Essays in Economics of the Family, Social Interactions
and Retirement Planning
Thèse
Finagnon Antoine Dedewanou
Doctorat en économique
Philosophiæ doctor (Ph. D.)
Essays in Economics of the Family, Social
Interactions and Retirement Planning
Thèse
Finagnon Antoine Dedewanou
Sous la direction de:
Vincent Boucher, directeur de recherche Charles Bellemare, codirecteur de recherche
Résumé
Cette thèse est un recueil de trois essais en économie de la famille, des interactions sociales et la planification de la retraite. Elle utilise plusieurs stratégies cohérentes d’estimation pour com-prendre et évaluer l’influence de l’environnement familial et social sur les rendements scolaires des élèves et la poursuite de leurs études. Elle examine aussi les méchanismes économiques qui expliquent la relation non-causale entre la littératie financière et la décision d’épargner pour la retraite chez les personnes âgées de 50 et plus aux États-Unis.
Le premier chapitre étudie l’impact des interactions sociales sur les croyances des élèves du secondaire aux États-Unis concernant leur participation au collège. Plus spécifiquement, nous nous sommes demandons si, le fait qu’une élève du secondaire décide par exemple de poursuivre ses études collégiales une fois graduée, est le reflet ou non des croyances ou des motivations de ses ami(e)s. Pour ce faire, nous supposons que les élèves évoluent dans un environnement d’apprentissage social à la DeGroot (1974) dans lequel ils mettent à jour quotidiennement leurs croyances en prenant la moyenne pondérée des croyances de leurs amis et leurs carac-téristiques observables. Nos résultats confirment la présence d’effets de pair significatifs dans l’apprentissage social des élèves de l’ordre de 12% en moyenne, mais avec une forte hétérogé-néité individuelle non observable allant de 8% à 73%. Nos résultats montrent également qu’il existe une forte hétérogénéité entre les écoles. Étant donné que cette hétérogénéité ne peut être approximée par des caractéristiques observables des élèves, nous suggérons que les politiques éducatives ciblées au niveau de l’école pourraient être plus efficaces.
Le deuxième chapitre fait une analyse économétrique de l’importance relative de l’implication des grands-parents dans l’éducation de leurs petits-enfants et de son effet sur le rendement scolaire des élèves. Plusieurs faits stylisés motivent cet essai. (i) Les personnes âgées vivent de nos jours de plus en plus longtemps et en parfaite santé. (ii) Les deux parents sont de plus en plus présents sur le marché du travail ; ce qui peut inciter les grands-parents à les remplacer, en partie ou du moins, à la maison. (iii) Également, l’importance croissante des mères mono-parentales présentes sur le marché du travail augmente la nécessité pour les grands-parents de jouer un rôle substantiel dans la vie de leurs petits-enfants. À partir des données américaines de panel de la “National Survey of Families and Households (NSFH)”, ce chapitre présente d’abord la relation non-causale entre le temps d’investissement des parents et l’implication
des grands-parents dans l’éducation de leurs petits-enfants. Comme l’augmentation de l’offre de travail des parents durant les années d’étude de leur enfant peut réduire leur temps d’inves-tissement et donc avoir probablement un effet négatif sur le développement de l’enfant, nous avons ensuite examiné si l’implication des grands-parents est susceptible de réduire cet effet négatif. Plus précisement, nous avons estimé la relation non-causale entre l’implication des grands-parents et le rendement scolaire de leurs petits-enfants. Enfin, nous avons présenté un modèle structurel à trois générations (grands-parents ; parents et petits-enfants) afin d’esti-mer l’impact causal de l’implication des grands-parents sur les performances académiques des enfants. Nos résultats montrent, entre autres, que l’implication des grands-parents (du côté de la mère) expliquent 15% du rendement scolaire des enfants.
Toutefois, une bonne implication des grands-parents dans la vie de leurs petits-enfants rime avec une bonne planification de la retraite et une bonne sécurité financière. Ceci nous amène donc à analyser dans le troisième chapitre de cette thèse, la relation non-causale entre la li-tératie financière et la planification de la retraite aux États-Unis. En effet, afin de pouvoir faire face au phénomène du vieillissement de la population et à cause des conséquences de la crise économique et financière que le monde a connus ces dernières années, plusieurs pays ont entamé la réforme de leurs systèmes de pension. Dans la plupart de ces nouveaux systèmes de pension, une bonne partie du risque et des responsabilités a été transférée du gouvernment, des employeurs et des fonds de pension vers les particuliers et les ménages (Prast and van Soest, 2016). Aux États-Unis par exemple, les régimes de pension à “prestation définie” qui garantissait un revenu donné après la retraite ont été largement remplacés par des régimes à “cotisation définie” où les primes sont fixes mais les revenus de pension dépendent des rende-ments des investisserende-ments et les particuliers doivent faire leurs propres choix d’investissement devant la multitude d’actifs financiers qui s’offrent à eux. Pour cette raison, plusieurs décideurs s’investissent à identifier les instruments de politique qui vont permettre à la population de prendre de bonnes décisions liées à leur pension et ainsi maximiser leur utilité espérée tout le long de leur cycle de vie. Du côté de la recherche, plusieurs études montrent dans plusieurs pays que le premier aspect le plus important est l’éducation financière chez les plus jeunes et la litératie financière chez les adultes (e.g., Lusardi and Mitchell, 2011;Bucher-Koenen and Lusardi, 2011). Ce chapitre analyse donc la corrélation non-causale entre la litératie finan-cière et la décision d’épargner pour la retraite chez les personnes âgées de 50 ans et plus aux États-Unis. Nos résultats montrent que chez les femmes, le manque de connaissance sur les questions relatives à l’inflation, au taux d’intérêt et à la diversification du risque est une cause de non-planification pour la retraite. Par contre chez les hommes, seulement le manque de connaissance sur la diversification du risque semble influencer leur décision d’épargner pour la retraite. Trois mécanismes permettent d’expliquer cette différence entre les hommes et les femmes : les besoins d’épargne différents, l’excès de confiance chez les hommes et l’utilisation beaucoup plus importante des services d’un conseiller financier chez les femmes.
Abstract
This thesis is a collection of three essays in economics of the family, social interactions and re-tirement planning. It uses coherent estimation strategies to ascertain the influence of friendship and family background on children educational attainment and their academic achievement. It also investigates the mechanisms underlying the interplay between financial literacy and retirement planning.
The first chapter studies the role of friendship in explaining teenagers’ subjective beliefs re-garding their college participation. More specifically, we wonder whether the fact that a high school student decides, for example, to continue her college studies after graduation, is due to her friends’ beliefs or motivations. For that purpose, we assume that students embedded in a social network à laDeGroot(1974) in which they update their beliefs by repeatedly taking the weighted average of their neighbor’s beliefs and their observable characteristics. Our results confirm the presence of significant peer effects in students’ social learning of 12% an average, but with a strong unobservable individual heterogeneity ranging from 8% to 73%. Our results also show that there is a strong heterogeneity between schools. Since this heterogeneity cannot be approximated by observable characteristics, we suggest that targeted at school level could be more effective.
The second chapter investigates grandparents’ involvement in grandchildren education. Three stylized facts motivates this chapter. (i) Seniors are living now progressively longer and health-ier lives. (ii) Both parents are increasingly present on the labor market; which may encourage grandparents to replace them, in part or at least, at home. (iii) The rise of working single mothers increases the potential for grandparents to play an important role in the life of their grandchildren. Using panel data from the US National Survey of Families and Households, we first present in this chapter, the non-causal correlation between parental time investment and grandparents’ involvement. Second, since increasing parents’ labour supply in a child school years can reduce their time investments and therefore cause a negative direct effect on child’s outcomes, we investigate whether grandparents’ involvement could reduce this negative effect. More precisely, we analyze the non-causal correlation between grandparents’ involvement and grandchildren academic achievement. Finally, we present a three-generational (grandparents-parents-grandchildren) model in order to estimate the causal effect between grandparents’
involvement and grandchildren academic achievement. Our results show that grandparents (from the mother-side) explains 15% of grandchildren academic achievement.
