Vincent Boucher
email: vincent.boucher@ecn.ulaval.ca

Yann Bramoullé
email: yann.bramoulle@univ-amu.fr

Binary Outcomes and Linear Interactions †

Keywords: Binary Outcomes, Linear Probability Model, Peer Effects, Econometrics of Games

first showed that binary outcomes are compatible with linear econometric models of interactions. This key insight was unduly discarded by the literature on the econometrics of games.

. Our reanalyses showcase the main interests of the linear framework and suggest that the estimations in these two studies suffer from endogeneity problems.

Introduction

In many contexts, researchers are interested in estimating interactions in agents' decisions and outcomes. A teenager's smoking may depend on whether their friends smoke. A firm's entry into a market may depend on the entry of its competitors. Social and strategic interactions likely play a key role in many important issues, including health, academic achievement, public good provision, consumption and imperfect competition. 1Obtaining credible causal estimates of social and strategic interactions, however, requires addressing some formidable econometric challenges. With binary outcomes, in particular, simultaneity in the behavior of interacting agents may yield multiple equilibria, see [START_REF] Brock | Discrete choice with social interactions[END_REF] and [START_REF] Tamer | Incomplete simultaneous discrete response model with multiple equilibria[END_REF]. Addressing multiplicity is a central objective of the econometrics of games, surveyed in [START_REF] Bajari | Game theory and econometrics: A survey of some recent research[END_REF] and De [START_REF] Paula | Econometric analysis of games with multiple equilibria[END_REF]. In the past 30 years, researchers have made great progress on this issue and have developed econometric frameworks that can, in principle, be used to analyze models with multiple equilibria. The ability to account for multiplicity, however, often comes at a significant cost in terms of practical implementation. At this stage, estimating interactions in binary outcomes under multiplicity may be computationally challenging, may require unrealistic amounts of data and cannot be done while controlling for unobserved heterogeneity with standard fixed effects.

To address these concerns, we propose to revisit linear models of interactions in binary outcomes. [START_REF] Heckman | A simultaneous equations linear probability model[END_REF] showed that binary outcomes are compatible with classical econometric models where an agent's outcome depends linearly on others' outcomes. This key insight, however, was discarded by the literature on the econometrics of games,2 perhaps because of a belief that these models lack microfoundations.

We show that this belief is unfounded and argue that this neglect is undue, given the well-known advantages of linear models. Estimation of linear models is straightforward, they have minimal data requirements, and these models can easily handle fixed effects. Moreover, we show that they can be embedded in models with multiple equilibria. We thus believe that linear models of interactions in binary outcomes should be rehabilitated, even if only as an intermediate step within a deeper analysis.

We consider a general model of linear interactions in binary outcomes. The model notably nests linear-inmeans models of peer effects in networks [START_REF] Bramoullé | Identification of peer effects through social networks[END_REF] and linear models of entry games [START_REF] Jovanovic | Observable implications of models with multiple equilibria[END_REF]. We develop our analysis in four stages. We first build on [START_REF] Heckman | A simultaneous equations linear probability model[END_REF] and characterize when this model is well defined (Theorem 1). Errors must have a specific discrete structure, imposed by the binary nature of the outcomes. The model then inherits well-known properties of linear interaction models with continuous outcomes. It generically has a unique solution, and identification is characterized by standard rank conditions, see Wooldridge (2010, Section 9), [START_REF] Bramoullé | Identification of peer effects through social networks[END_REF].

overidentification tests, and cannot handle large sets of fixed effects. In [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF], we can include fixed effects at the school-grade level, a natural feature missing from the original analysis. With or without these fixed effects, linear estimates of endogenous peer effects are positive and significant, as observed in the original study. The joint validity of the instruments is, however, strongly rejected by overidentification tests.

For [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF], we can include airline fixed effects, which were also absent from the original analysis. Results from our reanalysis are qualitatively different from the original results. Estimates of strategic interactions between airlines are generally positive and significant in a linear framework, whereas they are negative and significant in [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF]. Absent a proper means of testing one specification versus another, we can only speculate on the causes behind these differences. 3 As in many studies in the econometrics of games, the first step of [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF]'s estimation method is to obtain nonparametric estimates of conditional choice probabilities. These estimates capture how the probabilities of all possible market outcomes depend on all covariates. The estimation of conditional choice probabilities suffers from a well-known curse of dimensionality in practice, see [START_REF] Andrews | Confidence regions for parameters in discrete games with multiple equilibria, with an application to discount chain store location[END_REF]. This problem appears to be severe in [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF]'s application, as researchers must obtain nonparametric estimates of 63 functions of 20 variables with only 2, 742 observations. In contrast, estimates of conditional choice probabilities are not required to estimate linear models. More generally, linear estimations are not affected by a curse of dimensionality, and we suspect this plays an important role in explaining the different results. In addition, the joint validity of the instruments is also strongly rejected by overidentification tests. This suggests that the original analyses of [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF] and [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF], as well as our reanalyses, suffer from endogeneity problems.

Our analysis contributes, first, to a large and still-expanding literature on peer effects. 4 Early studies focused on group interactions. Population is then partitioned into groups; agents interact with all other members of their group and with no members of another group. [START_REF] Brock | Discrete choice with social interactions[END_REF] first proposed a microfounded econometric framework to analyze peer effects on binary outcomes. They consider a setup of incomplete information under group interactions. They show that the model has a unique equilibrium under moderate interactions and multiple equilibria under strong interactions. [START_REF] Soetevent | A discrete-choice model with social interactions: with an application to high school teen behavior[END_REF] analyze peer effects on binary outcomes under complete information and group interactions. They find that the game typically has a large number of equilibria. They propose a simulated maximum likelihood estimator based on the assumption that all Nash equilibria are equally likely. [START_REF] Nakajima | Measuring peer effects on youth smoking behaviour[END_REF] also analyzes peer effects on binary outcomes under complete information and group interactions. He considers a stochastic Markov process where agents sequentially and myopically play a best response. He assumes that the likelihood function is equal to the steady-state distribution of this process. Recent studies consider more complex network interactions. 5 Li and Zhao (2016) adapt partial identification approaches under complete information to the analysis of peer effects in networks and binary outcomes. [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF] extend the incomplete information framework of [START_REF] Brock | Discrete choice with social interactions[END_REF] to networks. They show that uniqueness holds under moderate interactions and propose an iterative simulated maximum likelihood estimator based on a subroutine that repeatedly computes the solution of a high-dimensional nonlinear fixed-point system.

All these studies develop nonlinear frameworks to analyze peer effects on binary outcomes.

In contrast, we show that linear models of peer effects, traditionally used to study continuous outcomes [START_REF] Manski | Identification of endogenous social effects: The reflection problem[END_REF]), can also be used for binary outcomes. We show that these models can be given proper microfoundations and maintain key properties when applied to binary outcomes. This concerns, in particular, the identification results and ideas of [START_REF] Bramoullé | Identification of peer effects through social networks[END_REF], which exploit holes in the network structure to solve the reflection problem. 6 We revisit the empirical analysis of peer effects and teenage smoking of [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF]. We obtain similar estimates of endogenous peer effects through a much simpler estimation procedure. In addition, we can control for school-grade fixed effects and verify whether the network-based instruments pass overidentification tests, two features absent from the original study.

Our analysis contributes, more generally, to the literature on the econometrics of games. Since the early work of [START_REF] Jovanovic | Observable implications of models with multiple equilibria[END_REF] and [START_REF] Bjorn | Modèles d'équations simultanées pour variables endogènes fictives: Une formulation par la théorie des jeux avec application à la participation au marché du travail[END_REF], researchers have made great progress on the empirical analysis of models with multiple equilibria. Applied researchers who wish to estimate interactions in binary outcomes under multiplicity can, notably, specify a flexible selection mechanism dependent on estimated parameters [START_REF] Bajari | Identification and estimation of a discrete game of complete information[END_REF], adopt a partial identification approach under complete information [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF], or assume that the same equilibrium is played across games under incomplete information [START_REF] Aguirregabiria | Sequential estimation of dynamic discrete games[END_REF]. Preference shocks are generally assumed to be logistically or normally distributed. Different assumptions yield different nonlinear econometric frameworks; a common first step is often to obtain flexible estimates of conditional choice probabilities. Despite this important methodological progress, however, two features may limit the usefulness of these approaches for obtaining credible causal estimates. First, and depending on the context, econometric frameworks developed to handle multiplicity may have unrealistic data requirements. We argue that this applies to the application in [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF]. Second, introducing unobserved heterogeneity in these frameworks is generally impractical or unfeasible. Due to the incidental parameter problem, nonlinear estimators usually cannot handle fixed effects, whose numbers grow at the same rate as sample size. While they could, in principle, account for a finite number of fixed effects, their introduction further intensifies the data requirements.7

To help reconcile the structural and reduced-form approaches, we propose to rehabilitate linear models of interactions in binary outcomes. While linear models cannot account for multiplicity, they provide a natural benchmark. They are very easy to implement, have minimal data requirements, and can easily handle fixed effects. They have transparent identification conditions and well-understood overidentification tests. [START_REF] Heckman | A simultaneous equations linear probability model[END_REF] first showed that classical linear models of simultaneous equations are compatible with binary endogenous variables. We build on this early work 8 and provide the first systematic analysis of the game-theoretic microfoundations of linear models of interactions. We show that they can be obtained as equilibria of games of complete or incomplete information with linear utilities and we characterize compatible preference shocks. Preference shocks can be independent and continuously distributed under complete information and iid and uniformly distributed under incomplete information. Discarding linear models of interactions simply because they do not emerge as equilibria of game-theoretic models with logistic or normal preference shocks seems rather extreme. Microfoundations and preference shocks are, in any case, not identified from the data generating process without making strong assumptions. We thus believe that linear models have a legitimate place in the toolkit of applied researchers interested in interactions in binary outcomes.

Finally, our analysis contributes to the literature on the econometrics of discrete variables and on linear probability models in particular. Researchers hold diverse views on the use of linear models to analyze binary outcomes. Following [START_REF] Angrist | Mostly Harmless Econometrics: An Empiricist's Companion[END_REF], applied economists focused on obtaining credible causal estimates generally estimate linear probability models. In contrast, researchers who are more theoretically or methodologically oriented often adopt models based on latent variables. This preference is perhaps explained by a belief that linear probability models lack microfoundations. This belief is incorrect. [START_REF] Heckman | Linear probability models of the demand for attributes with an empirical application to estimating the preferences of legislators[END_REF] provide microfoundations for linear probability models with a single decision-maker. A main contribution of our analysis is to clarify the game-theoretic microfoundations of linear probability models with outcome interactions. Notably, we show that linear models of interactions in binary outcomes are particular cases of models with latent variables. Given the many advantages of a linear framework, we believe that it could become a natural benchmark in empirical studies of binary outcome interactions.

