The Relationship from Friendship Links to Educational Achievement

1 Introduction

In the literature of peer effects, there are some recent papers investigating the effects of the number of friends on one’s own outcomes, such as labor productivity (Conti et al. 2013; Fletcher 2014), educational achievement (Mihaly 2009; Navy and Sand 2014), and health (Ho 2016). To distinguish the causal effect of the number of friends from unobserved heterogeneity, Mihaly (2009) uses an instrumental variables approach by excluding school composition variables from the achievement equation to obtain identification; Conti et al. (2013) and Ho (2016) use a structural approach to jointly model friendship formation and outcomes to account for selection; Fletcher (2014) use within-siblings estimation to wipe out family-specific unobservables across siblings; Navy and Sand (2014) exploit a random assignment of students to classrooms to examine the impact of classroom-friends on test scores. Except for Mihaly (2009), the other studies find a positive effect of the number of friends on labor productivity or educational achievement or health.

This paper uses a new empirical strategy to estimate the impact of the number of friends (sum of friendship links) on one’s own educational achievement. Exploiting the pseudo-panel structure of friendship links, this paper adopts a control function approach to deal with unobserved individual heterogeneity. In the first step, I exploit the pseudo-panel structure of friendship data, in which individuals make multiple binary friendship choices at a given period of time, to estimate individual effects in a linear dyad model. In the second step, the estimated individual effects are included as a control function in the education production, which controls for individual unobservables, so the effect of the number of friends on educational achievement is identified.

Unlike Conti et al. (2013) and Ho (2016), the identification of this paper does not rely on the non-linearity of a friendship model. In addition, this paper does not assume any parametric distribution of individual unobservables. Instead, the source of identification comes from the pseudo-panel structure of friendship links that is exploited to estimate individual effects. The estimated individual effects are not perfectly collinear in observable variables in the linear education production function, because the pseudo-panel (school size) varies across individuals. Excluding homophily variables in the dyad model from the education production further sharpens the identification.

Estimating models with data from the National Longitudinal Study of Adolescent Health (Add Health), this paper finds two main results. First, including individual effects estimated from the dyad model significantly increases the effects of the number of friends on test scores. This suggests that this paper’s empirical strategy helps to reduce a downward bias in the OLS estimate. Second, results show that having one additional reciprocal or out-degree friend increases about 2 % of a standard deviation in cognitive test scores.

The reminder of this paper is organized as follows. Section 2 describes the data. Section 3 outlines the empirical strategy. Section 4 discusses the empirical results, and Section 5 concludes.

2 Data

The data used in this paper come from the National Longitudinal Study of Adolescent Health (Add Health). Individuals were asked to nominate their friends (up to five males and five females) in the 1994–1995 in-school survey. Friends not included in school rosters cannot be identified and thus are excluded from the study, which implies that an individual’s friends are also his or her schoolmates. In the empirical analysis, I define out-degree friends as the individual’s schoolmates who were nominated by the individual as friends; I define in-degree friends as the individual’ schoolmates who nominate the individual as friends; I also define reciprocal friends as the individual’s schoolmates who nominate and are nominated by the individual as friends. In the in-home survey, individuals were asked to fill out a long-form questionnaire in which their individual, parental, and neighborhood particulars were collected. In addition, they were asked to take the Add Health Picture Vocabulary Test (AHPVT), which assessed their verbal ability and scholastic aptitude. The summary statistics of variables are given in Table 1.

3 Empirical Methodology

The education production function is structured as follows:

where Y_i is the educational achievement of individual i. The variable of interest is the number of friends F_i, where β captures the associated effect on the achievement. X_i is a vector of exogenous covariates and ηi is the i.i.d. error term. F_i is endogenous because it is correlated with the unobserved variable γi, i. e., ρ≠0.^[1] Thus, estimating eq. [1] by ordinary least-squares (OLS) gives a biased estimate of β.

The key to identifying β is to control for the unobservable γi. The traditional control function approach (Heckman and Robb 1985) is to run a regression of F_i on exogenous variables X_i and IV_i and then include the residual as a regressor to control for γi in eq. [1]. Successful identification requires IV_i to be correlated with F_i but excluded from eq. [1]; otherwise, the residual is perfectly collinear in X_i and F_i, which fails the rank condition. The novelty of this paper is to exploit the pseudo-panel structure of friendship links to help achieve identification.

