Peer Effects, Fast Food Consumption and Adolescent Weight Gain ∗

Peer Effects, Fast Food Consumption and Adolescent Weight Gain

Bernard Fortin^† Myra Yazbeck^‡ April 2014

Abstract

This paper aims at opening the black box of peer effects in adolescent weight gain. Using Add Health data on secondary schools in the U.S., we investigate whether these effects partly flow through the

eating habitschannel. Adolescents are assumed to interact through a friendship social network. We

first propose a social interaction model of fast food consumption. Our approach allows to control for correlated effects at the network level and to solve the simultaneity (reflection) problem. The model is estimated using maximum likelihood and generalized 2SLS strategies. We also estimate a panel dynamic weight gain production function relating an individual’s Body Mass Index z-score (zBMI) to his fast food consumption and his lagged zBMI. Results show that there are positive but small peer effects in fast food consumption among adolescents belonging to a same friendship school network.

Based on our preferred specification, the estimated social multiplier is 1.15. Our results also suggest that, after one year, an extra day of weekly fast food restaurant visits increases zBMI by 4.71% when ignoring peer effects and by 5.41%, when they are taken into account.

Keywords: Obesity, overweight, peer effects, social interactions, fast food, spatial models.

JEL Codes: C31 I10, I12

∗We wish to thank Christopher Auld, Charles Bellemare, Luc Bissonnette, Paul Frijters, Guy Lacroix, Paul Makdissi, Kevin Moran, Bruce Shearer and two anonymous referees for useful comments and Yann Bramoull´e, Badi Baltagi, Vincent Boucher, Rokhaya Dieye, Habiba Djeb- bari, Tue Gorgens, Bob Gregory, Linda Khalaf, Lung-Fei Lee, Xin Meng and Rabee Tourky for useful discussions. All remaining errors are ours.

Financial support from the Canada Research Chair in the Economics of Social Policies and Human Resources and le Centre interuniversitaire sur le risque, les politiques ´economiques et l’emploi is gratefully acknowledged.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.

†CIRP ´EE, IZA, CIRANO and Department of Economics, Universit´e Laval. E-mail: Bernard.Fortin@ecn.ulaval.ca

‡School of Economics, University of Queensland. Email: m.yazbeck@uq.edu.au

1 Introduction

For the past few years, obesity has been one of the major concerns of health policy makers in the U.S.

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It has also been one of the principal sources of increased health care costs. In fact, the increasing trend

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in children’s and adolescents’ obesity (Ogden et al., 2012) has raised the annual obesity-related medical

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costs to $147 billion per year. Obesity is also associated with increased risk of reduced life expectancy as

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well as with serious health problems such as type 2 diabetes (Maggio and Pi-Sunyer, 2003), heart disease

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(Calabr et al., 2009) and certain cancers (Calle, 2007), making obesity a real public health challenge.

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Recently, a growing body of the health economics literature has tried to look into the obesity problem

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from a new perspective using a social interaction framework. An important part of the evidence suggests

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the presence of peer effects in weight gain. Thus Christakis and Fowler (2007), Trogdon et al. (2008),

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Renna et al. (2008) and Yakusheva et al. (2014) are pointing to thesocial multiplieras an important element

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in the obesity epidemics. On the other hand, Cohen-Cole and Fletcher (2008b) found that there is no

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evidence of peer effects in weight gain. Also, results from a placebo test performed by the same authors

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(Cohen-Cole and Fletcher, 2008a) indicate that there are peer effects in acne (!) in the Add Health data

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when one applies the Christakis and Fowler (2007) method discussed later on. As long as it is strictly

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larger than one, a social multiplier amplifies, at the aggregate level, the impact of any shock that affects

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obesity at the individual level. This is so because the aggregate effect incorporates, in addition to the

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sum of the individual direct effects, positive indirect peer effects stemming from social interactions.

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While the presence of the social multiplier in weight gain has been widely researched, the literature

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on the mechanisms by which this multiplier flows is still scarce. Indeed, most of the relevant literature

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attempts to estimate the relationship between variables such as an individual’s Body Mass Index (BMI)

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and his average peers’ BMI, without exploring the channels at source of this potential linkage.¹ The aim

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1One recent exception is Yakusheva et al. (2011) and Yakusheva et al. (2014) who look at peer effects in weight gain and in weight management behaviours such as eating and physical exercise, using randomly assigned pairs of roommates in freshman year.

of this paper is to go beyond the black box approach of peer effects in weight gain and try to identify

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one crucial mechanism through which peer effects in adolescence overweight may flow:eating habits(as

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proxied by fast food consumption).

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Three reasons justify our interest in eating habits in analyzing the impact of peer effects on teenage

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weight gain. First of all, there is important literature that points to eating habits as an important com-

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ponent in weight gain (e.g., Niemeier et al., 2006; Rosenheck, 2008).² See also Cutler et al. (2003) which

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relates the declining relative price of fast food and the increase in fast food restaurant availability over

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time to increasing obesity in the U.S. Second, one suspects that peer effects in eating habits are likely

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to be important in adolescence. Indeed, at this age, youngsters have increased independence in general

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and more freedom as far as their food choices are concerned. Usually vulnerable, they often compare

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themselves to their friends and may alter their choices to conform to the behaviour of their peers. There-

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fore, unless we scientifically prove that obesity is a virus, it is counter intuitive to think that one can

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gain weight by simply interacting with an obese person.³ This is why we are inclined to think that the

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presence of real peer effects in weight gain can be estimated using behavioural channels such as eat-

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ing habits. Third, our interest in peer effects in youths’ eating habits is policy driven. There has been

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much discussion on implementing tax policies to address the problem of obesity (e.g.,Caraher and Cow-

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burn, 2007; Powell et al., 2013). As long as peer effects in fast food consumption is a source of externality

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that may stimulate overweight among adolescents, it may be justified to introduce a consumption tax

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on fast food. The optimal level of this tax will depend, among other things, on the social multiplier of

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eating habits, and on the causal effect of fast food consumption on adolescent weight.