However, good grandparents’ involvement goes hand-in-hand with good retirement planning and financial safety. This leads us to analyze in the third chapter of this thesis the non-causal correlation between financial literacy and retirement planning in the US. Indeed, to meet the challenges of an ageing population, and due to the economic and financial crisis that countries have experienced during the past decades, many countries have started reforming their pension systems. In most of these new pension systems, a substantial part of the risk and responsibility has been shifted from governments, employers and pension funds to individuals and private households (Prast and van Soest, 2016). In the US for instance, employer-provided “defined benefit” plans guaranteeing a given income after retirement have largely been replaced by “defined contribution”, where premiums are fixed but pension income depends on investment returns and individuals have to make their own investment choices. For this purpose, several decision-makers try to find policy instruments that could help people to take good decisions concerning their pension and then maximize their expected utility over their life cycle. On the research side, several papers show in many countries that the first important aspect is financial education among young people and financial literacy among adults (e.g., Lusardi and Mitchell,2011; Bucher-Koenen and Lusardi, 2011). This chapter therefore analyzes the non-causal correlation between financial literacy and the decision to save for retirement among people aged 50 and over in the US. Our results show that among women, lack of knowledge on inflation, interest rate and risk diversification is a cause of non-retirement planning. However, among men, only lack of knowledge in risk diversification seems to influence their decision to save for retirement. Three mechanisms explain this gender difference: different saving needs, overconfidence among men and an increasing use of services from financial advisors by women.
Keywords: Students’ beliefs, college participation, grandparents’ involvement, parental in-vestment, financial literacy, retirement planning.
Table des matières
Résumé iii
Abstract v
Table des matières vii
Liste des tableaux ix
Liste des figures xi
Remerciements xiii
Avant-propos xv
Introduction 1
1 Peer-Induced Beliefs Regarding College Participation 3
1.1 Résumé . . . 3 1.2 Abstract . . . 4 1.3 Introduction. . . 5 1.4 Model . . . 7 1.5 Data . . . 11 1.6 Results. . . 11 1.7 Conclusion . . . 23 1.8 Bibliographie . . . 24
2 Grandparents’ Involvement in Grandchildren Education 28 2.1 Résumé . . . 28
2.2 Abstract . . . 29
2.3 Introduction. . . 30
2.4 Data . . . 32
2.5 A Model of Parental Involvement . . . 39
2.6 Empirical Analysis . . . 43
2.7 Results . . . 46
2.8 Validating the Model . . . 47
2.9 Heterogeneity . . . 48
2.10 Determinants of Grandparents’ Involvement . . . 50
2.11 Conclusion . . . 51
3 Financial Literacy and Retirement Planning 56 3.1 Résumé . . . 56 3.2 Abstract . . . 57 3.3 Introduction. . . 58 3.4 Data . . . 60 3.5 Empirical Analysis . . . 62
3.6 Understanding the mechanisms . . . 66
3.7 Conclusion . . . 70 3.8 Bibliographie . . . 71 Conclusion 74 A Appendix to Chapter 1 75 A.1 Convergence. . . 75 A.2 Identification . . . 75 A.3 MCMC - Details . . . 76
A.4 Predictability of Friendship . . . 76
A.5 Additional Material. . . 76
A.6 Posterior Distributions . . . 81
A.7 Posterior distributions of the extended model . . . 84
A.8 Sampling more factors in one block . . . 89
A.9 Adding heterogeneity of the α’s in the extended model . . . 92
B Appendix to Chapter 2 96 B.1 Data Cleaning Procedures . . . 96
B.2 Tables . . . 96
B.3 Proof of Proposition 1 . . . 99
B.4 Computation of P (T1,i ≤ t1,i|T3,i= t3,i) . . . 100
Liste des tableaux
1.1 Summary statistics . . . 12
1.2 Estimation of homogeneous peer effects . . . 14
1.3 Heterogeneous peer effects in Wave 1 (School 77) . . . 17
1.4 Predictors of Group 1 . . . 18
1.5 Contribution of unobserved fixed effects for School 77 in Wave 1. . . 19
1.6 Robustness check : endogeneous network formation . . . 22
2.1 Distribution of Grandparents Involvement . . . 34
2.2 Summary Statistics. . . 35
2.3 Parental Time Investment and Grandparents’ Involvement . . . 37
2.4 Grandchild’s Academic Achievement and Grandparents’ Involvement . . . 39
2.5 Parameter Estimates of the Structural Model . . . 47
2.6 Selection on Observables . . . 48
2.7 Population percentiles of the structural model parameters (with heterogeneity) 49 2.8 Cross-section heterogeneity with respect to child’s characteristics . . . 49
3.1 Financial Literacy Patterns . . . 62
3.2 Probit estimates of retirement saving decision . . . 65
3.3 Distribution of responses to financial literacy questions for female and male . . 67
3.4 Financial knowledge and advisor from financial advisor. . . 68
3.5 Incorrect responses and demand for advisors by women respondents . . . 68
3.6 Proportion of respondents who report “don’t know ” to the financial literacy questions . . . 69
3.7 Confidence and financial advice seeking . . . 70
A.1 Network formation probit-marginal effects for all schools (×10−4). . . 77
A.2 Network formation probit-marginal effects in School 28 (×10−4). . . 77
A.3 Students’ beliefs and preferences (Wave 1) . . . 78
A.4 Heterogeneity between schools. . . 78
A.5 Endogeneous network formation–Heterogeneous Peer Effect . . . 95
B.1 Parental time investment and Parental hours worked . . . 97
B.2 Determinants of the Quality of Grandparents’ involvement . . . 98
B.3 Heterogeneous Effects of Maternal Education and Hours Worked, Grandparent’s Relationship and the Child Academic Achievement . . . 99
C.1 Column profiles of responses to interest rate and inflation questions . . . 101
C.3 Column profiles of responses to inflation and stock risk questions . . . 101
C.4 Row profiles of responses to inflation and stock risk questions . . . 102
C.5 Column profiles of responses to interest rate and stock risk questions . . . 102
C.6 Row profiles of responses to interest rate and stock risk questions . . . 102
C.7 Probit estimates of retirement saving decision—Female . . . 103
C.8 Probit estimates of retirement saving decision—Male . . . 104
Liste des figures
1.1 Heterogeneity among schools . . . 15
A.1 Distribution of beliefs for schools 58 and 77. . . 78
A.2 Posterior distributions for the estimation over the entire sample (continued). . 81
A.3 Posterior distributions for the estimation over the entire sample (continued). . 82
A.4 Posterior distributions for the estimation over the entire sample (continued). . 83
A.5 Posterior distributions for the estimation over the entire sample . . . 84
À ma fille Exaucée Mahuna et à mon épouse Rolande Carine.
Remerciements
J’ai une profonde gratitude envers mon directeur de recherche, Vincent Boucher, qui m’a fait confiance, m’a soutenu et guidé tout au long de ce cheminement de doctorat. Merci pour son dévouement à aider ses étudiants. Je remercie également mon co-directeur de recherche, Charles Bellemare, pour sa disponibilité et son enthousiasme. Merci de m’avoir permis de prendre du recul sur mes recherches, ce qui a beaucoup contribué à diminuer mon niveau de stress ! Je suis très heureux d’avoir bénéficé de leurs conseils multiformes et leur supervision. La rigueur et la gentillesse dont ils font preuve sont des valeurs qui continueront à me guider tant dans ma vie professionnelle que personnelle. Merci pour votre temps et votre confiance. Merci à Arnaud Dufays d’avoir accepté de co-écrire le chapitre1de cette thèse avec moi. Merci pour ton implication dans ce projet. J’ai appris beaucoup.
Ma reconnaissance va à l’endroit de tout le corps professoral du département d’économique de l’Université Laval pour tout ce que j’ai appris d’eux. Je remercie également le personnel administratif du département pour leur disponibilité sans cesse renouvellée.
