Abstract
This study analyzes peer influences in a variety of adolescent developmental outcomes and explores the robustness of the results with regard to alternative specifications of the interaction matrix and the model. I employ the spatial autoregressive (SAR) model with group fixed effects in Lee et al. (Econom J 13(2):145–176, 2010) to separately identify endogenous, contextual, and correlated effects. I find evidence for both endogenous and contextual effects for all outcomes under consideration, even after controlling for confounding effects. I also find that a simple spatial weights matrix with equal weight among friends will serve the estimation purpose satisfactorily for the standard SAR model with a single row-normalized weighting matrix. The results are sensitive to the friendship reciprocity assumption as well as alternative model specifications.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Take smoking as an example. There will exist endogenous effects if an individual’s smoking decision is affected by his/her peers’ smoking behaviors. Contextual effects will arise if the parents of peers with certain characteristics are more capable of lobbying for more learning resources for the whole group. There will be correlated effects if students are more likely to smoke due to low tobacco prices or their school’s smoking policy. Correlated effects will also occur if highly educated parents are more likely to send their children to a school where smoking is strictly prohibited.
Most studies that use the linear-in-means model simply assume away the “reflection problem” by including either endogenous effects or contextual effects, but not both, in the model.
This specification is convenient and is necessary in most cases, as many datasets do not have detailed information about the network to distinguish among peers.
Recently, a limited number of papers have started to incorporate similar information in their model specifications. Patacchini et al. (2011) study peer effect in education attainment, and they consider the heterogeneity of friends in terms of nomination order. However, their study is based on the Add Health in-home survey, with a small final sample size of 1,319, and their model specification is different from the current study. Lin (2014) explores the similar features of possible heterogeneity among friends, but the computational complications of the discrete-choice network model prevent the introduction of group fixed effects and the SAR disturbances.
Add Health provides information on the level of intensity of the four activities studied, making this model specification possible. For instance, for smoking, Add Health gives information on “how often” an adolescent smokes, not just whether he/she smokes or not.
As will be shown in Sect. 3, the group fixed effects will be eliminated through an eigenvalues eigenvectors decomposition technique, so the “incidental parameter” problem is not in issue for this model.
As will be discussed in Sect. 3, these exclusion restrictions are implied naturally by the network structure, instead of being imposed based on subjective assumption as in the several papers mentioned above.
Unlike the two-stage least squares procedure frequently used in previous studies, the ML procedure sidesteps the weak instrument-related issues.
In addition, the control of a rich set of individual characteristics and contextual effects could also help effectively reduce the unobserved within-group heterogeneity. Admittedly, since I cannot systematically model the friendship selection process, there may be some within-group unobservable effects left uncaptured by this strategy. Several recent papers, including Goldsmith-Pinkham and Imbens (2013) and Hsieh and Lee (2013), have proposed a simultaneous system to model the network formation and peer influences jointly. However, this joint modeling approach is computationally intensive and usually estimated by Bayesian method. It cannot handle such a large sample size as in the current study.
Specifically, the linear-in-means model is given as: \(y_{ir}=\lambda _0E(y_r|r)+\beta _{10}x_{ir}+\beta _{20}E(x_r|r)+\epsilon _{ir}\), which implies the reduced form \(y_{ir}=\beta _{10}x_{ir}+\frac{\lambda _0\beta _{10}+\beta _{20}}{1-\lambda _0}E(x_r|r)+\epsilon _{ir}.\) Therefore, only some combination of \(\lambda _{0}\) and \(\beta _{20}\), i.e., \((\lambda _{0}\beta _{10}+\beta _{20})/(1-\lambda _{0})\), is identified, hence the “reflection problem.” This inseparability between \(\lambda _0\) and \(\beta _{20}\) is due to perfect collinearity between these two terms \(E(y_r|r)\) and \(E(x_r|r)\).
