aa r X i v : . [ ec on . E M ] A ug PEER EFFECTS AND ENDOGENOUS SOCIAL INTERACTIONS
KOEN JOCHMANS ∗ UNIVERSITY OF CAMBRIDGE
Abstract
We introduce an approach to deal with self-selection of peers in the linear-in-meansmodel. Contrary to the existing proposals we do not require to specify a model forhow the selection of peers comes about. Rather, we exploit two restrictions that areinherent to many such specifications to construct intuitive instrumental variables.These restrictions are that link decisions that involve a given individual are notall independent of one another, but that they are independent of the link behaviorbetween other pairs of individuals. We construct instruments from the subnetworkobtained on leaving-out all one’s own link decisions in a manner that is reminiscentof the approach Bramoull´e, Djebbari and Fortin (2009) when the assignment of peersis assumed exogenous. A two-stage least-squares estimator of the linear-in-meansmodel is then readily obtained.
Keywords: instrumental variable, network, self-selection
JEL classification:
C31, C36 ∗ Address: University of Cambridge, Faculty of Economics, Austin Robinson Building, Sidgwick Avenue,Cambridge CB3 9DD, United Kingdom. E-mail: [email protected] . Financial support from the EuropeanResearch Council through grant n o ntroduction The importance of acknowledging the existence of social interactions between agents in theestimation of causal relationships is now widely acknowledged. In a program-evaluationproblem, for example, non-treated individuals can nonetheless benefit from the programthrough spillovers from treated units with whom they interact. Examples of this are detailedin Miguel and Kremer (2004), Sobel (2006), and Angelucci and De Giorgi (2009). A keyconcern when estimating models that feature peer effects is that agents may self-selecttheir peers, and do so based on (unobserved) factors that equally feature in the equation ofinterest, thus creating an endogeneity problem. Randomized assignment to peer groups hasproven useful in circumventing this threat to identification (Sacerdote 2001 contains anyearly application of this strategy) but this is, of course, not possible in many situations.The literature has worked on approaches to deal with the self-selection problem in thelinear-in-means model of social interactions as introduced in Manski (1993) and analyzedby Bramoull´e, Djebbari and Fortin (2009) for the case where individuals interact througha general, but fixed, network. The current paper is an addition to this growing body ofwork.We consider a setting where data on a large number of networks is available. This isthe conventional viewpoint. Goldsmith-Pinkham and Imbens (2013), Qu and Lee (2015),and Hsieh and Lee (2016) proposed control-function approaches to deal with endogeneityof the network. While the details of each of the proposals are different, the idea is tocomplement the linear-in-means model with a full specification of the network-formationprocess. The chief limitation of such an approach lies in the fact that it is not robust tomisspecification of the link-formation process. In the alternative setting where a singlelarge network is observed, Auerbach (2019) and, concurrently, Johnsson and Moon (2019)developed more flexible, semiparametric, control-function approaches. These require a less Manski (1993, p. 537) provides a discussion on the incompatibility of the model with data obtainedfrom randomly sampling individuals. Work on inference under snowball sampling, albeit in a differentsetting, is in Leung (2019).
Setup
Our asymptotics will involve data on many networks but, for now, it suffices to consider asingle network.
Model
Consider an undirected network involving n agents. Let A denote its n × n adjacency matrix. Then( A ) i,j = i and j are connected0 otherwise . When a link exists between ( i, j ) we say that they are neighbors. As usual, we do notconsider agents to be linked with themselves, so matrix A has only zeros on its maindiagonal. It will be useful to have a notational shorthand for the row-normalized adjacencymatrix, H , say. Its entries are( H ) i,j = ( A ) i,j .P nj ′ =1 ( A ) i,j ′ if P nj ′ =1 ( A ) i,j ′ >
00 otherwise . Recall that H corresponds to the transition matrix of a random walk through our network.Moreover, ( H ) i,j is the probability that, when taking a single step, starting at agent i , wearrive at agent j . In the same way, ( H ) i,j is the probability of arriving in two steps, andso on.Let y i and x i denote scalar variables, observable for each agent. Our baseline model is y i = α + βx i + γ (cid:16)P nj =1 ( H ) i,j x j (cid:17) + ε i , where ε i is a mean-zero unobserved variable. Taking the regressor to be a scalar is doneonly for notational convenience. Here, β captures the direct effect of x i on y i while γ reflectsan indirect, spillover, effect from the covariate values of the neighbors. In matrix form wecan succinctly write y = α ι n + β x + γ Hx + ε , where y = ( y , . . . , y n ) ′ , ι n = (1 , . . . , ′ is the n -vector of ones, x = ( x , . . . , x n ) ′ , and ε = ( ε , . . . , ε n ) ′ . An extension of the baseline specification that accommodates endogenous5eer effects, where y i also depends on P nj =1 ( H ) i,j y j , gives rise to what we will call the fullmodel, y = α ι n + δ Hy + β x + γ Hx + ε , which is the workhorse linear-in-means model on general networks as studied in Bramoull´e, Djebbari and Fortin(2009) and De Giorgi, Pellizzari and Redaelli (2010). It will be useful to first present ourapproach in the baseline model. The extension to the full model will then be intuitive. Restrictions