J’adresse mes sincères remerciements à Ismael Y. Mourifié pour m’avoir invité à passer 7 mois de séjour de recherche au département d’économie de l’Université de Toronto. J’ai énormément appris des discussions que nous avons eues, et celles-ci m’ont permis de compléter une grande partie du chapitre 2de cette Thèse.
Mes chaleureux remerciements vont à l’endroit de mes collègues et amis, Élysée A. Hounde-toungan, Marius Sossou, Jean-Louis Bago, Daouda Belem, Morvan Nongni, Ibrahima Sarr, et Koffi Akakpo pour les discussions instructives et les bons moments passés ensembles.
À ma merveilleuse épouse Rolande, je témoigne ma profonde gratitude pour avoir rêvé en-semble depuis notre second cycle universitaire d’aller aussi loin que possible et relevé enen-semble les défis qui se sont présentés à nous. Merci pour ta grande compréhension, ta patience et pour ton support sur lequel je pouvais m’appuyer quand tout semblait s’écrouler autour de moi. Merci également à ma fille Exaucée Mahuna qui, de part son sourire au visage, me donne au quotidien la force et le courage d’avancer.
d’aller à l’école mais qui ont pourtant compris la valeur de l’éducation mieux que quiconque. Malgré vos ressources limitées, vous avez tout sacrifié pour mon éducation. Il n’y a pas assez de mots pour exprimer mon admiration et ma gratitude envers vous.
Ce cheminement n’aurait pas été possible sans le soutien financier du Département d’éco-nomique de l’Université Laval, la Chaire de recherche Industrielle Alliance sur les enjeux économiques des changements démographiques, le Fonds de Recherche du Québec Société et Culture (FRQSC) et la Faculté des Sciences Sociales que je remercie ici.
Avant-propos
Cette thèse comporte trois chapitres. Le chapitre 1 est co-écrit avec Vincent Boucher (mon directeur de thèse) et Arnaud Dufays. Ce chapitre fait actuellement l’objet de quelques révi-sions pour être soumis dans une revue avec un comité de lecture. Le chapitre2 est un article sans co-auteur. Le chapitre3 est écrit conjointement avec mon co-directeur de thèse, Charles Bellemare. L’ordre des auteurs figurant sur les articles co-écrits est alphabétique. Chacun des co-auteurs y a contribué à parts égales.
Introduction
A growing literature in economics emphasizes the importance of family background and so-cial interactions in shaping student’s academic achievement. However many questions remain unanswered. In particular, very little is known about the effect of the extended family, spe-cially grandparents, on grandchildren’s educational success. In addition, the mechanisms by which peers affect education are unclear. This thesis uses quantitative analysis to fill this gap. In the chapter 1, we investigate friends’ influence on beliefs regarding college participation in the United States. To do so, we present a model of social learning in network where stu-dents update their beliefs by repeatedly taking the weighted average of their neighbor’s beliefs and their observable characteristics. We present a coherent estimation strategy to estimate social interaction models with ordered outcome. We use a very rich US longitudinal database in which teenagers were observed during the years 1994-2009. We find strong heterogeneity among schools and students. However, this heterogeneity cannot be approximated by obser-vable characteristics. In terms of policy implications, this implies that policies targeted at the level of the school are perhaps more sensible.
The focus of the chapter 2 is on grandparents’ involvement in grandchildren education. Two facts motivate this chapter. First, seniors in developed countries are living now progressi-vely longer and healthier lives. Second, the recent rise of maternal employment and single-parent families increase the potential for grandsingle-parents to play an important role in the life of their grandchildren. This chapter builds on the existing human capital production function of Chiappori, Salanié, and Weiss (2017) and introduces grandparents’ involvement as an addi-tional input. The model predicts that when the grandparents intervene in their grandchildren education, both parents whatever their level of education have the same marginal incentives to invest in their children human capital. As a result, this chapter finds that grandparents’ involvement (from the mother-side) explains 15% of the variation of the grandchild acade-mic achievement and this effect seems to be driven by the subgroup of mothers with a low educational level.
Finally, chapter 3 examines the extent to which the three financial literacy questions (infla-tion, interest rate and risk diversification) largely used in the literature to measure financial knowledge are relevant for predicting respondents’ retirement planning. We estimate the
ef-fect of both correct and incorrect responses on retirement planning using data from the 2004 Health and Retirement Survey. We find that among women, lack of financial knowledge is a cause of lack of retirement planning. However, among men only lack of knowledge in risk diversification is significantly associated with retirement planning. Three mechanisms explain this gender difference : different saving needs, overconfidence among men and an increasing use of services from financial advisors by women.
Chapitre 1
Peer-Induced Beliefs Regarding
College Participation
1.1
Résumé
Nous étudions dans cet article l’influence des amis sur les croyances concernant la partici-pation au collège des élèves. Pour ce faire, nous présentons un modèle d’apprentissage en réseau et montrons que celui-ci est identifié. Le modèle est ensuite estimé à partir des données sur les croyances des adolescent(e)s concernant leur participation au collège aux États-Unis, en controllant pour leurs préférences et leur performance académique. Nous trouvons qu’en moyenne, les croyances des amis représentent environ 12 % du processus de mise à jour des croyances des élèves. De plus, nous montrons qu’il existe une forte hétérogénéité entre les écoles et les élèves. En particulier, nous trouvons une hétérogénéité individuelle substantielle non observée, qui met en doute l’efficacité des politiques publiques ciblées sur les réseaux.
Classification JEL : D83, I20, C31, C11.
Keywords : Réseaux d’amis, Mise à jour des croyances, Participation au collège, Effets de pairs hétérogènes.
1.2
Abstract
We study peer effects on the formation of beliefs regarding college participation. To do so, we present a structural model of learning in friendship networks. We show that the model is identified and we present a Bayesian estimation procedure. The model is estimated using data on teenagers’ beliefs regarding college participation, controlling for preferences and academic achievement. On average, we find that friends’ beliefs account for about 12% of the updating process. In addition, we show that it exists a strong heterogeneity among schools and indi-viduals. In particular, we find substantial unobserved individual heterogeneity, which casts doubt on the efficiency of network-targeted public policies.
JEL Codes : D83, I20, C31, C11.
Keywords : Social networks, Beliefs updating, College participation, Heterogeneous peer ef-fects.
1.3
Introduction
It is not news that teenagers often have biased views about the world. Indeed students’ sub-jective expectations affect many of their choices, ranging from career choices (e.g.Ashby and Schoon,2010) to risky sexual behaviour (e.g.Sipsma et al.,2015). Typical predictors of teena-gers’ subjective biases include gender and the characteristics of their parents, such as parental education level (e.g. Attanasio and Kauffman,2010).
In this paper, we study the role of friendship in explaining teenagers’ subjective beliefs regar-ding college participation. We present a structural model of social learning in networks that we estimate using data from middle- and high-school teenagers in the US. A large set of obser-vable variables allows us to measure the impact of beliefs, controlling for (stated) preferences, academic achievement, and many other relevant socio-economic characteristics.
We find that the average belief of a student’s peers accounts for 12% of the belief upda-ting process, while the remaining percentage is due to individual characteristics. Interesupda-tingly, there exists a substantial heterogeneity among schools and individuals. In particular, we find substantial unobserved individual heterogeneity : the importance of peers’ beliefs on the upda-ting process ranges from 8% to 73%. Importantly, differences in the magnitude of peer effect cannot be well predicted using observable characteristics ; a result that has important policy implications. In particular, without accounting for unobserved heterogeneity, network-targeted policies such as key-players analysis (Ballester et al.,2006) are likely ineffective.
We present a model where, at each period, individuals update their beliefs according to their friends’ average beliefs (e.g.DeGroot,1974) and also according to their own particular charac-teristics (e.g. age, gender, parental education, etc.). Accordingly, the steady-state distribution of beliefs is affected by the shape of the social network, but it is not consensual (e.g. Friedkin and Johnsen,1990).
