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Journal of Business & Economic Statistics

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Comment

Matthew O. Jackson

To cite this article: Matthew O. Jackson (2013) Comment, Journal of Business & Economic Statistics, 31:3, 270-273, DOI: 10.1080/07350015.2013.794095

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270 Journal of Business & Economic Statistics, July 2013

REFERENCES

Chamberlain, G. (1985), “Heterogeneity, Omitted Variable Bias, and Duration Dependence,” inLongitudinal Analysis of Labor Market Data, eds. J. J. Heckman and B. Singer, Cambridge: Cambridge University Press, pp. 3– 38. [268]

Cox, D. R. (1958), “The Regression Analysis of Binary Sequences,”Journal of the Royal Statistical Society,Series B, 20, 215–241. [268]

Goodreau, S. M., Kitts, J. A., and Morris, M. (2009), “Birds of a Feather, or Friend of a Friend? Using Exponential Random Graph Models to Investigate Adolescent Social Networks,”Demography, 46, 103–125. [267]

Graham, B. S. (2011), “Econometric Methods for the Analysis of Assignment Problems in the Presence of Complementarity and Social Spillovers,” in Handbook of Social Economics(Vol. 1B), eds. J. Benhabib, A. Bisin, and M. Jackson, Amsterdam: North-Holland, pp. 965–1052. [267,268] ——— (2012), “Homophily and Transitivity in Dynamic Network Formation,”

Mimeo. [268]

Hahn, J. (2001), “The Information Bound of a Dynamic Panel Logit Model With Fixed Effects,”Econometric Theory, 17, 913–932. [268]

Heckman, J. J. (1981a), “Heterogeneity and State Dependence,” inStudies in Labor Markets, ed. S. Rosen, Chicago: University of Chicago Press, pp. 91–139. [267]

——— (1981b), “Statistical Models for Discrete Panel Data,” in Struc-tural Analysis of Discrete Data and Econometric Applications, eds. C. F. Manski and D. L. McFadden, Cambridge, MA: The MIT Press, pp. 114–178. [267]

——— (1981c), “The Incidental Parameters Problem and the Problem of Initial Conditions in Estimating a Discrete Time-Discrete Data Stochastic Process,”

Structural Analysis of Discrete Data and Econometric Applications, eds. C. F. Manski and D. L. McFadden, Cambridge, MA: The MIT Press, pp. 179–195. [267]

Honor´e, B. E., and Kyriazidou, E. (2000), “Panel Data Discrete Choice Models With Lagged Dependent Variables,”Econometrica, 68, 839–874. [268] Goldsmith-Pinkham, P., and Imbens, G. W. (2013), “Social Networks and the

Identification of Peer Effects,”Journal of Business and Economic Statistics, 31, 253–264. [267,268]

Jackson, M. O. (2008),Social and Economic Networks, Princeton, NJ: Princeton University Press. [267]

Kitts, J. A., and Huang, J. (2011), “Triads,” in Encyclopedia of Social Networks (Vol. 2), ed. G. Barnett, New York: Sage, pp. 873–874. [267]

Krivitsky, P. N., Handcock, M. S., Raftery, A. E., and Hoff, P. D. (2009), “Representing Degree Distributions, Clustering, and Homophily in Social Networks With Latent Cluster Random Effects Models,”Social Networks, 31, 204–213. [267]

Manski, C. F. (1987), “Semiparametric Analysis of Random Effects Linear Models From Binary Panel Data,” Econometrica, 55, 357– 362. [268]

——— (1993), “Identification of Endogenous Social Effects: The Reflection Problem,”Review of Economic Studies, 60, 531–542. [267]

McPherson, M., Smith-Lovin, L., and Cook, J. M. (2001), “Birds of a Feather: Homophily in Social Networks,”Annual Review of Sociology, 27, 415– 444. [267]

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Comment

Matthew O. J

ACKSON

Department of Economics, Stanford University, Stanford, CA 94305-6072, The Santa Fe Institute, and CIFAR (jacksonm@stanford.edu)

1. INTRODUCTION

Understanding peer effects is of first-order importance in a range of settings including education, labor markets, crime, voting, and consumer behavior. However, although casual in-trospection and some field experiments (e.g., Duflo and Saez

2003; Centola2010) suggest that the influence of friends and ac-quaintances on behavior can be substantial, there are formidable challenges in establishing peer effects using (increasingly avail-able) purely observational data that couple behaviors with social relationships.

