Troubled in school: does maternal involvement matter for adolescents?

We estimate the causal effect of mother’s involvement on the amount of trouble an adolescent experiences in school based on a sample of high school students in the USA. Our measure of mother’s involvement encompasses discussing school-related matters and providing help with school projects. We use multiple measures of trouble in school to construct a composite that we link to noncognitive skills. Using an instrumental variable based on a suitably chosen peer group, our main finding is that an increase in maternal involvement leads to a significant decrease in adolescents’ trouble in school. This result is robust across a large number of sensitivity tests aimed at detecting selection effects, shocks at the peer group level, and further potential violations of the exclusion restriction. Additionally, we present evidence suggesting adolescents’ college aspirations, mental health and the perception of parental warmth as potential channels through which the mother’s involvement effect operates.


Introduction
We study the causal effect of maternal involvement on adolescents' trouble in school, using data from the National Longitudinal Study of Adolescent to Adult Health (Add Health).1 Our focus on maternal involvement is motivated by existing evidence that parental efforts and investments during early childhood provide children with important and wide-ranging benefits (Carneiro et al. 2013; Heckman and Mosso 2014), including links between parental involvement and children's academic achievement (Jeynes 2007; Boonk et al. 2018).Yet, much less is known about the efficacy of parental investments during adolescence.
We use trouble in school as an outcome variable because it represents an important set of behaviors capturing both cognitive and noncognitive skills that affect a wide range of long-term outcomes.In particular, the importance of noncognitive skills, such as perseverance, impulse control, and empathy, has recently been established in the literature (Heckman and Kautz 2014).Evidence from adolescence is sparser, but such skills appear to remain malleable during this period, reflect a measure of adolescent development (Heckman and Mosso 2014; Hoeschler et al. 2018), and yield labor market returns in adulthood that have been rising as more occupations shift toward team work and soft skills (Deming 2017).
Much of the research on the impact of parental investments for the development of cognitive and noncognitive skills focuses on young children (Cunha and Heckman  2008; Cunha et al. 2010; Todd and Wolpin 2007; Aizer 2004; Welsch and Zimmer 2008;  Kalb and Van Ours 2014).Establishing the causal impact of parental investments, however, remains difficult.In observational data, the correlation between parental action and children's outcomes often disappears after controlling for family background characteristics (Avvisati et al. 2010), and it is not clear whether this pattern reflects the absence of an effect or bias from unobserved factors.A limited number of experimental studies among younger age groups, however, suggest that the effects of involvement are strong once endogeneity is removed (e.g., Avvisati et al. 2014; Attanasio et al.  2020).
A large literature in education and developmental psychology has studied the association between parental involvement and children's academic achievement.These studies, however, have generally not been able to address endogeneity and their estimates are not necessarily causal effects (e.g., Jeynes 2007; Boonk et al. 2018).An exception is a set of studies around the family check-up (FCU) intervention, which engaged parents and students and aimed to improve family management practices (Dishion et al. 2002).The FCU intervention was evaluated in a randomized controlled trial (RCT), conducted at three public middle schools.Evaluations of the FCU have found that it lowered the propensity for substance use (Dishion et al. 2002; Connell  et al. 2007; Stormshak and Dishion 2009), improved academic outcomes (Stormshak  et al. 2009), reduced problem behavior (Connell et al. 2007; Van Ryzin et al. 2012), and improved mental health and school engagement (Stormshak et al. 2010).While the FCU provides important evidence about the potential effectiveness of family-based interventions, the relatively small number of students and schools involved in the study raises questions about generalizability to a larger population of adolescents.
Within the non-RCT literature, studies on the development of cognitive and noncognitive skills generally find parental investments are important for skill production early in life.These studies tend to either take structural (Cunha and Heckman 2008; Cunha  et al. 2010; Todd and Wolpin 2007) or reduced-form approaches (Aizer 2004; Kalb  and Van Ours 2014; Price and Kalil 2019; Welsch and Zimmer 2008) and provide valuable information on the role of investments for skill production in early life. 2 Another strand of literature examines parental investment itself.This includes differences by gender (Baker and Milligan 2016) and factors affecting the investment choice (Doepke et al. 2019).However, to our knowledge, there is no evidence available that leverages quasi-random natural variation to generate instruments with less restrictive assumptions and isolate the average treatment effects of involvement.There is also less evidence on the role of parental involvement for adolescents.
The main contribution of our paper is to provide new causal evidence about the effect of maternal involvement, specifically maternal schooling-related involvement, on adolescent trouble in school.We use data from the Add Health study, which is nationally representative and covers students across the full age range of adolescence.Our empirical analysis leverages a novel identification strategy that addresses potential bias due to unobserved heterogeneity and selection effects.We focus on maternal schooling-related involvement for several reasons.First, the importance of maternal investments for child development has been stressed elsewhere (Heckman and Mosso  2014; Carneiro et al. 2013), and schooling-related involvement may more directly relate to experiences in school of which the child has less choice or preference as compared to recreational activities with the parent.Second, we use data from the Add Health parental survey, which focused primarily on mothers because they were expected to be the most involved in their children's day-to-day lives.Third, survey data was missing for fathers much more often than for mothers. 3e construct our school trouble measure from a factor analysis aimed at capturing a wide array of skills with multiple observed indicators.We also use follow-up waves of the survey and explore the association between school trouble and subsequent education and wage outcomes.As we discuss in Section 2.2, our results are similar to the associations between noncognitive skills and education and wages found elsewhere in the literature and suggest that trouble in school can have long-term consequences.
To address endogeneity in the relation between maternal involvement and school trouble, we propose an approach akin to that in Fruehwirth et al. (2019).They use variation within schools across an appropriately defined peer reference group to identify the effect of religiosity on mental health.In our study, we draw on evidence that parenting advice from social circles and families tends to be weighted more heavily than advice from experts (Kalil 2015).We expect that mothers are more likely to respond to a peer group of mothers who have similar education levels and children with the same exogenous characteristics (race, gender, school, and grade).This motivates our use of peer maternal involvement as an instrumental variable.
Our baseline estimates show that an increase in maternal involvement leads to a significant reduction in the adolescent's school trouble.This effect is obscured by a standard OLS regression, which yields a small effect estimate but one that may be biased toward zero by maternal responses to poor behavior (e.g., McNeal 2012).Our results provide new evidence that continued maternal involvement beyond early childhood remains important and, in particular, can support the development of noncognitive skills during adolescence.We conduct a large number of sensitivity analyses-aimed at detecting possible violations of the exclusion restriction-and find that our baseline estimate remains robust.
Our study further relates to a wider literature examining the role of parental beliefs in changing parenting style or the level of parental investments.This literature has found that parents' subjective beliefs about the child's skill production function may be distorted and sensitive to environments outside of the home.This can lead to lower investments among those lacking information and resources (Attanasio et al.  2019; Attanasio 2015; Cunha 2015; Kiessling 2021; Han 2022).Our results show that maternal investments can have a substantial influence on adolescent skill development.
Beliefs that change parental investments may therefore remain important throughout adolescence.
We additionally explore several mechanisms that may explain the impact of maternal involvement on school trouble.First, maternal involvement may change the adolescent's aspirations for future education.This is consistent with the theory that involvement is an effort to shift a child's choice set toward a more forwardlooking perspective (Doepke et al. 2019).Second, maternal involvement may affect the adolescent's mental health.Third, maternal involvement can affect the adolescent's perception of parenting style, which has been identified as an important factor determining child outcomes (Jeynes 2007; Doepke et al. 2019).To measure this, we use adolescent perceptions of parental warmth, control, and autonomy, which are three salient dimensions of parenting style (Steinberg et al. 1992; Marchant et al. 2001).We find that maternal involvement shifts adolescent aspirations, mental health, and, to a lesser extent, perceptions of warmth in the relationship with parents.
Finally, we conduct a descriptive analysis in which our measures of adolescent college aspirations, mental health, and parenting style are treated as potential mediators.Our findings suggest that these may act as significant mediators in the link between maternal involvement and school trouble.While our mediation analysis is descriptive, it provides further support to the importance of school trouble as an outcome measure and the role that maternal involvement continues to play during adolescence.
The remainder of this paper is organized as follows.Section 2 discusses the data and construction of the school trouble variable and maternal involvement.We outline our empirical strategy in Section 3 and present results in Section 4. We explore possible mechanisms in Section 5. Finally, Section 6 concludes.