As shown in Bramoullé et al. (2009), the necessary and sufficient condition for \(W_{r}^{2}X_{r}\) to be a valid instrument for \(W_{r}Y_{r}\) is that the matrices \(I_{r},W_{r},W_{r}^{2}\) are linearly independent, which is satisfied for most networks.
Note this transformation can be performed even to cross-sectional network data since the group dimension of the network data can play a similar role as the time dimension of the panel data.
Interested readers may refer to Lee et al. (2010) for more technical details.
As pointed out by the associate editor and two anonymous referees, self-report outcomes could contain potential measurement errors. In addition, the limit of five friends of each gender could introduce some mis-specifications in the interaction matrix for respondents who have more than five male or female friends. Fortunately, this nomination constraint only affects a small portion of our sample, as less than 10 % of the sample have listed five male or female friends. Therefore, this restriction should not have a significant impact on the results.
Jackson (2010) examines homophily patterns of race, gender, and grade using Add Health data and finds that homophily index for grade is the highest, implying students are most likely to form friendships within a school-grade.
Several alternative specifications of the spatial weights matrix incorporating friend heterogeneity will be explored in the next section.
As pointed out by the Associate Editor, some network links may be missing due to the sample construction. Some recent papers have started to investigate the challenges posed by missing data in peer effect estimations. In particular, for the conventional linear-in-means model, Sojourner (2013) develops a new way to handle missing peer data by relying on two mean independence conditions. The peer effects parameters can be identified or partially identified, depending on whether peer groups are formed by random assignment or not. For network data, addressing missing data is more complicated. Chandrasekhar and Lewis (2012) propose a two-step estimation procedure using graphical reconstruction to deal with missing data in the analysis of networks. In the first step, the observed portion of the network is used to predict the missing part of the network based on a potentially different network formation model for each network. In the second step, the conditional expectations of the regressors are then used to estimate the outcome equation. This procedure faces several technical challenges, one of which is the “incidental parameter” problem, as a parameter for the network formation model of every network must be estimated. Further, as the graphical reconstruction requires integrating over missing links, the researcher needs to have variables for each node that are predictive in the network formation models. Given the limited and immature development of the literature on the network missing data problem, I leave this issue for future research and do not pursue this direction in the current study.
Table 1 Sample summary statistics The standard deviations are 3.162, 8.649, 3.907, and 3.057, respectively.
As pointed out by the Associate Editor, this comparison does not rule out possible differences in unobservable attributes for the two samples.
Note that this specification would render the linear-in-means model unidentified due to the “reflection problem.”
The negative spatial correlation among the errors reveals negative selection in friendship formation process, implying that the formation of friendships is not necessary driven by the common interest in the outcome under studied. And if this negative selection effect is not controlled for, the peer effect estimation coefficient will be biased downward.
To save space, the results on own characteristics are not reported but are available upon request.
I thank the Associate Editor and an anonymous referee for suggesting this test.
The findings in this study are in general consistent with the existing literature, although the magnitudes of the estimates and those in previous studies are not directly comparable due to the differences in terms of model specification, control variables, peer groups definition, and identification strategy. As a matter of fact, most studies discussed in Sect. 2 find strong evidence for either endogenous effects or contextual effects in various activities of adolescents, even after controlling for the potential contamination bias using various identification strategies including fixed effects and instrumental variables.
Unfortunately, as mentioned above, due to the “reflection problem” in the linear-in-means model, these model specifications are common in the literature.
The related survey questions in the Add Health are, for example, “if you went to his (her) house in the last seven days,” “if you spent time with him (her) last weekend.”
I am grateful to the Associate Editor for suggesting this specification.