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In order to analyze the impact of peer effects in eating habits on weight gain, we propose a two-

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equation model. The firstlinear-in-meansequation relates an individual’s fast food consumption to his

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reference group’s mean fast food consumption (endogenous peer effect), his individual characteristics, and

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2An indirect evidence of the relationship between eating habits and weight gain come from the literature on the (negative) effect of fast food prices on adolescents’ BMI (see Auld and Powell, 2009; Powell, 2009; Powell and Bao, 2009).

3Of course, having obese peers may influence an individual’s tolerance for being obese and therefore his weight manage- ment behaviours.

his reference group’s mean characteristics (contextual peer effects). This equation also provides an estimate

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of the social multiplier in fast-food consumption. The second equation is a panel dynamic production

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function that relates an individual’s BMI adjusted for age (z-score BMI, or zBMI) to his fast food con-

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sumption, his lagged zBMI and and other control variables. The system of equations thus allows us to

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evaluate the impact of an eating habits’ exogenous shock on weight gain, when peer effects on fast food

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consumption are taken into account.

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Estimating our system of equations raises serious econometric problems. It is well known that the

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identification of peer effects (first equation) is a challenging task. These identification issues were pointed

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out by Manski (1993) and discussed among others by Bramoull´e, Djebbari and Fortin (2009) and Blume

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et al. (2010). On one hand, (endogenous + contextual) peer effects must be identified fromcorrelated(or

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confounding) factors. For instance, students in a same friendship group may have similar eating habits

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because they share similar characteristics (i.e.., homophily) or face a common environment (e.g., same

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school). On the other hand, simultaneity between an adolescent’s and his peers’ behaviour (referred to

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asthe reflection problemby Manski) may make it difficult to identify separately the endogenous peer effect

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and the contextual effects. This later task is important since the endogenous peer effect is the only source

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of a social multiplier.

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We use a new approach based on Bramoull´e et al. (2009) and Lee et al. (2010), and extended by Blume

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et al. (2013) to address these identification problems and to estimate the peer effects equation. First, we

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assume that in their fast food consumption decisions, adolescents interact through afriendship network.

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Each school is assumed to form a network. School fixed effects are introduced to capture correlated

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factors associated with network invariant unobserved variables (e.g., similar preferences due to self-

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selection in schools, same school nutrition policies, distance from fastfood restaurants). The structure of

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friendship links within a network is allowed to be stochastic and endogenous but is strictly exogenous,

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conditional on the school fixed effects and observable individual and contextual variables. The possi-

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bility that friends select each other using unobservable traits that may be correlated with their fast food

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consumption decisions is an important issue and is discussed later on.

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To solve the reflection problem, we exploit results by Bramoull´e et al. (2009) who show that if there

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are at least two agents who are separated by a link of distance 3 within a network (i.e., there are two

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adolescents in a school who are not friends but are linked by two friends), both endogenous and con-

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textual peer effects are identified. Finally, we exploit the similarity between the linear-in-means model

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and the spatial autoregressive (SAR) model with or without autoregressive spatial errors.⁴ The model is

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estimated using a maximum likelihood (ML) approach as in Lee et al. (2010) and Lin (2010). We also es-

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timate the model with a distribution free approach: generalized spatial two-stage least square (GS-2SLS)

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proposed in Kelejian and Prucha (1998) and refined in Lee (2003).

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The estimation of the production function (second equation) also raises serious econometric issues.

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First, fast food consumption is likely to be an endogenous variable correlated with the individual error

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term. Moreover, time-invariant unobserved effects potentially correlated with explanatory variables

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may influence the individual zBMI. Also, the short and the long term impacts of fast food consumption

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on zBMI may be different, suggesting the introduction of lagged zBMI as an explanatory variable. In

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order to deal with these problems, we estimate a dynamic AR(1) model using an IV fixed effect approach.

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Predetermined variables from previous periods are used as excluded instruments.

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To estimate our model, we use four waves of the National Longitudinal Study of Adolescent Health

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(Add Health).⁵ We define peers as the nominated group of individuals reported as friends within the

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same school. The consumption behaviour is depicted through the reported frequency (in days) of fast

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food restaurant visits in the past week. Results suggest that there is a positive but low endogenous peer

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effect in fast food consumption among adolescents in general. Based on our preferred specification, the

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estimated social multiplier is 1.15. Moreover, the production function estimates indicate that there is a

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positive significant impact of fast food consumption on zBMI. Combining these results, we find that,

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4Our approach is more general than the SAR model as the latter usually ignores contextual effects and spatial fixed effects.

5Note that for the first equation we use wave II and for the second equation we use all four waves.

after one year, an extra day of weekly fast food restaurant visits increases zBMI by 4.71% when ignoring

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peer effects and by 5.41%, when they are taken into account.

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The remaining parts of this paper will be laid out as follows. Section 2 provides a critical survey of

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the literature on peer effects in obesity as well as its decomposition into the impact of peer effects on

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fast food consumption and the impact of fast food consumption on obesity. Section 3 presents our two-

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equation model and our estimation methods. In section 4, we give a brief overview of the Add Health

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Survey and we provide descriptive statistics of the data we use. In section 5, we discuss estimation

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results. Section 6 concludes.

