The Causal Effect of Growing Up In A Two-Parent Household On Child’s Adult Earnings

The children who grew up in two-parent households earn more in adulthood than the other children reared in non-intact parental families. I identify the causal effect of growing up in a two-parent household on the child’s adult earnings by using the following specification strategies. First, I control the parental income and educational attainment, along with the child’s demographic characteristics (gender, age, race, region, and tenure), to disentangle the childhood family intactness effect from the other parental influence. Second, I consider the unobserved clan-specific heterogeneity, such as family traditions and genetic characteristics, by fitting the clan fixed effects (FE) models. Third, I use two instrumental variables (IVs), the state divorce rate and the no-fault divorce law effectiveness, for the endogenous childhood family structure and adopt the two-stage least squares (2SLS) approach. The estimates suggest that the adult children who grew up in a two-parent household earn about 16% more than their counterparts from non-intact families, other things equal. The effect goes through three channels. It varies with parental income, education, and child’s gender. Growing up in a two-parent household also encourages intergenerational relative-earnings improvement.


The Causal Effect of Growing up in a Two-Parent Household on Child's Adult Earnings
A stable parental marriage plays an essential role in the child's well-being.Unfortunately, the percentage of two-parent families has dropped over time in the United States, as illustrated in Figure 1.The Panel Study of Income Dynamics (PSID) surveys show that among all the families with at least one child under 18 years of age, the percentage of two-parent families declined gradually from 75% in 1968 to 64% in 2017.Would the childhood family structure influence the children's labor market outcomes?As far as the child's adult earnings are concerned, they are associated with whether the child grew up in a two-parent household.In the paper, I define the children growing up in a two-parent household as they lived with both parents for all 16 years of their childhood.If the parents had ever experienced divorce, separation, being widowed, or being a single parent at any time during the child's childhood, the child is not categorized as growing up in a two-parent household.Table 1 shows that the mean hourly earnings are significantly different between the two groups of adult children defined by whether growing up in a twoparent household.Take the male children as an example.Adult sons growing up in two-parent households for their entire childhood earned 11.03 dollars per hour more than the other adult sons in 2017.

Research Questions
The association or correlation between growing up in a two-parent family and the child's adult earnings does not necessarily mean causality.It could be due to a relevant factor, such as the parent's educational attainment, that influences both the child's adult earnings and her childhood family structure.In the paper, I intend to answer the following questions.First, could growing up in a two-parent household explain the child's adult earnings?And second, through which channels does childhood family structure affect the child's earnings?I need to tackle three identification issues to find a causal relationship.First, I should disentangle the two-parent household effect from the other parental influence.Second, I must consider the endogeneity problems due to the unobserved family heterogeneity.Third, I need to consider whether there is other omitted variable bias.

Background Information
There is a lack of consensus about the parent's presence effect on the child's adult earnings.Lang and Zagorsky (2001) find no evidence in the United States that parental presence affects the child's adult income after using a variety of background controls, including parental educational attainment and the child's race and regional dummies.Corak (2001) finds that in Canada, parental divorce lowers the adult earnings of sons by 3%, but there is no influence on daughters.Kane et al. (2010) find that in the United States, an absence of a biological parent for reasons other than death reduces a child's lifetime earnings between 3 and 12 percent.
It is generally accepted that parental divorce has a detrimental effect on the child's educational attainment and marital behavior (Keith and Finlay, 1988;Corak, 2001;Dronkers and Härkönen, 2008;Le Forner, 2020).Le Forner (2020) finds that parental separation is linked to poorer educational attainment for their children in France, from 32% to 12% of a standard deviation lower.The effect varies with age, and it is more detrimental to boys.Corak (2001) finds that children from divorced families are more likely to put off marriage and, once married, more likely to suffer separation and divorce.
As long as intergenerational earnings mobility is concerned, children of divorced parents are more likely to fall into a lower earnings distribution in adulthood.Bratberg et al. (2014) show that children of divorced parents tend to move downward in the earnings distribution compared to children from intact families in Norway.Couch and Lillard (1997) find that sons from families whose divorced parents had relatively low earnings have a greater chance of having low earnings themselves in the United States.
When it comes to identifying the parental marital effect, sibling fixed-effects models, difference-in-differences (DID) method, and instrumental variables (IVs) estimation are used to establish the causal relationship.Le Forner (2020) uses a sibling-differences model to estimate the divorce effect on the child's educational attainment.Corak (2001) treats parental death as an exogenous variable and uses DID method to estimate the true impact of divorce on the child's adult earnings and marital behavior.I use data from the Panel Study of Income Dynamics (PSID) to estimate the causal effect of growing up in a two-parent household on the adult earnings for children of prime working age (25 to 54 years old) in the United States.To the best of my knowledge, it is the first attempt to apply the clan fixed-effects (FE) model combined with the two-stage least squares (2SLS) method to the individual-level data and measure the causal effect of growing up in a two-parent household on the child's adult earnings.The paper provides four main contributions.First, I define growing up in a two-parent household by looking at their child's entire childhood instead of at a certain point in time.Second, it is the first time to make use of the clan-descendant structure of the PSID data and introduce the clan fixed effects (FE) method.Third, I use the state-level divorce rates and no-fault divorce law effectiveness as instrumental variables and employ the two-stage least squares (2SLS) approach.Fourth, it is the first attempt to look into the mechanism behind the effect and conduct the mediation analysis.I treat the child's education, health, and marriage as three channels through which growing up in two-parent household influences the child's adult earnings.I find a positive and significant two-parent household causal effect on the child's adult earnings after controlling for other parental influence and accounting for endogeneity problems.The purpose of this research is to add the causal effect of childhood family structure on the child's adult earnings to the relatively scarce intergenerational studies.It could be seen as a complement to the existing literature and helps better understand the parental factors behind the child's labor market success.