This seemingly counterintuitive result can be heuristically explained in the setting of SAR model with SAR disturbances. As pointed out in Footnote 22, the negative spatial correlation among the errors implies negative selection in friendship formation process, which will lead to downward bias on the peer effect coefficient if not properly controlled for. Therefore, for the model based on reciprocal friendship nominations, as only the effects of reciprocal friends are captured, and the negative selection effects among nonreciprocal friends are not controlled for, the peer effect coefficient in this model is downward biased. In contrast, the inflation of the endogenous effect coefficient in the model with the friendship reciprocity assumption imposed reflects the over correction of the negative selection effect, as the networks in this model are defined over a bigger (than the real nomination) set of friends.
The patterns of the robustness of the results with respect to heterogeneity among friends and friendship reciprocity assumptions are consistent with Lin (2014).
It is worth noting that all findings throughout the paper are based on the standard SAR model with a single row-normalized weighting matrix and, therefore, may not be generalized to other settings such as the higher-order SAR model that involves multiple weighting matrices and/or non-row-normalized weighting matrices.
References
Arcidiacono P, Nicholson S (2005) Peer effects in medical school. J Public Econ 89:327–350
Becker GS (1996) Accounting for tastes. Harvard University Press, Cambridge
Bifulco R, Fletcher JM, Ross SL (2011) The effect of classmate characteristics on post-secondary outcomes: evidence from the Add Health. Am Econ J Econ Policy 3:25–53
Bramoullé Y, Habiba D, Bernard F (2009) Identification of peer effects through social networks. J Econom 150:41–55
Calvó-Armengol A, Patacchini E, Zenou Y (2009) Peer effects and social networks in education. Rev Econ Stud 76:1239–1267
Card D, Giuliano L (2013) Peer effects and multiple equilibria in the risky behavior of friends. Rev Econ Stat 95(4):1130–1149
Chandrasekhar A, Lewis R (2012) Econometrics of sampled networks. Working paper, MIT
Clark AE, Lohéac Y (2007) “It wasn’t me, it was them!” Social influence in risky behavior by adolescents. J Health Econ 26:763–784
DeSImone J (2010) Drinking and academic performance in high school. Appl Econ 42(12):1–17
Duarte R, Escario JJ (2006) Alcohol abuse and truancy among Spanish adolescents: a count-data approach. Econ Educ Rev 25:179–187
Duncan GJ, Boisjoly J, Harris KM (2001) Sibling, peer, neighbor, and schoolmate correlations as indicators of the importance of context for adolescent development. Demography 38:437–447
Duncan G, Boisjoly J, Kremer M, Levy D, Eccles JS (2005) Peer effects in drug use and sex among college students. J Abnorm Child Psychol 33(3):375–385
Erwin P (1998) Friendship in childhood and adolescence. Routledge, New York
Espelage DL, Holt MK, Henkel RR (2003) Examination of peer-group contextual effects on aggression during early adolescence. Child Dev 74(1):205–220
Fletcher JM (2012) Peer influences on adolescent alcohol consumption: evidence using an instrumental variables/fixed effect approach. J Popul Econ 25:1265–1286
Gaviria A, Raphael S (2001) School-based peer effects and juvenile behavior. Rev Econ Stat 83(2):257–268
Goldsmith-Pinkham P, Imbens GW (2013) Social networks and the identification of peer effects. J Bus Econ Stat 31:253–264
Hanushek EA, Kain JF, Markman JM, Rivkin SG (2003) Does peer ability affect student achievement? J Appl Econom 18:527–544
Haynie DL (2001) Delinquent peers revisited: does network structure matter? Am J Sociol 106:1013–1057
Haynie DL (2002) Friendship networks and delinquency: the relative nature of peer delinquency. J Quant Criminol 18:99–134
Hawkins JD, Catalano RF, Miller JY (1992) Risk and protective factors for alcohol and other drug problems in adolescence and early adulthood: implications for substance abuse prevention. Psychol Bull 112(1):64–105
Hoxby C (2000) Peer effects in the classroom: learning from gender and rave variation. NBER working paper 7867