The paper proceeds as follows. We present the econometric framework in Section 2 and analyze its microfoundations in Section 3. We propose estimators and discuss their properties in Section 4. We revisit existing studies of interactions in binary outcomes in Section 5 and conclude in Section 6.

Econometric framework

A researcher has data on n agents and analyzes interactions affecting a binary outcome. Let y i ∈ {0, 1} denote agent i's outcome. Let y ∈ R n denote the vector of outcomes and x a matrix containing all observables.

heterogeneity when the unobservables have finite support. This is a potentially promising result. Its empirical applicability has not yet been demonstrated, however, and the assumption of finite support represents, in any case, a significant restriction.

8 One difference is that, unlike [START_REF] Heckman | A simultaneous equations linear probability model[END_REF], we do not impose linearity in observables.

For any well-defined data generating process on (x, y), we find it useful to distinguish between two types of stochastic terms. By convention, errors are defined directly from the data generating process, whereas preference shocks appear in underlying microfoundations. We define reduced-form errors as ν i = y i -E(y i |x),

leading to y i = E(y i |x) + ν i and E(ν i |x) = 0.
As is well known, the binary nature of the outcome imposes strong restrictions on the data generating process. In particular, P(y i = 1|x) = E(y i |x), and the reduced-form error ν i is a binary, Bernouilli stochastic variable: ν i = 1 -E(y i |x) with probability E(y i |x) and -E(y i |x) with probability 1 -E(y i |x).9 Thus, while preference shocks are often assumed to be continuously distributed, reduced-form errors are always binary.

For instance, consider a probit model without interactions. Let 1

(Y i ≥ 0) = 1 if Y i > 0 and 0 if Y i < 0.
Then, y i = 1(x i γ + e i ≥ 0), and preference shock e i follows a normal distribution N (0, 1) with cdf Φ, while reduced-form errors are such that ν i = 1 -Φ(x i γ) with probability Φ(x i γ) and -Φ(x i γ) with probability

1 -Φ(x i γ).
In our analysis, we consider the following general model of linear interactions

y i = f i (x, θ) + j β ij y j + ε i , (1)
under the exogeneity assumption, E(ε i |x) = 0. Outcome y i is affected by observables through function f i and parameters θ and by others' outcomes through linear interactions j β ij y j . Let β denote the interaction matrix, where β ii = 0, and β ij can potentially have any sign.

Whether there exists an error structure such that equation (1) holds with binary outcomes is not immediate. The interaction term j β ij y j can take up to 2 n-1 values and partly determines y i , which can take only 2 values. In this Section, we clarify the conditions under which this model is well defined. We analyze underlying microfoundations in the next Section.

Model (1) nests two important cases of interactions in binary outcomes: peer effects and entry games.

First, consider the benchmark linear-in-means model of peer effects in networks, see [START_REF] Bramoullé | Identification of peer effects through social networks[END_REF]. For each agent i, the researcher observes characteristics x i and set of peers N i . Peer relationships form a binary directed network. Let d i = |N i | denote i's degree, i.e., the number of peers of i.

Assume that no agent is isolated, d i > 0.10 Define G as the linear-in-means matrix of interactions:

g ij = 1 di if j ∈ N i
and 0 otherwise. The linear-in-means model of peer effects in networks can be written as

y i = α + x i γ + j g ij x j δ + β j g ij y j + ε i , (2)
under the assumption that E(ε i |x, G) = 0. In this model, outcomes can be affected by individual characteristics (individual effects, γ), peers' characteristics (contextual peer effects, δ), and peers' outcomes (endogenous peer effects, β). Model (2) is a case of model ( 1) with θ = (α, γ, δ), f i (x, G, θ) = α + x i γ + j g ij x j δ, and the interaction matrix β = βG. In this model, the structure of the interactions G is known but not their extent, β.

The assumption E(ε i |x, G) = 0 means that characteristics and the network are strictly exogenous and the problem of correlated effects has been solved. 11 This framework has generally been applied to study continuous outcomes. We show below that it is also compatible with binary outcomes.

Our second main application is about entry games. These games have been introduced to study competition between a small number of firms in a large number of markets. Firm i's decision to enter market m may depend on characteristics of the firm and the market and on the entry decisions of its competitors. In the literature, researchers generally consider nonlinear models of entry games, e.g., [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF].

In contrast, we consider the following linear model. Let y im ∈ {0, 1} denote the entry of firm i into market m. Then,

y im = α + x im γ + z m λ + j β ij y jm + ε im , (3)
under the assumption that E(ε im |x, z) = 0. 12 Firm i's entry depends on firm-market characteristics x im , on market characteristics z m , and on other firms' entries j β ij y jm . Observe that model ( 3) is a case of model

(1) applied to firm-market observations with θ = (α, γ, λ), f im (x im , z m , θ) = α + x im γ + z m λ, and under the assumption that the interaction matrix β is constant across markets.

We now characterize when binary outcomes are compatible with linear interactions, following arguments in [START_REF] Heckman | A simultaneous equations linear probability model[END_REF]. In what follows, the notation x refers to a matrix containing all observables, including the network in a peer-effect application and market characteristics in an entry game. Note that equation (1) defines a fixed-point system in the outcome profile y. In matrix notations,

y = f + βy + ε.
We assume that the matrix I-β is invertible. This holds generically and implies that this system has a unique solution. The reduced form of model (1), expressing outcomes y as a function of observables, parameters and errors is equal to

y = (I -β) -1 f + (I -β) -1 ε. Let P i = [(I -β) -1 f ] i and ν i = [(I -β) -1 ε] i . Here, P i = E(y i |x) = P(y i = 1|x
) is the conditional expected outcome and hence must lie between 0 and 1. Then, ν i = y i -E(y i |x) is the reduced-form error of the data generating process.

We have y i = P i + ν i , and y i = 1 and ν i = 1 -P i with probability P i , while y i = 0 and ν i = -P i with probability 1 -P i . Then, ε = (I -β)ν, leading to our first result.

11 [START_REF] Bramoullé | Peer effects in networks: a survey[END_REF] show that exogeneity of either a characteristic or the network can be sufficient to identify peer effects in model (2).

12 Nonlinear models of entry games generally assume that y * im = α + x im γ + zmλ + j β ij y jm + e im , where y * im is a latent variable such that y im = 1{y * im ≥ 0}.

Theorem 1. Assume that Iβ is invertible and that ∀i,

P i = [(I -β) -1 f ] i ∈ [0, 1].
Outcomes in the unique solution to model (1) are binary, y i ∈ {0, 1}, if and only if

ε i = ν i - j β ij ν j ,
where ν i = -P i with probability 1 -P i and 1 -P i with probability P i .

Theorem 1 clarifies the conditions under which binary outcomes are compatible with linear interactions.

The structural errors appearing in model (1) must have a specific discrete structure, induced by the binary nature of the outcomes and the linear framework. Note that these structural errors depend directly on the data generating process. In the next Section, we analyze the microfoundations of model ( 1) and find that preference shocks in underlying microfoundations can have very different properties. For instance, they can be iid and continuously distributed, see Theorem 3.

As in any data generating process with binary outcomes, reduced-form errors are binary and generally depend on observables and parameters. They can be correlated, and possible correlation patterns also depend on observables and parameters. 13 In the absence of interactions and when f i is linear, model (1) reduces to a standard linear probability model. In the presence of interactions, structural errors ε i usually take more than two values. In general, ε i can take up to 2 n values. Structural errors are also correlated even when reduced-form errors are uncorrelated. In model ( 2) of peer effects in networks, ε i can take up to 2 di+1 values.

Denote d(i, j) as the distance between i and j in the network, i.e., the number of links in a shortest path connecting i to j (or ∞ if there is no path connecting i to j). Then, ε i and ε j are generally correlated if i and j are peers or peers of peers. In contrast, cov(ε i , ε j ) = 0 if d(i, j) > 2 and reduced-form errors are uncorrelated.

Thus, the stochastic structure of structural errors generally depends on the network of interactions.

Probabilities must of course lie between 0 and 1. This is guaranteed in model ( 1) with binary outcomes when for any i,

[(I -β) -1 f ] i ∈ [0, 1]
. This condition depends both on interactions β and expected outcomes absent of interactions f . For instance, when n = 2 and f 1 , f 2 ∈]0, 1[, it holds for moderate interactions of any sign and for large negative interactions.14 Moreover, in any application we can easily compute the proportion of observations for which the estimated probability lies between 0 and 1. As with the standard linear probability model, this provides a simple measure of whether the estimated model is appropriate. We report these proportions in our estimations in Section 5.

A key property of the linear framework is that if Iβ is invertible, there is a unique solution to the fixedpoint system defined by model (1). In other words, the econometric model is both coherent and complete, see [START_REF] Tamer | Incomplete simultaneous discrete response model with multiple equilibria[END_REF] and [START_REF] Lewbel | The identification zoo: Meanings of identification in econometrics[END_REF]. In contrast, almost all existing studies of interactions in binary outcomes rely on a latent variable model of the following kind:

y i = 1(f i (x, θ) - 1 2 + j β ij y j + e i ≥ 0). (4)
The econometric model defined by equation ( 4) is generally incoherent or incomplete. The fixed-point system can have multiple solutions, leading to an incomplete model. The fixed-point system can also have no solution,

leading to an incoherent model. Interestingly, we show in the next Section that linear model ( 1) is a particular case of latent variable model ( 4) for specific assumptions on preference shocks e i .

Furthermore, identification in a linear framework follows from well-known results. For the linear-in-means model of peer effects in networks, in particular, the identification results of [START_REF] Bramoullé | Identification of peer effects through social networks[END_REF] apply when outcomes are binary under the assumptions of Theorem 1. This holds because their analysis does not impose restrictions on the nature of the outcome or on the error terms, other than the

exogeneity assumption E(ε i |x, G) = 0.
Corollary 1. [START_REF] Bramoullé | Identification of peer effects through social networks[END_REF] Consider the linear-in-means model of peer effects in networks with binary outcomes and under the assumptions of Theorem 1. Assume δ + βγ = 0. The model is identified if the matrices I, G, and G 2 are linearly independent.

Identification notably holds when the network's diameter is greater than or equal to 2 or under group interactions when groups have different sizes.

With entry games, model ( 3) is generally identified when the entry of firm i is affected by some firmmarket characteristic x im that does not directly affect the entry of other firms, a standard assumption in the literature. The entry of firm j can then be instrumented by x jm in equation ( 3), see e.g., [START_REF] Bajari | Game theory and econometrics: A survey of some recent research[END_REF]. More generally, model (1) defines simultaneous linear equations in outcomes. When the functions f i are also linear, classical rank conditions for identification apply, see e.g., Wooldridge (2010, Section 9).