Parental Utility Maximization Model and its Predictions
I follow the idea of the Becker-Tomes (1979) model and modify it to accommodate the parent's marital stability variable.The following utility maximization model describes the parent's marriage-specific investment behavior.The decision-making parent chooses her own composite consumption c and her child's future earnings capacity y to maximize her utility subject to the household resources constraint, where  is parent's marital status measure and  ∈ 0,1 .A higher value of  implies a more stable marriage.In the empirical part of this paper, I use a dummy variable to measure marital stability. 1 if parents maintained married or permanently cohabiting during their child's childhood, i.e., the child grew up in a two-parent household;  0 if parents have ever divorced or separated. is parent's investment in the child. represents parental household resources. is the accessibility parameter or the efficiency parameter of household resources. is a function of marital status , and / 0.  is the importance of the child's future earnings relative to the parent's own consumption.The choice of the utility function and the assumption are based on the following four property rights views on the marriage-specific investments in children.

Incentives to Invest in Children
The parent's incentives come from two sources: financial benefit and psychological wellbeing.The former is the returns to investment the parents expect to obtain after children grow up.The higher the returns, the stronger the parent's incentives.The parents can rely on their children in case they encounter financial difficulties in the future.In the parent's utility function, the child's future earnings capability becomes part of the parent's utility because parents will benefit from it.The incentives can also be seen as a result of altruistic behavior, as in the Becker-Tomes model (1979).That is, the parents care about their children's future income and happiness because of altruism.Parents often enjoy this type of psychological well-being.Both the financial benefit explanation and the altruistic assumptions lead to the same utility model setup, i.e., parents are concerned about both their own consumption and their child's future earnings capability, as shown in model (1).

Control over Household Resources
The ownership of and the access to household resources make it possible for the parents to invest in children's education and nutrition.Parents face resource constraints when they are making decisions.The more resources they control, the more investment they can make in their children.The household resources constraint equation reflects the marital relationship between spouses.The couples living together are more likely to have access to each other's resources than those living apart, i.e., / 0.

Transaction Cost of Household Production and Efficiency of Using Household Resources
The efficiency of using household resources could be different between married couples and divorced parents.Marriage as an arrangement of rights and power over household resources could reduce transaction costs of producing household goods.The most valuable household goods, without a doubt, are their children.Compared to divorced parents, married parents have lower monitoring and enforcement costs due to trust and proximity.A two-parent household can also exploit the comparative advantage and benefit from household-market specialization.This is the reason why the efficiency if using household resources is higher for married parents than for divorced parents.The parameter  can be seen as the efficiency parameter of the household resources.The transaction cost theory provides another justification for the assumption of / 0.

Risk of Losing the Returns to Investment and Outside-Marriage Options
A breakdown of marital relations leading to a divorce would prevent parents from investing in their children because the spouses know that they may lose the returns to investment after the divorce.The marriage-specific investment by one partner bears the risk of expropriation by the other partner if the divorce happens.The parents in marital difficulties face the uncertainty of the future, which decreases marriage-specific investment.Moreover, investment options outside the marriage are available for people experiencing a marital breakdown.These outsidemarriage options include the opportunity of starting a new relationship or raising children from a new family.The investment in original children may be shifted to those outside-marriage options.Thus, the effective household resources for the original children could be higher in intact families.It offers the third reason for / 0.
Child's adult earnings  are related to the parent's investment  as where  is the returns to parent's investment.I rewrite the household resources constraint using equations (2) as   /  . (3) The interior solution for the child's adult earnings is: The equilibrium child's adult earnings  * depend on a variety of factors, such as the parental household resources  , the household resources efficiency measure   , and the relative importance of child .Let's take a look at how the child's adult earnings change with whether the child grew up in a two-parent household.First, the marginal effect of growing up in a two-parent household on the child's adult earnings is positive. * /   / 0 because / 0 by assumption.It predicts that the children from intact parental families earn more than their counterparts reared by non-intact families.Second, this marginal effect increases with household resources  because   * / /  / 0. For highincome households, family intactness has a greater effect on the child's adult earnings than the low-income households.Since educational attainment is usually positively related to income, the effect should also be more significant for better-educated parents.Third, the two-parent household effect on the child's adult earnings is positively related to the relative importance of the child's well-being  because   * / /  / 0. If parents care about their children differently based on gender, the two-parent household effect on son's and daughter's adult earnings would be different in magnitude or significance.
The theoretical model predicts that childhood family structure is crucial in determining the child's adult earnings, and the effect may vary with the parental household income, parental education, and the child's gender.Based on the model's prediction, I propose the two-parent household hypothesis on the child's adult earnings.

