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Teenage dropouts and drug use: Does the specification of peer group structure matter?
Economics of Education Review 28 (2009) 497–504
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Economics of Education Review journal homepage: www.elsevier.com/locate/econedurev
Teenage dropouts and drug use: Does the specification of peer group structure matter? Darrell J. Glaser ∗ Department of Economics, United States Naval Academy, 589 McNair Road, Annapolis, MD 21402-5030, United States
a r t i c l e
i n f o
Article history: Received 29 August 2007 Accepted 10 November 2008 JEL classification: I10 I21 J24 Keywords: Economics of education Human capital Juvenile drug use Peer effects
a b s t r a c t Four alternative structures of peer groups are compared in an empirical analysis of teenage dropouts and recent drug use. In general, individual-specific covariates remain robust regardless of group structure specification in dropout models, but lose significance in models of drug-use. Estimates of correlated school effects depend on the specification of group structure. Contextual group effects have no influence on the probability that an individual uses drugs, but demonstrate some statistical significance, albeit ambiguous and strongly dependent on the specification of group structure. Endogenous peer effects do not influence the probability of dropping-out of school, but exhibit positive complementarities with respect to recent drug-use. Modeling the probabilities of leaving school and recent druguse within a jointly distributed empirical framework indicates that unobserved attributes bridging the two types of behavior demonstrate positive correlation. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The negative effects on the development of human capital in juveniles from drug use, as well as attendance at crime-ridden schools recently has received increasing attention, particularly as educators and policy-analysts seek to understand reasons for high dropout rates and substandard academic achievement. A stylized hypothesis suggests that youths face more difficult academic challenges and achieve substantially lower levels of human capital if they attend schools with higher rates of drug use or use drugs themselves. Implicitly this argument bases its foundation in research related to peer effects, but empirically identifying results can be difficult if not intractable. As Manski (1993) coined, reflection problems occur in research on peer effects when researchers attempt to infer how average behavior within a group affects the outcomes of individuals who comprise the group. Group outcomes
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associated with the reflection problem include endogenous, contextual and correlated peer effects. For the sake of discussion, endogenous effects capture changes in behavior caused by the behavior of the group. The effect is endogenous because researchers cannot readily determine which movement occurs first, the group behavior, or the behavior of the individual within the group. An endogenous effect in effort towards human capital achievement would exist if individual-level educational outcomes vary with average levels of educational outcomes in the school. This hypothesis of endogenous peer effects suggests that feedbacks exist within the entire structure of the group, which in essence implies the presence of a social multiplier. A contextual peer effect in effort towards the attainment of human capital measures how effort varies as exogenous attributes of a group vary. For example, well-educated parents in a neighborhood may generate positive contextual effects if youths within that neighborhood emulate parental behavior by exerting more effort in school. This paper will present results that consider the roles of peer effects on the probability that a youth leaves school (in the current year) and never returns to finish a degree
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within 6 years of the initial interview, and the probability that the youth has used drugs (in the past 30 days prior to the interview). Hypothetically, the jointly correlated unobserved factors (errors) that influence an individual’s proclivity to use drugs also influence their ability to succeed in school, implying that the two behaviors exhibit nonzero covariance. This assumption drives the basic empirical methodology employed in the paper. Results support this hypothesis, where the unobserved stochastic terms related to leaving school and recent drug-use appear positively correlated with strong statistical significance. While most previous empirical research concerning peer effects define contextual or endogenous group-level variables as averages of group behaviors or attributes, I also explore several different specifications of peer group structure. This has important ramifications for all empirical research of peer effects, since results appear quite sensitive to how researchers specify peer group structure. The remainder of the paper is organized as follows. Section 2 provides a brief overview of the various strands of relevant literature. Section 3 briefly reviews the twoequation discrete choice model with unobserved correlated disturbances across equations, while Section 4 introduces the four alternatives used to specify peer group structure and thereby weight peer effects. Section 5 covers the data and variables used in the estimations, including methodology for capturing probability weighted structures. Section 6 reviews the results, while Section 7 provides a brief conclusion. 2. Background and relevant literature The social sciences literature has addressed the emergence of behaviors from neighborhood effects through several alternative, if recently converging strands of thought. Sampson, Morenoff, and Gannon-Rowley (2002) summarize areas of focus in the sociological literature and identify four classes of neighborhood mechanisms that appear to affect the emergence of behaviors: social ties and patterns of interaction, measures of collective efficacy, localized institutional resources and patterns of land, and community resources and infrastructure. While evidence from sociological studies does not give a clear conclusion regarding specific neighborhood mechanisms, the majority indicate that concentrated poverty, and low neighborhood efficacy are linked to greater incidence of risky behavior. Each of these explanations point-towards the development of behavioral models in which peer effects, as well as the quality and quantity of community resources all matter. Recent economic models of juvenile behavior, particularly with respect to delinquent outcomes, have also framed contexts that allow for the emergence and effects of social interactions. Recently, Calvó, Petacchini, and Zenou (2006) explore the role of an individual’s centrality in education networks, while Weisberg (2006) outlines a framework for capturing endogenous group formation. Essentially in these papers, individuals in networks affect or are affected by each others’ behavior differently. Along with Akerlof and Kranton (2000), who build a model of educational attainment into a framework allowing for the utility effects of group structure, these most recent models have begun to
bridge a historical gap between economists and sociologists to model individual and group level endogenous behavior. The hypothesis proposed in this paper suggests that dropping-out of school and using drugs emerge jointly through a correlated unobserved component. For example, the same youths who fidget, hassle and generally engage in unwanted and disruptive behavior in school may more likely use drugs. Some support for this follows from a substantial body of research from the fields of criminology and psychology. Examples include Blumenstein, Farrington, and Moitra (1984) as well as Tremblay, Pihl, Vitaro, and Dobkin (1994), who in general indicate that shy children or children otherwise more risk averse are less likely to become juvenile delinquents. Other studies from Campbell (1990), Hechtman, Weiss, Perlman, and Amstel (1984), and Loney, Kramer, and Milich (1982) show that children with hyperactivity, attention deficit disorder or related ‘anti’-social behaviors during early childhood, tend to engage in crime at later stages of juvenile development. More recently Grossman, Kaestner, and Markowitz (2004), as well as Chatterji (2006) have empirically explored intrarelated teen behaviors in bivariate probit contexts. Their results indicate that anti-social behavior and possibly educational outcomes have an unobserved component that heretofore has remained empirically unexplored. A prevailing hypothesis within this research argues that the same individual traits (possibly independent of ability) that lead to anti-social behavior also increase the difficulty for children to succeed in school. Children might be excluded from standard tracking in school due to these traits find themselves placed unnecessarily in special education classes or non-placement into gifted or honors classes.1 The worst case outcomes for the worst malcontents include suspensions or expulsions for bad behavior. 3. A discrete choice empirical framework The results generate from a two-choice latent variable empirical framework, which extends from theoretical work outlined in Brock and Durlauf (2001a, 2001b).2 The binary choice specification facilitates the identification of parameters, since linear-in-means models of neighborhood effects mathematically cannot be identified (Manski, 1993).3 Individuals in this context will choose to use drugs or not, while simultaneously choosing to dropout of school or not. Implicitly, this defines latent variables for school acquired human capital, h∗i , and drug-use, di∗ as h∗i = zh,i h + h g(hj−i ) + i , di∗ = zd,i d + d g(dj−i ) + i ,
(1)
where each choice differently affects individual payoffs. Youths are heterogeneous with respect to a vector of exoge-
1
Compelling and related stories appear in Mooney (2007). Particularly under the assumption of uniform interaction weights, Brock and Durlauf (2001a, 2001b) demonstrate results for understanding the properties of self-consistent equilibria. Alternative group structure specifications, particularly Ioannides (2006) discussion of Walrasian-star interactions indicate similar results for equilibria. 3 This specification does not allow one to gauge the effects of variations in marijuana use or the comparative effects of marijuana relative to heroin. 2
D.J. Glaser / Economics of Education Review 28 (2009) 497–504
nous factors specific to each choice, zh,i and zd,i , which include individual-specific as well as contextual and correlated peer group effects. The vectors h and d represent parameters. The errors i and i are bivariate normal, with E[i ] = E[i ] = 0, Var[i ] = Var[i ] = 1 and Cov[i , i ] = . Empirical estimates follow from the observed binary vari∗ > 0]. The function ables hj−i = 1[h∗j−i > 0] and dj−i = 1[dj−i g( ) represents a weighting measure for peer effects conditional on all j individuals in the peer group who are not individual i (to be discussed in the following section). Each choice provides random utility to i according to additively separable specifications. Social utility from conforming to the peer group follows subjective expectations of peer behavior. The parameters h and d capture the weight of endogenous peer effects with respect to remaining in school and using drugs. Positive cross-partial derivatives and second derivatives in social utility functions indicate the presence of strategic complementarities and increasing marginal utility from conformist behavior, respectively. This framework subsumes all cross-behavior relationships into the covariance of the unobserved shocks such that cov(, ) = / 0.