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2 Previous literature

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In recent years, a number of studies found strong ”social networks effects” in weight outcomes. In

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a widely debated article, Christakis and Fowler (2007), using a 32-year panel dataset on adults from

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Framingham, Massachusetts and based on a logit specification, found that an individual’s probability of

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becoming obese increased by 57% if he or she had a friend who became obese in a given interval. How-

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ever, their analysis has been criticized for suffering from a number of limitations (see Cohen-Cole and

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Fletcher, 2008b; Lyons, 2011; Shalizi and Thomas, 2011).⁶ In particular, it ignores potential spurious cor-

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relations between two friends’ BMI resulting from the fact that they are exposed to a same environment.

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Both Shalizi and Thomas (2011) and Lyons (2011) show that the relying on link asymmetries does not rule

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out shared environment as it claims. Also, the simultaneity problem between these two outcomes is not

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directly addressed by allowing the peer’s obesity to be endogenous. Moreover, by introducing lagged

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obesity variables, it only partly takes into account the problem of selection that may occur as obese in-

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dividuals may have a higher probability to become friends (for details see pages 217-218 in Shalizi and

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Thomas, 2011). Finally, by focusing on dyads over time, it introduces an upward bias resulting from the

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6For a response to these criticisms and others, see Fowler and Christakis (2008), VanderWeele (2011) and Christakis and Fowler (2013).

unfriendingproblem as defined by Noel and Nyhan (2011). The basic idea behind this argument is that

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people who share similar characteristics are more likely to maintain social ties (homophily).⁷

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Using Add health data, Trogdon et al. (2008) include school fixed effects to account for the fact that

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students in a same school share a same surrounding. They also estimate their BMI peer model with

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an instrumental variable approach. They use information on friends’ parents’ obesity and health and

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friends’ birth weight as instruments for peers’ BMI. They find that an one point increase in peers’ average

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BMI increases own BMI by 0.52 point. Using a similar approach and based on Add Health dataset, Renna

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et al. (2008) also find positive peer effects. These effects are significant for females only (= 0.25 point).

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These analyses raise a number of concerns though. In particular, they assume no contextual variables

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reflecting peers’ mean characteristics. This rules out the reflection problem by introducing non-tested

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restriction exclusions. As a result, the peer effect estimates may be inconsistent. Moreover, it is not clear

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that their instruments is truly exogenous as peers’ parents obesity status or health may be correlated with

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unobserved variables influencing own BMI. Also, their instruments aread hocas they are not explicitly

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derived from the reduced form of the model. In our approach, we introduce school fixed effects as well

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as, for each individual variable, the corresponding contextual variable at the reference group level. We

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can thus identify both endogenous and contextual peer effects.

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Using the same dataset, Cohen-Cole and Fletcher (2008b) exploit panel information (wave II in 1996

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and wave III in 2001) for adolescents for whom at least one of same-sex friend is also observed over

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time. Compared with Christakis and Fowler’s approach, their analysis introduces time invariant and

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time dependent environmental variables (at the school level). Friendship selection is controlled for by

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individual fixed effects. The authors find that peer effects are no longer significant with this specifica-

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tion. As in Trogdon et al. (2008), their analysis ignores contextual variables, contrary to our approach.

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Moreover, the friendship network they used in estimations is incomplete, which may underestimate the

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7Steeg and Galstyan (2011) show that there is a test for ruling out homophily related to the use of longitudinal social networks. The intuition behind the test being that if an individual reproduces the same sequence of events as his friend, it is unlikely that homophily is a source of this replication.

endogenous peer effect (see Stinebrickner and Stinebrickner, 2006).

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All the studies discussed up to this point focus on peer effects in weight outcomes without analyzing

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quantitatively the mechanisms by which they may occur. The general issue addressed in this paper is

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whether the peer effects in weight gain among adolescents partly flow through theeating habitschannel.

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This raises in turn two basic issues: a) are there peer effects in fast food consumption ?, and b) is there a

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link between weight gain (or obesity) and fast food consumption ? In this paper, we address both issues.

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The literature on peer effects in eating habits (first issue) is recent and quite limited. In a recent natural

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experiment, Yakusheva et al. (2011) estimate peer effects in explaining weight gain among freshman

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girls using a similar set up but in school dormitories. Also, they test whether some of the student’s

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weight management behaviours (i.e., eating habits, physical exercise, use of weight loss supplements)

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can be predicted by her randomly assigned roommate’s behaviours. Their results provide evidence of

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the presence ofnegativepeer effects in weight gain. Their results also suggestpositivepeer effects in eating

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habits, exercise and use of weight loss supplements. In a subsequent paper, Yakusheva, Kapinos and

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Eisenberg (2014) investigate the presence of peer effects in weight gain exploiting random assignment

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of roommates during first year of college. The authors find evidence that suggests that peer effects

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in weight gain are predominantly significant among females. They also try to identify the possible

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mechanisms through which these effects occur. The authors conclude that eating related disorders and

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exercise can be excluded as potential mechanisms. However, they could not test whether these effects

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are mediated through eating habits.

