Victimization and Skill Accumulation

I. Introduction

According to psychologists, a bullying victim is a person who is repeatedly and intentionally exposed to injury or discomfort by others in an environment where an imbalance of power exists (Olweus 1997).¹ Sociologists suggest that bullying thrives in contexts where individuals need to show peer group status (Faris and Felmlee 2011). Not surprisingly, schools are the perfect setting for bullying. The combination of peer pressure, multidimensional heterogeneity of students, and juvenile lack of self-control makes schools a fertile ground for bullying. In 2013, 22 percent of U.S. students ages 12–18 reported being victimized in school (National Center for Education Statistics 2015).

Bullying is very costly. Eleven percent of urban American children miss school every day because of fear of being victimized (Kann et al. 2014). One of every ten students drops out or changes school because of it (Baron 2016). Homicide perpetrators are twice as likely as homicide victims to have been bullied previously by their peers (Gunnison, Bernat, and Goodstein 2016). Victims are between two and nine times more likely to consider suicide than nonvictims (Kim and Leventhal 2008; Kim et al. 2009). Surprisingly, economic literature has remained mostly silent on the topic. We know very little about its intermediate costs and long-term consequences. I contribute to bridging this gap by exploring the two-way relation between bullying and cognitive and noncognitive skills accumulation²—namely, how school bullying hampers the development of successful adults by impeding skill accumulation and the extent to which cognitive and noncognitive skills are themselves determinants of in-school victimization.³ To analyze this two-way relation, I extend the theoretical contributions of Cunha, Heckman, and Schennach (2010) to include peer-influenced events—like bullying—in the skills accumulation process. I allow future skills to depend on current skills, current investment choices, and victimization. I allow the likelihood of the bullying event to depend on own and peer observable and unobservable characteristics. Thus, I treat bullying as an event that may deplete the existing stock of skills, changing negatively the skill accumulation path for the people involved.

The model incorporates several desirable features. First, it acknowledges that social interactions like bullying are endogenous. Hence, the “treatment” is not randomly allocated across students. The way own characteristics relate to those of peers is critical in building up the social arena that determines victimization. Second, it recognizes that cognitive and noncognitive skill measures observed by the researcher are only approximations or functions of the true latent skills (Carneiro, Hansen, and Heckman 2003; Heckman, Stixrud, and Urzua 2006). Third, it does not assume that those true latent skills follow a normal distribution, thus guaranteeing the flexibility required to recreate appropriately the unobserved distributions in the estimation. Finally, the model allows me to simulate counterfactuals for each skill level, which I use to calculate the divergence in skill accumulation paths caused by bullying.

This paper contributes to the economic literature in several ways. First, it extends the literature on dynamic skill accumulation by introducing peer-influenced events as essential drivers of the process. In particular, it considers the role endogenous peer victimization has on skill formation. Second, it analyzes the consequences of bullying in school in terms of skill depletion. Third, it extends my previous work on school bullying (Sarzosa and Urzua 2021), where I found sizable consequences borne during adulthood, by providing additional insight into the channels through which high school bullying affects adult outcomes. Fourth, it allows the quantification of the long-run cost of bullying. That is, I can go beyond short- and medium-term outcomes, like school absenteeism or young adult health, and estimate skill endowments losses for life. In addition, this will open an auspicious research agenda on skill accumulation and negative social interactions.

Using detailed longitudinal data on a cohort of South Korean students, I find that kids with low initial stocks of skills and those who have uncommon traits among their peers are significantly more likely to be bullied. I also find that victimization depletes current skill levels and makes individuals more prone to experiencing bullying again in the future, creating a self-reinforcing mechanism that generates an enormous burden they will carry during adulthood.

In the following, after reviewing the scarce economic literature on the subject in Section II, I present the basic framework for the analysis of skill dynamics in Section III. Section IV defines the empirical strategy I use. Section V describes the data I use for the analysis and how I construct the cognitive and noncognitive skill measures. Section VI presents results. Section VII focuses on how, in light of my results, some policies regarding students allocation to school can reduce school bullying. Finally, Section VIII concludes.

II. Related Literature

Although economic literature on the consequences disruptive classmates have on their peers has grown in recent years (Carrell and Hoekstra 2010, 2012; Carrell, Hoekstra, and Kuka 2018), economic research on bullying is scarce.⁴ Two main reasons explain this sparseness: (i) a lack of adequate data and (ii) the nonrandomness of the selection into bullying. Regarding the former, there is little longitudinal data that inquire about bullying, so researchers can use very few sources that observe individuals before and after the event. Regarding the latter, bullying’s nonrandomness confounds the consequences of bullying with the intrinsic characteristics that made the person a victim in the first place. The scarce existing economic literature has focused on quantifying of the effects bullying has on short- and medium-term outcomes. Brown and Taylor (2008) find that being bullied and being a bully are correlated with lower educational attainment in the UK. Eriksen, Nielsen, and Simonsen (2014) find that being bullied decreases ninth grade marks among Danish students. They find causal estimates by instrumenting victimization with the proportion of classroom peers whose parents have a criminal conviction. Sarzosa and Urzua (2021) embed a similar empirical strategy (that is, instrumenting victimization with the proportion of classroom peers that come from violent families and the proportion who claim to be bullies) in a framework of unobserved heterogeneity in terms of cognitive and noncognitive skills. They find that bullying increases the probability of smoking and the likelihood of feeling sick, depressed, stressed, and unsatisfied with life during adulthood. It also reduces college enrollment and increases the dislike of school. Interestingly, they find that the detrimental effects of bullying are greater for people with low levels of noncognitive skills.

