Robustness Check 2: Conditional Logit Models

The core analysis of this chapter explores the effects of the mental health of adolescents on their ASB using equations (3.2) and (3.3). However, the estimated effects of adolescent’s mental health on their ASB may suffer from omitted variable bias if the error terms of the ASB models contain unobservables that also affect mental health. For example, unobserved family socioeconomic status and neighbourhood variables may affect both the mental health of adolescents and their ASB, thus leading to omitted variable bias. Consequently, to investigate the robustness of our initial findings, we control for family fixed effects using conditional logit models.77

Conditional logit models use within group variation in binary dependent variables, see Allison (2009). The conditional logit model shown by equation (3.6) utilises discordant reports of participation in ASB amongst adolescents who live in the same household. We present the case for households of 2 adolescents below:

𝐴𝑆𝐵2ℎ𝑡= 𝛽1(∆𝑆𝐷𝑄ℎ𝑡) + 𝜸(∆𝑿𝒉𝒕𝒅 ) + 𝜹(∆𝑿𝒉𝒕𝒔 ) + 𝜀ℎ𝑡 (3.6)

𝐴𝑆𝐵2ℎ𝑡 is a binary variable equal to 1 if adolescent 2 in household ℎ at time 𝑡

committed ASB, and 0 if adolescent 1 in household ℎ committed ASB. Households where both adolescents report the same behaviour are omitted from the estimation samples because there is no within household variation in participation in ASB. We also omit households containing 1 adolescent from the analysis: approximately 53% of households.

We estimate equation (3.6) using three dependent variables, each a separate measure of ASB, namely, fighting, vandalism, and truancy.78 For each of the three measures of ASB, we use a separate estimation sample. Firstly, for fighting behaviour, we estimate equation (3.6) using Sample 4(a), which contains 925 person-observations from 420 households. Sample 4(a) is composed of 343 households with 2 adolescents, 69 households with 3 adolescents, and 8 households with 4 adolescents.

Secondly, we estimate the effects of the mental health of adolescents on vandalising behaviour using Sample 4(b), containing 583 person-observations from 267 households. Sample 4(b) encompasses 225 households of 2 adolescents, 37 households of 3 adolescents, 3 households of 4 adolescents, and 2 households of 5 adolescents.

Finally, we explore the effects of the mental health of adolescents on participation in truancy using Sample 4(c). Sample 4(c) contains 339 person-observations from 155 77

We estimate the conditional logit model via the clogit Stata command, see Long and Freese (2006) for further details.

78

Unfortunately, conditional logit models exploring the effects of the mental health of adolescents on their participation in shoplifting suffer from problems of non-convergence that likely result from the small sample size. There are just 145 adolescents from 66 households where participation in shoplifting varies amongst adolescents within the same household.

93 households (131 households of 2 adolescents, 21 households of 3 adolescents, 1 household of 4 adolescents, and 2 households of 5 adolescents).79

∆𝑆𝐷𝑄ℎ𝑡 denotes the difference in the SDQ total difficulties scores of adolescent 2 and

adolescent 1 in household ℎ (∆𝑆𝐷𝑄ℎ𝑡 = 𝑆𝐷𝑄2ℎ𝑡− 𝑆𝐷𝑄1ℎ𝑡). In addition,

∆𝑿_𝒉𝒕𝒅 (∆𝑿_𝒉𝒕𝒔 ) indicates the difference in the demographic (social) characteristics between adolescent 2 and adolescent 1 in household ℎ at time 𝑡. We control for the adolescent's demographic characteristics that vary within their household, such as age and gender. We also control for the adolescent's social characteristics such as the number of close friends; the frequency that they have stayed out past 9pm in the past week, without their parents knowing their whereabouts; and whether the individual reports that it is very important to perform well in their GCSEs. We omit all explanatory variables that do not vary within households such as ethnicity, family income, the total number of children, the lower layer super output area of residence, and the rate of crime and ASB in the local neighbourhood. 𝜀ℎ𝑡 denotes an error term.

The parameter 𝛽1 represents the effect of the overall mental health of adolescents on

their ASB. However, as previously discussed, the tendency of adolescents to externalise and internalise may affect their ASB differently. As a result, we split the SDQ total difficulties scale into the externalising problems and internalising problems subscales, as shown by equation (3.7):

𝐴𝑆𝐵2ℎ𝑡 = 𝛼1∆𝐸𝑃ℎ𝑡+ 𝛼2∆𝐼𝑃ℎ𝑡+ 𝜸(∆𝑿𝒉𝒕𝒂 ) + 𝜹(∆𝑿𝒉𝒕𝒔 ) + 𝜀ℎ𝑡 (3.7)

∆𝐸𝑃_ℎ𝑡 (∆𝐼𝑃ℎ𝑡) indicates the difference in the externalising (internalising) problems

subscales scores between adolescent 2 and adolescent 1 in household ℎ at time 𝑡. We estimate the conditional logit models specified in equations (3.6) and (3.7) to explore the effects of the mental health of adolescents on participation in fighting, vandalism, and truancy. The advantage of this approach is that it is possible to account for unobserved household level variables that affect both the ASB of adolescents and their mental health, such as socioeconomic status. Arguably, this approach may therefore reduce the likelihood of omitted variable bias.

As previously discussed, the conditional logit model utilises variation in participation in the ASB of adolescents from within the same household. As a consequence, single adolescent households and households where each of the adolescents report the same behaviour are excluded from the analysis. Thus, a disadvantage of this approach is that whilst it may allow for improved conditioning upon unobserved confounders (family fixed effects) it makes use of a substantially reduced sample size. Consequently, the findings may not generalise to the wider population of adolescents.

79 _{Hence, there are 77, 42, and 24 households of 3 or more adolescents in Samples 4(a), 4(b), and 4(c),} respectively. This estimation method can be extended to such households.

94 For other applications of the conditional logit estimator using variation in outcomes between siblings, see Oettinger (2008) and Black et al. (2007). The former explore the effects of sex education on teenage pregnancy, whereas, the latter investigate the effects of birth-weight on infant mortality.

3.5 Results