The Quantity-Quality Tradeoff and the Formation ofCognitive and Non-cognitive Skills∗

Chinhui Juhn†

University of HoustonYona Rubinstein‡

London School of Economics

C. Andrew Zuppann§

University of Houston

April 21, 2014

Abstract

We estimate the impact of increases in family size on childhood outcomes usingmatched mother-child data from the National Longitudinal Survey of Youth 1979. Weuse the timing of sibling arrivals to estimate the effect of younger siblings on olderchildren. We find evidence that families face a substantial quantity-quality tradeoff:increases in family size decrease childhood cognitive abilities, increase behavioral prob-lems, and decrease parental investment. We also find evidence of heterogeneous effectsby race and mother’s AFQT score, with the effects being much larger for African-Americans and children of mothers with low AFQT scores. The negative effects oncognitive abilities are much larger for girls while the detrimental effects on behaviorare larger for boys.

JEL Classification: J13, J24

∗Acknowledgements. All remaining errors are our own.†University of Houston. E-mail: cjuhn@uh.edu‡London School of Economics. E-mail: y.rubinstein@lse.ac.uk§University of Houston. E-mail: cazuppann@uh.edu

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1 Introduction

Researchers have consistently found that larger families lead to poorer educational outcomes

for children in cross-sectional data (see Blake (1989) and Hanushek (1992) among others).

This empirical regularity is most often attributed to a quantity-quality trade-off in parental

investments as suggested by the well-known model of Becker (1960), Becker and Lewis (1973),

and Becker and Tomes (1976). A key feature of the quantity-quality model is the interaction

of child quality and child quantity in the household budget constraint; reductions in the

number of children reduces the marginal cost of investing in quality and an exogenous increase

in quality raises the shadow price of having more children. Becker and various co-authors

originally formulated this model to explain how rising incomes, by raising the demand for

quality, could lead to declining fertility, even when children are not inferior goods. The

negative reinforcing mechanism between quantity and quality also plays a central role in

macro growth models with endogenous fertility where higher fertility leads to less human

capital investment and lower levels of growth ((Becker and Barro 1988), (Becker et al. 1990),

(Moav 2005)).

Despite the pre-eminence of the quantity-quality model in many empirical and theoretical

papers, establishing a causal relationship between family size and education has been sur-

prisingly challenging. The key challenge is to correct for selection - that is, allow for the fact

that parents who choose to have more children may be inherently different from those who

choose to have fewer children. Rosenzweig and Wolpin (1980) first addressed this problem

by using twin births to instrument for quantity. They found that exogenous shocks in family

size do indeed lower average schooling of children in Indian families. Similar results have

also been found in more recent Chinese data in Rosenzweig and Zhang (2009). In the U.S.

context, Caceres-Delpiano (2006) and Conley and Glauber (2006) have found, using twin

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births and sibling composition to instrument for family size, that children in larger families

are less likely to attend private school.

In contrast to these studies, several prominent papers have found little evidence of the

quantity-quality tradeoff. Black et al. (2005) use large samples from Norway and find that the

negative relationship between family size and education disappears once birth order controls

are included or twin births are used to instrument for family size. Similarly, Angrist et al.

(2010) use the Israeli Census and alternative instruments, twin births and sex composition

of children, and find no relationship between family size and education.1

In this paper, we revisit these issues using detailed data of matched mothers and children

from the National Longitudinal Survey of Youth - 1979 (NLSY). Our paper contributes to

the quantity-quality tradeoff literature in two ways. First, we study how family size influ-

ences cognitive and non-cognitive skills during childhood as well as parental investments in

children. Most previous work has focused on adult outcomes such as education and earn-

ings. However, adult outcomes are likely to be a function of institutions as well as parental

investments. For example, the lack of family size effects in Black et al. (2005) may reflect the

existence of a strong public education system in Norway where at the margin, investments

in child quality may not result in variations in educational outcomes. By examining early

childhood outcomes, as well as measures of parental investment, we provide a more direct

1More recently, researchers have proposed alternative explanations for these ‘no evidence of a trade-off’ findings. Mogstad and Wiswall (forthcoming) show that imposing a linear specification where everyadditional family member has the same average effect on earlier children may obscure significant heterogeneityin the return from an additional children. Bagger et al. (2013) argue that conventional empirical specificationsfor testing the quantity-quality tradeoff suffer from a fundamental problem of being unable to simultaneouslyvary family size while holding the distribution of birth orders constant within the household. Thus if anadditional child changes the optimal tradeoff along the entire birth order distribution OLS regressions atthe individual level cannot correctly estimate the quantity-quality tradeoff. They propose an alternativeestimation strategy: estimate birth-order effects using family fixed effects, residualize to net out birth-ordereffects and then estimate the relationship between average family size and average outcomes, net of thesebirth-order effects.

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test of the original quantity-quality theory which is about parental choices and child skill

formation.

