Towards Causal Estimates of Children’s Time
Allocation on Skill Development
Gregorio Caetano, Josh Kinsler and Hao Teng⇤
May 2016
Abstract
Cognitive and non-cognitive skills are critical for a host of economic and social outcomes as an adult. While there is broad agreement that a significant amount of skill
acquisition and development occurs early in life, the precise activities and investments
that drive this process are not well understood. In this paper we examine how children’s time allocation affects their accumulation of skill. Children’s time allocation is
endogenous in a model of skill production since it is chosen by parents and children.
We apply a recently developed test of exogeneity to search for models that yield causal
estimates of the impact time inputs have on child skills. We show that the test, which
exploits bunching in time inputs induced by a non-negativity time constraint, has power
to detect endogeneity stemming from omitted variables, simultaneity, measurement error, and several forms of model misspecification. Results suggest that with a rich set
of controls we can consistently estimate the impact of time inputs on skills, though
there is significant heterogeneity in which controls matter for different skills at different
ages. For children aged 12 to 17, active time with adult family members appears to be
the most productive use of time in developing cognitive skills. In contrast, time spent
alone and passive time with the family are important to develop non-cognitive skills
at these ages. For children aged 5 to 11, with few exceptions time allocation does not
have a strong effect on cognitive skills. Moreover, sleep is the most productive input
for non-cognitive skill development at these ages.
Gregorio Caetano: Economics Department, University of Rochester. Josh Kinsler: Economics Department, University of Georgia. Hao Teng: Economics Department, University of Rochester. We would like
to thank Carolina Caetano, Vikram Maheshri, Daniel Ringo, and David Slichter for helpful discussions. All
errors are our own.
⇤
1
1
Introduction
There is a growing consensus among economists that skills acquired during childhood
have an important influence on later life outcomes.1 Cognitive skills measured as early as age
seven have been linked with educational attainment, employment, and wages as an adult.2
Recent research stresses the importance of early childhood non-cognitive skills on later life
labor market, marriage, health, and criminal outcomes.3 Additional analyses indicate that
adult labor market outcomes are largely determined by skills already in place by age 16.4
In light of the evidence linking children’s cognitive and non-cognitive skills to adult outcomes, it is important to understand the determinants of these skills. One can envision
a production technology for child skills where endowed ability is combined with various
environmental elements and time inputs (i.e., activities) contributed by parents, siblings,
grandparents, friends, etc. Households will choose time inputs optimally given their preferences and constraints. The challenge in estimating the causal effect of each time input on
skill production is two-fold. First, the time allocation of child activities is endogenous since
it reflects choices made by households. Second, researchers are typically unable to observe
the full vector of activities, and can often only observe one or two time inputs. Thus, even
if exogenous variation is available for a particular activity of interest, it is difficult to interpret the resulting coefficient without information on the substitution among all potential
activities.
These challenges are well known and have been widely discussed in the literature, yet
so far they have not been systematically addressed.5 On the one hand, studies such as
Dustmann and Schönberg (2012) and Bernal and Keane (2010) utilize quasi-experimental
policy variation that enables them to study how an increase in maternal time affects child
cognition. However, a lack of comprehensive data on all other time inputs prevents them
from understanding the substitution between activities, making it difficult to infer what
would be the effect of a reallocation of time inputs in circumstances other than the change
implied by the policy. On the other hand, studies such as Todd and Wolpin (2007), Fiorini
and Keane (2014), and Del Boca et al. (2013) estimate the impact of a more comprehensive
list of child inputs on skills, but lack quasi-experimental variation.6 Going forward, the
1
See Almond and Currie (2011) for a comprehensive review of the literature.
For instance, McLeod and Kaiser (2004) find that age 7 test scores and family background measures
predict 11% (12%) of the variability in the probability of high school (college) completion. Using similar
measures, Currie and Thomas (1999) are able to explain 4-5% of the variation in employment and 20% of
the variation in wages at age 33.
3
See for example Cunha et al. (2006), Deming (2009) and Heckman et al. (2013).
4
See Keane and Wolpin (1997) and Cameron and Heckman (1998) as examples.
5
See Todd and Wolpin (2003) and Keane (2010) for a discussion of these issues.
6
Todd and Wolpin (2007) choose their most preferred model using root mean-squared error as the selection
2
2
lack of exogenous variation in models with many inputs is unlikely to be solved through
an instrumental variables approach since each input requires its own instrument. Moreover,
running an experiment is also difficult, as the treatment arms would have to consist of fully
prescribed inputs for each child. If only some inputs are manipulated, parents can optimize
over the remaining ones to either reinforce or negate the intended effect. Either way, it will
be difficult to estimate the ceteris paribus causal impact of substituting one time input for
a well-defined alternative time input.
Our aim in this paper is to estimate the impact of children’s time allocation on cognitive
and non-cognitive skills while also investigating endogeneity concerns in a systematic manner.
To do this, we exploit a recently developed test of exogeneity (Caetano (2015); see also
Caetano and Maheshri (2015)) that provides an objective statistical criterion to determine
whether the parameters of interest can be interpreted as causal. We use the test to guide us
in the search for causal models without an explicit source of quasi-experimental variation.
Our empirical approach is essentially a model selection one, but with a key difference to
existing approaches: our criterion to select models speaks directly to causality, not to fit. In
the context of skill development, model selection occurs in a very large model space; indeed,
there are many ex ante equally plausible ways of formulating the relationship between time
inputs and skill. We show that a multivariate version of the exogeneity test in this context can
be a feasible tool to substantially reduce the set of models that can plausibly be considered
appropriate to make causal inference.
The intuition behind the test is as follows. Consistent with the rest of the literature, we
assume that cognitive and non-cognitive skills are continuous functions of time allocated to
various activities. Thus, conditional on controls, skill measures should not vary discontinuously when a child spends exactly zero minutes on any activity (in comparison to, say, just a
few minutes). If we find such discontinuities, then it is evidence that unobserved confounders
that are not absorbed by controls vary discontinuously at the zero minute threshold, and
hence the model suffers from endogeneity. For a test based on this logic to have power, we
need to ensure that unobservables vary discontinuously when time inputs equal zero. We
argue that this is the case in our context since children cannot spend negative time in any
activity, and therefore, may bunch at zero because of a “corner-solution”. For instance, consider children’s reading ability as an important unobserved confounder. Intuitively, children
criteria. However, as the authors point out, this measure speaks to fit but not necessarily to whether the
parameters of interest can be interpreted as causal. Fiorini and Keane (2014) approach the identification
problem by estimating multiple production functions that rely on different exogeneity assumptions. Stability
of the productivity ranking of inputs is cited as evidence that a reallocation of time use can enhance child
development. However, as the authors point out, it is difficult to claim these estimates are causal since
the models could suffer from similar biases. Similarly, Del Boca et al. (2013) assume inputs are exogenous
conditional on lagged skill, but it is difficult to assert whether these controls are enough to handle endogeneity.
3
who spend a few minutes per week reading choose (or let their parents choose) that amount
optimally given their constraints and preferences. In contrast, among children who spend
zero minutes per week reading, there are children who optimally choose exactly zero minutes
and children who would optimally choose negative amounts of reading time if possible. This
implies that the average reading ability among children who spend x minutes reading tends
to be discontinuous at x = 0, and thus the test would have power to detect endogeneity
stemming from unobservables related to reading ability. The more unobserved confounders
that vary discontinuously at zero minutes for a given time input, the more powerful the test
will be as it can detect endogeneity stemming from multiple sources.
One potential concern is that our exogeneity test might lack power: in spite of the intuition above, potential confounders might not vary discontinuously when children spend zero
minutes in an activity. If this is the case, then we would not be able to detect endogeneity
even if endogeneity exists. To allay this concern we first show that each activity of interest has a discontinuous mass of observations at zero minutes, providing direct evidence of
bunching (McCrary (2008)). Next, we show that a variety of key observed characteristics
are discontinuous at the zero threshold. While not a formal test, this is suggestive that unobservables are discontinuous as well.7 It also suggests that discontinuities when inputs are
zero are the norm rather than the exception. Finally, we are able to detect discontinuities in
the most parsimonious models. It is only when we add a richer set of controls that we fail
to reject exogeneity.
Even if the evidence discussed above establishes that the test has power, the test might
not have power to detect all types of endogeneity we are concerned about. In particular,
some potential confounders might be continuous when inputs are zero, or they might be
confounders only when inputs are not zero. We allay concerns about their existence in many
ways. First, we test for discontinuities at zero minutes for all time inputs jointly. This
increases the power of the test since a confounder that is undetected when one input is
zero can be detected when a different input is zero. Second, we provide empirical evidence
that key variables that elicit a comprehensive list of sources of potential endogeneity in this
context – omitted variables (e.g., child’s reading ability), simultaneity, measurement error,
and many forms of model misspecification – are discontinuous when inputs are zero. Third,
we implement a series of robustness checks designed to detect confounders that cannot be
detected by the test, and find that models that survive the exogeneity test also survive these
robustness checks. Finally, we show that estimates are similar across models that survive the
7
This logic is analogous to the one used in regression discontinuity (RD) designs. Support for the RD
identifying assumption is based on the idea that if observables vary continuously at a threshold, then unobservables are also likely to vary continuously at that same threshold.
4
test, although they are often different among models that do not survive the test. We track
the cumulative properties a confounder must have to bias our preferred estimates and be
undetectable by both the test of exogeneity and the further robustness checks we perform.
Ultimately, we conclude that interpreting our preferred estimates as causal is plausible, in
the sense that it is unlikely a confounder will possess all these properties at the same time.
We implement our approach using skill assessments and time diaries from the Child
Development Supplements of the Panel Study of Income Dynamics (PSID). With the help
of their primary caregiver, children filled out a detailed 24 hour time diary to record all
of their activities during the day, where each activity took place, and with whom they did
the activity. These time diaries were collected in 1997, 2002, and 2007, and covered one
weekday and one weekend day for each survey year. Cognitive and non-cognitive skills are
also assessed during each wave of the survey. We split the sample into older (12-17) and
younger (5-11) children and estimate separate production functions to allow for heterogeneity
in the accumulation of skill as children age. In addition to time use and skill assessments, the
PSID also includes a detailed list of child demographics, family background characteristics,
and other measures of the environment in which the child is raised.
Our search for a model whose parameters can be interpreted as causal proceeds in the
following manner. In our baseline models, we categorize child activities according to the
level of engagement – active (e.g., reading) or passive (e.g., watching TV) – and with whom
the activity is completed – mother, father, siblings, friends, grandparents, others, or no one.
We then relate the time devoted to these activities to skill measures (math, vocabulary,
comprehension, non-cognitive) using standard production functions, such as value-added.
We also consider models containing many other subclassifications of activities as suggested
in the previous literature. The activity excluded from all models is sleeping, so that the
results should be interpreted as a substitution between a given activity and sleeping time.
Every model also includes a series of indicator variables that reflect whether the time devoted
to each activity is zero. These indicator variables are included to absorb any discontinuous
change in the outcome variable when inputs are zero, conditional on controls. If we reject
the null hypothesis that the coefficients on the zero time input indicators are jointly equal
to zero, then we conclude that this particular model suffers from an endogeneity problem.
In our context, a model is defined by the outcome variable (skill), the main explanatory
variables (time inputs), controls (ranging from none to a detailed list of child, family, and
environmental observables) and a particular functional form establishing the relationships
between these variables.
For both younger and older children, our search ultimately identifies models for which
we are unable to reject exogeneity of children’s time inputs. Part of our success in this
5
endeavor is due to the detailed nature of the time use data. When we estimate the impact
of a particular time input, we are able to control for all other time inputs of the same child.
These alternative inputs absorb much of the endogeneity, as they elicit heterogeneity in preferences and constraints across children and activity partners in the sample. However, the
time use data alone is not sufficient to account for all endogeneity. For younger children,
child and family characteristics are crucial additional controls needed to absorb endogeneity
in all cognitive and non-cognitive skill models. In contrast, for older children lagged skills in
combination with different categories of controls are important in absorbing endogeneity for
different skill measures. For math skill models, child and family characteristics are crucial,
while for vocabulary and comprehension models, child characteristics, school characteristics
and school experience are important. Further, for models of non-cognitive skills, child characteristics, family environmental characteristics and school characteristics are important for
controlling for endogeneity.8 As surveys become increasingly onerous in their time demands
on respondents, our results can provide guidance regarding which observed variables are
critical to collect when studying child skill development.
Once we arrive at models for which we fail to reject exogeneity, we turn our attention to
the key parameters of interest. For older children, the activities that most promote cognitive
skill formation are active time with adult family members, such as parents and grandparents.
For example, one additional hour per week spent in active time with grandparents leads to
2.9% of a standard deviation increase in comprehension scores.9 Non-cognitive skills, on
the other hand, are increased most by passive time with parents and alone. For younger
children, the estimates of the impact of time inputs are quite different. Sleeping or napping
is the most productive time for the development of non-cognitive skills. For example, one
additional hour per week spent in sleeping or napping rather than in active time with friends
would increase non-cognitive skills by 1.8% of a standard deviation. Math, vocabulary, and
comprehension skills are less sensitive to alternative time allocations.
Overall, we find that active time is not necessarily better than passive time at improving
skills, skill improvement is more difficult to achieve through time re-allocation for younger
children as compared to older children, and that more time in school improves cognitive skills
but has a varying impact on non-cognitive skills according to child age. In the penultimate
section of the paper, we present a simple model of time allocation to help illustrate why
8
Our results help clarify the key finding in Fiorini and Keane (2014) that the ranking of time inputs across
various production function models is stable. Each of their preferred models, including contemporaneous,
lagged score, and lagged input, include controls similar to the ones we identify as being important for allaying
endogeneity.
9
This result aligns with studies in developmental psychology which emphasize the irreplaceable role of
grandparents in the development of grandchildren (see Smith (2003) for more details).
6
selection on observables is a valid assumption in this case. The model is also useful as a lens
through which to interpret our findings and place them in the literature. Finally, we discuss
promising avenues for further research.
The rest of the paper is organized as follows. In Section 2, we describe the PSID data.
Section 3 presents our approach and provides both theoretical and empirical evidence in
support of this approach. In Section 4, we present our main results. In Section 5, we
perform various robustness checks. In Section 6, we discuss why selection on observables is
sufficient to identify the parameters of interest in this context, before we conclude in Section
7.