Two-Parent Household Hypothesis on Child's Adult Earnings
H1. Positivity of two-parent household effect on child's adult earnings.
Growing up in a two-parent household for entire childhood has a positive effect on the child's adult earnings, holding other factors constant.

H2. Heterogeneity of two-parent household effect on child's adult earnings.
The two-parent household effect on the child's adult earnings is higher for wealthier or better-educated parents, and it could be different between sons and daughters.

H3. Mechanism of two-parent household effect on child's adult earnings.
The analysis of the parent's investment in the child implies that growing up in a twoparent household could affect the child's adult earnings through three observable channels.The first channel is the "child's education" channel.Parents living together invest more in their child's education, and better educational attainment results in higher earnings.The second channel is the "child's health" channel.The children who grow up in intact families receive better nutrition than the children raised by divorced or separated parents.Health in adulthood will affect earnings.The third channel is the "intergenerational marriage persistence" channel.The parent's marital status shapes the child's marital attitude and behavior.

Identification Challenges, Specification Strategies, and Summary Statistics
The econometric model that links the child's adult earnings to whether they grew up in two-parent households is given as (5).For the j descendant of the i clan, where    .
All the variables are measured at the individual level.The sample children were of the prime working-age (25 to 54 years old) in the United States who earned labor income in 2016 (reported in the survey year 2017).The dummy variable growing up in a two_parent household is equal to 1 if parents lived together and maintained married or cohabiting for all their offspring's 16 years of childhood.It is equal to 0 if parents have ever experienced the other marital status, such as divorce, separation, being widowed, or being single, during their offspring's childhood years.It is the key explanatory variable of interest in this research.The way to construct the dummy variable can be found in Appendix Figure 1.There are rare cases in which the parent reported "married" in the two consecutive surveys, but she got divorced, separated, or widowed and then remarried between these two consecutive surveys (PSID surveys were annual from 1968 to 1997 and biennial from 1997 to 2017).In these cases, their children were not growing up in a two-parent household for their entire childhood.After sample weights adjustment, 59.70% of the adult children grew up in a two-parent household for their entire childhood, and 40.30% of the adult children did not live with both parents for their entire childhood, as is shown in Table 2.The logarithm of the adult child's hourly earnings reported in the 2017 PSID survey is the outcome variable.The hourly earnings are the annual labor income divided by annual work hours.Annual labor income is the sum of wages and salaries, bonuses, overtime, tips, commissions, and other labor income.The first row in Table 3 confirms that the mean hourly earnings between the two groups of children defined by whether growing up with both parents are significantly different from each other.The vector  in equation ( 5) represents the control variables in the model, which includes parent's household income, parental educational level, child's gender, age, race, region, and job tenure.Notice that the parent's marital status and household income data were collected in real time every year (every two years after 1997) from 1968 (the first wave) to 2009, when the children were between 1 and 16 years old.The data are more accurate than the retrospective surveys.In Table 3, all the sample means have been adjusted using the sample weights to correct the oversampling of poor families in PSID that results in a disproportionately large number of lowincome households and African Americans.Using sample weights makes the statistics and estimates more representative of the US population.To identify the causal effect of growing up in a two-parent household on the child's adult earnings, I face at least three challenges.I come up with three specification strategies to alleviate the bias due to endogeneity issues.Does the childhood family structure or growing up in a two-parent household causally affect the child's adult earnings, or does the correlation reflect some other parental factors?If parental income and parental educational attainment are omitted from the model, it will lead to biased estimates because they are determinants of the child's adult earnings, and they are also correlated with parental marital status.From Table 3, we know that for the adult children from intact families, 46.41% of them have at least one parent who is a college graduate, while the percentage is only 30.34% for the children who did not always live with both parents in their childhood.The parents in non-intact families have a lower level of educational attainment than the parents who maintained an intact marriage during their offspring's childhood.The intact families also have more household resources on average.The household income consists of three components: the household taxable income, the household transfer income, and the household social security income.It is the average over the 16 years of the offspring's childhood.Each year's household income has been adjusted to 2017 US dollars using the personal consumption expenditures price index (PCE).I use per-parent household income because I assume that the consumption expenditures of two parents would be twice the expenditures of one parent.The household resources the parent can use for her own consumption and the investment in her child are the amount after taking away spouse's expenditures.I include parental household income and parental educational attainment in the model to isolate the effect of growing up in a two-parent family from the parental income and education effects.I also hold the child's demographic characteristics constant.The child's age, gender, race, current region, and tenure at the present job are not randomly distributed between the children living with both parents in their entire childhood and the other children.Table 3 illustrates the differences.For instance, in the sample, the adult children who grew up in two-parent families are about 1.5 years older than their counterparts ever reared in a one-parent household on average.I also find a disparity in the race composition between the two groups of adult children.There is a higher proportion of whites in the group of children who grew up in a two-parent household for their entire childhood.Whites compose 89.71% of them.The proportion is 71.83% in the other group of children.There is a 17.88% difference between them.The percentage of African Americans in the group of children who were not always living with both parents in their childhood is 25.57%, which is much higher than the entire sample average, 14.40%.Since the child's demographic characteristics could be correlated with both their earnings and their childhood family structure, leaving them out of the model could lead to bias.In addition to parental income, parental education, child's gender, age, and race, I include the average number of children in the parental household in the model to account for the concern that household investment in each child may decrease with more children.The child's current region is added to the model to control for the region earnings difference.The child's job tenure with the current employer would affect her hourly earnings too, so I put it in the model.