We estimate the model using data related to students’ beliefs regarding college participation. Specifically, each student is asked : “On a scale of 1 to 5, where 1 is low and 5 is high, how likely is it that you will go to college ?”.1 Since beliefs are only measured using an ordinal scale, we develop a multivariate ordered probit model. We show that all of the model’s parameters are identified and estimate the model using a Bayesian approach, which allows for greater flexibility regarding the heterogeneity of social learning.
Estimating the model over the entire sample (16 schools), we find an average peer effect on beliefs of 12%. We also separately focus on the two largest schools in our sample and find effects of 13% and 25%. In addition, we study this heterogeneity in greater depth by then estimating a latent class model. Specifically, students are sorted into two (unobserved) groups according to the strength of peer effect on their beliefs. Estimating the model for the largest school in our
sample (the school having an average effect of 13%), we observe marked differences between the two latent classes ; peer influence on beliefs are respectively 8% and 73%.
To study the source of the students’ heterogeneity, we try to find predictors for the probability of belonging to either class. We find that females and students having lower beliefs are slightly more likely to belong to the low-influence group. Overall, however, the predictive power is very weak. We believe that this is of great importance for policy making.
Indeed, one of the key implications of the literature on peer effects in social networks is that it allows the policy maker to target the individuals that would generate the largest spillovers (i.e. the key player(s), see for exampleBallester et al. (2006) andZenou(2016)). Importantly, however, this literature mostly assumes that peer effects are homogeneous.
Since we observe (1) a high variation in peer influence among individuals, and that (2) this heterogeneity cannot be predicted using observables, targeted policies will be likely ineffective. That is, the most central individual in the network is not necessarily the key player when accounting for unobserved heterogeneity. Since this heterogeneity cannot be approximated by observable characteristics, and since uniform policies are relatively cheap (Hoxby and Turner,
2013), policies targeted at the level of the school are perhaps more sensible.
We contribute to the literature on peer effects on students’ outcomes (e.g. Sacerdote et al.,
2011;Black et al.,2013;Burke and Sass,2013;Carrell et al.,2009;Hatami et al.,2015;Zabel,
2008). As discussed by Manski (2000), interactions can affect constraints, preferences, or ex-pectations. We estimate the impact of expectation interactions, and we show that they have a significant and heterogeneous impact on the steady-state beliefs regarding college participa-tion.
We also contribute to the literature on peer effects in networks (see Boucher and Fortin,2016;
de Paula,2017, for recent reviews). Our structural model is closely linked to the spatial autore-gressive model. Although identification conditions are well known (e.g.Bramoullé et al.,2009;
Lee et al.,2010), few papers study the implications of such models when the outcome variable is ordered.2 We present a Bayesian estimation procedure and show that the identification of the model is not impaired by the ordered nature of the outcome variable.
Finally, we also contribute to recent literature discussing the heterogeneity of peer effects. While some papers have assessed heterogeneous peer effects based on observable characteristics (e.g.Dieye et al.,2017;Arduini et al.,2016), few papers have studied unobserved heterogeneity on peer effects. One important exception is Peng (2016) who presented a model where the weight of the network is estimated using a LASSO technique. Since our model is non-linear, we cannot directly adapt the Lasso procedure ofPeng(2016). In this paper, we model individuals’ heterogeneity using a mixture model. In particular, the individuals belong to latent classes,
grouped according to the magnitude of their peer effects. We find substantial differences among individuals’ reliance on peers for belief formation.
The remainder of the paper is organized as follows. In Section 1.4, we introduce the model of social learning in networks, and we discuss the identification conditions and the estimation procedure. In Section 1.5, we present the data, while in the following Section1.6, we present our results. Section 1.7concludes.
1.4
Model
We consider a school of n > 1 students. Each student i ∈ N is characterized by a set of observable socio-economic variables xi and forms beliefs p∗i ∈ R regarding the likelihood that
they will go to college.
We assume that beliefs are formed as follows :
p∗i = xiβ + εi, (1.1)
where εi ∼ N (0, 1). A similar specification is used for instance in Bellemare et al. (2008).
The variables in xi are likely to influence the relative optimism (or pessimism) of the student regarding the likelihood that they will go to college, e.g. parents’ education, college preferences, racial group, gender, etc.
Given that students at the same school are likely to discuss their beliefs, we allow the students’ beliefs to evolve as a function of their peers’ beliefs. Students who have no peers are called isolated and their beliefs are simply given by (1.1). For non-isolated students, we assume that the beliefs are updated according a modified DeGroot learning process (DeGroot,1974) :
p∗(t)i = α ni
X
j∈Ni
p∗(t−1)j + (1 − α)xiβ + εi, (1.2)
where Ni ⊂ N represent the set i’s peers, i.e. the set of students from which i learns. We set
i /∈ Ni, ni = |Ni|, and we assume that α ∈ (0, 1).
At any period t, a non-isolated student’s belief is given by a convex combination between their type xiβ and the average beliefs of their peers. Note that here, the fixed term (1 − α)xiβ + εi
influences beliefs at every period t, which preserves the heterogeneity of beliefs at the steady state (see Equation (1.3) below). A similar approach is followed by Friedkin and Johnsen
(1990).
Although naive, DeGroot learning processes are widely used in the context of belief updating in social networks (e.g. Golub and Jackson,2010,2012).3 In particular, Chandrasekhar et al.
(2015) show, in a field experiment setting, that individuals’ beliefs are better described by DeGroot learning relative to Bayesian learning.
Note that we can write the learning process in matrix form for all students at time t as : p∗(t)= αTp∗(t−1)+ BαXβ + ε,
where Bαis a diagonal matrix taking the value Bii= (1 − α) if the student is not isolated, and Bii= 1 otherwise. The matrix T summarizes the learning network and is such that Tij = 1/ni
if i learns from j’s beliefs (i.e. j ∈ Ni), and Tij = 0 otherwise.
Importantly, note that T is generally not symmetric ; while isolated students do not learn from other students’ beliefs, a non-isolated student may learn from an isolated students’ beliefs. For the remainder of this paper, we will assume that the data is generated from the steady state of this learning process. (See the Appendix for a proof of convergence and of the invertibility of [I − αT].) :
p∗= [I − αT]−1BαXβ + [I − αT]−1ε. (1.3)
A particularity of the model is that p∗i is not observed. We assume that each student is asked to assess the likelihood that they will go to college using an ordinal scale. For student i, we therefore observe : pi = 1 if p∗i ≤ γ1, 2 if γ1< p∗i ≤ γ2, .. . K − 1 if γK−2< p∗i ≤ γK−1, K if p∗i > γK−1,
where we normalize γ1 = 0, and where γ = [γ2, ..., γK−1]0 must be estimated.
In the next section, we discuss the identification of α and β. We then detail the estimation procedure in Section 1.4.2.
1.4.1 Identification
If we were to observe p∗, identification of α and β would follow directly from the literature (e.g.
Bramoullé et al. (2009)).4 Here, however, we only observe the ordinal scale p. The likelihood of p is therefore given by a multivariate ordered probit model.
We show that under the typical restrictions where the variances of εi are normalized to 1, and γ1 is normalized to 0, the partial observability of p∗ does not affect the identification of α and
β. To understand this reasoning, one can assume that there are no isolated individuals and that individuals interact in pairs (A general treatment is provided in the Appendix).
Under these simplifying assumptions, the model reduces to a simple bivariate ordered probit model :
p∗i = αp∗j + (1 − α)xiβ + εi,
p∗j = αp∗i + (1 − α)xjβ + εj.
It can also be rewritten as : p∗i = √ 1 + α2 1 − α2 (1 − α) √ 1 + α2xiβ + α(1 − α) √ 1 + α2xjβ + νi , p∗j = √ 1 + α2 1 − α2 (1 − α) √ 1 + α2xjβ + α(1 − α) √ 1 + α2xiβ + νj , where " νi νj # ∼ N " 0 0 # , " 1 2α 2α 1 #! .