2
Data
To estimate the effect of time inputs on skill we use data from the Panel Study of Income
Dynamics (PSID) and the three waves of the Child Development Supplements (CDS-I, CDSII, and CDS-III).10 In 1997, the PSID started collecting data on a random sample of the
PSID families that had children under the age of 13. About 3,500 children aged 0-12 residing
in 2,400 households were interviewed in 1997, and then followed in two subsequent waves,
2002 and 2007. Rows 1-3 in Table 1 illustrate the age range and average age for each wave,
respectively.
Data collected in the CDS include measures of children’s cognitive and non-cognitive
skills, time use diaries, and information about child and family characteristics, such as parentchild relationships, child health, and home environment. We match the CDS children with
their PSID families to get additional information such as family annual income, mother and
father’s ages, mother and father’s education levels, and so on. We pool CDS children across
the three waves and divide all observations into two groups based on age: younger children
(5-11 years old) and older children (12-17 years old). Pooling the data in this manner
maximizes our sample size while still allowing for heterogenous production functions across
developmental stages. Rows 4 and 5 in Table 1 illustrate the age range and average age for
each group.
To the best of our knowledge, the only other data set combining information on child
skills and family background with time use diaries is the Longitudinal Study of Australian
Children (LSAC). Compared to the LSAC, the PSID-CDS has the advantage of focusing on
a larger age range of children (0-22 years old) and has richer time use data in terms of the
10
Panel Study of Income Dynamics is a US longitudinal survey of a nationally representative sample of
individuals and families, started in 1968 with a sample of 4800 families. It is funded by National Institute
of Child Health and National Development (NICHD).
7
number of children activities and with whom these activities were performed. As an example,
the PSID-CDS allows us to separate the time children spend with mothers and fathers, who
according to Del Boca et al. (2013), have differential impacts on skill development. The
ability to split activities according to detailed partner categories is also helpful in mitigating
endogeneity, an issue we discuss further in the next section.
2.1
Time Use Diaries
The time use diary from the CDS collects the details of child activities for two random
days of a week (one weekday and one weekend). Diary forms are mailed to each child’s
address, and each child (with the help of her primary caregiver if needed) fills out a detailed
24 hour time diary to record all of her activities during the day, such as where each activity
took place and with whom they did the activity.11 An interviewer then visits the household
to check/edit the diary that has been completed.12
Child activities are classified according to the type of activity (215 in CDS I , 317 in
CDS II, and 315 in CDS III), where the activity took place (14), and with whom (11) the
activity was completed. In CDS I, most diaries (80%) were completed by the child’s primary
caregiver or the child and her primary caregiver together. Sampled children are considerably
older in CDS II and CDS III, and as a result approximately half of the children in these
rounds completed the time diaries on their own.
We clean the time use data so that the diaries are as representative as possible. Time
diaries may have limited reliability since they are only a very small sample of a given child’s
days.13 To allay this concern, we first exclude cases where either the weekday or the weekend
diary was not returned. Second, we exclude diaries that describe a non-typical day. Third,
we keep only complete diaries and do not impute unassigned slots, with one exception: time
periods between 10 p.m. and 6 a.m. that are missing are recoded as sleeping or napping,
as in Fiorini and Keane (2014).14 As a result, we drop 4% of time diaries in CDS I, 3% in
CDS II, and 1% in CDS III. Thus, we are left with complete diaries – those such that the
duration of all the activities add up to 24 hours. The numbers of observations in our samples
are 2,807 in CDS I, 2,520 in CDS II, and 1,424 in CDS III.
Since we have over 200 variables corresponding to the type of activity the child per11
92% (90%) of the primary caregivers are mothers for the younger (older) cohort.
Some interviews were done via phone: 24% for the younger cohort and 9% for the older cohort.
13
Researchers have found that young children’s parents enjoy working with their child to complete the
child’s time diaries, and these diaries can adequately represent the child’s day (Timmer et al. (1985)), but
it is not clear whether that particular day is a representative sample.
14
We also follow them by recoding as sleeping or napping the time periods between 10 p.m. and 6 a.m.
originally filled as “refused to answer”.
12
8
formed and 11 variables corresponding to with whom the child performed the activity, it is
not feasible to estimate the effect of every single combination of these two variables given the
available sample size. Ideally, all else constant, more disaggregated categories of activities are
preferred, as it better exploits the heterogeneity in the data and estimates more interpretable
parameters. However, as the categories of activities become increasingly disaggregated, the
set of potential control variables gets exponentially larger. We choose to categorize children’s
activities into two general types of activities, namely active and passive. Active (passive)
activities include all activities in which the child actively (passively) participates. The activities that we recode as active are taking lessons (e.g., dancing), reading, socializing, active
leisure, household chores, jobs, school/day care, and organizational activities. In contrast,
the activities we recode as passive are obtaining goods and services, traveling/waiting, using
computers, watching TV, passive leisure, and personal needs and care.15
We categorize with whom the child performed the activities into seven groups of people:
mother, father, grandparents, siblings, friends, others (i.e., someone other than the first five
groups), and self. In reality, a child could perform an activity with many different people at
the same time. Whenever the child was with more than one person within the same time slot,
we assign the slot to what we consider to be the primary person according to: mother, father,
grandparent, sibling, friend, and others. Finally, we also add two other categories: refuse
to answer or do not know and sleeping or napping.16 We choose sleeping or napping as the
omitted time input in our estimation, so that all our reported results should be interpreted
as a substitution between a given activity and sleeping or napping.
Our decision to categorize child activities according to activity partners reflects two ideas.
First, prior research indicates that there is important heterogeneity in the productivity of
time depending on the partner (Del Boca et al. (2013)). Second, disaggregating activity partners allows us to better control for selection. Each of these partners has different preferences
and marginal costs of time, which may influence input choices and, for a given input choice,
the return of these inputs on skills (which likely depend on factors such as partner’s age, level
of education, intensity of engagement, child’s perceived authority of activity partner, etc).
Controlling for how time is allocated across all these potential partners helps minimize this
source of endogeneity. However, we also explore the sensitivity of our results to additional
controls that relate to the categorizations chosen by Fiorini and Keane (2014) and Del Boca
et al. (2013).
15
A full description of our recoding rules is available upon request.
Following Fiorini and Keane (2014), we distinguish “refused to answer or do not know” (which we include
in our sample) from the case where an activity is missing (which we exclude from our sample).
16
9
2.2
2.2.1
Summary Statistics
Children’s Time Allocation
In this section, we describe children’s time allocation using the recoded activity categories
as described above. We construct a weekly measure for each time input by multiplying the
weekday hours by 5 and the weekend day hours by 2, and then adding up the total hours.
Tables 2 and 3 show the weekly distributions of time (in hours) for younger and older
children, respectively. Sleeping or napping is the most popular activity in our sample, as
expected. Younger children also spend a lot of passive time with their mother, while older
children spend a lot of active time by themselves. This is not surprising as a five- to elevenyear-old child is likely to spend a lot of time on personal needs and care with her mother,
while an older, twelve- to seventeen-year-old child, is likely to spend a lot of time at school.17
Compared to younger kids, older children have smaller means across the 16 time categories
except for active and passive time with friends and with self. Specifically, as children age, the
average active time with the mother drops from 13 hours per week to 8 hours per week. Active
and passive time with friends and with self seem to reduce time with parents, grandparents
and siblings as children age. It is also evident that there is less variation in active time with
parents amongst older children, while there is more variation in passive time with parents.
Importantly, almost every input category has a sizable mass of respondents reporting zero
minutes.
2.2.2
Children’s Skills, Demographics, and Parental Background
In this section, we discuss other variables in the data that are relevant to our analysis. We start by describing the children’s skill variables that we use as outcomes in our
models. PSID-CDS children aged 3 and older were evaluated using the Woodcock-Johnson
Revised Tests of Achievement (WJ-R), Form B (Woodcock and Johnson (1989)). In 1997,
children aged 3-5 were administered Letter-Word Identification and Applied Problems subtests. Children aged 6 and above received Letter Word and Passage Comprehension sub-tests
as well as Applied Problems and Calculation sub-tests. In the 2002 and 2007 waves, these
tests were re-administered, with the exception of the Calculation sub-tests. Since the Calculation sub-test was only administered for the 1997 wave, we do not include it as one of our
skill measures. Thus, we use standardized versions of Letter Word, Applied Problems, and
17
90% (96%) of self active time for the older (younger) cohort consists of time at school. We incorporate
school time into self active time because it is difficult to determine with whom the child spends the bulk
of their time at school. As discussed in Section 5, our results are robust to splitting self active time into
two categories of inputs, one incorporating school activities and the other incorporating the other activities
comprising of self active time.
10
Passage Comprehension as our child cognitive skill measures.18 In the following sections we
refer to these scores as Vocabulary, Math, and Comprehension.
Non-cognitive skills are measured through parental assessment. In all three waves, the
primary caregiver was asked questions about the child’s behavioral problems. Twenty-six
questions are used to measure the child’s behavioral problem scale, and ten other questions
were asked about the positive aspects of children’s lives, including obedience/compliance,
social sensitivity, persistence and autonomy. With these thirty-six questions, we construct a
measure of non-cognitive skills by using iterated principal factor analysis, similar to Cunha
and Heckman (2008) and Fiorini and Keane (2014). In Table 21 in the Appendix we show
the rotated factor loadings. The factor loadings are all above 0.19 and stable across the
two age groups. The constructed measure is standardized to have mean zero and standard
deviation one and is ordered so that a higher score means better non-cognitive skills.
The PSID-CDS collects extensive information on the child, her household, as well as her
school environment. In Table 4, we present demographic and parental background statistics
for a few selected variables. Child characteristics are presented in rows 1 to 4, parental
characteristics are presented in rows 5 to 12, and environmental characteristics are presented
in rows 13 to 16. The table shows that the younger and older children are fairly similar in
terms of demographic and parental background characteristics. On average, children in the
sample are the second child to her mother, and more than 50% live with both biological
parents. The only sizable difference across age groups is the age of the children and their
parents. Parents of the younger children are on average in their late thirties and parents of
the older children are on average in their early forties. Also, households income is slightly
higher for the older children relative to their younger counterparts.
3
Empirical Approach
In this section, we discuss our empirical approach, paying special attention to how we
implement the test of exogeneity in our setting. A more technical discussion of the test in
a multivariate context can be seen in Caetano and Maheshri (2015); see Caetano (2015) for
the formal description of the test in the univariate context.
We are interested in assessing whether we can consistently estimate via OLS in the
following equation:
Skilli = Inputi + Controli ⇡ + Errori ,
18
(1)
We do not use the standardized scores provided by the PSID-CDS. Instead we standardize the raw score
of each skill measure to have mean 0 and standard deviation 1 for both older and younger children.
11
where i denotes a child. Skilli refers to a particular skill of the child (e.g., mathematics skill),
as measured by standardized assessment scores. Inputi refers to a vector of all activities done
by the child in hours per week, whose jth element is denoted as Inputji (e.g., active time spent
with the mother). Controli refers to covariates added to absorb confounding factors, and
Errori refers to the unobserved determinants of Skilli that are not absorbed by covariates.
In this context, a “model” is defined as a unique combination of (Skill, Input, Control) in
equation (1) for precise definitions of Skill, Input and Control. We can consistently estimate
:= ( 1 , ..., J )0 via OLS in model (Skill, Input, Control), described in equation (1), if:
0
1
B
C
Assumption 1. Cov @Errori , Inputji | Inputi j , Controli A = 0, for all j, where Inputi
|
{z
}
(Input1i , ..., Inputji 1 , Inputj+1
, ..., InputJi ).
i
j
:=
Covariatesji
Our approach consists of testing Assumption 1 (jointly for all j) in all feasible models. In
the models that survive the test, we conclude that, at the same time for all j, all confounding
factors that would bias j are absorbed by Covariatesji := Inputi j , Controli . Thus, we can
reasonably interpret the ˆOLS estimated from the models that survive the test as causal
effects of time inputs on skills. Of course, the credibility of this approach depends crucially
on the capability of the test to detect potential endogeneity. In the rest of this section, we
explain the test and discuss in detail which types of endogeneity the test can and cannot
detect in our context.
3.1
Testing the Exogeneity Assumption
The test of exogeneity relies on the assumption that unobserved confounders will be
discontinuous when at least one time input is zero; thus, when inputs are zero, unobserved
confounders are elicited in the equation. If these unobservables affect Skilli conditional on
⇥
⇤
covariates (i.e., if E Skilli |Inputji = x, Covariatesji is discontinuous at x = 0, for some j),
then they are not fully controlled for and hence cannot be consistently estimated via OLS
in this model. Thus, the test reflects whether covariates are able to absorb the confounding
unobservables which vary discontinuously when at least one input is equal to zero.
We explain the intuition of this test in Figure 1, which illustrates the correlation between
a generic time input and a generic child skill across all children in the sample. The goal
of the test is to understand whether part of this correlation can be interpreted as causal.
The discontinuity shown in Panel (a) must be the result of either the observed covariates or
unobserved confounders varying discontinuously when the time input is equal to zero.19 As
19
Of course, we rule out the possibility that the main effect is discontinuous at zero in equation (1). This
12
shown in Panel (b), conditional on all observed covariates Inputi j , Controli , the discontinuity remains. The remaining discontinuity in Panel (b) must be the result of unobserved
confounders that are not absorbed by the covariates, so Assumption 1 is rejected for this
model.
The statistical power of the test comes from the assumption that unobservables vary
discontinuously when a time input is zero. Below we show empirical evidence supportive of
that; but first, we discuss why this is the case. Unobservables are likely discontinuous at
zero in our context because observations are bunching at a threshold, leading to a “corner
solution” problem. For instance, consider a generic unobservable “mother type”, which helps
determine the skills of a child. Figure 2 illustrates how the average mother type varies
depending on the level of time spent reading books to the child. Mothers who read less
to their child tend to have a lower type, as illustrated in the figure. However, something
unique happens at zero. The mothers who read zero minutes to their child are discontinuously
different from the mothers who read a little to their child. The reason is that among mothers
who read zero, there are some whose type is so low that if possible they would have read
negative amounts of time to their child. In this example, if mother type is not fully absorbed
⇥
⇤
by covariates, then E Skilli |Inputji = x, Covariatesji will be discontinuous at x = 0, which
explains the discontinuity found in Panel (b) of Figure 1. Each time input can elicit many
such unobservable confounders; if covariates do not fully absorb them, then endogeneity will
be detected.