Second, our paper makes a methodological contribution to the quantity-quality literature

by providing alternative strategies for estimating the effect of family size. One common

criticism of using twin births as an instrument is that the exclusion restriction may be

violated, i.e. the birth of twins has a direct negative impact on siblings other than by

increasing family size. The panel structure of the NLSY allows us to make use of the timing

of expansions in family size to estimate this tradeoff controlling for selection into fertility

decisions. The simplest way to control for selection is by directly including individual child

fixed effects. By comparing children to themselves before and after an increase in family size

our estimates of a quantity-quality tradeoff control for potential selection into fertility at the

child level. Using fixed effects we find a strong negative impact of an additional child on

cognitive and non-cognitive outcomes as well as parental investment in older children. We

also find that the overall estimates mask important heterogeneity by gender, mother’s AFQT

score, and race. The negative effects on cognitive abilities are much larger for girls while the

detrimental effects on behavior are larger for boys. We also find evidence of heterogenous

effects by race and mother’s AFQT score, with the effects being much larger among blacks

and children of mothers with low AFQT scores.

However, fixed effects regressions may be biased in the presence of time varying individual

shocks. It may be the case that families are choosing to have additional children at the

best possible times which would tend to bias the negative effect of siblings towards zero.

Alternatively, families may be too optimistic when they decide to have additional children

and the family environment worsens after the arrival of an additional sibling due to regression

to the mean. To evaluate the bias arising from endogenous timing of births, we use the

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randomness of the timing of the NLSY survey with respect to a sibling’s birth. By comparing

children who were interviewed right before the sibling arrival to those interviewed right after

the sibling arrival we can estimate the impact of an additional child for families with similar

fertility timing. We find evidence that families time fertility to minimize the negative impact

of additional siblings.

Finding a strong tradeoff in the household during childhood ties our results to the broader

literature on the importance of early childhood investments and skill formation (Almond and

Currie 2011). Papers such as Cunha and Heckman (2007) and Cunha et al. (2010) have shown

that parental investments throughout childhood play an important role in adulthood cogni-

tive and non-cognitive skill. Our paper provides additional documentation of this channel as

we establish that lowered childhood investment, and consequently lower achievement scores,

is the likely mechanism for long-run quantity-quality tradeoffs.

The paper proceeds as follows. Section 2 discusses our empirical methodology. Section

3 describes the construction of the matched mother-child data set. Section 4 presents our

estimates. Section 5 discusses our results in context of previous research on twins and section

6 concludes.

2 Empirical Approach

2.1 Individual child fixed effects

A simple cross-sectional analysis of family size on child outcomes is confounded by differential

selection as larger families are likely to differ from smaller families in other important ways.

With panel data, we can control for these unobservable selection factors with individual child

fixed effects. By comparing older children’s outcomes before and after the birth of a younger

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sibling we can directly estimate how the presence of an additional sibling affected the older

child. Our baseline child fixed effects model is

Yijt = β0 + β11{after}ijt +Xitβ2 + λi + εijt (1)

where Y is an outcome of interest for child i with a sibling at birth parity j, 1{after} is a

dummy variable for whether the birth at parity j has occurred at time t, λi is a child fixed

effect and X is a vector of individual characteristics. Our sample for this specification is all

older children whose mother will have a birth at parity j.

2.2 Long-term effects

An advantage of having panel data is that we can look at the persistence of any effects over

time. To do this we estimate regressions that allow for different effects in the first 3 years

following a sibling birth and the remaining years after. The control period is the years prior

to the birth. Our estimating regression is thus

Yijt = β0 + β11{0-3 years after sibling k’s birth}it + β21{3+ years after sibling k’s birth}it

+Xitβ3 + λi + εit (2)

where i is the older sibling, k is a younger sibling, and t is time. This specification allows us

to test for a different effect in the short versus the long run by comparing β1 to β2.

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2.3 Endogenous birth timing

Our fixed effects estimates control for all unobservable but constant characteristics of children

in estimating the quantity-quality tradeoff. However, time-varying shocks may drive parents’

decision of when to have another child as well as changes in older sibling’s test scores. These

unobservable time-varying factors may be biasing our fixed effect estimate of the quantity-

quality tradeoff both positively and negatively. Two simple examples these possible biases.

For one, if household fertility decisions are responsive to income then a positive but

transitory income shock can lead to improvements in older children’s test scores as well

as parents deciding to have another child. As the transitory shock disappears the older

children’s test scores decline back to previous levels. There will therefore be a negative

correlation between the arrival of a younger sibling and older children’s test scores driven

solely by the transitory income shock. Our fixed effects estimator would be erroneously

biased towards overstating the extent of the quantity-quality tradeoff.

Alternatively, if parents seek to minimize possible negative effects on older children when

choosing when to have an additional child then our fixed effects estimates will be biased

towards zero. Parents may be able to foresee a time in their life – such as following an

expected promotion or when a grandparent retires – when they can have an additional child

while minimizing the tradeoffs they will face in devoting resources to the older children in

the household.

ANDY NOTE: should we add something about policy relevant versus literature relevant

effects here? Note that we can say something about both: FEs are about policy relevant

while windows are literature relevant

We explore whether endogenous timing is biasing our fixed effects results by taking

advantage of two plausibly random events: the timing of when the NLSY interviews a woman

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within a calendar year and the inability of parents to precisely time when their pregnancy

will occur.

