Social Spillovers of Parental Absence: The Classroom Peer Effects of ‘Left-behind’ Children on Household Human Capital Investments in Rural China

Abstract While the human capital consequences of rural-to-urban migration on left-behind children have been well-documented in developing countries, there is limited evidence regarding the social spillovers of parental migration on households without parent-child separation. This study investigates the effects of migration-induced left-behind children on household human capital investments in their non-left-behind peers. Leveraging the random student-class assignment within middle schools in rural China, we find that the share of left-behind children in class has significant negative impacts on household financial and time investments in non-left-behind classmates, especially out-of-school education expenditure. We also find heterogeneous effects demonstrating that the adverse spillovers are relatively larger among students who are boys, in grade nine, and from low socioeconomic status families. Further, our results suggest that exposure to left-behind classmates adversely affects non-left-behind students’ perceived quality of school life, cognitive and noncognitive skills, and their parents’ beliefs about returns of human capital investments. We interpret these findings as candidate mechanisms underlying the associations between parental absence and household investments in non-left-behind children. Our study sheds new light on the ‘costs’ of rural-to-urban migration in sending areas, which include not only welfare loss to families being left behind but negative spillover effects on non-left-behind households.


Introduction
Migration, either internationally or internally, is a global phenomenon with growing scope and impacts (Ratha, Mohapatra, & Scheja, 2011;United Nations, 2013).It is well accepted that migration has crucial consequences for not only receiving areas but sending communities (Antman, 2013, pp. 293-308).In particular, the massive labour migration has resulted in a substantial number of children being left behind at home while their parents migrate out for work, especially in developing countries (Nguyen, 2016;Zheng, Fang, Wang, & Fang, 2022).With rapid urbanization and economic development, China has witnessed an unprecedented scale of rural-to-urban labour migration over the past few decades.As reported by the National Bureau of Statistics (NBS) of China, the number of rural migrant workers has increased from 50 million in 1990 to approximately 296 million in 2022.Meanwhile, due to the institutional constraints caused by the household registration (hukou) system, which is often associated with access to local public resources and opportunities (e.g.access to urban public schools for children of migrant workers), a considerable number of children have been left behind in rural areas.According to China's 1% Population Sample Survey in 2015, there were about 40.5 million left-behind children (LBC) aged below 18 in rural villages, comprising 29.4% of all children in rural China (Lyu, Yan, Duan, & Cheng, 2018).
Given the crucial role of children's skill development in determining future labour quality, social stability, and economic growth (Boyden & Dercon, 2012;Mincer, 1984), the migrationinduced LBC phenomenon in China has attracted widespread attention across multiple disciplines.Past research has extensively studied the contemporaneous effects of parental migration on LBC's developmental outcomes, including academic achievement, cognitive and noncognitive skills, physical and mental health, and behavioral outcomes (Chang et al., 2019;Lei, Liu, & Hill, 2018;Wu & Zhang, 2017).Although some debates remain, a significant body of empirical evidence has suggested the negative impacts of parental absence on LBC's well-being (Zheng et al., 2022)  1 .
However, there is a dearth of studies that examine the social spillover effects of migration on non-LBC and their household behaviors.This research gap is important because the impacts of migration on home communities could be substantially underestimated if exposure to LBC has social externalities on non-LBC and their households, hence negatively influencing the formulation of optimal countermeasures.As such, this study shifts the focus from the impacts of parental absence on LBC's outcomes to household decisions among non-LBC, which complements the full picture of the consequences of rural-to-urban migration on source areas.In particular, using data from a nationally representative survey of rural middle school students, we investigate whether and how the presence of LBC due to parental migration affects household human capital investments in non-LBC in the same classroom.
One major challenge for our identification is the potential selection problem.For example, non-LBC with lower levels of household human capital investments might be more likely to be assigned to schools or classes with a higher share of LBC.To address this issue, we have controlled for within-school variations and focused on schools that employed a random studentclassroom assignment.In the case of randomization, our identification strategy is to compare the household human capital investments in non-left-behind students from two classes in the same grade of the same school who have similar characteristics of individuals, households, and classes, except for the fact that one class has a relatively higher proportion of LBC than the other due to the random assignment of students to classrooms in each school.
The results of our study show that the proportion of LBC in a class has significant negative effects on household financial and time investments in the human capital of their non-left-behind classmates.Specifically, a one-percentage-point increase in the share of LBC in a class leads to a reduction of 1.7% in non-LBC's total education expenditure, 2.9% in out-of-school education expenditure, and 0.2% in parents' time spent on tutoring students' homework.In terms of the heterogeneity of the impacts, we find that the adverse effects of LBC on household human capital investments in non-LBC are relatively larger among students who are boys, in grade nine, and living in low socioeconomic status families.
We have explored possible mechanisms underlying the negative social spillovers.The results demonstrate that the presence of LBC in the classroom has a detrimental impact on the perceived quality of school life and cognitive and noncognitive skills of non-LBC, suggesting the existence of negative peer effects arising from social interactions.We also find that the share of LBC in class negatively affects parental beliefs about child academic achievement and returns Social spillovers of parental absence 289 of human capital investments among non-LBC in the same classroom.Our analysis further takes the observability issue into account and the results show that the spillovers of LBC on non-LBC's parental beliefs and misbeliefs are more pronounced among parents who have observed the social interactions between their child and left-behind classmates.This suggests that exposure to LBC could not only indirectly affects non-LBC's parental beliefs through decreased student skills, but also directly distorts parental beliefs about non-LBC's academic achievement when they observe the social interactions between their child and left-behind peers.Additionally, we have explored an alternative explanation that parental absence may relieve the 'competition' in household human capital investments, but no evidence has been found to support this hypothesis.
To the best of our knowledge, this is the first study that examines the effects of migrationinduced LBC on household human capital investments in non-LBC.Our study contributes to the literature in the following aspects.First, we complement the paucity of literature on the external determinants of household human capital investments by focusing on the social interactions between LBC and non-LBC.Second, we add to the growing number of studies regarding the impacts of parental migration by investigating the social spillovers on the decisions of households without parent-child separation.Third, we contribute to the ongoing policy debates on the benefits and costs of rural-to-urban migration, highlighting the welfare loss to non-left-behind households.This bears significant policy implications not only for China but for other developing countries with a prevalent migration-induced LBC phenomenon.