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Nk . Connections within these small groups form edges Mij ∈ {0, 1}. Mathematically each Mij takes the value of 1 if an observed connection exists, while a 0 indicates no observed connection. Individuals cannot create connections with themselves, implying Mii = 0. Relationships are defined such that connections form an undirected graph, where Mij = Mji .4 The weighting function now appears as g(hj−i ) =
j
g(hj−i ) = 0
−1
Mij
(Mij jj−i )
if
j∈k
Mij > 0,
j∈k
if
(3) Mij = 0
j∈k
where M ≤ Nk represents the number of connections j ij in narrowly defined groups of the ith individual’s “closest” M = 0 are classified as isopeers. Individuals with j ∈ k ij lated. Isolated individuals within such a structure receive and transmit no endogenous or contextual effects.
4.2. Probability-weighted effects 4. Alternative interaction structures Typical empirical models with interactions most frequently weight peer effects using average behaviors and attributes of the group. While data availability may partially drive this, theoretical or empirical tractability also influences variable choices. This appears particularly apparent within the framework exposited by Brock and Durlauf (2001a, 2001b). Under assumptions of uniform interactions, researchers can theoretically map rational expectations behavior into readily interpretable results including the number of equilibria in the model and conditions for identification. 4.1. Average group effects The most common specifications in the literature use group-level averages for endogenous as well as contextual effects. For example, the weighting function for i’s academic outcome in a school of Nk individuals would appear as g(hj−i ) = (Nk − 1)
−1
hj−i
(2)
j∈k
This framework has readily available and interpretable theoretical equilibria, while also containing variables most easily accessible in data. While data availability may drive the use of neighborhood averages to define peer effects, a natural extension could use more narrowly defined groups. Conceivably data may reveal information about which members of a neighborhood interact with each other, capturing what Iyer and Weeks (2006) define as a small group effect. One might think of this as a group of “best friends”. To be more specific, if large group outcomes are defined from averages estimated over Nk individuals in the k th neighborhood, an individual within that group may M ≤ have “local” or “close” connections defined by j ∈ k ij
If the dominant social structure in neighborhoods appears as uniform connections between all individuals, it makes sense to use mean group variables for analyzing peer effects. The previous two structures are built on this premise of uniform interactions within peer groups, albeit describing these by different group sizes. If the nature of interactions varies either across or within groups, however, then a more general specification for group structure is appropriate. To generalize the earlier measure of peer effects into probability-weighted heterogeneous interactions, let Pij measure the probability of a connection between individuals i and j, where Pij ∈ [0, 1], Pij = Pji and Pii = 0. The Pij are known by each individual i and j. These probability weighted measures form the basis for two additional group structure specifications: connectivityweighted measures, which capture the popularity of the individual, and rank-weighted measures. The first probability weighted measure defines the strength of individuals connections within the context of the entire group. One might think of this as a measure of their overall connectivity to the group, or a measure of an individual’s popularity within the group. Individuals with stronger connections to the group defined by larger Pij for all j receive and transmit more peer effects. Group connectivity therefore defines the social utility function from the summed strength of likely peer connections. This is most similar to the Ioannides’ (2006) star-network mentioned above, with qualifications. First, each agent in the group has a different weight, rather than one principal star who attracts everyone equally. Secondly, the parameters h and d remain constant for members regardless of connectivity to others.
4 This set-up does not allow for differing the costs or benefits of connections, or whether the initiation of a connection matters more or less than its receipt.
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The weighting function for connectivity effects is defined
⎛
g(hj−i ) = ⎝
⎞⎛
Pij ⎠ ⎝
j∈k
⎞
hj−i ⎠ .
(4)
j∈k
In the rank-weighted structure, individuals weight interactions with every other individual separately and thereby rank interactions across each j in the group. Overall connectivity does not matter, but the relative weights of i placed onto each j does. Every individual has the same connection to their group, but they place greater weight on individuals with whom they most likely interact. The weight used in these estimates appears as
⎡ ⎤ (P˜ ij hj )⎦ , g(hj−i ) = ⎣ j∈k
(5)
P˜ = 1 for each i ∈ k. where P˜ ij = (Pij /( j ∈ k Pij )) and j ∈ k ij The P˜ ij capture the weighted rank of individual i’s connections to all other j’s in the group. In this measurement of peer group structure, even individuals who have very low estimated “popularity”, as defined by P , have a j ∈ k ij ranking for their interactions with all individuals in their school.5 5. Data and variables For the empirical analysis, I use The National Longitudinal Survey of Adolescent Health (Addhealth), a comprehensive and nationally representative data-set designed to capture youth behavior that also includes information gathered from parents and school administrators.6 Youths with severe learning disabilities are excluded from the analysis. The first wave of in-home interviews provides the bulk of the data used here, since it contains the largest, most comprehensive set of individuals. I use subsequent waves to establish which individuals ultimately and permanently dropped-out of school during the 1994–1995 school year, never to return.7
5
Large group average interactions discussed in Section 4.1 are a special case of rank-weighted effects if all individuals in a school share the same Pij weights. If Pij are uniform for all connections between all individuals, then rank-weighted effects of peer effects equal the average large group effect. 6 Addhealth is a program project designed by J. Richard Udry, Peter S. Bearman, and Kathleen Mullan Harris, and funded by a grant P01-HD31921 from the National Institute of Child Health and Human Development, with cooperative funding from 17 other agencies. Special acknowledgment is due Ronald R. Rindfuss and Barbara Entwisle for assistance in the original design. Persons interested in obtaining data files from Add Health should contact Add Health, Carolina Population Center, 123 W. Franklin Street, Chapel Hill, NC 27516-2524 ([email protected]). 7 Descriptive statistics are available from the author upon request. In addition to this, approximately 11% of the original sample that Addhealth follows in subsequent waves either disappear due to sample attrition or are omitted due to missing observations on explanatory variables. These missing observations tend to be from minority groups, male, more likely to have lived in single parent/adult households. They have used drugs prior to their sample disappearance at a rate 3 % higher than sample observations who are continuously tracked and used in the final regressions. These fac-
Models estimated in the paper use discrete indicators for youths dropping-out of school permanently or remaining in school for an additional year.8 When estimating specifications for large group, small group and connectivity weighted effects, the sample includes 14,892 individuals, of which 2.4% leave school.9 The matrix of friendship ties, M, draws from the first wave of in-home interviews as well as information provided in the in-school survey by each student who names their best friends. The surveys collected the friendship data ego-centrically, in that individuals answering the survey defined the presence of a link between themselves and others in the school. 5.1. Edge probabilities To estimate the Pij structures outlined in earlier sections, school networks form through a random process. More specifically, connections defined by Mij represent the edges (connections) between two individuals. These edges take the value of 1 if a connection is observed, while a 0 indicates that no observed connection exists. The combined undirected edges generate a vector of M observable connections within each of the k groups (schools) between individuals i and j of size (Nk (Nk − 1)/2) × 1. To generate the edge probabilities, define the vector A, which includes exogenous attributes such as gender, race, age, parental income, peabody achievement test scores, an indicator for whether the student ever repeated a grade in school, and indicators for whether the individuals have used drugs, alcohol or tobacco in years prior to the current sample. Following Udry and Conley (2004), differences in these variables between individuals i and j are modeled by |Ai − Aj |. Assume that Pij , the probability of connections between individuals forming in each of the k schools, differs across all schools in the sample.10 I estimate these probabilities assuming that connections Mij are random variables such that Prob {Mij = 1} = (˛k |Ai − Aj |) ≡ Pij ,
(6)
tors indicate that results explained later in the paper may understate the role of individual-specific variables on drug use. Given the unique methods used to construct the peer group structures, however, it is impossible to ascertain precisely how their omission affects parameters related to contextual and endogenous peer effects. 8 Additional analysis (not receiving discussion) extended this anti-social behavior to include all non-violent activity such as drug use, 5 or more drinks, frequent tobacco use, the deliberate damaging of property, theft and burglary. The results germane to the discussions in this paper did not notably change, vis-a-vis measuring only drug-use as an anti-social behavior. 9 At a constant dropout rate of 2.4% per year, 8th graders in 1995 have an overall graduation rate of 88.5% by 1999. After adjusting the dropout rate by grade from the data, the sample graduation rate after 4 years for an 8th grader drops to 88.3%. These sample estimates are only slightly higher than the national graduation rate of 86.5% for 18–24-year-olds in 2000. (Source: U.S. Department of Commerce, Bureau of the Census, Current Population Survey, October 2000.) Youths legally dropout of high school at the age of 16 in most states. A robustness check using a sub-sample of 9742 youths legally eligible to leave school generates no notable differences in results. 10 This allows for computational feasibility, but also produces heterogeneous measures of connections within and across schools.