I contribute to the literature of bullying by building on Sarzosa and Urzua (2021) and providing a possible explanation for the impacts they observe. While Sarzosa and Urzua (2021) estimate the average treatment effect (ATE) of middle school bullying on young adult outcomes, in the present study, I elucidate one mechanism behind the creation of these gaps, showing bullying as the triggering event that determines a divergence in skill accumulation paths. In the present paper, I show that the gaps between victims and nonvictims open up early in life by embarking victims in skill accumulation paths lower than those they would have been on in the absence of victimization. In consequence, the identification strategies in the two papers differ significantly. While Sarzosa and Urzua (2021) estimates a static Roy model with unobserved heterogeneity, here I estimate a model of skill formation. Although both papers use the allocation of students to classrooms as an exogenous variation for identification, in the present study, I use it as an exogenous shifter of how uncommon students’ traits are in the pool of traits available in the classroom.

This work also relates to recent contributions in skill development that have established that skills are dynamic and malleable (Cunha et al. 2006; Cunha, Heckman, and Schennach 2010). They depend on their past levels. They can be hindered, and they can be fostered. We know that skills beget skills, and, therefore, initial skill endowments and early accumulation are critical for the lifetime stock of skills (Cunha et al. 2006). Skills beget skills through the natural process of getting the stock available at time t to t + 1 and through investment. That is, skilled kids receive more skill investment and have higher returns to those investments than less-skilled kids (Skinner and Belmont 1993; Aizer and Cunha 2012; Espinoza, Sarzosa, and Urzua 2014).⁵ This self-reinforcing mechanism increases skill inequality as children age—those who start their childhood with high initial levels of skills accumulate skills three times faster than those who start their development with meager stocks of skills (Agostinelli and Wiswall 2016a). These dynamics give foundation to the call for early childhood development and preschool interventions (Knudsen et al. 2006; Doyle et al. 2009).

The claim that skills are malleable is backed up by a series of papers that show that skill-developing interventions modified the stock of skills of the treated population. For instance, the people treated by Perry Preschool Program have higher noncognitive skills—although similar levels of cognitive skills—than the controls (Heckman et al. 2010). The Socio-Emotional Learning programs have been widely reviewed as successful interventions that develop noncognitive skills such as goal setting, conflict resolution, and decision-making (Payton et al. 2008). Skill-developing interventions can compensate for low initial levels of both cognitive and noncognitive skills (Cunha, Heckman, and Schennach 2010). Furthermore, extensive literature finds that family background influences skill accumulation. Children whose parents are more engaged in their upbringing are likely to have higher skill levels.⁶ However, evidence shows that there are windows of opportunity outside of which skill malleability is lost (Knudsen et al. 2006) and that such windows close earlier for cognitive than for noncognitive skills (Cunha, Heckman, and Schennach 2006).

Besides the dynamism and malleability features of skills, recent literature has found that they strongly depend on the contexts in which the child grows.⁷ For instance, skill endowments depend on the level of stress a person was exposed to during childhood (McEwen and Seeman 2006) and the quality of school inputs, such as class size and teacher characteristics (Fredriksson, Ockert, and Oosterbeek 2013; Jackson 2013). One such context is the type of social interactions the child encounters in school. This paper includes interactions with peers as critical inputs in the development of own skills.

III. Skill Formation and Bullying

My framework needs to incorporate five facts that emanate from the skill formation literature: (i) skills beget skills, (ii) skill development can be affected by investment choices, (iii) past skills levels can affect next period skills indirectly by inducing skill investment, (iv) bullying (framed as a negative investment) can hamper skill development, and (v) bullying victimization depends also on the stock of cognitive and noncognitive skills of each person and those of their peers. Therefore, I propose to augment the dynamic structure in Cunha, Heckman, and Schennach (2010) to incorporate these five facts explicitly. Thus, I adopt the timeline proposed in Cunha, Heckman, and Schennach (2010), where student i who belongs to classroom c ∈ C starts the process with an initial stock of skills of type S ∈ {A, B}, θ_S,i∈c,τ, with A denoting noncognitive skills and B denoting cognitive skills. Parents observe θ_A,i∈c,τ and θ_B,i∈c,τ, receive an idiosyncratic investment-related shock ɛ_S,i∈c,τ+1, and decide I_S,i∈c,τ+1, the amount to invest between period τ and τ + 1 in each skill dimension. Analogously, θ_A,i∈c,τ and θ_B,i∈c,τ together with classroom characteristics determine the victimization that may occur between the skill measurements at τ and τ + 1, which I label M_i∈c,τ+1. Hence, in general, parental investment and victimization at a given moment in time t ∈ [τ, T] are simultaneous. Parents do not observe victimization at the time they decide their skill investment strategy. This responds to the ample evidence indicating that parents are not usually aware of their children’s victimization (deLara 2012; Waasdorp and Bradshaw 2015; Bjereld, Daneback, and Petzold 2017; Larranaga, Yubero, and Navarro 2018).⁸ Parental investment made between times t and t + 1 respond, however, indirectly to past victimization. That is, through the effect bullying that occurred between t − 1 and t has on the stock of skills at t. This relies on the results of psychological research that indicates that responsive and supporting parenting practices are related with lower levels of bullying (Flouri and Buchanan 2002). In particular, certain parental behaviors that hamper the development of locus of control on kids have been linked with in-school victimization (Ladd and Ladd 1998).

