity can be addressed by accounting for individual heterogeneity. In contrast to these studies, which focus on link formation, we develop a framework for group formation, where the unobserved factors driving endogenous groups can be pair-specific (e.g., student-school specific). We complement the methods in this literature by proposing an approach to correct for selection into groups.

Our work is also related to the literature on sample selection models heckman1979sample,das2003nonparametric and extends this literature to social interactions with endogenous group selection. Our framework is mostly related to that of das2003nonparametric, who examined selection into a single sample governed by multiple rules. In contrast, we focus on selection into multiple groups. We propose a novel approach to reduce the dimensionality of selection bias when the number of groups is moderately large, thereby providing a practical method for addressing selection bias despite dimensionality challenges.

The remainder of the paper is organized as follows. Section (ref) introduces the model. Section (ref) derives selection bias. Section (ref) investigates identification. Section (ref) proposes estimation methods and establishes asymptotic properties. Section (ref) evaluates our approach through simulations. Section (ref) studies social interactions in Chilean high schools. Section (ref) concludes the paper. All proofs are provided in Supplemental Appendix (ref).\footnote{All the numbered items designated with an \textquotedblleft S\textquotedblright are shown in the Supplemental Appendix.}

\protectModel

Social Interactions

Consider a set of individuals $\mathcal{N}=\{1,2,\dots,n\}$ who can join a set of groups $\mathcal{G}=\{1,\dots,G\}$. We assume that the number of groups $G$ is finite and each group $g\in\mathcal{G}$ has a predetermined capacity $n_{g}$ that is proportional to $n$. The groups are non-overlapping (e.g., colleges, neighborhoods), so one joins only one group. Let $g_{i}$ denote the group that individual $i$ joins and $\boldsymbol{g}\equiv(g_{1},\dots,g_{n})'$ the $n\times1$ vector that stacks $g_{i}$.

After the groups are formed, individuals interact with their groupmates following a linear-in-means social interactions model Manski1993,brock2001interactions

In this specification, $y_{i}\in\mathbb{R}$ represents the outcome of interest (e.g., GPA, earnings), $x_{i}\in\mathbb{R}^{d_{x}}$ is a vector of observed individual characteristics (e.g., parental education, family income), $\epsilon_{i}\in\mathbb{R}$ is an unobserved characteristic (e.g., ability). Let $\boldsymbol{y}$ denote the $n\times1$ vector that stacks $y_{i}$, $\boldsymbol{x}$ the $n\times d_{x}$ matrix that stacks $x'_{i}$, and $\boldsymbol{\epsilon}$ the $n\times1$ vector that stacks $\epsilon_{i}$. We assume that $i$'s outcome $y_{i}$ depends on $\sum_{j=1}^{n}w_{ij}y_{j}$ and $\sum_{j=1}^{n}w_{ij}x_{j}$, the weighted averages of outcomes and observed characteristics of $i$'s peers, where $w_{ij}\in\mathbb{R}$ denotes the weight of peer $j$ on individual $i$. Following the terminology in Manski1993, $\gamma_{1}$ represents the endogenous social effect, and $\gamma_{2}$ represents the exogenous/contextual social effect. The parameter of interest is $\gamma=(\gamma_{1},\gamma'_{2},\gamma'_{3})'\in\mathbb{R}^{2d_{x}+1}$.

Because individuals are influenced solely by their groupmates, the adjacency matrix $\boldsymbol{w}\equiv(w_{ij})\in\mathbb{R}^{n^{2}}$ exhibits a group structure, that is, $w_{ij}=0$ if $i$ and $j$ belong to different groups ($g_{i}\neq g_{j}$). A typical specification is given by $w_{ij}=\frac{1}{n_{g_{i}}}$ if $g_{i}=g_{j}$, where each group member has equal weight Manski1993. More generally, we can specify $\boldsymbol{w}$ to capture additional network structures in a group. For example, groupmates may form friendship ties, and only friends may have a nonzero influence.

If an individual $i$'s decision to join a group or form friendships is influenced by unobserved characteristics that are correlated with $\epsilon_{i}$, the adjacency matrix $\boldsymbol{w}$ would be correlated with $\boldsymbol{\epsilon}$ and become endogenous. In this paper, we treat the sorting of individuals into groups as endogenous but assume that link formation within groups (if any) is exogenous, as formulated in Assumption (ref) (in Section (ref)). This assumption is not a limitation but a simplification, which allows us to focus on the endogeneity in $\boldsymbol{g}$ that arises from selection into groups.

In the literature that relies on observational data---where random assignment into groups is not available---the issue of sorting into groups is predominantly addressed in the framework of correlated effects Manski1993,moffitt2001policy. It is often assumed that there exist unobserved characteristics that have common effects on the outcome of all individuals within a group (e.g., school resources, neighborhood amenities), and conditional on these group-level unobservables, an individual's group membership becomes exogenous lee2007identification,bramoulle2009identification,Sacerdote2011. While using group fixed effects to address sorting into groups may seem reasonable, it does not allow for unobserved characteristics that simultaneously affect both group selection and individual outcomes when these characteristics vary across individuals within the same group (bramoulle2009identification; patacchini2017heterogeneous).\footnote{Another limitation of using group fixed effects is that it requires sufficient variation in the regressors $\sum_{j=1}^{n}w_{ij}y_{j}$ and $\sum_{j=1}^{n}w_{ij}x_{j}$ within a group. If these regressors represent group averages that include oneself, we cannot control for group fixed effects due to perfect multicollinearity.}

In the following section, we develop a structural model of group formation to examine how individuals sort into groups and how to effectively account for endogenous sorting.

\protectGroup Formation

brock2001interactions,brock2002multinomial,brock2005multinomial, ioannides2008interactions, and gu2024peer have proposed various models of group formation to analyze the sorting of individuals into groups. These studies consider the framework where individuals can unilaterally decide whether to join a group or which group to join, resulting in a standard binary or multinomial discrete choice model. While this approach is well-suited for applications such as job training participation and neighborhood choice, there are scenarios where groups also actively decide whether to admit individuals, as seen in school choice, college admissions, medical residency placements, and nursing home admissions. In these contexts, the formation of groups is determined by the bilateral decisions of both individuals and groups.

We consider a model of group formation that accounts for two-sided decisions of individuals and groups. On one side, individuals choose among groups based on their preferences. On the other side, groups rank and admit individuals based on qualification criteria until reaching their capacity constraints.\footnote{For example, in college admissions, a student's qualifications reflect the preferences of colleges. In school choice, schools are typically assigned based on administrative rules that prioritize factors such as geographic proximity and sibling enrollment.} If groups have unlimited capacities and capacity constraints become non-binding, the framework reduces to one-sided decisions made by individuals, as in brock2005multinomial and ioannides2008interactions.

Our framework can be equivalently characterized as a two-sided many-to-one matching model without transfers, where individuals in a group are considered “matched with” the group (azevedo_supply_2016; HSS_twosided). Therefore, we follow this matching literature to specify the model primitives and analyze equilibrium outcomes. To the best of our knowledge, no previous studies have applied two-sided matching to analyze selection into groups.

\paragraph{Utility}

For individual $i\in\mathcal{N}$ and group $g\in\mathcal{G}$, let $u_{ig}$ denote $i$'s utility of joining group $g$ and $v_{ig}$ denote $i$'s qualification for group $g$:

where $z_{i}\in\mathbb{R}^{d_{z}}$ represents a vector of observed characteristics that are either individual-specific (e.g., test score, parental education, family income) or pair-specific (e.g., distance to school, a student's minority status interacted with the past minority composition of a school).\footnote{To illustrate that $z_{i}$ can flexibly include both individual-specific and pair-specific variables, consider a two-group example where $v_{ig}=s_{i}\delta_{g,s}+d_{ig}\delta_{g,d}+\eta_{ig}$, with $s_{i}$ being an individual-specific variable and $d_{ig}$ a pair-specific variable. This specification maps to equation ((ref)) by setting $z_{i}=(s_{i},d_{i1},d_{i2})'$, $\delta_{1}^{v}=(\delta_{1,s},\delta_{1,d},0)'$, and $\delta_{2}^{v}=(\delta_{2,s},0,\delta_{2,d})'$.} The components of $z_{i}$ that appear in $u_{ig}$ and $v_{ig}$ may differ; in such cases, the corresponding coefficients for absent components are set to zero. The group-specific coefficients $\delta_{g}^{u},\delta_{g}^{v}\in\mathbb{R}^{d_{z}}$ capture heterogeneous effects of $z_{i}$ across groups. The terms $\xi_{ig}\in\mathbb{R}$ and $\eta_{ig}\in\mathbb{R}$ represent pair-specific unobserved shocks to utility and qualification, respectively (e.g., family tradition, extracurricular activities).\footnote{Individuals may take anticipated outcomes into account when comparing between groups (e.g., students considering post-graduation job prospects). However, the utility specification in equation ((ref)) can be interpreted as a reduced form that implicitly accounts for these expectations. Developing a model that explicitly incorporates anticipated outcomes in the utility is beyond the scope of our analysis.} Let $\xi_{i}=(\xi_{i1},\dots,\xi_{iG})'$ and $\eta_{i}=(\eta_{i1},\dots,\eta_{iG})'$. We assume that the joint distribution of $(\epsilon_{i},\xi_{i},\eta_{i})$ is nonparametric, which has the advantage of allowing $\epsilon_{i}$ to have flexible dependence with $\xi_{i}$ and $\eta_{i}$.\footnote{The nonparametric specification implies that $z_{i}$ cannot include a constant or group-specific variables, as their effects cannot be separated from the nonparametric distribution of $(\xi_{i},\eta_{i})$. In Section (ref), we impose additional restrictions on the distribution of $(\xi_{i},\eta_{i})$, which then allows us to include group-level effects.}

To maintain a tractable framework, our model does not explicitly allow for peer effects in group formation, meaning that the utility and qualification functions in equation ((ref)) do not depend on prospective groupmates. However, this theoretical simplification does not necessarily preclude the empirical consideration of peer influences in group formation. In fact, peer effects can be proxied using the outcomes and characteristics of past group members (such as average test scores, gender or racial composition, and average family income), provided that these measures remain stable over time.

