Multivariate Choice and Identi cation of Social Interactions

Transcription

1 Multivariate Choice and Identi cation of Social Interactions Ethan Cohen-Cole Xiaodong Liu y Yves Zenou z October 17, 2012 Abstract We investigate the impact of peers on own outcomes where all individuals embedded in a network make a multitude of choices, many of which of dependent on each other. Peers can have multiple and sometimes opposite in uences on their friends. First, we generalize the standard local-aggregate model for multiple activities and give a condition that guarantees the existence and uniqueness of an interior Nash equilibrium in e orts. Second, we give a set of identi cation conditions of peer in uences in contexts of multiple decisions. Third, we show the empirical salience of these methods for education. In the empirics, as well as many others that we explored, including more than a single decision changed the signi cance and magnitude of estimated peer e ects. In some cases, the changes are qualitatively very important, including changing intuition about how peer e ects in education operate in high school environments. Our conclusion from this nding is that empirical peer e ects research should be carefully constructed to ensure that results re ect the full gamut of choices that individuals make. Key words: Social networks, identi cation, peer e ects. JEL Classi cation: C3, C21, I21, Z13. University of Maryland, College Park, USA. ecohencole@rhsmith.umd.edu. y University of Colorado at Boulder, USA. xiaodong.liu@colorado.edu. z Stockholm University, Research Institute of Industrial Economics (IFN) and GAINS. yves.zenou@ne.su.se. 1

2 1 Introduction Peer decisions and/or peer characteristics have been shown to be important in predicting di erent outcomes of individuals, ranging from education and crime to labor market (Ioannides and Loury, 2004; Sacerdote, 2011; Patacchini and Zenou, 2012). Most of this literature has, however, considered the e ect of peers on one speci c choice. For example, Calvó-Armengol et al. (2009) show that peers grades a ect own grades. Fletcher (2012) nds that a 10% increase in the proportion of classmates who drink increases the likelihood an individual drinks by ve percentage points. Cutler and Glaeser (2010) nd evidence for peer e ects in smoking, etc. In reality, individuals make a multitude of choices, many of which of dependent on each other. As a result, peers can have multiple and sometimes opposite in uences on their friends. For example, if a person s friends smoke, drink but perform well at school, how do these impact this person s choice across the three activities? This joint decision problem is what we study in the current paper. Our purpose is to help understand the impact of making more than one choice at a time; in particular, we discuss decision making with more than one choice in the context of peer in uences and social networks. To the best of our knowledge, this is the rst paper that analyzes this issue, at least from an econometric viewpoint. To be more precise, our purpose in this paper is threefold. One, we generalize the standard local-aggregate model for multiple activities and give a condition that guarantees the existence and uniqueness of an interior Nash equilibrium in e orts. Second, we show a set of methods for identi cation of peer in uences in contexts of multiple decisions. Third, we show the empirical salience of these methods for education. In the empirics below, as well as many others that we explored, including more than a single decision changed the signi cance and magnitude of estimated peer e ects. In some cases, the changes are qualitatively very important, including changing intuition about how peer e ects in education operate in high school environments. Our conclusion from this nding is that empirical peer e ects research should be carefully constructed to ensure that results re ect the full gamut of choices that individuals make. Even with the large gains made to date in understanding how interactions in uence individual decisions, a relatively large gap remains. To our knowledge, the existing literature on peer e ects has entirely focused on a single decision by each individual. In analyzing the decision to study, we ignore the related and complex decisions to smoke, drink, play sports or commit crimes. While evaluating decisions in isolation makes the challenging empirics of social networks more tractable, 2

3 it requires a host of unstated, and largely implausible, exclusion restrictions. Few people make important decisions about their life completely in isolation of other factors; however, this is precisely the assumption maintained by the literature. The inclusion of more than one choice in a individual optimization problem introduces at least two complexities. The rst is a standard simultaneity problem well known in the study of economics. Studying more means less time for sports and perhaps induces one to eat more, smoke less, etc. These generate a complementarity (or substitutability) of the decisions themselves, independent of social in uences. Playing sports, studying, participating in other extracurriculars, and spending time with family all require large time commitments, which at some point reaches the maximum available time in a day. In this sense, these activities are substitutes. The presence of substitutability may mean that a student who plays three sports, and has friends who study much more than he/she does, will respond di erently to the social in uence of studying than another student that does not play sports. That is, he/she will either choose not to respond to the studying in uence or will reduce his/her sports commitment. Estimation the studying decision in isolation of the sports decision means that we may incorrectly attribute his/her low responsiveness to the studying in uence as the lack of social in uence rather than the presence of other activities. The second feature relates to the interdependence of social spillovers themselves and the link between the social e ects of one action on the decision about the other action. For example, the decision to study is partially in uenced by other s choices to study. The same fact is true about the decision to participate in sports. However, it also appears reasonable that the social in uence of a student s peers on studying will also impact his/her decision to play sports. An encouragement to study more, whether tacit or direct, can lead to an in uence on the decision to play sports. After having a successful study session, the student may go home and need to decide whether to play soccer for the spring. The social in uence from the same group of friends can be quite distinct on separate decisions. This distinct in uence is the channel from the social in uence of friends playing soccer on studying and vice-versa. The other channel here is the determination in equilibrium of each of the two social in uences. The relevance of social in uence on studying can be quite di erent if students are also considering soccer at the same time, potentially reinforcing both e ects, minimizing both or trading o between the two. The current literature focuses on some single decision and the in uence of peers on that decision (Ballester et al., 2006; Bramoullé and Kranton, 2007; Bramoullé et al., 2012). The spillover occurs 3