Skills at τ + 1 are, thus, the product of the initial stock of skills θ_A,i∈c,τ and θ_B,i∈c,τ, the investment decisions and the victimization that took place between τ and τ + 1 I_S,i∈c,τ+1 and M_i∈c,τ+1 and the realization of a skill production shock η_i∈c,τ+1. See a diagram with the timeline of the model’s typical two-period cycle in Online Appendix Figure B.1.

IV. Empirical Strategy

B. Estimation

1. Identification and estimation steps

As shown in Online Appendix C, we can use Equation 3 to identify , and similarly, Equations 5 and 6 to identify and . Also, we can use Equation 4 to identify and consistently estimate and . In consequence, I am able to construct the vector 7

Taking advantage of the orthogonality and mutually independence between , , and η_t and of the nonlinearity of the skills production functions, I substitute them from Equation 1 into the measurement system for . For the sake of brevity, let me call the production function of skill S at time t + 1 for those whose victimization condition is M ∈ {0, 1}. Then, I can write Equation 7 as 8

which together with the victimization Equation 9 (that is, the empirical version of Equation 2) 9

builds a Roy-like potential outcomes model that endogenizes the bullying “treatment” and allows me to estimate treatment effects of victimization on skill formation (Heckman and Vytlacil 2007). Formally:

Equation 9 collects the facts that victimization not only depends on the potential victim’s characteristics, but also on the social arena the student faces (that is, the traits that classmates bring to the group). As explained in Section III, I introduce this feature in the model by creating a measure of how uncommon the traits of a given student are among their classmates. To identify empirically such social process, I require that the allocation of students—and therefore their traits—to classrooms be as good as random. That way, the social arena each student faces is random, and therefore the differences in the probability of being victimized given one’s traits depends on the differences in the traits’ distributions across classrooms. In the same way and as additional exclusion restrictions for the identification of Equation 9, I follow Sarzosa and Urzua (2021) and introduce two additional traits of the social arena of the classroom: the proportion of peers that report being bullies in the class and the proportion of peers in the classroom that come from a violent family.¹⁴

The measurement system requires several considerations. First, note that is a compounded error term, with and , whose diagonal elements are of the form , and its off-diagonal elements are of the form . It is straightforward to see that is identified from the fact that and are known from the first stage. Hence, I am effectively reducing the dimensionality of the computational task of estimating the model. It is now a four-dimensional unobserved heterogeneity problem: two dimensions of skills at t and the investment latent factor for each skill. Second, identification of this potential outcomes model and its associated treatment parameters requires that (a modified version of assumption (A-1c) in Heckman, Humphries, and Veramendi (2016). This assumption implies two underlying assumptions: and . The former is a mild assumption, as its violation would require a very unique kind of shock—one that jointly shifts the chances of victimization and scores recorded by the cognitive tests and noncognitive measures in t + 1, but does not affect the stock of skills at that moment in time. The latter assumption, which I alluded to in Section III.C, maintains independence of the shocks to the chances of victimization and the part of the variation in the latent variable θ_S,t+1 not explained by θ_A,t, θ_A,t, and I_S,t+1. Therefore, shocks that simultaneously alter the “sorting into victimization” and period t + 1 skills without going through are considered threats to the identification of the model. To deal with this concern, I estimated a version of the model that includes numerous shocks that could have the potential of violating the identifying assumption (that is, death of parent, parent failed in business, parent lost job, parent was hospitalized) both in the victimization Equation 9 and in the potential outcome Equation 8 as an additional observable control. Table 8 of the Web Appendix shows that although the shocks are significant determinants of victimization, the parameters of the production function remain unaltered. The fact that the results are robust to the introduction of the shocks illustrates that their scope is too small to be meaningful.