The challenges in establishing that peer effects are truly present include:

• Identification: A model of peer effects must be specified in a manner such that the channels through which peers influence one’s behavior can be identified as well as dis-tinguished from other sources of influence (Manski1993; Bramoull´e et al.2009; Blume et al.2011).

• Endogenous networks and homophily: Linked individuals are likely to be similar not only in terms of observed char-acteristics but also in terms of unobserved charchar-acteristics that could influence behavior. By failing to account for sim-ilarities in (unobserved) characteristics, similar behaviors might mistakenly be attributed to peer influence when they simply result due to similar characteristics (Aral et al.2009; Jackson2008). This is complicated by the richness of the set of possible characteristics that could matter, which not

only involve innate or exogenous ones that might influence preferences, but also correlated ones that include exposure to common stimuli (Manski1993).

• Computation: The possible set of networks of peer relation-ships is generally exponential in the size of the population and so having a model that allows for endogenous peer re-lationships can face significant computational challenges (Chandrasekhar and Jackson2012).

• Measurement error: Relationships can be difficult to ob-serve and quantify. This can reduce the power of a test, especially with self-reported relationships that are easily unobserved (Chandrasekhar and Lewis 2011). In particu-lar, this applies to data with caps on the numbers of friends that can be reported as in the Add Health dataset.

• Misspecification: Specifying the appropriate set of peers, which can be time- and context-dependent, is difficult, as in correctly specifying the ways in which they influence each other, including possible heterogeneity among a given individual’s friends.

The difficulties of convincingly demonstrating peer effects are complicated by the fact that the biases introduced by the various issues mentioned above can push in different directions. For

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example, not accounting for the potential similarity of friends along unobserved dimensions could lead one to overestimate peer effects, while some misspecifications could lead one to underestimate peer effects. Thus, without some confidence that we have adequately addressed the long list of issues mentioned earlier, we cannot be confident in estimating peer effects. Does this mean that we should simply give up? Of course not, given that understanding these effects are essential to designing effec-tive policies in so many applications, in both the developed and developing world.

In an ambitious and important article, Goldsmith-Pinkham and Imbens (2013) wrestle with each of these issues. This leads to a somewhat sprawling analysis, but the sprawl is inevitable given that these challenges are intertwined. Goldsmith-Pinkham and Imbens take a useful tack by grounding their analysis in the context of a specific setting: peer effects on grade point averages in U.S. high schools using the Add Health dataset. It is not their intention that this be taken seriously as an empirical analysis, as they trim down the data to make for a very clear methodological discussion. Thus, the empirics are illustrative rather than infor-mative. But the grounding of the problem is quite useful as ad-dressing the many intertwined issues in estimating peer effects is much easier to discuss in a concrete context than in the abstract. Goldsmith-Pinkham and Imbens’ main contribution is care-ful attention to the endogeneity of the network, and pointing out some consequences of failure to appropriately account for endogeneity. They also develop implications of the failure to account for an endogenous network and methods of testing for whether omitted endogenous network effects could be biasing peer influence measurements.

The concern that the network could be endogenous and strongly correlated with individual characteristics is well-founded. Fitting a network formation model with race and other characteristics in preferences in the Add Health data shows that individuals act as if they have a significant and strong pref-erence to interact with others who have similar characteristics (Currarini, Jackson, and Pin 2009,2010). This together with the fact that characteristics correlate with GPA means that not accounting for potential similarities in unobserved characteris-tics among peers could substantially bias the estimation of peer effects.

To tackle this issue, Goldsmith-Pinkham and Imbens model friendship formation in a manner that explicitly allows for sim-ilarity along an unobserved dimension. The important insight that they make clear from the network formation model is the following. If two individuals are friends, and hence peers, but differ in their observed characteristics, then it is more likely that the pair are similar on the unobserved characteristics. That is, given that friendships form based largely on similarities, if the observables are not similar then it must be (or is likely to be) that the unobserved characteristics are similar. If those unobserved characteristics relate to behavior, then that means that the pair’s behaviors should be more similar than one would estimate sim-ply based on their observed characteristics. Thus, the errors in predicted behaviors based on observables should be stronger when friends’ observed characteristics diverge. This also means that if one does not correctly model friendships and account for potentially unobserved characteristics, then one could inflate the peer effects.

Let us examine the details to be clear. In their model, an individuali’s GPA,Yi, is a function of the student’s observed

characteristics,Xi, the average peer GPA,Y(i), and the average peer characteristics,X(i), and the unobserved characteristic,ξi:

Yi =β0+βXXi+βYY(i)+βXX(i)+βξξi+ηi. (1)

The utility ofifor the relationship with jis proportional to the difference in their observed and unobserved characteristics, |Xi−Xj|and|ξi−ξj|, as well as whether they had a

relation-ship and/or a friend in common in the previous period,D0ijand

F0,ij:

Ui(j)=α0+αX|Xi−Xj| +αξ|ξi−ξj| +αdD0ij

+αfF0,ij+ǫij.