4 as an optimization process in which agents enjoy utility by choosing similarly to their peers. Other actions, other groups, and other in uences are assumed to be exogenous to this decision making process. Expanding to a set of two choices, in these models, individuals optimization process would involve forming expectations in isolation over each activity. Deciding whether to study more assumes that the choices and the peer decisions relating to sports are already known. Similarly, in deciding whether to join the tennis team, the amount of studying and the choices of friends over studying are given. We relax this strong assumption. If others also choose activities jointly, the expectations that a student forms over peers studying will depend on peers sports decisions as well. Thus, in addition to the complementarity / substitutability associated with multiple decisions, there are transmission channels between choices that pass through the social process as well. This paper will provide a structural model that helps to illustrate multivariate choice in a social interaction setting. As is normal in peer e ects contexts, we begin from the view that individuals enjoy utility as a function of the actions of others. Next, we allow agents to choose more than one activity. The model will allow for these activities to have an arbitrarily degree of complementarity or substitutability. As well, the model is general enough to handle arbitrary combinations of choices; that is, we make no assumption of the orderings of choice bundles. Indeed, this generality is essential; combining sets of choices in a social interactions context into bundles dramatically restricts the set of possible actions available to individuals. It is easy to construct examples of preference reversals in the bundled goods setting that comply with standard choice axioms in the general setting here. We develop the general theoretical model in Section 2.1. Section 3 begins our investigation of identi cation. Parallel to the economic drivers of the model are three identi cation issues. First, we have the same re ection problem (Manski 1993) that emerges from the coexistence of endogenous and contextual e ects as in most social interactions studies. 1 Second, the social spillovers across choices adds a new level of di culty. Finally, the simultaneity problem requires treatment as in many models. Here the social impacts make that problem more complex to solve. To better understand the layers of complexity added by these identi cation issues, we consider three models with increasing degrees of interdependence in decision making. The rst model we consider is the seemingly unrelated simultaneous equations model, where an individual s decision on an activity only depends on decisions of the peers on the same activity (the 1 One reason social networks matter in social interactions analysis is because they facilitate identi cation by breaking the re ection problem. This was originally recognized in Cohen-Cole (2006) and is systematically explored in Bramoullé et al. (2009). 4

5 endogenous e ect) and their characteristics (the contextual e ect). The second model we consider is the triangular system of simultaneous equations model, where we introduce a one-way cross-choice peer e ect besides the endogenous and contextual e ects such that an individual s decision on an activity can be a ected by the peers decision on related activities. Finally, we consider the square system of simultaneous equations model with the simultaneity e ect and two-way cross-choice peer e ect so that an individual s decision on an activity can be a ected by his/her own decision on related activities and the peers decision on related activities. We allow the decision errors of an individual to be correlated across related activities in all three models. We show that the general square simultaneous equations model with both simultaneity e ect and cross-choice peer e ect cannot be identi ed without any exclusion restrictions. To better understand this result, we consider three speci cations of the square model. In the rst one, only cross-choice peer e ects are considered. In the second one, only simultaneity e ects are taken into account. The last speci cation is the general square model where both simultaneity and cross-choice peer e ects are considered. We show that the two rst speci cations can be identi ed without any exclusion restriction on the exogenous variables. In other words, it is the coexistence of those two e ects that cause the nonidenti cation of the general square model. This is an interesting and somewhat surprising result since, usually, a simultaneous equations model with only simultaneity e ect needs IV (that is an exclusion restrictions on the coe cients of exogenous variables) for identi cation. Here we show that, for a network model, this is not the case. All these identi cation issues are fully studied in Section 3. Section 4 proceeds with the empirical implementation of our models. In Section 4.1, we describe our data. We use the widely known Add-Health data for adolescents in the United States. We then test the di erent models for two activities: education (i.e. grades) and time spent watching TV or videos or playing video games and expose the empirical results. We nd, in particular, that, keeping peers grades and screen activities xed, watching more TV could be bene cial to a student s grade. We conclude in Section 5. In Appendix A, we list the matrices whose rank conditions are used for the identi cation of the square model. In Appendix B, we present a more general econometric network model with m choices and give details of the generalized spatial 2SLS and 3SLS estimators. Appendix C collects all the proofs for the identi cation results in Section 3. In Appendix D, we discuss the identi cation and estimation of the network model with network-speci c xed e ects. 5

6 2 Theoretical Model 2.1 A general model There is a nite set of agents N = f1; :::; ng connected by a network g g N. We keep track of social connections in a network g through its adjacency matrix G = [g ij ], where g ij = g ij= j=1 g ij with g ij = 1 if i and j are friends and g ij = 0 otherwise. 2 We set g ii = 0. Observe that G is a row-normalized matrix such that each row of G sums to one. An example is given in Figure 1 for a tree network with four agents G = Figure 1: an example network and the corresponding adjacency matrix. As stated in the Introduction, agents choose e ort levels for more than one activity. We denote by y k = (y k1 ; ; y kn ) 0 the vector of e ort levels for the kth activity of agents in the network. Choices are made from elements of some set of possible choices (e ort levels), k. This set is both individual- and network-speci c. For every individual i, we must track the choices of other agents in the network. We express the model with a linear quadratic utility function. Consider a structural model where individual i in the network chooses m actions to maximize his/her utility. For the ease of presentation, we focus on m = 2 without loss of generality. Let u(y 1i ; y 2i ) = 1iy 1i + 2iy 2i 1 2 1y 2 1i 1 2 2y 2 2i + y 1i y 2i (1) + 11 j=1 g ijy 1i y 1j + 22 j=1 g ijy 2i y 2j + 21 j=1 g ijy 1i y 2j + 12 j=1 g ijy 2i y 1j ; where y 1i and y 2i are individual i s e orts in activities 1 and 2. As in the standard linear-quadratic utility model with one activity (Ballester et al., 2006), the rst ve terms of the utility function correspond to the impact of own characteristics on own e ort levels while the last four terms express 2 For ease of presentation, we assume that the network is undirected and no agent is isolated so that j=1 g ij 6= 0 for all i. The identi cation results of the paper hold for directed networks. 6