Conceptual framework
Theoretically, there are several potential channels through which rural-to-urban migration has spillover effects on household human capital investments in non-LBC.First, the presence of left-behind classmates has negative peer effects on the skill development of non-LBC in the same classroom, subsequently leading to reduced parental beliefs about returns of human capital investments.On the one hand, LBC may have negative peer effects on their non-left-behind classmates through social interactions.Previous studies have shown that peer effects in the classroom can happen for several reasons, including social norms, social learning, and social comparison (Sacerdote, 2011).Among them, social norms represent the unwritten rules that govern behavior in a group.Students may conform to these norms to fit in with their peers (Patacchini, Rainone, & Zenou, 2017).Social learning refers to the process by which students learn from each other, such as learning from peers who are more knowledgeable or skilled than they are (Lavy & Schlosser, 2011).Social comparison suggests that students compare themselves to their peers, including academic performance, behavior, or other characteristics (Barseghyan, Clark, & Coate, 2019).
According to Sacerdote (2011), having a 'bad apple' peer may affect students' development in the following ways: (i) they may cause commotion in the classroom to distract the teacher and students from productive tasks; (ii) they encourage additional raucous or disruptive behavior among other students; (iii) they have low ability and require extra teaching attention, thereby detracting from the experience of the other students.As aforementioned, ample evidence has supported that parental migration adversely affects LBC's human capital and behavioral outcomes.Thus, LBC might be disruptive peers and have negative spillovers on non-LBC's abilities in the same classroom (Wang & Zhu, 2021).
On the other hand, there is a load of empirical evidence to date suggesting that parental investments reinforce initial child endowments and abilities, especially among socioeconomically disadvantaged households (Almond & Mazumder, 2013;Fan & Porter, 2020;Restrepo, 2016).For resource-constrained families trying to maximize the returns of human capital investments, decisions on education inputs usually depend on parental perceptions about the returns of money and time spent on a given child and the child's abilities (Akresh, Bagby, De Walque, & Kazianga, 2012;Gr€ atz & Torche, 2016).Considering that rural households often face relatively tight budget constraints, it is reasonable to argue that they are more likely to adopt a reinforcement strategy on human capital investments in response to variations in child abilities.Consequently, if migrationinduced LBC have negative peer influences on non-LBC's skills, they will further adversely affect parental perceptions about child skills and subsequent household human capital investments.
Second, social interactions between LBC and non-LBC can distort parental beliefs about child skills through information friction even if their child's academic achievement remains unchanged.Recent research suggests that parental beliefs about their child's academic abilities are often not accurate due to information friction, which prevents parents from accessing the true information to make decisions, especially among low-income parents (Dizon-Ross, 2019).The presence of information friction between parents and children is because parents often obtain information about their children's school life relying on parent-child communications, which may not always be correct due to inaccurate or manipulated reports by children (Bergman, 2021).Kinsler and Pavan (2021) further find that parental beliefs about their child's cognitive skills relative to children of the same age are distorted by local environment, as measured by the average cognitive skills of children in the same school.The inaccurate parental beliefs due to local environment will further affect household investments and child skill distribution.Therefore, it is natural to expect that the presence of left-behind classmates influences parental beliefs about the academic skills of their non-LBC if parents have 'observed' the social interactions between their child and left-behind peers.As a result, low levels of academic performance of LBC may discourage the parents of non-LBC in the same classroom from human capital investments.
Third, another possible reason for the effects of LBC on non-LBC's household decisions is that parental absence reduces the 'competition' in household human capital investments.In Asian countries, parents place considerable emphasis on child education (Li & Xie, 2020).Due to the high levels of education returns under socioeconomic inequality, the school admission system forms a competitive environment and parents have economic incentives to participate in the competition in human capital investments for their child's better future (Chen, Dong, & Zhu, 2021;Chung & Lee, 2017).Meanwhile, in accordance with the Chinese belief that 'he who excels in study can follow an official career', parents in China often have high expectations of their child's educational attainment and are motivated to join in the competition of human capital investments to achieve upward mobility (Doepke, Sorrenti, & Zilibotti, 2019).A recent study has found that Chinese parents have significant and positive peer effects on household investments in shadow education and parenting styles among students with similar education achievement, while the 'competition' has limited impacts on child skills (Guo & Qu, 2022).In the context of our study, parental absence also denotes fewer parental time investments in LBC.Although the LBC's family economic status might be increased due to remittances from migrant parents and children being left behind often have alternative caregivers (Lei et al., 2018), the lack of parental care and supervision can be more easily discovered by other households.As such, as a signal of reduced human capital investment, the presence of LBC might lessen the competitive pressure on household investments in child's education.
However, whether the reduction in 'competition' plays a significant role in the negative peer effects on household human capital investments depends on the extent of 'competition' between non-LBC and LBC in the absence of parental migration, as rural households are more likely to face budget constraints than urban ones (D emurger & Wang, 2016).In addition, the feasibility of this explanation also lies in the degree that parental absence influences peer pressure and preferences on education investments in non-LBC (Barseghyan et al., 2019).