Unobserved Clan-Specific Heterogeneity and Clan Fixed-Effects (FE) models
There is something in common that runs in the blood through generations of descendants.It is the unobserved clan-specific heterogeneity, such as family traditions or genetic characteristics.They affect both the child's adult earnings and childhood family structure.For example, patience is an unmeasured virtue that is closely related to an intact family.Children inherit their parent's patience.The child's patience has a positive impact on her job performance and earnings.Patience is such an unobserved clan-specific heterogeneity that could bias the OLS estimates if we do not consider it.Genetic traits, such as attractiveness, could be passed down from parents to children.Attractiveness could be positively associated with both the intactness of parental household and the child's adult earnings.If patience or attractiveness is omitted from the model, it will lead to an upward bias.I take advantage of the clan-descendant structure of the PSID survey data and fit the clan fixed-effects (FE) models to take into account the unobserved clan-specific factors.

Family-Descendant Structure of the Sample
PSID surveys began in 1968 with 4,802 original families.The surveys traced children as they grew older and formed their own families, making it possible to link children's information to their parents'.The descendants of 1,337 original families reported their earnings in the 2017 PSID survey.I call all the descendants from the same original family a clan.The number of descendants of each clan ranges from 1 to 18, with an average of 2.8.I draw the family tree of a clan in Figure 2 to illustrate the structure of the data.In this clan, fourteen descendants were surveyed when they became the heads or the spouses of the heads in 2017.I put them into rectangles.The descendants could be siblings or cousins.They could be the same generation or different generations.PID11, for example, is the uncle of the other respondents.I assume that some unobserved factors are clan-specific and descendant-invariant, i.e., they are identical for all descendants within the same clan but varying across different clans.In the regression model ( 5), the composite error term  contains the clan fixed effects  and the idiosyncratic component  .In the clan FE model, we obtain the within-estimator by performing demeaning within clans over descendants so that the clan-specific heterogeneity  could be removed from the model.Thus, the clan FE model allows growing in a two-parent household to be correlated with the clan-specific heterogeneity  but we still have a consistent estimate of the two-parent household effect on the child's adult earnings.

Unobserved Idiosyncratic Factors and Instrumental Variable (IV) Estimation
Even after the clan fixed effects (FE) have been accounted for, the unobserved idiosyncratic factors  that affect the child's adult earnings could still be correlated with the childhood family structure and therefore lead to bias.An instance of such unobserved omitted variables could be the parent's social network.Parents with intact marriages are more likely to have a stronger network of friends who could help their children with their jobs.In this case, the effect of growing up in a two-parent household on the child's earnings will be overestimated using OLS because the parent's social network is positively correlated with childhood family intactness and the child's adult earnings.Another example of omitted variable bias comes from the "bad" kids' story.Parents could divorce because their kids are "bad" in the sense that they cause too much aggravation to the parents.These "bad" kids do poorly in the labor market and earn less than the "good" kids in adulthood.If the kid's quality is omitted from the model, it will cause upward bias because it is associated with both childhood family structure and child's adult earnings in the same direction.On the other hand, there could be omitted variables that result in a downward bias of the OLS estimates.Some child's personality traits, such as endurability or adversity quotient (AQ), develop from an unhappy childhood environment.Having a happy childhood with both parents is negatively related to the child's AQ formation, and AQ is usually positively associated with earnings capacity.As a result, the two-parent household effect on the child's adult earnings will be underestimated.Notice that these unobserved factors are different between individuals and therefore could not be eliminated by using the clan fixed effects model.To alleviate the bias, I find instrumental variables (IVs) for the endogenous childhood family structure and perform the two stage least squares (2SLS) estimation.The idea of 2SLS is that in the first stage, the endogenous variable, growing up in a two-parent household, is projected onto the instrumental variable space.The projected variation is exogenous in the sense that it is not correlated with the error term  , and therefore it leads to consistent estimates.In the fixed effects (FE) framework, the 2SLS model and the requirements for  as a valid IV is

OLS, Clan FE, and 2SLS Estimates of Two-Parent Household Effect on Child's Adult Earnings
I present the pooled OLS models, the clan fixed effects (FE) models, and the two-stage least squares (2SLS) models as an attempt to consistently estimate the causal effect of growing up in a two-parent household on the child's adult earnings.