7 the impact of direct peers on own e orts. Apart from the fact that there are two activities and di erent level of heterogeneities, the main di erence with the standard model is that there are new cross-e ects, which are expressed by the parameters, 21 and The standard cross-e ects between own and peer e orts for the same activity 2 u(y 1i ; y 2i 1j = g ij 11 u(y 1i ; y 2i 2j = g ij 22; (2) indicating strategic substitutability or complementarity depending on the signs of 11 and 22. The new cross-e ects are between own e orts for di erent activities, 2 u(y 1i ; y 2i 2i u(y 1i ; y 2i 1i = ; (3) and between own and peer e orts for di erent activities, 1i ; y 2i 2j = g ij 21 1i; y 2i 1j = g ij 12; (4) which also indicate strategic substitutability or complementarity depending on the signs of 21 and 12. The rst order conditions of utility maximization are given 1i ; y 2i 1i = 1i 1y 1i + y 2i + 1i ; y 2i 2i = 2i 2y 2i + y 1i + 22 j=1 g ijy 1j + 21 j=1 g ijy 2j = 0 (5) j=1 g ijy 2j + 12 j=1 g ijy 1j = 0: (6) From (5) and (6), we have the equilibrium best-reply functions as: y 1i = 1i + 1 y 2i + 11 j=1 g ijy 1j + 21 j=1 g ijy 2j (7) y 2i = 2i + 2 y 1i + 22 j=1 g ijy 2j + 12 j=1 g ijy 1j ; (8) where 1i = 1i = 1, 1 = = 1, 11 = 11= 1, 21 = 21= 1, 2i = 2i = 2, 2 = = 2, 3 Because, in the empirical part, we focus on the identi cation of peer e ects for the row-normalized adjacency matrix G, our utility function also has this feature. In terms of economic interpretation, this means that it is the average behavior of peers (local-average model) that matters for own e ort and not the sum of their e orts (localaggregate model). In terms of utility function, an interpretation of the local-average model is that individuals conform to the average behavior of their peers and that deviating from this social norm could be costly (Liu, Patacchini and Zenou, 2012; Patacchini and Zenou, 2012). 7

8 22 = 22= 2, and 12 = 12= 2. In matrix form, the equilibrium best-reply functions are: y 1 = y Gy Gy 2 (9) y 2 = y Gy Gy 1 ; (10) where y k = (y k1 ; ; y kn ) 0 and k = ( k1 ; ; kn ) 0 for k = 1; 2. The corresponding reduced form equations are: S y 1 = (I 22 G) 1 + ( 1 I + 21 G) 2 S y 2 = (I 11 G) 2 + ( 2 I + 12 G) 1 ; where I is the identity matrix and S = (1 1 2 )I ( )G ( )G 2 (11) Analogous to the single-activity network model (Ballester et al., 2006), if S is invertible, the peer e ect game with payo s (1) has a unique pure-strategy Nash equilibrium. Proposition 1 Assume that G is row-normalized. If j 1 2 j + j j + j j < 1; (12) then, the peer e ect game with payo s (1) has a unique interior Nash equilibrium in pure strategies given by: y 1 = S 1 [(I 22 G) 1 + ( 1 I + 21 G) 2 ] y 2 = S 1 [(I 11 G) 2 + ( 2 I + 12 G) 1 ] This is an important result that generalizes the main theorem of Ballester et al. (2006) for more than one activity. What is striking is that, as in Ballester et al. (2006), only parameters and! 1 (G), the largest eigenvalue of G, are needed for the existence and uniqueness of an interior Nash 8

9 equilibrium. In other words, we have again an upper bound on! 1 (G). This theoretical model helps us understand agents behavior when making more than one choice. First, this model helps analyze how a standard network model of social interactions can be generalized. Second, it provides economic fundamentals for the empirical work below. This will be very helpful when we will interpret our empirical results. 2.2 More speci c models Before moving to econometric analysis, we present a bit more on the behavioral foundation of the speci c models we study The seemingly unrelated simultaneous equations model Consider a model where = 12 = 21 = 0. In that case, the utility function can be written as: u(y 1i ; y 2i ) = 1iy 1i + 2iy 2i 1 2 1y 2 1i 1 2 2y2i j=1 g ijy 1i y 1j + 22 j=1 g ijy 2i y 2j : (13) This is a model where only the peer-e ects (2) within the same activity matter while the crosse ects (3) and (4) between the two activities are not taken into account. 4 In that case, equilibrium best-reply functions are equal to: y 1i = 1i + 11 j=1 g ijy 1j ; (14) y 2i = 2i + 22 j=1 g ijy 2j : (15) This is very close to the model of Ballester et al. (2006) where people are in uenced only by others within the same activity. Suppose y 1i is the time spent studying and y 2i is the time spent watching TV for individual i. Then, keeping 1i (say, family size, parental education, etc.) xed, individual i will spend more time studying (y 1i ) if his/her friends also spend more time studying (if 11 > 0). Similarly, keeping 2i xed, individual i will spend more time watching TV (y 2i ) if his/her friends also watch more TV (if 22 > 0). In this formulation, it is assumed that individual i choice on studying time is independent of her choice on TV time or the time that individual i 0 s friends watch, and vice versa. This form of the model is conformable to the assumption that TV time choices are exogenous to the studying decision, but this is clearly a very strong assumption. 4 This is a model that has been considered from a theoretical viewpoint by Belhaj and Deroïan (2012) for the speci c case when y 1i + y 2i = 1, 8i, 11 > 0 and 22 > 0. 9