\paragraph{Equilibrium}

Following the matching literature roth1992two, we assume that the matching outcome is stable. azevedo_supply_2016 showed that a stable matching exists and can be characterized by group cutoffs. For $g\in\mathcal{G}$, define the cutoff $p_{g}$ of group $g$ as the lowest qualification among the group members if the capacity constraint is binding; otherwise, the cutoff is set to $-\infty$. Namely, $p_{g}=\inf_{i:g_{i}=g}v_{ig}\text{ if }\sum_{i\in\mathcal{N}}1\{g_{i}=g\}=n_{g}$, and $p_{g}=-\infty$ otherwise.\footnote{Following azevedo_supply_2016, we assume that all individuals are acceptable to all groups. Therefore, if a group has vacancies remaining, any individual wishing to join can do so, resulting in a cutoff of negative infinity. This assumption can be relaxed by introducing an acceptance threshold whereby groups only admit individuals above a certain acceptability level HSS_twosided. We maintain this simpler specification as it does not affect the main conceptual framework.}

Given a vector of cutoffs $p=(p_{1},\dots,p_{G})'$, let $\mathcal{C}_{i}(p)=\{g\in\mathcal{G}:\:v_{ig}\geqslant p_{g}\}\subseteq\mathcal{G}$ denote individual $i's$ choice set---the subset of groups for which $i$ qualifies. Within $\mathcal{C}_{i}(p)$, $i$ chooses the group that yields the highest utility, $g_{i}=\arg\max_{g\in\mathcal{C}_{i}(p)}u_{ig}$. This is a multinomial discrete choice problem with the choice set $\mathcal{C}_{i}(p)$ determined endogenously by the cutoffs $p$.\footnote{For simplicity of exposition, we assume that individuals always prefer to join a group. This simplification can be relaxed by introducing a utility $u_{i0}=\xi_{i0}$ for the outside option (i.e., not joining any group). Such a relaxation requires no substantial modifications to our theoretical framework.} The structure highlights a key distinction between one-sided and two-sided group formation: in one-sided group formation, an individual's choice set is assumed to include all groups, whereas two-sided group formation allows the choice set to be latent and determined endogenously Barseghyan2021,Agarwal2022.

Individual $i$ joins group $g$ if and only if (i) $i$ qualifies for group $g$, and (ii) for any other group $h\ne g$, either $i$ does not prefer group $h$, or $i$ does not qualify for group $h$. This can be expressed as

From this expression, the group that $i$ joins depends on $i$'s observed and unobserved characteristics $(z_{i},\xi_{i},\eta_{i})$ as well as the cutoffs $p$. We write $g_{i}=g(z_{i},\xi_{i},\eta_{i};p)$.\footnote{In the one-sided setting, the capacities are infinite, the cutoffs $p_{g}$ are set to $-\infty$, and the consideration sets $\mathcal{C}_{i}(p)$ for all $i\in\mathcal{N}$ are equal to $\mathcal{G}$. The optimal decision in equation ((ref)) reduces to a multinomial discrete choice problem $1\{g_{i}=g\}=\prod_{k\neq g}1\{u_{ik}<u_{ig}\}=\prod_{k\neq g}1\{\xi_{ik}-\xi_{ig}<z'_{i}(\delta_{g}^{u}-\delta_{k}^{u})\}$. The group that $i$ joins depends on $(z_{i},\xi_{i})$ only, that is, $g_{i}=g(z_{i},\xi_{i})$.}

In a stable matching, a vector of cutoffs $p$ clears the supply and demand for each group.\footnote{An equilibrium cutoff vector $p$ satisfies the market-clearing equations: $\sum_{i\in\mathcal{N}}1\{g(z_{i},\xi_{i},\eta_{i};p)=g\}\leq n_{g}$ for all $g\in\mathcal{G}$, and $\sum_{i\in\mathcal{N}}1\{g(z_{i},\xi_{i},\eta_{i};p)=g\}=n_{g}$ if $p_{g}>-\infty$ azevedo_supply_2016.} Let $\boldsymbol{\boldsymbol{z}}$ denote the $n\times d_{z}$ matrix that stacks $z'_{i}$, $\boldsymbol{\xi}$ the $n\times G$ matrix that stacks $\xi_{i}$, and $\boldsymbol{\eta}$ the $n\times G$ matrix that stacks $\eta_{i}$. An equilibrium cutoff vector can be represented as $p(\boldsymbol{z},\boldsymbol{\xi},\boldsymbol{\eta})$.\footnote{There may exist multiple equilibrium cutoffs in a finite-$n$ market. We denote by $p(\boldsymbol{z},\boldsymbol{\xi},\boldsymbol{\eta})$ the equilibrium that is selected by nature.} Given $p(\boldsymbol{z},\boldsymbol{\xi},\boldsymbol{\eta})$, the equilibrium group that $i$ joins is given by equation ((ref)) and can be represented as $g_{i}=g(z_{i},\xi_{i},\eta_{i};p(\boldsymbol{z},\boldsymbol{\xi},\boldsymbol{\eta}))$. The equilibrium groups of all the individuals can be expressed as $\boldsymbol{g}(\boldsymbol{z},\boldsymbol{\xi},\boldsymbol{\eta};p(\boldsymbol{z},\boldsymbol{\xi},\boldsymbol{\eta}))$.

\protectSelection Bias

In this section, we use the model described in Section (ref) to derive the bias arising from selection into groups. This selection bias is highly complex due to the presence of equilibrium effects and the role of alternative groups. To address this, we propose novel methods to simplify the selection bias and improve tractability.

Throughout the paper, we maintain the following assumptions.

As previously stated, Assumption (ref) implies that endogenous selection occurs only during group formation. For an adjacency matrix $\boldsymbol{w}$ that represents group averages with binding capacities, this assumption is trivially satisfied because $\boldsymbol{w}$ is determined by $\boldsymbol{g}$. For a more general $\boldsymbol{w}$, the assumption requires that once groups are formed, how group members form additional connections must be exogenous. This assumption is satisfied if, for example, students in a school make friends independently of $\boldsymbol{\epsilon}$ or are randomly assigned to dorms or classes where they interact.\footnote{We can potentially relax the assumption by incorporating endogenous friendship formation within a group, following the setup in e.g., johnsson2021estimation. However, this added complexity would not yield additional insights, so we do not pursue it here.} Assumption (ref) imposes a set of regularity conditions that are standard in social interactions. It assumes that the observables and unobservables are i.i.d. across individuals, the unobservables have a smooth joint cdf, and the observables are exogenous.

Under Assumption (ref), equation ((ref)) exhibits a selection bias if

Recall that the equilibrium groups $\boldsymbol{g}$ can be expressed as $\boldsymbol{g}(\boldsymbol{z},\boldsymbol{\xi},\boldsymbol{\eta};p(\boldsymbol{z},\boldsymbol{\xi},\boldsymbol{\eta}))$, with each individual component given by $g_{i}=g(z_{i},\xi_{i},\eta_{i};p(\boldsymbol{z},\boldsymbol{\xi},\boldsymbol{\eta}))$, $i\in\mathcal{N}$. Suppose that the outcome unobservable $\epsilon_{i}$ is correlated with the unobservables in group formation $(\xi_{i},\eta_{i})$ (selection on the unobservables). Under Assumption (ref)(i)(iii), this correlation may lead to a selection bias through two channels. First, the equilibrium group that individual $i$ joins $g_{i}$ depends on $i$'s unobservables in group formation $(\xi_{i},\eta_{i})$, which are correlated with $\epsilon_{i}$. Therefore, $\epsilon_{i}$ can be correlated with $g_{i}$ directly through $(\xi_{i},\eta_{i})$, resulting in a selection bias. Second, observe that the equilibrium cutoffs $p(\boldsymbol{z},\boldsymbol{\xi},\boldsymbol{\eta})$ depend on the unobservable profile in group formation $(\boldsymbol{\xi},\boldsymbol{\eta})$ which includes $(\xi_{i},\eta_{i})$. Moreover, these equilibrium cutoffs affect not only the group that $i$ joins $g_{i}$, but also the groups that others join $g_{j}$, $j\neq i$. Consequently, $\epsilon_{i}$ can be correlated with the entire group structure $\boldsymbol{g}$, including both $g_{i}$ and $g_{j}$, $j\ne i$, indirectly through $p(\boldsymbol{z},\boldsymbol{\xi},\boldsymbol{\eta})$, again resulting in a selection bias.

We illustrate the selection bias arising through the first channel using college admissions as an example. Suppose that unobserved ability ($\epsilon_{i}$) that affects labor market outcomes ($y_{i}$) is positively correlated with unobserved ability ($\eta_{ig}$) that influences college qualifications ($v_{ig}$). Additionally, let $x_{i}$ and $z_{i}$ represent family income, which has a positive impact on both college qualifications and labor market outcomes. Through the college admission process, students with higher qualifications ($v_{ig}=z_{i}\delta_{g}^{v}+\eta_{ig}$) are sorted into more selective colleges. This sorting implies that students in more selective colleges are either more capable (high $\eta_{ig}$) or come from wealthier families (high $z_{i}$). As a result, the sorting creates a positive correlation between $\epsilon_{i}$ and the average family income of peers in $i$'s college ($\sum_{j}w_{ij}x_{j}$), leading to upward bias in estimates of peer effects dale_estimating_2002.

The general equilibrium effects through the second channel present a challenge in correcting for selection bias. Because the equilibrium cutoffs $p(\boldsymbol{z},\boldsymbol{\xi},\boldsymbol{\eta})$ depend on the characteristics of the $n$ individuals in a market, the selection bias in equation ((ref)) is a high-dimensional function that involves the observed characteristics of the $n$ individuals. Below, we propose an approach to address this dimensionality issue.

\protectLimiting Approximation

Our approach is to employ the limiting approximation of a market as its size $n$ approaches infinity. We show that, in a large market, the correlation between $\epsilon_{i}$ and $\boldsymbol{g}$ through equilibrium cutoffs vanishes, thereby reducing the dimensionality of selection bias.

To this end, let $p_{n}=(p_{n,1},\dots,p_{n,G})'$ denote the equilibrium cutoffs in a market with $n$ individuals. azevedo_supply_2016 demonstrated that, as the market size grows large ($n\to\infty$), the equilibrium cutoffs in a finite-$n$ market converge to a unique limit, denoted by $p^{*}=(p_{1}^{*},\dots,p_{G}^{*})'$, referred to as the limiting equilibrium cutoffs. Unlike the finite-$n$ cutoffs $p_{n}$, the limiting cutoffs $p^{*}$ are deterministic because they are determined by the distribution of characteristics.\footnote{The convergence result in azevedo_supply_2016 requires that the number of groups is finite and that the size of each group grows proportionally large. To apply their result, we adopt the same assumption. While extending the analysis to allow for a growing number of groups is an interesting direction, we leave this for future work.} Building on the cutoff convergence result established by azevedo_supply_2016, we show that the selection bias in a finite-$n$ market also converges to the limiting counterpart with $p^{*}$ in place of $p_{n}$.

The proposition indicates that we can approximate the finite-market selection bias using its limiting counterpart. Because the limiting cutoffs are deterministic, each individual's selection bias depends only on their own characteristics. This reduces the dimensionality of the selection bias from $O(n)$ to a finite number.