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Two caveats in Yakusheva et al. (2011) is its focus on girls and its limited sample (e.g., recruited par-

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ticipants, freshman level students). Moreover the estimates of both studies are likely to underestimate

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social interactions effects as co-eaters or roommates may not reflect thetruesocial network influencing

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students’ weight management behaviours (Stinebrickner and Stinebrickner, 2006). Finally, these studies

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do not estimate the causal links between behaviours and weight gain. Our paper finds its basis in this

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literature as well as the literature on peer effects and obesity discussed above. However, while works by

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Yakusheva et al. (2011) and Yakusheva et al. (2014) rely upon experimental data, we use observational

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non-experimental data. Thus, peers are not limited to assigned dyads. Rather, they are considered to

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have social interactions within a school network. This allows for the construction of a social interac-

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tion matrix that reflects how social interaction between adolescents in schools occurs in a more realistic

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setting (as in Trogdon et al., 2008; Renna et al., 2008). An additional originality of our paper lies in the

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fact that it relies upon a linear-in-means approach when relating an adolescent’s behaviour to that of

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his peers. Also, the analogy between the forms of the linear-in-means model and the spatial autoregres-

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sive (SAR) model allows us to exploit the particularities of this latter model, in particular the natural

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instruments that are derived from its structural form.

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Regarding the second issue, i.e., the relationship between weight gain (or obesity) and fast food

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consumption, it is an empirical question that is still on the debate table.⁸ There is no clear evidence in

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support of a causal link between fast food consumption and obesity. Nevertheless, most of the literature

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in epidemiology find evidence of a positive correlation between fast food consumption and obesity (see,

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Anderson et al., 2011; Rosenheck, 2008).

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The economic literature reveals to be conservative with respect to this question. It focuses the impact

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of “exposure” to fast food on obesity. Dunn et al. (2012), using an instrumental variable approach,

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investigates the relationship between fast food availability and obesity. He finds that an increase in

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the number of fast food restaurants has a positive effect on the BMI among non-whites. Alviola IV

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et al. (2014) using a similar approach provides evidence that the number of fast food restaurant has a

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significant impact on school obesity rates. Similarly, Currie et al. (2010) find evidence that proximity to

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fast food restaurants has a significant effect on obesity for 9th graders. On the other hand, Chen et al.

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(2009) found a small but statistically significant effect in favour of a relationship between BMI values

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and the density of fast food restaurants. Finally, Anderson and Matsa (2011), exploiting the placement

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8The literature on the impact of physical activity on obesity is also inconclusive. For instance, Berentzen et al. (2008) provide evidence that decreased physical activity in adults does not lead to obesity.

of Interstate Highways in rural areas to obtain exogenous variation in the effective price of restaurants,

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did not find any causal link between restaurant consumption and obesity.

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The factors underlying fast food consumption were also investigated. Chou et al. (2005) find a strong

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positive correlation between exposure to fast food restaurant advertising and the probability that chil-

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dren and adolescents are overweight. This effect seems to be stronger and more significant for girls

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(Chou et al., 2005). In fact, this influence can be clearly seen as children are more likely to pick up items

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that are in “Mac Donald’s” packaging (Robinson et al., 2007). More generally, Cutler et al. (2003) and

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Bleich et al. (2008) argue that the increased calorie intake (i.e., eating habits) plays a major role in ex-

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plaining current obesity rates. Importantly, weight gain prior to adulthood set the stage for weight gain

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in adulthood.

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While most of the economics literature analyses the relationship between adolescents’ fast food con-

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sumption and their weight gain using an indirect approach (i.e, effect to fast food exposure), we adopt a

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direct approach in this paper. More precisely, we estimate a dynamic model of weight gain as a function

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of fast food consumption and lagged weight gain.

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3 The econometric model

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In this section, we first propose a linear-in-means peer effects model of the adolescent’s fast food con-

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sumption (first equation) and discuss the econometric methods we use to estimate it. We then present

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our dynamic weight gain production function which relates the adolescent’s zBMI level to his fast food

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consumption (second equation).

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3.1 A model of peer effects in fast food consumption

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Assume a set of N adolescentsithat are partitioned in a set ofLnetworks. A network is defined as a

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structure (e.g., school) in which adolescents are potentially tied by a friendship link. Each adolescenti

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in his network has a set of nominated friendsN_iof sizen_ithat constitute his reference group (or peers).

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We assume thatiis excluded from his reference group. Since peers are defined as nominated friends,

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the number of peers will not be the same for every network member. LetGl(l = 1, . . . , L) be the social

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interaction matrix for a networkl. Its elementg_lij takes a value of _n¹

i wheniis friend withj, and zero

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otherwise. Therefore, theG_lmatrix is row normalized (assuming no isolated adolescent, that is, with no

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group of friends). We definey_lias the fast food consumed by adolescentiin networkl,x_lirepresents the

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adolescenti’s observable characteristics, y_lthe vector of fast food consumption in network l, andx_lis

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the corresponding vector for individual characteristics. To simplify our presentation, we look at only one

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characteristic (e.g., adolescent’s mother education).⁹The correlated effects are captured through network

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fixed effects (theα_l’s). They take into account unobserved factors such as preferences of school, school

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nutrition policies, or presence of fast food restaurants around the school. Theεli’s are the idiosyncratic

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error terms. They capture i’s unobservable characteristics that are not invariant within the network.