2. Overall mean shifts and the identification of the CES function

As in Cunha, Heckman, and Schennach (2010), my estimated factors’ distributions are centered at zero. In particular, E[θ_S,t] = E[θ_S,t+1] = 0 for S ∈ {A, B}. These normalizations are at odds with the fact that shifts with changes in ρ, as shown in Figure 1. It simulates 1,440 different combinations of γ₁, γ₂, and ρ to generate , where x, y, and z come from 5,000 random draws from independent normal distributions. Figure 1 evidences that estimating a model that fits = 0 greatly constrains the set of possible values that can take, and its combinations with the other parameters in the CES function. In other words, the normalizations limit the families of functions that can be estimated. This is consistent with the argument in Agostinelli and Wiswall (2016b), who point out that the normalizations bias the estimations towards finding a functional form consistent with a Cobb–Douglas. One way to fix this is to depart from the estimation procedure put forth by Cunha, Heckman, and Schennach (2010), as proposed by Agostinelli and Wiswall (2016a). Another way is to use the fact that the relation between and the CES parameters is predictable, as evidenced by Figure 1. In fact, Online Appendix Table D.1 shows that a flexible cubic polynomial in the CES parameters (that is, P₃(γ₁, γ₂, ρ)) captures 99.98 percent of the variation of . Hence, in order to avoid = 0 constraining the possible values of the CES parameters, I use P₃(γ_A, γ_B, ρ_∙) as a shifter of the mean of during estimation. That way, it counters the mean-shifting that mechanically occurs when ≠ 0. In practice, I am allowing .

Figure 1

Relation between the Mean of and ρ

Notes: The plotted are the results of 1,440 different combinations of γ₁, γ₂, and ρ parameters in the CES production function = [γ₁x^ρ + γ₂y^ρ + (1 − γ₁ − γ₂)z^ρ]^1/ρ, where x, y, and z come from 5,000 random draws from independent normal distributions.

The second implication of the normalizations is that the parameters estimated from Equation 1 will not respond to the overall mean changes in skills. However, given that I am comparing the skill trajectories of victims with those of nonvictims, being unable to measure overall mean shifts directly is an innocuous feature of the empirical strategy.¹⁵

V. Data and Institutional Context

I empirically estimate the described model using the Junior High School Panel (JHSP) of the Korean Youth Panel Survey (KYP). This choice is motivated by two main reasons: South Korea’s framework for allocating students to classrooms and critical data features available in the KYP-JHSP.

As explained in Section III, identification relies on the exogenous assignment of classmates. South Korea’s educational setting allows for that, thanks to a 1969 “leveling policy” regulating student placement. The law “requires that elementary school graduates be randomly (by lottery) assigned to middle schools—either public or private—in the relevant residence-based school district” (Kang 2007). The leveling policy also makes the grouping of students by ability and achievement levels “extremely rare.” Therefore, the “non-grouping (or ability mixing) in school exposes students to a classroom peer group that is nearly exogenously and randomly determined” (Kang 2007). Furthermore, the reader should note that, unlike in the United States, middle school students in South Korea have a fixed classroom—and hence, classmates—for all subjects. On top of this distinctive institutional feature, I take advantage of the fact that the KYP-JHSP has a sampling scheme that is critical for identifying the peer interactions that fuel the model. The data consist of a nationally representative sample of a cohort of middle-schoolers interviewed for the first time in 2003 when they were 14 years old. The importance of the sampling scheme relies on the fact that its sampling unit is the entire classroom. Hence, the KYP-JHSP permits a thorough inspection of the complete distribution of traits available in the classroom, a critical feature for identifying Equation 2. The panel consists of 3,449 youths (see descriptive statistics in Table 1). Subjects were consistently interviewed in six waves, one each year.¹⁷ During each wave, information was collected in two separate questionnaires: one for the teenager and another for the parents or guardians.

Another critical feature of the KYP-JHSP regarding this study is that it collects very detailed information on personality traits and behavioral responses through a comprehensive battery of personality questions consistent across waves. The KYP-JHSP inquires about academic performance, student effort, and participation in different kinds of private tutoring. The survey also asks about time allocation, leisure activities, social relations, attachment to friends and family, participation in deviant activities, and victimization in different settings, including bullying. While the survey often asks the children about their parents’ involvement in many aspects of their lives, parents and guardians answer only a short questionnaire covering household composition and their education, occupation, and income.

As with all other personal characteristics collected in the KYP-JHSP, bullying is self-reported by the students. It refers to events where they have been severely teased or bantered, threatened, collectively harassed, severely beaten, or robbed during the last year. Given that the KYP-JHSP collects its data during late November and the Korean school year runs from March to December, one can interpret the question as asking for bullying events during the school year that is about to end.

Even though psychologists define bullying to include more than physical violence (see its definition in the introduction above), due to the wording of the question in the KYP-JHSP, the kids in the study respond to its most direct and less subtle versions of bullying.¹⁸ This way of reporting about bullying is in line with the findings in several international studies that find that children “focus on the more obvious and less subtle forms of bullying such as direct verbal and physical abuse and overlook indirect aggression” (Naylor et al. 2010).¹⁹ In the same way, the reported incidence of bullying in the KYP-JHSP, presented in Table 1, is in line with other nationally representative studies (Kim, Koh, and Leventhal 2004) and with the incidence—and its year-to-year decline—reported in international studies (OECD 2017; Scheithauer et al. 2006; Ryoo, Wang, and Swearer 2015). Furthermore, it closely mirrors the victimization incidence found in the United States by the School Crime Supplement of the National Crime Victimization Survey (National Center for Education Statistics 2015). For this study, I use the bullying measured in Waves 1, 2, and 3.²⁰