Microfoundations

We now analyze the microfoundations of model ( 1). We assume in this Section that the outcome

y i ∈ {0, 1}
is a choice of agent i. We consider games of complete or incomplete information under the assumptions of Theorem 1 that Iβ is invertible and

P i = [(I -β) -1 f ] i ∈ [0, 1].
We adopt standard assumptions of the literature on the econometrics of games. We consider a classical random utility framework [START_REF] Mcfadden | Conditional logit analysis of qualitative choice behavior[END_REF]. Agent i derives utility v i (y i , y -i ) from playing y i when other agents play y -i . Utility v i (y i , y -i ) is the sum of deterministic utility u i (y i , y -i ) and of preference shock

e i (y i ). Let ∆u i (y -i ) = u i (1, y -i ) -u i (0, y -i
) denote the relative deterministic utility of playing 1, and let e i = e i (1) -e i (0) denote the relative preference shock in the utility of playing 1. Under complete information, the deterministic utilities and preference shocks of all agents are common knowledge. Under incomplete information, the deterministic utilities of all agents and the distribution of preference shocks are common knowledge. Agent i observes the realization of her own shock e i but not the realization of others' shocks e -i .

Existing studies of interactions in binary outcomes generally assume that relative utility is linear in others' actions:

∆u i (y -i ) = f i - 1 2 + j β ij y j . (5)
Interestingly, linear relative utility ( 5) is consistent with quadratic utility u i (y i , y -i ) = y i f i -1 2 y 2 i + j β ij y i y j . This quadratic utility has been well studied in network games with continuous actions, see [START_REF] Ballester | Who's who in networks. wanted: The key player[END_REF] and [START_REF] Bramoullé | Strategic interaction and networks[END_REF], and has been proposed as a microfoundation of the econometric model of peer effects with continuous outcomes, see e.g., [START_REF] Davezies | Identification of peer effects using group size variation[END_REF]. This provides a common game-theoretic framework for binary and continuous action games.

Complete information

Under complete information, outcomes y are assumed to form a Nash equilibrium of the game for any realization of preference shocks e. This means that for every agent i, if

y i = 1 then ∆u i (y -i ) + e i ≥ 0, whereas if y i = 0 then ∆u i (y -i ) + e i ≤ 0. Therefore, y i = 1(∆u i (y -i ) + e i ≥ 0
), and the system of equilibrium conditions is equivalent to a latent variable model with interactions, such as ( 4).

An important early finding is that multiple equilibria necessarily appear in games of complete information with linear relative utility when interactions are positive and preference shocks have full support over R, see [START_REF] Tamer | Incomplete simultaneous discrete response model with multiple equilibria[END_REF] and De [START_REF] Paula | Econometric analysis of games with multiple equilibria[END_REF]. To see why, consider the linear relative utility (5). Note that

v i (1, 0) -v i (0, 0) = f i -1
2 + e i . Then, (0, 0, ..., 0) is an equilibrium iff for every i, e i ≤ - and (1, 1, ..., 1) is an equilibrium iff for every i, e i ≥ -(f i -1 2 )j β ij . Therefore, both 0 and 1 are Nash equilibria iff for every i,

(f i -1 2 ). Similarly, v i (1, 1)-v i (0, 1) = f i -1 2 + j β ij +e i ,
-(f i -1 2 ) -j β ij ≤ e i ≤ -(f i -1 2 ).
If for every i, j β ij > 0 and the density of e i is everywhere strictly positive, then there is a strictly positive probability of multiple Nash equilibria. Multiplicity appears when preference shocks take intermediate values.

As our second main result, we characterize when the linear model of interactions can be microfounded with linear relative utilities. More precisely, denote by y * the unique solution to equation (1) under the conditions on parameters and errors described in Theorem 1. We provide necessary and sufficient conditions on preference shocks under which y * is a Nash equilibrium of the game with linear relative utilities and sufficient conditions under which this is the unique Nash equilibrium in dominant strategies.

Theorem 2. The unique solution to the linear model of interactions (1) is a Nash equilibrium of the game with linear relative utilities (5) and preference shocks e i if and only if

ν i > 0 ⇒ e i ≥ 1 2 -P i -j β ij ν j and ν i < 0 ⇒ e i ≤ 1 2 -P i -j β ij ν j .
It is the unique Nash equilibrium in dominant strategies if

ν i > 0 ⇒ e i > 1 2 -P i -j β ij ν j + j |β ij | and ν i < 0 ⇒ e i < 1 2 -P i -j β ij ν j -j |β ij |.
Theorem 2 describes precisely how preference shocks must be related to reduced-form errors in gametheoretic microfoundations to model (1). Intuitively, preference shocks must be large enough in situations where the agent plays 1 and low enough in situations where the agent plays 0. Theorem 2 has several noteworthy implications.

First, it shows that the linear model of interactions ( 1) is a particular case of the model with latent variables (4). When preference shocks satisfy the first set of inequalities described in Theorem 2, the linear model corresponds to one solution of the system of equations defined by model ( 4) among many possible solutions. When preference shocks satisfy the second set of inequalities, however, the linear model corresponds to the unique solution and hence the two models are formally equivalent.

Second, it shows that preference shocks in underlying microfoundations are not identified without further assumptions. Thus, very different kinds of preference shocks are consistent with the econometric model.

The linear model of interactions can notably be microfounded with preference shocks that are discrete and correlated, like structural errors, or independent and continuously distributed. To see why, note first that structural errors actually provide admissible preference shocks.15 However, underlying preference shocks can also be independent and continuously distributed. For instance, consider a situation where the ν i 's are uncorrelated. Assume that e i is uniformly distributed on

[-L i -1 + P i , -L i ] ∪ [M i , M i + P i ],
that the e i 's are independent, and that

ν i = 1(e i ≥ 0) for L i , M i ≥ 1 2 + 2 j |β| ij .
A direct application of Theorem 2 shows that in this case, the unique solution to the linear model of interactions is the unique Nash equilibrium of the corresponding game.

A third implication is that, in some cases, the unique solution to the linear model of interactions is the unique Nash equilibrium in dominant strategies. This happens when preference shocks are sufficiently dispersed: sufficiently high when high and sufficiently low when low. Preference shocks then do not take intermediate values, bypassing the multiplicity domain.

An intriguing consequence is that even in the presence of multiple Nash equilibria, the linear model of interactions becomes the unique equilibrium following specific changes in preference shocks. The linear model of interactions is then, in a sense, the only robust Nash equilibrium. We next develop these arguments formally. In our next result, we show that this reasoning holds for any deterministic utility and preference shocks.

Proposition 1. Suppose that the unique solution to model (1) is a Nash equilibrium of the game of complete information with deterministic utilities U i and preference shocks e i . Consider preference shocks e i where e i = e i + M i if ν i > 0 and e i -L i if ν i < 0 and L i , M i ≥ 0. Then, the unique solution to model (1) remains a Nash equilibrium of the game of complete information with deterministic utilities U i and preference shocks e i , and it is the unique Nash equilibrium in dominant strategies when L i and M i are sufficiently large.

Proposition 1 formalizes a natural idea: increasing preference shocks in situations where the agent plays 1 and decreasing preference shocks in situations where the agent plays 0 can only increase the incentives to play these actions. Because of the discreteness of the strategy space, we further show that these heightened incentives yield dominant strategies when these increases and decreases are sufficiently high. Say that a Nash equilibrium y for preference shocks e is robust to increases in shock dispersion when y remains an equilibrium for preference shocks e i where

e i = e i + L i if ν i > 0 and e i -M i if ν i < 0 and L i , M i ≥ 0. 16
Corollary 2. Suppose that the unique solution to model ( 1) is a Nash equilibrium of the game of complete information with deterministic utilities U i and preference shocks e i . Then, this is the unique equilibrium robust to increases in shock dispersion.

To summarize, outcomes in the linear model of interactions correspond to a Nash equilibrium of a game of complete information with linear relative utilities under conditions on preference shocks, which we characterize. These preference shocks are not identified and could be, for instance, independent and continuously distributed. Furthermore, outcomes in the linear model correspond to the unique Nash equilibrium that is robust to increases in shock dispersion. Together, these results provide game-theoretic microfoundations to the linear model of interactions under complete information. We next look at incomplete information.

Incomplete information

Under incomplete information, agent i observes her preference shock e i but not others' shocks e -i . Outcomes y are assumed to form a Bayes-Nash equilibrium of the game.

A first remark is that when actions are dominant under complete information, they are also dominant under incomplete information. Therefore in situations described in Theorem 2 and Proposition 1, where uniqueness in dominant strategies holds under complete information, the corresponding game of incomplete information also has a unique Bayes-Nash equilibrium in dominant strategies. As shown above, these results involve preference shocks with a dispersed distribution and non-convex support. This raises the question of whether shock dispersion and support non-convexity are necessary in microfoundations of the linear model of interactions.

In our third main result, we show that these features are not necessary. Under incomplete information and when reduced-form errors are uncorrelated, the linear model of interactions can be microfounded with preference shocks that are iid and uniformly distributed over an interval. Moreover, if preference shocks are independent between agents and independent of observables, they must be uniformly distributed.

Theorem 3. Assume that reduced-form errors ν i are uncorrelated. Consider the game of incomplete information with linear relative utilities (5) and preference shocks e i with e i ⊥ e -i , x. The unique solution to the linear model of interactions (1) is the unique interior Bayes-Nash equilibrium for all possible β, f if and only if the e i 's are iid and uniformly distributed over [-1 2 , 1 2 ]. If in addition ∀i j |β ij | < 1, this is the unique Bayes-Nash equilibrium.

Our proof in Appendix relies on the well-known property that a uniform distribution is the only distribution with a linear cumulative density function. 17 The uniqueness condition comes from a classical contraction mapping argument,18 and we show in Appendix that the game can have multiple Bayes-Nash equilibria when this condition is not satisfied. The uniqueness condition is easy to verify for estimated parameters, and we will see that it always holds in the empirical applications in Section 5. The fact that uniqueness holds under incomplete information when interactions are moderate is well known, see e.g., [START_REF] Brock | Discrete choice with social interactions[END_REF] and [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF]. However, existing applications typically assume that preference shocks have full support, thereby leading to nonlinear econometric models. For instance, Lee, Li, and Lin (2014) consider a logit framework, see Section 5.1. Theorem 3 clarifies the conditions under which a linear model of interactions in binary outcomes can be obtained as a Bayes-Nash equilibrium of a game of incomplete information.

Estimators

In this Section, we propose two simple estimators to analyze data with binary outcomes generated by model (1). We consider a many-groups asymptotic framework with independent groups of bounded size. This corresponds to a many-network asymptotic in a peer effect setting19 or to a many-market asymptotic for entry games. The number of groups thus goes to infinity with sample size, and consistency and the asymptotic normality of extremum estimators are guaranteed under standard assumptions, see, e.g., [START_REF] Cameron | Microeconometrics: methods and applications[END_REF], Section 5.3. 20 We explore the small sample properties of our proposed estimators through Monte Carlo simulations and apply these estimators to real data in Section 5.