Data
The data used in this study are drawn from the China Education Panel Survey (CEPS), which is a nationally representative longitudinal survey for middle school students since 2014.

Social spillovers of parental absence 291
The CEPS is administered by the National Survey Research Center (NSRC) at Renmin University of China.In 2014, the CEPS applied a stratified and multistage sampling design with probability proportional to size (PPS) 2 .A total of 19,487 seventh and ninth students in 438 classes across 112 middle schools in 28 counties were interviewed in the survey.In addition, students' parents, teachers, and school principals were also interviewed during the survey.
Three sample restrictions are applied to facilitate data analysis.First, we restrict our sample to the students in schools located in rural areas, highlighting the social spillovers of rural leftbehind children.Second, following previous studies (Gong, Lu, & Song, 2018;Huang & Zhu, 2020), we focus on schools that randomly assign students to classes to identify the causal effects of LBC on household human capital investments in their non-left-behind classmates.Specifically, according to the school principal's reported questions, we restrict our sample that meets two requirements: (i) students are randomly assigned to classrooms at the beginning of seventh grade; (ii) schools do not rearrange classes for grades eight and nine.
On the one hand, school principals were asked to report the assignment rules they applied to place students, including (i) placement exams before the beginning of students' first academic year, (ii) students' residential status, (iii) random assignment, and (iv) others.The Ministry of Education of China recommends a random classroom assignment to ensure equal and fair opportunities for students at the nine-year compulsory education stage (six-year primary school education and three-year middle school education), and this practice is widely used in China's middle schools.Indeed, 77.5% of all rural schools in our sample randomly placed students into classrooms at the beginning of the seventh grade in the CEPS data.On the other hand, the same school principals also reported whether their schools reorganize their classes for grades eight and nine.In the CEPS data, 80.8% of rural middle schools using a random classroom assignment do not rearrange classes in grades eight and nine.
Third, in case of principals' misreport, we further keep the schools where all head teachers in the same grade confirm that students are not assigned based on test scores.According to the head teachers' reports, 91.4% of rural schools that fulfill the two aforementioned randomness requirements are deemed less susceptible to misreporting by principals.Finally, the sample used for empirical analysis in this study consists of 4,276 students (1,013 LBC and 3,263 non-LBC) across 118 classes and 34 schools, with 2,302 seventh graders from 62 classes and 1,974 ninth graders from 56 classes.

Variables and descriptive statistics
Household human capital investments are measured using four indicators, namely total education expenditure, in-school expenditure, out-of-school expenditure, and parental time investment.These indicators encompass both financial and time investments in students' education (Wang, Cheng, & Smyth, 2022).The overall financial investment is measured by total household expenditure on child education during the surveyed semester.We further divide household education expenditure into two categories, namely in-school expenditure (expenditure on tuition, textbooks, teaching aids, school uniforms, accommodation, and insurance premiums) and out-of-school expenditure (expenditure on extracurricular classes, private tutoring, and hobby training).Parental time investment is measured using the average number of hours parents spent helping their children with their homework per day.
Consistent with previous studies (Wang & Zhu, 2021;Zheng et al., 2022), we use parent-child living arrangements to identify students' left-behind status.In particular, a student is considered to be a left-behind child if the student stays in rural areas while at least one parent migrates out to other counties or provinces for work.In the empirical analysis, we include a series of control variables in regressions, encompassing student characteristics (gender, age, only-child, ethnic minority, academic record rank in grade six), family characteristics (father's schooling years, mother's schooling years, family economic status), and class attributes (class size, head teacher's gender, age, working experience, and education level).
Table A1 reports the descriptive statistics of the main variables, as well as the results of group comparisons by left-behind status using two-sample t-tests.Overall, 23.7% of students are LBC in our sample.Regarding the differences in household human capital investments between LBC and non-LBC, it is evident that LBC receive significantly lower levels of parental financial and time investments.Moreover, LBC are more likely to be non-only-children and belong to ethnic minorities.They also tend to have poor learning performance in grade six and come from families with lower socioeconomic status in comparison to their non-LBC counterparts

