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Journal of Business & Economic Statistics
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Comment
Yann Bramoullé
To cite this article: Yann Bramoullé (2013) Comment, Journal of Business & Economic Statistics, 31:3, 264-266, DOI: 10.1080/07350015.2013.792265
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264 Journal of Business & Economic Statistics, July 2013
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Comment
Yann B
RAMOULLE´
Aix-Marseille University, Aix-Marseille School of Economics, CNRS & EHESS, Marseille 13002, France, and Department of Economics, Universit ´e Laval, Qu ´ebec G1K 7P4, Canada ([email protected])
In their article, Goldsmith-Pinkham and Imbens tackle three important issues in the study of peer effects in social networks: network endogeneity, network heterogeneity, and the structure of the error term. The endogenous nature of social links is prob-ably the most important and the most vexing of these three issues. In this note, I discuss first how network endogeneity is addressed. I then make a comment on network heterogeneity, a clarification on group interactions and a proposition of further research.
1. CORRECTING FOR NETWORK ENDOGENEITY
Goldsmith-Pinkham and Imbens’ approach can be viewed as a network version of the classical Heckman correction for sample selection. By studying network formation and outcome determination jointly, we may recover enough information on the error term of the outcome equation to solve the underlying endogeneity problem. Goldsmith-Pinkham and Imbens propose one of the first convincing applications of this idea to peer
ef-fects and networks, see Conti et al. (2012) and Hsieh and Lee (2012) for two recent studies in the same vein. The way identifi-cation works, however, is not very clear. With classical selection problems, the empirical importance of exclusion restrictions is well known. Without exclusion restrictions, identification relies on nonlinearities and may not be very robust. While peer effects in networks constitute a new setup, the question of the strength of identification remains key. In what follows, I highlight some properties and limitations of the model whose study may help clarify this issue.
Let us examine, first, the role played by exclusion restrictions in the analysis. Current linksDij are affected by past network
features and by covariates (Equation (6.2)). Past network fea-turesD0ij andF0ij do not affect outcomeYi (Equation (6.1)).
However, if unobserved characteristics are persistent, past
© 2013American Statistical Association Journal of Business & Economic Statistics
July 2013, Vol. 31, No. 3 DOI:10.1080/07350015.2013.792265
Bramoull ´e: Comment 265
network features will likely be correlated with the error term of the outcome equation. While this possibility is assumed away here, such persistence is likely. Current links and out-come are also affected by common covariates, but in different ways. Covariates affect links through absolute values of dif-ferences |Xi−Xj|, while they affect outcome directly. This
certainly constitutes a form of exclusion restriction, but I won-der about its identifying power in practice. In any case, finding variables affecting link formation but not outcome can only help identification in future applications.
Next, the precise way through which the model generates a bias in the outcome equation would deserve to be clarified. Sec-tion 6.2 describes the empirical content of network endogeneity. Under homophily, pairs of friends which are less similar in the observed dimension must be, on average, more similar in the unobserved dimension. And the same property holds for pairs of nonfriends. So the model generates a negative correlation be-tween observed and unobserved similarity in pairs. However, to correct for network endogeneity, the model must generate a cor-relationfor individualsbetween unobservableξiand peers’
ob-servables ¯X(i)in Equation (6.1). This is quite different. How this correlation is generated, precisely, is not obvious. And should we expect it to be positive or negative under homophily? I sus-pect that the distributions of observables and unobservables in the population matter. This could be studied through Monte Carlo simulations.
The interaction between observed and unobserved variables lies at the heart of the method, and I observe that the logit assumptions impose restrictions on this interaction. Formally, denote byXij = −|Xi−Xj|the observed proximity between
i and j and by ξij = −|ξi−ξj| their proximity in the
un-observed dimension. We can check that under homophily,
∂2E(D
ij)/∂Xij∂ξij >0 if p(Dij =1)<2/3 and ∂2E(Dij)/
∂Xij∂ξij <0 if p(Dij =1)>2/3. Thus, if network density
is relatively low, as in most empirical applications,unobserved and observed proximity are complement in their effects on links. An increase in unobserved proximity raises the marginal impact of observed proximity on the likelihood to form a link. This property needs not be satisfied in practice, and it could be inter-esting to develop a model allowing for substitution effects over a larger range of probabilities.