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Formally, one can write the linear-in-means model for adolescentias follows:

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y_li =α_l+β P

j∈N_iylj

n_i +γx_li+δ P

j∈N_ixlj

n_i +ε_li, (1)

where

P

j∈Niylj

ni and

P

j∈Nixlj

ni are respectively his peers’ mean fast food consumed and characteristics. In

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the context of our paper,β is theendogenous peer effect. It reflects how the adolescent’s consumption of

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fast food is affected by his peers’ mean fast food consumption. It is standard to assume that|β|<1. The

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contextual peer effectis represented by the parameter δ. It captures the impact of his peers’ mean char-

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acteristic on his fast food consumption. It is important to note that theGl matrix and thexl’ vector are

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allow to be stochastic but are assumed strictly exogenous conditional onα_l, that is,E(ε_li|x_l,G_l, α_l) = 0.

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This assumption is flexible enough to allow for correlation between the network’s unobserved common

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9Later on, in section 3.1.1, we will generalize the model to account for many characteristics.

characteristics (e.g., school’s cafeteria quality) and observed characteristics (e.g., mother’s education).

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Nevertheless, once we condition on these common characteristics, mother’s education is assumed to be

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independent of the idiosyncratic error terms. Let I_l be the identity matrix for a network l andι_l the

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corresponding vector of ones, the model (1) for networklcan be rewritten in matrix notation as follows:

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y_l = α_lι_l+βG_ly_l+γx_l+δG_lx_l+ε_l, for l= 1, ..., L. (2) Note that model (2) is similar to a SAR model (e.g., Cliff and Ord, 1981) generalized to allow for contex-

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tual and fixed effects (hereinafter referred to as the GSAR model). Since|β|<1,(I_l−βG_l)is invertible.

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Therefore, in matrix notation, the reduced form of the model can be written as:

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y_l = α_l/(1−β)ι_l+ (I_l−βG_l)⁻¹(γI_l+δG_l)x_l+ (I_l−βG_l)⁻¹ ε_l, (3) where we use the result that(I−βG_l)⁻¹ = P∞

k=0β^kG^k_l, so that the vector of intercepts isα_l/(1−β)ι_l,

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asuming no isolated adolescents.¹⁰

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Equation (3) allows us to evaluate the impact of a marginal shock in α_l (i.e., a common exogenous

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change in fast food consumption within the network) on an adolescenti’s fast food consumption, when

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the endogenous peer effect is taken into account. One has∂(E(y_li|·)/∂α_l = 1/(1−β). This expression

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is defined as the social multiplier in our model. When β > 0 (strategic complementarities in fast food

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consumption), the social multiplier is larger than 1. In this case, the impact of the shock is amplified by

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social interactions due for instance to a conformity or an imitation effect.

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We then perform a panel-likewithintransformation to the model. More precisely, we average equa-

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tion (3) over all students in networkland subtract it fromi’s equation. This transformation allows us to

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address problems that arise from the fact adolescents are sharing the same environment or preferences.

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LetKl = Il−Hl be the matrix that obtains the deviation from networkl mean withHl=_n¹

l(ι_lι⁰_l).The

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network within transformation will eliminate the correlated effectα_l. Pre-multiplying (3) byK_l yields

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10When an adolescent is isolated, his intercept isαl.

the reduced form of the model for networkl, in deviation:

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K_ly_l=K_l(I_l−βG_l)⁻¹(γI_l+δG_l)x_l+K_l(I_l−βG_l)⁻¹ε_l. (4)

3.1.1 Identification

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Our model raises two basic identification problems.

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- Simultaneity

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Simultaneity between individual and peer behaviour (thereflection problem) may prevent separating con-

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textual effects from endogenous effects. This problem has been analyzed by Bramoull´e et al. (2009) when

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individuals interact through social networks. Let us defineGthe block-diagonal matrix with theG_l’s

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on its diagonal. Assume first the absence of fixed network effects (i.e., αl = α for alll). In this case,

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Bramoull´e et al. (2009) show that the structural parameters of the model (2) are identified if the matrices

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I,G,G²are linearly independent. This condition is satisfied when there are at least two adolescents who

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are separated by a link of distance 2 within a network. This means that they are not friends but have

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a common friend (intransitive triad).¹¹The intuition is that this provides exclusion restrictions in the

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model. More precisely, the friends’ friends mean characteristics can served as instruments for the mean

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friends’ fast food consumption. Of course, when fixed network effects are allowed, the identification

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conditions are more restrictive. Bramoull´e et al. (2009) show that, in this case, the structural parameters

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are identified if the matricesI,G,G² andG³are linearly independent. This condition is satisfied when

263

at least two adolescents are separated by a link of distance 3 within a network, i.e., we can find two

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adolescents who are not friends but are linked by two friends. In this case,g_lij³ >0whileg_ij² = g_ij = 0.

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Hence, no linear relation of the formG³ = λ₀I+λ₁G+λ₂G² can exist. This condition holds in most

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friendship networks and, in particular, in the data we use.¹²

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11More generally, the model is identified when individuals do not interact in groups and are not included in their own reference group.

12Identification fails, however, for a number of non trivial networks. This is notably the case forcomplete bipartite networks.

In these graphs, the population of students is divided in two groups such that all students in one group are friends with all students in the other group, and there is no friendship links within groups. These include star networks, where one student, at the centre, is friend with all other students, who are all friends only with him.

- Correlated effects

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The presence of confounding unobservable variables affecting fast food consumption and correlated

269

with the explanatory variables raises difficult identification problems. First, since adolescents are not

270

randomly assigned into schools, endogenous self-selection through networks may be the source of po-

271

tentially serious biases in estimating (endogenous + contextual) peer effects. Indeed, if the variables

272

that drive this process of selection are not fully observable, correlations between unobserved network-

273

specific factors and the regressors are potentially important sources of bias. In our approach, we assume

274

that network fixed effects capture these factors. This is consistent with two-step models of link forma-

275

tion. Each adolescent joins a school in a first step, and forms friendship links with others in his school

276

in a second step. In the first step, adolescents self-select into different schools with selection bias due

277

to specific school characteristics. In a second step, link formation takes place within schools randomly

278

or based on observable individual characteristics only. Recall also that network fixed effects take into

279

account common unobservable variables at the school level that may influence fast food consumption

280

(e.g., availability of fast food restaurants).