More concretely, we implement the test by adding the vector Di to equation (1):
Skilli = Inputi + Controli ⇡ + Di + Error0i ,
(2)
where Di := d1i , ..., dJi , dji := 1{Inputj =0} . The vector Di allows for a discontinuity in
i
⇥
⇤
E Skilli |Inputji = x, Covariatesji at x = 0 for each j. We implement an F-test for whether
⇥
⇤
= 0, which tests for the null hypothesis that E Skilli |Inputji = x, Covariatesji is continuous
at x = 0 for all j jointly. This test is equivalent to testing whether Assumption 1 holds
(Caetano (2015); Caetano and Maheshri (2015)).
3.2
Evidence of Bunching
The test described above exploits the potential clustering of observations that results
from a non-negative time constraint. Here we show empirically that observations do indeed
implicit assumption, also made in all papers in the literature, is plausible in our context (e.g., a second
of reading a book should not affect the child’s skills that much). Another reason this assumption seems
innocuous in our context is that we cannot reject the null hypothesis of continuity for models with a detailed
enough list of covariates.
13
bunch at zero time input thresholds. Figure 4 shows the cumulative distribution function
(CDF) of various activities for both older and younger children. The fact that the CDFs
cross the vertical axis away from the origin is direct evidence of bunching, as it shows that
the probability density function is discontinuous at zero (McCrary (2008)). Moreover, the
CDFs are smooth away from zero, suggesting that there is not bunching elsewhere. We find
many other activities with similar distribution functions. Tables 2 and 3 show the proportion
of observations with zero time inputs for each input.
Note that the bunching of inputs is not necessary for the test to work; it is sufficient
that unobservables are discontinuous at the threshold for whatever reason. Nevertheless,
the evidence of bunching is suggestive of the “corner solution” intuition developed above:
unobservables should be discontinuous because people cannot choose negative amounts of
time for any child related activity.
3.3
Sources of Detectable Endogeneity
While the bunching evidence above indicates that the proposed test will have power, the
test may not have power to detect all sources of endogeneity. In this section, we provide
evidence concerning the sources of endogeneity the test has power to detect. To structure
the discussion, we write the following general model of skill production:
g i , Otheri )
Skilli = f (Input
(3)
g i is a vector of J˜ activities, defined at a very detailed level, and Otheri is a vector
where Input
g i in this generic
that includes all other inputs in the production function. Elements of Input
framework are defined precisely by a unique combination of all its features. For instance,
reading different books, or reading different pages of the same book, refer to different time
inputs. The production function f (·) is unrestricted in this general setting. This is a “general”
g i and Otheri , and if we
model in the following sense: if we observed all elements of Input
were able to estimate a non-restrictive f (·), then we would be able to identify the causal
g i on Skilli . This is a good benchmark, as it allows us to discuss a
partial effect of Input
comprehensive list of the sources of endogeneity that might show up in our application as
deviations from this ideal scenario.
Assumption 1 essentially combines all of the simplifying assumptions that are needed
to go from the general production function outlined above to the OLS specification we aim
to estimate. For example, Assumption 1 includes assumptions about linearity, additive
separability, and that Controli is sufficient to account for Otheri . A failure of any of these
assumptions will imply the existence of a variable wi that is excluded from Equation (2) and
14
that may be correlated with Inputi . If this variable wi is discontinuous when Inputji = 0 for
some j, then it is correlated to Inputi and the test has power to detect its presence.
Figure 5 shows a few examples of potentially omitted variables wi that are likely
elements of Otheri , where E[wi |Inputji = x] is discontinuous at x = 0 for some j.20 These
plots indicate that the exogeneity test has power to detect whether we incorrectly omitted
wi : if wi affects skill conditional on covariates then skill will be discontinuous at Inputji = 0.
As an example, Panel (a) of Figure 5 indicates that if maternal education affects child skill
conditional on covariates, then the test will detect endogeneity when maternal education
is excluded from Controli in (2). Panels (b)-(d) in Figure 5 indicate discontinuities in the
number of books at home, hours the mother works per week, and household income when
various time inputs are zero, expanding the list of potential elements of Otheri whose omission
the test has power to detect. Additional examples are provided in Figures 10 and 11 in the
Appendix.
The wi we consider in Figure 5 are observable, so they can in principle be incorporated
in Controli to avoid any endogeneity stemming from their exclusion. However, if observables
are discontinuous at zero, then unobservables are also likely to be discontinuous at zero. For
instance, the discontinuity in mother’s education level suggests that unobservables such as
whether the mother is well read, whether the mother pays attention to the academic progress
of the child, etc, might also be discontinuous.21 If these variables are not fully absorbed by
covariates, then their discontinuity will be captured by Di , leading to a rejection of the
model.
Another potential source of endogeneity is simultaneity. If time inputs are caused by
skills, rather than the other way around, Assumption 1 will be violated. For instance, children
with low comprehension skill may be less willing to read, which may generate a spurious
correlation between time spent reading and the comprehension skill measure. Because of
bunching, this spurious correlation should result in a discontinuity at zero, which our test
will detect. In contrast, the causal relationship of interest is plausibly continuous at zero:
for instance, spending time to read the first word of the title of a book has essentially the
same effect on comprehension skills as not reading it at all. In Figure 6, we show that strong
20
Each plot shows E[wi |Inputji = x] along with a local cubic fit, where wi is denoted in the title, and Inputji
is denoted in the horizontal axis. At x = 0, we also show the 95% confidence interval. For x > 0, the scatter
plot aggregates to the next hour of the time input. The shaded region represents the 95% confidence interval
for the fit with an out-of-sample prediction at zero minutes. For the local fit, we use the Epanechnikov
kernel with the rule-of-thumb bandwidth for the kernel and 1.5 times the rule-of-thumb bandwidth for the
standard-error calculation. Results are robust for different choices of kernel and bandwidths.
21
The argument that similar patterns of discontinuity in observables should also be found in unobservables
is analogous to the one made by researchers about continuity when implementing regression discontinuity
designs.
15
correlates of skill levels, such as birth weight, race, age and height, are in fact discontinuous
when various time inputs are zero, suggesting that we have power to detect simultaneity
bias.
g i 6= Inputi ) can also generate endogeneity if the degree
Measurement error (i.e., Input
of misreporting is correlated with the observed time input vector. For example, children
who spend more (active or passive) time alone may be more likely to fill out their own
time-use survey, and children might tend to overstate certain inputs relative to adults (e.g.,
they might overstate the amount of time they spend with friends or other family members
to conceal how often they are alone). In Figure 7, we show discontinuities at zero time
inputs for examples of wi that are likely correlates of misreporting, such as whether the child
completed the time diary alone. These discontinuities suggest that we have power to detect
endogeneity stemming from measurement error.
Our test is also useful for detecting misspecification errors. This is important in our
context since there are countless ways to group activities and model the relationship between
skill and time inputs. In particular, we make four key simplifying assumptions to arrive at
Equation (1). First, we aggregate many time activities into only a few categories, which may
induce endogeneity due to over-aggregation (i.e., J˜ > J). As an example, if a subcategory
of maternal active time, such as reading with the mother, increases disproportionately as
active time with the mother increases, then we may arrive at a biased estimate of maternal
active time. Figure 8 shows discontinuities for a few examples of wi that speak directly to this
potential issue. For example, Panel (a) shows that children who spend no passive time with
their father are likely to spend a discontinuously larger proportion of the active time they
spend with their father at home, relative to children who spend little passive time with their
father. This suggests that if active time with the father is differentially productive at home
(a type of heterogeneity precluded by our aggregation scheme) and results in endogeneity,
then the indicator variable for passive time with father would detect it.
g i , Otheri ) is separable, ruling out the presence of hetSecond, we assume that f (Input
erogeneous effects. For instance, mothers who read well may be more willing to read to
their children, and this activity may generate a higher return to their children’s skill relative to mothers who do not read well. The plots in Figure 5 (see also Figure 10) depict
discontinuities for examples of wi along which heterogeneity in returns of activities likely
occurs, suggesting that we can detect endogeneity resulting from heterogeneous treatment
effects. For instance, Panel (a) of Figure 5 shows that the level of education of the mother is
discontinuous when active time spent with the mother is zero. Thus, the test has power to
detect endogeneity from heterogeneous effects to the extent that any other time input (e.g.,
passive time with the mother) has a different effect on the child’s skills depending on the
16
level of education of the mother.
A third potential misspecification issue that will generate endogeneity is the presence
g i , Otheri ) 6= Input
g i + Controli ⇡). In this case, wi =
of non-linear effects (i.e., f (Input
g i , Otheri ) Input
g i
f (Input
Controli ⇡ might be discontinuous when inputs are zero. Panel
0
(a) of Figure 9 shows that E[Inputji |Inputji = x] is discontinuous at x = 0 for examples of
j 6= j 0 . Children who spend zero active time with their mother spend a discontinuously larger
amount of active time with their grandparents, an average increase from 2 to 5 hours per
week. This is direct evidence that the test has power to detect endogeneity from non-linear
0
0
0
effects (e.g., f j (5) f j (2) 6= (5 2) j ).22 Additionally, our linear model (2) may incorrectly
predict a discontinuous impact of inputs at zero because of non-linearities away from zero.
In this case, Di will be significantly different from zero in an attempt to correct for this
model misspecification. Regardless of the reason, the test has power to detect endogeneity
stemming from non-linear effects.
Finally, misspecification of controls may also lead to endogeneity. If wi is discontinuous at zero, then wi0 := g(wi ) is also discontinuous at zero for almost all functions g(·). Thus,
the test also has power to detect endogeneity due to misspecification of observed controls
wi , which can occur since it is unclear how they should be included in the equation.23
These examples of wi are just a small subset of observed variables for which we find
discontinuities at x = 0. Moreover, for ease of exposition, we have discussed in turn the
implications of each simplification that is needed to go from the general production function
outlined in equation (3) to the specifications we estimate. Our approach is agnostic about
the specific reason why Assumption 1 might fail, and in fact jointly tests for all sources of
detectable endogeneity, even ones we may not conceive. Of course, even among these sources
of endogeneity there may be confounders that cannot be detected by the test. For example,
some confounder wi implied by an aggregation choice may not be discontinuous when inputs
are zero. Next, we argue why these confounders are likely to be rare in our context.
22
In reality, there is heterogeneity across observations with the same value of Inputji , which enhances the
0
0
0
power of the test because it can detect endogeneity if f j (x1 ) f j (x2 ) 6= (x1 x2 ) j for other values of
x1 and x2 . For instance, Panel (b) of Figure 9 shows that the entire distribution is discontinuous at x = 0,
not only its first moment. Caetano and Maheshri (2015) discusses this point in more detail.
23
If wi enters the equation non-linearly, discontinuities in higher moments of the distribution will add
power to the test. Here we show only discontinuities in the first moment of the distribution, but we actually
find discontinuities in the whole distribution. For instance, the variance of wi is often discontinuously higher
when inputs are zero. This is intuitive, as observations tend to be discontinuously more heterogeneous at
that point because of bunching.
17
3.4
Which confounders cannot be detected by the test?
Consider the set of all potentially endogenous variables w, characterized by being correlated to both Inputi and Skilli . If any such w is not absorbed by covariates, Assumption 1
in the context of Equation (1) will be violated. These variables fall under two categories:
(a) those that vary discontinuously at Inputji = 0, and (b) those that vary continuously
at Inputji = 0. Thus far, we have focused our discussion on confounders of type (a) and
our ability to detect them with our test. However, endogenous variables of type (a) can
actually be further subdivided in two types: (a1) those that are correlated with Skilli when
Inputji = 0, and (a2) those that are uncorrelated with Skilli when Inputji = 0. These three
types of confounders, (a1), (a2), and (b), form a partition: any confounder wi must be of
one and exactly one of these three types. The exogeneity test described above can detect
all confounders of type (a1), but cannot detect confounders of types (a2) or (b). Note that
our multivariate setting adds redundancies that contribute to the power of the test since a
confounder of type (a2) or (b) for input j can be of type (a1) for input j 0 .24 In any case,
as we argue next, confounders of type (a1) should be the norm rather than the exception in
our context.
To help frame our argument, Figure 3 illustrates examples of confounders of types (a)
and (b). The solid black lines (and points) in the figure correspond to Inputji , while the
j?
dashed black lines correspond to Inputj?
i , where Inputi is the unrestricted optimal choice
j
j
j?
j
of input j by individual i. Of course, Inputj?
i = Inputi for Inputi > 0, but Inputi  Inputi
for Inputji = 0. This occurs because people cannot spend a negative amount of time on an
activity.25
Panel (a) of Figure 3 distinguishes between confounders of type (a1) and (a2), both of
which vary discontinuously at Inputji = 0. In this example, the red range along the right
side of the vertical axis is the support region of the confounder for the whole sample, while
the blue range along the left side of the vertical axis is the support region for the subsample
of observations such that Inputji = 0. A confounder, by definition, must be correlated to
Skilli in the red range. A confounder is of type (a1) if it is correlated to Skilli in the blue
range, while it is of type (a2) if it is not correlated to Skilli in the blue range. The evidence
of bunching shown above suggests that type (a2) is unlikely, since a significant portion of
the sample is such that Inputji = 0. Moreover, the redundancies implied by the multivariate
0
24
All observations with Inputji > 0 are such that Inputji = 0 for some j 0 , for all j, reducing the possibilities
0
of confounders of type (a2). Similarly, all observations with Inputji = 0 are such that Inputji = 0 for some
j 0 , for all j, reducing the possibility of confounders of type (b).
25
Note that Inputji , rather than Inputj?
i , should be included as inputs in the production function, since we
want to identify the effect of the actual (not the desired) time spent on activities. Thus, we do not have a
censored model, we only have a corner solution model. Wooldridge (2002) discusses this distinction in detail.
18
test is particularly helpful in this case because the blue range of the same confounder will
vary across inputs, allowing the test to cover more of the support of any confounder.