Since we do not observe the latent variables p∗i and p∗j, this is observationally equivalent to :5 p∗i = √(1 − α) 1 + α2xiβ + α(1 − α) √ 1 + α2xjβ + νi, p∗j = √(1 − α) 1 + α2xjβ + α(1 − α) √ 1 + α2xiβ + νj.
Thus, α and β are identified. In particular, α is directly identified from the variance of ν (e.g.
Graham (2008), Rose (2017)) and also from the ratio of the coefficients for xi and xj (e.g.
Moffitt et al. (2001),Bramoullé et al. (2009)). Notably, this implies that the identification of α does not follow from the normalization of the variance of νi.6
1.4.2 Estimation
The estimation of a multinomial ordered probit model is challenging as, among other things, the maximum likelihood estimator cannot be written in close form. In this paper, we adopt a Bayesian approach, although a classical approach could also be used.7
One advantage of the Bayesian approach is that the estimation can be performed separately for different schools, allowing for the study of heterogeneity among schools (see Section 2.7). In a classical setting, the dependence between observations often requires that the estimation be performed for many independent populations (e.g. schools) to obtain consistent estimates.8
5. This is due to the fact that we can rescale γ.
6. As usual, β is only scale-identified, i.e. its identification depends on the normalization of the variance of νi.
7. For instance, using a GHK (seeGeweke et al.(1994)) algorithm.
8. This is only a sufficient condition, see assumptions 2 and 3, and the discussion on page 1903 inLee(2004). An alternative is to endogenize the interaction matrix T and invoke limited spatial dependence arguments, as inQu and Lee(2015). In either case, the study of heterogeneity is not straightforward.
This is important since we find large heterogenous effects between schools (see Section 1.6.2) and within schools (see Section 1.6.3).
The model parameters are estimated via an Markov chain Monte Carlo (MCMC) algorithm. Specifically, we followAlbert and Chib(1993) and augment the targeted posterior distributions with two random variables, p∗ and γ (see Tanner and Wong (1987) for details). We use standard prior distributions that are set as ˜α ≡ ln1−αα ∼ N (0, 4) and β ∼ N (β0, Σ0) with
β0= (0 0 . . . 0)0 and Σ0 = IK. The MCMC scheme is documented in Algorithm 1.
Algorithm 1 MCMC algorithm Set initial values to β0, α0, p∗0, γ0
for j = 1 to M , where M denotes the number of MCMC iterations do [1] - Sample β, α ∼ β, α|p∗, p, γ :
— Draw ˜α∗ ∼ Nα˜j−1, ¯σj−12
in which the scale ¯σ2
j−1 is adapted over the MCMC
itera-tions using the approach ofAtchadé and Rosenthal(2005) and set α∗= eα˜∗/(1 + eα˜∗). — Sample β∗|α∗ ∼ N ( ¯βα∗, Σα∗), where ¯βα∗ = Σ−1[X0Bα∗[I − α∗T]p∗ + Σ−1 0 β0], and Σα∗ = [Σ−1 0 + X0B 2 α∗X]−1.
— Accept or reject the draw β∗, α∗ according to the Metropolis-Hastings ratio, χM H(β∗, α∗| ¯βj−1, αj−1) = Max{
f (p∗|α∗)f (α∗) f (p∗|αj−1)f (αj−1)
, 1},
where f (α) is the prior density function of α, and f (p∗|α) = f (p∗|α, ¯βα)f ( ¯βα)
f ( ¯βα|p∗,α)
. [2] - Sample γ ∼ γ|p∗, p, β, α :
— For k = 2, ..., K − 1, draw γk|γk−1, γk+1 ∼ U [u, ¯u], where u =
max {maxi{p∗i : pi= k} , γk−1}, and ¯u = min {mini{p∗i : pi= k + 1} , γk+1}.
[3] - Sample p∗|p, γ, β, α :
— For all i, draw p∗i from a truncated normal distribution having a unit variance and a mean equal to xiβ if i is isolated, and a mean αTip∗+ (1 − α)xiβ if i is non-isolated
(recall that Tii= 0). For pi = k, the left truncation is γk−1 and the right truncation
is γk.
end for
Steps [2] and [3] in Algorithm1are simple Gibbs steps where parameters are drawn from their conditional posterior distribution. Step [1] is a Metropolis–Hastings step where parameters α and β are both drawn before the accept/reject phase, which improves convergence.9
9. An alternative would be to update β through a Gibbs step and then draw α using a Metropolis–Hastings step.
1.5
Data
We use data from the National Longitudinal Study of Adolescent to Adult Health (Add Health). The study began in 1994 (Wave 1) with a representative sample of students from grades 7 to 12. For our main purpose, we use data from the first wave to evaluate peer effects on the students’ beliefs regarding the likelihood that they will go to college.10We also use the data from Wave 2 for our robustness analysis in sections 1.6.4 andA.4.
In addition to the information about the students’ beliefs, the survey contains information vis-à-vis the students’ preferences regarding college, as well as many relevant socio-economic and demographic characteristics. In particular, students are asked how much they want to go to college as well as how likely they think this will happen. Although the two questions are clearly correlated, there is substantial variation, which allows us to capture the role of beliefs, while controlling for preferences.11
For a subset of 16 schools, the survey contains a full description of the students’ friendship network. In the context of this paper, we assume that student i learns from student j if, and only if, i nominated j as a friend. Table 1.1presents the summary statistics.
1.6
Results
In this section, we present the results of the estimation for the entire sample as well as for a subset of two relatively large schools. We begin by discussing the interpretation of the parameter α : essentially, the identification of social learning using model (1.3).
1.6.1 Interpretation of α
Although α is identified from the structural econometric model, an important concern is that it may not necessarily capture social learning (e.g.Angrist(2014) orKline and Tamer(2014)). In fact, as for most of the existing literature, social interaction effects (here α) are picked up as residual correlations between outcomes (here, beliefs) after controlling for other potential sources of correlation.
Perhaps the most important source of correlation is the presence of common unobserved shocks.12 The beliefs of students i and j may be correlated because they share a common (unobserved) source of information ; for example, whether or not a career counselor is present in the school. In this paper, we control for these unobserved shocks using school-level fixed effects.
10. Estimations are robust for the use of data of the second wave (1995) for which a similar set of variables is available.
11. See TableA.3of the Appendix.
Table 1.1 – Summary statistics
Mean Std. Dev. Min Max Educational expectations
Belief (ordinal) 4.05 1.14 1 5 College preference (ordinal) 4.37 1.05 1 5 Individual characteristics Age 15.01 1.41 11 19 Female 0.49 0.50 0 1 Grades 7–8 0.22 0.41 0 1 Grades 9–10 0.48 0.50 0 1 Grades 11–12 0.30 0.46 0 1 White 0.61 0.48 0 1 Black or African–American 0.15 0.36 0 1 Hispanic 0.18 0.38 0 1 Mother education
Less than high school 0.07 0.26 0 1 High school 0.24 0.43 0 1 More than high school 0.29 0.45 0 1 Father education
Less than high school 0.07 0.25 0 1 High school 0.18 0.38 0 1 More than high school 0.25 0.43 0 1 Household characteristics
Household size 4.52 1.15 1 6 Network statistics
Average number of friends 3.33 2.75 0 10 Number of female friends 1.68 1.59 0 5 Number of male friends 1.65 1.60 0 5 Isolated students 595
Number of students 2,255 Number of schools 16
Note : The variables “Belief” and “Preference” are both on an ordinal scale. The specific questions read : “On a scale of 1 to 5, where 1 is low and 5 is high, how likely is it that you will go to college ?” and “On a scale of 1 to 5, where 1 is low and 5 is high, how much do you want to go to college ?”.
Another common source of correlation is introduced by the endogeneity of the network struc-ture. In particular, α can not only capture social learning, but it can also capture the corre-lation between beliefs due to self-selection into the friendship network. For example, students i and j may become friends because they share common beliefs. If this point is conceptually valid, we are not worried about this issue in our context.