281

Of course, one limitation of using network fixed effects is that it ignores the possibility that the

282

links formation within a network depend on omitted variables. The matrix G may be endogenous

283

even when controlling for the network fixed effects and observable characteristics. Thus friends may

284

select each other using unobservable traits that may be correlated with fast food consumption (e.g., im-

285

pulsivity, a specific taste for sugar- and fat-rich food,etc). Recently, some researchers (e.g., Hsieh and

286

Lee, 2011; Goldsmith-Pinkham and Imbens, 2013; Liu et al., 2013; Badev, 2013) have made attempts to

287

develop econometric models allowing for the joint estimation of network formation and network interac-

288

tions.¹³ However, empirical results using Add Health data and focusing on outcomes such as smoking,

289

sleeping behaviour, and scholar performance, do not seem to detect much difference in peer effects when

290

13Of course, strong assumptions are required to identify peer effects in these static models. For instance, the probability of a link between two students is assumed to be independent from the presence of links between other students. Note that strong assumptions are also needed even for dynamic networks (see Shalizi and Thomas, 2011).

networks are assumed exogenous and when they are allowed to be endogenous.¹⁴ One reason may be

291

that this data base includes a very large number of observable characteristics.

292

One specification of our model also allows the error terms to be (first-order) autocorrelated within

293

networks. Therefore its structure becomes analogous to that of a generalized spatial autoregressive

294

model with network autoregressive disturbances (hereinafter referred to as the GSARAR model). This

295

model implies that in addition to the endogenous and contextual effects, some unobserved characteris-

296

tics of the friends are also interdependent. In this case, the error terms in (2) can be written as:

297

ε_l = ρG_lε_l+ξ_l, (5)

where the innovations,ξ_l, are assumed to bei.i.d.(0, σ²I_l)and|ρ|<1. Given these assumptions, we can

298

write:

299

ε_l = (I_l−ρG_l)⁻¹ξ_l. (6) Allowing for many characteristics and performing a Cochrane-Orcutt-like transformation on the struc-

300

tural model in deviation, the latter is given by the following structural form:

301

K_lM_ly_l=βK_lM_lG_ly_l+K_lM_lX_lγ+K_lM_lG_lX_lδ+ν_l, (7) whereX_lis the matrix of adolescents’ characteristics¹⁵in thelth network,M_l= (I−ρG_l)andν_l=K_lξ_l.

302

The elimination of fixed network effects using a withintransformation leads to a singular variance

303

matrix such that E(ν_lν_l⁰ | X_l,G_l) = K_lK⁰_lσ² = K_lσ². To resolve this problem of linear dependency

304

between observations, we follow a suggestion by Lee et al. (2010) and applied by Lin (2010). Let[Q_l C_l]

305

be the orthonormal matrix ofKl, whereQlcorresponds to the eigenvalues of 1 andClto the eigenvalues

306

of 0. The matrixQ_lhas the following properties:Q⁰_lQ_l=I_n^∗

l,Q_lQ⁰_l =K_landQ⁰_lι= 0, wheren^∗_l =n_l−1

307

14Note that Badev (2013) obtains that neglecting the endogeneity of the friendship network leads to a downward bias of 10% to 15% in endogenous peer effects regarding adolescent smoking rates.

15Following the linear-in-means model, we allow the peers’ mean characteristic corresponding to each individual’s char- acteristic to have a potential effect on his fast food consumption. Therefore, we do not imposead hoc(identifying) exclusion restrictions to the model. Admittedly, these peers’ characteristics may also act as proxy variables that help to control for the selection effect.

with n_l being the number of adolescents in thelth network. Pre-multiplying (7) by Q⁰_l, the structural

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model can now be written as follows:

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M^∗_ly^∗_l =βM^∗_lG^∗_ly^∗_l +M^∗_lX^∗_lγ+M^∗_lG^∗_lX^∗_lδ+ν_l^∗, (8) whereM^∗_l =Q⁰_lM_lQ_l,y^∗_l =Q⁰_ly_l,G^∗_l =Q⁰_lG_lQ_l,X^∗_l =Q⁰_lX_l, andν_l^∗ =Q⁰_lξ_l. With this transformation,

310

our problem of dependency between the observations is solved, since we haveE(ν_l^∗ν_l^∗0|X_l,G_l) =σ²I_n^∗

l.

311

Following Lee et al. (2010), we propose two approaches to estimate the peer effects model (8): a max-

312

imum likelihood approach (ML) and a generalized spatial two stage least squares (GS-2SLS) approach.

313

The ML approach imposes more structure (normality) than GS-2SLS. Therefore, under some regularity

314

conditions, ML estimators are more asymptotically efficient than GS-2SLS ones when the restrictions it

315

imposes are valid.