Identification of the slope coefficients in our model is well-understood whenthe strict exogeneity condition E ( ε i | A , x ) = 0 (a.s.) holds. Here we relax this restrictionby allowing for dependence between the link decisions and the unobserved component inour model. We work with E ( ε i | A i , x ) = 0 (a.s.) , (1.1)where A i is the ( n − × ( n −
1) adjacency matrix of the subnetwork obtained from A ondeleting its i th row and its i th column. This condition implies unconditional moments thatcan be used in a two-stage least-squares procedure. For our instruments to be relevant wewill presume that E (( A ) i,j | A i , x ) = E (( A ) i,j | x ) (a.s.) . (1.2)This condition states that the link decisions of a given agent are not independent of oneanother, conditional on the covariate, and is natural in our context. Before turning to ourinstrumental-variable approach we provide motivation and justification for the conditionsin (1.1) and (1.2). Network formation
It is useful to start with the model for link decisions specified inAuerbach (2019) and Johnsson and Moon (2019). They stipulate that, for each pair ofagents ( i, j ), with i < j ,( A ) i,j = ( A ) j,i = h ( η i , η j ) > u i,j , (1.3)6here η , . . . , η n are independent scalar random variables, the u i,j are random shocks, and h is a conformable function. Link decisions are allowed to be endogenous because η i and ε i are allowed to be dependent. The shocks u i,j are independent of ( η i , η j ) and of ( ε i , ε j )The implication is that E ( ε i | A , x ) = E ( ε i | η i , x ) = E ( ε i | η i ) . Because ( A ) i,j depends on ( η i , η j ) all link decisions involving agent i correlate with ε i .However, for all i ′ = i and j = i , the link decision ( A ) i ′ ,j is independent of ε i . Consequently, E ( ε i | A i , x ) = E ( E ( ε i | A , x ) | A i , x ) = E ( E ( ε i | η i ) | η , . . . , η i − , η i +1 , . . . , η n ) = E ( ε i ) = 0 , meaning that our moment restriction in (1.1) holds in the model of Auerbach (2019) andJohnsson and Moon (2019). Furthermore, for all i ′ = i , ( A ) i,j and ( A ) i ′ ,j are dependent asthey are both functions of η j . Hence, there is predictive information about the former inthe latter, and (1.2) is satisfied.The stylized model in (1.3) can be generalized in a number of ways without jeopardisingthe validity of (1.1) and (1.2). The features that are embedded in it that are important forour purposes are that (i) η i only affects link decisions involving agent i ; (ii) the η , . . . , η n are independent conditionally on the x , . . . , x n ; and (iii) the decision to link depends oncharacteristics of both agents involved. These restrictions allow η i to be replaced by a vectorof agent-specific unobserved heterogeneity and permit link decisions to depend on a set ofadditional (observable or unobservable) variables. The function h could also be allowedto be pair-specific. These generalizations allow for excess heterogeneity and homophily ofunrestricted form.The specification in (1.3) is suitable when link formation is cooperative. A modifiedversion of it, suitable for non-cooperative situations, would be( A ) i,j = ( A ) j,i = h ( η i ) > u i,j and h ( η j ) > u j,i , (1.4) Auerbach (2019) and Johnsson and Moon (2019) also impose certain shape restrictions on the function h and an i.i.d. assumption on the u i,j to achieve identification in their setting but these are not importantfor our developments and, thus, not imposed here. u i,j and u j,i are pair-specific shocks. Clearly, such a specification also satisfiesall requirements for (1.1) and (1.2) to hold. Again, (1.4) can be extended in a variety ofways without compromising this.This discussion shows that (1.1) is implied by many commonly-used specifications fornetwork formation. The most important limitation of the requirements in (i)–(iii) is thatthey rule out situations where link decisions are interdependent. Transitivity, for example,where a pair of agents are more likely to be linked when they have more neighbors incommon, calls for a simultaneous-equation model. Such a design would violate (i), as ( A ) i,j will generally depend on all η , . . . , η n in such a case. Without access to panel data, as inGoldsmith-Pinkham and Imbens (2013), for example, dealing with such a design appearscomplicated. Start with the baseline