Data description
For this study we use the National Longitudinal Study of Adolescent to Adult Health (Add Health).Add Health began in 1994 as a nationally representative sample of adolescents in the USA.The study was split between an in-school survey and an inhome survey.The in-home survey is a subset of 20,745 adolescent students out of the 90,000 in-school participants.The in-home group has been followed through four waves, with the wave IV sample aged 26-32.
At wave I for the in-home sample, Add Health also conducted a parent survey.The mother was the targeted respondent.If the biological mother was not in the home, then the next mother figure was requested before the father.The expectation was that mothers would be more involved with the children's school and other activities and be able to provide more detail.We draw on this survey for several important measures on mothers.
The in-home sample provides rich information about the participants' home, social, and school life during the adolescent years.It also provides detailed information on young adult life outcomes.Key for our identification strategy is that, in wave I, we observe reference groups of "peer mothers."These are mothers who are similar along several dimensions and who have children with shared characteristics.For the analysis of mother's involvement and school trouble, we take advantage of random variation across groups of peer mothers to identify the effect of interest.

School trouble and skills
We conduct a factor analysis on observed school trouble measures, with a single latent variable (factor) to capture the underlying skills these trouble measures proxy.Our observed measures of latent skills are all self-reported and consist of grade point average, the number of unauthorized missed school days, reports on a zero to four scale of trouble with teachers, trouble with other students, and trouble getting homework done, a measure for the frequency one gets into fights, and an indicator for being suspended at any point during the school year. 4,5We take the negative of grade point average so that higher values imply greater trouble to be consistent with the rest of our measures; however, in a robustness check we omit GPA from the scale and find highly consistent, if marginally less efficient, results.
To create a single measure of skill, we estimate a basic latent factor structural equations model and predict the latent skill factor for each adolescent in the sample.For most observed measures, we use a linear measurement equation where M j is the j-th indicator, α j is the factor loading, θ is the latent skill factor, and j is measurement error.Following standard practice, we set the scale of θ by constraining the factor loading for one of the observed measures to 1.For school suspension we use a probit measurement equation where (•) is the CDF of the standard normal distribution. 6We also drop missing observations in our measures to ensure that the measurement equations are estimated on the same sample.Summary statistics for the measures are available in the online Appendix, Table A.1.The estimated factor loadings are given in column 1 of Table A.2, also in the online Appendix.Each measurement is strongly related to the latent skill variable θ .We standardized the scale to a mean of zero and a standard deviation of one.For ease of exposition, we often refer to the latent skill variable as the school trouble scale.
To test against significant heterogeneity in the loadings, we also report them split across gender and grade-levels.Columns 2-3 in the Appendix Table A.2 illustrate that the measures load onto our scale evenly across gender.Columns 4-9 illustrate the same by grade-level.The only exception is that days of skipping school loads more heavily at later grade-levels; otherwise, the loadings are consistent.We think this is sensible because skipping school may be easier when one is older.However, in all specifications to come we will control for the grade-level effect in a nonlinear manner. 7inally, we explore the relation between our composite school trouble scale and two future outcomes observed in wave IV: completed education level and wages.We report our results in the online Appendix, Section A.3.In terms of both completed education and wages, our school trouble scale closely follows the patterns reported by Heckman (2008) and Heckman et al. (2014) for noncognitive skills. 8Likewise, the picture vocabulary test score closely matches the patterns found for cognitive skills.
Our aim is to broadly capture school trouble through the skills that determine it.While our scale is strongly related to noncognitive skills, cognitive skills may also contribute.However, the results in the online Appendix, Section A.3 suggest that we have a reasonable proxy for noncognitive skills and that there are returns to these skills in the long run.Nevertheless, we focus on our scale as a broad measure of school trouble.

Mother's involvement
The Add Health survey contains a number measures for maternal involvement.Our set of interest involves responses to a series of questions about whether the adolescent has done a particular activity with their mother in the last four weeks. 9The full list with summary statistics is reported in the online Appendix, Table A.3.
We aggregate these using a principle component analysis (PCA) and reported rotated component loadings in the Appendix, Table A .4. 10 The PCA returns three reasonable components that each explain a greater share of variance than a single item.The strongest of these explains nearly three times as much variance as a single item. 11We then assign component interpretations based on loadings that are above 0.4.The first, and strongest, component loads on schooling-related involvement items.The second loads on activities -such as playing sports or going shopping.And, the third loads on communication items not directly related to school.We generate scales for each of the components based on the rotated loadings.
The three items loading on the first component focus on mother's involvement in school-related matters.These are: (1) talking about school work or grades, (2) working together on a school project, and (3) talking about other things you are doing in school.Our hypothesis is that these are the most directly related to school trouble, and it is these three items that drive the strongest component.Thus, we take the schooling-related involvement scale as our preferred scale, though we do report results for the other two scales at the baseline. 12We generally will refer to the schooling-related involvement scale simply as involvement and note where we use the alternative scales.

Sample selection and controls
We control for observable maternal characteristics, household characteristics, and adolescent individual characteristics drawn from the in-home wave I and the wave I parent survey.These include mother's education level indicators, mother's age, household income, the number of siblings in the home, an indicator for single parent homes, whether the adolescent is female, race and ethnicity, school-grade indicators, and school fixed effects.We also control for whether the interview took place during the summer months since some of the in-home surveys did not occur until this point. 13o construct our dependent variable, we dropped individuals who were not in school during wave I (395), who were older than 19 (85), who have missing values for any of the school trouble scale measures (412), or who are extreme outliers in the number of skipped school days (236).The full sample, after constructing the dependent variable, consists of 19,617 observations.For our final selected sample, we drop observations 9 Answers are no, yes (0,1).with missing values for mother's involvement or peer mothers' involvement. 14We also drop observations whose respondent to the parental survey is listed as male or as not the biological mother, when the biological mother, in fact, lives in the home.We do this because maternal education is taken from responses to the parental survey.This accounts for only a small percentage of observations that are dropped (384 total). 15ur final selected sample consists of 12,316 observations. 16e then check that the distribution of our key schooling-related maternal involvement variable is similar across the selected and unselected sample.The Appendix, Figure A.1 shows the comparison.The distributions are very similar across samples and indicate considerable variation across the scale. 17n the Appendix, Section A.2, we report summary statistics for the sample used to construct school trouble and for the final selected sample.Table A.5 shows that the mean differences are in some cases statistically significant; however, in all cases the magnitudes of these differences are very small, indicating that the full sample and the selected sample are very similar.We also show, in Appendix, Figure A.2, that in our selected sample our instrument has a strong degree of variation to identify first-stage effects even after removing school fixed effects.