The probability ofiandjbeing linked is then given by

Pr[linkij]=

This specification of Goldsmith-Pinkham and Imbens’ net-work formation model circumvents the central computational issues since an individual’s preference for a relationship with another person does not depend on any of the rest of the network structure except whether they had a friend in common in an ear-lier period. This means that the formation of the current network can be estimated link-by-link and instead of facing an exponen-tial number of calculations it is a quadratic one in the number of nodes. While this independence of links is violated in many set-tings (including Add Health), it allows Goldsmith-Pinkham and Imbens to concentrate on other issues for their methodological illustrations. Nonetheless, this means that this model of network formation is most likely (substantially) misspecified and so the endogeneity effects are likely to be biased and any inference has to be interpreted accordingly. Even with this computational sim-plification, likelihood calculations are still demanding because of the unobserved characteristics that have to be inferred for each student,ξi(so ann-dimensional vector, reintroducing the

expo-nential calculations), and Goldsmith-Pinkham and Imbens use MCMC sampling methods coupled with Bayesian estimation.

This specification makes it simple to see how failing to ac-count for the endogenous network would affect a peer effects analysis. If αξ is negative, then utilities for a relationship are

higher if the unobserved characteristicsξiandξj are similar. If

one then estimates the relationship (1) without accounting for the unobserved characteristics (so settingβξ =0), wheniandj

have similarξiandξjthey are not only likely to be peers, but they

are then also likely to have similarη’s since those will absorb the missingξ’s. Indeed, as Goldsmith-Pinkham and Imbens find, the residuals from estimations of Equation (1), when not including the unobserved characteristics, correlate with whetheriandjare linked. From their estimations, it also appears that the missing characteristics play a substantial and significant role in network formation. Very interestingly, however, despite this large role of unobserved similarities in link formation, Goldsmith-Pinkham and Imbens also find that accounting for these characteristics does not substantially impact the estimation of peer effects.

The significant effect of unobserved similarity on link forma-tion but the absence of any real impact on the estimaforma-tion of peer effects is surprising given that, for instance, race, wealth, and

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272 Journal of Business & Economic Statistics, July 2013

gender play significant roles in friendship formation in the Add Health data and also correlate with GPA. Thus, not properly ac-counting for such characteristics should presumably impact the peer influence estimates substantially and significantly. It is also surprising given that Goldsmith-Pinkham and Imbens found that the residuals in the peer equation estimation correlate signifi-cantly with friendship. One possible explanation for this pattern of findings lies in the functional specification of the unobserved characteristics and how they might influence both link forma-tion and behavior. Here, the specificaforma-tion of the unobserved variableξ is very stripped down and is just a binary variable. This could be enough to capture whether linked individuals are “similar” or “different” along unobserved characteristics, and hence enough to detect whether unobserved characteristics matter in link formation. However, it could be that the manner in which unobserved characteristics matter in peer effects is much more complicated and not at all captured by such a formulation. In particular, it could be that the sorts of missing character-istics that matter are things like study habits, whether or not students are involved in extracurricular activities, their socioe-conomic status, and so forth. Two people are more likely to be friends if they are “similar” rather than “different” on these char-acteristics, so the estimation picks up whether two students are similar or different on these dimensions. Thus, the specification can detect whether similarity on unobserved characteristics mat-ters significantly in forming a peer relationship. However, that does not encode the information about how many hours the stu-dents spend studying, or whether their parents attended college, and so forth. These are the sorts of missing characteristics that are likely to matter in GPA. Thus, the takeaway from this article by Goldsmith-Pinkham and Imbens should be the overarching methodological points, but not the particular specifications nor the particular findings when these are applied to the data.

There are several other interesting questions that Goldsmith-Pinkham and Imbens also explore regarding peer interactions: Do former friends matter? Do friends of friends matter? Does it matter whether both friends name each other or just one names the other? The answers to these questions paint an interesting picture. They find that the current GPAs of former friends matter almost as much as those of current friends. Friends of friends who are not current friends also provide a significant peer effect. Yet they find little difference between the impact of friendships as a function of whether friendship nominations are reciprocated or not.