model. Here, the self-selection of ones’ peers causes the spillovereffect P nj =1 ( H ) i,j x j to be endogenous because the weights ( H ) i, , . . . , ( H ) i,n correlate withthe unobserved component ε i . Because H is a row-normalized adjacency matrix, ( H ) i,j depends on all of ( A ) i, , . . . , ( A ) i,n . It does not depend on ( A ) i ′ ,j ′ for any of i ′ = i and j ′ = i , however. By (1.1), the link decisions that do not involve agent i are exogenous.Furthermore, by (1.2), these ( A ) i ′ ,j ′ are not independent of ( H ) i,j . This suggests theconstruction of instrumental variables by looking at linear combinations of x , . . . , x n , withweights coming from the leave-one-out network A i . A fruitful way of doing so is discussednext.For each i ′ we define the n × n matrix( H i ′ ) i,j = ( A ) i,j .P j ′ = i ′ ( A ) i,j ′ if i = i ′ and j = i ′ and P j ′ = i ′ ( A ) i,j ′ >
00 otherwise . This is the row-normalized version of the adjacency matrix A i ′ introduced previously,only complemented with one additional zero row and one additional zero column. Thisaugmentation is done for notational considerations, as it maintains the dimension of these8atrices to n × n . We stress that H i is not obtained from setting to zero the correspondingrow and column of H . We can interpret H i as the transition matrix on the networkobtained on ruling-out links that involve agent i . From (1.1), the entries of this matrix areuncorrelated with ε i . Furthermore, from (1.2), ( H i ) i ′ ,j ′ and ( H ) i,j are dependent for all( i, j ) and ( i ′ , j ′ ), conditional on the regressors.Recall that ( H ) i,j is the probability of arriving at agent j , from agent i , in a single stepin the network defined by the original adjacency matrix A . The entries of the n × n matrix( Q ) i,j = ( n − − P i ′ = i ( H i ) i ′ ,j , in contrast, give the probability of arriving at agent j in the network defined by A i , nomatter the starting point, in a single step. The average P nj =1 ( Q ) i,j x j is exogenous andwill correlate with the spillover term, P nj =1 ( H ) i,j x j . A simple intuition can be given byconsidering the example of network centrality: if agent j is involved in many links, it is likelythat she will also be linked with agent i . We may then predict the link decision betweenagents i and j by looking at the linking behavior of agent j with all the other agents inthe network. These choices are exogenous. A two-stage least-squares approach has justsuggested itself for the baseline model: we instrument the endogenous spillover effect Hx by Q x . The rank condition for identification here requires that x is not proportional to theunit vector ι n and that Q x covaries with Hx , after ι n and x have been projected-out fromit. This last condition is, of course, simply the usual relevance condition and requires that Q x has some predictive power for H x after controlling for x . It appears difficult to comeup with situations where this will fail—provided, of course, that (1.2) holds—except fortrivial cases. The empty network and the complete network are two such examples; clearly,in these examples identification would also fail under network exogeneity; Hx is a linearfunction of ι n and x , leading to a standard multicolinearity problem. Other instrumentsthan Q x can equally be constructed under our exclusion restriction; some examples followbelow. The current choice, however, has a natural extension to the full model, to which weturn next. 9n the full linear-in-means model, y = α ι n + δ Hy + β x + γ Hx + ε , the presence of Hy as a regressor would induce an endogeneity problem even if H wereexogenous. If − < δ <
1, and if all the agents in the network are linked to at least oneother agent, Hy = µ ι n + β Hx + λ ∞ X s =0 δ s H s +2 x + ∞ X s =0 δ s H s +1 ε , (2.1)where we write µ = α/ (1 − δ ) and λ = δβ + γ . The argument of Bramoull´e, Djebbari and Fortin(2009) and De Giorgi, Pellizzari and Redaelli (2010) is that H x , H x , and so on can beused as instrumental variables for H y when the network is exogenous, provided that λ = 0.The validity of these variables as instruments breaks down when the network is endogenous.On inspecting the expansion in (2.1) a natural extension to our approach in the baselinemodel presents itself. Because H i is a matrix of transition probabilities, it can be iteratedon, in the same way as H , to yield probabilities of arriving at each agent when takingmultiple steps through the network. In full analogy to Q , the entries of the n × n matrix( Q ) i,j = ( n − − P i ′ = i P nj ′ =1 ( H i ) i ′ ,j ′ ( H i ) j ′ ,j , give the probability of arriving at agent j in the network induced by A i , no matter thestarting point, in two steps. Under our moment condition in (1.1) these weights are, again,exogenous. This, then, allows to instrument the endogenous right-hand side variables, Hx and Hy , by the exogenous variables Q x and Q x . In light of the above the interpretationof this is immediate. Like in