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Table 1 Individual-specific marginal effects (Dropout model). Variable
Large group
Small group
Connectivity weighted
Rank weighted
Male Age (months) Age-squared Parents high school Parents income (1000s) Two-adult household Peabody score Repeated grade Prior suspension Prior drinking Prior drugs Prior tobacco Attend church Drugs in home
−9e−4 (7e−4) 0.002 ** (5e−4) −5e −6 ** (0.000) −0.004 ** (0.001) −9e−5 ** (2e−5) −0.001 † (8e−4) −1e−4 ** (3e−5) 0.004 ** (0.001) 0.006 ** (0.002) −3e−4 (7e−4) 0.002 * (0.001) 0.005 ** (0.001) −0.003 ** (7e−4) 0.001 (0.002)
−9e−4 (7e−4) 0.002 ** (5e−4) −5e −6 ** (0.000) −0.004 ** (0.001) −8e−5 ** (2e−5) −0.001 (8e−4) −6e−5 * (3e−5) 0.004 ** (0.001) 0.006 ** (0.002) −4e−4 (7e−4) 0.002 † (0.001) 0.006 ** (0.001) −0.003 ** (0.001) 0.002 (0.002)
−0.001 (7e−4) 0.002 ** (5e−4) −5e −6 ** (0.000) −0.004 ** (0.001) −9e−5 ** (2e−5) −0.001 (8e−4) −8e−5 ** (3e−5) 0.004 ** (0.001) 0.006 ** (0.002) −3e−4 (7e−4) 0.002 * (0.001) 0.005 ** (0.001) −0.003 ** (8e−4) 0.002 (0.002)
−9e−4 (7e−4) 0.002 ** (5e−4) −5e −6 ** (0.000) −0.004 ** (0.001) −9e−5 ** (2e−5) −0.001 † (8e−4) −1e−4 ** (3e−5) 0.004 ** (0.001) 0.006 ** (0.002) −2e−4 (7e−4) 0.002 * (0.001) 0.005 ** (0.001) −0.003 ** (7e−4) 0.002 (0.002)
Standard errors in parentheses. Marginal effects at Z¯ for continuous variables and 0–1 change for discrete variables. † Significance level: 10%. * Significance level: 5%. ** Significance level: 1%.
where represents the logistic distribution. Recovering the parameter vectors, ˛k , from these regressions generates an estimate of predicted probabilities for all connections between i and j.11 I generate the predicted probability of a connection using a two-step process. In the first step, all differences in A are regressed on the ((Nk − 1)N/2) possible undirected edges. Following model selection techniques using t-tests and likelihood ratio tests, extremely poorly performing variables (which differ across schools) are removed from the analysis, and the k regressions estimated again on the restricted models. Finally, parameters in these secondround regressions with p-values less than 0.10 construct the logits used to generate the predicted probability of the existence of an edge, Pij .12 6. Results The discussion follows with two-equation estimates of seemingly unrelated probits for the joint probability of leaving school during the current school year and taking illegal drugs in the 30 days prior to the interview. With some notational differences, this follows from the Greene (2003) outline for estimating two-equation discrete choice models with correlated and jointly normal error structures. Given this set-up, if > 0 then the covariance in unobserved attributes across behaviors appears positive,
11 Recovering Pij essentially amounts to the estimation of k = 130 logistic regressions (one for each school) each on (Nk × (Nk − 1)/2) conditionally independent observations. 12 While far too extensive to include all regression results here, summaries reveal the ratio of regressors which attain the previously highlighted levels of statistical significance. In 88% of all schools, age differences play a key role in the formation of friendships. Race differences appear significant in 44% of schools. Ability measured through peabody scores matters in 36% of schools, while the differences in whether individuals have repeated a grade matters in 22%. Gender matters 28% of the time, while parents’ income matters in 20% of schools. Differences in ability determined by peabody achievement scores determine connections in 36% of the schools. Prior drug use, drinking and smoking all statistically affect group formation in roughly 27%, 24% and 24% of schools, respectively.
implying that dropping-out and drug-use have a positive unobserved association.13 Tables 1, 2, 4 and 5 report estimated marginal effects as well as school-weighted robust standard errors from regressions specified with the six different neighborhood weighting mechanisms mentioned earlier in the paper. Marginal effects are calculated at the mean for continuous variables but measure the change in probability when the explanatory factor switches from a zero to a one for discrete variables. Table 3 reports likelihood ratio tests for various subsets of contextual variables. Sample sizes for large group, small group and connectivity weighted regressions are 14,982 observations.14 6.1. Individual-specific effects Table 1 shows results for individual specific variables with respect to the probability for a youth leaving school (while controlling for the unobserved correlation with the probability of recent drug use). Measures related to parental factors including parent education, income and church attendance, as well as ability proxies including peabody score and whether the student repeated a grade appear significant and robust across various specifications of weighting mechanisms. Other individual-specific controls, including previous behavior related to drug-use,
13 Estimations which include any non-violent delinquent behavior, rather than recent drug-use generated similar results in terms of statistical significance. Since the impact on results for dropout models does not change, and statistical significance of various weighting mechanisms does not change, the results from these regressions are not included. Additional regressions also tested location controls for urban, suburban and rural status. Dummy variables for school security measures also received consideration, but none demonstrated even a hint of statistical significance. The remaining marginal effects (coefficients) that appear in this paper remain statistically robust to the exclusion of the above list of controls. 14 Bivariate probit results where drug-use appears in the xh,i vector are not reported, since coefficients on these endogenous behaviors never appear statistically significant and tests indicate failure to reject the null hypothesis that = 0. These results appear for each specification of peer group structure. Quite likely this is due to identification issues from weak exclusion restrictions, similar to that reported in Chatterji (2006) and Grossman et al. (2004).
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Table 2 Individual-specific marginal effects (Drug use model). Variable
Large group
Small group
Connectivity weighted
Rank weighted
Male Age (months) Age-squared Parents high school Parents income (1000s) Two-adult household Peabody score Repeated grade Prior suspension Prior drinking Prior drugs Prior tobacco Attend church Drugs in home
0.005 (0.003) 0.005 * (0.002) −1e−5 * (1e−6) 0.004 (0.003) 3 e−5(2e−5) −0.005 (0.003) −2e−4 (1e−4) 0.002 (0.004) 0.030 ** (0.004) 0.041 ** (0.004) 0.219 ** (0.012) 0.032 ** (0.005) −0.008 * (0.004) 0.110 ** (0.018)
0.006 (0.004) 0.005 * (0.002) −1e−5 * (1e−5) 0.005 (0.003) 3 e−5(2e−5) −0.007 † (0.004) −2e−4 (1e−4) 0.001 (0.004) 0.029 ** (0.004) 0.039 ** (0.004) 0.220 ** (0.013) 0.031 ** (0.005) −0.007 * (0.003) 0.109 ** (0.017)
0.005 (0.003) 0.005 * (0.002) −1e−5 ** (1e−5) 0.005 (0.003) 3 e−5(2e−5) −0.006 † (0.004) −2e−4 (1e−4) 0.002 (0.004) 0.031 ** (0.004) 0.041 ** (0.004) 0.222 ** (0.012) 0.032 ** (0.005) −0.008 * (0.004) 0.110 ** (0.018)
0.006 † (0.003) 0.005 * (0.002) −1e−5 * (1e−5) 0.004 (0.003) 3 e−5(2e−5) −0.006 (0.004) −2e−4 (1e−4) 0.003 (0.004) 0.030 ** (0.004) 0.041 ** (0.004) 0.219 ** (0.014) 0.031 ** (0.005) −0.007 * (0.004) 0.110 ** (0.017)
Standard errors in parentheses. Marginal effects at Z¯ for continuous variables and 0–1 change for discrete variables. † Significance level: 10%. * Significance level: 5%. ** Significance level: 1%. Table 3 Correlated marginal effects. Variables
Large group
Small group
Connectivity weighted
Rank weighted
Dropout models Private school Per-capita ed. spending
−0.004 ** (0.001) −7e −6 * (0.000)
−0.004 ** (0.001) −4e −6 (0.000)
−0.004 ** (0.001) −5e −6 (0.000)
−0.005 ** (0.001) −6e −6 * (0.000)
Drug use models Private school Per-capita ed. spending
0.003 (0.008) −5e−7 (1e−5)
0.007 (0.007) −1e−5 (2e−5)
0.005 (0.006) 8e−6 (1e−5)
0.007 (0.008) 2e−7 (1e−5)
Standard errors in parentheses. * Significance level: 5%. ** Significance level: 1%.
tobacco-use, or a prior suspension also all appear robust across each specification. Included as a quadratic control variable, the effect of age also remains unchanged across different model specifications. Gender, the easy access to drugs in the home, two-adult households and prior drinking habits have essentially no statistical effect on dropout rates. Table 2 includes individual-specific marginal effects with respect to the probability of using drugs (while controlling for the probability of dropping-out). Unlike dropout models, parent-specific, and academic abilitybased controls have no statistical effect on drug-use with the exception of church attendance and the availability of drugs in the home. Not surprisingly, the probability of using drugs increases by 0.11 if youths have easy access to drugs at home. Prior behaviors all have quite strong effects as well. Prior drug use, the most important predictor, increases the probability of current drug use by 0.22. Prior suspensions, heavy drinking and tobacco use all increase the probability of recent drug use by 0.029–0.041. Similarly to dropout models, control variables for gender do not matter, while age has a statistically significant and quadratic effect .15
For comparison sake, the probability of dropping-out increases by six one-thousandths for students who have experienced a suspension in their past. The probability of drug-use increases by 0.03 for this same explanatory variable, five times larger. Marginal effects for other individual-specific variables demonstrate similar differences in economic magnitudes between the two models. In short, the results generally indicate that individualspecific factors have stronger explanatory effects on the probability of using drugs than the probability of droppingout.