OLS Estimates of Two-Parent Household Effect on Child's Adult Earnings
The OLS estimates for the effect of growing up in a two-parent household on the child's hourly earnings decline as more control variables are added to the model, as shown in Table 4.It is 0.326 when the "growing up in a two-parent household" dummy is the only explanatory variable.It reflects the gross correlation between childhood family structure and the child's adult earnings.It reduces to 0.257 when the number of children in the parental household and the child's demographic characteristics are added.As the parental household income and educational attainment are included in the model, the two-parent household effect declines to 0.168.It is statistically significant at the 1% level.The OLS estimate implies that the children who grew up in two-parent households earn 16.8% (or precisely 18.3%) more than the other children who did not grow up in a two-parent family for their entire childhood, holding relevant factors constant.In other words, for two children with the same age, gender, race, region, tenure, and they have parents with the same average household income during childhood and same educational level, the one from an intact family earns eighteen percent more than the other from a non-intact family.

Clan Fixed Effects (FE) and Random Effects (RE) Estimates of Two-Parent Household Effect on Child's Adult Earnings
The clan random-effects (RE) model gives an estimate of 0.143, while the clan fixedeffects (FE) model yields an estimate of 0.077.In the clan RE model, both within-clan variation and between-clan variation are used.In the clan FE model, only the within-clan variation is used.Using only within variation leads to less-efficient estimation with larger standard errors, smaller z-statistics, and lower significance.However, the clan FE model has its advantages.The clan FE model takes into account the unobserved clan-specific heterogeneity, i.e., the clan fixed effects  .If the clan fixed effects are correlated with whether the children grew up in two-parent households (usually they are), only the clan FE model provides a consistent estimate.The choice between RE and FE models is a trade-off between efficiency and consistency.In the research, the Breusch-Pagan test confirms the presence of clan-specific heterogeneity, and the Hausman test prefers the clan FE to RE.As shown in Table 5, on average, the children who grew up in two-parent households for their entire childhood earn 7.7% (or precisely 8.0%) more than their counterparts reared in non-intact families, holding both observed factors and unobserved clanspecific factors constant.The clan fixed effects estimate is lower than the pooled OLS estimate, which implies the presence of clan-specific omitted variables, such as patience and attractiveness.

2SLS Estimates of Two-Parent Household Effect on Child's Adult Earnings.
Two candidates of the instrumental variables (IVs) for the endogenous explanatory variable, growing up in a two-parent household, are the state divorce rate and the no-fault divorce law effectiveness at the parent's critical marital moment.The parent's critical marital moment is defined as the divorced or separated year for the ever-divorced or ever-separated parents.It is the child's birth year for the other parents.The first IV, the state-level divorce rate of each year, is calculated using data from the Current Population Survey (CPS).The second IV is a dummy variable of the no-fault divorce law that took effect in different years in different states.It is equal to 1 if the law was in effect at the parent's critical marital moment; it is equal to

Table 5. Clan Fixed-Effects (FE) and Random-Effects (RE) Estimates
0 if the law had not been passed.The two-stage least squares (2SLS) method is adopted to further alleviate the omitted variable bias.Column 1 of Table 6 uses one IV, while column 2 uses both IVs.I employ the 2SLS method in the fixed effects framework in columns 3 and 4. The first stage regressions verify a strong correlation between the instrumental variables and the childhood family structure.As the first column of Table 6 shows, a one percentage point increase in the state divorce rate is associated with a fifteen percentage point decrease in the probability of children growing up in a two-parent household in that state.The state-level divorce rate is highly correlated with the individual-level probability of growing up in a two-parent household.The second column shows that the no-fault divorce law effectiveness influences parent's marital behavior.The implementation of the no-fault divorce law reduces the probability of growing up in a two-parent household by 20.3% when it is jointly used as IVs with the state divorce rate.The first stage summary statistics confirm that the IVs are not weak instruments for three reasons.First, they are statistically significant, and the F statistics are high enough to reject any criteria of a weak instrument.Second, the IVs have decent prediction ability for childhood family structure.They correctly predict more than 60% of the observations alone, while the percentage of correct predictions is around 73% using all the regressors.Third, the IVs have explanatory power.Take the second column as an example.The two IVs explain 19.3% of the variation of the endogenous variable, while all the regressors together explain 30.7%.On the other hand, the state-level divorce rate and the no-fault divorce law effectiveness are not likely to affect the child's adult earnings directly or indirectly through any channels other than through parental marriage.This exclusion restriction requirement is very likely to be satisfied because the state-level divorce rates in the parent's generation have little to do with the individual-level child's adult earnings after they grew up.The child's unobserved qualities and characteristics would not change the state divorce rates at the time when they were children.The no-fault divorce law implementation is a government policy that was introduced not based on the child's adult earnings or the omitted variables that determine the child's earnings.The reduced form subset regressions provide support for this argument.I show the subset regression results in Appendix Table 3.I discuss the evidence of exclusion restriction in the robustness check and discussion section.The second stage gives the 2SLS estimates of the two-parent household effect on the child's adult earnings.It is 0.157 with one IV, and it is 0.161 with both IVs.An ideal specification is a combination of 2SLS and clan FE because childhood family structure is allowed to be correlated with both clanspecific and descendant-specific unobserved factors and the estimates are still consistent.The disadvantage, however, is its relatively larger standard errors and lower significance.It provides an estimate of 0.158 with one IV and 0.132 with both IVs.The 2SLS estimates are slightly smaller than the OLS estimates.The upward bias and the downward bias caused by omitted variables offset each other to some extent.After controlling for the observed parental and child factors, the unobserved clan heterogeneity, and the other omitted variables, I have identified the causal effect of growing in a two-parent household on the child's adult earnings.I summarize the estimates from OLS, clan FE, and 2SLS in Table 7.The positive effect of growing up in a twoparent household on the child's adult earnings is evidence of hypothesis H1.