10 2.2.2 The triangular simultaneous equations model Let us now consider a utility function for which = 12 = 0. We have: u(y 1i ; y 2i ) = 1iy 1i + 2iy 2i 1 2 1y 2 1i 1 2 2y 2 2i + 11 j=1 g ijy 1i y 1j + 22 j=1 g ijy 2i y 2j + 21 j=1 g ijy 1i y 2j : In this formulation, we still have peer e ects between own e orts for di erent activities, i.e. (2), but we have an asymmetry for the cross e ects between own and peer e orts for di erent activities. In other words, it is assumed u(y 1i;y 1i@y 2i u(y 1i;y 2i@y 2i@y 1j = 0. In this case, the best-reply functions are equal to: y 1i = 1i + 11 j=1 g ijy 1j + 21 j=1 g ijy 2j ; (16) y 2i = 2i + 22 j=1 g ijy 2j : (17) If we keep the same interpretation as above, keeping 1i xed, the more time his/her friends spend studying (if 11 > 0) and the less time his/her friends spend watching TV (if 21 < 0), then the more time individual i will spend studying (y 1i ). On the other hand, keeping 2i xed, the more his/her friends watch TV (if 22 > 0) the more individual i will watch TV (y 2i ). In this formulation, it is assumed that the time spent doing homework is in uenced by how much time one s friends are watching TV while the reverse is not true, i.e. the time spent watching TV is not directly a ected by how much time one s friends spend doing homework The square simultaneous equations model For the square system of simultaneous equations, we consider three speci cations. First, we consider a utility function for which = 0, so that (1) becomes u(y 1i ; y 2i ) = 1iy 1i + 2iy 2i 1 2 1y 2 1i 1 2 2y 2 2i + 11 j=1 g ijy 1i y 1j + 22 j=1 g ijy 2i y 2j + 21 j=1 g ijy 1i y 2j + 12 j=1 g ijy 2i y 1j : 10

11 In this case, the best-reply functions are y 1i = 1i + 11 j=1 g ijy 1j + 21 j=1 g ijy 2j ; (18) y 2i = 2i + 22 j=1 g ijy 2j + 12 j=1 g ijy 1j ; (19) For this speci cation, we assume how much time individual i will spend studying or watching TV would depend on how much time his/her friends spend studying and watching TV. However, this speci cation does not allow individual i s choice in one activity to be directly a ected by his own choice in a related activity. For the second speci cation, we let 12 = 21 = 0 in the utility function (1) so that u(y 1i ; y 2i ) = 1iy 1i + 2iy 2i 1 2 1y 2 1i 1 2 2y 2 2i + y 1i y 2i + 11 j=1 g ijy 1i y 1j + 22 j=1 g ijy 2i y 2j : In this case, the best-reply functions are y 1i = 1i + 1 y 2i + 11 j=1 g ijy 1j ; (20) y 2i = 2i + 2 y 1i + 22 j=1 g ijy 2j : (21) Suppose, as above, that y 1i is the time spent studying and y 2i is the time spent watching TV for individual i. Then, for this speci cation, how much time individual i will spend studying would depend on how much time he/she spends watching TV and how much time his/her friends spend studying. Similarly, how much time individual i will spend watching TV would depend on how much time he/she spends studying and how much time his/her friends spend watching TV. However, this speci cation assumes that how much time individual i will spend studying is not a ected by how much time his/her friends spend watching TV, and vice versa. Finally, we consider the most general model where we don t impose any restriction on the parameters. In that case, we end up with the utility function (1), where the equilibrium best-reply functions are given by (7) and (8). This is clearly the most interesting case since all activities in- uence each other. In other words, my time spent studying is a ected by how much time I spend watching TV and by how much time my friends spend studying and watching TV. Similarly, my 11

12 time spent watching TV is also a ected by how much time I spend studying and by how much time my friends spend studying and watching TV. 2.3 Model identi cation challenges As with most models in the peer e ects literature, a host of identi cation issues emerge. Here, we are concerned both with the re ection problem for each choice that is endemic in peer e ect research as well as with the complexities that emerge in estimating two or more equations simultaneously. Our speci cation of the econometric model follows closely from the equilibrium best response functions (7) and (8) for the general theoretical model. Indeed, denote x i = (x 1i ; ; x kx;i) 0 a 1k x vector of observable individual-speci c characteristics, j=1 g ij x j, a 1 k x vector of observable network-speci c characteristics (i.e. contextual e ects), and ki, a random individual-speci c characteristics associated with i. Let k and k be two k x 1 vectors of parameters. Then, ki in (7) and (8) can be written as ki = x i k + j=1 g ijx j k + ki ; for k = 1; 2. The econometric simultaneous equations model corresponding to (7) and (8) is then given by: y 1i = 1 y 2i + 11 j=1 g ijy 1j + 21 j=1 g ijy 2j + x i 1 + j=1 g ijx j 1 + 1i ; (22) y 2i = 2 y 1i + 22 j=1 g ijy 2j + 12 j=1 g ijy 1j + x i 2 + j=1 g ijx j 2 + 2i : (23) Our interest is to identify and estimate the various e ects in the model, which are: 5 Endogenous e ect and contextual e ect The endogenous e ect, where an individual s choice may depend on his/her peers choices on the same activity, is captured by the coe cients 11 and 22. The contextual e ect, wherein an individual s choice may depend on his/her peers exogenous characteristics, is captured by 1 and 2. The re ection problem (Manski, 1993) is well known and emerges from the coexistence of those two e ects. In Manski s linear-in-means model, individuals are a ected by all individuals belonging to their group and by nobody outside the group, and thus simultaneity in behavior of individuals in the same group introduces a perfect collinearity between the endogenous e ect and the contextual 5 For very complete overviews of identi cation problems in peer e ects and how to solve them, see Durlauf and Ioannides (2010), Blume et al. (2011) and Ioannides (2012), 12