Empirical strategy
We use a standard linear regression model to estimate the causal effect of mother's involvement on school trouble: Here, Y is is school trouble for adolescent i in school s; I is is our measure of maternal involvement; X is is a vector of covariates; α s is a school fixed effect and ε is represents unobserved heterogeneity.An obvious concern is that I is may be endogenous, for example due to reverse causality between Y is and I is .Becker and Tomes (1976) suggest that parents' involvement with their children may follow either an "enhancement model" or a "response model."In the enhancement model parents become more involved when their children do better and experience less school trouble, resulting in a negative correlation between I is and ε is .Assuming that α 1 in Eq. ( 3) is negative, the OLS estimator α1 will be biased away from zero and will overestimate the magnitude of the effect of involvement.Alternatively, in the 14 When one of the control variables is missing, we impute a value (the mean for a continuous variable and zero for a discrete variable) and add a missing indicator. 15The specific numbers of observations dropped at each stage of the sample selection process are given in Table A.5 in the online Appendix. 16Our sample selection is not unlike other studies who have used Add Health for similar analysis with the in-home data.For example, see Fruehwirth et al. (2019) who use Add Health and a similar identification strategy to ours to explore the effect of religiosity on mental health and have a very similar selected sample size. 17The variation clusters around four points which we might expect given that three items drive the scale.Nevertheless, the scale is approximately continuous as all items contribute some extent of variation based on the rotated loadings in the Appendix Table A.4.  response model parents increase their involvement in response to school trouble. 18onsequently, I is and ε is will be positively correlated.In this case -assuming again that α 1 is negative -the OLS estimator α1 will be biased toward zero and will underestimate the magnitude of the involvement effect.
To estimate the effect of mother's involvement on school trouble, we follow an identification strategy similar to the one proposed by Fruehwirth et al. (2019).They use peer religiosity as an instrument to estimate the effect of religiosity on mental health.In this paper, we use as an instrument the average of maternal involvement, excluding the individual, in a suitably chosen peer group. 19For a given adolescent, the peer reference group is defined as adolescents in the same school, grade, race and gender group and whose mothers have the same education level.20Thus, our instrument is the leave-one-out mean involvement among peer mothers who share the same school-grade-race-gender-mother's education (SGRGE).
The rationale behind this instrument is the idea that mothers with similar education levels and whose children are similar (in terms of the characteristics listed above) are more likely to interact with and influence each other.This idea is not new: Earlier studies by Carbonaro (1998); Sheldon (2002); McNamara et al. (2003), and Mullis  et al. (2003) have all found that parental networks can influence parents.Additionally, Kalil (2015) point out evidence suggesting parents, especially less educated parents, are more likely to take advice from their social circle than from experts.21Thus, by choosing a peer reference group at a level where the mothers are likely to interact, we expect the instrument to be relevant for mother's involvement.
If g(i) denotes the peer reference group and Īg(i)s is the leave-one-out average level of maternal involvement in that group, the first-stage model can be written as where β s is a school fixed effect and u is a residual.Our identifying assumptions are (1) where S i is an indicator for the school of adolescent i. Assumption (1) is the instrument relevance condition.Assumption (2) and the exclusion of Īg(i)s from Eq. ( 3) combine the exogeneity of the instrument and the exclusion restriction.
Our model accounts for selection at the school level.Selection implies that there are unobservables that are correlated with both the reference group and school trouble.An example is a case where more involved parents sort into schools that are better resourced, correlating peer group involvement with school resources, which may also determine school trouble.Our empirical strategy avoids such factors by isolating within school and between-cohort variation in maternal involvement conditional on school fixed effects.If parents select schools based on school-level characteristics, variation in maternal involvement between peer reference groups will be exogenous after controlling for school fixed effects. 22An example that would violate this would occur if parents within SGRGE cells share information on available school resources and at the same time maternal involvement is associated with better information.While this seems unlikely, we will consider a number of sensitivity analyses in Section 4.2.1, aimed at detecting possible selection on unobservable factors.We also test against the presence of within-cohort common teacher effects that could similarly violate this assumption.In all instances, the estimates are very similar to the baseline results.
The presence of peer effects in school trouble could also lead to a violation of Assumption (2).If a relevant measure of peer school trouble is incorrectly omitted from Eq. ( 3) but correlated with our instrument, the 2SLS estimator of α 1 will be biased.If, on the other hand, the instrument is uncorrelated with the peer effect, the 2SLS estimator remains unbiased.In this context, it is important to be specific about the notion of a peer effect, since different features of the peer school trouble distribution could affect an adolescent's own school trouble.A natural candidate is the (leave-onout) average of peer school trouble.Alternatively, it could be that the tails of the peer distribution (i.e., the low-trouble or high-trouble peers) drive the peer effect.
As part of our identification strategy, we assume that a peer effect, if present, operates through exposure to peers in the tails of the school trouble distribution.While peer mothers' involvement is likely correlated with the average of peer school trouble, we assume that the average is correctly omitted from Eq. ( 3) as a measure of a peer effect.In Section 4.2.2 we present several empirical analyses that support these assumptions.First, the estimated coefficient of the leave-one-out average of peer school trouble is very close to zero.This is consistent with a null or negligible peer effect in terms of the average.
In the case that peer average school trouble did have a positive effect, then including a peer effect measure that is also endogenous could lead to bias in all estimated coefficients.In the online Appendix, Section A.10, we examine this more formally and demonstrate clearly our necessary assumptions.We believe, however, that this possibility is unlikely for two main reasons.First, we find robust estimates of the effect of mother's involvement across all of our sensitivity checks for peer school trouble effects, and it is unlikely that all suffer from a more or less identical amount of bias.Second, we experiment with using an additional instrument, and in these cases, we fail to reject the null that the overidentifying restrictions are satisfied.This further suggests that the peer average is correctly omitted from Eq. ( 3).Finally, we present results that suggest that the extremes of the school trouble distribution, rather than the average, affect the adolescent's school trouble.The estimated coefficients are large and statistically significant, but do not alter our estimated effect of mother's involvement.In addition, we present evidence that our instrument is uncorrelated with these relevant peer effect measures, thereby lending further credibility to our baseline results.
Flexible checks on peer effects in school trouble also help us test against threats from common shocks within schools.The idea here is that violations via shocks within the school would likely correlate school trouble across individuals in our refined reference group.Thus, our inclusion of flexible forms of peer effects in school trouble in Section 4.2.2 should capture these and lead to sensitivity in our results if they represent violations of the IV assumptions.We again find that estimates for the effect of maternal involvement on school trouble remain essentially unchanged.
Finally, in Sections 4.2.3 and 4.2.4,we explore a range of additional sensitivity checks including concerns around fathers' involvement and a machine learning approach for instrument and control variable selection.We continue to find evidence consistent with our baseline result, lending further credibility to the assumption of instrument exogeneity and the exclusion restriction.Subsequently, we examine heterogeneity in Section 4.3 and explore some potential mechanisms that can explain the effect of mother's involvement on school trouble in Section 5.