The NLSY conducts interviews in several months throughout the calendar year of a

particular survey wave.2 Although the NLSY attempts to keep the month of interview

constant for a single respondent this is often not possible for many reasons. Most notably the

NLSY often changes the months of surveying from wave to wave (e.g., from June-December

in 1994 to April-October in 1996). The month of interview can also change from wave to wave

because of problems contacting the respondent or due to organizational/budgeting delays.

From the perspective of a respondent there is a great deal of uncertainty about when during

a given calendar year they will be interviewed.

ANDY NOTE: maybe add figure or table of distribution of interview months and changes

from wave-to-wave?

Another source of plausible randomness is that parents are unable to precisely time the

birth of a child to a particular month. Even if we assume that all pregnancies are planned

there is still a great deal of uncertainty about when the pregnancy will actually occur.

Parents often spend months and occasionally years trying to conceive. And after conceiving

there is very little parents can do to change the birth month. Of course parents are capable

of moving the date of birth by a few days or a week through induced labor or voluntary

Caesarean sections but this will change, at most, one month.

ANDY NOTE: should we add cites to medical literature on pregnancy duration?

Taken together, these two sources of uncertainty suggest a plausible quasi-experimental

treatment to test the quantity-quality hypothesis: compare families just prior to a birth to

families just after a birth. This explicitly conditions on the decision to have another child

2Please see https://nlsinfo.org/content/cohorts/nlsy79/intro-to-the-sample/interview-methods for com-plete details on the timing of interviews.

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and then uses the fact that the time between birth and the NLSY interview is plausibly

exogenously determined to define treatment status.3 We are therefore comparing older chil-

dren in households that are just about to expand to older children in households that just

expanded.

By using the randomness of the interview timing interacted with the lack of precise

pregnancy timing, this estimation is entirely (quasi-)experimental and there is no reason

why we need to include demographic controls or even individual fixed effects. Instead we

just restrict our sample to children observed between 12 months prior to a sibling’s birth and

12 months after a sibling’s birth. This is advantageous because focusing on narrow windows

around a birth while simultaneously including fixed effects is not possible given the biannual

nature of the NLSY. This method also controls for unobservable factors – both constant and

time-varying – that led parents to have another child.

Another advantage of using the randomness of pregnancy and interview timing in a

short window is that we can directly evaluate how unobservable factors may be biasing our

estimates. As we increase the time before a younger sibling’s birth that we include in our

sample – say by looking at children observed 24 months prior to a sibling’s birth – the

assumption of randomness becomes worse and unobservable factors become more important

in determining the assignment to a before or after treatment group. By comparing estimates

with a narrow window before a sibling birth to those with a wide window before a sibling

birth we can evaluate the direction of bias due to unobservable factors.

More formally, we are assuming that conditional on having a birth at a given parity, the

difference between the month that sibling is born and when the NLSY interviews the mother

(or mother-to-be) is independent of all unobservable mother, child, and time unobservables,

3Note that even if birth timing or NLSY interview timing is not random we are actually requiring theirdifference to be random to achieve causal identification.

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conditional on the interview happening within [−τ, 12] months of the birth. For a small value

of τ assignment to the before or after treatment is plausibly random. As τ increases this

assumption becomes worse as parents are more capable of choosing when to have a child

within, say, a five year window instead of a one year window. The change in estimates of

the quantity-quality tradeoff as we increase τ therefore tells us about the direction of bias

arising due to unobservable factors.4

ANDY NOTE: list disadvantages of this approach. some of those are: 1. can’t look at

long-lasting effects. 2. is this really the QQ tradeoff?

3 Data

The National Longitudinal Survey of Youth provides a unique opportunity to evaluate the

quantity-quality tradeoff as it contains information on childhood development as well as adult

education and labor market outcomes. We match mothers from the 1979 survey with all their

children from the Children and Young Adult survey. Children were surveyed biannually from

1986 to 2010. By matching children to their mothers and siblings we can identify siblings

as well as the precise timing of when family size expands. Table A.1 presents summary

statistics of the children in our matched mother-child sample.

The matched NLSY mother-child data contains detailed information about childhood

cognitive and non-cognitive abilities as well as longer term outcomes. Children aged 4

to 14 are given Peabody Individual Achievement Tests (PIATs) that measure cognitive

skills in mathematics, reading recognition, and reading comprehension. To measure non-

4Another possibility is that the unobservable factors that lead parents to decide to have another childare not prior to the birth but instead forecasted events that will occur after the arrival of the child. To testfor this we would instead expand the interval length to greater than 12 months, an exercise available uponrequest.

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cognitive abilities, the survey calculates a Behavioral Problem Index (BPI) and the subindices

which measure particular problems including antisocial behaviors, anxiety, dependence,

headstrongness, hyperactivity, and social problems. To measure parental investment, the

NLSY asks questions to construct a HOME (Home Observation Measurement of the Environment-

Short Form) score, “a unique observational measure of the quality of the cognitive stimula-

tion and emotional support provided by a childs family.” Examples of these questions include

how many books a child has, how often parents read to the child, and whether parents assist

with homework. HOME scores have been shown to be a significant determinant in a child’s

development.