13 e ect. Hence, those two e ects cannot be separately identi ed in the linear-in-means model. Among the many methods to resolve this problem, one can use the complexity of social networks to identify the peer e ect. In a social network, the assumption is that individuals are no longer impacted evenly by the full population in the sample; instead, they are in uenced by their friends or connections. Bramoullé et al. (2009) have studied this phenomenon and they have shown that one can identify the endogenous and contextual e ects if intransitivities exist in a network so that I; G; G 2 are linearly independent. Intuitively, if individuals i; j are friends and j; k are friends, it does not necessarily imply that i; k are also friends. Because of these intransitivities, the characteristics of an individual s indirect friends are not collinear with his/her own characteristics and the characteristics of his/her direct friends. Therefore, the characteristics of an individual s indirect friends can be used as instruments to identify the endogenous e ect from the contextual e ect. The seemingly unrelated simultaneous equations model only considers the endogenous e ect and the contextual e ect for each choice. Although we allow the choices for related activities to be correlated through unobservables (i.e. the error terms of simultaneous equations), the identi cation condition will be shown in Section 3 to be exactly the same as that in Bramoullé et al. (2009). While this does not yet address the core issue we study, it relaxes the assumption in baseline models in the literature that choices such as studying and smoking are uncorrelated. This has no impact on the Bramoullé et al. (2009) identi cation conditions as they simply use the mean of the reduced form equation, which is una ected by the cross-equation correlation in the error terms. Simultaneity e ect and cross-choice peer e ect The standard economic simultaneity e ect can be seen in the 1 and 2 coe cients of our general simultaneous equations model. Thus, an individual s choice on a certain activity may a ect his/her choices on related activities. In the absence of social in uence, the simultaneity problem is a well known problem for the identi cation of a simultaneous equations model. The usual remedy for this identi cation problem is to impose exclusion restrictions on the coe cients of exogenous variables x i. Furthermore, a central component of our paper is that we allow peers choices to impact an individual s decisions on related activities (e.g. more TV watching by friends may lead to less or more studying by an individual). We model this through the cross-choice peer e ect. The coe cients 21 and 12 precisely describe how the choices of peers over other activities in uences an individual s 13

14 decisions on a given activity. We illustrate the additional layer of complication in identi cation that emerges with these terms. We do so by considering a triangular model with a one-way cross-choice peer e ect, a square model with two-way cross-choice peer e ects, and a square model with simultaneity e ect. We will show below that to identify the simultaneity e ect or the cross-choice peer e ect from endogenous and contextual e ects, we need to further exploit the set of exclusion restrictions from the intransitivities that exist in a natural network of friendships. In Section 3, we formally show that the triangular model and the restricted square models can be identi ed when I; G; G 2 ; G 3 are linearly independent. The intuition of the identi cation condition is as follows. To identify one additional structural parameter ( 1 ( 2 ) or 21 ( 12 )), we need an additional exclusion restriction of some type. We draw then again on the fact that the social network has intransitivities of di erent degrees. Note that G 2 represents the second degree links in the network, and G 3 represents the third degree links. Therefore, the linear independence of I; G; G 2 ; G 3 implies that if individuals i; j are friends, j; k are friends and k; l are friends, k and l are not necessarily friends of i. In precisely the same way that the linear independence of I; G; G 2 provides an exclusion restriction to identify the endogenous e ect from the contextual e ect, the linear independence of I; G; G 2 ; G 3 provides an additional exclusion restriction to identify the simultaneity e ect or the cross-choice peer e ect from endogenous and contextual e ects. However, for the general square model with both simultaneity and cross-choice peer e ects, the intransitivities in the network structure would not be enough to identify the various social interaction e ects. Hence, to achieve identi cation, we may need to impose some exclusion restrictions on the coe cients of x i. Network correlated e ect and cross-choice correlated e ect The structure of the general simultaneous equations model is exible enough to allow us to incorporate two types of correlated e ects. First, the individuals in the same network may behave similarly as they have similar unobserved individual characteristics and they face similar institutional environment. We call this type of correlated e ect the network correlated e ect. The network correlated e ect can captured in our general simultaneous equations model by introducing a network xed e ect. The network xed e ect 14

15 can be interpreted as originating from a two-step link formation model, where individuals self-select into di erent networks in a rst step with selection bias due to network-speci c characteristics and, then, in a second step, link formation takes place within networks based on observable individual characteristics only. Therefore, network xed e ects serve as a (partial) remedy for the bias that originates from the possible sorting of individuals into networks. To estimate the simultaneous equations model with network xed e ects, we use a within transformation that eliminates the network xed e ects by subtracting the network average from the individual-level variables. The details of the estimator and identi cation results are presented in Appendix D. Second, the decisions of the same individual on related activities may be correlated. We call this type of correlated e ect the cross-choice correlated e ect. The cross-choice correlated e ect is introduced by allowing the error terms ki s to be correlated across equations. As our identi- cation results are based on the mean of reduce-form equations, they will not be a ected by the across-equation correlation in the error term. However, for estimation e ciency, it is important to consider the correlation structure of error terms. Following Kelejian and Prucha (2004), we adopt a generalized spatial 3SLS (GS3SLS) estimator for the estimation of our model. The details of the estimator are given in Appendix B.3. 3 Identi cation of Econometric Model 3.1 General setup Consider a data set containing n agents, partitioned into r networks such that each network has n r agents (r = 1; ; r) and P r r=1 n r = n. 6 Links between agents are captured by an n r n r zero-diagonal row-normalized adjacency matrix G r = [gij;r ] de ned as in the previous section. The equilibrium best response functions (22) and (23) can be rewritten in matrix form as 7 y 1;r = 11 G r y 1;r + 1 y 2;r + 21 G r y 2;r + X r 1 + G r X r 1 + 1;r (24) y 2;r = 22 G r y 2;r + 2 y 1;r + 12 G r y 1;r + X r 2 + G r X r 2 + 2;r : (25) In this model, for k = 1; 2, y k;r = (y k1 ; ; y knr ) 0 is an n r 1 vector of cross sectional observations on the kth decision. X r = (x 0 1; ; x 0 n r ) 0 is an n r k x matrix of exogenous variables. 8 k;r = 6 For simplicity, we assume no agent is isolated. 7 For ease of presentation, we rst present the model without the network xed e ect. The identi cation results for the model with network xed e ects are collected in Appendix D. 8 Without loss of generality, henceforth we assume k x = 1 so that there is a unique exogenous variable in the model 15