15 Although results of simpler models without neighborhood effects are not reported, the individual-specific coefficients remain robust to the complete exclusion of neighborhood effects. This supports the conclusion that individual-specific coefficients in models without peer effects will
not suffer omitted variable biases related to peer outcomes and contexts. The same conclusion cannot be made with respect to correlated neighborhood variables, which appear sensitive to the exclusion of contextual and endogenous peer effects in most specifications.
6.2. Correlated effects The results shown in Table 3 for correlated neighborhood variables appear strongly dependent on specifications of peer group structure, in spite of the fact that correlated variables are not directly defined by the peer group weighting mechanism. Attendance at a private school consistently has a negative and statistically significant impact, lowering the probability of a youth dropping-out by approximately 0.004. The same variable has no statistical
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Table 4 Likelihood ratio statistics for sets of contextual variables. Large group
Small group
Connectivity weighted
Rank weighted
Dropout models Family contextuals Ability contextuals Prior behavior contextuals
12.57 * 11.82 ** 6.87
2.32 0.52 15.28 **
4.23 3.62 4.33
13.3 ** 19.7 ** 8.64 †
Drug use models Family contextuals Ability contextuals Prior behavior contextuals
4.10 1.92 7.07
6.48 5.81 † 6.41
0.78 0.16 5.43
4.88 4.14 2.14
Sets of “contextuals” include: “Family”: parents’ h.school, parents’ inc., two adult h-holds, church and drugs in home.; “Ability”: peabody score and grade repeaters.; “Prior behavior”: suspensions, heavy drinking, drug use and tobacco use. † Significance level: 10%. * Significance level: 5%. ** Significance level: 1%.
effect on drug use. Community education spending varies in size, and subsequently statistical significance depending on weighting specifications, particularly in dropout models. It never appears to matter with respect to drug use. 6.3. Contextual effects Table 4 reports likelihood ratio tests for sets of contextual variables, grouping these into sets based on family-level contextual variables, ability-based contextual variables and prior-behavior contextual variables. Family-level contextual variables include the weighted education, income, household structure, church attendance and home-drug availability for all households in the child’s peer group. Ability-based contextual variables include weighted peabody scores and grade repeaters for the child’s peer group. Prior-behavior contextual variables include weighted prior suspensions, drug use, heavy drinking and tobacco use among other children of each child’s peer group. Several things of note arise upon inspection of these statistics. First and perhaps most surprisingly, the probability of using drugs does not appear predictable by any measure of any contextual effect. The story for dropout model specifications appears more convoluted. In rankweighted models, all types of contextual effects matter, but this seems an exception. Each specification generates different results, enough to conclude most importantly that
contextual coefficients are sensitive to the specification of peer group structure. 6.4. Endogenous effects Table 5 reports marginal endogenous effects. In dropout models, the coefficients are negative, which would indicate the presence of multiple equilibria if true (Brock & Durlauf, 2001b), but also statistically not significant indicating that endogenous effects on the probability of dropping-out of school do not exist. With respect to the probability of drug-use, however, endogenous effects appear statistically significant and quantitatively notable across all specifications. To unpack the equilibrium results of an endogenous peer-effect, I refer to the discussion in Durlauf and Cohen-Cole (2005) concerning how neighborhood endogenous effects amplify incentives and produce “social multipliers”. In the druguse model reported in Table 5 with large-group weights, school-wide average drug use generates a marginal effect of 0.149 (column 1). For example, this indicates that an exogenous decrease in the rate of school drug-use by 0.05 decreases the probability that an individual uses drugs by 0.0075. Extending this result into the limit to allow for endogenous feedbacks between all individuals in a school generates a social multiplier of (1/(1 − 0.149)) = 1.176. This social multiplier indicates that an exogenous shock to school-level drug-use, such as a raid on local drug suppliers which decreases the rate of drug use throughout the
Table 5 Endogenous effects Large group
Small group
Connectivity weighted
Rank weighted
Dropout models Marginal effect Standard error
−0.008 (0.032)
0.005 (0.004)
−0.002 (0.007)
−0.001 (0.021)
Drug use models Marginal effect Standard error
0.149 * (0.071)
0.034 ** (0.011)
0.047 * (0.019)
0.090 † (0.051)
Standard errors in parentheses. Marginal effects measured at Z¯ for continuous variables. † Significance level: 10%. * Significance level: 5%. ** Significance level: 1%.
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Table 6 Correlation in unobservables.
ˆ Standard error Sample size **
Large group
Small group
Connectivity weighted
Rank weighted
0.125 ** (0.040) 14892
0.111 ** (0.041) 14892
0.124 ** (0.038) 14892
0.122 ** (0.039) 14892
Significance level: 1%.
school by 0.05, decreases the equilibrium probability that any given individual uses drugs by approximately 0.059.16 6.5. Covariance in unobservables Table 6 includes sample sizes and estimates for . These indicate a statistically significant relationship between the stochastic components of the two behaviors, where Wald tests cannot reject the null for the jointly specified model. This lends support to the aforementioned hypothesis that maladjusted children may share an unobserved attribute for both the propensity to dropout of school and engage in anti-social behavior such as drug-use. This also indicates a need for further explorations into empirical methodologies which assume correlated multivariate error distributions. 7. Conclusion This paper compares statistical and qualitative results for school dropouts and drug use using four alternative structures of peer effects. Several key conclusions arise from reviewing the results of these different specifications. First, individual-specific coefficients remain robust, regardless of specification. Secondly, family and group-level contextual factors demonstrate effects on the probability of dropping-out of school inconsistently across the specifications of peer group structure. Contextual effects do not matter in any specification of drug-use. Regardless of peer group structure, endogenous effects appear statistically not significant for dropout models, but results indicate that using drugs have strong endogenous peer effects for large group, small group and probability weighted models. Furthermore, results from all peer group specifications indicate that correlated group effects matter with respect to the probability of dropping-out of school but not using drugs. The results also indicate further empirical support for estimating juvenile human capital decisions jointly with detrimental behaviors such as drug use. These conclusions follow from a simple two-equation specification where two outcomes (or choices) link via an unobserved parameter. This unobserved component not only affects the propensity to use drugs, but also the propensity to remain in school. A review of results that remain robust leads to policy implications related to reducing the probability of using drugs. Particularly, a focus in drug prevention efforts that relate to the individual and individual-specific factors may
16 The results cannot be compared across the other peer-group specifications, since the effects of coefficients would depend on an infinite number of probability weights for peer groups.
generate lower rates of drug use. For example, youths with factors conducive to drug use (prior tobacco, alcohol or drug use, a prior suspension, drugs easily available in the home, etc.) can be targeted with school resources aimed at the mitigation of these negative forces. Any shock that exogenously decreases group-level drug use generates spill-overs reducing the likelihood of drug use for all individuals within the group. References Akerlof, G., & Kranton, R. (2000). Economics and identity. Quarterly Journal of Economics, 105(3), 715–753. Blumenstein, A., Farrington, D., & Moitra, S. (1984). Delinquency careers: Innocents, amateurs and persisters. Crime and Justice: A Review of Research, 6, 187–219. Brock, W. A., & Durlauf, S. N. (2001a). Discrete choice with social interactions. Review of Economic Studies, 68(2), 235–260. Brock, W. A., & Durlauf, S. N. (2001b). Interactions-based models. In J. Heckman & E. Leamer (Eds.), Handbook of econometrics. Amsterdam: North-Holland. Calvó, A., Patacchini, E., & Zenou, Y. (2006). Peer effects and social networks in education. Working paper. Campbell, S. (1990). Longitudinal studies of active and aggressive preschoolers: Individual differences in early behavior and in outcome. In D. Cicchetti & S. Toth (Eds.), Internalizing and Externalizing Expressions of Dysfunction. Hillsdale, NJ: Erlbaum. Chatterji, P. (2006). Does alcohol use during high school affect educational attainment? Evidence from the National Education Longitudinal Study. Economics of Education Review, 25(5), 482–497. Durlauf, S., & Cohen-Cole, E. (2005). Social interactions models. In K. Kempf-Leonard (Ed.), Encyclopedia of social measurement (pp. 517–522). Elsevier Inc. Greene, W. H. (2003). Econometric analysis (5th Ed.). Upsaddle River NJ: Prentice-Hall. Grossman, M., Kaestner, R., & Markowitz, S. (2004). Get high and get stupid: The effect of alcohol and marijuana use on Tenn sexual behavior. Review of Economics of the Household, 2, 413–441. Hechtman, L., Weiss, G., Perlman, T., & Amsel, R. (1984). Hyperactives as young adults: Initial predictors of adult outcome. Journal of the American Academy of Child Psychiatry, 23, 250–260. Ioannides, Y. (2006). Topologies of social interactions. Economic Theory, 28, 559–584. Iyer, S. & Weeks, M. (2006). Multiple social interaction and reproductive externalities: An investigation of fertility behaviour in Kenya. Cambridge working papers in economics, Cambridge University. Loney, J., Kramer, J., & Milich, R. (1982). The hyperactive child grows up: Predictors of symptoms, delinquency, and achievement at follow-up. In K. Gadow & J. Loney (Eds.), Psychosocial aspects of drug treatment for hyperactivity. Boulder, CO: Westview. Manski, C. (1993). Identification of endogenous social effects: The reflection problem. Review of Economic Studies, 60, 531–542. Mooney, J. (2007). The short bus: A journey beyond normal. New York, NY: Henry Holt and Company. Sampson, R. J., Morenoff, J. D., & Gannon-Rowley, T. (2002). Assessing neighborhood effects: Social processes and new directions in research. Annual Review of Sociology, 28, 443–478. Tremblay, R., Pihl, R., Vitaro, F., & Dobkin, P. (1994). Predicting the early onset of male antisocial behavior from preschool behavior. Archives of General Psychiatry, 51, 732–739. Udry, C. R. & Conley, T. G. (2004). Social networks in Ghana. Discussion paper no. 888, Economic Growth Center, Yale University. Weisberg, B. A. (2006). Social interactions and endogenous association. Working paper, Ohio State University.