We estimate the impact of the younger sibling on the older sibling using a fixed effects

strategy. We construct a “sibling-pair sample” where we match test scores, HOME and

behavioral index measures of older siblings with the birthdate of each subsequent younger

sibling. As our estimation methods rely on before-after comparisons of test scores for the

same child and the NLSY PIAT tests are only administered to children aged 4-14 this

imposes a stringent restriction on the children in our “sibling-pair sample.” Specifically,

older siblings in our “sibling-pair sample” who have valid test scores before and after the

arrival of a younger sibling are significantly older than the average child at the birth of a

younger sibling. Since the children in our “sibling-pair sample” will clearly differ on these

dimensions it is worth checking to see if there are other important differences between the

families. Table 1 shows summary statistics of demographic characteristics of mother and

children in both the full sample of older siblings and our more restricted sample of older

siblings who have valid PIAT test scores before and after the arrival of a younger sibling.

Not surprisingly, children in the restricted sample have substantially greater spacing between

births (26 months). However on other dimensions these children appear similar to the full

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sample. Mothers have similar cognitive and non-cognitive ability measures, have a similar

distribution of race, and start their fertility at similar ages. Children in the restricted sample

are of similar birthweight and, surprisingly, are less likely to be living in a blended family.

However they are more likely to be living with a single mother.

4 Results

Table 2 presents estimates from the fixed effects model of the impact of an additional child

on older children’s outcomes at second birth parity.5 Column 1 estimates the average decline

in test scores and parental investments after the second child is born. We see that older

siblings are generally worse off in both cognitive and non-cognitive outcomes following the

birth of a younger child. We combine math and reading recognition scores by taking a simple

average of the two.6 Cognitive test scores fall by 2.1 percentile points, or nearly one-tenth

of a standard deviation.

We also find evidence of a potential channel by which these outcomes are worsening:

parental investment in the older child, as measured by the HOME Inventory index, falls by

one-tenth of a standard deviation after the birth of a younger child. Similarly, behavioral

problems also increase, with the index increasing by more than one-tenth of a standard

deviation.

We might expect that the birth of an additional child is a disruptive but transitory

shock to the home environment and that the negative effects of an additional sibling would

dissipate over time as the household adjusts. Column 2 of Table 2 tests this hypothesis

by showing estimates of the negative effects of an additional sibling both in the short and

5Results at third parity are reported in Table A.3. We find that the negative results for HOME score andbehavioral problems remain while cognitive test results disappear at third parity.

6We find somewhat stronger results for math and weaker results for reading when we disaggregate.

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long run. Not only do we fail to find evidence that the impact is transitory, effects appear

to substantially worsen over the longer run. Test scores and parental investments are both

worse off at the longer horizon than in the short run. Only in behavioral problems do we

find possible evidence of the additional sibling being a temporary disruption.

4.1 Addressing endogeneity of timing of births

To see how endogenous timing of fertility may be biasing our results we estimate the effect

of an additional sibling among children who were interviewed within a narrow window of

a sibling birth. Column (1) of Table 3 shows estimates with τ = 12 where we restrict the

sample to observation/births that were within one year of each other. In Column (2) we

broaden the window in the “before” period to 24 months while keeping the “after” period

to 12 months.

Column 1 of Table 3 shows that even when we narrow the window to 12 months, esti-

mates of the sibling effect remains negative for all three measures. Given the small samples,

however, we lose precision and the effect on cognitive scores are no longer significant. The

effect on HOME index and behavioral problems, however, remain. When we extend the

“before” period to 24 months, the sibling effect becomes less negative which suggests that

families time fertility decisions as to minimize the negative impact on older children. To

further illustrate this pattern Figure 1 to Figure 3 show estimates of the effect of an addi-

tional sibling on outcomes for a range of different choices of τ . While we lack precision, the

figure shows that the sibling effect becomes smaller as we expand the window in the “before”

period, thereby allowing for more endogenous timing of fertility.

As a check on whether the before and after assignments for our narrowest window, τ = 12,

are random Table A.5 shows average mother and child characteristics of the before and

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after groups in the years before they entered the sample. If previous shocks to child test

scores influenced the timing of births within our narrow window then we would expect to

find differences across these groups. Looking at Table A.5 it does not appear there are

any significant differences in either maternal characteristics or child test scores across these

groups.

4.2 Heterogenous effects across gender, race, and mother’s AFQT

One advantage of the NLSY sample is that we have detailed characteristics on the mothers

of these children and can investigate whether there is heterogeneity in the impact of an

additional child across different types of mothers and families. To do this we interact mother’s

characteristics with the after dummy in our fixed effects regressions.

Table 4 reports the effects separately by mother’s AFQT group. There are strikingly

different results by mother’s ability. For children with low ability mothers, the arrival of

younger siblings have large and significantly negative effects on cognitive skills while for

children with high ability mothers, the effects of siblings are often significant and positive.

The one caveat is the differential effect for ”Behavioral Problems” index which suggests

that children of high ability mothers are more adversely impacted by the arrival of younger

siblings.

Table 5 show differential effects of family size by race. The table shows that the detri-

mental effects of family size are largest for black families. It is possible that the differential

effect by race is confounded with mother’s AFQT effect.