Indeed, multiple studies have shown repeatedly that the impact of self-selection is virtually nonexistent in the context of the particular database that we use (e.g. Goldsmith-Pinkham and Imbens,2013;Boucher,2016;Hsieh and Lee,2016;Boucher and Fortin,2016).13 A likely explanation is the extremely weak predictability of friendship relations. As presented in detail in Appendix A.4, the marginal effects of most identified explanatory variables is very weak (on the order of 10−5 for the probability of creating a friendship relation). This is expected as friendship relations are rare events : for any two randomly selected individuals, the probability of friendship is very small.
13. AlthoughHsieh and Lee(2016) find moderate changes in the point estimate after controlling for network endogeneity, the change is not statistically significant (see their Tables II and IV). See also Hsieh and Lin
The only variable that has some meaningful predictive power is the preexistence of a friendship relation. For the entire sample of 16 schools, the existence of a friendship relation in Wave 1 only increases the probability of friendship in Wave 2 by less than 2% (see Table A.1 of Appendix A.4). However, it may have some predictive power when we restrict the sample to specific schools. For example, for School 28, having a friendship relation in Wave 1 increases the probability of a friendship relation in Wave 2 by 42% (see Table A.2 of Appendix A.4). Regardless, this analysis suggests that any unobserved predictor of friendship relations is likely to have a very weak bias on the estimated value of α (see below for additional verifications of robustness).
A final source of correlation is due to omitted individual variables. Here, an important comment should be made : we make no causal claim for the estimated parameters β. Indeed, many variables included in X are likely endogeneous. For example, unobserved effort is likely to be correlated with the students’ beliefs, academic achievements, or college preferences. That being said, including academic achievement and college preference variables is nonetheless crucial to the interpretation of α as a social learning effect.
Many studies have shown the presence of peer effects on academic achievement (e.g.Sacerdote et al., 2011) or preferences for college (e.g. Giorgi et al.,2010). Thus, if we were to exclude achievement and preference variables, the within-network correlation of beliefs, picked up by α, would not only capture social learning, but it would also capture peer effects for academic achievement and college preferences.
Regardless, we also present, as a robustness check in Section 1.6.4, a version of the model estimated in deviations, which account for unobserved individual fixed effects. The fact that we obtain very similar results therefore reduces the likelihood that the measured value for α is due to unobserved common shocks, endogenous friendship selection, or missing individual variables.
A final concern for the interpretation of α comes from a potential mis-specification of the theoretical model. Although this is always a possibility, previous studies have preferred mo-dels similar to (1.3) to the Bayesian alternative (Chandrasekhar et al., 2015). In addition, a particular feature of social learning is that beliefs can be thought as being directly transmitted from a student to their peers (e.g. by talking), as opposed to other outcomes, such as body mass index (BMI). Indeed, since a student’s BMI does not directly affect their peers’ BMI, the social transmission necessarily goes through the students’ (unobserved) effort, which also has to be modeled to interpret α as capturing social interactions (e.g. Boucher and Fortin,2015).
1.6.2 Results and Heterogeneity Across Schools
We first perform the estimation for the entire sample (i.e. for the 16 schools). Results are presented in Table 1.2 and the histograms for all posterior distributions are presented in
FigureA.2of AppendixA.6. We find that, on average, the beliefs of a student’s peers account for about 12% of the updating of their beliefs. Female students have a more positive attitude toward college, as do students having educated parents.
Table 1.2 – Estimation of homogeneous peer effects
Mean Std. Err. Endogenous effect α 0.121∗∗∗ 0.026 Child’s characteristics Female 0.292∗∗∗ 0.055 Grades 9–10 0.189 0.129 Grades 11–12 0.328∗∗ 0.165 Age -0.052 0.038 White -0.090 0.095 Black or African–American -0.028 0.106 Hispanic -0.079 0.089 Mother education
Less than high school -0.100 0.121
High school 0.073 0.089
More than high school 0.210∗∗∗ 0.092 Father education
Less than high school -0.018 0.120
High school -0.040 0.088
More than high school 0.276∗∗∗ 0.085 Household characteristics Household size -0.017 0.016 Urban 0.287 1.350 Constant 0.260 1.854 γ2 0.855∗∗∗ 0.053 γ3 2.047∗∗∗ 0.047 γ4 2.961∗∗∗ 0.044
College preferences yes
Grade in mathematics, English, history, and science yes
School fixed effect yes
Observations 2,255
Note : We let our chain run for 20, 000 iterations, discarding the first 9, 999 iterations ;∗p < 0.1 ;∗∗p < 0.05 ;∗∗∗p < 0.01.
To explore the heterogeneity of social learning, we also ran the model, separately, for the two largest schools in our sample. Figure 1.1 presents a kernel density estimate of the posterior distribution of α for both schools. The full results are presented in TableA.4in AppendixA.5. We see that social learning is more important in School 58. This is interesting as the distri-bution of beliefs is similar for both schools (see Figure A.1 of Appendix A.5). Thus, from a policy point of view, information campaigns are likely to be more effective in School 58 since beliefs are more dependent on social learning and less on factors that are difficult to influence through policies (e.g. parents’ education).
Figure 1.1 – Heterogeneity among schools
1.6.3 Individual Heterogeneity
In this section, we explore in greater detail the role of heterogeneity in social learning. In the previous section, we found that the weight associated to friends’ beliefs can vary among schools. Here, we assume that this weight can vary among students. Specifically, we assume that the learning process of student i is as follows :
p∗(t)i = α (i) ni X j∈Ni p∗(t−1)j + (1 − α(i))xiβ + εi,
where α(i)= 0 if i is isolated, and where α(i) ∈ α ≡ {α1, α2} if i is not isolated. We therefore
assume that non-isolated students can be classified into two (unobserved) groups.
The approach developed in this section could be generalized to more than two groups, at some important computational cost. However, in this section, we do not aim to fully characterize the unobserved heterogeneity. We only seek to gain a measure of its importance. As we will see, focusing on two groups is sufficient for our aim.
Using the same argument as before, we can show that the steady-state beliefs are given by : p∗ = [I − AαT]−1[I − Aα] Xβ + [I − AαT]−1ε, (1.4)
where Aα is a diagonal matrix taking Aii= α(i) for all i. One can easily verify that (1.3) is a
special case of (1.4).14
14. In a classical setting, this model would be called a finite mixture (or latent class) model and estimated using an EM (expectation-maximization) algorithm.
To estimate the parameters (and to divide the students into two groups), we augment the model with a latent component S = {s1, s2, . . . , sN}. The variable si ∈ {1, 2} represents the
group to which the student i belongs and exhibits a prior distribution given by P [si= 1] = q.
Given this new state variable, the model for non-isolated students can be restated as : p∗i|p∗−i, si, β, α ∼ N (αsiTp
∗
+ (1 − αsi)xiβ, 1). (1.5)
Equation (1.5) highlights that sampling the state is straightforward since, for k ∈ {1, 2}, the conditional probability function of the variable is given by P (si = k|p∗, β, α) ∝ f (p∗i|p∗−i, si =
k, β, α)P [si = k]. The MCMC steps used to estimate the model parameters are presented in
Algorithm 2.15
Algorithm 2 MCMC algorithm
Set initial values to β0, α0, p∗0, γ0, S0, q0
for j = 1 to M , where M denotes the number of MCMC iterations do [1] - Same as in Algorithm1 but also conditional on S.
[2] - Same as in Algorithm1. [3] - Same as in Algorithm1. [4] - Sample S ∼ S|p∗, p, γ, β, α : for i = 1 to N do
— Draw u ∼ U [0, 1] and fix si = 1 if u ≤ P (si = 1|p∗, β, α). Set si = 2 otherwise. Note
that P (si = 1|p∗, β, α) ∝ fN(p∗i|α1Tp∗+ (1 − α1)xiβ, 1)q, where fN(x|µ, σ2) stands
for the normal density function with expectation µ and variance σ2 evaluated at x. end for
[5] - Sample q|S ∼ Beta(1 + n1, 1 + n2), where nk=PNi=11 {si = k}.
end for
Due to the computational burden of Algorithm 2, we only present results for School 77 (the largest school in our sample). Our results can be found in Table1.3.