Baseline results
We report our baseline results in Table 1. 23All specifications control for school fixed effects, our controls and, where applicable, missing indicators for the control variables.Standard errors are clustered at the school level.In the first row, we report estimates for the schooling-related involvement scale.The OLS estimate of mother's involvement in column 1 is negative and significant but may be either over-or underestimated.Under the response model, where mothers respond to poor behavior in school with more involvement, this estimate is biased toward zero.
Next, we turn to 2SLS.The first-stage estimate in column 2 shows that peer mothers' involvement is positively and significantly related to maternal involvement, suggesting that the instrument is indeed relevant.In column 3, we report the second-stage estimate.Based on this, a standard deviation increase on our scale of maternal (schoolingrelated) involvement translates into nearly half a standard deviation decrease in school trouble.This effect is larger in magnitude than the OLS estimate and suggests that endogeneity leads to a substantial attenuation bias. 24o provide some context for the magnitude of the 2SLS estimate, consider the difference in average involvement and school trouble between mothers with no high school degree and those with post-college training.From column (2) of Table A.8 in the online Appendix, the conditional mean difference in involvement is about 0.35 standard deviations.Our 2SLS estimate predicts that this leads to a difference of about 0.35×0.47≈ 0.16 standard deviations on the school trouble scale.Given the betweengroup difference in school trouble of about 0.43 standard deviations (see column (4) of the Appendix Table A.8), our 2SLS estimate shows that about 39% of the difference in school trouble between mothers without a high school degree and mothers with post-college training can be explained by the difference in mother's involvement.25The Kleibergen-Paap F statistic (K-P F) is 13.451, suggesting that the instrument is reasonably strong, yet weak instrument bias is a concern.We follow the advice of Andrews et al. (2018) and report the Anderson-Rubin (AR) weak instrument robust test for the null hypothesis that γ = 0.26 The AR test rejects the null with a p-value of 1.5%.Thus, our IV estimate does not appear to be driven by weak instrument bias.
As demonstrated in the online Appendix, Table A.7, the school trouble scale is strongly associated with future education and wages.Depending on the specification chosen from Table A.7 and based on a simple translation, a standard deviation increase in mother's involvement is associated with a 1.9%−6.3%increase in future wages.Together with the 2SLS estimate, this result implies that maternal involvement can have a long-lasting impact.
Our primary baseline result is the estimate for schooling-related maternal involvement; however, in columns 4 and 5 we replace this scale with the activities related scale (column 4) and the non-schooling-related communication scale (column 5).Our aim is to explore the relationships between school trouble and different available measures of involvement.For each measure, we define the instrument as the average of that measure in our reference group. 27e find a similar effect on the activities scale, suggesting there are benefits from wider types of involvement.However, the first stage is weaker (K-P F: 8.174); while it passes the AR test, we pursue our robustness checks and further analysis around the school-related scale, which exhibits a stronger first stage and is more directly related to school.Next, on the communication scale, we find a point estimate that is not significant.
We do not claim that other measures of involvement are irrelevant; rather, the schooling-related measures seem particularly important.Thus, in the remainder of this paper we use our preferred measure of mother's involvement.Of course, the reliability of our baseline estimate rests on the validity of the exclusion restriction for the instrument.In the following sections, we explore several robustness checks aimed at detecting potential violations of that restriction.

Robustness checks
In this section, we present a number of robustness checks aimed at detecting potential violations of the exclusion restriction.We organize these checks around several different channels.First, we consider the possibility that there are unobserved effects that are not accounted for by school fixed effects and that correlate with both peer mothers' involvement and adolescent school trouble.An example of this is the presence of teacher effects that vary within the school.If a specific teacher has a significant impact on a student's school trouble and also encourages parental involvement, then the exclusion restriction would not hold.Second, we investigate the possibility of peer effects in school trouble.If average school trouble in the peer group affects the adolescent, then the average involvement of mothers in the peer group is an invalid instrument. 28Third, we explore whether our instrument affects other forms of parental involvement, which in turn can affect school trouble.Fourth and final, we consider robustness with respect to the choice of instruments, control variables and functional form by employing a lasso-based 2SLS estimator.

Robustness to selection
We consider the inclusion of a variety of additional controls that would reasonably be associated with a selection mechanism, if one is present.Table 2 reports our results.In columns 1-3, we control for peer maternal involvement in different peer groups that get progressively closer to the group that defines our instrument.We control for peer maternal involvement at the same school and grade level in column 1, at the same school, grade and race level in column 2, and at the same school, grade, race and gender level in column 3. We expect that if unobservables are correlated with both our instrument and school trouble, then controlling for maternal involvement in different peer groups should result in sensitive estimates.For example, if the added control variables for mothers' involvement are correlated with an unobserved teacher effect, we expect estimates of our treatment effect to be sensitive to their inclusion.
We find that the estimated effect of mother's involvement remains robust and significant at the 5% level in all cases of controlling for peer maternal involvement at different reference groups (columns 1-3).For a common teacher effect to be completely missed here, this effect would have to be strictly demarcated along the exact definition of our peer reference group used for the instrument, i.e., school-gradegender-race-mother's education.This seems unlikely, and the evidence here supports our identifying assumptions.
In column 4, we include the Add Health Peabody picture vocabulary test (AH PVT) score as a control for the adolescent's cognitive ability in case school fixed effects have not adequately captured selection on ability.We find this has little impact on the estimated effect of mother's involvement, nor does it affect the strength of the instrument.
Next, in columns 5-6 we include school trends.Our first approach is to interact each school indicator with a grade-level variable (column 5), allowing across grade trends.The estimated coefficient for maternal involvement is similar to our baseline estimate, though it, and its standard error, slightly increase in magnitude.Our second approach is to interact each school indicator with the same school-grade peer average maternal involvement to control for school trends at the school-grade level in peer mothers' involvement.In column 6, controlling for differences in peer mothers' involvement between schools and grades, the estimate remains similar to our baseline result.
In columns 7 and 8, we again consider potential teacher effects.We use information from the parent survey about membership of a parent-teacher organization (PTO).In columns ( 7) and ( 8), we restrict the sample to observations where the parents were  school fixed effects.Inv. is involvement; SG is school-grade; SGR is school-grade-race; SGRG is school-grade-race-gender.Each of these refers to the definition of the peer group level used in controlling for the peer mean.AH PVT is the Add Health Peabody Picture Vocabulary test score administered to the adolescent.SG-peer trend includes an interaction between grade-level and each school indicator.SG-peer mothers' involvement trend includes an interaction between school-grade level average peer mother involvement and each school indicator.PTO is an indicator for whether the parent reports being apart of a parent-teacher organization.We report results restricting to those who are not (column 7) and then to those who are (column 8) a member.Block and country controls are as follows: the census block-level percent of children above the age of three in private school (elementary or high school), the census block-level percent of adults with a college degree, and the county total juvenile arrests per 100,000 of the population not members of a PTO, or were members of a PTO, respectively.If a common teacher effect drives our results, we expect our estimates to be mostly determined by parents who are more likely to interact with teachers, thereby correlating their involvement with less school trouble.We find no evidence for this; in fact, the results suggest the opposite. 29The estimates in columns ( 7) and ( 8), while more noisy because of the smaller sample size, suggest that those who are not members of a PTO drive the effect of mother's involvement.This pattern is consistent with a story that parents who are less engaged with teachers rely more on other parents for advice.Relatedly, Kalil  (2015) indicates that less educated parents rely more on their networks for parenting advice.Indeed, in our data, parents who are not members of a PTO have on average less education. 30n column 9, we extend our check against concerns over interactions with teachers.Here we restrict both to mothers who are not PTO members and restrict the peer reference group to only those peer mothers who are also not in a PTO.The idea here is to remove those mothers from the peer reference group who are more likely to interact with teachers.Here the 2SLS point estimate for the effect of maternal involvement is significant and quite similar to the baseline.We see this as entirely consistent with our expectations given the previous discussion and suggestive that a correlation between our instrument and teachers is not creating a violation of the exclusion restriction.Again, it appears that mothers less involved at the school most affect each others involvement.
Finally, we include census controls aimed to capture features that may predict common shocks around involvement and school trouble.In doing so, we aim to capture neighborhood effects that may vary within school and correlate with our instrument and outcome.In column (10), we add controls for the census block-level percentage of children above the age of three in private school (as a proxy for block-level parental investments), the block-level percentage of adults with a college degree, and countylevel juvenile arrests per 100,000 population.We again find that the effect of maternal involvement on school trouble remains highly robust.
Overall the results in Table 2 support our claim that selection into schools is largely based on factors fixed at the school level.These are accounted for by the school fixed effects.To test this further, we also explore balancing tests in Section A.5, Table A.11 of the online Appendix.In these tests, we regress the observable controls that are not part of our peer group definition on our instrument.Moreover, we supplement this set with some additional characteristics: being the first born and birth weight.If selection effects are removed conditional on school fixed effects, then we do not expect much correlation to exist between these variables and our instrument.We find no evidence that our instrument is related to these controls, further suggesting that any selection effects have been removed.