Two further limitations of the current model are the absence of unobserved contextual peer effects and its pairwise nature. If we think that unobserved and observed covariates are of the same kind, we should add a termαξ¯ξ¯(i) to the outcome equa-tion. This would complicate identification and estimation but would probably not radically alter the method. In addition, the assumption of statistical independence between current links is quite strong. The empirical estimates show the importance of structural effects related to past links; these effects probably also matter for current links. Two recent studies, Mele (2011) and Hsieh and Lee (2012), tackle this issue by adding struc-tural effects directly to Equation (6.2). For instance, here, we could add a termα′fFij to capture the possibility that having
friends in common in the current network affects the probabil-ity to form links. However, the analysis of network formation with structural effects raises some serious econometric diffi-culties. An alternative, unexplored, and potentially promising possibility would be to work with structural models of the kind
developed by Jackson and Rogers (2007) and Bramoull´e et al. (2012).
Finally, let me comment briefly on the estimation techniques. Faced with the computational complexity of maximizing the likelihood, Goldsmith-Pinkham and Imbens adopt Bayesian methods. An interesting, and unexploited, feature of their esti-mation procedure is that it generates individual-level estimates of the unobservedξi’s. This could be used in at least two ways.
First, to test the assumptions made on the unobservable distribu-tion. Second, to see how these estimated unobservables correlate with relevant characteristics.
The choice of Bayesian techniques raises its own computa-tional issues. An alternative way forward could be to adapt al-gorithms developed in computational graph theory to obtain ap-proximate solutions to the maximum likelihood problem. How is the likelihood maximized here? The objective is to find the best partition of the population in two groups,ξi =0 andξi =1,
combining the following two objectives. First, we want to add students with real grade above their predicted grade to the group
ξi=1. And second, we want to have many links within and few
links between the two groups. This problem is reminiscent of classical graph theoretical problems such as MAX CUT, and I suspect that classical approximation algorithms could be suc-cessfully imported here.
2. HOW TO MODEL HETEROGENEOUS
PEER EFFECTS?
In a second step, Goldsmith-Pinkham and Imbens analyze heterogeneity in peer effects. Teenagers may well be affected in different ways by their male and female friends, younger and older friends, or former and current friends as in Section 7.2. To study these differentiated effects, Goldsmith-Pinkham and Imbens consider a linear-in-double meansformulation where the mean characteristics of both types of peers affect outcomes, see Equation (7.1). One issue is that this formulation does not nest the regular linear-in-means model with undifferentiated peers. The reason is that a linear combination of the two means is not proportional to the overall mean. In particular, if both networks have the same effectsβy,A¯ =βy,B¯ , it does not mean that all peers have the same impact. If individual i has more A-friends, change in the outcome of oneA-friend has a lower impact than for aB-friend.
There is only one way to address this issue within a linear framework. The solution is to weight the respective means by the proportions of friends of each typepi,A =MA,i/(MA,i+MB,i) in-means model with equal peers. Of course, we should be able to test whether the data are better represented by a linear-in-double means or a linear-in-weighted means formulation.
266 Journal of Business & Economic Statistics, July 2013
3. A CLARIFICATION ON GROUP INTERACTIONS
Most existing studies of peer effects assume that peer groups partition the sample, as in Manski (1993). This is an important special case with specific features and properties. Goldsmith-Pinkham and Imbens note that under group interactionsGG=
G. In this case, Proposition 1 in Bramoull´e, Djebbari, and
Fortin (2009) confirms Manski (1993)’s earlier result. The linear-in-means model is not identified because of the reflection problem.