Data and institutional requirements aside, it is worth noting that—as in the United States and many other countries globally—bullying is a critical issue in the South Korean society, usually characterized by ultra-competitive academic environments that praise scholastic achievement.²¹ Not surprisingly, such environments foster unhappiness and aggressiveness in the classrooms, a fertile ground for bullying. Given the link between bullying and suicides (Kim and Leventhal 2008; Kim et al. 2009), and the striking suicide rate among young people in South Korea,²² the government has deployed active policies aimed at curbing these phenomena.²³

VI. Results²⁸

As explained in Section IV.B, estimation was divided into two stages.²⁹ For that reason, I used the limited information maximum likelihood (LIML) technique to correct the second stage’s standard errors (Greene 2000). Common controls to all the equations in the structural model were age, gender, family composition (number of older and younger siblings, urban status, broken home status, father’s education), and per capita household monthly income. Below, I will explain the additional variables specific to each equation, that is, exclusion restrictions.

A. Model Fit

In the first step, I estimate the initial distribution of skills (age 14) from Model 10, which incorporates the structure proposed by Hansen, Heckman, and Mullen (2004) to address the possible problems of joint causality. Figure 1 in the Web Appendix presents the estimated initial distribution of skills. As expected noncognitive and cognitive skills are positively correlated: corr (θ_A,t, θ_B,t) = 0.450.

Table 2 and Figure 2 show that the model fits the actual data extremely well. The former shows that the model matches the incidence of bullying almost exactly and that the means and standard deviations of the simulated scores are very close to the ones obtained from the actual cognitive and noncognitive measures for each victimization state (that is, bullied or not bullied). I cannot reject the null of equality of means in any of the 12 cases.

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Table 2

Goodness-of-Fit of the Model

Figure 2

Actual versus Predicted Scores Cumulative Distributions Conditional on Victimization at t = 1

Notes: Actual (diamond) and predicted (line) cumulative distributions plotted of the following manifest variables: (Panel A) locus of control, (Panel B) irresponsibility, (Panel C) self-esteem, (Panel D) language and social studies, (Panel E) math and sciences, and (Panel F) year exam. The predicted values come from simulations based on the estimated parameters of the model.

Figure 2 plots the predicted values of the manifest variables provided by the model against the actual CDF of each cognitive test or noncognitive measurement observed in the data. The figures show a remarkable fit in all scores, regardless of the victimization condition. I corroborate this by performing a Kolmogorov–Smirnov (K-S) test on the predicted and actual measurement distributions. The results (Table 2) indicate that the predicted scores come from a distribution that is not different from the one the actual scores describe in ten out of the 12 comparisons. The only measurement for which I fail the K-S test is self-esteem, even though the model closely matches its first and second moments. I suspect this is the case because the actual self-esteem distribution has a kink or jump close to the median, which is difficult to fit with smooth and continuous latent factors. When I smooth the observed distributions of self-esteem using kernel approximations, the K-S test statistics get closer to the nonrejection values.

B. Results from the Model of Skill Formation

1. Incidence of victimization

Column 1 in Table 3 shows the relation between skills and selection into bullying. Kids with less noncognitive skills are significantly more likely to be bullied. A one standard deviation decrease in noncognitive skills increases the likelihood of being victimized by 2.26 percentage points (, where ). It represents an increase in the probability of being victimized by about one-fifth. Column 1 in Table 3 also shows the importance of the relation between own and peer characteristics has in determining peer victimization. Controlling for their observable characteristics and skill levels, kids placed in a school in which their noncognitive skills are uncommon are significantly more likely to be bullied. The results indicate that the average student’s victimization likelihood drops by one percentage point with each additional classmate with similar noncognitive skill endowments. Interestingly, uncommonness in terms of income also encourages victimization. Bullying probability falls by half a percentage point for each additional classmate with a family income level similar to the one of the prospective victim. These results are in line with the psychological literature that links victimization with those considered weird or unlikeable (for example, Hodges, Malone, and Perry 1997) and remarkably robust to the inclusion of the percentage of classmates that come from troubled families and the percentage of bullies in the classroom.³⁰

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Table 3

Estimating the Model of Skill Formation

The fact that the model relies on identifying unobserved heterogeneity allows me to quantify the victimization probability for the average student and for every combination of skills at a given point in time. Figure 3 shows striking differences in the likelihood of being bullied depending on the level of noncognitive skills. Kids in the first decile of noncognitive skills are twice more likely to be bullied than those in the tenth decile and are 36 percent more likely to be bullied than the average student. In addition, Figure 3 shows that among those with low noncognitive skills, the ones with higher cognitive skills are three percentage points more likely to be victimized than those at the bottom of the cognitive skill distribution. These results reflect the widely held notion that socially awkward smart children face greater chances of being victimized in school.

Figure 3

Probability of Being Bullied

Notes: Results based on 40,000 simulations based on the estimated parameters of the model of skill formation.