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Economics of Education Review journal homepage: www.elsevier.com/locate/econedurev
Teenage dropouts and drug use: Does the specification of peer group structure matter? Darrell J. Glaser ∗ Department of Economics, United States Naval Academy, 589 McNair Road, Annapolis, MD 21402-5030, United States
a r t i c l e
i n f o
Article history: Received 29 August 2007 Accepted 10 November 2008 JEL classification: I10 I21 J24 Keywords: Economics of education Human capital Juvenile drug use Peer effects
a b s t r a c t Four alternative structures of peer groups are compared in an empirical analysis of teenage dropouts and recent drug use. In general, individual-specific covariates remain robust regardless of group structure specification in dropout models, but lose significance in models of drug-use. Estimates of correlated school effects depend on the specification of group structure. Contextual group effects have no influence on the probability that an individual uses drugs, but demonstrate some statistical significance, albeit ambiguous and strongly dependent on the specification of group structure. Endogenous peer effects do not influence the probability of dropping-out of school, but exhibit positive complementarities with respect to recent drug-use. Modeling the probabilities of leaving school and recent druguse within a jointly distributed empirical framework indicates that unobserved attributes bridging the two types of behavior demonstrate positive correlation. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The negative effects on the development of human capital in juveniles from drug use, as well as attendance at crime-ridden schools recently has received increasing attention, particularly as educators and policy-analysts seek to understand reasons for high dropout rates and substandard academic achievement. A stylized hypothesis suggests that youths face more difficult academic challenges and achieve substantially lower levels of human capital if they attend schools with higher rates of drug use or use drugs themselves. Implicitly this argument bases its foundation in research related to peer effects, but empirically identifying results can be difficult if not intractable. As Manski (1993) coined, reflection problems occur in research on peer effects when researchers attempt to infer how average behavior within a group affects the outcomes of individuals who comprise the group. Group outcomes
∗ Tel.: +1 410 293 6879; fax: +1 410 293 6899. E-mail address: [email protected]. 0272-7757/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.econedurev.2008.11.001
associated with the reflection problem include endogenous, contextual and correlated peer effects. For the sake of discussion, endogenous effects capture changes in behavior caused by the behavior of the group. The effect is endogenous because researchers cannot readily determine which movement occurs first, the group behavior, or the behavior of the individual within the group. An endogenous effect in effort towards human capital achievement would exist if individual-level educational outcomes vary with average levels of educational outcomes in the school. This hypothesis of endogenous peer effects suggests that feedbacks exist within the entire structure of the group, which in essence implies the presence of a social multiplier. A contextual peer effect in effort towards the attainment of human capital measures how effort varies as exogenous attributes of a group vary. For example, well-educated parents in a neighborhood may generate positive contextual effects if youths within that neighborhood emulate parental behavior by exerting more effort in school. This paper will present results that consider the roles of peer effects on the probability that a youth leaves school (in the current year) and never returns to finish a degree
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within 6 years of the initial interview, and the probability that the youth has used drugs (in the past 30 days prior to the interview). Hypothetically, the jointly correlated unobserved factors (errors) that influence an individual’s proclivity to use drugs also influence their ability to succeed in school, implying that the two behaviors exhibit nonzero covariance. This assumption drives the basic empirical methodology employed in the paper. Results support this hypothesis, where the unobserved stochastic terms related to leaving school and recent drug-use appear positively correlated with strong statistical significance. While most previous empirical research concerning peer effects define contextual or endogenous group-level variables as averages of group behaviors or attributes, I also explore several different specifications of peer group structure. This has important ramifications for all empirical research of peer effects, since results appear quite sensitive to how researchers specify peer group structure. The remainder of the paper is organized as follows. Section 2 provides a brief overview of the various strands of relevant literature. Section 3 briefly reviews the twoequation discrete choice model with unobserved correlated disturbances across equations, while Section 4 introduces the four alternatives used to specify peer group structure and thereby weight peer effects. Section 5 covers the data and variables used in the estimations, including methodology for capturing probability weighted structures. Section 6 reviews the results, while Section 7 provides a brief conclusion. 2. Background and relevant literature The social sciences literature has addressed the emergence of behaviors from neighborhood effects through several alternative, if recently converging strands of thought. Sampson, Morenoff, and Gannon-Rowley (2002) summarize areas of focus in the sociological literature and identify four classes of neighborhood mechanisms that appear to affect the emergence of behaviors: social ties and patterns of interaction, measures of collective efficacy, localized institutional resources and patterns of land, and community resources and infrastructure. While evidence from sociological studies does not give a clear conclusion regarding specific neighborhood mechanisms, the majority indicate that concentrated poverty, and low neighborhood efficacy are linked to greater incidence of risky behavior. Each of these explanations point-towards the development of behavioral models in which peer effects, as well as the quality and quantity of community resources all matter. Recent economic models of juvenile behavior, particularly with respect to delinquent outcomes, have also framed contexts that allow for the emergence and effects of social interactions. Recently, Calvó, Petacchini, and Zenou (2006) explore the role of an individual’s centrality in education networks, while Weisberg (2006) outlines a framework for capturing endogenous group formation. Essentially in these papers, individuals in networks affect or are affected by each others’ behavior differently. Along with Akerlof and Kranton (2000), who build a model of educational attainment into a framework allowing for the utility effects of group structure, these most recent models have begun to
bridge a historical gap between economists and sociologists to model individual and group level endogenous behavior. The hypothesis proposed in this paper suggests that dropping-out of school and using drugs emerge jointly through a correlated unobserved component. For example, the same youths who fidget, hassle and generally engage in unwanted and disruptive behavior in school may more likely use drugs. Some support for this follows from a substantial body of research from the fields of criminology and psychology. Examples include Blumenstein, Farrington, and Moitra (1984) as well as Tremblay, Pihl, Vitaro, and Dobkin (1994), who in general indicate that shy children or children otherwise more risk averse are less likely to become juvenile delinquents. Other studies from Campbell (1990), Hechtman, Weiss, Perlman, and Amstel (1984), and Loney, Kramer, and Milich (1982) show that children with hyperactivity, attention deficit disorder or related ‘anti’-social behaviors during early childhood, tend to engage in crime at later stages of juvenile development. More recently Grossman, Kaestner, and Markowitz (2004), as well as Chatterji (2006) have empirically explored intrarelated teen behaviors in bivariate probit contexts. Their results indicate that anti-social behavior and possibly educational outcomes have an unobserved component that heretofore has remained empirically unexplored. A prevailing hypothesis within this research argues that the same individual traits (possibly independent of ability) that lead to anti-social behavior also increase the difficulty for children to succeed in school. Children might be excluded from standard tracking in school due to these traits find themselves placed unnecessarily in special education classes or non-placement into gifted or honors classes.1 The worst case outcomes for the worst malcontents include suspensions or expulsions for bad behavior. 3. A discrete choice empirical framework The results generate from a two-choice latent variable empirical framework, which extends from theoretical work outlined in Brock and Durlauf (2001a, 2001b).2 The binary choice specification facilitates the identification of parameters, since linear-in-means models of neighborhood effects mathematically cannot be identified (Manski, 1993).3 Individuals in this context will choose to use drugs or not, while simultaneously choosing to dropout of school or not. Implicitly, this defines latent variables for school acquired human capital, h∗i , and drug-use, di∗ as h∗i = zh,i h + h g(hj−i ) + i , di∗ = zd,i d + d g(dj−i ) + i ,
(1)
where each choice differently affects individual payoffs. Youths are heterogeneous with respect to a vector of exoge-
1
Compelling and related stories appear in Mooney (2007). Particularly under the assumption of uniform interaction weights, Brock and Durlauf (2001a, 2001b) demonstrate results for understanding the properties of self-consistent equilibria. Alternative group structure specifications, particularly Ioannides (2006) discussion of Walrasian-star interactions indicate similar results for equilibria. 3 This specification does not allow one to gauge the effects of variations in marijuana use or the comparative effects of marijuana relative to heroin. 2
D.J. Glaser / Economics of Education Review 28 (2009) 497–504
nous factors specific to each choice, zh,i and zd,i , which include individual-specific as well as contextual and correlated peer group effects. The vectors h and d represent parameters. The errors i and i are bivariate normal, with E[i ] = E[i ] = 0, Var[i ] = Var[i ] = 1 and Cov[i , i ] = . Empirical estimates follow from the observed binary vari∗ > 0]. The function ables hj−i = 1[h∗j−i > 0] and dj−i = 1[dj−i g( ) represents a weighting measure for peer effects conditional on all j individuals in the peer group who are not individual i (to be discussed in the following section). Each choice provides random utility to i according to additively separable specifications. Social utility from conforming to the peer group follows subjective expectations of peer behavior. The parameters h and d capture the weight of endogenous peer effects with respect to remaining in school and using drugs. Positive cross-partial derivatives and second derivatives in social utility functions indicate the presence of strategic complementarities and increasing marginal utility from conformist behavior, respectively. This framework subsumes all cross-behavior relationships into the covariance of the unobserved shocks such that cov(, ) = / 0.