Table 6 reports the effects by gender. We report estimates of a pooled specification where

age effects are restricted to be equal across gender. The table shows strikingly different

family size effects by gender. The effects on cognitive scores are small and insignificant for

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boys. They are large and significant for girls. In contrast, the family size effect is large

and significant for boys for ”Behavioral Problems” but not significant for girls. One possible

explanation for these gender differences in cognitive and non-cognitive outcomes is that girls

have to do more babysitting with the arrival of younger siblings while boys receive less

parental supervision and are more likely to have behavioral problems.

The differential effects by mother’s AFQT score and race provide a possible insight as to

why empirical papers in this area have found a quantity-quality trade-off in some, but not in

all, cases. In countries with a comprehensive welfare system and a strong public education

system, parental resource constraints may not be binding for educational outcomes. In a

country such as the U.S. with limited income support programs and limited quality public

education, lower ability mothers with limited monetary resources may face a real trade-off

between quantity of children and the resources she can devote to invest in their skills.

To test this hypothesis that mothers of different races and cognitive ability face different

resource constraints and that this effect leads to the heterogeneity in the quantity-quality

tradeoff we look at how maternal labor supply adjusts in response to increases in family

size. Table 7 presents estimates of the effect of an additional sibling on whether the mother

worked last week and whether she typically works full-time. Looking first at working last

week, we see that there are differential responses by both race and mother’s AFQT. Mothers

who have below median AFQT scores do not respond to an additional child by reducing

work while mothers above the median do. There is also substantial heterogeneity across

races: black mothers do not reduce their workforce participation while white and hispanic

mothers do. This evidence is consistent with maternal resources being an important factor

in whether there is a quantity-quality tradeoff among children.

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5 What about twins?

So far we have shown that having an additional child has a substantial negative impact on

older children in the same household. This finding is in contrast with many findings in the

literature which fail to find a tradeoff between larger families and outcomes for older children.

In this section we present estimates using previous approaches which find imprecise negative

effects.

Due to reasons of data availability, most previous work on the quantity-quality tradeoff

has been cross-sectional and focused on long-term adult outcomes such as years of education

or labor market earnings. Unfortunately, our methods rely on panel methods with children

observed both before and after the arrival of a younger sibling. Our methods are thus

incapable of directly estimating the impact of an additional sibling on long-term outcomes

for which we only have cross-sectional data.7

Since panel methods are not applicable, a standard approach to estimating the impact

of an additional child is to use twin births as an instrumental variable for family size and

look at children born before the birth of twins. This strategy is concisely described in Black

et al. (2005) by the following pair of instrumental variable equations:

Y = β0 + β1FAMSIZE +Xβ2 + ε (3)

FAMSIZE = α0 + α11{TWIN}+Xα2 + ν (4)

where Y is an outcome of interest, eg. years of education, FAMSIZE is the number of

children in the family, and X is vector of individual characteristics. 1{TWIN} is a dummy

7It is worth noting that we do find persistence in the negative impact of the arrival of a sibling throughoutchildhood.

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variable indicating whether the family has any twins. For households with twins present,

we restrict the sample to children who were born prior to the twin birth. As a conceptual

example of the source of identification, we are comparing outcomes of firstborn children across

families where the second birth was either a singleton or twins. Families with twin births

have had a plausibly exogenous increase in family size. Differences in outcomes between

firstborns in households with twins versus households without twins can thus be interpreted

as a causal estimate of how family size affects outcomes.

One potential concern with using the NLSY sample in our empirical IV strategy is the

relatively small number of households with twins in the sample. A mere 142 children in the

matched dataset are born prior to the arrival of twins. Table 8 shows the distribution of

households of various family sizes as well as the parity at which twin births occurred in the

NLSY matched mother-children data.8

Table 9 presents estimates using both OLS and the twin IV of the impact of family size

on years of education.9 Columns 1 and 2 are OLS estimates with and without birth order

controls and Column 3 uses the presence of a younger twin birth as an instrument for family

size as specified in equations 1 and 2.

As reported in the first column, the family size effect without maternal controls produces

a large and highly significant negative effect of family size. The coefficient implies that

raising final family size by an additional child reduces average schooling of children by -.13.

Since we add additional maternal controls, the coefficient is slightly smaller than the -0.18

8Twins are not directly identified in the NLSY. However, there are data on the month and birth of eachchild. We identify twins (and triplets and quadruplets) as two (or more) siblings who have the same motherand share the same year and month of birth. Out of 11,476 children respondents living in 4,510 householdswe identify 117 pairs of twins.

9In order to ensure the highest grade completed in our sample represents completed education we restrictour sample to children who were surveyed at least once after age 24. We include demographic controls for thechild such as age, gender, and race and maternal controls such as mother’s race, mother’s AFQT, mother’sage at first birth, and mother’s non-cognitive measures such as Rosenberg score and Rotter scale score.

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reported in Black et al. (2005) and -.20 reported by (Blake 1989) using U.S. data. Similar

to Black et al. (2005), we also find that the addition of birth order controls in Column 2

reduces the size and significance of the family size effect considerably.