Evidence on the randomness of classroom assignment
In line with previous studies (Gong et al., 2018;Sacerdote, 2011), we use students' classmates as the primary peer group.A key challenge to estimating the influences of classmates on students' outcomes is the selection problem due to self-selection and confounding heterogeneity.There are mainly two approaches, with tradeoffs, to address such a concern.On the one hand, conducting experiments that randomly assign peer groups can rule out the selection problem in the assignment process.However, randomised controlled trials often have externality invalidity and scale-up problems.Meanwhile, with a slight change in the experiment design, the social interaction patterns could be greatly different (B ed ecarrats, Gu erin, & Roubaud, 2020).On the other hand, although using observed data does not have such problems, it is not clear whether the regression results are causal impacts as experiments even after controlling for fixed effects.Due to the non-experimental data feature in this study, the second strategy is adopted and we believe that the causal effects should be convincing if the randomness of classroom assignment is verified for our sample.
As mentioned in Section 3.1, we restrict our sample to students from middle schools who have been randomly assigned to classes according to the responses of school principals and head teachers.As such, our identification strategy relies on the idiosyncratic variations in the proportion of LBC across the classes within the same grade of each school.To ensure that our empirical results are causal impacts, it is necessary to confirm the randomness of the classroom assignment, particularly ruling out the possibility that the left-behind peer composition is an outcome of school and family choice.Therefore, we conduct several randomisation checks for our sample.
First, if the assignment process is truly random, the characteristics of students in different classes within the same grade of each school should be statistically independent.Given that the CEPS covers two classrooms in the same grade of each school, we check whether the random student-classroom assignment is balanced using these paired classrooms.Specifically, we follow Ammermueller and Pischke (2009) and perform Pearson's v 2 tests to compare a rich set of baseline variables in the survey between the paired classes, including characteristics of students, families, and classes.Table 1 displays the p values of Pearson's v 2 tests for the class assignment of grades seven and nine, respectively.The results show that students' left-behind status and other predetermined variables are not significantly associated with which class they are assigned, suggesting that there is no statistically significant evidence to reject the null hypothesis of random classroom assignment.
Second, if the random classroom assignment assumption holds, it is expected that statistical evidence would support the notion of random assignment of student peers.That is, whether a student is a left-behind child should not be correlated with the share of LBC among other students in the classroom.Following the randomisation check by Guryan, Kroft, and Notowidigdo (2009), we control for the grade-level leave-me-out share of LBC to correct for the mechanical negative relationship between student left-behind status and the proportion of Social spillovers of parental absence 293 LBC among other students 3 .The balance test results in Table 2 show that the class-level leaveone-out proportion of LBC is not significantly associated with students' left-behind status after controlling for grade-by-school fixed effects.Such results also support that the classroom assignment is random within schools in our sample.
Third, in the presence of randomisation, the proportion of LBC in class should not be significantly correlated with the student, parent, and class characteristics.In particular, the allocation of educational resources (e.g. the capability of head teacher) may not be random across classes and correlates with the share of LBC in the classroom even if classes are formed randomly.Therefore, consistent with Gong et al, (2018) and Chen et al. (2021), we perform another balance test and separately regress each baseline variable on class-level LBC proportion.Table 3 reports the regression results and column 1 demonstrates that several variables (e.g.family economic status) are significantly associated with the share of LBC in the class, which is possibly attributed to the variations among different grades and schools.After controlling for grade-byschool fixed effects, the results in column 2 suggest that the correlations between the share of LBC in class and all the predetermined variables are statistically insignificant, suggesting that these characteristics are well-balanced across classes with different class-level proportions of LBC.
To summarize, the results of the above balance tests indicate that we are unable to reject the hypothesis of idiosyncratic variations in the proportion of LBC across the classes within schools for our sample.

Regression model
To investigate the impact of migration-induced LBC on household human capital investments in non-LBC, we employ a reduced-form regression model as follows: where Y ics represents the household human capital investments, including financial and time investments in education, of a non-left-behind student i in class c of school s: propLBC cs denotes the proportion of LBC in class c of school s: X ics represents a vector of control variables, including student, family, and class characteristics.Given that the randomisation of

Social spillovers of parental absence 295
Following Bietenbeck (2020), we use the non-LBC sample for our estimations.Under the random student-classroom assignment assumption, the nature of our identification strategy is to compare the household human capital investments of non-LBC from two classes within the same grade of the same school who share similar characteristics except that one class has a relatively higher share of LBC 4 .As our randomisation checks suggest that students are randomly assigned to classrooms, the share of LBC in a class should be orthogonal to the error term and the estimated results are unlikely to be driven by school/class selections and other unobserved confounders.As a result, the estimates of b can be interpreted as causal effects.To allow for heteroskedasticity and any arbitrary correlations across students within the same classroom, standard errors are clustered at the class level.