Table 6. 2SLS Estimates of Two-Parent Household Effect on Child's Adult Earnings
employed in the clan fixed-effects specification.The positive parental effect of growing up in a two-parent household and the negative parental divorce effect are in agreement with each other in size.

OLS, Clan FE, and 2SLS Estimates of Effect of Years of Living with Both Parents
When I use the "years of living with both parents during childhood" as the key explanatory variable instead of the dummy variable of whether growing up in a two-parent household, I find consistent results.Table 9 shows that one more year of living with both parents during childhood increases the child's adult earnings by 1.2% on average, according to the OLS model.For the children who grew up in two-parent households, they lived with both parents for all 16 years of childhood.For the other children, they spend an average of 5.48 years with both parents.The difference between these two groups of children is 10.52 years.Thus, the adult earnings gap is 13% (0.012 10.52 0.126) between them, which is comparable to the estimates using growing up in a two-parent household dummy as the explanatory variable of interest.

Table 8. OLS, Clan FE, and 2SLS Estimates of Parent's Divorce Effect on Child's Adult
effect of growing up in a two-parent household on the child's adult earnings through this channel can be calculated as γ β .

"Investment in Child's Health" Channel Model
The coefficient  is the partial effect of growing up in a two-parent household on the child's health.It represents the "investment in child's health" channel through which childhood family structure affects the child's earnings.The composite indirect effect of growing up in a two-parent household on the child's earnings through this channel can be calculated as   .

"Intergenerational Marriage Persistence" Channel Model
The intergenerational marriage persistence coefficient λ reflects how strong childhood family structure shapes their child's attitude and behavior towards marriage.The composite indirect effect of growing up in a two-parent household on the child's earnings through this channel can be calculated as λ β .

Decomposition of Growing Up in a Two-Parent Household Effect on Child's Earnings
It turns out that the indirect effects through the three observable paths account for more than half of the total effect of growing up in a two-parent household on the child's adult earnings.Take the clan fixed effects (FE) specification as an example (see the last column of Table 10).First, after controlling for child's education, health, and marital status, as well as all the parental and child demographic control variables, the direct effect of growing up in a twoparent household on the child's earnings is 0.023, which is positive but not significant.Most of the effect has been mediated through the child's educational attainment, health, and marital status.Second, according to the "investment in child's education" channel regression, the children from intact families have 0.437 more years of schooling on average than those from non-intact families, other variables being fixed.From the direct effect regression, one more year of schooling increases hourly earnings by 9.9%.Therefore, the indirect effect of growing up in a two-parent household on the child's earnings through the "investment in child's education" channel is equal to the partial effect of growing up in a two-parent household on the child's schooling times the partial effect of the child's education on her earnings, i.e., 0.437 0.099 0.043.It suggests that a 4.3% earnings gap between the children from intact families and the children from non-intact families can be explained by the "investment in child's education" mechanism.Third, the "investment in child's health" channel model implies that children who grew up in two-parent families are 4.1 percentage points more likely to have excellent or very good health in adulthood than the children who did not always live with both parents.Having good health increases hourly earnings by 9.4%.Therefore, the indirect effect of growing up in a two-parent household on the child's adult earnings through the investment in the child's health channel is equal to 0.041 0.094 0.004.Fourth, according to the "intergenerational marriage persistence" regression, the children with parents maintaining married during their childhood are 5.5 percentage points more likely to maintain their marriage in adulthood than those whose parents got divorced or separated.It is consistent with previous findings that children whose parents divorced are more likely to divorce themselves as adults (Dronkers and Härkönen, 2008).From the direct effect regression, married workers have 12.3% more earnings than unmarried workers.Thus, the indirect effect of growing up in a two-parent household on the child's earnings through the "intergenerational marriage persistence" channel is equal to the partial effect of growing up in a two-parent household on the child's marriage times the partial effect of the child's marriage on earnings, i.e., 0.055 0.123 0.007.Finally, the sum of indirect effects from the three channels is 0.054, which means 5.4% of the earnings gap can be explained by these three mechanisms.The total effect of growing up in a two-parent household on the child's earnings is 0.077.The share of the indirect effects that are mediated through the three intergenerational transmission mechanisms is 70.1% (0.054/0.077=70.1%).In the OLS specification, the indirect effects also account for more than half of the total effect.The decomposition of the total two-parent household effect on the child's adult earnings provides evidence for hypothesis H3.The effect of growing up in a two-parent household on the child's adult earnings is neither identical for all parental families nor for all children.It varies with different parental factors and child characteristics.The theoretical model predicts that it increases with parental income, and it may be different between sons and daughters.In this part, I test the predictions by including the interaction terms of childhood family structure and parental or child factor in the model.The heterogeneous pattern of the two-parent household effect on the child's earnings is in agreement with the theoretical model's prediction and hypothesis H2.