Column 3 shows that the first stage coefficient for the IV is -1.81, which suggests that

having a twin birth increases the final family size by more than one child. One potential

concern is that families with higher desired fertility are more likely to experience a twin

birth. One solution is to conduct the analysis separately at each parity. Given the small

number of twin births in our data we lose considerable precision and we adopt a parity-

pooled approach suggested by Angrist et al. (2010) in Column 4. The IV estimate using this

approach leads to even larger negative impact of family size on years of education. While

we find a large negative effect,we are wary of the small number of twin births in the NLSY

matched mother-children data.

6 Conclusion

Using a variety of approaches we have documented a significant tradeoff between quantity

and quality of children for NLSY mothers and their children. On average, children in larger

families have lowered parental investment and worse cognitive and non-cognitive outcomes.

We found this tradeoff in two different specifications that control for possible confounds.

First, we compared older children to themselves before and after the arrival of a younger

sibling. Second, we compared older children who were surveyed right before the birth of a

younger sibling to those surveyed right after.

These results differ substantially from previous research so it is worthwhile to consider

why our estimates are so different. One potential explanation for our results in contrast to

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(Black et al. 2005) may be institutional differences between Norway and the U.S. In partic-

ular, at the margin, parental investments may matter more in the U.S. where a substantial

fraction of young men and women, particularly from lower income backgrounds, are at risk

of not finishing high school. A recent paper by (Black et al. 2010) offers some support for

the idea that the particular country and the particular cohort examined matters. In contrast

to their earlier paper which examined an older cohort, their recent paper finds a negative

impact of family size on IQ among younger birth cohorts in Norway. Likewise, Li et al.

(2008) also find that the negative family size effect on schooling is particularly strong in

rural areas of China where the public education system is less well developed. Finally, child

allowances and subsidies may have mitigated the impact of family size in the Israeli data

examined by Angrist et al. (2010) although the authors minimize the importance of these

subsidies.

Although we consistently find, on average, the presence of a quantity-quality tradeoff

in the NLSY sample, the strong differences across mother race and AFQT score provides

suggestive evidence of the importance of resource availability or institutions in determining

the extent of the quantity-quality tradeoff. High AFQT mothers of all races appear to face

less of a trade-off than mothers with low AFQT scores. Differences in mother’s AFQT scores

are correlated with a wide set of lifestyle differences that could explain these differences. For

instance, having worse child care coverage, maternity policies, or flexibility in household labor

supply could all make the presence of an additional child more detrimental to other children

in the household. Another possible explanation is that some women are more cognizant of

the possibility of a negative impact that an additional child may have and so do a better

job timing their fertility to minimize these effects. If a mother’s AFQT score is correlated

with their conscientiousness in optimally choosing the timing of their pregnancies then we

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would expect to find the observed gradient. Understanding the mechanisms underlying

these differentials in the quantity-quality tradeoff remains an important question for future

research.

References

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22

Table 1: Summary statistics: Sibling-pair sample

Full Sample FE SampleMother’s AFQT pctile 37.98 38.55

(27.02) (26.98)

Mother’s self-esteem index 21.67 21.74(4.083) (4.094)

Mother’s locus-of-control index 8.915 8.915(2.515) (2.501)

Age of mother at birth of child 20.34 20.77(2.946) (2.788)

Black 0.197 0.190(0.398) (0.393)

Hispanic 0.0994 0.107(0.299) (0.309)

Birth order 1.403 1.423(0.716) (0.724)

Birth weight of child (oz) 116.4 116.6(20.60) (20.32)

Spacing to next birth (months) 302.6 328.0(58.60) (45.10)

Current number of siblings 2.564 2.222(1.255) (1.105)

Living in blended family 0.0964 0.0786(0.295) (0.269)

Living with single mother 0.309 0.338(0.462) (0.473)

Observations 5330 2309

Data from Children of the NLSY79, 1986-2010. Our siblng-pair sample includes all older children-younger siblings pairs where the older child has a valid PIAT test score. PIAT tests were adminis-tered to children aged 4 to 14. Standard deviations in parentheses.

23

Table 2: Individual fixed effects: 2nd parity

(1) (2)Individual FEs Individual FEs

After sibling birth Cognitive Ability -2.126*(1.113)

0-3 years after -1.889*(1.122)

3+ years after -3.408**(1.258)

N 9936 9936

After sibling birth HOME Inventory -2.910**(1.063)

0-3 years after -3.032**(1.064)

3+ years after -5.093***(1.401)

N 13217 13217

After sibling birth Behavioral Problems 3.091**(1.299)

0-3 years after 3.127**(1.301)

3+ years after 2.049(1.544)

N 11119 11119

Each row presents estimates for a different outcome of interest. Controls are child’s age at test andchild fixed effects. Robust standard errors are clustered at the individual level. Data from Childrenof the NLSY79 sample, 1986-2010.∗: significant at 10% level. ∗∗: significant at 5% level. ∗ ∗ ∗: significant at 1% level.