We consider the following variant of model (1). We assume that f is linear in observables and that the interaction matrix β depends linearly on a fixed number of parameters to estimate. Formally, f = Xθ and

β = K k=1 β k G k ,
where K is finite and independent on the sample size. This yields

y = Xθ + K k=1 β k G k y + ε. ( 6
)
While not essential, the linearity of f is sufficient for most applications and facilitates exposition. The assumption on the interaction matrix ensures that the number of parameters to estimate does not grow with sample size 21 and allows us to provide a common framework for our two applications. Denote by

β (K) = (β 1 , ..., β K )
and similarly for G (K) .

In the benchmark linear-in-means model of peer effects in networks, there is only one interaction parameter to estimate. In that model, K = 1 and β = βG, where G ij = 1/d i if i and j are linked and 0 otherwise.

Extended versions of the model with heterogeneous peer effects, as in [START_REF] Nakajima | Measuring peer effects on youth smoking behaviour[END_REF], [START_REF] Soetevent | A discrete-choice model with social interactions: with an application to high school teen behavior[END_REF], and [START_REF] Dieye | Gender peer effects heterogeneity in obesity[END_REF], are also cases of model ( 6). For instance, when men and women can be differentially affected by their male and female peers, there are K = 4 interaction parameters to estimate. In that case,

β = β F F G F F + β F M G F M + β M M G M M + β M F G M F , where, for example, G F M
models the structure of interactions between female individuals and male peers.

In the linear model of entry game, there are K = n(n -1) interaction parameters to estimate, where n is the number of firms competing across markets. Here, β = i,j β ij G ij , where G ij has 1 at entry (i, j) and 0's elsewhere. In the next Section, we estimate a version of this model where the entry of a firm has a common impact on the entry of other firms, i.e., β ij = β j . In that version, there are K = n parameters to estimate, and β = N j=1 β j G j , where G j = i G ij has 1's in its jth column and 0's elsewhere. We assume that the model to be estimated is identified, see Section 2. The linear-in-means model of peer effects is identified under conditions reported in Corollary 1. Similar conditions hold when peer effects are heterogeneous, see e.g., [START_REF] Dieye | Gender peer effects heterogeneity in obesity[END_REF]. In the linear model of entry games, identification holds under the exclusion restrictions that some characteristics of firm j, affecting its profit, do not affect the profit of firm i.

Our first proposed estimator is a Two-Stage Least Squares (2SLS) estimator, building on the 2SLS estimation strategies proposed by [START_REF] Kelejian | A generalized spatial two-stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances[END_REF] and [START_REF] Bramoullé | Identification of peer effects through social networks[END_REF].

Since E(ε|X, G (K) ) = 0, we have

E(G k y|X, G (K) ) = G k Xθ + K l=1 β l G k G l (I -β) -1 Xθ.
In particular, variables in G k X that are not already in X provide natural instruments for G k y in equation ( 6).

In the linear-in-means model (2), we have X = [1, x, Gx] and hence GX = [G1, Gx, G 2 x], so G 2 x can be used as an instrument for Gy. This instrument is valid under the conditions described in Corollary 1. Intuitively, characteristics of peers of peers who are not peers affect individual outcome only through their impact on peers' outcomes, see [START_REF] Bramoullé | Identification of peer effects through social networks[END_REF]. In the linear model of entry game (3), for any market m, we have

X m = [1, x m , z m ] and, hence, G ij X m = [G ij 1, G ij x m , G ij z m ]. Here, [G ij y] i = y j can be instrumented by [G ij x m ] i = x jm .
The impact of the entry of firm j on i's entry can be instrumented by the characteristics of firm j. 21 Peng (2019) proposes a penalized regression strategy that depends on the weaker assumption that K ≤ c √ n ln n for some c as the sample size n goes to infinity. However, his analysis depends on the errors being independent and sub-Gaussian.

The validity of this IV strategy relies on the exogeneity condition E(ε i |X, G (K) ) = 0 but not on the specific structure of the errors. This strategy is thus valid when errors have the discrete structure uncovered in Theorem 1. Error structure may of course matter for inference. In particular, errors in the linear-in-means model (2) with binary outcomes are heteroscedastic and correlated among peers and peers of peers, see Section 2. In a many-groups asymptotic framework, we propose to use group-level cluster-robust standard errors for inference since they allow for arbitrary within-group correlations, see [START_REF] Cameron | A practitioner's guide to cluster-robust inference[END_REF]. In other asymptotic frameworks, spatial Heteroscedastic and Autocorrelation Consistent (HAC) variance estimators could be appropriate (e.g., [START_REF] Conley | Gmm estimation with cross sectional dependence[END_REF], [START_REF] Kelejian | Hac estimation in a spatial framework[END_REF], [START_REF] Leung | Inference in models of discrete choice with social interactions using network data[END_REF]). 22Our second proposed estimator is a Nonlinear Least Squares (NLS) estimator, exploiting the structure of the reduced-form equations. Recall that y i = P i + ν i with E(ν i |x) = 0, and P(β

(K) , θ) = (I - K k=1 β k G k ) -1 Xθ.
The model can thus be consistently estimated by the following Nonlinear Least Squares estimator:

( β(K) , θ) = arg min β (K) ,θ [y -P(β (K) , θ)] [y -P(β (K) , θ)]. (7)
Conveniently, the estimator in ( 7) can be concentrated around β (K) . Indeed, taking the first-order conditions with respect to θ, conditional on β (K) , we obtain

θ(β (K) ) = [Z (β (K) )Z(β (K) )] -1 Z (β (K) )y,
where Z(β (K) ) = (I-

K k=1 β k G k ) -1 X.
Substituting in the objective function (7), we obtain the concentrated NLS estimator:

β(K) = arg min β (K) y I -Z(β (K) )[Z (β (K) )Z(β (K) )] -1 Z (β (K) ) y. ( 8
)
While the objective function in (8) may not be convex, numerical optimization of the concentrated NLS is relatively straightforward when K is small.

To analyze interactions in binary outcomes, the 2SLS and NLS estimators are natural and easy to implement with standard statistical software. There are, of course, other estimators that can be used to estimate model ( 6) and its variants. Following [START_REF] Bramoullé | Identification of peer effects through social networks[END_REF], many studies proposed alternative strategies to estimate model (2) with continuous outcomes. Some of these strategies can be applied, or extended, to binary outcomes. Notably, these include moment-based estimators (e.g., [START_REF] Kelejian | A generalized spatial two-stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances[END_REF] and Lee and Liu (2010)) and different ways to compute instruments (e.g., [START_REF] Kelejian | Estimation of spatial models with endogenous weighting matrices, and an application to a demand model for cigarettes[END_REF]).

In applied studies, researchers often instrument average peers' outcome by the average characteristics among a subset of peers of peers who are not peers, see [START_REF] Nicoletti | The family peer effect on mothers' labor supply[END_REF] and De Giorgi, Frederiksen, and Pistaferri (2020). In contrast, Theorem 1 indicates that quasi-maximum likelihood approaches based on independence and homoscedasticity, as in [START_REF] Lee | Specification and estimation of social interaction models with network structures[END_REF], cannot be used with binary outcomes.

Known estimators developed to estimate standard linear probability models can also be extended to account for social and strategic interactions. For instance, if the ν i 's are uncorrelated, Maximum Likelihood and (feasible) Weighted Nonlinear Least Squares provide valid alternatives. While these estimators are more efficient, their implementation in practice raises empirical and computational issues that will likely be aggravated by interactions. 23 In the absence of interactions, these practical considerations and the fact that efficiency gains appear to be small in practice have led researchers to focus on Ordinary Least Squares, see Section 3.4.1 in [START_REF] Angrist | Mostly Harmless Econometrics: An Empiricist's Companion[END_REF]. Our proposed 2SLS and NLS estimators provide natural counterparts of Ordinary Least Squares in a setup with interactions.

We next compare the small-sample performances of those two estimators using Monte Carlo simulations based on linear-in-means model (2). We let x i ∼ U [0, 1]. Note that setting α, β, γ, δ > 0, with α+β+γ+δ < 1,

ensures that P i ∈ [0, 1] when x i ∈ [0, 1]
. We assume that the population is partitioned into M > 0 groups of size N > 0 such that g ij = 0 whenever i and j belong to different groups. For any two agents i and j belonging to the same group, we let g ij = 1 with a probability p = 0.1. The overall network is thus composed of M = 500 disjoint instances of Erdős-Renyi subnetworks connecting N agents.

Results for two sets of parameters (high and low β), N = 30, M = 100 and 300-corresponding to N M = 3, 000 and 9, 000 observations-are presented in Table 1. Both estimators display moderate smallsample bias. Bias and estimates' dispersion tend to be smaller for the NLS and when β is high. In the Appendix, we report results for the same parameters, N = 20 and 50, M = 500-corresponding to 10, 000 and 25, 000 observations-in Table 5. Bias is then very low in all scenarios. Estimates' dispersion is also low, and dispersion is lower for the NLS and at larger group sizes. Overall, these simulations show that the 2SLS and NLS estimators perform well in small samples of artificial data where binary outcomes are subject to endogenous peer effects. The NLS appears to outperform the 2SLS, especially when β is high.

Nonetheless, the 2SLS estimator has an important appealing feature for empirical applications: it can easily handle group fixed effects. Formally, suppose that α varies across groups r = 1, ..., M . We have

y r = α r 1 + X r θ + K k=1 β k G k,r y r + ε r .
Since group size is bounded, the number of groups-and hence the number of parameters α r to estimategoes to infinity as the same rate as sample size. This is known as the incidental parameter problem and can notably yield inconsistent estimates of θ and β (K) , see [START_REF] Lancaster | The incidental parameter problem since 1948[END_REF] for a review.

For linear models, however, a standard workaround is to rewrite the model in deviation from the group average (see [START_REF] Cameron | Microeconometrics: methods and applications[END_REF], Section 21.6). Let J r = I r -1 r 1 r . We obtain

J r y r = J r X r θ + K k=1 β k J r G k,r y r + J r ε r ,
which does not depend on α r . The 2SLS strategy can then easily be adapted to estimate the model in deviation. 24 This issue may be critical in practice, in contexts where common unobservables may generate spurious correlations in outcomes. Note: For each simulation, M networks are generated among the N = 30 individuals using iid Bernoulli trials with a probability p = 0.1. Thus, the expected number of links for each individual is 0.1(N -1). Values represent the average (standard deviation) of the 1000 simulations.

Applications

We now apply the linear framework to real data. To highlight differences with existing approaches, we revisit two studies: [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF] on peer effects in teenage smoking and [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF] on entry into airline markets. We reanalyze the same data as that of the original studies. These reanalyses show the usefulness of the linear model ( 1) and of our proposed estimators for analyzing interactions in binary outcomes. They illustrate the main advantages of the linear framework: ease of implementation, readily available overidentification tests, and the ability to handle fixed effects. In contrast, existing nonlinear approaches are generally computationally demanding, lack overidentification tests, and cannot handle large sets of fixed effects. These reanalyses also help assess the robustness of existing results.