316

3.1.2 Maximum Likelihood (ML)

317

Assuming thatν_l^∗ is an^∗_l-dimensional normally distributed disturbance vector, the log-likelihood func-

318

tion is given by:

319

lnL = −n^∗

2 ln (2πσ²) +

L

X

l=1

ln|I_n^∗

l −βG^∗_l| +

L

X

l=1

ln|I_n^∗

l −ρM^∗_l| − 1 2σ²

L

X

l=1

ν_l^∗0ν_l^∗, (9)

wheren^∗ =

L

P

l=1

n^∗_l =N−L, and, from (8),ν_l^∗ =M^∗_l(y_l^∗ − βG^∗_ly^∗_l − X^∗_lγ − G^∗_lX^∗_lδ). Maximizing (9)

320

with respect to(β,γ⁰,δ⁰, ρ, σ)yields the maximum likelihood estimators of the model. Interestingly, the

321

ML method is implemented after the elimination of the network fixed effects. Therefore, the estimators

322

are not subject to the incidental parameters problem that may arise since the number of fixed effects

323

increases with the the size of the networks sample.

324

3.1.3 Generalized spatial two stage least squares (GS-2SLS)

325

To estimate the model (8), we also adopt a generalized spatial two-stage least squares procedure pre-

326

sented in Lee et al. (2010). This approach provides a simple and tractable numerical method to obtain

327

asymptotically efficient IV estimators within the class of IV estimators. In the case of our paper this

328

method will consist of a two-step estimation.¹⁶ To simply the notation, Let X^∗ be a block-diagonal

329

matrix withX^∗l on its diagonal, G^∗ be a block-diagonal matrix with G^∗l on its diagonal, and y^∗ the

330

concatenated vector of they^∗_l’s over all networks.

331

Now, let us denote by X˜^∗ the matrix of explanatory variables such thatX˜^∗ = [G^∗y^∗ X^∗ G^∗X^∗].

LetPbe the weighting matrix such thatP=S(S⁰S)⁻¹S⁰, andSa matrix of instruments such thatS = X^∗ G^∗X^∗ G^∗2X^∗

. In the first step, we estimate the following 2SLS estimator:

θˆ1 = (X˜^∗0P ˜X^∗)⁻¹X˜^∗0Py^∗,

whereθˆ₁ is the first-step 2SLS vector of estimated parameters(ˆγ₁⁰,δˆ₁⁰,βˆ₁)of the structural model. This

332

estimator is consistent but not asymptotically efficient within the class of IV estimators.

333

Now, in the second step, we estimate a 2SLS using a new matrix of instrumentsZˆ given by:

Zˆ = [G^∗ˆy^∗ X^∗ G^∗X^∗],

whereG^∗ˆy^∗is computed from the first-step 2SLS reduced form (pre-multiplied byG^∗):

334

G^∗ˆy^∗ =G^∗(I−βˆ₁G^∗)⁻¹(X^∗ˆγ₁+G^∗X^∗δˆ₁). (10) We then estimate:

θˆ₂ = (Zˆ⁰X˜^∗)⁻¹Zyˆ ^∗.

This estimator can be shown to be consistent and asymptotically best IV estimator. Its asymptotic vari-

335

ance matrix is given byN[Z⁰X˜^∗R⁻¹X˜^∗Z]⁻¹. The matrixRis consistently estimated byRˆ =s^2ˆZ_N^0ˆZ, where

336

16Note that for this particular case we imposeρ= 0and thusMl=Il.

s² =N⁻¹PN

i=1uˆ_i²anduˆ_i are the residuals from the second step. It is important to note that, as in Kele-

337

jian and Prucha (1998), we assume that errors are homoscedastic. The estimation theory developed by

338

Kelejian and Prucha (1998) under the assumption of homoscedastic errors does not apply if we assume

339

heteroscedastic errors (Kelejian and Prucha, 2010).

340

3.2 A weight gain production function

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In this section, we propose a dynamic weight gain production function that relates an individual’s zBMI

342

in timetto his lagged zBMI, his lagged fast food consumption as well as his own characteristics in period

343

t. Lety^b_itbe an individual i’s zBMI level at timet, andy^f_it−1 be the individual’s fast food consumption

344

at timet−1. Then, for a given vector of characteristicsx˜_it, the weight gain production function can be

345

formally expressed as follows (for notational simplicity we suppressl):

346

y^b_it=π0+π1y_i,t−1^b +π2y_it−1^f +π3x˜it+ηit, (11)

where

ηit=µi+ζit,

withµ_irepresenting the individuali’s time-invariant error component (fixed effect) andζ_it, his idiosyn-

347

cratic error that may change across t. It is assumed that |π₁| < 1 and thatζit is serially uncorrelated

348

(A test of this latter assumption is provided in the empirical section). As discussed earlier, our interest

349

in this production function goes beyond a mere association between fast food consumption and weight

350

gain. We are particularly interested to analyze the magnitude of a change in zBMI resulting from a

351

common exogenous shock on fast food consumption within the network, when peer effects are taken

352

into account. Our two equation model allows us to compute this result. Partially differentiating (11)

353

with respect toy_it−1^f and using the social multiplier [= 1/(1−β)]yields the magnitude of a short run

354

change in zBMI (i.e., fory^b_i,t−1given) resulting from a common marginal shock on fast food consumption:

355

∂E(y_it^b|·)/∂α_l = _1−β^π2 . This expression entails two components: the impact of the fast food consumption

356

on the zBMI (=π₂) and the multiplier effect (= _1−β¹ ).