Large-market approximation has been widely used in the matching literature choo_who_2006,menzel_large_2015,azevedo_supply_2016,fack_beyond_2019,galichon2022cupid,HSS_twosided. We follow the literature and assume that selection bias takes the limiting form.\footnote{azevedo_supply_2016 demonstrated that the finite-$n$ cutoffs converge to the limiting cutoffs at the rate of $\sqrt{n}$. However, addressing the approximation error in the subsequent asymptotic analysis remains an open question for future research.}

\protectGroup-Invariant and Symmetric Selection Function

We now derive the limiting selection bias in ((ref)) in an explicit form. Let $\tau_{ig}\equiv z'_{i}\delta_{g}^{u}\in\mathbb{R}$ and $\iota_{ig}\equiv z'_{i}\delta_{g}^{v}-p_{g}\in\mathbb{R}$ denote the preference and qualification (net cutoff) indices of individual $i$ for group $g\in\mathcal{G}$, respectively. By equation ((ref)) and the exogeneity of $(x_{i},z_{i})$ (Assumption (ref)(iii)), we can represent the limiting selection bias in ((ref)) as

where $\lambda_{g}(\cdot):\mathbb{R}^{2G}\rightarrow\mathbb{R}$ is a nonparametric function of $(\tau_{ig},\iota_{ig}){}_{g=1}^{G}$, referred to as the selection function.\footnote{A group's qualification index is relevant only when its capacity constraint binds. For groups with non-binding capacity constraints, their qualification indices can be omitted. Specifically, let $\overline{\mathcal{G}}\subseteq\mathcal{G}$ denote the subset of capacity-binding groups, with cardinality $\overline{G}$. Then $\lambda_{g}(\cdot)$ is a function of $(\tau_{ig})_{g\in\mathcal{G}}\in\mathbb{R}^{G}$ and $(\iota_{ig})_{g\in\overline{\mathcal{G}}}\in\mathbb{R}^{\overline{G}}$.} The expression of the selection bias in ((ref)) supports the approach proposed by dale_estimating_2002, who overcame selection bias by matching students who applied to and were accepted by comparable sets of colleges. In our context, however, such application and admission information is not observed. Equation ((ref)) extends the results in standard sample selection models (heckman1979sample; das2003nonparametric) to social interactions with endogenous group selection. In heckman1979sample's model, there is a single “group” (the sample) and individuals decide whether to join that group (be selected into the sample) through a binary choice. das2003nonparametric extended heckman1979sample's framework by considering a binary choice determined by multiple rules under nonparametric specifications. Our approach shares a similar structure, as it can also be viewed as a binary choice (i.e., $1\{g_{i}=g\}$) determined by multiple rules (e.g., $1\{u_{ig}>u_{ih}\}$ and $1\{v_{ig}>p_{g}\}$), as shown in equation ((ref)). However, das2003nonparametric focused on selection into a single sample governed by multiple rules, while our approach addresses selection among multiple groups.\footnote{das2003nonparametric represented selection bias as a nonparametric function of the propensity scores corresponding to each selection rule. In our setting, however, the propensity scores associated with the selection rules in equation ((ref)) (e.g., $1\{u_{ig}>u_{ih}\}$ and $1\{v_{ig}>p_{g}\}$) are not available, as we do not observe individuals' rankings of the groups or whether they qualify for a group. Instead, we express selection bias as a function of indices.} In social interactions, brock2001interactions,brock2002multinomial,brock2005multinomial, ioannides2008interactions, and gu2024peer studied social interactions under one-sided group formation with parametric unobservables. We generalize their results to two-sided group formation with nonparametric unobservables.

The subscript $g$ in $\lambda_{g}(\cdot)$ indicates that its functional form can differ across groups. Below we illustrate $\lambda_{g}(\cdot)$ in the case of three groups.

The selection bias in equation ((ref)) present two features that complicate its correction. First, the selection function $\lambda_{g}(\cdot)$ may be group-specific if the distribution of the unobservables $(\xi_{ig},\eta_{ig})$ differs across groups.\footnote{Equation ((ref)) also indicates that the indices for group $g$ play a distinct role in the selection function compared to those for the other groups. Nevertheless, this group-specific feature can be addressed by separating the indices for group $g$ from those for the other groups, as formulated in equation ((ref)).} A group-specific selection function poses a challenge in identifying the effects of group-level variables, as these effects cannot be distinguished from nonparametric selection bias that is specific to each group.\footnote{Existing literature (e.g., brock2005multinomial) does not encounter this issue because with parametric assumptions, the functional form of $\lambda_{g}(\cdot)$ is known.} This problem resembles an issue in panel data models, where the effects of time-invariant variables cannot be distinguished from individual fixed effects. In our context, there could be variables of interest that are at group level. For example, let $w_{i}$ denote the $i$th row of $\boldsymbol{w}$. if $w_{i}\boldsymbol{y}$ and $w_{i}\boldsymbol{x}$ are group averages that include individual $i$ as in Manski1993, they are invariant within a group.\footnote{If $w_{i}\boldsymbol{y}$ and $w_{i}\boldsymbol{x}$ are group averages that exclude $i$, they converge to including-oneself group averages as group size goes to infinity (see footnote (ref)). Hence, the variation of $w_{i}\boldsymbol{y}$ and $w_{i}\boldsymbol{x}$ within a group vanishes to zero as the group size grows.} Moreover, in certain applications group fixed effects may themselves be the parameters of interest. For instance, school fixed effects may be used as measures of school effectiveness in school choice. Second, the selection function depends on the indices of all groups, which may lead to a dimensionality issue if the number of groups is moderately large.

We propose a novel method to overcome these problems. Note that by appropriately arranging the indices, the selection function can be made group-invariant and symmetric in alternative groups, provided that the distribution of unobservables exhibits symmetry across groups. Motivated by this insight, we assume that the joint distribution of unobservables is exchangeable across groups.

Assumption (ref) requires that the joint distribution of unobservables remains invariant under the relabeling of groups. The concept of exchangeability has been widely used in various contexts, such as differentiated product markets gandhi2019measuring, panel data altonji_matzkin, matching fox_unobserved_2018, and network formation menzel2021central, although the specific methodologies vary. Exchangeability can accommodate a complex dependence structure among the unobservables of an individual. Specifically, it allows for (i) the dependence between the unobserved preference $\xi_{ig}$ and the unobserved qualification $\eta_{ig}$ for each group $g\in G$, and (ii) the dependence between the unobservables $(\xi_{ig},\eta_{ig})$ for group $g$ and the unobservables $(\xi_{ih},\eta_{ih})$ for another group $h\neq g$. However, because exchangeability imposes symmetry across groups in the distribution of unobservables, it rules out scenarios where the variance of unobservables differs across groups or where the correlation between the unobservables of two groups differs across pairs of groups.\footnote{For example, in college admissions, students' unobserved preferences may be more strongly correlated between elite colleges than between elite and non-elite colleges. Similarly, elite colleges may share unobserved evaluation criteria for qualitative factors (e.g., extracurriculars) that differ from those used by non-elite colleges.} In such scenarios, we can relax Assumption (ref) by categorizing groups into distinct types (e.g., public and private schools) and imposing exchangeability only among groups of the same type.\footnote{The subsequent results can be extended to exchangeability conditional on group type. However, for notational simplicity, we retain the basic setting of exchangeability across all groups in the main text. In our empirical application, we impose exchangeability separately for public and private schools. See Section (ref) for more details.} Furthermore, note that we can include group fixed effects in $u_{ig}$ to account for unobserved heterogeneity across groups, but the individual-varying unobservables must not exhibit group heterogeneity.\footnote{While we could also include group fixed effects in $v_{ig}$, they become redundant when groups rank individuals based on $v_{ig}$ and cannot be distinguished from the cutoffs.} Assumption (ref) is satisfied under the usual logit or probit specifications.\footnote{For example, $\xi_{ig}=a_{i}+\tilde{\xi}_{ig}$, where $a_{i}$ represents an individual effect that is i.i.d. across $i$, and $\tilde{\xi}_{ig}$ follows a type I extreme value or Gaussian distribution, and is i.i.d. across both $i$ and $g$.}

Observe from equation ((ref)) that only utility differences matter. For $g\in\mathcal{G}$, we define the utility difference $\Delta_{g}\tau_{ih}\equiv\tau_{ih}-\tau_{ig}$ for $h\neq g$. By separating the index $\iota_{ig}$ for group $g$ from those for the other groups, the selection bias in equation ((ref)) can be reformulated as

Under exchangeability, the selection function $\lambda_{g}^{e}(\cdot)$ becomes invariant across groups, that is, there exists $\lambda^{e}(\cdot)$ such that $\lambda_{g}^{e}(\cdot)\equiv\lambda^{e}(\cdot)$ for all $g\in\mathcal{G}$. Furthermore, exchangeability implies that the ordering of the index pairs $(\Delta_{g}\tau_{ih},\iota_{ih})$ across all $h\neq g$ is irrelevant -- the selection function is symmetric in the indices for groups other than $g$.

Proposition (ref) has important implications. First, the group invariance in Proposition (ref)(i) implies that we can use a single selection function for all the groups. Second, the symmetry established in Proposition (ref)(ii) can further reduce the dimensionality of the selection function. Note that the selection function depends not only on the index of the group an individual joins $\iota_{ig_{i}}$, but also on the indices of all other groups $(\Delta_{g_{i}}\tau_{ih},\iota_{ih})$, $h\neq g_{i}$. Because the selection function is symmetric in the indices $(\Delta_{g_{i}}\tau_{ih},\iota_{ih})$ for $h\neq g_{i}$, we can equivalently express the selection bias in equation ((ref)) using elementary symmetric functions of these indices.

Specifically, we denote the elementary symmetric functions of $(\Delta_{g_{i}}\tau_{ih},\iota_{ih}){}_{\forall h\ne g_{i}}\in\mathbb{R}^{2(G-1)}$ as $\pi_{i,-g_{i}}$, which can be represented by the coefficients of the polynomial function $\prod_{h\neq g_{i}}(1+(\Delta_{g_{i}}\tau_{ih},\iota_{ih})t)$ in the indeterminates $t=(t_{1},t_{2})'$ Weyl1946.\footnote{The first two orders of the elementary symmetric functions are given by the sums of all individual terms ($\sum_{h\neq g_{i}}\Delta_{g_{i}}\tau_{ih}$ and $\sum_{h\neq g_{i}}\iota_{ih}$) and the sums of all pairwise products ($\sum_{(h_{1},h_{2})\neq g_{i}}\Delta_{g_{i}}\tau_{ih_{1}}\Delta_{g_{i}}\tau_{ih_{2}}$, $\sum_{(h_{1},h_{2})\neq g_{i}}\iota_{ih_{1}}\iota_{ih_{2}}$, and $\sum_{(h_{1},h_{2})\neq g_{i}}\Delta_{g_{i}}\tau_{ih_{1}}\iota_{ih_{2}}$, where $\sum_{(h_{1},h_{2})\neq g_{i}}$ denotes the sum over all combinations of distinct $h_{1}$ and $h_{2}$ in $\mathcal{G}\backslash\{g_{i}\}$). The higher-order functions can be derived similarly.} Define $\pi_{i}\equiv(\iota_{ig_{i}};\pi_{i,-g_{i}})$. By the fundamental theorem of symmetric functions in conjunction with the Weierstrass approximation theorem, any symmetric function can be approximated arbitrarily closely by a polynomial function of the elementary symmetric functions altonji_matzkin. Therefore, there exists a function $\lambda(\cdot)$ such that