The difference between the two groups is striking. For some students (Group 1), social learning is small (around 8%). However, for other students (Group 2), social learning is very important (close to 75%). The composition of those two groups, as well as the strength of social learning within each group, is determined entirely by the data. Therefore, this method allows us to capture the unobserved heterogeneity in social learning.
The estimation technique also provides, for each student, an estimate of the probability of belonging to one of the two groups.16We can therefore search for predictors of this probability. In Table 1.4, we present the results of a (censored) regression for the probability of belonging to Group 1.
First, note that there are few significant predictors. This is important as it means that the strength of social learning cannot be directly predicted by observable variables in our sample.
15. Alternatively, see Algorithm4in Appendix 6.4
Table 1.3 – Heterogeneous peer effects in Wave 1 (School 77)
Mean Std. Err. Endogenous peer effects
α1 0.079∗∗ 0.038 α2 0.735∗∗∗ 0.185 Child’s characteristics Female 0.340∗∗∗ 0.084 Grades 9–10 -0.331 0.465 Grades 11–12 -0.439 0.491 Age -0.016 0.053
Born in the USA 0.064 0.117
White -0.153 0.106
Black or African–American 0.042 0.122
Hispanic -0.109 0.111
Mother education
Less than high school 0.087 0.179
High school 0.041 0.141
More than high school 0.199 0.139 Father education
Less than high school -0.140 0.170
High school 0.020 0.162
More than high school 0.139 0.130 Household characteristics Household size -0.016 0.022 Constant 0.318 0.762 γ2 0.813∗∗∗ 0.075 γ3 2.136∗∗∗ 0.057 γ4 3.004∗∗∗ 0.060
College preferences yes
Grade in mathematics, English, history and science yes
Observations 920
Note : We let our chain run for 20, 000 iterations, discarding the first 9, 999 iterations ;∗p < 0.1 ;∗∗p < 0.05 ;∗∗∗p < 0.01.
The sorting of the students between the two groups therefore truly captures unobserved hete-rogeneity.
That being said, one of the variables that appears to predict group membership is the students’ beliefs. All else being equal, students having higher beliefs are less likely to belong to Group 1 and are therefore more likely to rely on social learning when forming their beliefs. This is somewhat intuitive. More pessimistic students (those having low beliefs, everything else being equal) are less likely to discuss their beliefs with their friends and therefore to update their beliefs as a result of social interactions.
The importance of unobserved heterogeneity has powerful policy implications. The results of Table 1.4 imply that the heterogeneity of peer effects captured using our structural model (which is computationally and data intensive) cannot be easily predicted using observable characteristics.
This is important as the literature widely promotes the key player principle : policies should target the most central individual(s) in the network (often using Bonacich’s centrality, see
Ballester et al. (2006) orZenou (2016) for details). However, when peer effects are heteroge-neous, those centrality measures must be adjusted using a weighted network analysis. If the
Table 1.4 – Predictors of Group 1 Est. Std. Err. Individual characteristics Intercept 0.126 0.354 Female 0.048∗ 0.023 Grades 11–12 -0.043 0.032 Born in the USA 0.057 0.031
Age 0.036 0.022 White -0.042 0.030 Black or African–American 0.002 0.035 Hispanic -0.002 0.029 College preferences 3 0.010 0.067 College preferences 4–5 -0.064 0.065 Belief 3 −0.118∗ 0.055 Belief 4–5 -0.095 0.054 Network structure Betweenness centrality 2.739 3.030 Mother education
Less than high school 0.021 0.048 High school -0.061 0.040 More than high school -0.063 0.036 Father education
Less than high school -0.057 0.047 High school 0.004 0.044 More than high school -0.001 0.034 Household characteristics
Household size 0.010 0.006
Observations 630
policy maker cannot easily capture the heterogeneity, their policies will be targeted toward the wrong individuals.
To be clear, we are not saying that the key-player strategy cannot be used in the context of unobserved heterogeneity. However, to do so, the policy maker would have to observe the network structure and be able to fully characterize the unobserved heterogeneity (which is outcome dependent). That is, estimate an extension of the model presented in this section (with potentially more than two groups) over the whole population of interest.17
Thus, in the context of strong unobserved heterogeneity, and where large-base information policies are relatively cheap (Hoxby and Turner, 2013), we argue that policies not targeted toward networks should be privileged. Moreover, the fact that we capture strong social learning is also an indication that those policies are likely to have a significant impact as they may act as a substitute to friends’ subjective beliefs.
1.6.4 Robustness : Unobserved Fixed Effect
As described in Section 2.4, we observed students’ beliefs for two consecutive years (waves 1 and 2). In this section, we use this feature of the data to account for the possibility of
17. Note also that since the model is non-linear, the approach inPeng(2016) cannot be directly applied to our setting.
unobserved individual fixed effects. Specifically, we extend (1.3) as follows :
pw∗= [I − αTw]−1Bα[Xβw+ µδw] + [I − αTw]−1εw, (1.6)
where µ is unobserved and where the parameters βwand δware allowed to vary for each wave, w = 1, 2. We also normalize δ1= 1.
Isolating µ for w = 1 and then substituting, we easily find :
p2∗= [I − αT2]−1[BαX(β2− δ2β1) + δ2p1∗− δ2αT1p1∗] + [I − αT2]−1˜ε,
where ˜ε = ε2− δ2ε1. Here, as the latent variable p1∗is not observed, we use the mean of the
posterior distribution from Section 1.6.2 as a proxy.18
The estimated values are presented in Table 1.5. Results for α are remarkably close to the baseline estimates to add further credibility to the interpretation of α as social learning. Also, the small estimated effect for δ2 implies that the contribution of the previous year’s beliefs is small, which increases the probability that the learning process effectively converged in the data.
Table 1.5 – Contribution of unobserved fixed effects for School 77 in Wave 1.
Mean Std. Err. Endogenous effect α 0.113∗∗∗ 0.033 Unobserved fixed effect δ2 0.068∗∗∗ 0.026
Child’s characteristics
Female 0.406 0.087
Grades 9–10 -0.336 0.354
Grades 11–12 -0.127 0.303
Age 0.021 0.040
Born in the USA -0.127 0.114
White -0.030 0.110
Black or African–American 0.229∗ 0.122
Hispanic 0.121 0.108
Mother education
Less than high school -0.087 0.186
High school -0.223 0.145
More than high school 0.127 0.137 Father education
Less than high school 0.103 0.169
High school 0.327∗∗ 0.162
More than high school 0.298∗∗ 0.133
Household characteristics Household size -0.021 0.021 Constant -0.902 0.751 γ2 0.798∗∗∗ 0.109 γ3 2.088∗∗∗ 0.156 γ4 2.883∗∗∗ 0.156
College preferences yes
Grade in mathematics, English, history, and science yes
School fixed effect yes
Observations 920
Note : We let our chain run for 20, 000 iterations, discarding the first 9, 999 iterations ;∗p < 0.1 ;∗∗p < 0.05 ;∗∗∗p < 0.01.
18. An alternative would be to estimate the models jointly. However, the computational burden makes this approach impractical.
1.6.5 Robustness : Endogenous Network Formation
In this section, we check the robustness of our results with respect to a potential endogeneity of the network formation. Indeed, in the presence of student-level unobservables that drive both peer choice and students’ beliefs, parameters from the benchmark model (1.3) will be biased due to the network endogeneity.