the exogenous case, we require Q and Q to be sufficientlydifferent. Contrary to Bramoull´e, Djebbari and Fortin (2009), it is more difficult here togive simple primitive conditions for instrument relevance, however. In their case (2.1)implies a linear reduced form from which such conditions can be derived. This does notapply here.Because the parameters in our model are overidentified we can consider additionalinstruments. One natural way to do so is by taking additional steps through the network.10etting ( Q s ) i,j = ( n − − P i ′ = i P nj =1 · · · P nj s − =1 ( H i ) i ′ ,j ( H i ) j ,j · · · ( H i ) j s − ,j , for any integer s it is apparent that Q s x is a valid instrumental variable. Instruments soconstructed play a role analogous to H s x in the approach of Bramoull´e, Djebbari and Fortin(2009). Of course, like there, as s increases the transition matrix H si will tend to its steady-state distribution, so that higher iterations will provide increasingly less (additional) in-formation. We also note that other instruments are equally possible. For example, notingthat P nj =1 ( Q s ) i,j x j = ι ′ n H si x / ( n − H si x , it would be natural to consider second moments like x ′ H ′ i H i x and x ′ H ′ i H i x , and so on.The condition that λ = 0 is crucial to the approach when the network is exogenous.It requires that x affects y , either directly or through Hx , and also that endogenous andexogenous peer effects do not exactly cancel each other out in the reduced form. Moreover,when λ = 0 we would have E ( H y | H , x ) = µ ι n + β Hx when link formation is exogenous; the H s x , for all s >
1, no longer contain predictiveinformation about Hy , conditional on Hx . The situation is different when link formationis endogenous. Indeed, here, there will generally still be information on Hy in Q s x comingfrom the fact that E ( x ′ Q ′ s H p +1 ε ) = 0 , for any pair of integers p, s , in this case. This isso because, while the entries of H i x do not correlate with ε i , they do correlate with all ε j for j = i when link decisions are endogenous. The implication is that endogeneity of thenetwork can yield identification in settings where no exogenous variables are present in thelinear-in-means model. 11 Inference
We now consider a collection of G independent networks, of size n , . . . , n G , respectively,and rechristen n = P Gg =1 n g , the total number of observations. Each of these networkscomes with its associated adjacency matrix, A g —and, thus, its row-normalized version H g —as well as with the variables y g = ( y g, , . . . , y g,n g ) ′ and x g = ( x g, , . . . , x g,n g ) ′ , whichfollow the linear-in-means model. We may write y g = X g ϑ + ε g , where we combine all regressors in X g = ( ι n g , H g y g , x g , H g x g ) and collect all parametersto estimate in ϑ = ( α, δ, β, γ ) ′ . The estimator of ϑ that we consider here is the two-stageleast-squares estimator using the instruments introduced above.On denoting the instrument matrix as Z g the estimator can be written in its usual form ϑ n = ( P g X ′ g Z g ( P g Z ′ g Z g ) − P g Z ′ g X g ) − ( P g X ′ g Z g ( P g Z ′ g Z g ) − P g Z ′ g y g ) . Its large-sample properties can be deduced from the work of Hansen and Lee (2019) onclustered data sets. Their results are particularly well-suited for the problem at hand. Inparticular, they allow for arbitrary dependence within each network, for networks to growin size, and do not require data to be identically distributed. The following four conditionsneed to be imposed:(i) For some 2 ≤ r < ∞ , (cid:16)P g n rg (cid:17) /r n ≤ c < ∞ , max g n g n n ↑∞ −→ , where c is an arbitrary constant.(ii) For some s with r < s , sup g,i E ( | y g,i | s ) < ∞ and sup g,i E ( | x g,i | s ) < ∞ . (iii) The matrices P g E ( Z ′ g ε g ε ′ g Z g ) n , P g E ( Z ′ g Z g ) n . have minimum eigenvalue bounded away from zero.12iv) The matrix P g E ( Z ′ g X g ) n has maximal column rankThese four conditions are intuitive. A discussion on (i)–(ii) is in Hansen and Lee (2019).Conditions (iii)-(iv) are nothing else than the usual rank conditions that are needed in anyinstrumental-variable problem.By Hansen and Lee (2019, Theorems 8 and 9), the estimator ϑ n is consistent as n → ∞ and has a normal limit distribution. The robust estimator of its asymptotic variance equals V n = ( S ′ n W n S n ) − ( S ′ n W n Ω n W n S n )( S ′ n W n S n ) − , where we use the shorthand notation S n = P g Z ′ g X g , W n = ( P g Z ′ g Z g ) − , Ω n = P g Z ′ g ˆ ε g ˆ ε ′ g Z g , and ˆ ε g = y g − X g ϑ n are the residuals from the two-stage least-squares procedure. We thenhave V − / n ( ϑ n − ϑ ) d → N ( , I )as n → ∞ . The procedure was evaluated in a Monte Carlo experiment. We generated networks viathe link formation process ( A ) i,j = η i + η j > c , where the η i are independent standard-normal variates and we set c = −√ Φ − ( . Φ the standard-normal distribution function. In this way, the unconditional link-formation13robability is .