We find similar qualitative results as those of [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF]. Furthermore, we find that these results are robust to the inclusion of school-grade fixed effects. By contrast, we find opposite qualitative results from those found by [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF]. While we cannot rule out that these differences are induced by the different assumptions, we observe that the econometric method proposed by [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF] suffers from a severe curse of dimensionality. This curse does not affect estimations based on model (1). Moreover, overidentification tests are rejected in both applications, which suggests that these existing analyses suffer from problems of endogeneity.

Peer effects in teenage smoking

In this Section, we revisit the analysis of peer effects in teenage smoking of [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF]. 25 This study is based on data from the National Longitudinal Survey of Adolescent to Adult Health, or Add Health, which provides rich information on the outcomes, behaviors, and characteristics of middle and high school students in the US. The data notably include detailed information on self-reported friendship relationships and has been widely used to analyze peer effects in networks. For the sake of comparison, we focus on the same sample, outcomes, characteristics, and networks as [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF].

The data come from Wave I of the In-School Add Health survey, collected from 1994 to 1995. The sample contains information on the smoking behavior of 74,783 students in 127 schools. [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF] classify a student as a non-smoker if they declared having never smoked or smoked only once or twice in the past twelve months. A student's peers are his or her self-reported friends in the same school and grade.

There are 532 school-grade groups, and hence the overall network is composed of 532 disjoint subnetworks. To analyze peer effects on binary outcomes, [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF] develop an incomplete information framework, extending [START_REF] Brock | Discrete choice with social interactions[END_REF] to networks. Their econometric framework can be microfounded as follows, see also [START_REF] Liu | Simultaneous equations with binary outcomes and social interactions[END_REF]. Assume that agents have linear deterministic relative utilities ( 5) and preference shocks -e i . Assume further that the e i 's are independent and identically distributed with a cumulative distribution function F . The expected relative utility of playing

Summary statistics are presented in

y i = 1 for agent i is equal to E(u i (1, y -i ) -u i (0, y -i )|e i , x) = f i (x, θ) -1 2 + β j g ij E(y j |e i , x
) -e i . Recall, P i = P(y i = 1|x). By independence, E(y j |e i , x) = E(y j |x) = P j in a Bayes-Nash equilibrium. Therefore, Bayes-Nash equilibria are characterized by the fixed-point equation

P i = F (f i - 1 2 + β j g ij P j ), (9)
which corresponds to equation (1) in [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF].

By contrast, model (2) yields P i = f i + β j g ij P j . Theorem 3 clarifies the conditions under which model ( 2) can be viewed as a particular case of this framework: reduced-form errors must be uncorrelated, interactions must be moderate, and preference shocks must be uniformly distributed. [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF] consider a logit framework in their empirical analysis. They assume that the probability that student i smokes tobacco is equal to

P i = exp(α + x i γ + j g ij x j δ + β j g ij P j ) 1 + exp(α + x i γ + j g ij x j δ + β j g ij P j ) . ( 10
)
They propose to estimate the model via an iterative simulated maximum likelihood. Each iteration has two steps: solving for P i 's in the nonlinear fixed-point equation ( 10), conditional on parameter values, and then re-estimating parameters through (simulated) maximum likelihood, conditional on these P i 's. These two steps are repeated until convergence. Their preferred specification includes contextual and endogenous peer effects, fixed effects at the school level, and random effects at the school-grade level. They find evidence of statistically significant, positive endogenous peer effects, with estimates of β ranging from 0.598 to 0.665.

Their approach has two drawbacks. First, it is computationally demanding and involves a series of relatively high-dimensional nonlinear optimizations and fixed-point computations. This will likely limit the application of the method to other data and may make estimation unfeasible for larger data sets. Second, and as discussed by [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF] in Section IV.B, the model cannot be estimated in deviations. This complicates the inclusion of group fixed effects, a main means of controlling for correlated effects. Simply including group dummies may be computationally unfeasible and may bias the estimates due to the incidental parameter problem.

In contrast, these drawbacks are absent from the 2SLS estimation of the linear model (2). Fixed effects can be eliminated by taking deviations from the group average. As well, efficient computation of 2SLS estimates are pre-programmed in standard statistical software and can be computed quickly even for massive data sets.

We therefore reanalyze the same data assuming that model (2) holds.

Our estimation results are presented in Table 2. We consider specifications without fixed effects (NLS estimates in Column 1 and 2SLS estimates in Column 2), with school fixed effects (2SLS estimates in Column 3), and with school-grade fixed effects (2SLS estimates in Column 4). We see that estimates of the endogenous peer effects are remarkably similar to those obtained in [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF]: 0.568 in our preferred specification, compared to 0.666 in theirs. Standard errors have a similar magnitude, and this coefficient is very precisely estimated. Incorporating school-grade fixed effects only slightly decreases the estimate of the endogenous peer effect. The proportion of observations having a predicted probability between 0 and 1 lies 6 of Appendix 7.1. The number of observations is 74,783, the number of schools is 127, and the number of school-grades is 532. Standard errors are clustered at the grade-school level for the 2SLS estimators and are heteroscedastic-robust for the NLS estimator. Instruments for 2SLS estimations are generated using second-degree friends: G 2 X. The weak instrument tests are based on first-stage F-tests. The test statistic under the null hypothesis that all instruments are weak follows a non-central χ 2 distribution (see [START_REF] Stock | Testing for weak instruments in linear iv regression[END_REF]).

The null hypothesis for all specifications is rejected at a confidence level < 1%. The null hypothesis of the Sargan test is that all instruments are exogenous. The test statistic follows a χ 2 distribution under the null hypothesis. The null hypothesis is rejected at < 1% for all specifications.

between 96% and 97%.26 Furthermore, in all specifications, the estimated coefficient satisfies the uniqueness condition of Theorem 3.27

The characteristics of the students and their peers also affect smoking behavior. Another well-known advantage of a linear formulation is that the marginal impact of a characteristic on the outcome is simply equal to the characteristic's estimated coefficient. 28 For instance, being Black rather than White is associated with a 0.14 decrease in the likelihood to be a smoker. Students living with both parents and those with a high school-educated mother are less likely to smoke. Interestingly, these beneficial effects appear to spill over to students' friends because a student having friends who live with both parents or with a high school-educated mother is also less likely to be a smoker.

Finally, we see that the joint validity of the instruments is rejected by overidentification tests. This is perhaps not surprising since there are 17 instruments here for one endogenous variable. Even though the instruments, jointly, appear to be strong, some instruments are likely to be weak. How best to estimate IV regressions and test overidentification in the presence of many weak instruments is an active area of research, see e.g., [START_REF] Davidson | Bootstrap tests for overidentification in linear regression models[END_REF], [START_REF] Carrasco | Efficient estimation with many weak instruments using regularization techniques[END_REF], and Tchuente (2019).

Entry games and the airline industry

In this Section, we revisit the analysis of entry into airline markets of [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF]. As done in the previous Section, and for the sake of comparison, we analyze the same sample, variables, and data as [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF] Each firm i is either present or absent from market m, y im ∈ {0, 1}. The data include 10 variables assumed to be exogenous: 8 market-level variables z m and 2 firm-market-level variables, x im : "airport presence" and "cost". We present summary statistics of these variables in Table 7 of Appendix 7.1. [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF] develop an econometric framework allowing for multiple equilibria. They consider static games of complete information based on variants of relative utility (5). The payoff for one firm of entering into a market depends linearly on the entry decisions of other firms. A key distinguishing assumption is that preference shocks are continuous and independent from observables in their framework.29

Equilibrium multiplicity can then generally not be avoided, and allowing for such multiplicity is central to their approach.

In their framework, a main object of interest is P(y|x, z), the probability of observing entry decisions y ∈ {0, 1} n conditional on all market and firm-market observables. The authors derive sharp bounds on P(y|x, z) implied by equilibrium behavior. The lower bound captures situations where y is the unique Nash equilibrium. The higher bound captures situations where y is one Nash equilibrium among possibly many. In turn, these bounds define the identified set of parameters, i.e., the set of parameters for which the inequalities are satisfied almost everywhere. They propose a two-step estimation procedure. In a first step, the researcher must obtain a consistent estimate of the 2 n conditional probabilities P(y|x, z). In a second step, parameters θ, β are obtained by minimizing a distance from the identified set, built from this consistent estimate and simulated bounds. [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF] take the first step as a given when developing their method. However, this step suffers from a curse of dimensionality in practice. 30 The problem is that P(y|x, z) is a high-dimensional object.

In the airline application, there are n = 6 firms and hence 2 6 = 64 possible market structures. This object is thus composed of 63 functions of 20 observed variables: 8 market variables and 2 * 6 = 12 firm-market variables for all firms. Some of these variables take continuous values. Obtaining reliable nonparametric estimates of these 63 functions requires massive amounts of data. In contrast, there are on average only 2742/63 ≈ 44 observations available to estimate each function of 20 variables in the airline data.31 A usual solution, applied by [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF], is to discretize the observable space. Given the curse of dimensionality, however, discretization in this context leads to a severe loss of information.

In contrast, our proposed estimators based on linear model (3) do not require estimating P(y|x, z) and do not suffer from a curse of dimensionality. We next compare estimates based on models with linear interactions to the original estimates. We consider two specifications: one with homogeneous interactions, 3 in [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF], and one with heterogeneous interactions, β ij = β j , corresponding to Column 3 in their Table 3. We report the original estimates in Column 1, estimates from a 2SLS estimation of model (3) in Column 2, and estimates from a 2SLS estimation of model ( 3) with airline fixed effects in Column 3 in Table 3 for homogeneous interactions and in Column 3 in Table 4 for heterogeneous interactions.

β ij = β, corresponding to Column 2 in Table
We see that estimated interactions are generally positive and significant under linear formulations, whereas they are negative and significant in [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF]. Both approaches thus appear to yield qualitatively different results. Overidentification tests show that the joint validity of the exclusion restrictions is strongly rejected in the absence of airline fixed effects for both specifications. We then assess the effect of controlling for airline fixed effects, absent from the specifications analyzed in [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF].

In the homogeneous specification, the estimated interaction parameter is lower but remains positive and significant. The joint validity of the instruments is now not rejected at the 10% level. In the heterogeneous specification, adding airline fixed effects has a strong effect on interaction estimates, an indication that airline unobservables matter. The validity of the instruments is strongly rejected in this more general specification.

The impact of airline fixed effects on estimates and the results from overidentification tests both suggest that endogeneity is a serious concern in the empirical analysis of [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF].

The proportion of observations having a predicted probability between 0 and 1 ranges between 78% and 85%, depending on the specification. Predicting a probability outside [0, 1] is strongly correlated with the variable "airport presence", however, and is sensitive to how this variable is measured. 32 For example, we can replace this variable by a dummy equal to 0 when airport presence is lower than the median and 1 otherwise, as in [START_REF] Chen | Monte carlo confidence sets for identified sets[END_REF]. When we re-estimate linear specifications with a binary airport presence, the interaction estimates are qualitatively similar, and the proportion of predicted probabilities between 0 and 1 now ranges between 85% and 93%.33 Note that airport presence is a function of outcomes, and its inclusion can only be justified by making strong separability assumptions, see Footnote 27 in [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF].