Robustness to peer effects
In this section, we consider the potential impact of peer effects in school trouble on our estimates.First, we note that even if adolescent peer effects are present, these do not necessarily invalidate our instrument.A number of studies do not find evidence that the mean of outcomes-for example, ability-drives the relevant margins of peer effects in school based reference groups (Sacerdote 2014).The findings of Lavy et al. (2012)  on ability peer effects suggest that cohort peer effects may be driven by "extreme" peers.In our case, this would be those on either the high or low end of the school trouble spectrum, rather than by peers near the average.Key for our study is that to the extent that peer mothers' involvement shifts school trouble within the peer group, it must not shift the relevant margin for peer effects in school trouble, as we discuss in more detail in the online Appendix, Section A.10.
Regarding instrument validity, we first check the conditional correlation between our instrument and the percentage of peers (within the SGRGE reference group) in the bottom decile (high peer quality) and the top decile (low peer quality) of the schoolgrade, school trouble distribution.These results are reported in the online Appendix, Table A.11.We find no relation between our instrument and these measure of high and low school trouble among peers.Next, we test for sensitivity in our second-stage estimate for maternal involvement after introducing both average school trouble at our reference group level and the shares of high and low trouble among these peers.
In column 1 of Table 3, we add to our baseline controls only the leave-one-out average of school trouble in the peer group defined by the school, grade, race, gender and mother's education level, and in column 2, we add to this the leave-one-out average of peer ability (AH PVT scores) defined at the same level. 31In column 3, we further add the percentage of high-and low-quality peers in terms of school trouble and ability in the SGRGE reference group, and in column 4, we further supplement the controls with means of peer characteristics.In all of these specifications, our effect of interest on maternal involvement remains remarkably consistent and very near the baseline estimate.
The estimated coefficient for average peer school trouble is small and negative in columns 1-2.This could be due to exclusion bias (Caeyers and Fafchamps 2020).Exclusion bias is mechanical and arises because individuals cannot be their own peer.
If the leave-one-out average of peers' outcomes is high, the outcome for the individual is more likely to be low, and vice versa.Consequently, a regression with the leaveone-out average as a control variable yields a coefficient estimate that is confounded by negative correlation and contains a negative bias.In columns 3-4, where we add the high-and low-quality peer controls, the effect on the peer average is now positive but it is small in magnitude and does not change our estimate for the effect of maternal involvement.As we show through a simulation in the Appendix, Figure A.6, our estimated treatment effect for maternal involvement can tolerate relatively sizable effects from average peer school trouble and our evidence here is consistent with that.Finally, for peer quality, the estimates are intuitive: High-quality peers (very low school trouble) decrease school trouble, while low-quality peers increase it.Since we found no relation between the low-quality and high-quality peer measures and our instrument, as noted earlier, it is not surprising that including these peer measures here does not fundamentally change our baseline 2SLS estimate.Additionally, when we omit the leave-one-out peer averages in column 8, in order to focus on the lowand high-quality peer measures, we continue to find no real change in the estimated maternal involvement effect.
In column 5, we supplement the specification estimated in column 3 with a quadratic in the leave-one-out mean of school trouble to allow for nonlinearities at the peer mean.We find no evidence for nonlinearities on this margin and our effect estimate for maternal involvement remains unchanged.In column 6, we shift to the school-grade peer group level and again control for peer skills and peer averages in the control variables. 32Again, our estimate of the effect of maternal involvement remains stable.
A related concern that would violate the exclusion restriction is that the mothers of an adolescent's peers may have some direct effect on the outcome.We believe this to be unlikely, particularly in high school and given that we are not using actual friend groups but exogenously defined peer groups; however, we consider this possibility in columns 4 and 6 by including controls for a wide range of peers' parental background characteristics.While these characteristics only serve as a proxy for peer mothers' involvement, we again find no sensitivity in the estimated effect of mother's involvement.
Next, in column 7, we add an indicator for whether the mother works outside of the home and control for the percent of mothers who work in our main peer reference group (SGRGE).Olivetti et al. (2020) find that exposure to the labor force participation of their peers' mothers increases participation among girls, potentially through identity formation.Thus, given that identity formation could plausibly influence school trouble here we check that our instrumenting strategy with peer mothers is not sensitive to this factor.Again, we find our results are robust.
For a second set of sensitivity checks, we develop an additional instrument by redefining the peer group based on another potentially relevant dimension for mothers, namely religious denomination. 33To sort denominations, we follow the same approach as Fruehwirth et al. (2019). 34We list the categories in the online Appendix, Table A.12 and provide the frequency distribution.
In Table 4 we report the first and second stage, using as an instrument only the average of peer mothers' involvement from the new peer group definition.We first condition on observations that are non-missing in this variable.The first stage (column 1) is similar to the baseline first-stage estimate, although the instrument is slightly weaker with a K-P F of 9.698.However, the estimated effect of maternal involvement (column 2) is similar to our baseline estimate and significant at the 5% level.stages from redefining the reference group to the same school-grade-race-gender-mother's religious denomination (SGRGR).We omit observations missing peer mother involvement at this reference group definition.Columns 3 and 4 report the first and second stages using peer mother involvement at both our original reference group and redefined group as instruments.Columns 5 and 6 report results after setting missings in peer mother involvement for the redefined level to the mean and controlling for a missing indicator in both stages.Again we have two instruments of peer mother involvement at two definitions of the reference group In columns 3-4, we use both our new and original instrument, conditioning on the sample that is non-missing in either instrument (N = 10, 670).In the first stage, each instrument remains significantly correlated with maternal involvement, and our second-stage estimate again remains stable and statistically significant.Moreover, we do not reject the null hypothesis that the overidentifying restrictions are valid.
Finally, in columns 5-6, we return to our original selected sample by imputing missing observations in SGRGR peer mothers' involvement to the mean and including an indicator for missingness.We include the missing indicator in both stages but maintain our instrument set.The estimated effect of maternal involvement is about −0.48 and significant at the 1% level, even when using the weak instrument robust AR test.
In summary, we checked for possible violations of the exclusion restriction that may run through the adolescent's peer group but find no evidence consistent with this concern.The estimates presented in Tables 3 and 4 are similar to the baseline estimate in Table 1.Next, we consider robustness with respect to using alternative forms of parental involvement, and to selection of instruments and control variables.