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Nk . Connections within these small groups form edges Mij ∈ {0, 1}. Mathematically each Mij takes the value of 1 if an observed connection exists, while a 0 indicates no observed connection. Individuals cannot create connections with themselves, implying Mii = 0. Relationships are defined such that connections form an undirected graph, where Mij = Mji .4 The weighting function now appears as g(hj−i ) =
j
g(hj−i ) = 0
−1
Mij
(Mij jj−i )
if
j∈k
Mij > 0,
j∈k
if
(3) Mij = 0
j∈k
where M ≤ Nk represents the number of connections j ij in narrowly defined groups of the ith individual’s “closest” M = 0 are classified as isopeers. Individuals with j ∈ k ij lated. Isolated individuals within such a structure receive and transmit no endogenous or contextual effects.
4.2. Probability-weighted effects 4. Alternative interaction structures Typical empirical models with interactions most frequently weight peer effects using average behaviors and attributes of the group. While data availability may partially drive this, theoretical or empirical tractability also influences variable choices. This appears particularly apparent within the framework exposited by Brock and Durlauf (2001a, 2001b). Under assumptions of uniform interactions, researchers can theoretically map rational expectations behavior into readily interpretable results including the number of equilibria in the model and conditions for identification. 4.1. Average group effects The most common specifications in the literature use group-level averages for endogenous as well as contextual effects. For example, the weighting function for i’s academic outcome in a school of Nk individuals would appear as g(hj−i ) = (Nk − 1)
−1
hj−i
(2)
j∈k
This framework has readily available and interpretable theoretical equilibria, while also containing variables most easily accessible in data. While data availability may drive the use of neighborhood averages to define peer effects, a natural extension could use more narrowly defined groups. Conceivably data may reveal information about which members of a neighborhood interact with each other, capturing what Iyer and Weeks (2006) define as a small group effect. One might think of this as a group of “best friends”. To be more specific, if large group outcomes are defined from averages estimated over Nk individuals in the k th neighborhood, an individual within that group may M ≤ have “local” or “close” connections defined by j ∈ k ij
If the dominant social structure in neighborhoods appears as uniform connections between all individuals, it makes sense to use mean group variables for analyzing peer effects. The previous two structures are built on this premise of uniform interactions within peer groups, albeit describing these by different group sizes. If the nature of interactions varies either across or within groups, however, then a more general specification for group structure is appropriate. To generalize the earlier measure of peer effects into probability-weighted heterogeneous interactions, let Pij measure the probability of a connection between individuals i and j, where Pij ∈ [0, 1], Pij = Pji and Pii = 0. The Pij are known by each individual i and j. These probability weighted measures form the basis for two additional group structure specifications: connectivityweighted measures, which capture the popularity of the individual, and rank-weighted measures. The first probability weighted measure defines the strength of individuals connections within the context of the entire group. One might think of this as a measure of their overall connectivity to the group, or a measure of an individual’s popularity within the group. Individuals with stronger connections to the group defined by larger Pij for all j receive and transmit more peer effects. Group connectivity therefore defines the social utility function from the summed strength of likely peer connections. This is most similar to the Ioannides’ (2006) star-network mentioned above, with qualifications. First, each agent in the group has a different weight, rather than one principal star who attracts everyone equally. Secondly, the parameters h and d remain constant for members regardless of connectivity to others.
4 This set-up does not allow for differing the costs or benefits of connections, or whether the initiation of a connection matters more or less than its receipt.
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The weighting function for connectivity effects is defined
⎛
g(hj−i ) = ⎝
⎞⎛
Pij ⎠ ⎝
j∈k
⎞
hj−i ⎠ .
(4)
j∈k
In the rank-weighted structure, individuals weight interactions with every other individual separately and thereby rank interactions across each j in the group. Overall connectivity does not matter, but the relative weights of i placed onto each j does. Every individual has the same connection to their group, but they place greater weight on individuals with whom they most likely interact. The weight used in these estimates appears as
⎡ ⎤ (P˜ ij hj )⎦ , g(hj−i ) = ⎣ j∈k
(5)
P˜ = 1 for each i ∈ k. where P˜ ij = (Pij /( j ∈ k Pij )) and j ∈ k ij The P˜ ij capture the weighted rank of individual i’s connections to all other j’s in the group. In this measurement of peer group structure, even individuals who have very low estimated “popularity”, as defined by P , have a j ∈ k ij ranking for their interactions with all individuals in their school.5 5. Data and variables For the empirical analysis, I use The National Longitudinal Survey of Adolescent Health (Addhealth), a comprehensive and nationally representative data-set designed to capture youth behavior that also includes information gathered from parents and school administrators.6 Youths with severe learning disabilities are excluded from the analysis. The first wave of in-home interviews provides the bulk of the data used here, since it contains the largest, most comprehensive set of individuals. I use subsequent waves to establish which individuals ultimately and permanently dropped-out of school during the 1994–1995 school year, never to return.7
5
Large group average interactions discussed in Section 4.1 are a special case of rank-weighted effects if all individuals in a school share the same Pij weights. If Pij are uniform for all connections between all individuals, then rank-weighted effects of peer effects equal the average large group effect. 6 Addhealth is a program project designed by J. Richard Udry, Peter S. Bearman, and Kathleen Mullan Harris, and funded by a grant P01-HD31921 from the National Institute of Child Health and Human Development, with cooperative funding from 17 other agencies. Special acknowledgment is due Ronald R. Rindfuss and Barbara Entwisle for assistance in the original design. Persons interested in obtaining data files from Add Health should contact Add Health, Carolina Population Center, 123 W. Franklin Street, Chapel Hill, NC 27516-2524 ([email protected]). 7 Descriptive statistics are available from the author upon request. In addition to this, approximately 11% of the original sample that Addhealth follows in subsequent waves either disappear due to sample attrition or are omitted due to missing observations on explanatory variables. These missing observations tend to be from minority groups, male, more likely to have lived in single parent/adult households. They have used drugs prior to their sample disappearance at a rate 3 % higher than sample observations who are continuously tracked and used in the final regressions. These fac-
Models estimated in the paper use discrete indicators for youths dropping-out of school permanently or remaining in school for an additional year.8 When estimating specifications for large group, small group and connectivity weighted effects, the sample includes 14,892 individuals, of which 2.4% leave school.9 The matrix of friendship ties, M, draws from the first wave of in-home interviews as well as information provided in the in-school survey by each student who names their best friends. The surveys collected the friendship data ego-centrically, in that individuals answering the survey defined the presence of a link between themselves and others in the school. 5.1. Edge probabilities To estimate the Pij structures outlined in earlier sections, school networks form through a random process. More specifically, connections defined by Mij represent the edges (connections) between two individuals. These edges take the value of 1 if a connection is observed, while a 0 indicates that no observed connection exists. The combined undirected edges generate a vector of M observable connections within each of the k groups (schools) between individuals i and j of size (Nk (Nk − 1)/2) × 1. To generate the edge probabilities, define the vector A, which includes exogenous attributes such as gender, race, age, parental income, peabody achievement test scores, an indicator for whether the student ever repeated a grade in school, and indicators for whether the individuals have used drugs, alcohol or tobacco in years prior to the current sample. Following Udry and Conley (2004), differences in these variables between individuals i and j are modeled by |Ai − Aj |. Assume that Pij , the probability of connections between individuals forming in each of the k schools, differs across all schools in the sample.10 I estimate these probabilities assuming that connections Mij are random variables such that Prob {Mij = 1} = (˛k |Ai − Aj |) ≡ Pij ,
(6)
tors indicate that results explained later in the paper may understate the role of individual-specific variables on drug use. Given the unique methods used to construct the peer group structures, however, it is impossible to ascertain precisely how their omission affects parameters related to contextual and endogenous peer effects. 8 Additional analysis (not receiving discussion) extended this anti-social behavior to include all non-violent activity such as drug use, 5 or more drinks, frequent tobacco use, the deliberate damaging of property, theft and burglary. The results germane to the discussions in this paper did not notably change, vis-a-vis measuring only drug-use as an anti-social behavior. 9 At a constant dropout rate of 2.4% per year, 8th graders in 1995 have an overall graduation rate of 88.5% by 1999. After adjusting the dropout rate by grade from the data, the sample graduation rate after 4 years for an 8th grader drops to 88.3%. These sample estimates are only slightly higher than the national graduation rate of 86.5% for 18–24-year-olds in 2000. (Source: U.S. Department of Commerce, Bureau of the Census, Current Population Survey, October 2000.) Youths legally dropout of high school at the age of 16 in most states. A robustness check using a sub-sample of 9742 youths legally eligible to leave school generates no notable differences in results. 10 This allows for computational feasibility, but also produces heterogeneous measures of connections within and across schools.