16 ( k1 ; ; knr ) 0 is an n r 1 vector of disturbances. We assume ( 1;r ; 2;r ) = V r 1=2, where V r = [v ki;r ] is an n r 2 matrix of I.I.D. innovations with zero mean and unit variance and = is a positive semi-de nite symmetric matrix. Thus, we allow the error terms of decisions made by the same agent to be correlated, which is captured by 12. For all r networks in the sample, we have, for k = 1; 2, y k = (y 0 k;1 ; ; y0 k;r )0, k = ( 0 k;1 ; ; 0 k;r )0, X = (X 0 1; ; X 0 r) 0, and G = diagfg r gr=1. r Then, y 1 = 11 Gy y Gy 2 + X 1 + GX (26) y 2 = 22 Gy y Gy 1 + X 2 + GX ; (27) with reduced form equations SE(y 1 ) = X( ) + GX( ) + G 2 X( ) (28) SE(y 2 ) = X( ) + GX( ) + G 2 X( ); (29) where S = (1 1 2 )I ( )G + ( )G 2. As ( 1 ; 2 ) = V 1=2, where V is an n 2 matrix of I.I.D. innovations with zero mean and unit variance, E( 0 ) = I n for = ( 0 1; 0 2) 0. Let Z 1 = [Gy 1 ; y 2 ; Gy 2 ; X; GX], Z 2 = [Gy 2 ; y 1 ; Gy 1 ; X; GX], and Q denote the IV matrix based on G; X and their functions. 9 Then, the model is identi ed if the following condition is satis ed. Identi cation Assumption lim n!1 1 n Q0 E(Z k ) is a nite matrix which has full column rank for k = 1; 2. This identi cation assumption implies the rank condition that E(Z k ) has full column rank and that the column rank of Q at least as high as that of E(Z k ), for large enough n. In the rest of this section, we provide some su cient conditions for the identi cation assumption to hold. This simultaneous equation model not only incorporates the endogenous e ect (through 11 as in Bramoullé et al. (2009). 9 In this paper, we assume G; X are nonstochastic. This assumption can be easily relaxed and the results will be conditional on G; X. 16

17 and 22 ) and the contextual e ect (through 1 and 2 ) as in a standard single equation network model, but also includes the simultaneity e ect (through 1 and 2 ), wherein an individual s decision depends on his/her decisions on related activities, the cross-choice peer e ect (through 21 and 12 ), wherein an individual s decision depends on his/her peers decisions on related activities, and the cross-choice correlated e ect through the correlation in 1 and 2. The general model structure will allow us to illustrate how the various e ects operate on a empirical basis, and to show the identi cation requirements for each e ect. In the following subsections, we articulate identi cation conditions of three di erent versions of the general simultaneous equations model, corresponding to the three speci c theoretical models discussed in the previous section, which amount to increasing degrees of interdependence in decision making. 3.2 The seemingly unrelated regressions (SUR) model First, we consider the econometric model under the restrictions 1 = 2 = 12 = 21 = 0, such that (26) and (27) reduce to y 1 = 11 Gy 1 + X 1 + GX (30) y 2 = 22 Gy 2 + X 2 + GX : (31) In this case, an individual s decision is still allowed to be correlated with his/her other decisions through the error term. Without loss of generality, we focus on the identi cation of equation (30). The identi cation condition for (31) can be analogously derived. Let Z 1 = [Gy 1 ; X; GX]. For the identi cation assumption to hold, E(Z 1 ) needs to have full column rank. The following proposition shows that this rank condition relies on some topological properties of the network. 10 Proposition 2 Suppose = 0. Then E(Z 1 ) of (30) has full column rank if and only if I; G; G 2 are linearly independent. From the reduced form equation of (30), we have 10 All proofs can be found in Appendix C. E(y 1 ) = X 1 + P 1 j=0 j 11 Gj+1 X( ): (32) 17