2. Skills production

Columns 4–7 in Table 3 present the results of estimating the system described by Equation 1. They contain the parameters that, together with the ones related to selection into bullying and the distributions of the unobserved heterogeneity, govern the process of skill formation between ages 14 and 15.³¹ Two main results stand out—the massive importance of self-productivity and the relatively low productivity of parental investments in skill development.

Self-productivity. Figures 4A and 4B show that high noncognitive skills produce high future noncognitive skills and that marginal increments of those initial skills are very productive (that is, noncognitive skills self-productivity ∂θ_A,t+1/∂θ_A,t > 0 for the entire (θ_A,t, θ_B,t) space). Table 3 shows that the noncognitive skills’ input shares in the production of future cognitive skills amount to 0.904 and 0.952, depending on the victimization status. These results align well with the estimates found in existing literature.³² Cunha, Heckman, and Schennach (2010) report that input share to be 0.868 among white American children between the ages of 7 and 13. Likewise, Figures 4C and 4D show that cognitive skills production relies heavily on past levels of cognitive skills. My estimates indicate that the cognitive skills’ input shares in the production of future cognitive skills among Korean adolescents is 0.838 and 0.864, depending on the victimization status. Cunha, Heckman, and Schennach (2010) and Agostinelli and Wiswall (2016a) report that input share to be 0.902 and 0.910 among America preadolescents.

Figure 4

θ_{S,t +1} as a Function of θ_A,t and θ_B,t

Notes: Results based on 40,000 simulations based on the estimated parameters of the model of skill formation.

Cross-productivity. Figures 4A and 4B also demonstrate that cognitive skills are unimportant in the noncognitive skill production process, except that higher initial cognitive skills make the marginal increments of the initial noncognitive skills more productive (that is, ∂²θ_A,t+1/∂θ_A,t∂θ_B,t > 0). Likewise, Figures 4C and 4D show that although the existing levels of noncognitive skills contribute to the cognitive skills’ production process, their contribution is small compared to that of the existing cognitive skills stock. For instance, going from decile one to decile ten in the noncognitive skills distribution has the same effect on the production of cognitive skills as increasing the cognitive skills input by one decile.

Productivity of investment. Table 3 indicates parental investments are relatively unimpactful in the production of skills.³³ The parental investment’s input shares range between 0.03 and 0.057. These meager input shares among older children are also found in Cunha, Heckman, and Schennach (2010) (0.02 and 0.055 in the cognitive and noncognitive skills’ production functions) and Agostinelli and Wiswall (2016a) (0.087 in the production of the cognitive skills).

My results indicate a strong path dependence in which skills produce skills, setting a high cost in terms of future stock of skills for those who start the accumulation process in the lower quantiles of the skill distribution. My results also show that investment choices do not reverse this path dependence. Columns 2 and 3 in Table 3 show that investment choices in noncognitive skills depend greatly on the past level of noncognitive skills, and investment choices in cognitive skills depend greatly on past levels of that skill in the first place. Hence, people with high skills pass their high stock on to the next period and are more prone to invest in their development.³⁴

3. Effects of bullying on skill production and future bullying

Table 4 shows the effect of bullying on the accumulation of cognitive and noncognitive skills. To calculate this, I compare the next period skills of those who would be selected into bullying with those who would not, given a particular level of period skills. That is, . In Table 4, I present two summarizing estimates of the effect bullying has on skill accumulation. First, I present the mean average treatment effect: , where I aggregate the treatment affects across all levels of period t skills. Second, I show the average treatment effect for the average student:

for S ∈ {A, B}, where represents skill S mean. I find that, on average, bullying impedes noncognitive skills accumulation by −0.249. That is equivalent to a reduction in noncognitive skill accumulation of 39.9 percent of a standard deviation, a sizable effect. It implies a reduction of 33.6 percent of a standard deviation in the language test score and a reduction of 28.9 percent of a standard deviation in the math test score. These skill losses imply that the average kid would be 19 percentage points more likely to report being sick recently, 5.5 percentage points more likely to smoke, and 10.5 percentage points more likely to drink alcoholic beverages. The stock of skills lost also translates to setbacks in mental health. They equate to increases of 48.77 percent of a standard deviation in the depression symptom scale, 38.1 percent of a standard deviation in the levels of stress caused by insecurities regarding their image, and a third of a standard deviation in the levels of stress caused by issues regarding school.³⁵

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Table 4

ATE of Being Bullied on Next Period Skills

The same estimation shows there is no statistically significant effect of bullying on cognitive skill accumulation. These results indicate that, as expected, bullying is much more costly in the noncognitive dimension than in the cognitive one. Although victims might skip school, their learning ability is not affected as gravely as their ability to self-regulate, overcome obstacles, see themselves positively, or relate with others. Note that even if cognitive skills are unaffected, grades drop because of the effect noncognitive skills have on them. Note that the fact that bullying does not affect the accumulation of cognitive skills could be due to the nature of the victimization itself or that cognitive skills are less malleable than noncognitive skills during adolescence (Walsh 2004; Kautz et al. 2014). Then, as a robustness check, I estimate a version of the model in which cognitive skills are allowed to evolve but are not subject to the effects of victimization. The results collected in Web Appendix 3.2 show that the impact of victimization on noncognitive skill accumulation remains unchanged.