Using the symmetric representation reduces the number of nuisance parameters in a sieve approximation of the selection function. For instance, if we use linear basis functions, $\lambda^{e}(\cdot)$ has $2G-1$ approximating functions, whereas $\lambda(\cdot)$ has only $3$, including one for group $g_{i}$ and two for the remaining $G-1$ groups combined. If we consider basis functions of order two, $\lambda^{e}(\cdot)$ has $(2G-1)G$ approximating functions, whereas $\lambda(\cdot)$ has $9$.\footnote{Follow the discussion in footnote (ref). For $\lambda^{e}(\cdot)$, we have $(2G-1)G$ functions of order two: $2G-1$ squared indices and $(2G-1)(G-1)$ pairwise interactions. For $\lambda(\cdot)$, we have the following $9$ functions of order two: $(\iota_{ig_{i}})^{2}$, $\iota_{ig_{i}}\sum_{h\neq g_{i}}\Delta_{g_{i}}\tau_{ih}$, $\iota_{ig_{i}}\sum_{h\neq g_{i}}\iota_{ih}$, $(\sum_{h\neq g_{i}}\Delta_{g_{i}}\tau_{ih})^{2}$, $(\sum_{h\neq g_{i}}\iota_{ih})^{2}$, $(\sum_{h\neq g_{i}}\Delta_{g_{i}}\tau_{ih})(\sum_{h\neq g_{i}}\iota_{ih})$, $\sum_{(h_{1},h_{2})\neq g_{i}}\Delta_{g_{i}}\tau_{ih_{1}}\Delta_{g_{i}}\tau_{ih_{2}}$, $\sum_{(h_{1},h_{2})\neq g_{i}}\iota_{ih_{1}}\iota_{ih_{2}}$, and $\sum_{(h_{1},h_{2})\neq g_{i}}\Delta_{g_{i}}\tau_{ih_{1}}\iota_{ih_{2}}$.} Aggregating the indices for groups other than $g_{i}$ by elementary symmetric functions has the advantage that the number of nuisance parameters in a sieve approximation for a given order does not depend on the number of groups. This reduces the dimensionality of a sieve approximation if the number of groups is moderately large. gandhi2019measuring employed a similar strategy to reduce the dimensionality of instruments in a BLP model with many products.

\paragraph{Discussion.}

Selection bias $\lambda(\pi_{i})$ is individual-specific because it depends on $\pi_{i}=\pi(z_{i},g_{i})$, which involves individual characteristics $z_{i}$. While endogenous selection occurs only during group formation, individuals with different values of $z_{i}$ may be subject to varying levels of selection bias. This feature mirrors the results in standard sample selection models. For instance, the selection bias in heckman1979sample---the inverse Mills ratio---depends on individual characteristics included in the selection equation. This finding provides empirical guidance that contrasts with the group-fixed-effect approach, which assumes constant selection bias within a group. Altonji2018 demonstrated that selection bias can be captured by group averages of covariates. However, their result relies on specific linear restrictions and may not be generalized to nonparametric settings. In our setting, neither group fixed effects nor group averages in the outcome equation adequately addresses group selection. To effectively correct for selection bias, it is essential to exploit the individual-level information contained in $z_{i}$.

Another key feature of selection bias is that it depends not only on the indices of the group an individual joins, but also on those of alternative groups. This finding highlights the importance of accounting for the impacts of alternative groups, which can be challenging due to dimensionality issues. Dahl2002 imposed an assumption that controlling for the probability of an individual's first-best choice exhausts the impacts of all groups. In contrast, we aggregate the impacts of alternative groups by leveraging the symmetry of the selection function under exchangeability. Related to our work, abdulkadirouglu2020parents proposed a selection correction method in centralized school choice, where selection occurs only through unobserved preferences. Their selection correction aggregates the impacts of alternative schools, constructed using rank-order lists under multinomial logit unobservables, which resembles the linear components of the elementary symmetric functions in our selection bias derived under exchangeability. We provide a microfoundation for the aggregation and derive the exact form of selection bias under more general nonparametric unobservables. Unlike their approach, ours is applicable to decentralized markets with selection on both unobserved preferences and qualifications, where rank-order lists are unavailable.

In addition, the literature on social interactions has long struggled with disentangling two distinct sources of correlated effects: those arising from group-level unobservables and those arising from self-selection into groups based on individual-level unobservables. Distinguishing between these sources is crucial, as they have distinct policy implications. For example, in the context of school choice, should a policymaker focus on improving school effectiveness or altering self-selection among students? By imposing the exchangeability assumption, our approach allows for the inclusion of group fixed effects alongside selection correction, thereby enabling the separate identification of school effectiveness and self-selection.

\protectIdentification

We now turn to the identification of the social effects $\gamma$. HSS_twosided established identification results for the group formation parameters, including the slope parameters $\delta\equiv(\delta_{g}^{u},\delta_{g}^{v})_{g\in\mathcal{G}}$ and the cutoffs $p$. We thus proceed by treating $\pi_{i}$ as known.

Let $\nu_{i}\equiv\epsilon_{i}-\lambda(\pi_{i})$ represent the residual of $\epsilon_{i}$ after eliminating selection bias. Write equation ((ref)) as

where $X_{i}\equiv(w_{i}\boldsymbol{y},w_{i}\boldsymbol{x},x'_{i})'\in\mathbb{R}^{d_{X}}$ denotes a vector of regressors with dimension $d_{X}\equiv2d_{x}+1$. This is a partially linear model Robinson1988. To eliminate $\lambda(\pi_{i}),$ we take the expectation of equation ((ref)) conditional on $\pi_{i}$ and subtract it from equation ((ref)):

The following rank condition guarantees the identification of $\gamma$.

This assumption was imposed by cosslett1991semiparametric and discussed by newey2009two. The rank condition is satisfied if and only if there is no linear combination of $w_{i}\boldsymbol{y}$, $w_{i}\boldsymbol{x}$, and $x_{i}$ that is a function of $\pi_{i}$ almost surely.\footnote{To see the equivalence between the two statements in Assumption (ref), note that $\mathbb{E}[(X_{i}-\mathbb{E}[X_{i}|\pi_{i}])(X_{i}-\mathbb{E}[X_{i}|\pi_{i}])']$ is singular if and only if there exists $a\neq0$, $a\in\mathbb{R}^{d_{X}}$, such that $a'\mathbb{E}[(X_{i}-\mathbb{E}[X_{i}|\pi_{i}])(X_{i}-\mathbb{E}[X_{i}|\pi_{i}])']a=0.$ This can be rewritten as $\mathbb{E}[((X_{i}-\mathbb{E}[X_{i}|\pi_{i}])'a)^{2}]=0$, which holds if and only if $(X_{i}-\mathbb{E}[X_{i}|\pi_{i}])'a=0$ with probability one. Since $\mathbb{E}[X_{i}|\pi_{i}])'a$ is a measurable function of $\pi_{i}$ (call it $h(\pi_{i})$), we have $X_{i}\prime a=h(\pi_{i})~$with�probability�1. Conversely, if there exists some $a$ such that $X'_{i}a=h(\pi_{i})$, we have $(X_{i}-\mathbb{E}[X_{i}|\pi_{i}])'a=X'_{i}a-\mathbb{E}[X'_{i}a|\pi_{i}]=h(\pi_{i})-h(\pi_{i})=0$ with probability one.} A sufficient condition of it is that (i) $w_{i}\boldsymbol{y}$, $w_{i}\boldsymbol{x}$, and $x_{i}$ are linearly independent; and (ii) the conditional distribution of $\pi_{i}$ given $X_{i}$ has an absolutely continuous component with conditional density that is positive on the entire real line for almost all $X_{i}$ newey2009two. The violation of condition (i) is referred to as the reflection problem Manski1993,blume2011identification, which we will discuss in Section (ref). For condition (ii), recall that $\pi_{i}=\pi(z_{i},g_{i})$. Suppose $z_{i}$ contains at least one component that is not present in $x_{i}$ and has an absolutely continuous density that is positive on its support. This excluded variable would provide sufficient variation in $\pi_{i}$ (conditional on $X_{i}$) to satisfy the rank condition.

\protectThe reflection problem

To investigate the reflection problem, we consider the social equilibrium in equation ((ref)). Let $\boldsymbol{\lambda}=\boldsymbol{\lambda}(\boldsymbol{\pi})$ denote the $n\times1$ vector that stacks $\lambda(\pi_{i})$, where $\boldsymbol{\pi}\equiv(\pi'_{1},\dots,\pi'_{n})'$, and $\boldsymbol{\nu}$ the $n\times1$ vector that stacks $\nu_{i}$. Write equation ((ref)) in a matrix form

Assume $|\gamma_{1}|<1$ and $\interleave\boldsymbol{w}\interleave_{\infty}=\max_{i\in\mathcal{N}}\sum_{j=1}^{n}|w_{ij}|=1$, so $\boldsymbol{s}\equiv I_{n}-\gamma_{1}\boldsymbol{w}$ is invertible and $\boldsymbol{s}^{-1}=(I_{n}-\gamma_{1}\boldsymbol{w})^{-1}=\sum_{k=0}^{\infty}\gamma_{1}^{k}\boldsymbol{w}^{k}$. The social equilibrium is \[ \boldsymbol{w}\boldsymbol{y}=\boldsymbol{s}^{-1}(\boldsymbol{w}^{2}\boldsymbol{x}\gamma_{2}+\boldsymbol{w}\boldsymbol{x}\gamma_{3}+\boldsymbol{w}\boldsymbol{\lambda}+\boldsymbol{w}\boldsymbol{\nu}). \] Its $i$th equation is given by

where $w_{i}^{k}$ denotes the $i^{th}$ row of $\boldsymbol{w}^{k}$. The expression implies that $w_{i}\boldsymbol{y}$, $w_{i}\boldsymbol{x}$, and $x_{i}$ are linearly independent if (i) the support of $x_{i},w_{i}\boldsymbol{x},w_{i}^{2}\boldsymbol{x},w_{i}^{3}\boldsymbol{x},\dots$ is not contained in a proper linear subspace of $\mathbb{R}^{2d_{x}+1}$ and $\gamma_{1}\gamma_{3}+\gamma_{2}\neq0$, or (ii) the support of $x_{i},w_{i}\boldsymbol{x},w_{i}\boldsymbol{\lambda},w_{i}^{2}\boldsymbol{\lambda},\dots$ is not contained in a proper linear subspace of $\mathbb{R}^{2d_{x}+1}$.

Sufficient conditions have been established in the existing literature for case (i). For example, $w_{i}^{2}\boldsymbol{x}$, $w_{i}\boldsymbol{x}$, and $x_{i}$ are linearly independent if there is an intransitive triad in each group bramoulle2009identification, or if there is variation in group sizes when we consider group averages that exclude oneself lee2007identification,graham2008identifying,Davezies2009. However, this identification strategy fails for group averages that include oneself because $\boldsymbol{w}^{2}=\boldsymbol{w}$ Manski1993,bramoulle2009identification.\footnote{In large groups, the difference between group averages that include or exclude oneself vanishes (see footnote (ref)). Consequently, identification through variation in group sizes becomes less effective as groups grow large.}

The presence of selection offers an alternative method of identification through case (ii). It is evident from equation ((ref)) that $w_{i}\boldsymbol{y}$, $w_{i}\boldsymbol{x}$, and $x_{i}$ are linearly independent if $w_{i}\boldsymbol{\lambda}$, $w_{i}\boldsymbol{x}$, and $x_{i}$ are linearly independent. This identification strategy is applicable regardless of whether group averages include or exclude oneself, or whether there are networks within groups. The result is consistent with the insight of brock2001interactions that identification can be achieved through self-selection, provided that there is variation in the selection within a group. Because selection bias acts as an individual-level variable whose average is not included in the contextual effect, its presence precludes $w_{i}\boldsymbol{y}$ and $w_{i}\boldsymbol{x}$ from being linearly dependent Manski1993,brock2001interactions.