Robustness to alternative forms of parental involvement
Different types of parental involvement, for example focused on other activities or coming from fathers, might impact school trouble.If these are affected by the instrument, it would violate the exclusion restriction.We start by exploring the relationship between the instrument and alternative forms of mother's involvement (activities and communication).Based on the estimates reported in Section A.7 of the online Appendix, Table A.13, we find no evidence of such a relationship.This suggests that peer mothers' involvement in schooling-related matters is indeed strongly related to mother's involvement of the same type but not to other types of involvement.While we cannot precisely disentangle how mothers in the peer reference group interact and influence each other, these results further support the relevance of the instrument.
Next, we consider father's involvement as an additional form of parental involvement.If fathers respond to peer mothers' involvement, we would again have a potential violation of the exclusion restriction.We examine this in Table 5.We use the average of peer mothers' schooling-related involvement as the instrument and consider different ways of controlling for mother's and father's involvement in the school trouble equation.First, we form a combined involvement measure that is the sum of the mother's and father's schooling-related involvement.When data on the father is missing, which frequently occurs, we use mother's involvement instead.The estimate in column 1 is similar to our baseline result.The same is true for the first-stage K-P F statistic.
Second, the estimates in column 2 are based on instrumenting maternal involvement while controlling for the father's involvement.We impute missing fathers to the mean and use a missing indicator as a control variable.Column 2 shows that our estimate for mother's involvement is somewhat larger but still yields the same conclusions as our baseline model.
Finally, in columns 3-4 we report results from regressing father's involvement on maternal involvement, our instrument, and baseline control set.As long as fathers respond to the mother but not directly to peer mothers, there is no threat to the All specifications include the base set of controls, missing indicators for controls, and school fixed effects.Inv. is involvement.Column 1 uses a combined mother/father standardized scale of the sum of mother and father involvement (equal to mother if father missing and vice versa).Column 2 instruments mother's involvement and controls for father involvement and missingness in father involvement.Columns 3 reports on a father involvement specification where we maintain our analytic sample via imputation to the mean and controlling for missingness in father involvement.Columns 4 reports on a specification removing imputation and dropping observations missing in father involvement exclusion restriction.To maintain our selected sample, we maintain the imputation for missing fathers in column 3 and control for the missing father indicator.Father's and mother's involvement are highly correlated, as expected, but we find no significant correlation between peer maternal involvement and father's involvement.To ensure that this result is not driven by data imputation for missing fathers, we restrict the sample to non-missing fathers in column 4. Again, we find no correlation between our instrument and the father's involvement.While maternal involvement is endogenous in these regressions, this evidence is consistent with peer mothers' schooling-related involvement affecting the mother directly but not the father.

Selecting instruments and controls
Our choice of instrument is based on a homophily argument: mothers are more likely to interact with and be influenced by other mothers who have similar education levels and whose children have similar characteristics.This still allows for several different ways to define peer groups.Beforehand, it is not necessarily clear what the most relevant grouping will be.A second issue is functional form: can we use a linear model for the relation between the instrument and maternal involvement, or should we account for possible nonlinearities (e.g., through polynomials or interactions)?If a large number of nonlinear transformations of the instrument are used, however, the instrument set overall may be weak and can lead to familiar bias problems.
To address these issues and assess the robustness of our baseline results further, we consider several approaches.First, we employ the lasso-based method of Belloni et al.  (2012) to select the optimal instruments among a large set of candidate instruments.The selected instruments are then used to calculate the standard 2SLS estimator.Second, variable selection issues also affect the school trouble equation.Inclusion of too many controls reduces the efficiency of the estimator, therefore we apply the post-double-selection (PDS) lasso approach proposed by Belloni et al. (2014).In this approach, the lasso is used twice to select two sets of control variables: one that predicts school trouble and one that predicts mother's involvement.A third lasso step is employed for instrument selection, whereby the selected controls from the mother's involvement lasso are always included in the model.The final step calculates the 2SLS estimator with the union of selected controls from first two lasso steps and the selected instruments from the third step.The advantage is that we can reduce the dimensions of the control set and can also explore including many controls at once.
Results are reported in Table 6. 35In column 1, we only allow selection of instruments.The instruments are the average of peer mothers' involvement at six different configurations of the peer group. 36For each definition of the instrument, we also include second and third degree polynomials to capture possible nonlinearities.The total number of included instruments is 18.In column 2, we repeat the exercise but also allow the controls to be selected.There are a total of 21 variables in our original control set.In column 3, we add new reference groups based on a mother's religious denomination and again include a third degree polynomial. 37Finally, in column 4, we maintain our baseline instrument but include all controls from our robustness check sections-including school-grade trends and school-grade mothers' involvement trends-and follow the PDS method for selection on controls.In this case we have 354 possible controls. 38n all cases, we find that the 2SLS estimate is close to our baseline estimate.In columns 1-3, the lasso always selects just one instrument, our original baseline instrument.This is consistent with our expectation that mothers with similar children and who share similar education levels will be the most likely to influence each other, and thus, the most relevant choice for our peer reference group.In column 2 only 8 controls are selected.Not surprisingly, the first-stage relevance increases, as the value of the F statistic increases.In column 3, the sample size falls because we include the religious denomination based reference groups, but our effect estimate remains similar.Finally, in column 4, we find that out of the 354 possible controls, 40 are selected by the PDS method.Again, the second-stage estimate is close to our baseline estimate and The instrument set is always based on peer mother involvement.We include this at different versions of the peer reference group and include up to a third degree polynomial in peer mother involvement at each reference group level.SGRGE is the same school-grade-race-gender-mother's education reference group that we use in our baseline estimates.The SGRGE selected IV is the leave-one-out SGRGE average mothers' involvement.SGRGR is the same school-grade-race-gender-mother's religious denomination reference group used in Section 4.2.2.In column 4, we include all base controls plus all controls from our robustness checks.The number of controls does not include the school fixed effects.These are always included as they are crucial for the identification assumption remains significant.Overall, the results in Table 6 show that our baseline estimate is not sensitive to different choices of instruments or control variables.