Panel (b) of Figure 3 depicts a confounder of type (b). In this case, the average of the
confounder when Inputj?
i  0 has to be equal to the corresponding average for observations
j?
where Inputi = 0. This is implausible because the confounder is by definition correlated
to Inputji and there are many observations such that Inputj?
i < 0 (as per the bunching
evidence shown in Section 3.2). Indeed, the discontinuity plots discussed in the previous
section suggest that confounders of type (a) are much more likely to occur.
Despite the improbability of confounders of types (a2) and (b), we pursue in Section 5
an extensive set of robustness checks designed to detect them. However, prior to presenting
these checks, we report our main findings.
Remark 1. Our approach is not more helpful than standard approaches in verifying whether
there is important unobserved heterogeneity in a given model. Rather, our approach is
particularly helpful in verifying whether this unobserved heterogeneity leads to endogeneity.
For instance, lack of endogeneity due to heterogeneous effects (i.e., Inputi and Otheri are
non-separable) does not imply lack of heterogeneous effects. It only implies that our estimate
should be an unbiased estimate of the weighted average of different heterogeneous effects,
where weights are given by the distribution of Otheri in the data. For instance, it may be
that an additional hour of passive time alone has a positive effect for a wealthier child and a
negative effect for a poorer child (e.g., the TV program the child is actually watching may be
completely different for these two types of children). In this case, the estimates of a model
where endogeneity cannot be detected by the test of exogeneity could be zero. Indeed, they
should be an unbiased weighted average of one additional hour on passive time alone across
all children, where weights are given by the proportion of wealthier and poorer children.
4
Main Results
We start by proposing a set of regression models that we can plausibly estimate given
the data described in Section 2. As explained, a “model” is defined as a unique combination
of (Skill, Input, Control) in equation (1), where Skill 2 {math, vocabulary, comprehension,
non-cognitive skills}, Input 2 {sleeping or napping, active time with “companion”, passive
time with “companion”, don’t know or refuse to answer}, and companion 2 {mother, father,
grandparents, siblings, friends, others, self}. The set of potential Controls will vary by age
group because of data restrictions. We now describe the models we consider and present our
results for each age group in turn.
19
4.1
4.1.1
Linear Treatment Effects
Older Children (12-17 Year Olds)
For older children, we consider only value-added models, which have been standard in
the literature, by including in all specifications the value of Skilli observed in the previous
wave.26 We consider a sequence of seven specifications of the value-added model, where each
specification includes a richer set of controls than the previous one. We take this approach
for two reasons. First, it illustrates that our exogeneity test has power to detect endogeneity
in the most parsimonious models. Second, it helps identify the key controls that absorb
important sources of endogeneity.
The details of each specification are as follows. Specification (1) has no controls other
than the corresponding lagged skill. Specification (2) adds child characteristics, such as
age, gender, and race. Specification (3) adds mother demographic characteristics, such as
age, education level, and age at child’s birth. Specification (4) adds family demographic
characteristics, such as father’s age, whether the child lives with biological parents, and
household annual income. Specification (5) adds family environmental characteristics, such
as whether the child has a musical instrument at home and whether the child’s neighborhood
is safe. Specification (6) adds school characteristics, such as whether the child is in a public
or private school and the school’s teacher-student ratio. Specification (7) adds the child’s
school experience, such as whether the child has ever repeated a grade, and whether the
child has ever attended a gifted program.27
Table 5 shows the exogeneity test F-statistic and corresponding p-value for each model
we consider. The F-statistics and p-values in bold represent the surviving specifications,
i.e., specifications that we are not able to reject exogeneity at the 10% significance level. In
specification (1), we reject exogeneity irrespective of the dependent variable, which provides
26
We also show results for non value-added models in the appendix, for completeness.
Here is a full list of the control variables included for each category. Child characteristics: child’s
age, child’s age squared, child’s gender, child’s race indicators, birth order to mother, born in the US
indicator, child’s grade indicator, and child’s BMI. Mother demographic characteristics: mother’s education
in years, mother’s current age, mother’s current age squared, and mother’s age at child birth. Family
demographic characteristics: father’s education in years, father’s current age, father’s age at child birth,
mother’s marital status at child birth, household annual income (in $1,000s), number of siblings child lives
with, indicators of whether child lives with biological parents, and indicator of whether child lives with
grandparents. Family environmental characteristics: spending on tutoring programs, spending on extracurricular lessons, indicator for whether caregivers spent on school supplies for the child, indicator for
whether child has a musical instrument at home, indicator for whether child has a desk at home, and rating
of neighborhood quality. School characteristics: indicator for whether school is public, indicators for school
enrollment criteria (e.g., based on geography), school’s teacher-student ratio, and child’s teacher’s full-time
teaching experience. Child’s school experience: indicator for whether child has ever skipped grade, indicator
for whether child has ever attended a gifted program, and indicator for whether child has ever repeated a
grade.
27
20
direct evidence that our test has power to detect endogeneity in the basic value-added model
we consider, complementing the evidence shown in Section 3.3. For different skill measures,
the specification of Control that makes the test no longer able to reject exogeneity is different.
For example, the child’s observed characteristics, together with their lagged skill and all
time inputs, are enough to absorb any confounder in the production function of math,
vocabulary, and non-cognitive skills. In contrast, comprehension seems to be a more
complex production process, as we fail to reject only models that include observed child,
family, and school characteristics. School characteristics (i.e. specification (6)) lead to a
jump in p-value for comprehension skills (i.e. p-value goes from 0.079 to 0.175), which
is suggestive of the importance of school characteristics in absorbing endogeneity. Thus,
Table 5 suggests that, for older children, different groups of control variables are playing
very different roles in absorbing endogeneity depending on the skill in question.28 We are
unable to reject exogeneity in specifications (6) and (7) for all four skills.29
Table 6 presents the estimated coefficients of time inputs from a surviving specification
(specification (7)) for all four skill measures. We find that for math skills, active time with
mother, active time with father, active time with grandparents, active time with friends, self
active time, passive time with mother, passive time with siblings, and self passive time are
statistically significant. Active time with grandparents is the most productive input: one
more hour a week spent on active time with grandparents rather than on sleeping or napping
would increase the math test score by 2.4% of a standard deviation, while one more hour
a week spent on passive time with mother rather than sleeping or napping would increase
test score by about 0.6% of a standard deviation. It is also noteworthy that active time
with friends is as productive as active time with mother for older children. Although we
find that parental inputs have an impact on math skills for older children, there is little to
no effect on vocabulary skills. In contrast, we find that active time with grandparents has
a statistically significant effect on child cognitive skills generally (i.e. math, vocabulary
28
Note that in exercises not reported here, we vary the order in which we add controls. The importance of
each group of variables in accounting for endogeneity is similar to what is observed in Table 5. We complete
a similar exercise for the younger cohort as well. A full description of the permutation exercises we performed
is available upon request.
29
Our identification strategy is based on the premise that we should expect that any confounder will be
absorbed as we add controls, otherwise the test of exogeneity would detect its presence. However, this might
not be the case if hthe standard ierrors of ˆ also increased with the addition of controls. In that case, a
discontinuity in E w|Inputji = x at x = 0 would be wrongly interpreted as continuous; i.e., confounders
of type (1) would be erroneously understood to be confounders of type (3). To check if this is the case,
we present the distribution of the standard errors of all elements of ˆ for all combinations of specifications,
skills and age group in the online appendix. In practice, the standard errors do not seem to increase as more
detailed controls are added in the specifications that we consider. This is not surprising since the addition
of controls is simply an addition of incidental parameters to the regression, so it does not necessarily affect
inference on the parameters of , which remain fixed across all specifications.
21
and comprehension). We also find that passive time with mother, self active time, and
self passive time have similar effects (0.7%-0.8% of a standard deviation) on child’s noncognitive skills.
The coefficients in Table 6 indicate the impact of each input on skills relative to sleeping.
However, by comparing the coefficients with each other we can comment on the relative
effectiveness of various inputs. For example, substituting an additional hour per week of
active time with the father for active time with others would increase math scores by 1.4%
of a standard deviation (with a standard error of 0.2%) . The effect on skills of substituting
one input for another among older children could be quite different from the effect on younger
children, a group we now turn to.
4.1.2
Younger Children (5-11 Year Olds)
Data restrictions prevent us from considering value-added models for the younger set of
respondents. For a child in this age group to have Letter Word or Applied Problem scores
in the previous wave, she has to be at least 8 years old in the current wave, and for her to
have a Passage Comprehension score in the previous wave, she has to be at least 11 years
old in the current wave.30 As a result, if we want to estimate the value added model for
younger children in the same way as for older children, we would be left with 172 observations
only. Given that we have 15 time inputs and a large number of control variables (i.e. 11
in specification (2), 15 in specification (3), 24 in specification (4), 30 in specification (5), 36
in specification (6), and 39 in specification (7)), there would be very few degrees of freedom
left. Thus, we consider only models that exclude lagged skills. So in the baseline model (i.e.
specification (1)) for young children, we include only time inputs. Specifications (2) through
(7) add the same controls used when estimating the production for older children.31
Table 7 presents the F-statistics and corresponding p-values for the test of exogeneity
performed in each model we consider.32 Differently from the case of older children, child
characteristics are no longer enough to absorb confounders with regard to math skills.
Instead, mother’s demographic characteristics are now pivotal for absorbing endogeneity in
math skills. For non-cognitive skills, family demographic characteristics are important
to absorb confounders amongst younger children. Family demographic characteristics are
also crucial for absorbing confounders for the vocabulary skills, but for comprehension
30
As described in Section 2, Letter Word and Applied Problems were not administered for children below
3, and Passage Comprehension was not administered for children below 6.
31
See footnote 27 for a full description of the control variables.
32
For both age groups, the surviving specifications are the same irrespective of which input is omitted.
This is not surprising, as any unobservable that might be correlated to sleeping or napping will necessarily
have to be correlated to one of the other 15 inputs that are included in the regression.
22
skills, mother demographic characteristics seem to be enough. We fail to reject exogeneity
in specifications (4)-(7) for all skill measures.33 This result is somewhat surprising given the
fact that lagged test scores are not included as controls, as in the case of older children.
This could be explained in part by the fact that younger children have fewer opportunities
to choose different activities from each other relative to older children, reducing the scope
for endogeneity.34
Table 8 shows the estimated coefficients from specification (7) for the four skill measures.
The estimates for children ages 5-11 in Table 8 are quite different from the estimates in Table
6 for children ages 12-17. Time inputs, in the way we categorize them, do not seem to play
a critical role in improving younger children’s math or vocabulary skills. When we control
only for child characteristics, we find significant impacts of maternal time on cognitive skill
formation, however these models are rejected by the exogeneity test. Once we add controls
for parental characteristics, the impact of parental time on skill formation vanishes and we
fail to reject the model. Comprehension skills, on the other hand, are affected by time use
at younger ages. Passive time with siblings, passive time with mother, as well as self active
time have a statistically significant influence on comprehension skills. For non-cognitive
skills, the estimates for younger children are mostly negative or zero, which suggests that
spending time sleeping or napping appears to be the most productive way to improve younger
children’s non-cognitive skills. Among the negative coefficients, active time with friends
is the most unproductive one: one more hour a week spent in sleeping or napping rather
than in active time with friends would increase the child’s non-cognitive skills by 1.8% of
a standard deviation. Comparing the estimates for the non-cognitive skills between the
younger and older children, we find that in order to improve a child’s non-cognitive skills,
it is more productive to spend time sleeping or napping when the child is young, and more
time with herself or with her parents as the child gets older.
Similar to older children, we can compare various activities to each other using the
coefficients in Table 8. Because many of these coefficients are small and insignificant, the
differences will also be small and insignificant. One exception would be to substitute one
additional hour per week in active time with mother for active time with friends. This would
33
For both age groups, the surviving models do not change even when we explicitly include napping as an
input in the regression (leaving only night sleeping as the input of reference), along with its corresponding
indicator variable, and implement a stronger test of whether the 16 coefficients of the indicator variables are
equal to zero.
34
In the appendix, we present estimation results without including lagged skills for older children (Tables
22 and 23). Not surprisingly, without the lagged scores more controls are generally needed in order for us
to fail to reject exogeneity. However, we are able to arrive at specifications where the coefficients can be
interpreted causally. Reassuringly, the surviving models for children aged 12-17, with or without lagged test
scores, provide similar estimates for all time inputs (see Tables 23 and 6).
23
yield a 1.7% of a standard deviation (with a standard error of 0.3%) increase in non-cognitive
skills. While variation in time inputs has relatively little impact for younger children, family
background variables are quite strong predictors of math and verbal skills. We suspect that
the structured nature of younger children’s days makes it difficult to identify the impact of
alternative time uses.35
4.2
Non-Linear Treatment Effects
Our main specifications assume that the effects of time inputs on child skills are linear, but
there can be interesting hidden heterogeneity in the results.36 In this section, we re-estimate
our models using a linear B-spline in order to allow for non-linear treatment effects:
Skilli =
X
f j (Inputji ) + Controli ⇡ + Di + Error0i ,
(4)
j
where f j (·) is a linear B-spline function of Inputji with parameters jk , k = 1, 2, 3, representing the linear effect within equally frequent intervals of the distribution of Inputji .
4.2.1
Older Children (12-17 Year Olds)
The exogeneity test results for children ages 12-17 are presented in Table 9. For comparison, we show in bold the surviving specifications according to the linear model (2). It is
useful to check if the models that survive the linear exogeneity test also survive the non-linear
exogeneity test. As discussed at the end of Section 3.3, the coefficients of Di in these linear
models can capture endogeneity from either discontinuous confounders or from a failure of
the linearity assumption. The results show that the specifications that survive the exogeneity test in the linear model also tend to survive the exogeneity test in the B-spline model,
and vice-versa. The only exception is specification (5) for comprehension, which survives the
non-linear test but does not survive the linear test, suggesting that the linear test detects
endogeneity partly due to misspecification of the production function. Overall, most of the
power of the test seems to stem from discontinuous unobservables, otherwise the B-spline
models would fail to reject in even the most parsimonious specifications. From specifications
(6) onwards, all models survive both exogeneity tests for all skills.