\protectEstimation

We now discuss how to estimate the social effects $\gamma$. Denote the true value of $\gamma$ by $\gamma_{0}\in\mathbb{R}^{d_{X}}$. Following equation ((ref)), we propose a three-step sieve OLS estimator for $\gamma_{0}$. Recall that the indices $\pi_{i}=\pi(z_{i},g_{i},\theta_{0})$ in selection bias depend on the parameters in group formation, $\theta_{0}\equiv(\delta'_{0},p^{*\prime})'$. In the first step, we estimate these indices by the estimator $\hat{\pi}_{i}=\pi(z_{i},g_{i},\hat{\theta})$, where $\hat{\theta}=(\hat{\delta}',\hat{p}')'$ is an estimator of $\theta_{0}$. Using the estimated indices, we estimate $\gamma_{0}$ using sieve OLS in the next two steps.

Specifically, let $b^{K}(\pi_{i})=(b_{1K}(\pi_{i}),\dots,b_{KK}(\pi_{i}))'$ be a $K\times1$ vector of approximating functions for individual $i$, and $B_{K}(\boldsymbol{\pi})=(b^{K}(\pi_{1}),\dots,b^{K}(\pi_{n}))'$ the $n\times K$ matrix of all approximating functions, where $\boldsymbol{\pi}=(\pi'_{1},\dots,\pi'_{n})'$. By replacing $\boldsymbol{\pi}$ with its estimator $\hat{\boldsymbol{\pi}}=(\hat{\pi}'_{1},\dots,\hat{\pi}'_{n})'$, we obtain an estimate of the approximating functions $\hat{B}_{K}=B_{K}(\hat{\boldsymbol{\pi}})=(b^{K}(\hat{\pi}_{1}),\dots,b^{K}(\hat{\pi}_{n}))'$. Next, we estimate the conditional expectation $\mu_{0}^{X}(\pi_{i})\equiv\mathbb{E}[X_{i}|\pi_{i}]$ using the sieve estimator $\hat{\mu}^{X}(\hat{\pi}_{i})\equiv\boldsymbol{X}'\hat{B}_{K}(\hat{B}'_{K}\hat{B}_{K})^{-1}b^{K}(\hat{\pi}_{i})$, where $\boldsymbol{X}$ is the $n\times d_{X}$ matrix that stacks $X'_{i}$. Define the matrix $\hat{M}_{K}\equiv I-\hat{B}_{K}(\hat{B}'_{K}\hat{B}_{K})^{-1}\hat{B}'_{K}$. The three-step sieve OLS estimator of $\gamma_{0}$ is given by

With the addition of the following assumptions, we show in Theorems (ref) and (ref) that $\hat{\gamma}$ is $\sqrt{n}$ consistent and asymptotically normal.

Assumption (ref) assumes that the covariates $z_{i}$ and $x_{i}$ are bounded. Assumption (ref)(i) requires that the group formation parameter $\theta_{0}$ lies in a compact set.\footnote{Although we set a cutoff to $-\infty$ if the capacity is not binding, such a cutoff does not affect individuals' choices and is excluded from $\theta_{0}$ (see footnote (ref)). Strictly speaking, we also need to assume that any market-clearing cutoff is bounded away from $-\infty$. Although the demand and supply for a group may be equal at a cutoff of $-\infty$, the group's capacity must take a particular value for that to occur. This is because a cutoff of $-\infty$ no longer makes the group selective, and the demand is solely determined by the number of individuals who do not prefer or qualify for any other group. Such a solution is non-generic (i.e., it requires a precise alignment of parameters that rarely occurs in practice), and we assume that this unlikely case is ruled out.} These boundedness assumptions are standard in the literature. Assumption (ref)(ii) requires that the estimator $\hat{\theta}$ has an asymptotically linear representation, which can be satisfied by semiparametric estimators lee1995semiparametric,Sun2019 or parametric estimators (e.g., the constrained maximum likelihood estimator proposed in Supplemental Appendix (ref)). Assumption (ref) imposes standard regularity conditions for the sieve estimation. Assumption (ref)(i) is a normalization.\footnote{Alternatively, we can assume that the smallest eigenvalue of $\mathbb{E}[b^{K}(\pi)b^{K}(\pi)']$ is bounded away from zero uniformly in $K$. Assuming this, let $Q_{0}=\mathbb{E}[b^{K}(\pi)b^{K}(\pi)']$ and $Q_{0}^{-1/2}$ the symmetric square root of $Q_{0}^{-1}$. Then $\tilde{b}^{K}(\pi)=Q_{0}^{-1/2}b^{K}(\pi)$ is a nonsingular transformation of $b^{K}(\pi)$ that satisfies $\mathbb{E}[\tilde{b}^{K}(\pi)\tilde{b}^{K}(\pi)']=I_{K}$. Notably, nonparametric series estimators are invariant under nonsingular transformations of $b^{K}(\pi)$: let $\tilde{\beta}^{Z}=Q_{0}^{1/2}\beta^{Z}$ then $\tilde{b}^{K}(\pi)'\tilde{\beta}^{Z}=b^{K}(\pi)'\beta^{Z}$. Furthermore, $\tilde{b}^{K}(\pi)$ satisfies Assumption (ref)(iii)(iv) if and only if $b^{K}(\pi)$ does. Therefore, all parts of Assumption (ref) are satisfied with $b^{K}(\pi)$ replaced by $\tilde{b}^{K}(\pi)$ liracine2007.} Assumption (ref)(ii)-(iv) impose rate conditions on the basis functions, similar to those used in the literature (newey1994asymptotic; liracine2007). Assumption (ref)(i) requires that $\epsilon_{i}$ have a finite eighth moment. Assumption (ref)(ii) is used to account for the estimation errors in the first and second steps, which we establish following the approach of Hahn2013. Assumption (ref) imposes crucial restrictions on the adjacency matrix $\boldsymbol{w}$, which are essential for establishing the asymptotic theorems. Assumption (ref)(i) is the usual row normalization, and the remaining parts of Assumption (ref) will be discussed below.

We propose an OLS estimator instead of an IV estimator for $\gamma_{0}$, which might seem counter-intuitive given that the regressor $w_{i}\boldsymbol{y}$ could be endogenous due to simultaneity. Nevertheless, we show that the endogeneity of $w_{i}\boldsymbol{y}$ vanishes asymptotically under the assumption that the adjacency matrix $\boldsymbol{w}$ is dense (Assumption (ref)(ii)), which implies that each component of $\boldsymbol{w}$ is bounded by $O_{p}(n^{-1})$. This assumption is satisfied in large groups if $\boldsymbol{w}$ is specified by group averages or dense networks within each group (see Supplemental Appendix (ref) for examples of $\boldsymbol{w}$). The intuition is that if an individual has a growing number of peers, the average in $w_{i}\boldsymbol{y}$ will converge to a population expectation, which is no longer endogenous. Our result aligns with the findings in Lee2002, who demonstrated the consistency and efficiency of OLS estimators for peer effects under a deterministic adjacency matrix.

The asymptotic analysis of $\hat{\gamma}$ is complicated by two sources of randomness in the adjacency matrix $\boldsymbol{w}$. First, $\boldsymbol{w}$ depends on the group memberships $\boldsymbol{g}$, which are random and can be correlated with $\boldsymbol{\epsilon}$. Second, $\boldsymbol{w}$ may include additional randomness arising from the networks within each group. These features imply that traditional asymptotic methods, which often assume a deterministic adjacency matrix, are not applicable. Consequently, we develop new methods to establish the asymptotic properties of $\hat{\gamma}$.

We begin by observing that the leading terms in $\hat{\gamma}$ take the form of weighted $U$-statistics of order 2, with weights given by the $(i,j)$ components of $\boldsymbol{w}$ as well as its polynomials and series, denoted by $\boldsymbol{q}$. These polynomials and series arise because $w_{i}\boldsymbol{y}$ is included as a regressor. It is worth noting that our case is more challenging than the standard weighted $U$-statistics analyzed in Lee1990, where weights are assumed to be fixed. In our setting, the weights can be stochastic as they depend on group memberships and within-group connections. Furthermore, these random weights may introduce network dependence, which complicates the asymptotic analysis. To address this problem, we extend the methods in Lee1990 by imposing additional restrictions on the network dependence in $\boldsymbol{q}$, conditional on $\boldsymbol{\psi}=(\boldsymbol{x},\boldsymbol{z},\boldsymbol{g})$, as specified in Assumption (ref)(iii)-(v).

Specifically, we establish the consistency of the sieve estimation in the second step using the following facts: (a) $\boldsymbol{q}$ is dense in the same way as $\boldsymbol{w}$, which follows from row normalization and the density of $\boldsymbol{w}$ (Assumption (ref)(i)-(ii)), (b) the conditional dependence between network connections $q_{ij}$ and $q_{kl}$ on disjoint nodes $\{i,j\}$ and $\{k,l\}$ diminishes sufficiently fast, and (c) the difference between conditioning on global information ($\boldsymbol{\psi}$) and local information ($\psi_{i}$ and $\psi_{j}$) becomes negligible at an appropriate rate (Assumption (ref)(iii)). Building on these results, we establish the consistency of $\hat{\gamma}$ under a similar limited dependence condition on $\boldsymbol{q}$ (Assumption (ref)(iv)).

To derive the asymptotic distribution of $\hat{\gamma}$, we extend the Hoeffding decomposition for standard weighted $U$-statistics Lee1990 by (a) conditioning on all individual-level variables, including group memberships and, if applicable, individual fixed effects in network formation within each group (see Example (ref) in Supplemental Appendix (ref)), and (b) assuming that, conditional on such individual-level variables, the dependence between network connections $q_{ij}$ and $q_{ik}$ with a shared node $i$ diminishes sufficiently fast (Assumption (ref)(v)). Under this assumption, the Hoeffding decomposition remains valid despite random weights, enabling us to derive the asymptotic distribution of $\hat{\gamma}$.

We verify in Supplemental Appendix (ref) that Assumption (ref) is satisfied for group averages (both including and excluding oneself) and for dyadic networks with fixed effects Graham2017. There might be sufficient conditions on $\boldsymbol{w}$ that achieve the desired asymptotic results while accommodating sparsity and/or strategic interactions in network formation within groups (leung2015two; ridder2020estimation; menzel2021central; Leung_Moon_2023), but we leave these directions for future research.