Hsieh CS, Lee L (2013) A social interaction model with endogenous friendship formation and selectivity. Working paper, Ohio State University
Jackson MO (2010) An overview of social networks and economic applications. In: Benhabib J, Bisin A, Jackson MO (eds) The handbook of social economics. North Holland Press, New York
Krauth BV (2006) Simulation-based estimation of peer effects. J Econom 133:243–271
Laursen B (1993) Close friendships in adolescence. Jossey-Bass, San Francisco
Lee L, Liu X, Lin X (2010) Specification and estimation of social interaction models with network structures. Econom J 13(2):145–176
Lin X (2010) Identifying peer effects in student academic achievement by spatial autoregressive models with group unobservables. J Labor Econ 28:825–860
Lin X (2014) Peer effects in adolescents’ delinquent behaviors: evidence from a binary choice network model. Reg Sci Urban Econ 46:73–92
Loeber R, Hay D (1997) Key issues in the development of aggression and violence from childhood to early adulthood. Annu Rev Psychol 48:371–410
Lundborg P (2006) Having the wrong friends? Peer effects in adolescent substance use. J Health Econ 25:214–223
Manski C (1993) Identification of endogenous social effects: the reflection problem. Rev Econ Stud 60:531–542
Manski C (2000) Economic analysis of social interactions. J Econ Perspect 14(3):115–136
Nakajima R (2007) Measuring peer effects on youth smoking behaviour. Rev Econ Stud 74:897–935
Patacchini E, Rainone E, Zenou Y (2011) Dynamic aspects of teenage friendships and educational attainment. University of Rome (manuscript)
Powell LM, Tauras JA, Ross H (2005) The importance of peer effects, cigarette prices and tobacco control policies for youth smoking behavior. J Health Econ 24:950–968
Rivkin SG (2001) Tiebout sorting, aggregation and the estimation of peer group effects. Econ Educ Rev 20(3):201–209
Roebuck MC, French MT, Dennis ML (2004) Adolescent marijuana use and school attendance. Econ Educ Rev 23:133–141
Sacerdote B (2001) Peer effects with random assignment: results for dartmouth roommates. Q J Econ 116(2):681–704
Scheier LM, Botvin GJ, Baker E (1997) Risk and protective factors as predictors of adolescent alcohol involvement and transitions in alcohol use: a prospective analysis. J Stud Alcohol Drugs 58(6):652–667
Soetevent AR, Kooreman P (2007) A discrete-choice model with social interactions: with an application to high school teen behavior. J Appl Econom 22:599–624
Sojourner A (2013) Identification of peer effects with missing peer data: evidence from Project STAR. Econ J 123(569):574–605
Vaquera E, Kao G (2008) Do you like me as much as I like you? Friendship reciprocity and its effects on school outcomes among adolescents. Soc Sci Res 37:55–72
Zimmerman DJ (2003) Peer effects in academic outcomes: evidence from a natural experiment. Rev Econ Stat 85(1):9–23
Author information
Authors and Affiliations
Corresponding author
Additional information
I thank the Editor, the Associate Editor, and four anonymous referees for valuable comments. This research uses data from Add Health, a program project directed by Kathleen Mullan Harris and designed by J. Richard Udry, Peter S. Bearman, and Kathleen Mullan Harris at the University of North Carolina at Chapel Hill, and funded by grant P01-HD31921 from the Eunice Kennedy Shriver National Institute of Child Health and Human Development, with cooperative funding from 23 other federal agencies and foundations. Special acknowledgment is due to Ronald R. Rindfuss and Barbara Entwisle for assistance in the original design. Information on how to obtain the Add Health data files is available on the Add Health Web site (http://www.cpc.unc.edu/addhealth). No direct support was received from grant P01-HD31921 for this analysis.
Rights and permissions
About this article
Cite this article
Lin, X. Utilizing spatial autoregressive models to identify peer effects among adolescents. Empir Econ 49, 929–960 (2015). https://doi.org/10.1007/s00181-014-0897-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00181-014-0897-4