357

At this point it is important to recall that the first differences OLS estimator of eq.(11) to get rid of

358

the fixed effects is inconsistent becausey^b_i,t−1 is correlated withζi,t−1 whiley_it−1^f may also be correlated

359

withζi,t−1. However, an IV variant leads to consistent estimates. Formally, assuming that all character-

360

istics are time-invariant (except age whose coefficient is proportional to the intercept of the transformed

361

model), the first differences model can be written as follows:

362

∆y^b_it= ˜π₀+π₁∆y^b_i,t−1+π₂∆y_it−1^f + ∆ζ_it, (12)

where∆is the first difference operator. To resolve the problem of correlation between the right hand side

363

variables and the error term one can instrument for∆y_i,t−1^b and∆y^f_i,t−1. Following Anderson and Hsiao

364

(1981), we instrument∆y_i,t−1^b by the lagged first difference variable∆y^b_i,t−2. To instrument∆y_i,t−1^f , we

365

use lagged changes in time use variables (work hours, sleep hours, leisure hours), in cigaret consumption

366

and in income, between wave II and wave I. The latter variables are predetermined in the model and

367

therefore are weakly exogenous (this is tested in our empirical section). Moreover they are likely to

368

be correlated with change in fast food consumption between wave III and wave II (also tested in our

369

empirical section).¹⁷ The model as laid out in equation (12) is estimated using generalised method of

370

moments (GMM).

371

4 Data and Descriptive Statistics

372

The Add Health survey is a longitudinal study that is nationally representative of American adoles-

373

cents in grades 7 through 12. It is one of the most comprehensive health surveys that contains fairly

374

exhaustive social, economic, psychological and physical well-being variables along with contextual data

375

on the family, neighbourhood, community, school, friendships, peer groups, romantic relationships,etc.

376

In wave I (September 1994 to April 1995), all students (around 90 000) attending the randomly selected

377

high schools were asked to answer a short questionnaire. An in-home sample (core sample) of approx-

378

17As reported earlier, Jeffery and French (1998) provides evidence that leisure time, such as hours of TV viewing, positively influence the BMI of women.

imately 20 000 students was then randomly drawn from each school. These adolescents were asked

379

to participate in a more extensive questionnaire where detailed questions were asked. Information on

380

(but not limited to) health, nutrition, expectations, parents’ health, parent-adolescent relationship and

381

friends nomination was gathered.¹⁸ This cohort was then followed in-home in the subsequent waves in

382

1996 (wave II), 2001 (wave III) and 2008-2009 (wave IV). The extensive questionnaire was also used to

383

construct the saturation sample that focuses on 16 selected schools (about 3000 students). Every student

384

attending these selected schools answered the detailed questionnaire. There are two large schools and

385

14 other small schools. All schools are racially mixed and are located in major metropolitan areas except

386

one large school that has a high concentration of white adolescents and is located in a rural area. Con-

387

sequently, fast food consumption may be subject to downward bias if one accepts the argument that the

388

fast food consumption among white adolescents is usually lower than that of black adolescents.

389

In this paper we use the saturation sample of wave II in-home survey to investigate the presence of

390

peer effects in fast food consumption.¹⁹ One of the innovative aspects of this wave is the introduction

391

of the nutrition section. It reports among other things food consumption variables (e.g., fast food, soft

392

drinks, desserts,etc.). This allows us to depict food consumption patterns of each adolescent and relate

393

it to that of his peer group. In addition, the availability of friend nomination allows us to retrace school

394

friends and thus construct friendship networks.To estimate the weight gain production function, we

395

considered information from wave I, wave II, wave III and wave IV.

396

We exploit friends nominations to construct the network of friends. Thus, we consider all nominated

397

friends as network members regardless of the reciprocity of the nomination. If an adolescent nominates

398

a friend then a link is assigned between these two adolescents (directed network with non symmetric

399

links).

400

18Adolescents were asked to nominate up to five female friends and five male friends.

19It includes all meals that are consumed at a fast food restaurant such as McDonald’s, Burger King, Pizza Hut, Taco Bell and other fast food outlets.

4.1 Descriptive statistics

401

In our peer effects model, the dependent variable of interest is fast food consumption, as approximated

402

by the reported frequency (in days) of fast food restaurant visits in the past 7days. Table 1 reports respec-

403

tively the mean and the standard deviation of the endogenous variable, the covariates used and other

404

relevant characteristics. We note that on average, adolescents’ fast food consumption is fairly within

405

the range of 2.33 times/week. This is consistent with the frequency reported by the Economic Research

406

Service of the United States Department of Agriculture. Around 62% of the adolescents consumed fast

407

food twice or more in the past week and 44% of the adolescents who had consumed fast food did so 3

408

times in the past week.

409

The covariates of the fast food peer effect equation include the adolescent’s personal characteristics,

410

family characteristics as well as the corresponding contextual social effects.The personal characteristics

411

are gender, age, ethnicity (white or other) and grade. We observe that 50% of the sample are females,

412

that the mean age is 16.3 years and that 57% are white. Family characteristics are dummies for mother

413

and father education. We observe that around 45% of mothers and fathers have at least some college

414

education. To control further for parents’ income we use child allowance as a proxy. An adolescent’s

415

allowance is on average 8.28 $ per week, around 50% of the adolescents in our the sample have a weekly

416

allowance. At this point, it is important to highlight that since we use cross section data, we do not

417

have to control for fast food prices as they are taken into account by network fixed effects. As for the

418

weight gain production function, the dependent variable that we use is the variation in the zBMI between

419

waves III and IV. The zBMI variables for each wave, the fast food variables for wave II and III and the

420

instruments are detailed in Table 4.²⁰A large part of changes in the variables between waves is explained

421

by age.

422

20It is important to note that information on fast food consumption was not collected in wave I.