To understand the identification, a dyadic model of friendship links is set up as follows:

where Fij=1 if i forms a friendship link with j and zero otherwise. Friendship links are either directed (i. e., F_ij might not equal F_ji) or undirected (i. e., Fij=Fji). Links are directed for out-degree friends and in-degree friends, but undirected for reciprocal friends. Self-ties are ruled out (i. e., Fii=0). Z_ij are dyad observables that characterize homophily; I parameterize Z_ij as |Zi−Zj|, which measures absolute demographic differences in race, gender, and school grade between i and j, namely, |Minorityi−Minorityj|,|Malei−Malej|, and |Gradei−Gradej|, respectively. ^[2]γi captures individual characteristics that determine friendship decisions, such as race, gender, parental background, cognitive ability, and personality traits, etc. αj captures peer effect: individual j is attractive to others and then highly valued to be friends. ϵij is a random shock of match quality that is independently and identically distributed across dyads. γi and αj are treated as parameters to be estimated, so the joint distribution of unobserved heterogeneity and observed variables of homophily is left unrestricted. In particular, observable covariates, Xi, are absorbed by γi. I normalize ∑γi=0 and ∑αj=0. I further assume that ηi⊥(γi,Fi,Xi),γi⊥(Zij,Xi),ϵij⊥(Zij,Xi,γi,αj), and ϵij∼i.i.d. (0, σ2).

From eq. [3], an individual’s number of school friends is the sum of his or her friendship links: Fi=∑j=1JsFij, where J_s is the number of schoolmates in school s. It is clear that F_i is endogenous in eq. [1] because γi jointly affects friendship decisions and achievement. Owing to multiple observations of binary friend choices of each individual, γi can be identified in eq. [2]. In a linear model, it follows that γˆi=∑j=1JsFijJs−∑j=1JsZ′ijΠˆJs=FiJs−Z˜i′Πˆ, where Z˜i=∑j=1JsZijJs. The following are the details of the two-step estimation strategy:

1. The linear probability model is used to estimate γˆi in eq. [2]. ^[3]

2. γˆi is included as a control function in eq. [1].

In a linear model, the estimated individual effects, γˆi=FiJs−Z˜i′Πˆ, are not perfectly collinear in X_i and F_i in eq. [1] because of two identifying sources. First, J_s is not constant: school sizes are heterogeneous. Second, Z˜i could be excluded from the achievement equation. Z˜i includes average absolute differences between an individual and his or her schoolmates in race, gender, and school grade. Conditional on school fixed effects, which subsume demographic characteristics of schoolmates, Z˜i might not directly affect an individual’s achievement. Nevertheless, exclusion restrictions are not required for identification. Thus, I construct two different control functions. Control function 1 is constructed as γ^i1=Fi/Js without Z˜i′∏ˆ. Control function 2 is constructed as γˆi2=FiJs−Z˜i′∏ˆ, where homophily variables averaged at the individual level Z˜i are excluded from eq. [1].

4 Results

Table 2 reports estimation results. To ease interpretations, test scores are standardized with a mean of zero and a standard deviation of one. To take into account of estimation errors of the coefficients in the linear probability model, bootstrapped standard errors are used in the model using control function 2. The specification in column 1 includes school fixed effects, grade fixed effects, individual, parental, and neighborhood covariates, as well as missing-variable dummies. It shows that having one out-degree friend increases one’s own test score by 1.8 % of a standard deviation.

To deal with unobserved individual heterogeneity, the specification in column 2 adds control function 1 to the model. It shows that the coefficient of interest significantly changes from 0.018 to 0.027. The coefficient of control function 1 is −0.051 and statistically significant. It means that a one-standard-deviation increase in the individual unobservable lowers one’s own test score by 5.1 % of a standard deviation. The result is interpreted as follows: First, there is a tradeoff between leisure and study hours – individuals who spend more time in social activities like making friends have less time and motivation in their studies. Hence, omitting such factors in the education production generates a downward bias in the OLS estimate. This result is in line with Conti et al. (2013) who find that the individual unobservable of friendship choices has a negative impact on labor productivity. Second, measurement errors of the number of friends shrink the OLS estimate toward zero, but the control function approach helps to correct the downward attenuation bias. For a robustness check, control function 2 is used in column 3. The estimated coefficient of control function becomes smaller, but the estimated effect of the number of out-degree friends is unchanged.

Table 3 shows results from a model in which three types of number of friends are jointly included. Control functions 1 and 2 are conditioned in columns 1 and 2, respectively. In column 1, the result shows that having one additional reciprocal or out-degree friend increases about 2 % of a standard deviation in cognitive test scores. However, having in-degree friends has no effect. The results are robust to using control function 2 in column 2.

In the empirical analysis, individuals were restricted from nominating more than five male and five female friends. This is not a critical issue because only 8 % of individuals nominated five male friends, 12.5 % of individuals nominated five female friends, and 2.5 % of individuals nominated ten friends. For a robustness check, friendship dummies are defined such that the values are equal to one when individuals have one or more friends and zero otherwise. These alternative friendship variables are not constrained by the upper bound of friendship nominations. Table 4 shows that the qualitative results do not change – out-degree and reciprocal friends matter for individual achievement. However, the magnitude of the estimates is larger using friendship dummies.

5 Conclusion

This paper uses a control function approach to estimate the impact of the number of friends (sum of friendship links) on one’s own educational achievement. Results show that the empirical approach helps to reduce a downward bias in the OLS estimate. The main finding is that having one additional reciprocal or out-degree friend increases about 2 % of a standard deviation in cognitive test scores.