\protectSimulations

Setup

In this section, we evaluate our approach through a simulation study. We generate a market of $2,000$ individuals. Each individual is assigned i.i.d. $x_{i}\sim N(5,25)$ and $\epsilon_{i}\sim N(0,1)$, where $x_{i}$ is independent of $\epsilon_{i}$. The individuals interact according to the linear-in-means model in equation ((ref)). We consider two scenarios: one without endogenous effects and one with endogenous effects. In the absence of endogenous effects, we set the parameter values to $\gamma=(0,1,1)$. In the presence of endogenous effects, we set $\gamma=(0.5,1,1)$.

The market consists of five groups with capacities of 280, 340, 200, 460, and 400, resulting in a total of 1,680 seats. Individuals choose which group to join based on the model described in Section (ref). The utility of individual $i$ when joining group $g$ is specified as $u_{ig}=\zeta_{g}+\delta_{1}^{u}z_{1,ig}^{u}+\delta_{2}^{u}z_{2,i}+\xi_{ig},$ where $\zeta_{g}$ is a group-specific fixed effect, $z_{1,ig}^{u}$ is a pair-specific characteristic that is i.i.d. across $i$ and $g$, following $N(0,9)$, and $z_{2,i}$ is an individual-specific characteristic that is i.i.d. following $N(2,1)$. We allow $z_{2,i}$ to be correlated with $x_{i}$ such that $\text{Cov}(z_{2,i},x_{i})=2$. An individual may also choose not to join any group, in which case their utility is given by $u_{i0}=\xi_{i0}$. The unobserved preference $\xi_{ig}$ is i.i.d. across $i$ and $g=0,1,\dots,5$, following the type I extreme value distribution. Individual $i$'s qualification for joining group $g$ is specified as $v_{ig}=\delta_{1}^{v}z_{1,ig}^{v}+\delta_{2}^{v}z_{2,i}+\eta_{ig},$ where the pair-specific characteristic $z_{1,ig}^{v}$ follows $N(0,9)$ and the unobservable $\eta_{ig}$ follows $N(\epsilon_{i},1)$, both i.i.d. across $i$ and $g$. Note that $\eta_{ig}$ is correlated with $\epsilon_{i}$, thereby leading to endogenous groups. We set the group fixed effects to $\zeta=(\zeta_{1},\zeta_{2},\zeta_{3},\zeta_{4},\zeta_{5})=(9,6,4,2,0)$ and the parameter values $(\delta_{1}^{u},\delta_{2}^{u},\delta_{1}^{v},\delta_{2}^{v})=(-1,1,1,1)$. Based on these model primitives, the stable groups are determined through the individual-proposing Deferred-Acceptance algorithm gale_college_1962. The capacity constraints in all markets are binding.

Given the groups, we consider two specifications for the adjacency matrix $\boldsymbol{w}$. In the first specification, we use group averages that exclude the individual themselves. In the second specification, we average over friends within a group, where the friendships are generated independently with a constant probability of $0.5$. For each specification, we estimate $\gamma$ using data from a single market. The experiment procedure is repeated independently 200 times and we report the average bias, standard errors, and root mean squared errors (RMSE) of the 200 estimates of $\gamma$.\footnote{The group formation parameters are estimated using constrained maximum simulated likelihood, where the cutoffs are treated as auxiliary parameters that satisfy market-clearing conditions. See Supplemental Appendix (ref) for more details on the estimation method and results.}

Estimation Results

Table (ref) presents the estimation results in the absence of endogenous effects ($\gamma_{1}=0$). For the specification with group averages (Panel A), the OLS estimate of $\gamma_{2}$ is biased upward, indicating the presence of selection bias (Column 1). Including group fixed effects (FE) does not mitigate this bias (Column 2). In Column 3, we control for a second-order polynomial series of the elementary symmetric functions of utility difference and qualification indices, as detailed in Section (ref). The sieve OLS estimate of $\gamma$ is unbiased, demonstrating the effectiveness of the selection correction. The specification with networks produces similar results (Panel B). Both OLS and OLS with group FE yield biased estimates (Columns 4 and 5), whereas sieve OLS provides unbiased estimates (Column 6). In addition, sieve OLS exhibits smaller standard errors and RMSE compared to OLS and OLS with group FE.

The estimation results in the presence of endogenous effects ($\gamma_{1}\neq0$) are presented in Table (ref). For the specification with group averages (Panel A), OLS yields biased estimates of $\gamma_{1}$ and $\gamma_{2}$ (Column 1). Including group FE significantly exacerbates the bias, likely due to multicollinearity between the group averages and group dummies (Column 2). By applying the polynomial selection correction, sieve OLS yields unbiased estimates (Column 3). The specification with networks shows a similar pattern (Panel B). The estimates obtained from OLS and OLS with group FE are biased (Columns 4 and 5), whereas sieve OLS yields unbiased estimates (Column 6). Consistent with the results in Table (ref), sieve OLS has smaller root mean squared errors compared to both OLS specifications.

In sum, the simulation results indicate that including group FE is insufficient to correct for selection bias, whereas sieve OLS provides an effective approach for selection correction.

\protectSocial Interactions in Chilean High Schools

Data

In this section, we apply our approach to analyze social interactions among high school students in Chile. We use data from the SIMCE, provided by the Agency for the Quality of Education in Chile simce.\footnote{SIMCE is an abbreviation for Sistema de Medici�n de la Calidad de la Educaci�n (Education Quality Measurement System).} The SIMCE dataset provides information on math and language scores, as well as information on parental education, parental income, and other student and family characteristics collected through a parental questionnaire sent home with students. To track a student's academic performance in subsequent years, we merge the SIMCE dataset with Ministry of Education administrative records, which provide detailed information on educational attainment, ranging from high school completion to college graduation.

Our sample consists of 6,872 tenth-grade students enrolled in 53 high schools in the Biob�o Region in 2006. Of these schools, 23 are public and 30 are private. Public schools are required to accept any student willing to enroll, while private schools can select students based on their admission criteria.\footnote{Although public schools cannot select students, peer effect estimates based on these schools may be biased because students' unobserved preferences for schools ($\xi_{i}$) may be correlated with $\epsilon_{i}$.} Panel A of Table (ref) provides descriptive statistics for the students in our sample. On average, their mothers have 9.58 years of education, 51% of the students are female, and 87% of the students are enrolled in Fonasa.\footnote{Fonasa (Fondo Nacional de Salud) is Chile's tax-funded public health insurance system that provides free or subsidized healthcare for those unable to afford private insurance.} We use academic performance across various educational stages as our measures of outcomes. Tenth-grade math and language scores come from the SIMCE standardized test, measured as percentile ranks ranging from 0 to 1. High school graduation is an indicator that equals 1 if a student completes high school on time at the end of twelfth grade and 0 otherwise. In our sample, 72% of the students graduate from high school on time. Additionally, we consider four long-run outcomes: post-secondary enrollment, college enrollment, post-secondary graduation, and college graduation. Among the students in our sample, 63% enroll in post-secondary education, with 30% attending college. Furthermore, 50% complete post-secondary education, and 24% graduate from college.

We construct the peer measures in equation ((ref)) using class averages that exclude the student themselves. It is documented that a large fraction of peer effects in education arises at the classroom level ammermueller2009peer. In our sample, the average class size (37) is approximately one quarter of the average school size (130). Therefore, classroom averages provide a dense adjacency matrix that satisfies Assumption (ref). Furthermore, this specification allows us to distinguish peer influence from school effectiveness, the latter of which is captured by school FE.

Panels B and C of Table (ref) present summary statistics for peer and school averages, respectively. If students were randomly assigned to schools, school averages would be relatively homogeneous, with standard deviations significantly smaller than those of individual characteristics. However, the standard deviations of school averages reported in Panel C remain comparable to those of individual characteristics in Panel A, indicating sorting across schools.\footnote{If students are randomly assigned to schools, the variance of the average of a characteristic $x_{i}$ in school $s$ is given by $\text{Var}(\bar{x}_{s})=\text{Var}(\frac{1}{n_{s}}\sum_{i=1}^{n_{s}}x_{i})=\frac{1}{n_{s}}\text{Var}(x_{i})$, where $n_{s}$ represents the school size. In our sample, the average school size is 130, so the ratio between the standard deviation of an individual characteristic and the standard deviation of its school average under random assignment should be approximately $\sqrt{130}\approx11.4$, which is much larger than what we observe in the sample.} This finding aligns with the evidence on high levels of socioeconomic segregation between schools valenzuela2019acrecentando.

Table (ref) decomposes the total variances of peer characteristics and outcomes into the variances within and between schools. We find that 88% and 91% of the variation in peer mother\textquoteright s education and peer Fonasa enrollment, respectively, occur between schools, while the fraction is a bit lower for peer fraction female (73%). Peer outcomes have similar fractions of variation occurring between schools: 73-74% for test scores, 66-76% for college-related outcomes, and the lowest fraction of 48% for high school graduation. These results suggest substantial sorting across schools, as the majority of the variation in peer characteristics and outcomes takes place between schools rather than within them. In Section (ref), we provide further evidence that classroom assignment within schools does not follow systematic patterns, implying that Assumption (ref) is satisfied in our context.

Estimation Results

We estimate the parameters in high school admissions using the MPEC (Mathematical Programming with Equilibrium Constraints) algorithm su2012constrained, as described in Supplemental Appendix (ref). We specify preferences and qualifications as linear functions of individual characteristics (logarithm of family income, eighth-grade composite score calculated as the sum of math and language scores, mother's education, and distance to school) and their interactions with school characteristics (logarithm of tuition, average composite score, and average mother's education).\footnote{Our empirical model of high school admissions follows closely that in HSS_twosided. However, instead of the Bayesian approach used in HSS_twosided, we adopt a frequentist approach to estimate the parameters. Our approach has two main advantages: it reduces the computational burden and, more importantly, it facilitates the adjustment of standard errors in the second stage, where we estimate social interactions by frequentist methods.} Based on these estimates (provided in Supplemental Appendix (ref)), we estimate social interactions among tenth graders and compute standard errors using the methodology developed in Section (ref).

Our basic specification of equation ((ref)) includes individual characteristics (female, mother's education, and Fonasa enrollment), the peer averages of these characteristics, and the peer outcome. We consider both short-run outcomes (tenth-grade math and language scores and high school graduation) and long-run outcomes (post-secondary and college enrollment and graduation). For each outcome, we estimate equation ((ref)) by (a) OLS, and then progressively control for (b) school FE, and (c) selection correction. The selection correction is constructed using a second-order polynomial series of the elementary symmetric functions of utility difference and qualification indices in high school admissions.

To account for institutional differences between public and private schools, we relax the framework in Section (ref) by imposing exchangeability conditional on school type (public or private) rather than across all schools. Conditional exchangeability by school type requires only that schools of the same type have exchangeable unobservables, while allowing for systematic differences in unobservables between public and private schools.\footnote{Conditional exchangeability by school type implies that public and private schools have two distinct selection functions, with each selection function depending on two sets of elementary symmetric functions: one for the indices of public schools and the other for the indices of private schools. See Supplemental Appendix (ref) for details on sieve estimation in this case.} By leveraging symmetric functions, our approach significantly reduces the dimensionality of sieve estimation, requiring only 35 basis functions compared to 3,485 basis functions if symmetric functions were not used (see Supplemental Appendix (ref)). Additionally, by exchangeability our approach can separate two sources of correlated effects within a school: (i) school effectiveness, captured by the school FE, and (ii) self-selection into schools, captured by the selection correction.