35
In the decade since these households were interviewed, there has been a significant focus both in academia
and the public media on early childhood investments. It would be interesting to explore these same questions
for a more recent cohort of children.
36
In general, our approach is not capable of addressing the question of whether there is substantial,
interesting heterogeneity which is not explained in the current linear model; our approach only aims at
addressing the question of whether this unexplained heterogeneity generates endogeneity.
24
Table 10 shows estimates for all four skill measures in our preferred model of specification
(7). We find that maternal active time has a significant positive effect on math and noncognitive skills only when it is more than 15 hours per week, and maternal passive time
only has a significant positive effect on math and comprehension skills when it is below 17
hours per week, and in fact has a negative, significant effect in comprehension skills when it
is above 29 hours per week. A large amount (above 36 hours per week) of active self time
seems to be productive for math, while a little (up until 1 hour per week) passive time with
the father seems to be productive for vocabulary. These results are consistent with the linear
results, but provide further details about the production function of skills.
4.2.2
Younger Children (5-11 Year Olds)
The exogeneity test results for younger children are presented in Table 11. Again, models
that survive the linear test of exogeneity tend to survive the non-linear one and vice-versa,
with a few exceptions, suggesting that the linear test of exogeneity detects endogeneity partly
stemming from misspecification of the skill production function.
Table 12 reports the estimation results for children ages 5-11. Active time with the mother
has a positive effect on non-cognitive skills, if in moderation (between 6 and 15 hours per
week), but a negative effect when it is more than 15 hours per week. A lot of passive time
with the mother or with siblings seem to be productive for vocabulary and comprehension.
In contrast, passive time with the father seems to be counterproductive for the same skills.
Moreover, up to 32 hours per week of self active time (mostly due to school activities, as
discussed previously) has a positive effect on cognitive skills.
Remark 2. As discussed in Remark 1, the fact that the treatment effect estimates are not
linear is not evidence that our surviving specifications in the linear models suffer from endogeneity. Indeed, the results suggest that the linear estimates in our preferred models are
a weighted average of the corresponding non-linear estimates. For example, the coefficient
of passive time with the mother on mathematics skills is 0.006 for older children, which is
similar to a weighted average of the three coefficients of passive time with the mother from
specification (7) shown in Table 10 (i.e. 0.06⇡1/3(0.016+0.000+0.004)). In general, an Ftest for whether each coefficient of the linear model is the same as the weighted average of
the corresponding coefficients of the B-spline model for all 15 time inputs yields a p-value of
0.6526.
25
5
Sensitivity Analysis
Thus far, we have chosen appropriate models for causal inference purely based on the
exogeneity test described in Section 3. However, there can be confounders that are not
detectable by the test. As discussed in Section 3.3, there are two potential categories of
confounders: (a) confounders that are discontinuous at Inputji = 0, and (b) confounders
that are continuous at Inputji = 0. Among type (a) confounders, there are two subtypes:
(a1) those that are correlated with skill at Inputji = 0, and (a2) those that are not. The
exogeneity test introduced in Section 3.1 is capable of detecting all unobservables of type
(a1), but is incapable of detecting unobservables of types (a2) or (b).
As discussed in Subsection 3.4, there are a number of reasons to believe that the class of
variables included in types (a2) and (b) is small in our context. Regardless of how implausible
the existence of these variables might be, this section provides robustness checks that can in
principle detect them if they exist.
5.1
Comparing Surviving and Non-Surviving Specifications
In this section, we compare estimates of across specifications, irrespective of whether
the specification survives or does not survive the test, as shown in Section 4. This comparison is often done in empirical studies, where, heuristically, a good model is one that
provides estimates that are robust to added controls (which might be omitted variables in
the model).37 This “test of stable coefficients” is in principle capable of detecting endogeneity
from the two undetectable sources of endogeneity discussed above. Indeed, added controls
may partly absorb (both at Inputji = 0 and at Inputji > 0) confounders of type (a2) or (b),
leading to a change in the main estimates. If a model survives the test of exogeneity, but
does not survive this test, then it is evidence that the test did not detect some important
source of endogeneity.
We test for whether the fifteen elements of in each specification (1)-(6) from Section
4 are jointly significantly different from the corresponding coefficients in specification (7),
our preferred model. We present the p-value of this test for each skill measure for older
and younger children in Tables 13 and 14, respectively. Numbers in bold refer to those
specifications that survive the exogeneity test at the 10% level of significance. In general,
specifications that survive the exogeneity test (in bold) also survive the test of stable coefficients (p-value > 10%). Across all models of both tables, only one model that survives
the exogeneity test is rejected by the other test: specification (2) for math in Table 5. This
37
For instance, Fiorini and Keane (2014) implement a somewhat weaker version of this test whereby they
compare whether the ranking of the magnitude of each coefficient is the same across specifications.
26
suggests that confounders from the undetectable sources of endogeneity discussed above
are only controlled for after mother characteristics are added as controls (specification (3)).
Conversely, a few models do not survive the exogeneity test but survive the other test (e.g.,
specification (4) for comprehension in Table 5, specification (3) for non-cognitive skill in
Table 7). In these cases, the test of stable coefficients is unable to detect some confounders
that are discontinuous when inputs are zero because they are not correlated to the full list of
controls of specification (7). From specification (5) onwards, all specifications survive both
tests for all skills and both cohorts. Overall, these results are consistent with the idea that,
as we add controls from specifications (1) to (7) in Section 4, we converge to the true causal
estimates.
Tables 15 and 16 show analogous results for the non-linear models discussed in Section
4.2. For each cohort and each specification (1)-(6), we show the p-value from a test of
whether the 27 coefficients jk are significantly different from the corresponding ones in
specification (7).38 We show in bold the specifications that survive the exogeneity test. For
older children, all surviving specifications according to the exogeneity test also survive the
other test, but the reverse is not true. For younger children, in two cases a specification
survives the exogeneity test but does not survive the other test, and in one case the reverse
happens. As in the linear models, all specifications from specification (5) onwards survive
both tests for all skills and both age groups.
In an online appendix, we present the actual estimates for specifications (1)-(7) for each
age group and for each skill, for both the linear and the B-spline cases, illustrating more
explicitly how the estimates are virtually unchanged for the surviving specifications but often
change for the non-surviving ones.39
5.2
Alternative Specifications
In this section, we perform many additional robustness checks on specification (7) from
Section 4. Tables 17 and 18 report the p-value of a test for whether the coefficient of
changes as we add controls to specification (7) from Section 4. Each specification in these
tables contain additional controls of two types: (a’) variables that are discontinuous when
some input is zero (some of which are shown in the plots presented in Section 3), and (b’)
variables that are continuous when each input is zero, for all inputs.40 These variables might
be correlated to undetectable confounders, as discussed above. For instance, observables of
38
Some inputs did not allow for more than one or two B-spline terms.
The online appendix is available at http://bit.ly/1KOy1aj.
40
Of course, these variables may not be confounders of type (b), because they may not be correlated to
inputs at all.
39
27
type (a’) (type (b’)) might be correlated to unobservables of type (a2) (type (b)). If they
are, then they will partly absorb confounders that are undetectable by the exogeneity test,
which would tell us that the test is unable to detect important sources of endogeneity. The
p-values in Tables 17 and 18 provide clear evidence that our estimates of specification (7)
are statistically unchanged in all alternative specifications for both age groups.
Specifications (1’)-(3’) are particularly useful to allay further concerns about omitted
variables and simultaneity. In specification (1’), we add more control variables related to
child characteristics, family demographic characteristics, and environmental characteristics.41
In specification (2’), we add the 15 lagged (i.e., from the previous wave) time inputs.42 In
specification (3’), we add the other three lagged skill measures as well as the interactions
between any two of the four lagged skills.43
In specification (4’), we add controls related to misreporting of time diaries (12 additional
controls)44 , to allay further concerns about measurement error. Specifications (5’)-(11’) are
included to check for undetectable confounders from over-aggregation. Active time activities
are further subcategorized in the data as educational, social, and school activities, while
passive time activities are further subcategorized in the data as general care and media
activities.45 In specification (5’), we add one more time input by separating school time
from self active time, and test whether any of the 15 original coefficients change.46 In
specification (6’), we add the proportions of each active time input spent in educational
activities (7 additional controls).47 In specification (7’), we add the proportions of each
41
Here is the full list of added controls in specification (1’): child’s birth weight, child’s current height,
mother’s race indicators, father’s race indicators, birth order to father, mother’s working hours in a week,
mother’s working days in a week, indicator for whether mother’s working schedule is a regular (vs. night)
shift, number of books mother read last year, and indicator for whether caregivers spent on clothes for the
child last year. We show in Section 3.3 that some of these controls are discontinuous when input is zero (e.g.
child’s birth weight and hours mother works per week).
42
This specification is referred to as the “cumulative model” by Todd and Wolpin (2007) and Fiorini and
Keane (2014).
43
We do not present younger cohort’s results for specification (3’) for lack of data, as discussed in Section
4.
44
The list includes whether the diary was self-administered, whether the diary was reviewed face-to-face,
whether the diary was reviewed via phone, and indicators of who completed the diaries. Some of these
variables (e.g. indicators of who completed the diaries) are shown to be discontinuous when inputs are zero
in Section 3.3.
45
Fiorini and Keane (2014) stratifies activities according to these five types, depending on whether the
activity involves parents.
46
School activities, originally fully included in self active time, comprise attending classes for full-time
students, and daycare or nursery school for children not in school. They represent about 19% (18%) of all
activities, 59% (54%) of all active activities and 90% (96%) of the self active time activity for the older
(younger) cohort.
47
Educational activities include helping adults doing household chores, taking extracurricular lessons, and
reading. They represent about 6% (5%) of all activities and 20% (15%) of all active activities for the older
(younger) cohort.
28
passive time input spent in general care (7 additional controls).48 In specification (8’), we
add the proportions of each passive time input spent watching TV (7 additional controls).49
In specification (9’), we add the proportions of each time input spent at home as opposed
to elsewhere (14 additional controls).50 In specification (10’), we add the proportions of
each time input spent in activities with someone participating (14 additional controls).51
In specification (11’), we add the proportions of each time input spent during weekends
(14 additional controls). In specification (12’), we check if our results are robust to the
definition of age groups. In this specification, the younger group refers to five- to twelveyear-old children (rather than five- to eleven-year old children) and the older group refers to
thirteen- to seventeen-year-old children (rather than twelve- to seventeen-year old children).
Analogously, Tables 19 and 20 present the same robustness checks for the non-linear models discussed in Section 4.2, with the aim of allaying further concerns about non-linearities.
We test whether the coefficients jk , for all j and k (27 coefficients) change as we change
specification (7). The results show that, similarly to the linear models, the estimates do not
change.
Given the evidence presented in this section, it is difficult to conceive of a confounder
that may be biasing our estimates. It needs to be of type (a2) or (b) for all inputs and at
the same time be undetectable by all the robustness checks provided in this section. For
instance, it is difficult to conceive of variables (of type (a2) or (b) for all inputs) correlated
to both Skilli and Inputi observed in the current wave, and yet uncorrelated to both Skilli
and Inputi observed in the previous wave.
6
Discussion
6.1
Why Does Selection on Observables Work in This Context?
The results for the linear and non-linear models discussed in the prior sections indicate
that with rich enough controls we are able to arrive at specifications for which we fail to
48
General care include obtaining goods and services, personal needs and care (e.g. having meals), and
traveling/waiting. They represent about 16% (15%) of all activities and 55% (60%) of all passive activities
for the older (younger) cohort.
49
Watching TV represents about 8% (8%) of all activities and 29% (31%) of all passive activities for the
older (younger) cohort.
50
Time spent at home accounts for about 26% of children’s total time in a week for both cohorts.
51
When filling out the time diaries, the respondents were asked not only about with whom each activity
was performed, but also whether the partner actually participated in the activity (versus being just around
while the child performed the activity). Participation time accounts for about 18% (21%) of children’s total
time in a week for the older (younger) cohort. This variable was used in Del Boca et al. (2013) to categorize
inputs.
29
reject exogeneity. Moreover, as discussed in detail in the past sections, this does not appear
to result from a lack of power. A natural question to ask at this point is why a selection on
observables approach seems to be appropriate in the context of this application.
While the richness of the available controls in the PSID is certainly helpful for mitigating
endogeneity, incorporating the full set of inputs into the production function is also quite
useful. To see this, consider the following simple model of input choices and skill formation
where, for simplicity, we treat the child as the sole decision-maker. Skill for individual i is
determined according to
Skilli = f (Inputi , ✓i ),
where Inputi is a vector of J time inputs and ✓i is a vector of other inputs (i.e., Otheri
in equation (3)) impacting skill which reflects any heterogeneity in the production function
across children (e.g., how much attention the child pays when reading). Children choose
Inputi to maximize utility
Ui = g(Inputi , ✓i , !i )
P
subject to Inputji
0 and Jj=1 Inputji = T , where T is the total available time (i.e., 24
hours per day). !i is a vector denoting heterogeneity in utility that is not associated with
heterogeneity in skill production (e.g., how much the child enjoys reading).52 In this general
formulation, skill and time inputs can in principle affect utility directly, as can the other
inputs influencing the production of skill, ✓i .
Given this maximization problem, the chosen vector of time inputs, Input⇤i , is implicitly
defined by the levels of ✓i and !i :53
Input⇤i = h(✓i , !i )
so that individuals with different levels of (✓i , !i ) tend to choose different levels of the vector
of inputs. For a given ✓i , the variation in inputs due to !i is not endogenous and is in fact
precisely the type of variation we want to exploit when estimating the production function.
Of course, although the component of !i that is orthogonal to ✓i would make ideal instruments to identify the effect of interest, it is difficult to know ex ante which source of variation
is included in !i and which source of variation is included in ✓i , hence our need to develop
52
!i can also include additional constraints in the maximization problem. For instance, a child whose
father is not in the picture cannot interact with him.
53
g in equation (3). For simplicity in the exposition, we assume no measurement
Input⇤i represents Input
i
error in this section.
30
an alternative identification strategy in this paper.