25. We then drew x i ∼ N (1 ,
1) and generated outcomes from the full model,inducing endogeneity in link formation by generating ε i = ϕ ( η i ) + u i , u i ∼ N (0 , , for different choices of the function ϕ . The parameters were set as α = 0, β = 1, γ = . δ = .
5. Data were generated for 250 groups, each consisting of 25 agents. Results arepresented for the estimator of Bramoull´e, Djebbari and Fortin (2009) (TSLS-X) and forour proposal (TSLS-E). The former instruments Hx by itself and Hy by H x , . . . , H x .This approach is valid when ϕ ( η ) does not depend on η . The latter instruments H x by Q x and Hy by Q x , . . . , Q x . We use two overidentifying moments to discipline thesampling distribution of the estimators—ensuring that their first two moments exist—sothat we can meaningfully report on their bias and standard deviation (see, e.g., Mariano1972).Table 1 contains the bias and standard deviation of the estimators, the mean andstandard deviation of the implied t -statistics, as well as the empirical rejection frequencyof two-sided t -tests (at the 5% significance level). We do not report results for the estimatorof the intercept. Four different specifications for ϕ were considered: (i) a constant, (ii) alinear function, (iii) an exponential function, and (iv) a sine function. All results wereobtained over 5,000 Monte Carlo replications and all the variables were redrawn in eachiteration.The estimator of Bramoull´e, Djebbari and Fortin (2009) does well when link formationis exogenous. Otherwise, the coefficient estimates are biased, except for those of β . Thelatter observation can be explained by the fact that link formation is independent of thecovariates in our design here. The (estimated) standard error (not reported) also tends tosubstantially underestimate the true variability in the point estimates. Together with thepresence of bias, this implies that the t -statistics constructed from TSLS-X have a meanthat is far from zero and a variance that greatly exceeds unity. Consequently, the t -testdisplays large overrejection rates. Using instruments constructed from the leave-one-outnetworks delivers estimators that are virtually unbiased for all the designs in Table 1. The14able 1: Simulation results TSLS-X TSLS-E ϑ n − ϑ V − / n ( ϑ n − ϑ ) ϑ n − ϑ V − / n ( ϑ n − ϑ )bias std mean std rate bias std mean std rate ϕ ( η ) = 0 β γ -0.0002 0.0414 -0.0077 1.0084 0.0542 -0.0002 0.1638 -0.2083 1.0287 0.0600 δ ϕ ( η ) = ηβ -0.0081 0.0181 -0.4580 1.0244 0.0816 0.0002 0.0180 0.0134 1.0049 0.0532 γ -0.0833 0.1336 -1.2585 2.0672 0.4930 -0.0003 0.1406 -0.1060 1.0261 0.0572 δ -0.0833 0.0651 7.7740 3.1595 0.9596 -0.0004 0.0459 0.1075 1.0224 0.0548 ϕ ( η ) = exp(3 Φ ( η )) β -0.0073 0.0661 -0.1140 1.0071 0.0538 0.0007 0.0669 0.0083 1.0009 0.0524 γ δ ϕ ( η ) = sin(3 Φ ( η )) β -0.0017 0.0130 -0.1336 0.9951 0.0496 -0.0003 0.0133 -0.0209 0.9899 0.0468 γ -0.0315 0.0436 -0.7777 1.0663 0.1406 -0.0139 0.1552 -0.2240 1.0319 0.0652 δ t -statistics have a mean that is close to zero and a standard deviation that isclose to unity. Furthermore, the empirical rejection frequencies are close to their nominalsize of 5%, and this for all parameters and for all designs. Hence, the normal approximationdoes well for TSLS-E.The findings discussed here were confirmed in a larger set of Monte Carlo designs, wherethe η i were drawn from asymmetric distributions and network formation also depends oncovariates. These results were similar in spirit to those reported here and, hence, are notdiscussed further here. References
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