24

Table 3: Controlling for endogeneity of timing through smaller sampling windows aroundbirths: 2nd parity

(1) (2)0-12 months before 0-24 months before

Cognitive Ability -2.823 -0.741(2.165) (1.752)

N 427 554

HOME Index -2.630* -2.418*(1.565) (1.298)

N 1392 1926

Behavioral Problems 8.060*** 4.415**(2.398) (2.059)

N 626 796

Each row presents estimates for a different outcome of interest. Column (1) restricts sample tointerviews taken 0-12 months before and 0-12 months after the birth of a younger sibling. Column(2) restricts sample to interviews taken 0-24 months before and 0-12 months after the birth ofa younger sibling. Regressions include controls for child age, gender, mother’s age at first birth,AFQT, race, Rosenberg and Rotter scores. Robust standard errors are clustered at the individuallevel. Data from Children of the NLSY79 sample, 1986-2010.∗: significant at 10% level. ∗∗: significant at 5% level. ∗ ∗ ∗: significant at 1% level.

25

Table 4: Fixed effects estimates by mother’s AFQT quartile - 2nd parity

(1)Individual FEs

Below median Cognitive Ability -5.415**(1.871)

Above median -0.346(1.328)

N 9936

Below median HOME Inventory -3.922**(1.650)

Above median -2.536**(1.183)

N 13217

Below median Behavioral Problems 0.779(1.953)

Above median 4.253**(1.647)

N 11119

Each row presents estimates for a different outcome of interest. Regressions include controls forchild age and child fixed effects. Robust standard errors are clustered at the individual level. Datafrom Children of the NLSY79 sample, 1986-2010.∗: significant at 10% level. ∗∗: significant at 5% level. ∗ ∗ ∗: significant at 1% level.

26

Table 5: Fixed effects estimates by race - 2nd parity

(1)Individual FEs

Hispanic Cognitive Ability -2.424(1.816)

Black -4.657**(1.739)

White -1.288(1.463)

N 9936

Hispanic HOME Inventory -2.497(1.839)

Black 0.523(1.698)

White -3.564**(1.189)

N 13217

Hispanic Behavioral Problems 1.501(2.454)

Black 4.022**(1.836)

White 3.030*(1.681)

N 11119

Each row presents estimates for a different outcome of interest. Regressions include controls forchild age and child fixed effects. Robust standard errors are clustered at the individual level. Datafrom Children of the NLSY79 sample, 1986-2010.∗: significant at 10% level. ∗∗: significant at 5% level. ∗ ∗ ∗: significant at 1% level.

27

Table 6: Fixed effects estimates by gender - 2nd parity

(1)Individual FEs

Boys Cognitive Ability -0.464(1.367)

Girls -4.208**(1.755)

N 9936

Boys HOME Inventory -2.919**(1.276)

Girls -2.901**(1.325)

N 13217

Boys Behavioral Problems 3.975**(1.799)

Girls 2.157(1.795)

N 11119

Each row presents estimates for a different outcome of interest. Regressions include controls forchild age and child fixed effects. Robust standard errors are clustered at the individual level. Datafrom Children of the NLSY79 sample, 1986-2010.∗: significant at 10% level. ∗∗: significant at 5% level. ∗ ∗ ∗: significant at 1% level.

28

Table 7: Impact on maternal labor supply

(1) (2) (3)Whole sample By AFQT By race

Worked last week (CPS) -0.0238*(0.0139)

Below median 0.00996(0.0136)

Above median -0.0316**(0.0135)

Hispanic -0.00536(0.0160)

Black 0.0198(0.0122)

White -0.0255**(0.0128)

Usually FT (CPS) -0.0656**(0.0332)

Below median -0.0867**(0.0368)

Above median -0.0680**(0.0274)

Hispanic -0.103**(0.0449)

Black -0.0588(0.0391)

White -0.0716**(0.0270)

Each row presents estimates for a different outcome of interest. Regressions include controls forchild age and child fixed effects. Robust standard errors are clustered at the individual level. Datafrom Children of the NLSY79 sample, 1986-2010.∗: significant at 10% level. ∗∗: significant at 5% level. ∗ ∗ ∗: significant at 1% level.

29

Table 8: Distribution of family size and twin births

Family size Birth order of twins within family# of HHs # of HHs

1 1179 442 1966 353 1126 194 430 105 135 66 49 37 228 79 410 211 2Total 4922 117

Data from Children of the NLSY79, 1986-2010.

30

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31

Figure 1: Estimated impact of an additional sibling on cognitive ability for different windowlengths

−40 −30 −20 −10 0−8

−6

−4

−2

0

2

4

Months included in ‘before’ sample

Est

imat

edeff

ect

(per

centi

lep

oints

)

-2.83

-0.74

32

Figure 2: Estimated impact of an additional sibling on HOME index for different windowlengths

−40 −30 −20 −10 0

−6

−4

−2

0

2

Months included in ‘before’ sample

Est

imat

edeff

ect

(per

centi

lep

oints

)

-2.63-2.42

33

Figure 3: Estimated impact of an additional sibling on behavioral problems for differentwindow lengths

−40 −30 −20 −10 0

0

2

4

6

8

10

12

Months included in ‘before’ sample

Est

imat

edeff

ect

(per

centi

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)

8.06

4.42

34

Table A.1: Children of NLSY79: Summary Statistics

Mean Std. Dev.Age in 2010 23.9 6.39At least 25 years old in 2010 0.51 0.50Female 0.49 0.50Black 0.28 0.45Hispanic 0.19 0.39Birth weight of child (oz) 116.1 22.7Family size 2.92 1.42Twins in family 0.038 0.19