We can write Input⇤,j
as
i
⇤, j
j
Input⇤,j
).
i = h (✓i , !i , Inputi
In our context, endogeneity arises if an input is correlated with ✓i across individuals, condi⇤, j
j
tional on covariates: Cov Input⇤,j
, Controli 6= 0, i.e., if hj (·, Input⇤,
, Controli )
i , ✓i |Inputi
i
varies with ✓i .
We conjecture that we are able to identify the effects of interest with our data because
j
of two reasons. First, to the extent that other inputs Input⇤,
absorb elements of ✓i , adding
i
them as covariates can substantially reduce the potential for endogeneity, requiring less of the
vector Controli . Second, as we add Controli we are able to shut down any correlation between
j
✓i and Input⇤,j
(conditional on Input⇤,
) before we shut down the correlation between !i
i
i
⇤,j
and Inputi . The full set of controls incorporated in the empirical model must be unable
to thoroughly absorb !i , otherwise there would be no independent variation remaining in
Inputi to estimate the production function. !i reflects tastes and household constraints,
which are likely quite heterogeneous across people, while ✓i is bound by technical features of
the skill production technology. Thus, it is not surprising that covariates can fully control
for ✓i without fully controlling for !i .
The above discussion illustrates a largely under-appreciated benefit of modeling the full
vector of inputs in skill production. The inclusion of a comprehensive list of time activities
not only helps the interpretability of the production parameters, but also can substantially alj
lay endogeneity concerns. Indeed, all else constant, Input⇤,
helps absorb more confounders
i
the more disaggregated inputs are.
6.2
What Can (and Cannot) be Inferred from Our Estimates?
In this paper, we estimate the average marginal productivity of each input on each skill,
across all children of each age group. It is useful to interpret these estimates with the aid
of the simple framework described above. Using that notation, we estimate E[fj (Input⇤i , ✓i )]
for each j, where fj refers to the first derivative of the production function f with respect
to its jth input, and the expectation is taken across all children i of each age group.
When E[fj (Input⇤i , ✓i )] > 0, we can interpret that on average children should see an
increase in their skill if they decide to spend more time on activity j (relative to sleeping),
in comparison to their current time. However, that does not necessarily imply that these
children should spend more time on that activity. Indeed, children and their families typically
consider maximizing not only skills when they make time allocation decisions. To see this,
31
we show how different children and their parents might choose different levels of time inputs,
and how these different choices might lead to different estimates of fj (Input⇤i , ✓i ). Assume
that children and their parents care about skill (f ), non-skill (u), and costs (c) such that
Ui = f (Inputi , ✓i )
nc(Inputi , ✓i , !i )
where nc(Input⇤i , ✓i , !i ) := c(Input⇤i , ✓i , !i ) u(Input⇤i , ✓i , !i ) represent the utility cost net
of non-skill benefits, which is allowed to be heterogenous across different time investments.
Intuitively, one can think of c as representing the component of utility related to “costs” and
u as representing the component of utility related to “fun”, although u can be interpreted
more generally to also encompass any mistake in optimization.54 The first order condition
for an optimum in the interior is then
fj (Input⇤i , ✓i )
ncj (Input⇤i , ✓i , !i ) = fj 0 (Input⇤i , ✓i )
ncj 0 (Input⇤i , ✓i , !i )
where ncj is defined analogously to fj . In words, there should be a one-to-one relationship
between differences in marginal product across two positive inputs j and j 0 and their corresponding net costs. If time input j is observed to have a greater marginal product than input
j 0 , the reason must be that input j is commensurately more costly (net of non-skill utility
0
benefits). In addition, consider a situation where Input⇤,j
= 0 and Input⇤,j
> 0. Then it
i
i
must be the case that
fj (Input⇤i , ✓i )
ncj (Input⇤i , ✓i , !i )  fj 0 (Input⇤i , ✓i )
ncj 0 (Input⇤i , ✓i , !i ).
That is, if the optimal choice for input j is zero, then the marginal net return of input j
0
should be lower than the marginal net return of input j 0 , for Input⇤,j
> 0.
i
Given the discussion above, it is difficult to know ex ante what one should expect the
distribution of fj (Input⇤i , ✓i ) to be. These effects depend implicitly on the distribution across
children of the marginal net costs of each activity, ncj (Input⇤i , ✓i , !i ), which are in turn
functions of the joint distribution of (✓i , !i ).55 This framework is useful to understand the
54
For instance, if children and their parents want to maximize the true skill but perceive the production function to be f˜(Inputi , ✓i , !i ) instead of f (Inputi , ✓i ), u can be written as u := f˜(Inputi , ✓i , !i )
f (Inputi , ✓i ), where in this case !i is interpreted as the vector representing the heterogeneity of this misperception across children and their family. If instead they maximize just fun, then u := u0 (Inputi , ✓i , !i )
f (Inputi , ✓i ) where u0 represents the actual component of the utility representing “fun”.
55
Moreover, non-linarities can exacerbate any difference in this joint distribution. If f (Input⇤i , ✓i ) is nonseparable between Input⇤i and ✓i , then our estimate E[fj (Input⇤i , ✓i )] should depend crucially on the joint
distribution of Input⇤ (✓i , !i ) and ✓i , and thus on the joint distribution of (✓i , !i ). Further, if f (·, ✓) is nonlinear in inputs, as it appears to be according to our results in Section 4, then children with different values
of (✓i , !i ) should choose different levels of Input⇤ (✓, !), leading them to have potentially different values of
fj (Input⇤ (✓, !), ✓). Remarks 1 and 2 discuss this topic in more detailed.
32
role of heterogeneity in shaping our estimates of the effect of time allocation on skills. As
discussed in Remark 1, the estimates of our surviving models should represent an unbiased
average of the distribution of fj (Input⇤i , ✓i ) across all children of each age group. The fact
that (say) for younger children we find that sleeping has a positive return on non-cognitive
skills suggest that on average, an increase in the time a child spends sleeping should increase
his or her non-cognitive ability. This does not necessarily mean that all, or even the majority
of children at that age group are sleep deprived.
6.3
Relationship with the Previous Literature
It is widely believed that child outcomes might improve if more of their time is spent
in productive activities. However, evaluating this conventional wisdom is difficult because
it is not clear which activities are productive and what these productive activities might
substitute for. This paper adds to the literature by examining how child cognitive and
non-cognitive skills are impacted by time use, where time is categorized into comprehensive
and precisely defined activities. We find that active time with parents or other activity
partners does not help younger children, and helps older children but only in developing
math skills. Additional passive time does not seem to hurt and sometimes helps with skill
development. Schooling helps develop cognitive skills, but additional time in school can
inhibit the development of non-cognitive skills for younger children. An increase in sleep
helps non-cognitive skills for younger children, but does not seem to matter for older children.
Finally, the impact of time inputs are lower for younger children than for older children.
Although there is an extensive literature in economics on child skill development, there
are only two studies, Del Boca et al. (2013) and Fiorini and Keane (2014), that estimate
the effect of children’s time allocation on skill formation. Del Boca et al. (2013) use the
same dataset as we do, but do not incorporate all child activities, making it difficult to
compare our results to theirs even if both papers provided unbiased estimates. In contrast,
Fiorini and Keane (2014) incorporates a comprehensive list of activities as we do, but use
data from Australia rather than the US. Thus, it is difficult to make comparisons between
our estimates and theirs even if both papers provided unbiased estimates. Indeed, one can
think of institutional differences across countries that may lead to different estimates of
E[fj (Input⇤i , ✓i )] across countries purely because wi is distributed differently for children
with the same value of ✓i (e.g., child care costs, female labor supply elasticity, social norm
about how children should be raised, etc).
Nevertheless, for completeness we compare our main findings with those from Fiorini
33
and Keane (2014).56 This discussion is limited to younger children, since Fiorini and Keane
(2014) do not have data for older children. A common finding across the two studies is that
non-cognitive skills of young children are relatively unresponsive to parental time inputs.
Additionally, the fit of the non-cognitive skill regressions in both papers tends to be small,
suggesting that much of the variation in child non-cognitive skills remains unexplained. Both
studies also find that sleeping is one of the more important activities for non-cognitive skill
production. While our findings regarding the production of non-cognitive skills is similar to
Fiorini and Keane (2014), our results relating to the production of cognitive skills is quite
diferent. In particular, we find that active time with parents or others has little to no effect
on cognitive skill formation, while Fiorini and Keane (2014) find that educational time with
parents or others is quite productive. The source of this difference is difficult to pin down.
In addition to the issues cited above, our aggregation scheme for time inputs and the set of
controls included in our models are different from theirs.
To truly understand the differences in findings, and ultimately the role of time allocation
in skill development more broadly, much richer models of skill production and time allocation
are needed. It is not enough to simply estimate more flexible production functions, since as
noted above it is difficult to interpret the results without a formal model of time allocation.57
Such a model would require specifying a utility function, determining the costs associated
with each time input, and assessing the information available to children and their parents
as they consider these input choices. While such a model is beyond the scope of this paper,
we believe our approach and estimates of skill production are an important first step in this
broader framework.
7
Conclusion
Cognitive and non-cognitive skills are critical for a host of economic and social outcomes
as an adult. While there appears to be a consensus view that a significant amount of skill
acquisition and development occurs early in life, the precise activities and investments that
drive this process are not well understood. In this paper we examine how children’s time
allocation affects the accumulation of skill.
56
To be sure, even if both estimates are unbiased, they are not directly comparable simply because the
joint distribution of (✓i , !i ) in the Australian data is completely different from that in the American data.
This can be inferred by the difference in the distribution of Input⇤i (✓, !) across these two countries as seen
in the summary statistics in both papers. For instance, American children spend more passive time and less
active time with mother than Australian children on average.
57
The non-linearity we incorporate in our models was of a relatively modest form, a limitation imposed
by the size of our sample. Fiorini and Keane (2014) faced similar issues when attempting to incorporate
non-linearities in their models.
34
To do this, we apply a recently developed test of exogeneity to search for models that yield
causal estimates of the impact time inputs have on child skills. The test exploits bunching
in time inputs induced by a non-negativity time constraint. We provide evidence that the
test is able to detect endogeneity arising from omitted variables, simultaneity, measurement
error, and a host of misspecification errors. There are potential sources of endogeneity that
the test is unable to detect. However, our robustness exercises, which are designed to detect
them, suggest that our rich set of controls, together with a comprehensive list of time inputs,
are able to absorb them. The test indicates that with a sufficient set of controls, already
available in the most detailed datasets, we are unable to reject exogeneity of time inputs for
both younger and older children.
For younger children, we find that sleeping is critical for the development of non-cognitive
skills while maternal passive time is important for cognitive skill development. For older
children, active time with adults is relatively valuable in developing cognitive skills, while
passive time with parents and alone are important for non-cognitive skills. However, these
effects are likely to be heterogeneous across families, children within families and activities
within our time input categories. As better data become available, a similar approach to
the one implemented here can be used to uncover causal estimates at a more disaggregated
level.
An additional benefit of our approach is that it can be used as a first stage in a broader
model aimed at understanding household decisions about work, leisure, and investments in
children. Typically, papers that are interested in such questions embed a skill production
function in a more detailed household behavioral model (e.g., Del Boca et al. (2013)). Using
our estimates would reduce the computational burden as well as ensure that endogeneity
concerns have been considered.
Finally, our approach to estimating how children’s time allocation affects skill development can be utilized to study the consequences of other similar resource allocation decisions.
Examples include understanding the impact of watching violent media on violent behavior
or the productivity benefits of time spent exercising. In both examples, the activity of interest is endogenous, it is unclear which activity is being substituted for,58 and individuals
are likely to bunch at zero as a result of non-negativity constraints. As time diaries become
more ubiquitous, the methodology employed here provides researchers with a potential tool
to study causality without an ex-ante source of exogenous variation.
58
DellaVigna and Ferrara (2015) discusses these endogeneity issues in the context of the economic and
social impacts of the media.
35
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37
Table 1: Summary of Ages
CDS I: 1997
CDS II: 2002
CDS III: 2007
Younger Children
Older Children
Age Range
0-12 years old
5-17 years old
10-22 years old
5-11 years old
12-17 years old
Average Age
6 years and 9 months
11 years and 9 months
16 years and 9 months
8 years and 4 months
14 years and 6 months
Table 2: Weekly Time in Each Activity (in Hours), Younger Children
Younger Children
Mean
SD
Active time with mother
Passive time with mother
Active time with father
Passive time with father
Active time with grandparents
Passive time with grandparents
Active time with siblings
Passive time with siblings
Active time with friends
Passive time with friends
Active time with others
Passive time with others
Self active time
Self passive time
Sleeping or napping
Refused to answer or do not know
13.04
23.21
1.93
2.66
1.25
1.90
2.48
2.95
2.44
1.13
2.41
3.60
30.45
7.72
70.49
0.28
9.77
12.09
4.28
5.58
4.12
5.98
4.77
5.62
5.45
3.18
4.98
7.61
11.52
4.62
7.67
1.88
Proportion
of Zero
0.09
0.04
0.69
0.57
0.85
0.79
0.62
0.52
0.70
0.71
0.69
0.39
0.08
0.00
0.00
0.96
Note: The third column shows the proportion of children who spend zero minutes in a week on the corresponding time category.
38
Table 3: Weekly Time in Each Activity (in Hours), Older Children
Older Children
Mean
SD
Active time with mother
Passive time with mother
Active time with father
Passive time with father
Active time with grandparents
Passive time with grandparents
Active time with siblings
Passive time with siblings
Active time with friends
Passive time with friends
Active time with others
Passive time with others
Self active time
Self passive time
Sleeping or napping
Refused to answer or do not know
7.69
22.13
1.22
2.56
0.46
1.38
1.60
3.49
4.42
5.19
1.91
2.58
34.95
10.98
64.55
2.90
8.21
14.45
3.80
5.62
2.16
5.69
4.07
6.74
7.51
7.77
5.18
7.89
13.67
8.11
10.01
6.65
Proportion
of Zero
0.23
0.07
0.81
0.62
0.92
0.86
0.75
0.55
0.56
0.37
0.81
0.58
0.06
0.00
0.00
0.67
Note: The third column shows the proportion of children who spend zero minutes in a week on the corresponding time category.