Effects of LBC on household human capital investments in non-LBC
Table 4 presents the regression results on the associations between LBC and household human capital investments in non-LBC.We first use students from rural schools that randomly assign students to classrooms as the sample for empirical analysis.The results in Panel A show that the presence of left-behind classmates has significant negative impacts on household financial and time investments in non-LBC's human capital.Specifically, a one-percentage-point increase in the proportion of LBC in class reduces non-LBC's total education expenditure, out-of-school education expenditure, and parental time investment by 1.7, 2.9, and 0.2%, respectively.However, the share of LBC in class is not significantly associated with in-school education expenditure.A possible reason is China's implementation of the free compulsory education reform in 2006, which aimed to alleviate students' in-school education costs.This reform provided a school subsidy package, including tuition fee exemptions, free textbooks, and living subsidies (Xiao, Li, & Zhao, 2017).As a result, the classroom peer effects of exposure to LBC on household education expenditure could be mainly attributed to the reduction of out-ofschool education spending (e.g.private tutoring) rather than in-school expenditure.
In panel B, we further use the sample of students from schools that do not adopt a random student-classroom assignment as a comparison.The estimated results using non-randomised assignment observations help to inform the potential direction of estimation bias if we do not account for the selection problem of the proportion of LBC in class.The estimates in Panel B demonstrate that the coefficients for non-randomly assigned students in classes are substantially larger than those reported in Panel A, especially for family spending on education.In consequence, if we have not excluded the observations that are not randomly selected into classrooms, the pooled sample estimates will overestimate the negative spillovers of LBC on household human capital investments in non-LBC.Therefore, we only use the sample consisting of students from rural schools that meet the requirements of random classroom assignment of this study in the following empirical analysis.

Heterogeneity analysis
In Table 5, we investigate the heterogeneous effects of LBC on household human capital investments in non-LBC.Specifically, we examine whether the social spillovers of LBC differ across subgroups by student gender, grade, parental education, and family economic status.First, the results in Panel A demonstrate that compared with their female counterparts, the share of LBC in class has relatively larger adverse effects on monetary and time investments in non-leftbehind boys.One possible reason is that rural households often allocate more resources to males' education due to son preference, and boys are more susceptible to being influenced by their peers to develop delinquent behaviors (Cameron, Meng, & Zhang, 2022;Knight, Shi, & Deng, 2010).However, the between-group tests show that the coefficients between different genders are not statistically significant 5 , implying moderate heterogeneous effects by gender.Second, the estimates in Panel B suggest that the proportion of LBC in class has significant negative impacts on household human capital investments in non-LBC who are ninth graders, Social spillovers of parental absence 297 while it is not statistically significant for seventh graders.The between-group differences are statistically significant for both financial and time investments.A feasible explanation is that ninth graders experience greater academic pressures due to high school enrollment, leading parents to reinforce support for high-ability children and reduce investment in education for those with 'disruptive' left-behind classmates (Celhay & Gallegos, 2022).Third, we generate two subgroups by parental education, with a threshold indicating whether at least one parent has obtained a high school degree.The results in Panel C reveal that the spillover effects of LBC on the household human capital investments in non-LBC are slightly larger for students whose parents are not well-educated.Similarly, the heterogeneous effects by family economic status in Panel D indicate that the adverse effects of LBC are more significant for household human capital investments among low economic status families.Despite that the between-group differences are not statistically significant, these results suggest that facing greater budget constraints, households with disadvantaged socioeconomic status could be more likely to reduce human capital investments when the proportion of LBC in the class increases.

Possible mechanisms
As aforementioned, there are three potential mechanisms underlying the effects of LBC on household human capital investments in non-LBC, including student skill development, parents' beliefs about returns of human capital investments, and potential 'competition effect' on household human capital investments (see variable definitions for mechanism analysis in Table A2).
First, given that students' social interactions usually take place in the school setting, we examine whether the presence of LBC in the classroom affects their non-left-behind classmates' perceived quality of school life.The results can preliminarily inform us how left-behind peers influence regular students' skills.In Panel A of Table 6, the estimates show that the proportion of LBC in class is statistically significant for all dependent variables of student quality of school life.Specifically, the presence of left-behind classmates is adversely associated with positive peer interactions and class climate, while it is positively linked to students' negative feelings at school.This suggests that students exposed to left-behind classmates are more likely to have negative emotions about school life quality due to peer-to-peer interactions.
Second, we directly investigate the effects of LBC on the cognitive and noncognitive skills among their non-left-behind classmates.The results in Panel B and Panel C of Table 6 demonstrate that a one-percentage-point increase in the share of left-behind classmates adversely affects non-left-behind students' cognitive test scores by 0.234 standard deviations.In addition, having left-behind classmates is significantly associated with lower levels of students' responsiveness ('give quick responses', 'fast learner') and emotional stability (higher levels of negative emotions).
Third, we examine the impacts of LBC on parental beliefs and misbeliefs among non-LBC in the same classroom.Parental beliefs are measured by parental education expectation and confidence in their child's future.Following Wang et al. (2022), we compare parental beliefs about their child's academic achievement with the child's true test scores and compute the gap between the perceived and actual student academic performance as the proxy of parental misbelief.In addition, we create a dummy variable for parents' underestimation of the child's academic ability based on the degree of biased parental beliefs.Columns 1 and 2 of Panel A in Table 7 show that the proportion of LBC in class significantly and negatively affects parental expectations on child education attainment and confidence in the child's future.The results in columns 3 and 4 demonstrate that exposure to left-behind classmates is significantly associated with a downward bias of parental beliefs on non-left-behind students' academic performance.This suggests that the negative effects of LBC on non-LBC's parental perceptions of child skills could not solely be attributed to the adverse impacts of LBC on non-LBC's skill development, but also the distorted parental beliefs induced by peer interactions between LBC and non-LBC.