Figure 5A presents the effect of bullying on the next period noncognitive skills for each initial skills level. It shows that the kids who suffer the greatest negative impact come into the process with low stocks of skills. Victims with low levels of skills lose almost half of a standard deviation of noncognitive skills, while victims with high stocks of skills lose a third of a standard deviation. In particular, those who start with low cognitive skills face harsher consequences. However, due to the positive correlation between cognitive and noncognitive skills, those with low cognitive skills are very likely to be those with low levels of noncognitive ones. Such treatment effect heterogeneity based on the initial levels of skills and the fact that victimization also depends on them yield a very interesting result—kids with low initial levels of skills are not only more likely to be bullied, but also its consequences are stronger on them.³⁶ Table 5 attests to that. It shows how bullying shifts students to lower deciles of the next period skills distribution. It shows that if students from the lower 40 percent of the noncognitive skill distribution at t were to be victimized, they would end up belonging to the lowest noncognitive skills decile at t + 1. Furthermore, if students from the bottom 80 percent of the noncognitive skill distribution at t were to be victimized, they would end up belonging to the lowest half of the noncognitive skills distribution in the next period. Notably, those who start with abundant stocks of skills fall closer to their original place in the skills distribution. Victims from the top decile at t end up in the ninth decile at t + 1.

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Table 5

Decile of θ_A,t+1 and Victimization Probability in t + 2 that Students Would End up Facing if Victimized in t + 1, by Skills Decile in t

Figure 5

E[θ_{S,t + 1}|θ_A,t, θ_B,t, M_t+1 = 1] − E[θ_{S,t + 1}|θ_A,t, θ_B,t, M_t+1 = 0]

Notes: Results based on 40,000 simulations based on the estimated parameters of the model of skill formation.

Such skill depletion between t and t + 1 due to bullying increases the chances of being bullied again in t + 2. The bottom row of Table 5 shows evidence of that. The likelihood of being bullied in t + 2 for those bullied in t + 1 exceeds 4.62—the unconditional probability of being victimized at that period. Using the dynamic features of my model, I can calculate the ATE of prior bullying on the chances of being victimized again. To do so, the model exploits two facts: (i) that victimization in t + 2 depends on t + 1 skills, as indicated in Figure 6A and (ii) that those victimized have their skills t + 1 depleted. In consequence, I find that those bullied in t + 1 are, on average, 1.65 percentage points more likely to be bullied again next period. That effect is not only statistically significant but economically meaningful. It represents a massive 34.6 percent increase relative to the overall victimization incidence in t + 2. When disaggregating the effect by the initial level of skills, Figure 6B shows that the effect is significant for the students who start the process with relatively low skills. For instance, for students whose initial skill endowments place them in the first decile of the noncognitive skill distribution, being bullied in t + 1 increases the probability of being victimized in t + 2 by 2.04 percentage points. That amounts to a 42.8 percent increase relative to the overall victimization incidence in t + 2.

Figure 6

Victimization in t + 2

Notes: Results based on 40,000 simulations based on the estimated parameters of the model of skill formation. Panel A presents the probability of being bullied in t + 2 by skill levels at t + 1. Panel B presents how the probability of being victimized at t + 2 changes due to having been victimized in t + 1 for every initial level of noncognitive skills. Namely, E[M_t+2|M_t+1 = 1, θ_A,t] − E[M_t+2|M_t+1 = 0, θ_A,t]. The spikes represent the 90 percent confidence intervals.

The channel through which these effects materialize is, of course, skill depletion. Low noncognitive-skilled students are more likely to be bullied in t + 1. Due to that, they accumulate fewer noncognitive skills during that period relative to what they would have if they had not been bullied. Now, with substantially less noncognitive skills, they face a higher probability of being victimized in t + 2. These results show the importance of the model’s dynamics. Even though the overall incidence of bullying drops dramatically from year to year, victimization becomes more selective, as described in the psychological literature by Nylund et al. (2007) and Reijntjes et al. (2010). Those who end up being bullied are most likely those who were bullied before.

All the evidence presented in this paper confirms the existence of a self-reinforcing mechanism: kids who start the process with low levels of skills are more likely to be bullied and thus have their stock of skills depleted. These forces send them in a downward spiral by making them even more at risk of being victims of bullying in the future.³⁷ Subsequent bullying events will be much more harmful, preventing them from acquiring the noncognitive skills they lack.

4. Complementarities

As explained in Section III, an essential feature of the model is that it allows the analysis of complementarities between skills and bullying. Namely, the measurement of how much a marginal change in previous period skills modifies bullying’s effect; that is,

According to Figure 7A, marginally increasing the initial levels of noncognitive skills will result in small reductions in bullying’s negative effect on future period skills. This result attests to the fact that the impact of bullying is relatively constant across the entire noncognitive skills distribution. The palliation of the negative effect due to a marginal increase in noncognitive skills is larger for those with above the mean initial noncognitive skills.