Yrs of education — At least 25 years old 12.7 2.34HS grad — At least 25 years old 0.74 0.442009 Wage + salary income 22916.3 22251.3Log 2009 wage + salary income 9.92 0.98Worked 35+ hours at 1st job, 2010 0.76 0.43Child was convicted of a crime 0.25 0.43Child had a teenage pregnancy indicator 0.075 0.26

Math test (percentile) 51.2 28.0HOME Inventory score (percentile) 46.3 29.4Behavioral Problems Index (percentile) 59.1 28.0

Data from Children of the NLSY79, 1986-2010. Adult outcomes in the middle box are conditionalon being at least 25 years old in the 2010 survey wave.

35

Table A.2: Maternal characteristics by twin and non-twin households

HH w/o twins HH w/ twins DifferenceMothers AFQT Score (percentile) 40.2 44.0 -3.779SELF ESTEEM SCORE 80 22.1 22.6 -0.554Mother’s Rotter (locus-of-control) score (percentile) 8.82 8.74 0.0745Age of mother at birth of child 22.9 23.3 -0.436Black 0.26 0.26 0.00234Hispanic 0.17 0.17 0.0000758

HH w/o twins HH w/ twins DifferenceCognitive ability 55.0 48.0 7.007∗∗∗

HOME Inventory score (percentile) 46.8 42.8 3.937∗∗

Behavioral Problems Index (percentile) 58.9 61.3 -2.347Birth weight of child (oz) 116.4 116.7 -0.286

HH w/o twins HH w/ twins DifferenceHighest grade completed 12.7 11.9 0.810∗∗

12+ years of schooling completed 0.75 0.57 0.175∗∗∗

Log 2009 wage + salary income 9.92 9.61 0.308Child had a teenage pregnancy indicator 0.076 0.22 -0.147∗∗∗

Data from Children of the NLSY79 sample, 1986-2010. For households with twins present werestrict the sample to children born prior to the birth of the twins. The last column reportsp-values of an unpaired t-test for difference in means between households with and without twins.

36

Table A.3: Individual fixed effects: 3rd parity

(1) (2)Individual FEs Individual FEs

After sibling birth Cognitive Ability 0.0773(1.496)

0-3 years after 0.0988(1.499)

3+ years after -0.120(1.784)

N 4504 4504

After sibling birth HOME Inventory -1.941(1.465)

0-3 years after -1.974(1.467)

3+ years after -2.684(1.947)

N 6451 6451

After sibling birth Behavioral Problems 5.169**(1.580)

0-3 years after 5.117**(1.571)

3+ years after 7.132***(2.045)

N 5114 5114

Each row presents estimates for a different outcome of interest. Controls are child’s age at test andchild fixed effects. Robust standard errors are clustered at the individual level. Data from Childrenof the NLSY79 sample, 1986-2010.∗: significant at 10% level. ∗∗: significant at 5% level. ∗ ∗ ∗: significant at 1% level.

37

Table A.4: Controlling for endogeneity of timing through smaller sampling windows aroundbirths: 3rd parity

(1) (2)0-12 months before 0-24 months before

Cognitive Ability -1.554 -1.249(2.376) (1.494)

N 631 936

HOME Index -3.064 -2.882**(1.983) (1.372)

N 1139 1791

Behavioral Problems -0.587 1.752(2.597) (1.693)

N 762 1151

Each row presents estimates for a different outcome of interest. Column (1) restricts sample tointerviews taken 0-12 months before and 0-12 months after the birth of a younger sibling. Column(2) restricts sample to interviews taken 0-24 months before and 0-12 months after the birth of ayounger sibling. Regressions include controls for child age age, gender, mother’s age at first birth,AFQT, race, Rosenberg and Rotter scores. Robust standard errors are clustered at the individuallevel. Data from Children of the NLSY79 sample, 1986-2010.∗: significant at 10% level. ∗∗: significant at 5% level. ∗ ∗ ∗: significant at 1% level.

38

Table A.5: Comparing the pre-trend characteristics of the 12 month sampling window

Before group After sample DifferenceMothers AFQT Score (percentile) 39.6 38.0 1.598∗

Age of mother at birth of child 23.0 23.1 -0.101Cognitive ability 52.6 51.8 0.816HOME Inventory score (percentile) 48.5 50.1 -1.610∗

Birth weight of child (oz) 115.8 116.0 -0.169Highest grade completed 14.0 13.8 0.21112+ years of schooling completed 0.73 0.75 -0.0165Log 2009 wage + salary income 9.37 9.37 -0.00192Child had a teenage pregnancy indicator 0.072 0.063 0.00832

Panel (a) shows child characteristics for children who will be interviewed either one year before orone year after a younger sibling’s birth and Panel (b) shows similar characteristics for two years.We omit responses taken during the last six month’s of a mother’s pregnancy. Data from Childrenof the NLSY79 sample, 1986-2010.∗: significant at 10% level. ∗∗: significant at 5% level. ∗ ∗ ∗: significant at 1% level.

39