18 If = 0, then E(Gy 1 ) = GX 1. In this case, model (30) can not be identi ed because the matrix E(Z 1 ) = [E(Gy 1 ); X; GX] does not have full column rank. This corresponds to the case where the endogenous e ect and exogenous e ect exactly cancel out. Lee et al. (2010) have shown, when = 0, the reduced form of (30) becomes a simple regression model with neither endogenous nor contextual e ects. Interdependence across individuals goes through unobservables (correlated disturbances) instead of observables. A special case of = 0 would be 1 = 1 = 0. In this case, E(Gy 1 ) = 0, and the model cannot be identi ed as there is no relevant exogenous covariate in the model that can be used as an instrument for the endogenous e ect. On the other hand, if = 0, Bramoullé et al. (2009) have shown that the single-equation network model can be identi ed if intransitivities exist in a network so that I; G; G 2 are linearly independent. In this case, G 2 X is not perfectly collinear with X and GX and thus can be used as an instrument for the endogenous e ect. For the SUR model, although the error term of (30) is allowed to be correlated with the error term of (31), the identi cation condition for (30) remains the same as that for the single-equation network model given by Bramoullé et al. (2009). The presence of two equations with correlate errors does not inhibit identi cation. This is because the identi cation condition given by Bramoullé et al. (2009) is based on the mean of the reduced form equation (32), which is not a ected by the correlation structure of the error terms. 3.3 The triangular simultaneous equations model Next, we will consider the econometric model under the restrictions 1 = 2 = 12 = 0. This shuts down the simultaneity e ect and one direction of the cross-choice peer e ect. It retains the endogenous and contextual e ects associated with each choice and retains the need to solve for the cross-choice peer e ect in one direction. In this case, (26) and (27) become y 1 = 11 Gy Gy 2 + X 1 + GX (33) y 2 = 22 Gy 2 + X 2 + GX : (34) From Proposition 2, the unknown parameters in equation (34) can be identi ed if = 0 and I; G; G 2 are linearly independent. We can next consider the identi cation condition for equation (33). Let Z 1 = [Gy 1 ; Gy 2 ; X; GX]. For the identi cation of equation (33), we need to nd a su cient condition for E(Z 1 ) to have full column rank. 18

19 Proposition 3 Suppose = 0 and ( )( ) + 21 ( ) 6= 0: (35) Then E(Z 1 ) of model (33) has full column rank if and only if I; G; G 2 ; G 3 are linearly independent. A special case of (35) is of particular interest. If 21 = 0 and 11 = 22 so that the data generating process (DGP) is a pair of SUR equations with identical endogenous e ects, then, according to (35), (33) cannot be identi ed. Indeed, if 21 = 0 in the DGP, E(Z 1 ) = [GE(y 1 ); GE(y 2 ); X; GX], where GE(y 1 ) = GX 1 + (I 11 G) 1 G 2 X( ); GE(y 2 ) = GX 2 + (I 22 G) 1 G 2 X( ): Thus, when 11 = 22, E(Z 1 ) does not have full column rank and the triangular model cannot be identi ed. On the other hand, if 11 6= 22, = 0, and = 0, the triangular model can be identi ed if I; G; G 2 ; G 3 are linearly independent. (a) (b) (c) (d) Figure 2: networks where the SUR model can be identi ed but the triangular model cannot. By comparing Proposition 3 with Proposition 2, we can see that the identi cation of the crosschoice peer e ect in the triangular model has higher requirement in terms of intransitivities than the SUR model. Suppose i and j are friends, j and k are friends and k and l are friends. Then, the identi cation of the SUR model requires that k is not a friend of i and the identi cation of the triangular model requires that k is not a friend of i and l is not a friend of i or j. Figure 2 lists a few examples from Bramoullé et al. (2009) where the SUR model can be identi ed but the triangular model cannot. It is easy to check that for all four networks in Figure 2, I; G; G 2 are linearly independent. However, for networks (a) and (c), G 3 = G; for network (b), G 3 = 1 4 I G + G2 ; and for network (d), G 3 = 1 8 I G G2. That is, for those networks, no one has a third order friend that is not his/her direct or second order friend. Therefore, it follows by Propositions 2 and 19

20 3 that, for the networks in Figure 2, the SUR model can be identi ed but the triangular model cannot. 3.4 The square simultaneous equations model To better understand the identi cation challenges brought by the simultaneity e ect and the crosschoice peer e ect, we consider three speci cations of the square model. In the rst case we only include the cross-choice peer e ect and in the second case we only include the simultaneity e ect. The third case incorporates both simultaneity and cross-choice peer e ects The square model with cross-choice peer e ects First, we will consider the model under the restrictions 1 = 2 = 0. Thus, (26) and (27) reduce to: y 1 = 11 Gy Gy 2 + X 1 + GX (36) y 2 = 22 Gy Gy 1 + X 2 + GX ; (37) The following proposition gives a su cient condition for the identi cation of (36). The su cient condition for the identi cation of (37) can be analogously derived. Let Z 1 = [Gy 1 ; Gy 2 ; X; GX]. Proposition 4 If I; G; G 2 ; G 3 are linearly independent and A 1 given by (42) has full rank, then E(Z 1 ) of model (36) has full column rank. Similarly to the triangular model with a one-way cross-choice e ect, Proposition 4 shows that the two-way cross-choice peer e ect in a square model can be identi ed through intransitivities that exist in a network The square model with simultaneity e ects Next, we will consider the model under the restrictions 12 = 21 = 0: In this case, (26) and (27) become: y 1 = 11 Gy y 2 + X 1 + GX (38) y 2 = 22 Gy y 1 + X 2 + GX ; (39) The following proposition gives a su cient condition for the identi cation of (38). The su cient condition for the identi cation of (39) can be analogously derived. Let Z 1 = [Gy 1 ; y 2 ; X; GX]. 20

