The Quantity-Quality Tradeoff and the Formation of
Cognitive and Non-cognitive Skills∗
Chinhui Juhn†
University of Houston
Yona 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 using
matched mother-child data from the National Longitudinal Survey of Youth 1979. We
use the timing of sibling arrivals to estimate the effect of younger siblings on older
children. We find evidence that families face a substantial quantity-quality tradeoff:
increases in family size decrease childhood cognitive abilities, increase behavioral problems, and decrease parental investment. We also find evidence of heterogeneous effects
by race and mother’s AFQT score, with the effects being much larger for AfricanAmericans and children of mothers with low AFQT scores. The negative effects on
cognitive abilities are much larger for girls while the detrimental effects on behavior
are larger for boys.
JEL Classification: J13, J24
∗
Acknowledgements. All remaining errors are our own.
University of Houston. E-mail: [email protected]
‡
London School of Economics. E-mail: [email protected]
§
University of Houston. E-mail: [email protected]
†
1
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 surprisingly 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
2
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 influences 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 earnings. 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
1
More recently, researchers have proposed alternative explanations for these ‘no evidence of a tradeoff’ findings. Mogstad and Wiswall (forthcoming) show that imposing a linear specification where every
additional family member has the same average effect on earlier children may obscure significant heterogeneity
in the return from an additional children. Bagger et al. (2013) argue that conventional empirical specifications
for testing the quantity-quality tradeoff suffer from a fundamental problem of being unable to simultaneously
vary family size while holding the distribution of birth orders constant within the household. Thus if an
additional child changes the optimal tradeoff along the entire birth order distribution OLS regressions at
the individual level cannot correctly estimate the quantity-quality tradeoff. They propose an alternative
estimation strategy: estimate birth-order effects using family fixed effects, residualize to net out birth-order
effects and then estimate the relationship between average family size and average outcomes, net of these
birth-order effects.
3
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
4
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 cognitive 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
2.1
Empirical Approach
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
5
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 + β1 1{af ter}ijt + Xit β2 + λi + εijt
(1)
where Y is an outcome of interest for child i with a sibling at birth parity j , 1{af ter} 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 + β1 1{0-3 years after sibling k ’s birth}it + β2 1{3+ years after sibling k ’s birth}it
(2)
+ Xit β3 + λi + εit
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 .
6
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 quantityquality 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
7
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
2
Please see https://nlsinfo.org/content/cohorts/nlsy79/intro-to-the-sample/interview-methods for complete details on the timing of interviews.
8
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 children 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,
3
Note that even if birth timing or NLSY interview timing is not random we are actually requiring their
difference to be random to achieve causal identification.
9
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 non4
Another possibility is that the unobservable factors that lead parents to decide to have another child
are not prior to the birth but instead forecasted events that will occur after the arrival of the child. To test
for this we would instead expand the interval length to greater than 12 months, an exercise available upon
request.
10
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 EnvironmentShort Form) score, “a unique observational measure of the quality of the cognitive stimulation 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
11
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
5
Results at third parity are reported in Table A.3. We find that the negative results for HOME score and
behavioral problems remain while cognitive test results disappear at third parity.
6
We find somewhat stronger results for math and weaker results for reading when we disaggregate.
12
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, estimates 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 additional 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
13
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 detrimental 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
14
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.
15
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 + β1 F AM SIZE + Xβ2 + ε
F AM SIZE = α0 + α1 1{T W IN } + Xα2 + ν
(3)
(4)
where Y is an outcome of interest, eg. years of education, F AM SIZE is the number of
children in the family, and X is vector of individual characteristics. 1{T W IN } is a dummy
7
It is worth noting that we do find persistence in the negative impact of the arrival of a sibling throughout
childhood.
16
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
8
Twins are not directly identified in the NLSY. However, there are data on the month and birth of each
child. We identify twins (and triplets and quadruplets) as two (or more) siblings who have the same mother
and share the same year and month of birth. Out of 11,476 children respondents living in 4,510 households
we identify 117 pairs of twins.