Tables (ref) and (ref) report the estimates for short-run and long-run outcomes, respectively. Across all outcomes, simple OLS yields the largest estimates of endogenous peer effects (coefficients of peer outcomes). Including school FE reduces these OLS estimates by 12-46%. Additionally controlling for our selection correction further lowers the estimates; the selection-corrected estimates of endogenous peer effects are 10-45% lower than those obtained using school FE alone. These findings demonstrate that the OLS estimates of endogenous peer effects are biased upward due to sorting into schools. While the inclusion of school FE mitigates selection bias to some extent, it does not completely eliminate it. In contrast, our selection correction effectively addresses the selection issue and provides unbiased estimates of endogenous peer effects.

The selection-corrected estimates in Tables (ref) and (ref) show that peer outcomes have positive and statistically significant effects across all outcomes. In high school, one standard deviation increases in peer math and language scores (0.204 and 0.185) raise a student's math and language scores by 11.1 and 7.5 percentile ranks, respectively. This peer effect persists in later outcomes with a similar magnitude. For example, one standard deviation increases in peer college enrollment and graduation (0.302 and 0.260) raise a student's college enrollment and graduation rates by 14.2 and 10.6 percentage points, respectively. These are substantial effects compared to the mean rates (30.3% and 23.8%).

In addition, Tables (ref) and (ref) provide similar results for exogenous peer effects (coefficients of peer characteristics). Across all outcomes, peer mother's education has positive and significant effects in OLS regressions with school FE. The estimates decrease when we correct for selection, suggesting upward selection bias. The selection-corrected estimates remain positive and significant for all outcomes (except for tenth-grade math and language scores). Peer fraction female shows a similar pattern across short-run outcomes and college enrollment and graduation: the selection correction further reduces the estimates compared to OLS with school FE. The selection-corrected estimates are positive and significant for tenth-grade math score, high school graduation, and post-secondary enrollment and graduation, albeit with a negative effect for college graduation. This finding is consistent with the documented evidence that the fraction of female peers is positively correlated with academic achievement Sacerdote2011. For peer Fonasa enrollment, the selection correction also lowers the OLS estimates with school FE, leading to larger negative effects.\footnote{This suggests that peer Fonasa enrollment may be positively correlated with the selection correction, conditional on other controls. } As expected, the selection-corrected estimates are negative and significant for all outcomes (except for tenth-grade math and language scores).

In a nutshell, OLS regressions with school FE only partially account for selection and tend to overestimate peer effects. The F-statistics for the selection correction are statistically significant across all outcomes (except for college enrollment), underscoring the importance of our selection correction approach. The selection-corrected estimates provide evidence that both peer outcomes and peer characteristics have significant effects on tenth graders.

Peer Influence, School Effectiveness, and Self-Selection

In our framework, a student's outcome are shaped by three factors related to the school they attend: (i) the influence of their peers within the school, (ii) the academic effectiveness of the school, including aspects such as teacher quality and school resources and investments; and (iii) unobserved factors that affect self-selection into the school. In this section, we aim to answer the following questions: To what extent can a student's academic performance be attributed to peer influence, school effectiveness, and self-selection? How would the evaluation of peer influence and school effectiveness be impacted if self-selection were ignored?

Using the selection-corrected estimates, we calculate the portion of a student's predicted outcome that depends on the school they attend. This portion is defined as the sum of three components: (i) peer influence, calculated as $\sum_{j=1}^{n}w_{ij}y_{j}\hat{\gamma}_{1}+\sum_{j=1}^{n}w_{ij}x_{j}\hat{\gamma}_{2}$, where $\hat{\gamma}_{1}$ and $\hat{\gamma}_{2}$ are the estimates of $\gamma_{1}$ and $\gamma_{2}$, (ii) school effectiveness, represented by the estimated school FE for the school the student attends, and (iii) self-selection, calculated as $\hat{\lambda}(\hat{\pi}_{i})$, the estimated polynomial series used for the selection correction.\footnote{Our measure of peer influence captures both peer quality ($\sum_{j=1}^{n}w_{ij}y_{j}$ and $\sum_{j=1}^{n}w_{ij}x_{j}$) and peer effects ($\gamma_{1}$ and $\gamma_{2}$).} We then decompose the total variance of the school-dependent predicted outcome into the variances of peer influence, school effectiveness, and self-selection, as well as the covariances between any two of these components.

The decomposition results are presented in Table (ref). We find that self-selection accounts for the largest fraction of the total variation in school-dependent predicted outcomes, except for post-graduate enrollment and college graduation, where it accounts for the second largest fraction. Peer influence and school effectiveness contribute comparably to short-run outcomes; however, peer influence becomes more important for long-run outcomes.\footnote{There is evidence that peer interactions during adolescence have a lasting impact in later years. For example, LlerasMuney2024 found that friendships formed during adolescence have significant influence on labor market outcomes.} In contrast, school effectiveness accounts for the smallest fraction of the explained variation in long-run outcomes. These findings underscore the critical role of self-selection in explaining a student's predicted outcomes. Failing to account for self-selection can lead to significant bias when measuring the contributions of peer influence and school effectiveness.

The last three rows of Table (ref) present covariances among peer influence, school effectiveness, and self-selection. The correlation between self-selection and peer influence is generally positive, suggesting that students and their peers are sorted into schools in a positively assortative manner. In contrast, the correlation between self-selection and school effectiveness is negative, indicating that more selective schools may provide lower value-added. This result aligns with the existing literature that documents limited school effectiveness for highly selective schools abdulkadirouglu2014elite,dobbie2014impact.

To understand the impact of ignoring selection on the evaluation of peer influence and school effectiveness, we further examine these correlations at the school level. Figure (ref) plots the average selection in each school against the average peer influence in the school, with peer influence calculated using both biased estimates (OLS with school FE) and unbiased estimates (sieve OLS). Both biased and unbiased estimates demonstrate that peer influence is positively correlated with selection at the school level, consistent with the result in Table (ref). Figure (ref) plots the average selection in each school against school effectiveness, with school effectiveness similarly calculated using both biased estimates (OLS with school FE) and unbiased estimates (sieve OLS).\footnote{Since we measure school effectiveness using school fixed effects, we include the dummies for all schools in our specifications of school FE and sieve OLS. To achieve identification, we drop the constant term and normalize $\mathbb{E}[\epsilon_{i}]=0$. By construction, $\mathbb{E}[\nu_{i}|\pi_{i}]=\mathbb{E}[\epsilon_{i}-\lambda(\pi_{i})|\pi_{i}]=\lambda(\pi_{i})-\lambda(\pi_{i})=0$ and therefore $\mathbb{E}[\lambda(\pi_{i})]=\mathbb{E}[\epsilon_{i}]-\mathbb{E}[\nu_{i}]=0$ by the law of iterated expectations. In the sieve OLS specification, we implement the normalization by demeaning all the basis functions in a sieve approximation.} The biased estimates suggest a misleading positive correlation between average selection and school effectiveness, potentially leading to the incorrect conclusion that more selective schools are more effective. In contrast, the unbiased estimates reveal a negative correlation between average selection and school effectiveness, reaffirming the earlier finding that more selective schools tend to provide lower value-added.

Because peer influence and school effectiveness are correlated with selection in opposite directions, and they are also correlated with each other in varying directions (Table (ref)), it is challenging to predict the directions of selection bias in these variables. Instead, we directly compare their biased and unbiased estimates in Figures (ref) and (ref). Figure (ref) plots the distributions of school-average peer influence, calculated using both biased and unbiased estimates across all outcomes. Compared to the unbiased distributions, the biased ones are shifted to the right, indicating an overestimation of peer influence, and exhibit increased dispersion across schools. Peer influence is overestimated in all schools, with larger upward bias for more selective schools (Figure (ref)). Figure (ref) plots the distributions of school effectiveness, calculated using both biased and unbiased estimates. The biased estimates of school effectiveness tend to be smaller and less dispersed across schools. School effectiveness is primarily underestimated for less selective schools (Figure (ref)).

These findings highlight the significant impact of selection bias on the evaluation of peer influence and school effectiveness. Addressing the selection problem is critical for deriving accurate implications for policymakers. Specifically, school performance should not be assessed solely based on students' outcomes, as these outcomes are heavily influenced by self-selection. While peer effects are substantial, they are often overestimated when selection is not adequately addressed, and the overestimation is more pronounced in highly selective schools. Moreover, selection tends to distort the evaluation of school effectiveness, primarily by underestimating the value-added of less selective schools, which often serve less advantaged student populations. Correcting for selection is essential to accurately recognize school performance and ensure equitable resource allocation.

\protectTesting for Random Assignment into Classrooms

There may be potential concerns about additional sorting at the classroom level. To check whether classroom assignment is random, we apply a regression-based test originally proposed by sacerdote2001peer and later refined with size correction by jochmans2023testing. The underlying intuition of the test is straightforward: under random assignment, a student's characteristics should not be correlated with the average characteristics of their classroom peers, after controlling for sorting across schools.

We apply jochmans2023testing's test to a variety of student characteristics that schools may consider when assigning students into classrooms: female, mother's education, Fonasa enrollment, family income, and test scores. For each characteristic, we perform (i) a baseline test that controls for school FE, and (ii) a selection-corrected test that controls for both school FE and selection correction. Table (ref) reports the p-values of these tests. In the baseline tests, the p-values for mother's education and eighth-grade composite score are significant, suggesting potential sorting at the classroom level based on these variables. However, after controlling for our selection correction, none of the characteristics yield significant p-values. These results provide evidence that classroom assignment is random, as long as that selection into schools is properly controlled for.

\protectConclusion

In this paper, we study social interactions in endogenous groups. We develop a model of group formation to analyze how individuals select into groups and how to account for the impact of group selection. Our model accommodates two-sided decision-making, where individuals choose groups based on their preferences, while groups admit individuals based on qualifications until reaching their capacities. This framework mirrors many admission processes in the real world.

We make significant contributions to the literature in several aspects. First, we characterize group formation using a two-sided many-to-one matching model with nonparametric unobservables. Building on this framework, we demonstrate that endogenous selection into groups leads to selection bias in the estimation of peer effects. Second, we propose innovative methods to address the dimensionality challenges in correcting for this selection bias. Specifically, we employ the limiting approximation of a market as it grows large to reduce the high dimensionality due to equilibrium effects. Additionally, we impose an exchangeability assumption, under which the selection bias can be expressed through a group-invariant selection function that remains tractable even with a moderately large number of groups. Third, we propose a sieve OLS estimator for the social interaction parameters, which achieves $\sqrt{n}$-consistency and asymptotic normality. These asymptotic properties are established using novel asymptotic methods under additional conditions (e.g., a dense adjacency matrix, limited network dependency). We verify that these conditions hold for commonly used specifications of adjacency matrices, including group averages and dyadic networks with fixed effects.