Figure 7

Static Complementarity

Notes: Results based on 40,000 simulations based on the estimated parameters of the model of skill formation. The scatter plot presents the static complementarity measures at 750 points along the skill distributions. The line represents a local polynomial approximation.

On the other hand, Figure 7B shows that marginal increases in initial levels of cognitive skills have larger effects in palliating the negative effect of bullying on noncognitive skills. In fact, those adverse effects would shrink by four percentage points or 16 percent for the average kid. For those in the sixth and seventh decile, the palliation effect is even larger, reducing the negative effect of bullying on noncognitive skills by about a fourth.

Even though I showed that investment in skills during middle school years is often unproductive, the static complementarity results suggest that even a tiny bit of skill accumulation during earlier years would have an immense impact not only in deterring bullying but also in lessening its consequences among those that are more at risk.

VII. Policy Implications

Several antibullying campaigns have been deployed all around the world in an ambitious effort to eliminate this unwanted phenomenon.³⁸ My findings indicate there are at least two fronts on which policymakers can work. First is the development of noncognitive skills. Noncognitive-skilled kids will be less likely to be victimized. Moreover, if they happen to be bullied, the impact on their skill accumulation path is much lessened. The strong dependence of current skill levels on past skill levels heightens the importance of developing noncognitive skills at young ages.

The second implication of my results relates to classroom assignment. Column 1 in Table 3 shows that, given skill levels and observable characteristics, children with uncommon traits are more likely to be targeted by bullies. This finding leads to a policy-relevant question: to what extent can allocating children to more homogenous classrooms deter victimization? To answer this, I simulate the model with an extreme—unfeasible in practice—mechanism of allocating students to classrooms, Consider it a benchmark scenario that places students in classrooms with kids with similar stocks of noncognitive skills, as measured by the self-esteem score. This exercise ignores geographical distances. It sorts the universe of students with respect to their self-esteem scores and splits them into classrooms according to the typical classroom size in South Korea.

Figure 8 presents the results of these simulations. As in Figure 3, it plots the likelihood of being bullied for every skill level. A comparison between these two figures shows the massive impact of reducing in-classroom noncognitive skill heterogeneity on the likelihood of being victimized. The benchmark case in Figure 8 shows that by arranging students with classmates that have similar levels of noncognitive skills, the overall likelihood of victimization falls from 11.5 percent to 2.8 percent. This dramatic reduction is across the entire skills domain to the point that almost everyone has a probability of being victimized that is not statistically different from zero. Only those who start the period with very low noncognitive skills would still face a nonzero likelihood of being bullied at around 4 percent. However, they would face a sizable reduction in their hazard of being bullied on the order of 11 percentage points.

Figure 8

Classroom Allocation Simulations: Benchmark

Notes: Results based on 40,000 simulations based on the estimated parameters of the model of skill formation.

Of course, this exercise ignores all other possible consequences that the homogenization of classrooms along skill lines might have. Being in a skill-diverse classroom might be beneficial to students—in particular, those not victimized—in other domains. My simulation cannot specify whether the benefit of reduced bullying due to the homogenization of classrooms outweighs the potential positive implications of a skill-diverse classroom. The exercise shows that one implication of homogenization along skill lines is less bullying.

VIII. Conclusions

This work develops and estimates a structural model of skill accumulation that introduces endogenous social interactions as drivers of the skill formation process. The model uses several dimensions of unobserved heterogeneity and in-classroom variation of student characteristics to identify the endogenous selection of bullying victims. My findings indicate the existence of a vicious cycle between victimization and skill depletion. I find that bullying is disproportionately suffered by students who lack socio-emotional skills, and among those, the smart students are more likely to be victimized. In line with psychological studies, my findings suggest that conditional on the level of skills, kids with uncommon characteristics relative to those of their classmates are more likely to be victimized.

The estimation showed that bullying is very costly in terms of the skills lost from one period to the next. Bulling at age 15 reduces noncognitive skill accumulation by a 40 percent of a standard deviation for the average kid. That effect is a third greater for kids with low initial levels of skills. Static complementarity shows that the current stock of cognitive skills greatly influences the “negative productivity” of the bullying event.

These results show the existence of a self-reinforcing mechanism, in which initial levels of skill become crucial, suggesting that policies aimed to foster noncognitive skills at early ages will greatly reduce victimization occurrence. My model also indicates that allocating students to more homogeneous classrooms might reduce victimization by preventing kids with uncommon characteristics from being isolated and targeted by bullies.

This paper intends to contribute to the human development literature in economics by exploring how school-aged kids’ victimization may hamper the development of successful adults. In the process, this paper contributes to the skill formation literature by introducing endogenous social interactions as triggers of phenomena that have long-lasting consequences. This opens a promising research agenda. For instance, researchers can use the model to analyze other types of disruptive behaviors, the role that gender plays in classroom dynamics vis-á-vis those social interactions, or—data permitting—the introduction of physical traits as determinants of victimization. Furthermore, given the importance of initial levels of skills, we should inquire about how these negative social interactions affect younger children’s skill accumulation.