Table 4: Demographics and Parental Background
Child’s age (months)
Child’s gender
Birth order to mother
Born in US
Younger
Mean
100.50
0.51
1.92
0.98
Children
SD
24.33
0.50
1.09
0.14
Older Children
Mean
SD
174.10 20.57
0.50
0.50
1.95
1.06
0.98
0.14
Mother’s age
Father’s age
Mother’s age at child birth
Father’s age at child birth
Mother has only high school degree
Mother has college degree
Father has only high school degree
Father has college degree
35.20
38.02
27.53
30.54
0.29
0.20
0.17
0.27
6.37
6.95
6.02
6.45
0.45
0.40
0.37
0.45
41.36
44.27
27.91
31.02
0.31
0.21
0.15
0.27
6.01
6.65
5.71
6.31
0.46
0.41
0.36
0.45
Number of siblings child lives with
Lives with two biological parents
Lives with grandparent
Household annual income (in $1,000s)
1.67
0.60
0.08
107.95
1.86
0.49
0.27
128.00
2.21
0.55
0.07
117.71
2.66
0.50
0.26
129.68
39
Table 5: Exogeneity Test Results: Older Children
Controls
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Lagged Score
Child Chrs.
Mother Demog. Chrs.
Family Demog. Chrs.
Family Environ. Chrs.
School Chrs.
School Experience
Math
F-stat
p-Value
2.920
0.000
1.312
0.187
1.256
0.223
1.071
0.379
1.065
0.385
0.951
0.506
0.879
0.588
Vocabulary
F-stat
p-Value
2.519
0.001
1.262
0.219
1.246
0.230
1.160
0.297
1.078
0.373
1.075
0.375
1.020
0.431
Comprehension
F-stat
p-Value
2.923
0.000
2.103
0.008
1.857
0.023
1.694
0.046
1.557
0.079
1.331
0.175
1.376
0.151
Non-cognitive
F-stat
p-Value
1.900
0.019
1.315
0.185
1.312
0.186
1.341
0.169
1.267
0.215
1.225
0.245
1.232
0.240
Note: Entries in bold are “surviving specifications” for which we cannot reject exogeneity at 10% of significance. Each specification contains different control variables: (1) no controls, except for the lagged corresponding input; (2) child characteristics;
(3) mother demographic characteristics; (4) family demographic characteristics; (5) Family environmental characteristics; (6)
School characteristics; (7) Child’s school experience. See footnote 27 for a full description of the control variables. All standard
errors are corrected for heteroskedasticity.
40
Table 6: Effects of Children’s Time Allocation: Older Children
Active time with mother
Passive time with mother
Active time with father
Passive time with father
Active time with grandparents
Passive time with grandparents
Active time with siblings
Passive time with siblings
Active time with friends
Passive time with friends
Self active time
Self passive time
Active time with others
Passive time with others
Don’t know or refuse to answer
R-Squared
Observations
Exogeneity test F-statistic
Exogeneity test p-value
Math
Vocabulary
Comprehension
Non-cognitive
0.008**
(0.003)
0.006**
(0.002)
0.016**
(0.006)
-0.001
(0.004)
0.024**
(0.012)
-0.003
(0.004)
-0.002
(0.005)
0.006*
(0.004)
0.008**
(0.003)
0.004
(0.003)
0.007**
(0.002)
0.005*
(0.003)
0.002
(0.006)
0.003
(0.003)
0.003
(0.003)
0.604
1453
0.879
0.588
0.002
(0.003)
0.002
(0.002)
0.007
(0.006)
-0.001
(0.004)
0.025**
(0.012)
-0.003
(0.005)
-0.006
(0.005)
0.003
(0.003)
0.005
(0.003)
-0.004
(0.003)
-0.001
(0.002)
0.002
(0.002)
-0.007
(0.005)
-0.004
(0.003)
0.001
(0.003)
0.608
1455
1.020
0.431
0.001
(0.003)
-0.001
(0.003)
-0.000
(0.007)
0.007
(0.004)
0.029**
(0.013)
-0.002
(0.005)
-0.011*
(0.007)
0.000
(0.004)
-0.002
(0.004)
-0.001
(0.003)
0.002
(0.002)
-0.001
(0.003)
-0.005
(0.006)
-0.005
(0.003)
-0.000
(0.004)
0.568
1453
1.376
0.151
0.005
(0.004)
0.007**
(0.003)
0.008
(0.007)
0.009*
(0.005)
0.019
(0.018)
-0.000
(0.006)
0.010
(0.007)
0.005
(0.005)
0.002
(0.005)
0.002
(0.004)
0.007**
(0.003)
0.008**
(0.003)
0.003
(0.007)
0.001
(0.006)
-0.000
(0.004)
0.400
1454
1.232
0.240
Note: All estimates are for specification (7). See footnote 27 for a full description of the control variables. Standard errors
corrected for heteroskedasticity are in parentheses. * Significant at the 10% level. ** Significant at the 5% level.
41
Table 7: Exogeneity Test Results: Younger Children
Controls
(1)
(2)
(3)
(4)
(5)
(6)
(7)
No controls
Child Chrs.
Mother Demog. Chrs.
Family Demog. Chrs.
Family Environ. Chrs.
School Chrs.
School Experience
Math
F-stat
p-Value
11.065
0.000
2.020
0.011
0.979
0.475
0.794
0.685
0.719
0.768
0.685
0.802
0.657
0.828
Vocabulary
F-stat
p-Value
11.327
0.000
2.761
0.000
1.675
0.049
1.094
0.356
1.126
0.326
1.104
0.346
1.116
0.335
Comprehension
F-stat
p-Value
8.001
0.000
2.497
0.001
1.338
0.171
1.034
0.416
1.025
0.426
0.958
0.498
0.891
0.574
Non-cognitive
F-stat
p-Value
2.774
0.000
2.427
0.002
1.644
0.056
1.434
0.122
1.396
0.140
1.389
0.143
1.333
0.173
Note: Entries in bold are “surviving specifications” for which we cannot reject exogeneity at 10% of significance. Each specification contains different control variables: (1) no controls; (2) child characteristics; (3) mother demographic characteristics;
(4) family demographic characteristics; (5) family environmental characteristics; (6) school characteristics; (7) child’s school
experience. See footnote 27 for a full description of the control variables. All standard errors are corrected for heteroskedasticity.
42
Table 8: Effects of Children’s Time Allocation: Younger Children
Active time with mother
Passive time with mother
Active time with father
Passive time with father
Active time with grandparents
Passive time with grandparents
Active time with siblings
Passive time with siblings
Active time with friends
Passive time with friends
Self active time
Self passive time
Active time with others
Passive time with others
Don’t know or refuse to answer
R-Squared
Observations
Exogeneity test F-statistic
Exogeneity test p-value
Math
Vocabulary
Comprehension
Non-cognitive
0.001
(0.002)
0.001
(0.002)
0.001
(0.003)
-0.003
(0.003)
0.002
(0.003)
-0.004
(0.003)
-0.001
(0.003)
0.002
(0.002)
-0.000
(0.003)
0.000
(0.003)
0.003
(0.002)
0.002
(0.002)
0.000
(0.003)
-0.001
(0.002)
0.007
(0.009)
0.805
2443
0.657
0.828
0.001
(0.002)
0.003*
(0.001)
0.000
(0.003)
-0.005**
(0.003)
0.001
(0.003)
0.002
(0.002)
-0.002
(0.003)
0.003
(0.002)
-0.001
(0.003)
0.000
(0.003)
0.001
(0.002)
0.002
(0.002)
0.004
(0.003)
-0.002
(0.002)
0.003
(0.006)
0.807
2449
1.116
0.335
0.004
(0.002)
0.004*
(0.002)
0.004
(0.005)
-0.004
(0.004)
0.008
(0.005)
0.003
(0.004)
-0.003
(0.004)
0.007**
(0.003)
-0.003
(0.003)
0.004
(0.005)
0.005**
(0.003)
0.003
(0.003)
0.007
(0.004)
-0.001
(0.003)
0.010
(0.009)
0.673
2085
0.891
0.574
-0.001
(0.004)
-0.004
(0.003)
0.008
(0.007)
-0.003
(0.006)
-0.004
(0.008)
-0.006
(0.005)
-0.007
(0.006)
-0.007
(0.005)
-0.018**
(0.008)
-0.007
(0.009)
-0.009**
(0.004)
-0.003
(0.005)
-0.014**
(0.006)
-0.005
(0.004)
-0.018
(0.020)
0.131
2548
1.333
0.173
Note: All estimates are for specification (7). See footnote 27 for a full description of the control variables. Standard errors
corrected for heteroskedasticity are in parentheses. * Significant at the 10% level. ** Significant at the 5% level.
43
Table 9: Exogeneity Test Results: Older Children, B-spline
Controls
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Lagged Score
Child Chrs.
Mother Demog. Chrs.
Family Demog. Chrs.
Family Environ. Chrs.
School Chrs.
School’s Experience
Math
F-stat
p-Value
1.959
0.015
1.109
0.342
1.100
0.351
1.049
0.401
0.853
0.618
0.827
0.648
0.805
0.673
Vocabulary
F-stat
p-Value
2.214
0.005
1.366
0.156
1.438
0.121
1.402
0.138
1.322
0.180
1.305
0.191
1.270
0.214
Comprehension
F-stat
p-Value
2.217
0.005
1.641
0.057
1.753
0.036
1.698
0.045
1.434
0.123
1.339
0.170
1.320
0.182
Non-cognitive
F-stat
p-Value
1.832
0.026
1.400
0.139
1.383
0.147
1.383
0.147
1.255
0.224
1.239
0.235
1.228
0.243
Note: All specifications in this table are in the form of a linear B-Spline with 2 knots placed at 33rd and 67th percentiles of
each time input, whenever possible. Entries in bold are “surviving specifications” for which we cannot reject exogeneity at 10%
of significance in the linear model . Each specification contains different control variables: (1) no controls, except for the lagged
corresponding input; (2) child characteristics; (3) mother demographic characteristics; (4) family demographic characteristics;
(5) family environmental characteristics; (6) school characteristics; (7) child’s school experience. See footnote 27 for a full
description of the control variables. All standard errors are corrected for heteroskedasticity.
44
Table 10: B-spline Estimation Results: Older Children
Math
Vocabulary
Comprehension
Noncognitive
Active time with mother (0,5.8)
Active time with mother (5.8,15)
Active time with mother (15,.)
Passive time with mother (0,17.4)
Passive time with mother (17.41,28.7)
Passive time with mother (28.7,.)
Active time with father (0,.)
Passive time with father (0,1.2)
Passive time with father (1.2,.)
Active time with grandparents (0,.)
Passive time with grandparents (0,.)
Active time with siblings (0,.)
Passive time with siblings (0,1.7)
Passive time with siblings (1.7,.)
Active time with friends (0,.)
Passive time with friends (0,0.8)
Passive time with friends (0.8,.)
0.024
0.013
0.016
-0.007
(0.016)
(0.015)
(0.016)
(0.018)
-0.009
-0.004
-0.005
-0.001
(0.008)
(0.007)
(0.008)
(0.008)
0.020**
0.006
0.003
0.014**
(0.005)
(0.005)
(0.006)
(0.006)
0.016**
0.008
0.010*
0.007
(0.006)
(0.005)
(0.006)
(0.007)
0.000
0.001
0.005
0.011*
(0.006)
(0.005)
(0.006)
(0.007)
0.004
0.000
-0.010**
0.004
(0.003)
(0.003)
(0.004)
(0.004)
0.014**
0.007
0.000
0.007
(0.006)
(0.006)
(0.007)
(0.007)
0.131
0.263*
0.119
-0.075
(0.122)
(0.146)
(0.161)
(0.143)
-0.001
-0.003
0.008
0.010*
(0.005)
(0.004)
(0.005)
(0.005)
0.026**
0.025**
0.031**
0.018
(0.012)
(0.012)
(0.012)
(0.018)
-0.003
-0.002
-0.001
-0.000
(0.004)
(0.005)
(0.005)
(0.006)
-0.001
-0.005
-0.010
0.010
(0.006)
(0.005)
(0.007)
(0.007)
-0.064
-0.098
-0.047
-0.100
(0.084)
(0.075)
(0.080)
(0.104)
0.008**
0.005
0.002
0.006
(0.004)
(0.004)
(0.004)
(0.005)
0.009**
0.005
0.000
0.003
(0.003)
(0.003)
(0.004)
(0.005)
0.104
0.305
-0.006
-0.061
(0.376)
(0.442)
(0.331)
(0.269)
0.004
-0.004
-0.000
0.001
45
(0.003)
(0.003)
(0.003)
(0.004)
0.003
-0.001
0.004
0.009
(0.004)
(0.004)
(0.005)
(0.006)
0.019
-0.005
-0.012
0.013
(0.014)
(0.013)
(0.015)
(0.017)
0.008**
0.001
0.004
0.005
(0.003)
(0.003)
(0.003)
(0.004)
-0.013
0.002
-0.013
-0.016
(0.028)
(0.027)
(0.031)
(0.035)
0.007
0.015
0.004
0.029
(0.015)
(0.014)
(0.016)
(0.018)
0.005*
0.000
0.001
0.006
(0.003)
(0.003)
(0.003)
(0.004)
0.002
-0.007
-0.004
0.003
(0.006)
(0.005)
(0.006)
(0.007)
-0.059
0.014
-0.062
-0.025
(0.039)
(0.036)
(0.043)
(0.049)
0.006*
-0.004
-0.002
0.001
(0.003)
(0.004)
(0.004)
(0.007)
0.004
0.001
0.001
-0.001
(0.003)
(0.003)
(0.004)
(0.004)
R-squared
0.609
0.612
0.575
0.404
Observations
1,453
1,455
1,453
1,454
Exogeneity test F-statistic
0.805
1.270
1.320
1.228
Exogeneity test p-value
0.673
0.214
0.182
0.243
Self active time (0,31.5)
Self active time (31.5,36.3)
Self active time (36.3,.)
Self passive time (0,5.1)
Self passive time (5.1,9)
Self passive time (9,.)
Active time with others (0,.)
Passive time with others (0,2.5)
Passive time with others (2.5,.)
Don’t know or refuse to answer
Note: All estimates are for specification (7). See footnote 27 for a full description of the control variables. In the first column,
the parentheses shown after each time input indicates the time intervals. For example, (0,2.5) means between 0 hours and 2.5
hours per week. Depending on the distribution, some time inputs have less than three time intervals because the time input
was not complex enough to accommodate two knots. Standard errors corrected for heteroskedasticity are in parentheses. *
Significant at the 10% level. ** Significant at the 5% level.