We then verify that the uniqueness condition of Theorem 3 holds for the estimated interaction parameters in all linear specifications. Under homogeneity, this condition is equivalent to |β| < 1/5 = 0.2, and here β = 0.098 without airline fixed effects and 0.080 with airline fixed effects. Under heterogeneity, uniqueness holds if j |β j | < 1. From Table 4, we see that j | βj | = 0.637 without airline fixed effects and 0.501 with airline fixed effects.34

Endogeneity may notably be caused by market-level unobservables, i.e., unobserved characteristics of the markets that affect firms' entry decisions and are correlated with firm-market characteristics. Interestingly, interaction parameters may be partly identified in variants of linear model (3) with market fixed effects α m .

We show in Appendix that the model with heterogeneous interactions (β ij = β j ) and market fixed effects, while unidentified, only has one degree of underidentification. In this model, interaction parameters are identified conditional on some normalization, and their ranking is identified under some slight sign restriction.

We leave a full exploration of identification in linear entry models with market fixed effects for future research.

Conclusion

We consider a general model of linear interactions in binary outcomes. Building on [START_REF] Heckman | A simultaneous equations linear probability model[END_REF], we first characterize the conditions under which the econometric model is well defined. Errors must have a specific discrete structure, imposed by the binary nature of the outcomes. We then analyze the game-theoretic microfoundations of the model. We characterize the conditions on preference shocks under which the linear model of interactions corresponds to a Nash (Bayes-Nash) equilibrium of a game of complete (incomplete) information with linear utilities. Many different preference shocks are compatible under complete information, whereas under incomplete information and independence, preference shocks must be iid and uniformly distributed. We also obtain conditions that guarantee equilibrium uniqueness. Overall, this clarifies the game-theoretic microfoundations of the linear model of interactions. We then propose two simple estimators and discuss their properties. Finally, we revisit the analysis of teenage smoking and peer effects of [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF] and that of entry into airline markets of [START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF]. These reanalyses showcase the advantages of the linear framework and suggest that these previous analyses suffer from endogeneity problems.

We do not claim, of course, that data with binary outcomes are always best represented by a linear model of interactions. We do claim that linear models provide a useful benchmark that has been unduly discarded by the literature on the econometrics of games. In some contexts, controlling for unobserved heterogeneity may be more important than accounting for multiplicity. In other contexts, the data requirements of existing methods that account for multiplicity are too demanding. In all contexts, a linear framework provides a useful benchmark. Its estimation is straightforward, its data requirements are minimal, and it is well suited for exploring critical identification issues. Moreover, Theorem 2 and 3 show that the linear model of interactions can be microfounded as an equilibrium of games that can have multiple equilibria. It can thus be embedded in models with multiple equilibria. Under incomplete information and independence, in particular, the uniqueness condition is easy to verify. This uniqueness condition holds for the estimated interaction parameters in our empirical reanalysis. We thus propose to rehabilitate linear models of interactions in binary outcomes.

Our proposition and analysis raise a number of issues that could be addressed in future research. Under incomplete information, it would be interesting to know whether, and if so how, the linear model of interactions can be microfounded when reduced-form errors and preference shocks are correlated. In general, one challenge is to assess whether a linear model represents the data well or whether estimating a nonlinear model is necessary. Relatedly, we still lack tests of multiplicity that do not rely on strong parametric assumptions.

Researchers often interpret the presence of bunching and clustering in the data as a sign of multiple equilibria.

However, bunching and clustering can also be explained by the presence of common unobservables. Thus, a central challenge is to better understand what can be identified when both multiplicity and unobserved heterogeneity may matter and to develop appropriate estimation frameworks.

Appendix

7.1 Tables Note: For each simulation, M = 500 networks are generated using iid Bernoulli trials with a probability p = 0.1. Thus, the expected number of links for each individual is 0.1(N -1). Values represent the average (standard deviation) of the 1000 simulations. Note: Summary statistics using the same sample as in [START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF]. The number of observations is 74,783, the number of schools is 127, and the number of school-grades is 532.

is a function of an agent's shock into their action set, i.e., y i (e i ) ∈ {0, 1}. Payoffs are given by the agent's expected utilities:

Eu i (1|e i ) -Eu i (0|e i ) = f i -1 2 + e i + j β ij P(y j = 1|e i ).

By independence, P(y j = 1|e i ) = P(y j = 1). Then, agent i with private information e i chooses y i = 1 iff f i -1 2 + e i + j β ij P(y j = 1) ≥ 0.

Introduce ēi such that

f i - 1 2 + ēi + j β ij P(y j = 1) = 0.
Then, y i = 1 ⇔ e i ≥ ēi and P(y i = 1) = P(e i ≥ ēi ) = 1 -F i (ē i ).

(1) Assume, first, that the profile y i = 1 with probability P i and 0 with probability 1 -P i is a Bayes-Nash equilibrium, where P i = [(I-β) -1 f ] i for every possible f and β. This means that P i = P(y i = 1) = 1-F i (ē i ).

Since P i = f i + j β ij P j , 1 -F i (ē i ) = f i + j β ij P j . By the definition of ēi , f i + j β ij P j = 1 2 -ēi and hence

F i (ē i ) = ēi + 1 2 .
As f and β take all possible values, ēi takes all values in [-1 2 , 1 2 ]. This shows that F i is the cdf of the uniform distribution on [-1 2 , 1 2 ].

(2) Conversely, assume that ∀i, e i is uniformly distributed on [-1 2 , 1 2 ]. Then, F i (e) = min{max{e + 1 2 , 0}, 1}. This implies that P(y i = 1) = min{max{0, (1/2 -ēi )}, 1}. Next, consider a situation where for every i, ēi ∈ [-1 2 , 1 2 ]. This corresponds to the following fixed-point equation:

ē = -f + 1 2 1 - 1 2 β1 + βē, or ē = -(I -β) -1 [f - 1 2 1 + 1 2 β1],
and this yields ē = -(I -β) -1 f + 1 2 1.

Since P i = [(I -β) -1 f ] i ∈ [0, 1], we indeed have ēi ∈ [-1/2, 1/2] and hence P(y i = 1) = P(e i ≥ ēi ) =

1 2 + [(I -β) -1 f ] i - 1 2 = P i ,
and hence the stochastic profile y i = 1 with probability P i and 0 with probability 1 -P i is a Bayes-Nash equilibrium.

  . The main data come from the 2001 Airline Origin and Destination Survey, a 10% sample of tickets collected by the US Department of Transportation. A market is defined as the trip between two airports, irrespective of intermediate transfer points and the direction of the flight. The sample includes 2,742 markets. Six firms are assumed to compete across all markets: American (AA), Delta (DL), United (UA), Southwest (WN), and two "composite" firms: Medium Airlines (MA) and Low-Cost Carriers (LCC).

Table 1 :

 1 Monte Carlo Simulations -Number of Groups

			High β
		M = 100			M = 300
	Parameters	2SLS	NLS	Parameters	2SLS	NLS
	α = 0.1	0.098	0.098	α = 0.1	0.099	0.100
		(0.031)	(0.029)		(0.017)	(0.015)
	β = 0.7	0.712	0.705	β = 0.7	0.704	0.700
		(0.139)	(0.086)		(0.077)	(0.045)
	γ = 0.05	0.053	0.053	γ = 0.05	0.050	0.050
		(0.042)	(0.041)		(0.024)	(0.023)
	δ = 0.1	0.088	0.096	δ = 0.1	0.096	0.101
		(0.126)	(0.079)		(0.070)	(0.042)
			Low β
		M = 100			M = 300
	Parameters	2SLS	NLS	Parameters	2SLS	NLS
	α = 0.1	0.099	0.098	α = 0.1	0.099	0.099
		(0.027)	(0.027)		(0.015)	(0.015)
	β = 0.25	0.284	0.262	β = 0.25	0.263	0.254
		(0.242)	(0.188)		(0.134)	(0.104)
	γ = 0.2	0.199	0.201	γ = 0.2	0.199	0.200
		(0.040)	(0.040)		(0.024)	(0.023)
	δ = 0.3	0.081	0.095	δ = 0.3	0.093	0.098
		(0.130)	(0.102)		(0.070)	(0.056)

  Table 6 of Appendix 7.1. The proportion of smokers among students is 23%.

Table 2 :