9
In order to ensure the highest grade completed in our sample represents completed education we restrict
our sample to children who were surveyed at least once after age 24. We include demographic controls for the
child such as age, gender, and race and maternal controls such as mother’s race, mother’s AFQT, mother’s
age at first birth, and mother’s non-cognitive measures such as Rosenberg score and Rotter scale score.
17
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 paritypooled 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
18
(Black et al. 2005) may be institutional differences between Norway and the U.S. In particular, 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
19
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
37.98
(27.02)
FE Sample
38.55
(26.98)
Mother’s self-esteem index
21.67
(4.083)
21.74
(4.094)
Mother’s locus-of-control index
8.915
(2.515)
8.915
(2.501)
Age of mother at birth of child
20.34
(2.946)
20.77
(2.788)
Black
0.197
(0.398)
0.190
(0.393)
Hispanic
0.0994
(0.299)
0.107
(0.309)
Birth order
1.403
(0.716)
1.423
(0.724)
Birth weight of child (oz)
116.4
(20.60)
116.6
(20.32)
Spacing to next birth (months)
302.6
(58.60)
328.0
(45.10)
Current number of siblings
2.564
(1.255)
2.222
(1.105)
Living in blended family
0.0964
(0.295)
0.0786
(0.269)
Living with single mother
0.309
(0.462)
5330
0.338
(0.473)
2309
Mother’s AFQT pctile
Observations
Data from Children of the NLSY79, 1986-2010. Our siblng-pair sample includes all older childrenyounger siblings pairs where the older child has a valid PIAT test score. PIAT tests were administered to children aged 4 to 14. Standard deviations in parentheses.
23
Table 2: Individual fixed effects: 2nd parity
After sibling birth
(1)
Individual FEs
-2.126*
(1.113)
Cognitive Ability
0-3 years after
3+ years after
N
After sibling birth
9936
HOME Inventory
3+ years after
After sibling birth
13217
Behavioral Problems
-3.032**
(1.064)
-5.093***
(1.401)
13217
3.091**
(1.299)
0-3 years after
3+ years after
N
-1.889*
(1.122)
-3.408**
(1.258)
9936
-2.910**
(1.063)
0-3 years after
N
(2)
Individual FEs
11119
3.127**
(1.301)
2.049
(1.544)
11119
Each row presents estimates for a different outcome of interest. Controls are child’s age at test and
child fixed effects. Robust standard errors are clustered at the individual level. Data from Children
of 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 around
births: 2nd parity
Cognitive Ability
N
HOME Index
N
Behavioral Problems
N
(1)
0-12 months before
-2.823
(2.165)
427
(2)
0-24 months before
-0.741
(1.752)
554
-2.630*
(1.565)
1392
-2.418*
(1.298)
1926
8.060***
(2.398)
626
4.415**
(2.059)
796
Each row presents estimates for a different outcome of interest. Column (1) restricts sample to
interviews 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
a 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 individual
level. 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
Below median
Cognitive Ability
Above median
N
Below median
HOME Inventory
-3.922**
(1.650)
-2.536**
(1.183)
13217
Behavioral Problems
0.779
(1.953)
4.253**
(1.647)
11119
Above median
N
Below median
(1)
Individual FEs
-5.415**
(1.871)
-0.346
(1.328)
9936
Above median
N
Each row presents estimates for a different outcome of interest. Regressions include controls for
child age and child fixed effects. Robust standard errors are clustered at the individual level. Data
from 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
Hispanic
Cognitive Ability
Black
White
N
Hispanic
HOME Inventory
-2.497
(1.839)
0.523
(1.698)
-3.564**
(1.189)
13217
Behavioral Problems
1.501
(2.454)
4.022**
(1.836)
3.030*
(1.681)
11119
Black
White
N
Hispanic
(1)
Individual FEs
-2.424
(1.816)
-4.657**
(1.739)
-1.288
(1.463)
9936
Black
White
N