We apply our approach to investigate social interactions among tenth graders in Chile. We find that including school fixed effects is insufficient to account for selection into high schools, whereas our selection correction method yields unbiased estimates of peer effects. A variance decomposition shows that self-selection accounts for the largest share of explained variance in most outcomes, highlighting the importance of properly accounting for endogenous selection in school evaluation. Ignoring selection into high schools tends to overestimate peer influence, particularly in highly selective schools. Moreover, the selection-corrected estimates suggest that more selective schools may actually provide lower value-added. These results underscore the significant impact of selection bias on the evaluation of peer influence and school effectiveness. Properly correcting for selection is crucial for policymakers to achieve equitable resource allocation.

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\protectEstimating Group Formation Parameters

To establish the empirical relevance of our estimation approach, we study a parametric simulation setting of group formation process. We assume known distributions for the unobserved heterogeneity terms $\xi_{ig}$ and $\eta_{ig}$. Let $z_{i}$ be the vector of observable characteristics, which contains the individual-group specific variables$\{z_{1,ig}^{u},z_{1,ig}^{v}\}_{g=1}^{G}$ and individual specific variable $z_{2i}$. For each group $g$, the conditional probability of an individual joining $g$ can be written as

where $U_{ig}=\delta_{1}^{u}z_{1,ig}^{u}+\delta_{2}^{u}z_{2,i}$ and $V_{ig}=\delta_{1}^{v}z_{1,ig}^{v}+\delta_{2}^{v}z_{2,i}$ represent the deterministic part of the utility and qualification respectively, $\eta_{i}=(\eta_{i1},\dots,\eta_{iG})'$, and $\mathbf{1}\left(\cdot\right)$ is an indicator function. Let $\boldsymbol{\sigma}(\delta,\alpha,p;z_{i})=(\sigma_{1}(\delta,\alpha,p;z_{i}),\dots,\sigma_{G}(\delta,\alpha,p;z_{i}))'$ denote the vector of conditional probabilities across all groups.

We estimate the group formation parameters $\delta$ and $\zeta$ by maximizing log-likelihood, where $p$ is treated as auxiliary parameters that satisfy the market clearing condition. The estimator solves the constrained optimization problem: \[ \max_{\delta,\zeta,p}\;\mathcal{L}_{n}(\delta,\zeta,p;z_{i})\qquad\text{s.t.}\quad D_{n}(\delta,\zeta,p;z_{i})=S_{n}, \] where the log likelihood function is given by:

the demand is given by \[ D_{n}(\delta,\zeta,p)=\frac{1}{n}\sum_{i=1}^{n}\boldsymbol{\sigma}(\delta,\zeta,p;z_{i}), \] and $S_{n}$ is the vector of group capacities.

Since the match probabilities involve an integral without a closed form, we approximate it through simulation. For each individual $i$, we generate 300 independent random draws of the vector $\eta_{i}$ from its distribution to approximate the probabilities through Monte Carlo integration. To address the numerical challenges posed by non-smooth indicator functions, we adopt the smoothed A-R simulator train2009discrete. Specifically, we replace the indicator function $\mathbf{1}(p_{g}-(V_{ig}+\eta_{ig})<0)$ with a smoothed logistic function:$\left(1+\exp\left(\frac{p_{g}-(V_{ig}+\eta_{ig})}{\kappa}\right)\right)^{-1}$, where the scale parameter $\kappa$ is set to 0.05. This smooth approximation improves numerical properties while maintaining estimation accuracy and converges to the original indicator function as $\kappa\rightarrow0$.

aitchison1958maximum has established the $\sqrt{n}$ consistency and asymptotic distribution of a constrained maximum likelihood estimator like ours. We calculate our estimator using the MPEC (Mathematical Programming with Equilibrium Constraints) algorithm su2012constrained. Table (ref) shows the estimation results. We find that the estimator works well, with minimal bias across parameters and reasonable standard errors.

\protectEstimating Parameters in High School Admissions

We model the group formation process empirically by analyzing the matching between students and high schools (grades 9-12) in Chile's Biob�o Region during 2005.\footnote{The Biob�o Region includes twelve municipalities: Los �ngeles, Nacimiento, Yumbel, Laja, Cabrero, Yungay, Mulch�n, Negrete, Tucapel, Pemuco, Quilleco, and Santa B�rbara. The high school admission market in Biob�o Region is relatively independent, with only $2.19\%$ of tenth graders in Biob�o Region schools residing outside the market area and $3.96\%$ of tenth graders living in the Biob�o Region attending schools elsewhere in 2006.} Table (ref) summarizes the student and school characteristics. Note that school average composite score and average mother's education was calculated from student characteristics of the tenth graders in a school in 2003, and thus are pre-determined in the 2005 admissions that we study.

Our empirical framework for high school admissions follows closely HSS_twosided, while our estimation approach deviates by employing frequentist methods rather than the Bayesian approach in HSS_twosided. Specifically, we allow student preferences to be school-type-specific. For student $i$, the utility of attending school $g$ of type $t$ $\in$ \{public, private\} is

where $\xi_{ig}$ is i.i.d.\ extreme value type I; and $Z_{ig}^{u}$ are student-school-specific variables, including a constant term, and

Each student has an outside option, $u_{i0}=\xi_{i0}$, with $\xi_{i0}$ being extreme value type I.

As for school preferences, public schools do not have a utility function because they cannot select students. For private school $g$, its qualification function is

where $\eta_{ig}$ is i.i.d. standard normal; and the vector $Z_{ig}^{v}$ includes a constant term, and

We estimate the model by maximizing a simulated log-likelihood, subject to equilibrium constraints, as illustrated through Monte Carlo simulations (Appendix (ref)). The estimation results are summarized in Table (ref). Panel A shows student preference estimates, with expected coefficient signs. Both distance and tuition negatively impact student preferences, though the latter effect diminishes with parental income through interaction term. Students of more educated mothers like schools with higher average mother\textquoteright s education, regardless of school type. Panel B shows admission preferences among private schools. Schools with higher average mother's education increasingly prefer students with higher educated mother. The impact of student score on school preferences depends critically on school quality - positive preferences for high-achieving students emerge only in schools whose existing student body maintains an average composite score above 0.8.

\protectConditional Exchangeability by School Type

In our empirical application, we impose exchangeability conditional on school type (public or private). In this case, while the selection function is invariant across schools of the same type, it has distinct functional forms for public and private schools. Moreover, the selection function is symmetric in the indices of schools within the same type, rather than across all schools.

Given these properties, in our empirical setting, when constructing the elementary symmetric functions, instead of aggregating over the indices of all schools, we aggregate over the indices of public and private schools separately. Moreover, in our sieve estimation, the type-specific selection function translates into estimating separate coefficients for the polynomial basis functions for each school type. That is, we interact the basis functions with indicator variables for each school type.

Specifically, let $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ be a partition of $\mathcal{G}$, where $\mathcal{G}_{1}$ refers to the set of private schools and $\mathcal{G}_{2}$ refers to the set of public schools. Below we list the basis functions up to order 2 used in our empirical application:

Note that the five terms containing $\iota_{ig_{i}}$ are included only for students who attend a private school ($g_{i}\in\mathcal{G}_{1}$), as only private schools have a qualification index, while public schools cannot select students. For the same reason, the sum of $\iota_{ih}$ is calculated over private schools only. Overall, this implementation results in 20 basis functions for students attending private schools and 15 basis functions for students attending public schools, requiring a total of 35 sieve coefficients to be estimated. The reduction in nuisance parameters is substantial compared to a specification that does not use symmetric functions, which would require 82 basis functions of order one ($G+G_{1}-1=53+30-1$) and 3,403 basis functions of order two $((G+G_{1}-1)+(G+G_{1}-1)(G+G_{1}-2)/2)$ given our empirical setting with 23 public schools and 30 private schools.

\protectProofs

\paragraph{Notation}

Let $x=(x_{1},\dots,x_{n})'\in\mathbb{R}^{n}$ denote an $n\times1$ vector, and let $A=(a_{ij})\in\mathbb{R}^{n^{2}}$ denote an $n\times n$ matrix. We use $\|\cdot\|$ to denote the Frobenius norm, that is, $\|x\|\equiv(\sum_{i=1}^{n}x_{i}^{2})^{1/2}$ and $\|A\|\equiv(\text{tr}(AA'))^{1/2}=(\text{\ensuremath{\sum}}_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^{2})^{1/2}$. For a matrix $A$, we denote the maximum row sum norm as $\interleave A\interleave_{\infty}\equiv\max_{1\leq i\leq n}\sum_{j=1}^{n}|a_{ij}|$ and the maximum column sum norm as $\interleave A\interleave_{1}\equiv\max_{1\leq j\leq n}\sum_{i=1}^{n}|a_{ij}|$. Note that $\interleave A'\interleave_{\infty}=\interleave A\interleave_{1}$. Furthermore, for both vectors and matrices, we denote the $l_{\infty}$ and $l_{1}$ norms as $\|\cdot\|_{\infty}$ and $\|\cdot\|_{1}$ respectively. Specifically, we have $\|x\|_{\infty}\equiv\max_{1\leq i\leq n}|x_{i}|$, $\|x\|_{1}\equiv\sum_{i=1}^{n}|x_{i}|$, $\|A\|_{\infty}\equiv\max_{1\leq i,j\leq n}|a_{ij}|$, and $\|A\|_{1}\equiv\sum_{i,j=1}^{n}|a_{ij}|$. It follows that $\|A\|_{\infty}\leq\min\{\interleave A\interleave_{\infty},\interleave A\interleave_{1}\}$, and $\max\{\interleave A\interleave_{\infty},\interleave A\interleave_{1}\}\leq n\max_{1\leq i,j\leq n}|a_{ij}|=n\|A\|_{\infty}$. For matrices $A$ and $B$ and vectors $x$ and $y$, we can derive $\|AB\|_{\infty}\leq\interleave A\interleave_{\infty}\|B\|_{\infty}$, $\|AB\|_{1}\leq\interleave A\interleave_{1}\|B\|_{1}$, $\|Ax\|_{\infty}\leq\interleave A\interleave_{\infty}\|x\|_{\infty}$, and $\|Ax\|_{1}\leq\interleave A\interleave_{1}\|x\|_{1}$. Moreover, we have $|x'Ay|\leq\interleave A\interleave_{\infty}\|x\|_{\infty}\|y\|_{1}\leq n\interleave A\interleave_{\infty}\|x\|_{\infty}\|y\|_{\infty}$.\footnote{These results can be found in Horn and Johnson (1985, Section 5.6) or proved similarly.} Finally, let $0<C<\infty$ denote a universal constant.

Proofs in Sections (ref) and (ref)

Proofs in Section (ref)

\protectConsistency of $\hat{\gamma}$

Theorem (ref) is proved based on several lemmas to be presented later in this section. Table (ref) states the relationships between Theorem (ref) and these lemmas.

Asymptotic Distribution of $\hat{\gamma}$

Theorem (ref) is proved based on several lemmas presented in Section (ref) and later in this section. Table (ref) states the relationships between Theorem (ref) and these lemmas.

\protectExamples of the Adjacency Matrix

In this section, we verify Assumption (ref) for several adjacency matrices that are commonly used in the literature.