 2 Peer Effects on Smoking

		NLS	2SLS		2SLS	2SLS
	Endogenous effect	0.545 (0.038)	0.608 (0.062)	0.588 (0.058)	0.568	(0.058)
	Individual effects
	Constant	-1.043 (0.089)	-0.950 (0.141)	-	(-)	-	(-)
	Age	0.159 (0.012)	0.147 (0.020)	0.131 (0.018)	0.102	(0.027)
	Age 2 /10	-0.044 (0.004)	-0.040 (0.007)	-0.034 (0.006)	-0.024	(0.009)
	Years in school	0.001 (0.002)	0.001 (0.002)	-0.001 (0.002)	0.000	(0.002)
	Male	0.005 (0.004)	0.007 (0.005)	0.006 (0.005)	0.005	(0.005)
	Black	-0.172 (0.006)	-0.157 (0.007)	-0.141 (0.007)	-0.142	(0.007)
	Asian	-0.080 (0.007)	-0.070 (0.008)	-0.056 (0.008)	-0.059	(0.008)
	Hispanic	-0.080 (0.006)	-0.071 (0.009)	-0.036 (0.007)	-0.036	(0.007)
	Other race	0.029 (0.007)	0.022 (0.007)	0.028 (0.007)	0.027	(0.007)
	Live with both parents	-0.046 (0.004)	-0.039 (0.004)	-0.042 (0.004)	-0.042	(0.004)
	Sports club	-0.040 (0.003)	-0.036 (0.004)	-0.039 (0.004)	-0.040	(0.004)
	Mom education less than high school	0.010 (0.006)	0.007 (0.006)	0.010 (0.006)	0.009	(0.006)
	Mom education more than high school	-0.012 (0.004)	-0.008 (0.004)	-0.007 (0.004)	-0.007	(0.004)
	Mom education missing	-0.030 (0.005)	-0.026 (0.004)	-0.021 (0.004)	-0.021	(0.004)
	Mom job is professional	0.022 (0.004)	0.019 (0.005)	0.018 (0.005)	0.019	(0.005)
	Mom other jobs	0.024 (0.004)	0.021 (0.004)	0.021 (0.004)	0.021	(0.004)
	Mom on welfare	0.027 (0.017)	0.025 (0.016)	0.027 (0.016)	0.024	(0.016)
	Mom job is missing	0.014 (0.006)	0.008 (0.006)	0.009 (0.006)	0.009	(0.006)
	Contextual effects
	Age	-0.007 (0.002)	-0.010 (0.003)	-0.009 (0.003)	-0.010	(0.003)
	Age 2 /10	-0.002 (0.001)	-0.001 (0.002)	-0.001 (0.002)	0.000	(0.002)
	Years in school	-0.003 (0.002)	-0.003 (0.002)	-0.005 (0.002)	-0.004	(0.003)
	Male	-0.006 (0.005)	-0.015 (0.006)	-0.018 (0.006)	-0.017	(0.006)
	Black	0.062 (0.010)	0.065 (0.014)	0.062 (0.014)	0.059	(0.014)
	Asian	0.001 (0.010)	0.000 (0.012)	0.012 (0.013)	0.010	(0.013)
	Hispanic	-0.010 (0.009)	-0.004 (0.011)	0.026 (0.011)	0.024	(0.011)
	Other race	0.028 (0.011)	0.016 (0.013)	0.022 (0.014)	0.024	(0.014)
	Live with both parents	-0.033 (0.007)	-0.022 (0.009)	-0.027 (0.008)	-0.028	(0.008)
	Sports club	-0.004 (0.005)	-0.003 (0.007)	-0.009 (0.007)	-0.010	(0.007)
	Mom education less than high school	0.009 (0.009)	0.011 (0.011)	0.011 (0.011)	0.011	(0.011)
	Mom education more than high school	-0.028 (0.006)	-0.020 (0.008)	-0.018 (0.008)	-0.018	(0.008)
	Mom education missing	0.000 (0.009)	0.017 (0.011)	0.023 (0.011)	0.023	(0.011)
	Mom job is professional	0.011 (0.007)	0.001 (0.009)	0.003 (0.009)	0.004	(0.009)
	Mom other jobs	0.008 (0.006)	-0.001 (0.007)	0.002 (0.008)	0.004	(0.008)
	Mom on welfare	-0.025 (0.026)	0.005 (0.031)	0.015 (0.031)	0.010	(0.031)
	Mom job is missing	0.031 (0.010)	0.010 (0.011)	0.012 (0.011)	0.013	(0.011)
	School fixed effects				X
	School-grade fixed effects						X
	Weak instruments		39.376		46.510		46.750
	Sargan		44.449		57.190		60.431
	Fraction predicted in [0, 1]	0.973	0.971		0.960
	Note: Estimated coefficients and associated standard errors (in parenthesis). Estimation of linear model (2). Outcome is smoking. Summary
	statistics are presented in Table

Table 3 :

 3 Market Structure of the Airline Industry: Homogeneous Effects Note: Estimated coefficients and associated standard errors (in parenthesis). Column (1) is reproduced from[START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF]. Standard errors for columns (2) and (3) are clustered at the market level. Predicted values for columns (2) and (3) give the proportions of markets for which the observed structure is equal to the structure of highest likelihood. For all columns, there are 2,742 markets and 6 firms in each market. The weak instrument tests are based on first-stage F-tests. The test statistic under the null hypothesis that all instruments are weak follows a non-central χ 2 distribution (see[START_REF] Stock | Testing for weak instruments in linear iv regression[END_REF]). The null hypothesis is rejected at < 1% for both specifications. The null hypothesis of the Sargan test is that all instruments are exogenous. The test statistic follows a χ 2 distribution under the null hypothesis. The null hypothesis is rejected at < 1% for the specification without fixed effects but is not rejected at 10% for the specification with airline fixed effects.

		Ciliberto and Tamer (2009)	2SLS	2SLS
	Endogenous Effect	[-14.151,-10.581]	0.098 (0.002)	0.080	(0.003)
	Individual Effects
	Airport presence	[3.052,5.087]	1.504 (0.013)	1.877	(0.020)
	Cost	[-0.714,0.024]	-0.044 (0.006)	-0.022 (0.004)
	Market Controls
	Wright	[-20.526,-8.612]	-0.096 (0.011)	-0.105	(0.013)
	Dallas	[-6.890,-1.087]	0.035 (0.007)	0.040	(0.009)
	Market size	[-0.972,2.247]	0.008 (0.001)	0.009	(0.001)
	Market distance	[4.356,7.046]	0.001 (0.007)	0.035	(0.006)
	Close airport	[4.022,9.831]	-0.004 (0.011)	-0.020	(0.011)
	U.S. center distance	[1.452,3.330]	0.003 (0.005)	-0.024	(0.005)
	Per capita income	[0,568,2.623]	0.010 (0.005)	0.010	(0.006)
	Income growth rate	[0.370,1.003]	0.002 (0.001)	0.002	(0.001)
	Constant	[-13.840,-7.796]	-0.345 (0.018)	-	(-)
	Airline fixed effects				X
	Correctly predicted	0.328	0.395	0.433
	Weak instruments		5389.230	5464.494
	Sargan		72.490	0.462
	Fraction predicted in [0, 1]		0.846	0.838

Table 4 :

 4 Market Structure of the Airline Industry: Heterogenous Effects Note: Estimated coefficients and associated standard errors (in parenthesis). Column (1) is reproduced from[START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF]. Standard errors for columns (2) and (3) are clustered at the market level. Predicted values for columns (2) and (3) give the proportions of markets for which the observed structure is equal to the structure of highest likelihood. For all columns, there are 2,742 markets and 6 firms in each market. The weak instrument tests are based on first-stage F-tests. The test statistic under the null hypothesis that all instruments are weak follows a non-central χ 2 distribution (see[START_REF] Stock | Testing for weak instruments in linear iv regression[END_REF]). The null hypothesis is rejected at the < 1% level for both specifications. The null hypothesis of the Sargan test is that all instruments are exogenous. The test statistic follows a χ 2 distribution under the null hypothesis. The null hypothesis is rejected at the < 1% level for both specifications.

		Ciliberto and Tamer (2009)	2SLS	2SLS
	Endogenous Effect
	Presence of AA	[-10.914.-8.822]	0.188 (0.010)	0.065	(0.008)
	Presence of DL	[-10.037,-8.631]	0.250 (0.009)	0.133	(0.006)
	Presence of UA	[-10.101,-4.938]	0.075 (0.011)	0.090	(0.010)
	Presence of MA	[-11.489,-9.414]	-0.007 (0.008)	0.074	(0.007)
	Presence of LCC	[-19.623,-14.578]	-0.055 (0.014)	0.079	(0.012)
	Presence of WN	[-12.912,-10.969]	0.063 (0.009)	0.060	(0.007)
	Individual Effects
	Airport presence	[11.262,14.296]	1.631 (0.014)	1.894	(0.020)
	Cost	[-1.197,-0.333]	-0.048 (0.006)	-0.022	(0.004)
	Market Controls
	Wright	[-14.738,-12.556]	-0.034 (0.018)	-0.081	(0.013)
	Dallas	[-1.186,0.421]	0.005 (0.014)	0.034	(0.008)
	Market size	[0.532,1.245]	0.011 (0.002)	0.010	(0.001)
	Market distance	[0.106,1.002]	-0.046 (0.008)	0.030	(0.006)
	Close airport	[4.022,9.831]	-0.019 (0.015)	-0.019	(0.011)
	U.S. center distance	[1.452,3.330]	0.043 (0.007)	-0.022	(0.005)
	Per capita income	[-0.080,1.010]	0.014 (0.008)	0.007	(0.006)
	Income growth rate	[0.078,0.360]	-0.005 (0.002)	0.001	(0.001)
	Constant	[-1.362,2.431]	-0.401 (0.027)	-	(-)
	Airline fixed effects			X
	Predicted	0.326	0.342	0.437
	Weak instruments AA		1,350.680	1,090.872
	Weak instruments DL		1,373.260	1,274.265
	Weak instruments UA		1,023.390	926.734
	Weak instruments MA		1,360.520	896.077
	Weak instruments LCC		455.520	424.383
	Weak instruments WN		1,521.330	1,474.447
	Sargan		119.28	93.923
	Fraction predicted in [0, 1]		0.780	0.837

Table 5 :

 5 Monte Carlo Simulations -Group Sizes

			High β
		N = 20			N = 50
	Parameters	2SLS	NLS	Parameters	2SLS	NLS
	α = 0.1	0.099	0.099	α = 0.1	0.100	0.100
		(0.014)	(0.012)		(0.015)	(0.012)
	β = 0.7	0.706	0.701	β = 0.7	0.700	0.699
		(0.071)	(0.039)		(0.054)	(0.034)
	γ = 0.05	0.051	0.051	γ = 0.05	0.050	0.050
		(0.021)	(0.020)		(0.015)	(0.014)
	δ = 0.1	0.094	0.099	δ = 0.1	0.101	0.101
		(0.069)	(0.038)		(0.044)	(0.029)
			Low β
		N = 20			N = 50
	Parameters	2SLS	NLS	Parameters	2SLS	NLS
	α = 0.1	0.099	0.099	α = 0.1	0.100	0.100
		(0.012)	(0.012)		(0.013)	(0.012)
	β = 0.25	0.254	0.252	β = 0.25	0.252	0.251
		(0.070)	(0.063)		(0.060)	(0.053)
	γ = 0.2	0.201	0.201	γ = 0.2	0.200	0.200
		(0.022)	(0.022)		(0.016)	(0.016)
	δ = 0.3	0.297	0.299	δ = 0.3	0.297	0.298
		(0.056)	(0.050)		(0.039)	(0.036)

Table 6 :

 6 Summary Statistics: Teenagers' Smoking Decisions

	Statistic	Mean	St. Dev.	Min	Pctl(25)	Pctl(75)	Max
	Age	15.068	1.685	10	14	16	19
	Years in school	2.493	1.407	1	1	3	6
	Male	0.488	0.500	0	0	1	1
	Black	0.183	0.386	0	0	0	1
	Asian	0.066	0.248	0	0	0	1
	Hisp.	0.139	0.346	0	0	0	1
	Other race	0.056	0.230	0	0	0	1
	Live with both parents	0.730	0.444	0	0	1	1
	Sports club	0.524	0.499	0	0	1	1
	Mom education less than high school	0.101	0.301	0	0	0	1
	Mom education more than high school	0.412	0.492	0	0	1	1
	Mom education missing	0.107	0.309	0	0	0	1
	Mom job is professional	0.262	0.440	0	0	1	1
	Mom other jobs	0.358	0.479	0	0	1	1
	Mom on welfare	0.009	0.093	0	0	0	1
	Mom job is missing	0.090	0.286	0	0	0	1
	Smoke	0.231	0.421	0	0	0	1

See, e.g.,[START_REF] Fadlon | Family health behaviors[END_REF] on health, Sacerdote (2011) on academic achievement,[START_REF] Foster | Learning by doing and learning from others: Human capital and technical change in agriculture[END_REF] on public good provision,Kuhn et al. (2011) on consumption, and[START_REF] Berry | Estimation of a model of entry in the airline industry[END_REF] on imperfect competition.

None of the articles and surveys in our references cite[START_REF] Heckman | A simultaneous equations linear probability model[END_REF] or estimate a linear model of interactions.