21 Proposition 5 If I; G; G 2 ; G 3 are linearly independent and A 2 given by (43) has full rank, then E(Z 1 ) of model (38) has full column rank. Without social interaction e ects, the simultaneity problem is a well known problem for the identi cation of a simultaneous equations model. The usual remedy for this identi cation problem is to impose exclusion restrictions on the coe cients of exogenous variables X. Proposition 5 shows that, in the absence of cross-choice peer e ects, the simultaneity e ect in a simultaneous equations network model can be identi ed through intransitivities in G instead of imposing exclusion restrictions on the coe cients of X The general square model Finally, we consider the identi cation of the model with both simultaneity and cross-choice peer e ects. The following proposition show that, without imposing any exclusion restrictions, the general square model cannot be identi ed. Proposition 6 The system of simultaneous equations (26) and (27) cannot be identi ed. Proposition 6 shows that, for a simultaneous equations model with both simultaneity and crosschoice peer e ects, exploiting the exclusion restrictions from the intransitivities that exist in a natural network is not su cient for the identi cation. One possibility to achieve identi cation is to impose exclusion restrictions on the coe cients of exogenous variables. To clarify ideas, we consider the following model y 1 = 11 Gy y Gy 2 + X GX (40) y 2 = 22 Gy y Gy 1 + X GX ; (41) where X 1 and X 2 are column vectors and X 1 6= X 2. The following proposition gives a su cient condition for the identi cation of (40). The su cient condition for the identi cation of (41) can be analogously derived. Let Z 1 = [Gy 1 ; y 2 ; Gy 2 ; X 1 ; GX 1 ] and 0 ; 1 ; 2 be generic constant terms. Proposition 7 If (i) I; G; G 2 ; G 3 are linearly independent and A 3 given by (44) has full rank, or (ii) I; G; G 2 are linearly independent, G 3 = 0 I + 1 G + 2 G 2 and A 3 given by (45) has full rank, then E(Z 1 ) of (40) has full column rank. 21

22 Note that a su cient condition for A 3 in (44) to have a full rank is that 2 ; 2 are not both zeros, = 0 and = 0. 4 Empirical Application 4.1 Data We use for this analysis a unique and now widely used data set provided by National Longitudinal Survey of Adolescent Health (AddHealth). The AddHealth data set provides detailed information on a large number of high school students including self-reported friendship networks. 11 This database was designed to study the impact of the social environment on adolescent behavior. The program collected national representative information on 7th-12th graders in both public and private school settings. The survey was conducted in and was designed to capture information on friends, family, school and neighborhood in uences on students behaviors, including academic performance, social decisions, extracurriculars, dangerous behaviors and more. Every student attending schools on the sampling day was provided with a questionnaire that covered topics on demographics, behavioral characteristics, education, family background and critically for our purposes, friendships. This inschool survey sample included more than 90,000 students. A follow-up, more focused study covered a subset of the total sample with approximately 20,000 students. This more detailed survey included an in-home questionnaire as well as a supplemental survey with more sensitive questions. Our empirical study focus on the 20,000 students in the in-home survey study. As illustrated above, we will mainly focus here on two activities: education (i.e. GPA) and screen activities. The former (GPA) is the average of the student most recent grades in English, science, math and history. The latter is the response to the question: During the past week, how many times did you watch television or videos, or play video games?, coded as 0 if not at all, 1 if one or two times, 2 if three or four times, and 3 if ve times or more. The adjacency matrix G = [g ij ], where g ij = g ij= j=1 g ij, is constructed based on the friendnomination information of the AddHealth data. In the questionnaire, students were asked to identify their best friends from a school roster, and then list up to ve boys and ve girls from this list. We 11 This research uses data from Add Health, a program project designed by J. Richard Udry, Peter S. Bearman, and Kathleen Mullan Harris, and funded by a grant P01-HD31921 from the National Institute of Child Health and Human Development, with cooperative funding from 17 other agencies. Special acknowledgment is due Ronald R. Rindfuss and Barbara Entwisle for assistance in the original design. Persons interested in obtaining data les from Add Health should contact Add Health, Carolina Population Center, 123 W. Franklin Street, Chapel Hill, NC (addhealth@unc.edu). No direct support was received from grant P01-HD31921 for this analysis. 22

23 assume friendship is reciprocal. Thus, for students i and j (i 6= j), g ij = 1 if either i nominates j or j nominates i as a friend and g ij = 0 otherwise. After removing students with no friends, the sample consists of 9,065 students distributed over 1,964 networks, with network size ranging from 2 to Because the strength of peer e ect may vary with network size (see Calvó-Armengol et al., 2009), we exclude networks at the extremes of the network size distribution and focus our analysis on moderate-sized networks with network size between 10 and 200. Our selected sample consists of 1,806 students distributed over 56 networks, with network size ranging between 10 and 145. The mean and the standard deviation of network size are and Furthermore, in our sample, the average number of friends of a student is 3.18 with the standard deviation A detailed data description is available in Table 1. [Insert Table 1 here] 4.2 The SUR case First, we model a student s GPA and screen activities using the SUR equations (30) and (31) with network-speci c xed e ect. 12 To avoid the incidental parameter problem induced by the network xed e ect dummy variables, we perform a within transformation to get rid of network dummy variables by subtracting the network average from the individual-level variables. Then, we estimate the model using Kelejian and Prucha s (2004) generalized spatial 2SLS (GS2SLS) with the IV matrix Q = [X; GX; G 2 X]. The GS2SLS approach estimates the simultaneous equations model equation by equation. It is not e cient as it does not take into account the cross-equation correlation in the error term. To utilize the full system information, we also consider the generalized spatial 3SLS (GS3SLS) estimator, which jointly estimate the simultaneous equations (30) and (31). The GS3SLS estimator can be considered as a generalized least squares version of the GS2SLS estimator, which take the cross-equation correlation of the disturbances into account. The details of the estimators are given in Appendix B.3. Table 2 reports the estimation results. For both GS2SLS and GS3SLS estimators, the well documented impact of peer GPA is observed to be quite strong. In other words, the higher is the average grades of an individual s friends, the higher is his/her own grade. Ceteris paribus, a student s GPA increases by 0.7 if the average peer GPA increases by 1 point. This result is in line with most 12 The details on the speci cation and identi cation of the simultaneous equations model with network xed e ects are given in Appendix D. 23