Each row presents estimates for a different outcome of interest. Regressions include controls for
child age and child fixed effects. Robust standard errors are clustered at the individual level. Data
from 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
Boys
Cognitive Ability
Girls
N
Boys
HOME Inventory
-2.919**
(1.276)
-2.901**
(1.325)
13217
Behavioral Problems
3.975**
(1.799)
2.157
(1.795)
11119
Girls
N
Boys
(1)
Individual FEs
-0.464
(1.367)
-4.208**
(1.755)
9936
Girls
N
Each row presents estimates for a different outcome of interest. Regressions include controls for
child age and child fixed effects. Robust standard errors are clustered at the individual level. Data
from 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)
Whole sample
Worked last week (CPS)
-0.0238*
(0.0139)
Below median
Above median
(2)
By AFQT
0.00996
(0.0136)
-0.0316**
(0.0135)
Hispanic
-0.00536
(0.0160)
0.0198
(0.0122)
-0.0255**
(0.0128)
Black
White
Usually FT (CPS)
(3)
By race
-0.0656**
(0.0332)
Below median
-0.0867**
(0.0368)
-0.0680**
(0.0274)
Above median
Hispanic
-0.103**
(0.0449)
-0.0588
(0.0391)
-0.0716**
(0.0270)
Black
White
Each row presents estimates for a different outcome of interest. Regressions include controls for
child age and child fixed effects. Robust standard errors are clustered at the individual level. Data
from 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
1
2
3
4
5
6
7
8
9
10
11
Total
Family size
# of HHs
1179
1966
1126
430
135
49
22
7
4
2
2
4922
Birth order of twins within family
# of HHs
44
35
19
10
6
3
117
Data from Children of the NLSY79, 1986-2010.
30
31
7931
0.139
(2)
+ Birth order controls
-0.0182
(0.0557)
-0.606***
(0.123)
-0.682**
(0.265)
-1.469***
(0.300)
-1.954***
(0.748)
-2.098
(1.492)
-2.645***
(0.300)
7931
0.154
First Stage Coeff.
1.805***
(0.148)
7316
0.132
(3)
IV, full sample
-0.487***
(0.178)
-0.482***
(0.146)
-0.109
(0.362)
-0.461
(0.488)
-0.827
(0.985)
1.781
(1.389)
(4)
IV, pooled parity sample
-0.883**
(0.447)
-0.350*
(0.208)
0.399
(0.638)
0.346
(0.975)
0.296
(1.490)
2.941
(1.810)
0.585
(1.658)
7320
0.0482
All results include controls for child age and gender. Maternal controls are mother’s age at first birth, race, AFQT,
Rosenberg and Rotter scores. Robust standard errors are clustered at the individual level. Data from Children of the
NLSY79 sample, 1986-2010.
∗: significant at 10% level. ∗∗: significant at 5% level. ∗ ∗ ∗: significant at 1% level.
N
r2
7th+ Child
6th Child
5th Child
4th Child
3rd Child
(1)
Maternal controls
Family Size
-0.126**
(0.0544)
2nd Child
Table 9: Effect of an additional child on years of education | age ≥ 25
Estimated effect (percentile points)
Figure 1: Estimated impact of an additional sibling on cognitive ability for different window
lengths
4
2
0
-0.74
−2
-2.83
−4
−6
−8
−40
−30
−20
−10
Months included in ‘before’ sample
32
0
Estimated effect (percentile points)
Figure 2: Estimated impact of an additional sibling on HOME index for different window
lengths
2
0
−2
-2.42
-2.63
−4
−6
−40
−30
−20
−10
Months included in ‘before’ sample
33
0
Estimated effect (percentile points)
Figure 3: Estimated impact of an additional sibling on behavioral problems for different
window lengths
12
10
8
8.06
6
4.42
4
2
0
−40
−30
−20
−10
Months included in ‘before’ sample
34
0
Table A.1: Children of NLSY79: Summary Statistics
Age in 2010
At least 25 years old in 2010
Female
Black
Hispanic
Birth weight of child (oz)
Family size
Twins in family
Yrs of education — At least 25 years old
HS grad — At least 25 years old
2009 Wage + salary income
Log 2009 wage + salary income
Worked 35+ hours at 1st job, 2010
Child was convicted of a crime
Child had a teenage pregnancy indicator
Math test (percentile)
HOME Inventory score (percentile)
Behavioral Problems Index (percentile)
Mean
23.9
0.51
0.49
0.28
0.19
116.1
2.92
0.038
Std. Dev.