Growing Up in Two-Parent Household Effect and Parental Household Income
Including the interaction term between childhood family structure and parental household income in the pooled OLS model helps me find out how the two-parent household effect changes with parental income.I plot the fitted child's hourly earnings against the parental household income for the two groups of children of different childhood family structures in the left panel of Figure 4.The vertical distance between the two lines is the hourly earnings gap between them.It becomes larger and more significant as parental household income increases.This pattern can also be seen in the right panel of Figure 4.The two-parent household effect on the child's adult earnings increases with the parental household income.It is 0.134 for children who grew up in families with the median annual income (55,598 in 2017 USD).For those families in the 75th income percentile (83,543 in 2017 USD), the two-parent household effect rises to 0.177.For the very wealthy parental households in the 90th income percentile (116,111 in 2017 USD), childhood family structure plays a much bigger and more significant role in determining their child's adult earnings.There is a 21.4% earnings gap between the two groups of adult children.The finding that the positive effect of growing up in a two-parent household on the child's earnings increases with parental household income is parallel to the finding by Bernardi and Boertien (2016).They find that the negative parental separation effects on the child's educational attainment are stronger for high-income families.The childhood family stability and the parental income reinforce each other in raising the child's adult earnings.

Growing Up in Two-Parent Household Effect and Parental Educational Attainment
I add the interaction term between childhood family structure and parental educational attainment to the model.As is shown in the left panel of Figure 5, the earnings gap between the two groups of children is larger and more significant for the children who have better-educated parents.In other words, growing up in a two-parent household has a more distinct effect on the child's adult earnings if either of the parents has received a college education.Thus, the childhood family structure effect on the child's earnings differs by parental socioeconomic status measured by income and education.It is evidence of hypothesis H2. son who grew up in a two-parent household whose parent is a college graduate earns 21.6% more on average than his counterparts with the same background but grew up in divorced or separated families.By contrast, a daughter from an intact family whose parent is a high school graduate does not significantly earn more than her counterparts from non-intact families.
Growing Up in a Two-Parent Household and Intergenerational Relative-Earnings Change Intergenerational Relative-Earnings Change The relative earnings are measured by the quintile in the earnings distribution of the sample.The first quintile is the lowest quintile of the earnings distribution, while the fifth quintile is the highest of the earnings distribution.Each individual is at an earnings quintile based on the relative position of their earnings in the earnings distribution.The intergenerational relative-earnings change is defined as the change in the earnings quintiles between parents and children.An intergenerational relative-earnings improvement occurs when the child's relative earnings are at a higher quintile than her parents.An intergenerational relative-earnings worsening occurs when the child's relative earnings are at a lower quintile than her parents.Figure 7 depicts the intergenerational relative-earnings change between parents and children in the sample.Around 43.7% of the children are at a higher earnings quintile than their parents, implying an intergenerational relative-earnings improvement.About 42.2% of the children experience intergenerational relative-earnings worsening.The relative earnings for the rest of the children are the same as their parents.

Effect of Growing Up in a Two-Parent Household on Intergenerational Relative-Earnings Change
There are three categories of the dependent variable: intergenerational relative-earnings improvement, stagnancy, and worsening.Growing up in a two-parent household is the key explanatory variable of interest.Including parental household income, parental educational attainment, number of children in the parental household, child's age (and its squared term), gender, region, race, and tenure in the model, I fit an ordered probit regression to estimate the two-parent household effect on the intergenerational relative-earnings change.The result can be better interpreted using a graph.Figure 8 shows that the children who grew up in two-parent households always have a higher probability than their counterparts of improving the intergenerational relative-earnings.Children from intact families are also less likely to become worse in the relative earnings position than their parents.Having an intact family in childhood lowers the probability of the intergenerational relative-earnings worsening by 6.72% and increases the relative-earnings improvement by 6.67%, holding relevant factors constant (see Table 11).The finding agrees with Bratberg et al. (2014) in that children of divorced parents tend to move downward in the earnings distribution compared to children from intact families.Growing up in a two-parent household not only increases the child's adult earnings in absolute values, but it also has a positive and significant effect on the intergenerational relative-earnings improvement.

Robustness Check and Discussion
In this section, I first present the evidence that the IV is very likely to satisfy the exclusion restriction requirement.Then, I consider the child's self-selection into the labor force and compare the two-parent household effect among children at different earnings quantiles.I also look at the effects in three childhood periods.Finally, I discuss the concern about the possible endogeneity of parental income and education.