Notes:
The dependent variables in the first two columns are standardized to zero mean and unit variance.The original values of the dependent variable in column 3 range from À4 to 4 indicating the direction and degree of parental misbelief.To facilitate result interpretation, we have also standardized it to zero mean and unit variance.The dependent variable in column 4 indicates whether parents underestimate their child's academic achievement (yes ¼ 1).Standard errors in parentheses are clustered at the class level.Each regression includes all control variables and controls for grade-by-school fixed effects.Ã p < 0.1; ÃÃ p < 0.05; ÃÃÃ p < 0.01.

Social spillovers of parental absence 299
To validate this argument, we further restrict our sample to non-LBC whose parents know some or all of their child's friends at school.The rationale for this sample restriction is that if parental beliefs stem from both changes in true student skills and distorted beliefs resulting from students' social interactions, the spillover effects of LBC on non-LBC's parental beliefs would be more pronounced for those parents who have observed such interactions.The estimates in Panel B report consistent results regarding the LBC's impacts on non-LBC's parental beliefs and misbeliefs.Comparatively, the effect sizes are relatively larger than those in full sample estimates, revealing that parental belief about child skills could be directly distorted by the skills of students' peers (Kinsler & Pavan, 2021).
Fourth, another possible explanation for our main findings is that the presence of LBC may relieve 'competition' in household human capital investments.If this is true, then the effects of LBC on household human capital investments in non-LBC are expected to be larger among peers with similar academic performance.As such, we re-define the peer group of each student as classmates whose ranking distance is within 5, according to students' academic record rank in grade six 6 .We then compute the share of LBC based on the newly defined peer group and re-run the baseline regressions.The results in Panel A of Table A3 also suggest the negative effects of LBC on household human capital investments in non-LBC.However, compared with baseline regressions, the effect sizes of the proportion of LBC in the new peer group are smaller.We have additionally re-defined the ranking distance of a peer group to 10 and 20.The results presented in Panel B and Panel C of Appendix Table A3 demonstrate that the effect sizes of the LBC escalate, rather than diminish, with the increase of student ranking distance.Consequently, the social spillovers of LBC on household investments in non-LBC's human capital are not significantly driven by peers with similar academic performance but rather by peer networks based on actual interactions.