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Table 1 Individual-specific marginal effects (Dropout model). Variable
Large group
Small group
Connectivity weighted
Rank weighted
Male Age (months) Age-squared Parents high school Parents income (1000s) Two-adult household Peabody score Repeated grade Prior suspension Prior drinking Prior drugs Prior tobacco Attend church Drugs in home
−9e−4 (7e−4) 0.002 ** (5e−4) −5e −6 ** (0.000) −0.004 ** (0.001) −9e−5 ** (2e−5) −0.001 † (8e−4) −1e−4 ** (3e−5) 0.004 ** (0.001) 0.006 ** (0.002) −3e−4 (7e−4) 0.002 * (0.001) 0.005 ** (0.001) −0.003 ** (7e−4) 0.001 (0.002)
−9e−4 (7e−4) 0.002 ** (5e−4) −5e −6 ** (0.000) −0.004 ** (0.001) −8e−5 ** (2e−5) −0.001 (8e−4) −6e−5 * (3e−5) 0.004 ** (0.001) 0.006 ** (0.002) −4e−4 (7e−4) 0.002 † (0.001) 0.006 ** (0.001) −0.003 ** (0.001) 0.002 (0.002)
−0.001 (7e−4) 0.002 ** (5e−4) −5e −6 ** (0.000) −0.004 ** (0.001) −9e−5 ** (2e−5) −0.001 (8e−4) −8e−5 ** (3e−5) 0.004 ** (0.001) 0.006 ** (0.002) −3e−4 (7e−4) 0.002 * (0.001) 0.005 ** (0.001) −0.003 ** (8e−4) 0.002 (0.002)
−9e−4 (7e−4) 0.002 ** (5e−4) −5e −6 ** (0.000) −0.004 ** (0.001) −9e−5 ** (2e−5) −0.001 † (8e−4) −1e−4 ** (3e−5) 0.004 ** (0.001) 0.006 ** (0.002) −2e−4 (7e−4) 0.002 * (0.001) 0.005 ** (0.001) −0.003 ** (7e−4) 0.002 (0.002)
Standard errors in parentheses. Marginal effects at Z¯ for continuous variables and 0–1 change for discrete variables. † Significance level: 10%. * Significance level: 5%. ** Significance level: 1%.
where represents the logistic distribution. Recovering the parameter vectors, ˛k , from these regressions generates an estimate of predicted probabilities for all connections between i and j.11 I generate the predicted probability of a connection using a two-step process. In the first step, all differences in A are regressed on the ((Nk − 1)N/2) possible undirected edges. Following model selection techniques using t-tests and likelihood ratio tests, extremely poorly performing variables (which differ across schools) are removed from the analysis, and the k regressions estimated again on the restricted models. Finally, parameters in these secondround regressions with p-values less than 0.10 construct the logits used to generate the predicted probability of the existence of an edge, Pij .12 6. Results The discussion follows with two-equation estimates of seemingly unrelated probits for the joint probability of leaving school during the current school year and taking illegal drugs in the 30 days prior to the interview. With some notational differences, this follows from the Greene (2003) outline for estimating two-equation discrete choice models with correlated and jointly normal error structures. Given this set-up, if > 0 then the covariance in unobserved attributes across behaviors appears positive,
11 Recovering Pij essentially amounts to the estimation of k = 130 logistic regressions (one for each school) each on (Nk × (Nk − 1)/2) conditionally independent observations. 12 While far too extensive to include all regression results here, summaries reveal the ratio of regressors which attain the previously highlighted levels of statistical significance. In 88% of all schools, age differences play a key role in the formation of friendships. Race differences appear significant in 44% of schools. Ability measured through peabody scores matters in 36% of schools, while the differences in whether individuals have repeated a grade matters in 22%. Gender matters 28% of the time, while parents’ income matters in 20% of schools. Differences in ability determined by peabody achievement scores determine connections in 36% of the schools. Prior drug use, drinking and smoking all statistically affect group formation in roughly 27%, 24% and 24% of schools, respectively.
implying that dropping-out and drug-use have a positive unobserved association.13 Tables 1, 2, 4 and 5 report estimated marginal effects as well as school-weighted robust standard errors from regressions specified with the six different neighborhood weighting mechanisms mentioned earlier in the paper. Marginal effects are calculated at the mean for continuous variables but measure the change in probability when the explanatory factor switches from a zero to a one for discrete variables. Table 3 reports likelihood ratio tests for various subsets of contextual variables. Sample sizes for large group, small group and connectivity weighted regressions are 14,982 observations.14 6.1. Individual-specific effects Table 1 shows results for individual specific variables with respect to the probability for a youth leaving school (while controlling for the unobserved correlation with the probability of recent drug use). Measures related to parental factors including parent education, income and church attendance, as well as ability proxies including peabody score and whether the student repeated a grade appear significant and robust across various specifications of weighting mechanisms. Other individual-specific controls, including previous behavior related to drug-use,
13 Estimations which include any non-violent delinquent behavior, rather than recent drug-use generated similar results in terms of statistical significance. Since the impact on results for dropout models does not change, and statistical significance of various weighting mechanisms does not change, the results from these regressions are not included. Additional regressions also tested location controls for urban, suburban and rural status. Dummy variables for school security measures also received consideration, but none demonstrated even a hint of statistical significance. The remaining marginal effects (coefficients) that appear in this paper remain statistically robust to the exclusion of the above list of controls. 14 Bivariate probit results where drug-use appears in the xh,i vector are not reported, since coefficients on these endogenous behaviors never appear statistically significant and tests indicate failure to reject the null hypothesis that = 0. These results appear for each specification of peer group structure. Quite likely this is due to identification issues from weak exclusion restrictions, similar to that reported in Chatterji (2006) and Grossman et al. (2004).
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Table 2 Individual-specific marginal effects (Drug use model). Variable
Large group
Small group
Connectivity weighted
Rank weighted
Male Age (months) Age-squared Parents high school Parents income (1000s) Two-adult household Peabody score Repeated grade Prior suspension Prior drinking Prior drugs Prior tobacco Attend church Drugs in home
0.005 (0.003) 0.005 * (0.002) −1e−5 * (1e−6) 0.004 (0.003) 3 e−5(2e−5) −0.005 (0.003) −2e−4 (1e−4) 0.002 (0.004) 0.030 ** (0.004) 0.041 ** (0.004) 0.219 ** (0.012) 0.032 ** (0.005) −0.008 * (0.004) 0.110 ** (0.018)
0.006 (0.004) 0.005 * (0.002) −1e−5 * (1e−5) 0.005 (0.003) 3 e−5(2e−5) −0.007 † (0.004) −2e−4 (1e−4) 0.001 (0.004) 0.029 ** (0.004) 0.039 ** (0.004) 0.220 ** (0.013) 0.031 ** (0.005) −0.007 * (0.003) 0.109 ** (0.017)
0.005 (0.003) 0.005 * (0.002) −1e−5 ** (1e−5) 0.005 (0.003) 3 e−5(2e−5) −0.006 † (0.004) −2e−4 (1e−4) 0.002 (0.004) 0.031 ** (0.004) 0.041 ** (0.004) 0.222 ** (0.012) 0.032 ** (0.005) −0.008 * (0.004) 0.110 ** (0.018)
0.006 † (0.003) 0.005 * (0.002) −1e−5 * (1e−5) 0.004 (0.003) 3 e−5(2e−5) −0.006 (0.004) −2e−4 (1e−4) 0.003 (0.004) 0.030 ** (0.004) 0.041 ** (0.004) 0.219 ** (0.014) 0.031 ** (0.005) −0.007 * (0.004) 0.110 ** (0.017)
Standard errors in parentheses. Marginal effects at Z¯ for continuous variables and 0–1 change for discrete variables. † Significance level: 10%. * Significance level: 5%. ** Significance level: 1%. Table 3 Correlated marginal effects. Variables
Large group
Small group
Connectivity weighted
Rank weighted
Dropout models Private school Per-capita ed. spending
−0.004 ** (0.001) −7e −6 * (0.000)
−0.004 ** (0.001) −4e −6 (0.000)
−0.004 ** (0.001) −5e −6 (0.000)
−0.005 ** (0.001) −6e −6 * (0.000)
Drug use models Private school Per-capita ed. spending
0.003 (0.008) −5e−7 (1e−5)
0.007 (0.007) −1e−5 (2e−5)
0.005 (0.006) 8e−6 (1e−5)
0.007 (0.008) 2e−7 (1e−5)
Standard errors in parentheses. * Significance level: 5%. ** Significance level: 1%.