However, the property that GG=G only holds when the
mean is inclusive and is computed over everyone in the peer group including i. Assuming that an individual is one of his own peers seems a bit strange, and applied researchers typically considerexclusive means, where the average is computed over everyone in the peer group excepti. As it turns out, this minor distinction has key implications for identification. Lee (2007) show that a linear-in-exclusive means model with group fixed effects is generally identified if there is variation in group sizes. Boucher et al. (in press) provide the first empirical application of this result and clarified the intuition behind identification. In essence, identification relies on mechanical effects. Better students have worse peers; this reduces the dispersion in out-comes, and this dispersion reduction decreases with group size at a decreasing rate. These mechanical effects hold in nonlinear models as well. Given the paucity of network data and the im-portance of the issue, I think that this idea deserves to be further investigated.
4. WHAT IF THE OUTCOME ALSO AFFECTS THE NETWORK?
To conclude, let me highlight a limitation of Goldsmith-Pinkham and Imbens’ model, which is common to all studies on the topic. Here, outcome attis affected by the network at
t but the network at t is not affected by outcome at t. This asymmetry may not hold in reality, and the way social links are formed during periodt may well be affected by the outcome of interest. To capture this possibility, we would need to de-velop an econometric model with simultaneous determination of outcome and links. This would undoubtedly raise interesting econometric challenges.
ACKNOWLEDGEMENTS
The author thanks Vincent Boucher, Habiba Djebbari, Xavier Joutard, and Michel Lubrano for helpful comments and discussions.
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Boucher, V., Bramoull´e, Y., Djebbari, H., and Fortin, B. (in press), “Do Peers Affect Sudent Achievement? Evidence from Canada Using Group Size Vari-ation,”Journal of Applied Econometrics. [266]
Bramoull´e, Y., Currarini, S., Jackson, M., Pin, P., and Rogers, B. (2012), “Ho-mophily and Long-Run Integration in Social Networks,”Journal of Eco-nomic Theory, 147, 1754–1786. [265]
Bramoull´e, Y., Djebbari, H., and Fortin, B. (2009), “Identification of Peer Effects Through Social Networks,”Journal of Econometrics, 150, 41–55. [266] Conti, G., Galeotti, A., Mueller, G., and Pudney, S. (in press), “Popularity,”
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Lee, L. F. (2007), “Identification and Estimation of Econometric Models With Group Interactions, Contextual Factors and Fixed Effects,” Journal of Econometrics, 140, 333–374. [266]
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Comment
Bryan S. G
RAHAMDepartment of Economics, University of California, Berkeley, CA 94720, and NBER ([email protected])
Consider a group of three, potentially connected, individuals (i=1,2,3). Available is a large random sample of such groups. For each group sampled, we observe all social ties among its con-stituent members in eacht =0,1,2,3 periods. LetDij t =1 if
individualiis “friends” (i.e., connected) with individualjin pe-riodtand zero otherwise. Ties are undirected so thatDij =Dj i
fori=j. We rule out self-ties so thatDii=0. The network
ad-These comments were prepared for theJournal of Business and Economic StatisticsInvited Address delivered by Guido Imbens on January 7that the 2012 annual meeting of the American Economic Association. I am grateful to Kei Hirano and Jonathan Wright, in their capacity as coeditors, for the opportunity to comment on Paul Goldsmith-Pinkham and Guido Imbens’ article.
jacency matrix in periodtis denoted byDtwith typical element
Dij t. The sampling process asymptotically reveals
f(d3,d2,d1,d0)=Pr(D3=d3,D2=d2,D1=d1,D0=d0).
LetFij t =1 if iandj have any friends in common during
periodtand zero otherwise. For example, ifiandk, as well as
© 2013American Statistical Association Journal of Business & Economic Statistics
July 2013, Vol. 31, No. 3 DOI:10.1080/07350015.2013.792261