It seems that there is a common explanation to all of these ob-servations: unobserved friendships and thus measurement error. Past friendships could still be friendships, but ones that are sim-ply not named on the second survey. Given that the Add Health data have students typically only having five to ten friends, many peers go unnamed. Similarly, it could be that pairs of individuals who are at distance two in the measured network are actually friends, but the link is not mistakenly unobserved. Indeed, as estimated by Jackson, Barraquer, and Tan (2012) in a different setting, pairs of individuals who have friends in common are roughly five times more likely to be friends than not. In fact, they estimate that small amounts of measurement error could account for many of the pairs who have friends in common but appear not to be friends. Finally, failures in reciprocation can be due mistaken observations in the data. Indeed, as found

by Banerjee et al. (2012) reciprocation rates can be low even when individuals are asked to name relatives; and so many re-lationships are simply not named by the subjects in surveys. Thus, it is clear that there are contexts with high rates of fail-ure of reciprocation pfail-urely from measfail-urement error and that many relationships go unreported in survey data. In summary, the mismeasuring of current friendships makes it appear as if past friendships that are no longer friendships matter, or having friends in common matter, when it could be that neither of these are really causal, but instead just correlating with measurement error in observing current friendships.

It is important to emphasize one thing in closing. There is a temptation to write down simple econometric models in the abstract that a researcher could take off-the-shelf and adapt to a particular setting. What should be taken away from the anal-ysis here, and developed further, is an understanding of how to address issues of specification, endogeneity, omitted variables, measurement errors, etc., butnotthe particular functional forms for doing this. There are many channels for peer effects even restricting ourselves to an education setting: students study to-gether, they may exert pressures on each other not to study, they might engage in illicit activities together, they may serve as role models for each other, they may feel pressures to conform, they may be searching for an identity and sense of belonging, they may experience parental influences, they may experience com-plementarities in their payoffs from the educational decisions, they may be exposed to stimuli that they react to communally, and so forth. Clearly, this long and still incomplete list of po-tential peer effects is unlikely to be all captured in a simple, for example linear, manner based on some measurement of a grade or other outcome. While this requires adding complexity to the modeling, being very explicit about the specific type of effect that one is seeking to identify is much more likely to yield a useful model. This is illustrated by Banerjee et al. (2012) who dissected peer effects in the context of word-of-mouth spread of a new product (microfinance in rural Indian villages). By explicitly modeling information passing in the network and fitting that model to data, they found that what appears to be a strongly positive peer/endorsement effect becomes insignificant (and even slightly negative) once one models information pass-ing. An interpretation of their findings is that a given individual is more likely to participate when more friends do, is not due to some complementarity, peer influence, or endorsement effect, but rather because having more friends take up the product sim-ply makes one more likely to be aware of the product. While the particulars of the finding are relative to the specifics of the set-ting, the broader message is that more explicit modeling of the details of peer interaction can lead to very different conclusions from what one might infer based on standard linear-in-means style regressions. More generally, such an approach will require models that account for the wide multitude of different forms of learning, peer influences, and peer interactions.

These challenges become even more substantial when build-ing models of peer effects that also incorporate network for-mation, as there can be complex feedbacks between behavior and friendship choices. While strategic network formation mod-els have been studied in some detail in the theory literature (e.g., Jackson and Wolinsky1996; Bala and Goyal2000; Jack-son2008), models designed for empirical implementation of

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strategic network formation are still in their infancy (Christakis et al.2010; Mele2011; Chandrasekhar and Jackson2012; Leung

2013). The very important lesson that we should take from the analysis of Goldsmith-Pinkham and Imbens is that accounting for the endogeneity of relationships in analyses of peer effects is feasible, and can provide substantial new insights, for example, into unobserved characteristics that might correlate both with behavior and friendship formation. The specifications that are needed to properly model both network formation and peer ef-fects require careful additional analysis in context, and provide us with a rich agenda going forward.

REFERENCES

Aral, S., Muchnik, L., and Sundararajan, A. (2009), “Distinguishing Influ-ence Based Contagions from Homophily Driven Diffusion in Dynamic Net-works,”Proc. Natl. Acad. Sci., 106, 21544–21549. [270]

Bala, V., and Goyal, S. (2000), “A Noncooperative Model of Network Forma-tion,”Econometrica, 68, 1181–1229. [272]

Banerjee, A., Chandrasekhar, A., Duflo, E., and Jackson, M. (2012), “Dif-fusion of Microfinance,” NBER Working Paper 17743. Available at http://www.stanford.edu/jacksonm/diffusionofmf.pdf. [272]