tobacco-use, or a prior suspension also all appear robust across each specification. Included as a quadratic control variable, the effect of age also remains unchanged across different model specifications. Gender, the easy access to drugs in the home, two-adult households and prior drinking habits have essentially no statistical effect on dropout rates. Table 2 includes individual-specific marginal effects with respect to the probability of using drugs (while controlling for the probability of dropping-out). Unlike dropout models, parent-specific, and academic abilitybased controls have no statistical effect on drug-use with the exception of church attendance and the availability of drugs in the home. Not surprisingly, the probability of using drugs increases by 0.11 if youths have easy access to drugs at home. Prior behaviors all have quite strong effects as well. Prior drug use, the most important predictor, increases the probability of current drug use by 0.22. Prior suspensions, heavy drinking and tobacco use all increase the probability of recent drug use by 0.029–0.041. Similarly to dropout models, control variables for gender do not matter, while age has a statistically significant and quadratic effect .15
For comparison sake, the probability of dropping-out increases by six one-thousandths for students who have experienced a suspension in their past. The probability of drug-use increases by 0.03 for this same explanatory variable, five times larger. Marginal effects for other individual-specific variables demonstrate similar differences in economic magnitudes between the two models. In short, the results generally indicate that individualspecific factors have stronger explanatory effects on the probability of using drugs than the probability of droppingout.
15 Although results of simpler models without neighborhood effects are not reported, the individual-specific coefficients remain robust to the complete exclusion of neighborhood effects. This supports the conclusion that individual-specific coefficients in models without peer effects will
not suffer omitted variable biases related to peer outcomes and contexts. The same conclusion cannot be made with respect to correlated neighborhood variables, which appear sensitive to the exclusion of contextual and endogenous peer effects in most specifications.
6.2. Correlated effects The results shown in Table 3 for correlated neighborhood variables appear strongly dependent on specifications of peer group structure, in spite of the fact that correlated variables are not directly defined by the peer group weighting mechanism. Attendance at a private school consistently has a negative and statistically significant impact, lowering the probability of a youth dropping-out by approximately 0.004. The same variable has no statistical
D.J. Glaser / Economics of Education Review 28 (2009) 497–504
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Table 4 Likelihood ratio statistics for sets of contextual variables. Large group
Small group
Connectivity weighted
Rank weighted
Dropout models Family contextuals Ability contextuals Prior behavior contextuals
12.57 * 11.82 ** 6.87
2.32 0.52 15.28 **
4.23 3.62 4.33
13.3 ** 19.7 ** 8.64 †
Drug use models Family contextuals Ability contextuals Prior behavior contextuals
4.10 1.92 7.07
6.48 5.81 † 6.41
0.78 0.16 5.43
4.88 4.14 2.14
Sets of “contextuals” include: “Family”: parents’ h.school, parents’ inc., two adult h-holds, church and drugs in home.; “Ability”: peabody score and grade repeaters.; “Prior behavior”: suspensions, heavy drinking, drug use and tobacco use. † Significance level: 10%. * Significance level: 5%. ** Significance level: 1%.
effect on drug use. Community education spending varies in size, and subsequently statistical significance depending on weighting specifications, particularly in dropout models. It never appears to matter with respect to drug use. 6.3. Contextual effects Table 4 reports likelihood ratio tests for sets of contextual variables, grouping these into sets based on family-level contextual variables, ability-based contextual variables and prior-behavior contextual variables. Family-level contextual variables include the weighted education, income, household structure, church attendance and home-drug availability for all households in the child’s peer group. Ability-based contextual variables include weighted peabody scores and grade repeaters for the child’s peer group. Prior-behavior contextual variables include weighted prior suspensions, drug use, heavy drinking and tobacco use among other children of each child’s peer group. Several things of note arise upon inspection of these statistics. First and perhaps most surprisingly, the probability of using drugs does not appear predictable by any measure of any contextual effect. The story for dropout model specifications appears more convoluted. In rankweighted models, all types of contextual effects matter, but this seems an exception. Each specification generates different results, enough to conclude most importantly that
contextual coefficients are sensitive to the specification of peer group structure. 6.4. Endogenous effects Table 5 reports marginal endogenous effects. In dropout models, the coefficients are negative, which would indicate the presence of multiple equilibria if true (Brock & Durlauf, 2001b), but also statistically not significant indicating that endogenous effects on the probability of dropping-out of school do not exist. With respect to the probability of drug-use, however, endogenous effects appear statistically significant and quantitatively notable across all specifications. To unpack the equilibrium results of an endogenous peer-effect, I refer to the discussion in Durlauf and Cohen-Cole (2005) concerning how neighborhood endogenous effects amplify incentives and produce “social multipliers”. In the druguse model reported in Table 5 with large-group weights, school-wide average drug use generates a marginal effect of 0.149 (column 1). For example, this indicates that an exogenous decrease in the rate of school drug-use by 0.05 decreases the probability that an individual uses drugs by 0.0075. Extending this result into the limit to allow for endogenous feedbacks between all individuals in a school generates a social multiplier of (1/(1 − 0.149)) = 1.176. This social multiplier indicates that an exogenous shock to school-level drug-use, such as a raid on local drug suppliers which decreases the rate of drug use throughout the
Table 5 Endogenous effects Large group
Small group
Connectivity weighted
Rank weighted
Dropout models Marginal effect Standard error
−0.008 (0.032)
0.005 (0.004)
−0.002 (0.007)
−0.001 (0.021)
Drug use models Marginal effect Standard error
0.149 * (0.071)
0.034 ** (0.011)
0.047 * (0.019)
0.090 † (0.051)
Standard errors in parentheses. Marginal effects measured at Z¯ for continuous variables. † Significance level: 10%. * Significance level: 5%. ** Significance level: 1%.
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Table 6 Correlation in unobservables.
ˆ Standard error Sample size **
Large group
Small group
Connectivity weighted
Rank weighted
0.125 ** (0.040) 14892
0.111 ** (0.041) 14892
0.124 ** (0.038) 14892
0.122 ** (0.039) 14892
Significance level: 1%.
school by 0.05, decreases the equilibrium probability that any given individual uses drugs by approximately 0.059.16 6.5. Covariance in unobservables Table 6 includes sample sizes and estimates for . These indicate a statistically significant relationship between the stochastic components of the two behaviors, where Wald tests cannot reject the null for the jointly specified model. This lends support to the aforementioned hypothesis that maladjusted children may share an unobserved attribute for both the propensity to dropout of school and engage in anti-social behavior such as drug-use. This also indicates a need for further explorations into empirical methodologies which assume correlated multivariate error distributions. 7. Conclusion This paper compares statistical and qualitative results for school dropouts and drug use using four alternative structures of peer effects. Several key conclusions arise from reviewing the results of these different specifications. First, individual-specific coefficients remain robust, regardless of specification. Secondly, family and group-level contextual factors demonstrate effects on the probability of dropping-out of school inconsistently across the specifications of peer group structure. Contextual effects do not matter in any specification of drug-use. Regardless of peer group structure, endogenous effects appear statistically not significant for dropout models, but results indicate that using drugs have strong endogenous peer effects for large group, small group and probability weighted models. Furthermore, results from all peer group specifications indicate that correlated group effects matter with respect to the probability of dropping-out of school but not using drugs. The results also indicate further empirical support for estimating juvenile human capital decisions jointly with detrimental behaviors such as drug use. These conclusions follow from a simple two-equation specification where two outcomes (or choices) link via an unobserved parameter. This unobserved component not only affects the propensity to use drugs, but also the propensity to remain in school. A review of results that remain robust leads to policy implications related to reducing the probability of using drugs. Particularly, a focus in drug prevention efforts that relate to the individual and individual-specific factors may
16 The results cannot be compared across the other peer-group specifications, since the effects of coefficients would depend on an infinite number of probability weights for peer groups.
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