[START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF] develop and implement a partial identification approach. Although no direct specification tests are available, measures of goodness of fit are quantitatively similar across the two specifications, with a slight advantage to the linear models.

See, for instance,[START_REF] Manski | Economic analysis of social interactions[END_REF],Kline and Tamer (Forthcoming), and Angrist (2014) for a critical review.

About 10 years ago, four studies independently understood that the reflection problem[START_REF] Manski | Identification of endogenous social effects: The reflection problem[END_REF]) is naturally solved by network interactions(Bramoullé, Djebbari, and Fortin (2009), De Giorgi, Pellizzari, and[START_REF] De Giorgi | Identification of social interactions through partially overlapping peer groups[END_REF],[START_REF] Lin | Identifying peer effects in student academic achievement by spatial autoregressive models with group unobservables[END_REF], and Laschever (2013)). Since then, the literature on peer effects in networks has rapidly grown and extended in many directions, see[START_REF] Boucher | Some challenges in the empirics of the effects of networks[END_REF], De[START_REF] Paula | Econometrics of network models[END_REF] Bramoullé, Djebbari, and[START_REF] Bramoullé | Peer effects in networks: a survey[END_REF]. Relatively few studies, however, analyze peer effects in networks and binary outcomes.

The linear framework can also, of course, be applied to analyze peer effects in binary outcomes with group interactions, under appropriate identification conditions. For instance,[START_REF] Soetevent | A discrete-choice model with social interactions: with an application to high school teen behavior[END_REF] and[START_REF] Nakajima | Measuring peer effects on youth smoking behaviour[END_REF] assume that an agent's outcome does not depend directly on their peers' characteristics (no contextual peer effects). We can easily estimate linear interaction models of their data under the same assumption.

[START_REF] Aguirregabiria | Identification of games of incomplete information with multiple equilibria and unobserved heterogeneity[END_REF] show that identification may hold under incomplete information, multiplicity, and unobserved

This further implies that V(y i |x) = E(y i |x)(1 -E(y i |x)). The conditional variance and, more generally, higher moments of the conditional outcome distribution do not contain extra information with respect to the conditional expectation.

The model can easily be extended to account for isolated individuals.

For instance, one possible stochastic structure with correlation is as follows. Suppose that u is uniformly distributed on [0, 1], and for every i set ν i = 1 -P i if u < P i and -P i if u > P i .

Note that for (almost) any β and p ∈ [0, 1] n , the system (I -β) -1 f = p always has a unique solution f . Thus, the conditionP i = [(I -β) -1 f ] i ∈ [0, 1]for all i is always possible, irrespective of the strength of the interactions β.

A direct application of Theorem 2 shows that when e i = ε i , the unique solution to model (1) is always a Nash equilibrium of the corresponding game and is the unique Nash equilibrium in dominant strategies when ∀i, j |β ij | < 1 2 .

There exists an extensive game-theoretic literature on robust equilibria, proposing various definitions of robustness. A common, central idea, as here, is that an equilibrium is robust when it remains an equilibrium following perturbations of the underlying game (e.g., Trembling Hand perfection,[START_REF] Selten | Reexamination of the perfectness concept for equilibrium points in extensive games[END_REF]).

Unexpectedly, our proof is quite similar to the proof of Proposition 3.2 in[START_REF] Bloch | Pricing in social networks[END_REF] in a context of monopoly pricing under network externalities.

We show in Appendix that uniqueness holds when β < 1 for any submultiplicative matrix norm • .

Population is then partitioned into groups, and agents can only be affected by others in their own group. The overall network is composed of disjoint subnetworks with a block diagonal interaction matrix.

The analysis of estimators' properties in a single network asymptotic framework is an active area of research, see e.g.,[START_REF] Lee | Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models[END_REF],[START_REF] Menzel | Inference for games with many players[END_REF][START_REF] Leung | Treatment and spillover effects under network interference[END_REF].

Cluster-robust standard errors can still yield valid inferences in some situations where groups are not independent, see[START_REF] Bester | Inference with dependent data using cluster covariance estimators[END_REF].

Assuming independence of the errors,[START_REF] Amemiya | Some theorems in the linear probability model[END_REF] shows that weighted least squares is as efficient as the MLE. Moreover, since the MLE is maximized using a numerical algorithm, one needs to ensure that all proposed parameters are such that P i (θ) ∈ (0, 1). A similar issue arises for the implementation of the feasible weighted least squares: predicted probabilities used to weigh the estimator must fall between 0 and 1.

Identification may of course be affected by the presence of fixed effects.[START_REF] Bramoullé | Identification of peer effects through social networks[END_REF] derive identification conditions in variants of model (2) in the presence of group fixed effects.

We are grateful to the authors for providing the replication codes.

Note that this proportion cannot be computed when the model is estimated in deviations.

, this condition is equivalent to |β| < 1.

[START_REF] Lee | Binary choice models with social network under heterogeneous rational expectations[END_REF], this corresponds to their "naive" estimation of the marginal effects. Our estimates and theirs have similar signs, although their estimated marginal effects are generally larger in absolute value.

See Assumption 1 p. 1799 in[START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF].

The assumption that P(y|x, z) can be estimated nonparametrically from the data is prevalent in the literature (e.g.,[START_REF] Beresteanu | Sharp identification regions in models with convex moment predictions[END_REF],[START_REF] Chesher | Simultaneous equations models for discrete outcomes: coherence, completeness, and identification[END_REF],[START_REF] Manski | Inference on regressions with interval data on a regressor or outcome[END_REF],[START_REF] Galichon | Set identification in models with multiple equilibria[END_REF][START_REF] Galichon | Set identification in models with multiple equilibria[END_REF][START_REF] Tamer | Incomplete simultaneous discrete response model with multiple equilibria[END_REF]). See[START_REF] Andrews | Confidence regions for parameters in discrete games with multiple equilibria, with an application to discount chain store location[END_REF] for a discussion of the associated curse of dimensionality.

To put this into perspective, suppose that there is only one binary firm-market characteristic. The matrix of observables x can then take 2 6 = 64 values. Estimating P(y|x) may then require the estimation of 63 × 64 = 4032 parameters.

"Airport presence" computes the average proportion of other markets served by a carrier out of its departure and arrival airport, see the Supplementary Appendix of[START_REF] Ciliberto | Market structure and multiple equilibria in airline markets[END_REF].

This proportion is equal to 90% in the specification with heterogeneous interactions and airline fixed effects.

The null hypothesis j |β j | ≥ 1 is rejected at a significance level of < 1% for all specifications.

We thank Habiba Djebbari, Arthur Lewbel, the participants at various seminars and conferences for their helpful comments, and Aureo de Paula for invaluable feedback. This work was supported by the French National Research Agency Grant ANR-17-EURE-0020. This research uses data from Add Health, a program directed by Kathleen Mullan Harris and designed by J. Richard Udry, Peter S. Bearman, and Kathleen Mullan Harris at the University of North Carolina at Chapel Hill, and funded by Grant P01-HD31921 from the Eunice Kennedy Shriver National Institute of Child Health and Human Development, with cooperative funding from 23 other federal agencies and foundations. Special acknowledgment is given to Ronald R. Rindfuss and Barbara Entwisle for assistance in the original design. Information on how to obtain Add Health data files is available on the Add Health website (http://www.cpc.unc.edu/addhealth). No direct support was received from Grant P01-HD31921 for this research. The replication codes are available at https://github.com/vincentboucherecn/linearbinaryinteractions.

Substituting in the inequalities yields the first part of the result, characterizing preference shocks for which y * is a Nash equilibrium.

Next, derive a sufficient condition for uniqueness in dominant strategies. If

Here,

Substituting again ε i by its expressions yields the result. QED.

Proof of Proposition 1

First, show that y * is still a Nash equilibrium for preference shocks e i . Consider a realization of errors ν and y * the unique solution to equation (1). Consider i such that ν i > 0 and y * i = 1. Since y * is a Nash equilibrium for shocks e i , u i (1, y * -i ) -u i (0, y * -i ) + e i ≥ 0. Since e i = e i + L i and L i ≥ 0, u i (1, y * -i ) -u i (0, y * -i ) + e i ≥ 0. Similarly, if ν i < 0 and y * i = 0, u i (1, y * -i ) -u i (0, y * -i ) + e i ≤ 0. Since e i = e i -M i and M i ≥ 0, then u i (1, y * -i ) -u i (0, y * -i ) + e i ≤ 0. And hence y * is a Nash equilibrium for shocks e i . Next, assume that

The right-hand sides of these inequalities are well defined because y -i takes a finite number of values.

From the first inequality, we have for every

Playing 1 is a dominant strategy for agent i. From the second inequality, we have for every y -i , M i > u i (1, y -i ) -u i (0, y -i ) + e i . If ν i < 0, then e i = e i -M i and hence u i (1, y -i ) -u i (0, y -i ) + e i < 0. Playing 0 is a dominant strategy for player i. QED.

Proof of Theorem 3

Suppose that the ν i 's are uncorrelated and recall that P i = [(I -β) -1 f ] i = P(y i = 1|x). Consider preference shocks e i , that are independent and independent of observables, and let F i denote the cdf of e i . A strategy Can the game have other Bayes-Nash equilibria? Any Bayes-Nash equilibrium corresponds to the following fixed point equation:

Rewrite the fixed-point problem in matrix form:

, where h(ē) i = min{max{0, (1/2ēi )}, 1}. We have

for any submultiplicative norm • . If β < 1, the fixed-point function is a contraction mapping and thus has a unique fixed point. From the argument above, this fixed point is interior.

With n = 2, we can easily verify that when f 1 = f 2 = -1 and β 12 = β 21 = 3, the game has 3 Bayes-Nash equilibria: the one corresponding to model (1), P(y 1 = 1) = P 1 = 1 2 and P(y 2 = 1) = P 2 = 1 2 , as well as two others, P(y 1 = 1) = P(y 2 = 1) = 0 and P(y 1 = 1) = P(y 2 = 1) = 1. QED.

Partial identification in linear entry games with market fixed effects

Consider the following variant of model ( 3) with market fixed effects:

The effect of market-level characteristics is of course not identified here, as these characteristics are absorbed in the market fixed effects. These fixed effects must be eliminated. Consider the model in deviation with respect to y 1m :

This equation can be estimated by instrumenting y 1m on the right-hand side by x 1m . This implies that the composite parameters γ/(1 + β i ) and b i = (β 1 -β i )/(1 + β i ) = (1 + β 1 )/(1 + β i ) -1 are identified, and hence the ratios (1 + β i )/(1 + β j ) are identified. The β's are not identified without further assumptions.

However, β i for i = 1 is identified when β 1 is known (except when b i = -1), which shows that there is one degree of underidentification. The β's are thus identified conditional on some normalization. Furthermore, the ordering of the β's is identified under some slight restriction on signs, for instance that 1 + β 1 > 0. QED.