Several recent papers have developed a number of structural frameworks to address this issue. The common approach consists in including a network formation model and using bayesian techniques to estimate the parameters of interest (see for instance, Goldsmith-Pinkham and Imbens, 2013; Hsieh and Lee, 2016). Indeed, Goldsmith-Pinkham and Imbens (2013) first proposed to introduce an unobserved characteristic which may affect outcome directly and network formation. They analyze grade-point average using AddHealth data. In their esti-mation, they assume that the unobserved characteristic is binary. They find that accounting for this common unobserved characteristic appears to matter for network estimation but not for peer effects.Hsieh and Lee (2016) have extendedGoldsmith-Pinkham and Imbens(2013)’s analysis by introducing multidimensional and continuous unobserved characteristics. They also analyze grade-point average using AddHealth data and consider one, two and three unobser-ved characteristics. They find that the endogeneous peer effect is unaffected when including one unobservable, drops when including two unobservables and unaffected by a third one. Their results therefore suggest to include mutliple common unobservables when investigating network endogeneity in peer effect regressions.
We follow the strategy of Hsieh and Lee (2016) and consider two continuous unobserved factors, z1 and z2, that drive both students’ beliefs (i.e., the benchmark model 1.3) and students’ friendship. Specifically, we estimate simultaneously the following models :
p∗ = αTp∗+ (1 − α)Xβ + δ1z1+ δ2z2+ η (1.7) and Tij = ( 1 if Tij∗ ≥ 0 0 otherwise, where : Tij∗ = λ0+ M X m=1 λm|Xim− Xjm| + φ1|z1,i− z1,j| + φ2(z2,i+ z2,j) + ζi, (1.8)
where η and ζ denote error terms. Note that if one of the parameters φ1 or φ2is different from
zero, then the network T is endogeneous and estimates of model (1.3) are biased. As inHsieh and Lee (2016), identification in the extended model (1.7)-(1.8) requires some restrictions on the covariance parameters of z in the network formation model. To see which restriction we can impose, let us assume that the unobserved factors are (z1− z1,i1N) and (z2+ z2,i1N) for
student i. We have the following computation : f = φ1(z1− z1,i1N) + φ2(z2+ z2,i1N) = φ1[IN −1Ne(i)0]z1+ φ2[IN +1Ne(i)0]z2, where e(i) = (0, 0, . . . , 0 | {z } i−1 , 1, 0, . . . , 0)0, E(f ) = 0,
V(f ) = φ21[IN −1Ne(i)0][IN− e(i)1N0 ] + φ22[IN +1Ne(i)0][IN + e(i)10N],
= (φ21+ φ22) | {z } κ1 [IN +1N10N] + (φ22− φ21) | {z } κ2 [1Ne(i)0+ e(i)10N], = κ1[IN +1N10N] + κ2[1Ne(i)0+ e(i)10N].
Thus, κ1and κ2are identified through the correlation structure of Tij∗. To impose a one-to-one
mapping between κk and φk, k ∈ {1, 2}, one can simply view the equation φ21+ φ22 = κ1 as a
circle of radius κ1. In other words, a sufficient condition to fully point identified parameters φk
is : φ1 < 0 and φ2> 0. We estimate the extended model (1.7)-(1.8) using the Gibbs sampling
algorithm. The unobservable variables z are thus generated according to the joint likelihood of link formation and outcome. We provide in AppendixA.7the posterior distributions of the parameters of interest. The MCMC steps used to estimate the extended model parameters are presented in the Algorithm3 below :19
Table 1.6 collects the results that are obtained when estimating equation (1.7)-(1.8) simulta-neously for school 77. It reveals that φ2 is significantly different from zero, suggesting that
the network matrix T is endogeneous. However, the endogeneous peer effect α is closed to what has been estimated with our benchmark model in section1.6.2 for school 77 : ˆα = 0.129 and ˆσα = 0.040 without accounting for network endogeneity (see Figure 1.1) ; ˆα = 0.136
and ˆσα = 0.038 when accounting for the network endogeneity (see Table 1.6). This finding
is consistent with Goldsmith-Pinkham and Imbens (2013) and Hsieh and Lee (2016) who also find that network endogeneity does not affect the endogeneous peer effect when using AddHealth data.
19. We also provide in the AppendixA.9the full characterisation of the MCMC steps to estimate heteroge-neous peer effect with a potential endogeneity of the network.
Algorithm 3 MCMC algorithm
Set initial values to β0, α0, p∗0, T0∗, γ0, φ0, λ0, δ0, z0
for j = 1 to M , where M denotes the number of MCMC iterations do
[1] - Sample β, α, δ ∼ β, α, δ|p∗, p, γ, z : Same as in Algorithm 1. See AppendixA.7 for more details.
[2] - Sample γ ∼ γ|p∗, p, β, α : Same as in Algorithm 1.
[3] - Sample p∗|p, γ, β, α : Same as in Algorithm1. The full details are provided in the AppendixA.7.
[4] - Sample T∗|z, φ, δ, λ :
— For all i, j, draw Tij∗ from N (Xiλ − (zi1− z j
1)φ1− (zi2+ z j
2)φ2, 1)1(−1)TijT∗ ij<0.
[5] Sample λ, φ from N (¯µ, ¯Σ), where Σ¯ = PN i=1X0iXi+ Σ−10 −1 and µ¯ = ¯ Σ PN i=1XiTi∗+ Σ −1 0 µ0
. To ensure the constraints φ1 < 0 and φ2 > 0 :
— Draw from a truncated univariate normal distribution φ2 ∼ N (¯µφ2, ¯Σφ2,φ2).
— Compute the conditional variance and expectation : ˜µφ1 = ¯µφ1 +
¯ Σφ2,φ1 ¯ Σφ2,φ2.
[6] Sample z = (z1, z2)|α, β, δ, φ, T∗ :
— Draw z from N (¯µz, ¯Σz), where ¯Σz = (I2N + ∆0∆ + PNi=1Φ0iΦi)−1 and ¯µz =
¯
Σz[∆0Q−1(p∗− QBXβ) +PNi=1Φ0i(Ti∗− Xiλ)].
— Details concerning ∆0∆ and PN
i=1Φ0iΦi are presented in the Appendix A.8.
end for
Table 1.6 – Robustness check : endogeneous network formation
Estimates Standard Error
α 0.136*** 0.038 δ1 0.085 0.130 δ2 0.200*** 0.052 φ1 -0.058 0.011 φ2 0.232*** 0.013 Observation 920
Note : This table presents the results for school 77. The regression includes the same controls as in the Table1.2. We let our MCMC run for 80,000 iterations, discarding the first 69,999 iterations. *** p<0.01, ** p<0.05, * p<0.1.
1.7
Conclusion
In this paper, we study peer effects on students’ beliefs regarding college participation. To do so, we specify a structural model in which the students’ beliefs are updated with respect to their friends’ beliefs and other socio-economic characteristics. The steady state represents a multivariate ordered probit that is estimated using a Bayesian approach. Given that few papers take the discreetness of the outcome into account, we discuss identification issues and show that our model parameters can be identified. In addition, we also discuss the interpretations of parameters, and we estimate multiple models to ensure the robustness of our results. When the 16 schools are taken together, we find a peer effect on beliefs that amounts to 12% on average. However, we highlight that the average effect may be meaningless since we find much network heterogeneity. If we estimate, separately, the same model for the two largest schools, we observe peer effects that reach 13% and 25%, respectively. While this result emphasizes some heterogeneity among the different schools, we also explore the unobserved heterogeneity within schools. Importantly, we find two groups of students. On one hand, we have students that have strong beliefs and that are hardly influenced by others. On the other hand, we also uncover a group that exhibits a very strong social learning effect (around 73%). This finding, coupled with individual socio-economic characteristics being unable to explain group affiliation, leads us to believe that policies targeting key player(s) are likely ineffective. In particular, before finding the key player(s), one must first understand the heterogeneity of the network and then target those who are sensitive to peers’ beliefs so as to adapt the key player’s argument.
As a final note, we see this work as a first step toward a better understanding of network heterogeneity. One important research avenue lies in capturing the variable that can predict the level of students’ social learning.