Robustness checks
To ensure the validity of our main findings, we have conducted a variety of robustness checks (see variable definitions and results in Supplementary Materials).First, we test the sensitivity of our results to alternative measures of household human capital investments, including students' time spent on tutoring courses and parenting styles (Guo & Qu, 2022).The estimation results in Table S1 show a significant negative association between the presence of left-behind classmates and the time non-LBC spent on tutoring courses during weekdays and weekends.While the share of LBC in class has no significant effect on parental 'demandingness' for non-LBC, it is significantly associated with lower levels of parental 'responsiveness' 7 .
Second, we check the sensitivity of our results using three alternative measures of LBC, including (i) left behind by fathers only, (ii) left behind by mothers only, and (iii) left behind by both parents 8 .Panel A of Table S2 shows that no matter using which alternative measure of LBC, the proportion of left-behind classmates also significantly and negatively affects household investments in non-LBC's human capital.It is worth noting that the share of classmates left behind by both parents has a greater adverse impact compared with other definitions of LBC, which could be attributed to the cumulative effect of parent-child separation (Zheng et al., 2022).
Third, we include additional controls to account for potential contextual characteristics.Following Eisenberg, Golberstein, and Whitlock (2014), we additionally control for average characteristics of LBC and non-LBC in our estimations.The contextual characteristics include the proportion of boys, average paternal schooling years, average maternal schooling years, and average family economic status among LBC and non-LBC in class, respectively.The regression estimates in Panel B of Table S2 demonstrate that despite declining effect sizes, the results still support the negative spillovers of LBC on household human capital investments in non-LBC.
Fourth, although the initiators and recipients of spillover effects are clearly separated in our setting, we still perform regressions for the sample including LBC to test whether such a sample selection influences the validity of our results.In Panel C of Table S2, the results also confirm the negative effects of LBC on household investments in non-LBC's human capital.Comparatively, the effect sizes of the LBC with the inclusion of the LBC sample are slightly larger than our baseline estimates, indicating that social interactions among LBC may have stronger detrimental spillovers on household human capital investments.
Fifth, we employ Lewbel (2012)'s instrumental variable (IV) approach using a two-stage least squares (2SLS) estimator.Lewbel (2012)'s approach serves to identify structural parameters in regression models with endogenous regressors in the absence of external instruments.Previous studies have shown that the estimation results using Lewbel (2012)'s IV strategy are very close to those using traditional validated IVs (Churchill & Smyth, 2017).In Panel D of Table S2, the results of the Breusch-Pagan test for heteroskedasticity confirm the satisfaction of the precondition for identification using Lewbel (2012)'s approach.The Kleibergen-Paap rk Wald F statistics, well above 10, indicate that the internally generated instruments are unlikely to suffer from a weak instrument problem.Moreover, the 2SLS estimates remain qualitatively similar to our baseline results.
Sixth, to account for the sensitivity of the baseline estimates to the unobservables, we perform Oster (2019)'s bounding analysis for omitted variable bias based on the selection of observables in the linear regression framework.In particular, we follow Zheng et al. (2022) and assume a R max ¼1.3(R 2 ), where R max is the goodness of fit when controlling for all observed and unobserved factors correlated with household human capital investments and R 2 is from linear regressions with observed controls.Panel E of Table S2 shows that the upper bounds of the estimates for household human capital investment outcomes (except in-school expenditure) do not include zero.Meanwhile, the values of the proportional degree of selection on unobservables to selection on observables (d) range from 1.455 to 4.487 when the observed effects are eliminated.This suggests that there are at least partial causal effects between LBC and household human capital investments in their non-left-behind classmates.
Finally, considering that the estimated results might be contaminated by the inclusion of observations from schools that claimed to use randomisation design but did not, we implement another sensitivity test to verify our identification assumption.Following the approach used by Gong et al. (2018), we randomly exclude approximately 10% of schools (3 schools) to obtain a reduced sample and corresponding regression estimates.This process is repeated 1,000 times to generate the coefficient distribution for each human capital investment outcome.The results in Figure S3 demonstrate that the distribution of coefficients using reduced samples centers around our baseline estimates and the upper and lower bounds of the coefficient distribution share the same negative sign for household human capital investments (except in-school expenditure).Therefore, our results are not likely to be driven by potential bias due to the possible inclusion of schools that non-randomly assigned students to classrooms, suggesting that our baseline results are robust.