Blume, L., Brock, W., Durlauf, S., and Ioannides, Y. (2011), “Identification of Social Interactions,” inThe Handbook of Social Economics, ed. J. Benhabib, A. Bisin, and M. Jackson, San Diego: North Holland. [270]

Bramoull´e, Y., Djebbari, H., and Fortin, B. (2009), “Identification of Peer Effects Through Social Networks,”Journal of Econometrics, 150, 41–55. [270]

Centola, D. (2010), “The Spread of Behavior in an Online Social Network Experiment,”Science, 32, 1194–1197, doi: 10.1126/science.1185231. [270] Chandrasekhar, A., and Jackson, M. (2012), “Tractable and Consis-tent Random Graph Models,” SSRN Working Paper. Available at http://ssrn.com/abstract=2150428. [270,273]

Christakis, N., Fowler, J., Imbens, G., and Kalyanaraman, K. (2010), “An Em-pirical Model for Strategic Network Formation,” NBER Working Paper. [273]

Currarini, S., Jackson, M., and Pin, P. (2009), “An Economic Model of Friend-ship: Homophily, Minorities, and Segregation,”Econometrica, 77, 1003– 1045. [271]

Currarini, S., Jackson, M., and Pin, P. (2010), “Identifying the Roles of Race-Based Choice and Chance in High School Friendship Network Formation,” Proceedings of the National Academy of Sciences, 107, 4857–4861. [271] Duflo, E., and Saez, E. (2003), “The Role of Information and Social Interactions

in Retirement Plan Decisions: Evidence From a Randomized Experiment,” Quarterly Journal of Economics, 118, 815–842. [270]

Goldsmith-Pinkham, P., and Imbens, G. (2013), “Social Networks and the Iden-tification of Peer Effects,”Journal of Business and Economic Statistics, 31, 253–264. [271]

Jackson, M. (2008),Social and Economic Networks, Princeton, NJ: Princeton University Press. [270,272]

Jackson, M., Barraquer, T., and Tan, X. (2012), “Social Capital and Social Quilts: Network Patterns of Favor Exchange,”American Economic Review, 102, 1857–1897. [272]

Jackson, M., and Wolinsky, A. (1996), “A Strategic Model of Social and Eco-nomic Networks,”Journal of Economic Theory, 71, 44–74. [272] Leung, M. (2013), “Two-Step Estimation of Network-Formation Models With

Incomplete Information,” Working Paper, Stanford University. [273] Manski, C. (1993), “Identification of Endogenous Social Effects: The Reflection

Problem,”The Review of Economic Studies, 531–542. [270]

Mele, A. (2011). “A Structural Model of Segregation in Social Networks,” Working Paper, Johns Hopkins University. [273]

Comment

Charles F. M

ANSKI

Department of Economics and Institute for Policy Research, Northwestern University, Evanston, IL 60201 (cfmanski@northwestern.edu)

To begin, I will express mixed feelings about the decision by Goldsmith-Pinkham and Imbens to attach my name to the linear-in-means models that are the concern of their article. I will not object strenuously to their invention of the acronym MLIM model, if only because I am aware of the adage that “any publicity is good publicity, as long as they spell your name right.” However, I will note that Manski (1993) did not origi-nate linear-in-means models, whose use in empirical research goes back at least to the 1960s and perhaps earlier. Nor did my article advocate empirical application of these models to study social interactions—it also studied nonparametric models and parametric nonlinear ones, taking no position on the realism of any of them. My objective in writing the article was to an-alyze identification of models with endogenous social effects, recognizing that the outcomes in such models solve equilibrium conditions. For this purpose, linear-in-means models provided an analytically tractable case that illustrates well some basic issues.

I similarly interpret Goldsmith-Pinkham and Imbens not to be advocating linear-in-means models but rather to be focus-ing on these models to illustrate some basic issues. Their article extends the literature in several directions. One that I find partic-ularly interesting is their development of some relatively simple

economic models of network formation and integration of these models with study of interactions within the formed networks. Another is their use of Bayesian inferential methods to circum-vent conceptually and technically difficult issues that arise in performing frequentist inference in settings where a population does not partition into a large set of separate networks. When making these contributions, the authors maintain parametric modeling assumptions that may be too restrictive to provide a realistic basis for credible empirical analysis. Yet, as with the linear-in-means model itself, I can appreciate the illustrative value of focusing attention on tractable special cases.

SPECIFYING THE OBJECT OF INTEREST

I will focus my comment on one passage in the article by Goldsmith-Pinkham and Imbens, where they consider the object of interest for empirical analysis of social interactions. They write