Child's Self-Selection into Labor Force
Only the adult children who participate in the labor force can earn labor income and appear in the earnings regression.Some children choose not to work in adulthood.Nearly 14% of the adult children were unemployed in the sample.If the childhood family structure has a very different effect between the employed children and the unemployed children, the estimates could not represent the entire population of adult children.The solution is to follow the Heckman twostep procedure.In the first step, the adult child's probability of working is regressed on all the regressors in the earnings regression and two additional exogenous variables, the adult child's family non-labor income and the number of her children.I obtain the inverse Mills ratio from the selection regression and then add it into the earnings regression in the second step.Thus, the child's willingness to working can be held constant, and the self-selection bias can be corrected.The estimate of the childhood family intactness effect on the child's adult earnings is 0.166, which is slightly smaller than the polled OLS estimate without the self-section correction, 0.168.The inverse Mills ratio is not significant in the earnings regression (see Table 13).In the clan fixed effects (FE) model, the estimate after self-section correction also becomes a little bit lower.The sample selection problem is either negligible or tiny in this research.

Quantile Regression
I run a simultaneous-quantile regression for the 25th, 50th, and 75th percentiles of the child's hourly earnings.The estimates are 0.114, 0.122, and 0.148, respectively (see Table 14).The higher the hourly earnings the children have, the higher the parent's marital stability effect seems to be.However, the null hypothesis that the effects of growing up in a two-parent household are equal to each other at the three quantiles is not rejected (F-statistics=0.58,p-value=0.56),implying that the effect is not significantly different between the high-earnings children and the low-earnings children.

Exogeneity of Parental Income and Parental Education
I treat parental income and parental educational attainment as exogenous explanatory variables.Parents usually complete their education before they have children or even before they have a marriage.The factors that influence their child's adult earnings are not likely to affect the educational level they obtain.This is true, especially in the clan fixed effects model when the unobserved genetic characteristics have been controlled.The parental household income during the offspring's childhood is related to the parent's marital status and child's factors.It is more likely to be endogenous.There are two solutions to this issue.First, I can drop it because I have parental educational attainment in the model, which can be seen as a proxy for parental income.However, the effect of growing up in a two-parent household will absorb part of the parental income effect and become larger.For the purpose of disentangling the childhood family intactness effect from the other parental influence, I tend to include both parental income and education in the model and use the second method to alleviate the possible endogeneity of the parental income.The second method is using the two instrumental variables, the state divorce rate and the no-fault divorce law effectiveness, for the two possible endogenous explanatory variables, the childhood family structure and the parental income.The 2SLS specification gives an estimate of 0.155.It is close to the estimate of 0.161 when parental income is treated as an exogenous variable, as shown in Table 16.

Table 16. 2SLS Estimates of Two-Parent Household Effect on Child's Earnings Conclusion
The marriage-specific investment in children is affected by parent's marital status because the access to household resources and the incentive to invest could be different between the parents maintaining an intact marriage and the parents with an unstable marriage.The efficiency of the use of household resources also differs between them.Based on the utility maximization model, I propose the two-parent household hypothesis on the child's earnings, emphasizing the positivity, the heterogeneity, and the mechanism of the childhood family intactness effect on the child's adult earnings.The causal effect of growing up in a two-parent household on the child's adult earnings remains positive and significant for various specifications and approaches, such as OLS, clan FE, and 2SLS.It can be identified because I have alleviated the endogeneity problems using three specification strategies.First, I disentangle the childhood family intactness effect from the parent's income and education influence and control for the child's demographic characteristics.Second, I fit the clan fixed-effects (FE) models to account for the unobserved clan-specific heterogeneity.Third, I adopt the two-stage least squares (2SLS) method as an attempt to account for other omitted variables.The empirical results are consistent with the theoretical model's prediction and the hypothesis.The effect of growing up in a two-parent household on the child's adult earnings is 0.168 in OLS, 0.077 in clan FE, 0.161 in 2SLS, and 0.132 in 2SLS+FE.Taking the estimate from 2SLS as an example, the children of prime working ages in the United States who grew up in two-parent households earn 16.1% (or precisely 17.5%) more than their counterparts reared in non-intact families, holding relevant factors constant and correcting the omitted variable bias.Childhood family intactness not only increases the child's hourly earnings in absolute values, but it also has a positive and significant effect on the intergenerational relative-earnings improvement.The disparity and mobility in the child's adult earnings demonstrate the benefit of growing up in a two-parent household.

Table 1 .
Mean Hourly Earnings by Growing Up in a Two-Parent Household for Male and Female Workers, 2017.
Figure 1.Construction of Dummy Variable "Growing Up in a Two-Parent Household"

Figure 2 .
Figure 2. Family Tree of a Clan.

Table 10 .
Direct and Indirect Effects of Growing Up in a Two-Parent Household on Child's Earnings Heterogeneous Pattern of Growing Up in a Two-Parent Household Effect on Child's Earnings

Figure 4 .
Figure 4. Predicted Child's Hourly Earnings by Childhood Family Structure and Parental Household Income (Left Panel); Two-Parent Household Effect on Child's Hourly Earnings by Parental Household Income (Right Panel)

Figure 8 .
Figure 8. Probability of Intergenerational Relative Earnings Change by Child's Age and Childhood Family Structure