6.39
0.50
0.50
0.45
0.39
22.7
1.42
0.19
12.7
0.74
22916.3
9.92
0.76
0.25
0.075
2.34
0.44
22251.3
0.98
0.43
0.43
0.26
51.2
46.3
59.1
28.0
29.4
28.0
Data from Children of the NLSY79, 1986-2010. Adult outcomes in the middle box are conditional
on 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
Mothers AFQT Score (percentile)
40.2
SELF ESTEEM SCORE 80
22.1
Mother’s Rotter (locus-of-control) score (percentile)
8.82
Age of mother at birth of child
22.9
Black
0.26
Hispanic
0.17
HH w/ twins
44.0
22.6
8.74
23.3
0.26
0.17
Difference
-3.779
-0.554
0.0745
-0.436
0.00234
0.0000758
Cognitive ability
HOME Inventory score (percentile)
Behavioral Problems Index (percentile)
Birth weight of child (oz)
HH w/o twins
55.0
46.8
58.9
116.4
HH w/ twins
48.0
42.8
61.3
116.7
Difference
7.007∗∗∗
3.937∗∗
-2.347
-0.286
Highest grade completed
12+ years of schooling completed
Log 2009 wage + salary income
Child had a teenage pregnancy indicator
HH w/o twins
12.7
0.75
9.92
0.076
HH w/ twins
11.9
0.57
9.61
0.22
Difference
0.810∗∗
0.175∗∗∗
0.308
-0.147∗∗∗
Data from Children of the NLSY79 sample, 1986-2010. For households with twins present we
restrict the sample to children born prior to the birth of the twins. The last column reports
p-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
After sibling birth
(1)
Individual FEs
0.0773
(1.496)
Cognitive Ability
0-3 years after
3+ years after
N
After sibling birth
4504
HOME Inventory
3+ years after
After sibling birth
6451
Behavioral Problems
-1.974
(1.467)
-2.684
(1.947)
6451
5.169**
(1.580)
0-3 years after
3+ years after
N
0.0988
(1.499)
-0.120
(1.784)
4504
-1.941
(1.465)
0-3 years after
N
(2)
Individual FEs
5114
5.117**
(1.571)
7.132***
(2.045)
5114
Each row presents estimates for a different outcome of interest. Controls are child’s age at test and
child fixed effects. Robust standard errors are clustered at the individual level. Data from Children
of 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 around
births: 3rd parity
Cognitive Ability
N
(1)
(2)
0-12 months before 0-24 months before
-1.554
-1.249
(2.376)
(1.494)
631
936
HOME Index
N
Behavioral Problems
N
-3.064
(1.983)
1139
-2.882**
(1.372)
1791
-0.587
(2.597)
762
1.752
(1.693)
1151
Each row presents estimates for a different outcome of interest. Column (1) restricts sample to
interviews 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 a
younger 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 individual
level. 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
Mothers AFQT Score (percentile)
Age of mother at birth of child
Cognitive ability
HOME Inventory score (percentile)
Birth weight of child (oz)
Highest grade completed
12+ years of schooling completed
Log 2009 wage + salary income
Child had a teenage pregnancy indicator
Before group
39.6
23.0
52.6
48.5
115.8
14.0
0.73
9.37
0.072
After sample
38.0
23.1
51.8
50.1
116.0
13.8
0.75
9.37
0.063
Difference
1.598∗
-0.101
0.816
-1.610∗
-0.169
0.211
-0.0165
-0.00192
0.00832
Panel (a) shows child characteristics for children who will be interviewed either one year before or
one 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 Children
of the NLSY79 sample, 1986-2010.
∗: significant at 10% level. ∗∗: significant at 5% level. ∗ ∗ ∗: significant at 1% level.
39