Heterogeneity
We explore heterogeneity across three dimensions.First, there is evidence in the literature that parental influence on skill development declines as a child ages (Doepke  et al. 2019; Heckman and Mosso 2014).Our sample includes 7th and 8th graders, so we aim to test whether a mother's response to peer mothers' involvement and the effect of mother's involvement are driven by the youngest adolescents in our sample.Second, we investigate whether a mother's response to peer mothers' involvement and the efficacy of mother's involvement varies by education level.Third, we test for heterogeneity by gender since in the Add Health data, males generally experience more trouble in school.
Our ability to explore heterogeneity is limited.First, our sample size precludes many refined cuts of the data.Second, the instrument may not be strong enough to disentangle multiple layers of heterogeneity.Third, some of the heterogeneity questions may be substantive.How parents choose to invest and their subsequent influence along differing dimensions of socio-economic status, neighborhoods, and other characteristics may depend on a number of factors that are beyond the scope of this study and that deserve careful theoretical and empirical attention.39Thus, our analysis here is exploratory in nature and provides a direction for further work.
We report our analyses in the online Appendix, Section A.8.We find no evidence for heterogeneity by grade level.This applies to both the mother's response to peer mothers' involvement and to the efficacy of mother's involvement.
On maternal education level, we do find some evidence in the first stage that less educated mothers have the strongest response to peer maternal involvement.As noted in Section 3, this result is consistent with evidence that parents, especially less educated parents, put more weight on parenting advice from their social relationships, communities, and families (Kalil 2015).
Next, we find that the effect of mother's involvement on school trouble appears to be driven by mothers who did not complete college.Less educated parents have lower overall involvement, as we show in the online Appendix Table A.8, thus small increases in their involvement level may matter a lot.This is sensible if there are diminishing returns to involvement.Given that more highly educated parents tend to invest more in their children, interventions attempting to boost maternal involvement will likely be focused on those with lower levels of education (Heckman and Mosso  2014).Our evidence implies that such targeted interventions can indeed be beneficial.
Finally, school trouble exhibits substantial variation by gender.In the online Appendix, Figure A.5, we plot the estimated density of school trouble by gender.The distribution of school trouble for males is substantially shifted to the right, compared to females.This can be partly explained by the fact that male noncognitive development at early ages lags behind that of girls (Bertrand and Pan 2013).In our results, however, we do not find evidence that the effect of mother's involvement varies substantially by gender (Appendix, Table A.15).

Mechanisms
In this section, we discuss three potential mechanisms that we can explore empirically in the Add Health data.The first mechanism is the transfer of educational values and expectations from parents to children.Fan and Chen (2001), Hill and Tyson (2009), Jeynes (2007) and Castro et al. (2015) show that parental expectations and aspirations for their children's academic achievement are significant predictors of academic outcomes.If maternal involvement coincides with communicating and transferring values, expectations, and aspirations to adolescents, then this may be one channel through which maternal involvement reduces school trouble.
The second mechanism is adolescent mental health.Wang and Sheikh-Khalil (2014) present evidence that parental involvement reduces adolescent symptoms of depression.This may occur because involvement provides parents an opportunity to give emotional support to their children.Involvement may also foster a feeling of connectedness between parents and children that improves emotional and mental well-being.In turn, this can facilitate the transfer of values and aspirations between parents and adolescents and increase academic engagement in school (Wang and Sheikh-Khalil  2014).
The third mechanism we consider is parenting style.Parenting style reflects the relation between parents and children and is a strong predictor of academic achievement (Jeynes 2007).Steinberg et al. (1992) identifies three salient dimensions of style: parental warmth and responsiveness, behavioral supervision and strictness, and granting psychological autonomy.The empirical results of Dornbusch et al. (1987); Steinberg et al. (1992); Deslandes et al. (1997) and Marchant et al. (2001) show that an "authoritative" parenting style, characterized by high levels of emotional responsiveness and parental supervision but without being overly strict, is associated with higher academic achievement.Experimental studies of the family check-up (FCU) intervention have also pointed to parenting style as a mechanism that can explain its success.Dishion et al. (2003); Stormshak et al. (2010); Fosco et al. (2013) find evidence that parental style and monitoring are associated with greater adolescent self-regulation, which in turn leads to a range of better outcomes.
An authoritative style may also take greater effort to implement by the parent.For instance, it may require more involvement to habituate the child toward a more forward-looking perspective (Doepke et al. 2019).Greater involvement, in turn, may alter an adolescent's perception of parenting, boosting their aspirations or providing protective emotional support.These predictions suggest we should find a link between involvement and some measure of parenting style.
We constructed several measures from the Add Health survey to explore these mechanisms.Details about the construction of each measure can be found in the online Appendix, Section A.9.One measure represents college aspirations, three measures represent mental health (depression, self-esteem and suicidal ideation) and three measures reflect parenting style (warmth and responsiveness, behavioral supervision and strictness, and autonomy).For depression, we use the 19 item scale from the Center for Epidemiological Studies Depression (CES-D) scale (CES-D, Radloff 1977).This scale is available in Add Health and is a widely accepted screening tool for depressive symptoms used in psychiatric epidemiology. 40able 7 reports the estimated impact of mother's involvement on each of our mechanism variables, where involvement is instrumented as before and we include our baseline set of controls and school fixed effects.An increase in mother's involvement leads to a statistically significant increase in the level of college aspirations and a decrease on the depression scale, while self-esteem and suicidal ideation do not appear to be affected.Turning to the parental style measures, mother's involvement is significantly related to the perceived warmth of the parents (column 5) but not to perceived parental control and autonomy (columns 6-7).The link with warmth is nevertheless consistent with the notion that involvement and parenting style are intertwined.
Next, we conduct an exploratory mediation analysis in Table 8.There are a number of intuitive reasons to suspect a link between mother's involvement, the mechanism variables, and school trouble.For instance, in a model where students hold beliefs over uncertain capabilities and future opportunities, maternal involvement may act to improve these beliefs, translating into improved aspirations and mental health. 40Our construction of the CES-D scale and the self-esteem scale is the same as in Fruehwirth et al. (2019).Notes: * p < 0.1; ** p < 0.05; *** p < 0.01.Column 1 contains coefficient estimates from regressing school trouble on each of the row variables plus our baseline set of controls and school fixed effects.Column 2 contains estimates from the auxiliary regression for the association of each mediator with school trouble when all included in one regression along with instrumenting maternal involvement and including the baseline controls as our auxiliary specification outlines.Column 3 reports the indirect effect estimates and column 4 reports these as shares of the estimated total effect from maternal involvement to school trouble.Standard errors are in parentheses and are clustered at the school level Involvement may also alter the adolescent's perception of the relationship with their mother, as proxied by our style measures such as warmth.Indeed, Doepke et al.  (2019) think of involvement as effort to shift children toward a more forward-looking perspective, while beliefs about opportunity represents an important mechanism for shifts in depressive symptoms (Quidt and Haushofer 2017).In turn, greater patience, more positive emotional symptoms, and improved parent-child relationships may translate into less trouble in school.Our mediation analysis is not necessarily causal and should primarily be viewed as descriptive.It does, however, reveal the associations between school trouble and the mechanism variables, and we can combine this with the causal evidence from Table 7.To do this, we follow a decomposition approach from Gelbach (2016).We assess how much of our treatment effect from maternal involvement on school trouble runs through each of our mechanism variables, using the following specifications:

Table 7
(5) 123 The first equation is estimated for each mediator variable m isk (k = 1, . . ., K ) with 2SLS and reported in Table 7.The second equation is our baseline model with school trouble as the outcome.The third and final equation is an auxiliary 2SLS regression where we instrument for maternal involvement (I is ) while controlling for the all mediators (m isk ).We then calculate the indirect effect of maternal involvement on school trouble through mediator k as I E = α m 1k × γ aux k , and the fraction of the total effect that is mediated through m isk as IE/α y 1 .For this mediation analysis to be causal, the mediators would have to be uncorrelated with the error term in the auxiliary regression.Since this is unlikely, we reiterate that this is mostly a descriptive exercise.
First, in column 1, we present the association between each of our mechanism variables and school trouble.Each coefficient estimate is from a regression of school trouble on the row variable plus our baseline controls and school fixed effects omitting maternal involvement. 41While these are not causal estimates, they have the expected sign: college aspirations, self-esteem, parental warmth, and autonomy are all related to less school trouble, while depressive symptoms and suicidal ideation are related to more school trouble.
Next, in column 2, we present associations for the mediators with school trouble from a single regression corresponding to the our auxiliary specification.We then use these estimates along with the estimates in Table 7 to calculate the indirect effect reported in column 3. Finally, column 4 contains the share, dividing the indirect effect by our baseline "total effect" estimate for maternal involvement in Table 1.
The mediation estimates are consistent with our intuition and the results from Table 7. College aspirations and depressive symptoms (CES-D scores) each account for about 24% of the effect of mother's involvement on school trouble, while perceptions parental warmth accounts for a smaller 6% share.
In summary, the estimates presented here suggest that the beneficial effect of maternal involvement may operate through shifting beliefs and improving well-being.This, in turn, may protect the adolescent from experiencing trouble in school.Our evidence points to these as useful avenues for future work and further strengthens the case that maternal involvement during adolescence is linked to important features of adolescent development and well-being.

Conclusion
Over the past few decades parental involvement has been promoted by policy makers and educators as an important factor that can help drive student success.The No Child Left Behind Act of 2002 and the Every Student Succeeds Act of 2015 both 41 Here we are including the mediators one at a time to look at their simple association with school trouble conditional on the baseline controls.
required states to formulate strategies to promote parental involvement at home and in the school.Part of this policy focus has been driven by a large body of research, emanating from education and developmental psychology, that has pointed to a positive association between parental involvement and student outcomes.
Very few studies have been able to estimate the causal effect of parental involvement on academic achievement and noncognitive outcomes using observational data.Recent evidence has emerged about the causal link between parental investments and skill formation during early childhood but much less is known about the period of adolescence.The main contribution of this paper is to provide new evidence in this area.Specifically, we estimate the causal effect of maternal involvement on adolescent trouble in school.
We construct a measure of adolescent school trouble and link it with noncognitive skills.We identify the causal effect of maternal involvement on adolescent school trouble by using the average of mothers' involvement in an appropriately chosen peer group as an instrument.The peer group of mothers is not self-selected but rather defined as the group of mothers who have a number of exogenous characteristics in common (the child's race, gender, school and grade, and the mother's education level).We then leverage within-school, across-cohort variation as an approach to eliminate selection factors and satisfy instrument exogeneity.Our baseline estimates point to a statistically significant and substantial effect of mother's involvement: an increase of 1 standard deviation leads to a reduction in school trouble of about 0.5 standard deviations.The richness of the Add Health data allows us to conduct a wide range of robustness checks around the exclusion restriction.We find our result to be remarkably stable, lending further credibility to our baseline results.
Finally, we explore a number of mechanisms that may explain the causal effect of maternal involvement on school trouble.These include the impact of involvement on the adolescent's college aspirations, mental health and perceptions of parenting style.We find that an increase in involvement is associated with higher college aspirations, lower levels of depression, and a higher perceived level of warmth in the relationship with parents.What mothers do may shift how adolescents feel about themselves and their family, and this mechanism can operate as a protective device that prevents subsequent poor choices by the adolescent at school.A more thorough study of processes within the family remains a promising topic for future study.appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
01.Standard errors are in parentheses and are clustered at the school level.All specifications include our baseline controls and 01.Standard errors are in parentheses and are clustered at the school level.Inv. is involvement.All specifications control for school fixed effects, our base set of controls, missing indicators for our controls, and indicators for the mother's religious denomination.Columns 1 and 2 report the first and second 01.Standard errors are in parentheses and are clustered at the school level.Column headers indicate the dependent variable for the specification.In each specification, we drop additional observations that are missing in the column header.Sample sizes remain very similar to our baseline.All specifications include our baseline set of controls and school fixed effects 123 Standard errors are in parentheses and are clustered at the school level.All specifications include school fixed effects, our base set of controls, and missing indicators for missing observations in our control set.Mother's involvement (our preferred scale) is extracted from the PCA first component rotated loadings and is defined by the schooling-related items.Mother's involvement (Act.) is based on the second component which loaded on activities, and mother's involvement (Comm.) is based on the third component, which loaded on non-schooling-related communication.All scales predicted based on the rotated loadings and are standardized to mean zero and standard deviation of one.Column 2 reports the first stage of average peer mother's involvement (schooling-related scale) at the school-grade-race-gendermother education level on mother's involvement.The Anderson-Rubin (AR) weak IV robust p-values are reported at the 95% level and 250 gridpoints.These report a weak instrument robust test of the null that

Table 2
Robustness to selection: additional controls Standard errors are in parentheses and are clustered at the school level.All specifications control for school fixed effects, our base set of controls, and missing indicators for our controls.Column 1 reports 2SLS estimates for our preferred schooling-related involvement index after controlling for the SGRGE peer leave-one-out mean of school trouble, and column 2 adds the SGRGE leave-one-out mean of AH PVT scores.Column 3 is similar but adds high-and low-quality peers defined by the percent of SGRGE peers in the bottom (top) decile of the school-cohort school trouble distribution.We also include SGRGE peers' AH PVT scores, the percent of SGRGE peers in the bottom (top) decile of the school-cohort AH PVT distribution, and own AH PVT scores.In column 4 we further add SGRGE peer average characteristics: number of siblings, parental age, single parent homes, and household income.We cannot include peer controls on the variables used to define the peer reference group.Column 5 reports results supplementing the specification in column 3 with a quadratic in the leave-one-out mean of peer school trouble.Column 6 reports 2SLS estimates controlling for peer level averages at the school-grade level.Column 7 repeats the specification in 1 and adds mother's and peer (SGRGE) mothers' labor force participation.Column 8 repeats the specification in column 3 but omits the leave-one-out mean peer variables.Missing indicators and imputation are included throughout where needed

Table 6
Machine learning and 2SLS

Table 8
Descriptive mediation