Conclusion
Using data from the China Education Panel Survey, this study sheds light on social spillovers of the presence of left-behind classmates on household human capital investments in students living with both parents in rural China.We focus on China's rural middle schools that implement a randomised student assignment to mitigate concerns of selection bias due to the nonrandom selection of students into classrooms.To the best of our knowledge, this is the first study that extends the consequences of parental absence caused by rural-to-urban migration from the well-being of LBC to household behaviors among their non-left-behind classmates.Additionally, our study expands the understanding of peer effects within classrooms, which has Social spillovers of parental absence 301 predominantly centered on student outcomes, by examining the adverse influences on household human capital investments.
The findings of our study suggest that the share of LBC in class has negative impacts on household financial and time investments in non-LBC's human capital, especially out-of-school education expenditure.These findings are robust to a series of robustness checks, including alternative measures of household human capital investments, additional controls for contextual characteristics, Lewbel (2012)'s IV approach, sensitivity to unobservables, and tests for the randomised assignment assumption.Our heterogeneity analysis shows that the detrimental effects of LBC on non-LBC's household human capital investments are comparatively more pronounced for students who are boys, in grade nine, and living in households with low socioeconomic status.We have also investigated potential mechanisms underlying the spillover effects of LBC on household human capital investments in non-LBC.The results indicate that the share of LBC in class adversely affects student quality of school life, cognitive and noncognitive skills, thereby influencing parental beliefs about their child's skills and returns on human capital investments.Moreover, apart from variations in parental beliefs driven by changes in student skills, negative social interactions between LBC and non-LBC can also directly distort parental beliefs, even if their child's skills remain unchanged.
Our findings have important policy implications for the human capital development of rural children.As suggested by our results, the total impacts of parental migration on home communities could have been substantially underestimated without accounting for the spillovers of LBC on non-LBC and their household behaviors.Previous studies have shown that parental migration has adverse impacts on short-and long-term human capital accumulation and labour market outcomes (Liu, Zheng, Parker, & Fang, 2020;Zheng et al., 2022).Our results further indicate that parental absence also negatively affects household human capital investments in non-LBC, which could subsequently affect their human capital accumulation and urban-rural inequality of socioeconomic status in the long run.Considering the widespread prevalence of rural-to-urban migration not only in China, our findings are also relevant to other developing countries such as India, Indonesia, and Vietnam.Consequently, evidence-based interventions, such as reducing migration restrictions and implementing parenting programs for caregivers of rural children, should be pursued to mitigate the adverse effects of involuntary parent-child separation, not only for LBC but those non-left-behind ones.
This study has several limitations due to data constraints.First, although we have explored the social spillovers of parental absence using different definitions of LBC, we are unable to investigate the heterogeneous effects based on the duration of parental migration.Second, the lack of information prevents us from considering time spent by other child caregivers (e.g.grandparents) when measuring household time investments in human capital.Third, as our sample comprises middle school students rather than children of all ages, caution should be exercised in generalising the findings of this study.Despite these limitations, our study offers novel evidence on the 'costs' of rural-to-urban migration in sending areas, particularly focusing on the decision-making of human capital investments among non-left-behind households.Notes 1. See Zheng et al. (2022) for a literature review on the effects of parental migration on the short-and long-term human capital of left-behind children.2. See Supplementary Materials for more details about the sampling design of the CEPS.3. Given that the individual himself is removed from the 'urn' (the same grade) from which his peers are chosen, the peers for LBC are selected from a group with a slightly lower proportion of LBC than the peers for non-LBC.As a result, there is a mechanical negative relationship between one's left-behind status and the share of LBC of randomly-assigned peers.According to Guryan et al. (2009), such a bias could be corrected by simply controlling for the proportion of LBC of all individuals in the 'urn'.4. The kernel distribution and descriptive statistics of class-level LBC proportion are shown in Supplementary Materials Figure S1.
5. We conduct seemingly unrelated estimations for the subsamples and statistically test the between-group differences in coefficients of the LBC variable (the proportion of LBC in a class).6.Using the predetermined academic record in grade six, instead of contemporary test scores, helps address potential simultaneity issues.We have also used contemporary test scores on subjects of Chinese, mathematics, and English to compute students' ranking distance.The results remain qualitatively similar, which are available upon request.7. We have also examined the associations between the proportion of LBC in class and each item of the parental 'demandingness' index and 'responsiveness' index.As shown in Supplementary Materials Table S3 and Table S4, the estimation results remain qualitatively similar.8.The kernel distribution and descriptive statistics of the proportion of LBC in class using alternative definitions of LBC are shown in Supplementary Materials Figure S2.
Table A2.Social spillovers of parental absence 307 the child's academic achievement Biased parental belief in child achievement is measured by comparing their beliefs regarding academic achievement with their child's true performance in three subjects: Chinese, English, and mathematics.Parents are asked to report their beliefs about their child's academic rank in their class (1¼ very low, 5 ¼ very high).Students' actual academic achievement is measured by Chinese, English, and mathematics test scores in the mid-term examination in the autumn of 2013.We group the average test scores of the three subjects within the class into five categories based on equal percentiles.Parental biased belief is measured by the gap between the perceived and actual student academic performance.variable regarding parents' biased belief in their child's academic achievement, we further generate a dummy variable indicating whether parents underestimate their child'All variables are standardized to zero mean and unit variance besides dummy variables in the empirical analysis.

Table 1 .
Pearson's v 2 tests for the class assignment, p values student-class assignment is largely dependent on the variations of grades and schools, we have included grade-by-school fixed effects D gs in the regression model to control for grade-byschool level factors that may affect students' school choices and subsequently confound with the associations between class-level proportion of LBC and non-LBC's household human capital investments.e ij denotes the unobserved error term.

Table 3 .
Balancing test: class-level LBC proportion and characteristics of student, family, and class Standard errors in parentheses are clustered at the class level.Ã p < 0.1; ÃÃ p < 0.05; ÃÃÃ p < 0.01.

Table 4 .
Effects of LBC on household human capital investments in non-LBC Notes: Standard errors in parentheses are clustered at the class level.Each regression includes all control variables and controls for grade-by-school fixed effects.ÃÃ p < 0.05; ÃÃÃ p < 0.01.

Table 5 .
Heterogeneity effects of LBC on household human capital investments in non-LBC Notes: Standard errors in parentheses are clustered at the class level.Each regression includes all control variables and controls for grade-by-school fixed effects.The coefficients and corresponding standard errors in each cell come from one regression.In Panel D, if the answers to the parent-reported question 'how is the financial condition of your family at present' are 'somewhat rich' or 'very rich', we then treat the households as high economic status families.Otherwise, they are regarded as low economic status families.Group differences (p value) are from between-group tests of difference in coefficients after seemingly unrelated estimations (SUEST).Ã p < 0.1; ÃÃ p < 0.05; ÃÃÃ p < 0.01.

Table 6 .
Possible mechanisms: student quality of school life, cognitive and noncognitive skills Notes: The dependent variable in each regression is standardized to zero mean and unit variance.Standard errors in parentheses are clustered at the class level.Each regression includes all control variables and controls for grade-by-school fixed effects.Ã p < 0.1; ÃÃ p < 0.05; ÃÃÃ p < 0.01.

Table A3 .
Possible mechanisms: peer competition Standard errors in parentheses are clustered at the class level.Each regression includes all control variables and controls for grade-by-school fixed effects.Ã p < 0.1; ÃÃ p < 0.05; ÃÃÃ p < 0.01.308X.Zheng and Y. Zhou