46
Table 11: Exogeneity Test Results: Younger Children, B-spline
Controls
(1)
(2)
(3)
(4)
(5)
(6)
(7)
No Controls
Child Chrs.
Mother Demog. Chrs.
Family Demog. Chrs.
Family Environ. Chrs.
School Chrs.
School’s Experience
Math
F-stat
p-Value
6.918
0.000
1.502
0.096
0.981
0.472
0.815
0.661
0.661
0.825
0.625
0.857
0.642
0.842
Vocabulary
F-stat
p-Value
8.080
0.000
1.358
0.159
1.052
0.398
0.841
0.632
0.785
0.696
0.766
0.717
0.837
0.637
Comprehension
F-stat
p-Value
6.055
0.000
1.059
0.390
0.764
0.719
0.675
0.811
0.554
0.911
0.519
0.932
0.570
0.900
Non-cognitive
F-stat
p-Value
2.142
0.006
1.610
0.063
1.459
0.112
1.615
0.062
1.474
0.106
1.474
0.106
1.460
0.112
Note: All specifications in this table are in the form of a linear B-Spline with 2 knots placed at 33rd and 67th percentiles of
each time input, whenever possible. Entries in bold are “surviving specifications” for which we cannot reject exogeneity at 10%
of significance in the linear model . Each specification contains different control variables: (1) no controls, except for the lagged
corresponding input; (2) child characteristics; (3) mother demographic characteristics; (4) family demographic characteristics;
(5) family environmental characteristics; (6) school characteristics; (7) child’s school experience. See footnote 27 for a full
description of the control variables. All standard errors are corrected for heteroskedasticity.
47
Table 12: B-spline Estimation Results: Younger Children
Math
Vocabulary
Comprehension
Noncognitive
Active time with mother (0,5.8)
Active time with mother (5.8,15)
Active time with mother (15,.)
Passive time with mother (0,17.4)
Passive time with mother (17.41,28.7)
Passive time with mother (28.7,.)
Active time with father (0,.)
Passive time with father (0,1.2)
Passive time with father (1.2,.)
Active time with grandparents (0,.)
Passive time with grandparents (0,.)
Active time with siblings (0,.)
Passive time with siblings (0,1.7)
Passive time with siblings (1.7,.)
Active time with friends (0,.)
Passive time with friends (0,0.8)
Passive time with friends (0.8,.)
-0.015
-0.001
0.010
-0.029
(0.011)
(0.011)
(0.015)
(0.023)
0.002
0.002
0.005
0.019**
(0.004)
(0.004)
(0.006)
(0.008)
0.002
0.000
0.002
-0.009**
(0.002)
(0.002)
(0.003)
(0.004)
0.005
0.000
0.001
0.002
(0.004)
(0.004)
(0.005)
(0.008)
0.001
0.003
0.003
-0.009
(0.003)
(0.003)
(0.004)
(0.006)
0.000
0.004*
0.006*
-0.000
(0.002)
(0.002)
(0.004)
(0.005)
0.001
0.000
0.005
0.008
(0.003)
(0.003)
(0.005)
(0.007)
-0.020
0.046
0.039
-0.013
(0.066)
(0.068)
(0.093)
(0.121)
-0.002
-0.006**
-0.004
-0.003
(0.003)
(0.003)
(0.004)
(0.006)
0.002
0.002
0.008
-0.004
(0.004)
(0.003)
(0.005)
(0.008)
-0.003
0.002
0.002
-0.005
(0.003)
(0.002)
(0.004)
(0.005)
-0.001
-0.003
-0.002
-0.007
(0.003)
(0.003)
(0.004)
(0.006)
-0.017
-0.034
-0.051
-0.130*
(0.036)
(0.036)
(0.050)
(0.075)
0.003
0.004
0.009**
-0.004
(0.003)
(0.002)
(0.003)
(0.005)
0.001
-0.001
-0.002
-0.017**
(0.003)
(0.003)
(0.003)
(0.008)
0.003
-0.070
-0.107
0.496**
(0.125)
(0.115)
(0.193)
(0.234)
0.001
0.000
0.004
-0.011
48
(0.004)
(0.003)
(0.005)
(0.010)
0.006**
0.006**
0.009**
-0.007
(0.003)
(0.002)
(0.003)
(0.005)
-0.004
-0.008
-0.000
-0.022*
(0.007)
(0.007)
(0.009)
(0.013)
-0.000
-0.003
0.001
-0.004
(0.005)
(0.004)
(0.007)
(0.009)
0.035**
0.018
0.061**
0.023
(0.013)
(0.012)
(0.020)
(0.025)
-0.001
0.006
0.005
-0.004
(0.008)
(0.008)
(0.011)
(0.015)
0.000
-0.002
-0.003
-0.007
(0.003)
(0.003)
(0.004)
(0.007)
-0.000
0.004
0.006
-0.014**
(0.003)
(0.003)
(0.004)
(0.006)
-0.009
-0.001
-0.002
-0.003
(0.018)
(0.018)
(0.025)
(0.034)
-0.001
-0.001
0.000
-0.004
(0.002)
(0.002)
(0.003)
(0.004)
0.007
0.002
0.009
-0.017
(0.009)
(0.006)
(0.009)
(0.020)
R-squared
0.807
0.808
0.676
0.138
Observations
2,443
2,449
2,085
2,548
Exogeneity test F-statistic
0.642
0.837
0.570
1.460
Exogeneity test p-value
0.842
0.637
0.900
0.112
Self active time (0,31.5)
Self active time (31.5,36.3)
Self active time (36.3,.)
Self passive time (0,5.1)
Self passive time (5.1,9)
Self passive time (9,.)
Active time with others (0,.)
Passive time with others (0,2.5)
Passive time with others (2.5,.)
Don’t know or refuse to answer
Note: All estimates are for specification (7). See footnote 27 for a full description of the control variables. In the first column,
the parentheses shown after each time input indicates the time intervals. For example, (0,2.5) means between 0 hours and 2.5
hours per week. Depending on the distribution, some time inputs have less than three time intervals because the time input
was not complex enough to accommodate two knots. Standard errors corrected for heteroskedasticity are in parentheses. *
Significant at the 10% level. ** Significant at the 5% level.
49
Table 13: p-Values for Comparing Surviving and Non-surviving Specifications: Older Children
Controls
(1)
(2)
(3)
(4)
(5)
(6)
Lagged Score
Child Chrs.
Mother Demog. Chrs.
Family Demog. Chrs.
Family Environ. Chrs.
School Chrs.
Math
0.000
0.040
0.185
0.585
0.932
0.958
Older Cohort
Vocabulary Comprehension
0.000
0.000
0.463
0.006
0.540
0.072
0.491
0.303
0.958
0.675
0.998
0.957
Non-cognitive
0.975
0.974
0.984
0.999
1.000
0.998
Note: This table shows the p-values of a joint test for whether the 15 coefficients of Inputi for each specification are the same
as the corresponding ones from specification (7) in Table 5. Entries in bold are “surviving specifications” with respect to the
exogeneity test, i.e., those for which we cannot reject exogeneity at 10% of significance. Each specification contains different
control variables: (1) no controls, except for the lagged corresponding input; (2) child characteristics; (3) mother demographic
characteristics; (4) family demographic characteristics; (5) family environmental characteristics; (6) school characteristics. See
footnote 27 for a full description of the control variables. All standard errors are corrected for heteroskedasticity.
Table 14: p-Values for Comparing Surviving and Non-surviving Specifications: Younger
Children
Controls
(1)
(2)
(3)
(4)
(5)
(6)
No controls
Child Chrs.
Mother Demog. Chrs.
Family Demog. Chrs.
Family Environ. Chrs.
School Chrs.
Math
0.000
0.000
0.364
0.570
0.374
0.830
Younger Cohort
Vocabulary Comprehension
0.000
0.000
0.000
0.001
0.151
0.350
0.555
0.533
0.431
0.441
0.682
0.828
Non-cognitive
0.004
0.011
0.531
0.904
0.942
1.000
Note: This table shows the p-values of a joint test for whether the 15 coefficients of Inputi for each specification are the same
as the corresponding ones from specification (7) in Table 7. Entries in bold are “surviving specifications” with respect to the
exogeneity test, i.e., those for which we cannot reject exogeneity at 10% of significance. Each specification contains different
control variables: (1) no controls; (2) child characteristics; (3) mother demographic characteristics; (4) family demographic
characteristics; (5) family environmental characteristics; (6) school characteristics. See footnote 27 for a full description of the
control variables. All standard errors are corrected for heteroskedasticity.
50
Table 15: p-Values for Comparing Surviving and Non-surviving Specifications: Older Children, B-spline
Controls
(1)
(2)
(3)
(4)
(5)
(6)
Lagged Score
Child Chrs.
Mother Demog. Chrs.
Family Demog. Chrs.
Family Environ. Chrs.
School Chrs.
Math
0.000
0.182
0.465
0.734
0.996
1.000
Older Cohort
Vocabulary Comprehension
0.000
0.000
0.857
0.148
0.844
0.428
0.869
0.591
0.998
0.956
1.000
0.999
Non-cognitive
0.990
0.998
0.996
0.999
1.000
1.000
Note: This table shows the p-values of a test for whether the 26 coefficient estimates of Inputi for each specification are
statistically the same as the corresponding ones from Specification (7) in Table 9. Entries in bold are “surviving specifications”
for which we cannot reject exogeneity at 10% of significance. Each specification contains different control variables: (1) no
controls, except for the lagged corresponding input; (2) child characteristics; (3) mother demographic characteristics; (4)
family demographic characteristics; (5) family environmental characteristics; (6) school characteristics. All standard errors are
corrected for heteroskedasticity. See footnote 27 for a full description of the control variables. All standard errors are corrected
for heteroskedasticity.
Table 16: p-Values for Comparing Surviving and Non-surviving Specifications: Younger
Children, B-spline
Controls
(1)
(2)
(3)
(4)
(5)
(6)
No Controls
Child Chrs.
Mother Demog. Chrs.
Family Demog. Chrs.
Family Environ. Chrs.
School Chrs.
Math
0.000
0.002
0.331
0.660
0.760
0.943
Younger Cohort
Vocabulary Comprehension
0.000
0.000
0.000
0.005
0.120
0.378
0.583
0.610
0.813
0.694
0.872
0.899
Non-cognitive
0.007
0.035
0.898
0.979
0.999
1.000
Note: This table shows the p-values of a test for whether the 27 coefficient estimates of Inputi for each specification are
statistically the same as the corresponding ones from Specification (7) in Table 11. Entries in bold are “surviving specifications”
for which we cannot reject exogeneity at 10% of significance. Each specification contains different control variables: (1) no
controls; (2) child characteristics; (3) mother demographic characteristics; (4) family demographic characteristics; (5) family
environmental characteristics; (6) school characteristics. See footnote 27 for a full description of the control variables. All
standard errors are corrected for heteroskedasticity.
51
Table 17: Alternative Specifications: Older Children
(1’)
(2’)
(3’)
(4’)
(5’)
(6’)
(7’)
(8’)
(9’)
(10’)
(11’)
(12’)
Alternative Specifications
(7) + more controls
(7) + lagged inputs
(7) + lagged skills
(7) + measurement error controls
(7), school time as a separate input
(7) + prop. educational activities
(7) + prop. general care
(7) + prop. watching TV
(7) + prop. at home
(7) + prop. participation time
(7) + prop. weekend time
(7), change age of cohorts
Math
0.855
0.997
0.798
0.534
0.998
0.984
0.973
0.506
0.781
0.863
0.965
0.953
Vocabulary
0.969
0.996
0.452
0.660
0.999
0.960
0.723
0.827
0.920
0.629
0.822
0.571
Comprehension
0.992
0.990
0.791
0.597
0.958
0.723
0.988
0.822
0.907
0.873
0.699
0.462
Non-cognitive
0.966
0.979
0.826
0.995
1.000
0.671
0.995
0.999
0.498
0.874
0.689
0.243
Note: This table shows the p-values of a test for whether the 15 coefficient estimates of Inputi for each alternative specification
are statistically the same as the corresponding ones from Specification (7) in Table 5. Alternative specifications: (1’) the full
list of added controls can be seen in footnote 41; (2’) lagged time inputs of all 15 activities; (3’) lagged skill measures of other
types and interactions of any two skills; (4’) full list of added controls can be seen in footnote 44; (5’) 16 time inputs (15 original
time inputs plus school activities), whereby the p-value refers to a test of whether the 15 coefficients of the original time inputs
are statistically unchanged with respect to specification (7); (6’) proportions of each time input spent in educational activities
(e.g. reading): 7 additional covariates; (7’) proportions of each time input spent in general care (i.e having meals): 7 additional
covariates; (8’) proportions of each time input spent watching TV: 7 additional covariates; (9’) proportions of each time input
spent at home: 14 additional covariates; (10’) proportions of each time input that partner actually participates in the activity:
12 additional covariates; (11’) proportions of each time input spent during weekends: 15 additional covariates; (12’) the age
range of the older cohort is changed to be 13-17 years old.
52
Table 18: Alternative Specifications: Younger Children
(1’)
(2’)
(4’)
(5’)
(6’)
(7’)
(8’)
(9’)
(10’)
(11’)
(12’)
Alternative Specifications
(7) + more controls
(7) + lagged inputs
(7) + measurement error controls
(7), school time as a separate input
(7) + prop. educational activities
(7) + prop. general care
(7) + prop. watching TV
(7) + prop. at home
(7) + prop. participation time
(7) + prop. weekend time
(7), change age of cohorts
Math
0.902
0.991
0.836
1.000
0.918
0.997
0.999
0.663
0.955
0.976
0.954
Vocabulary
0.717
0.874
0.877
1.000
0.934
0.968
0.999
0.097
0.991
0.874
0.554
Comprehension
0.389
0.981
0.504
1.000
0.922
0.982
0.999
0.518
0.882
0.906
0.492
Non-cognitive
0.898
0.942
0.989
0.988
0.842
1.000
1.000
0.727
0.919
0.942
0.833
Note: This table shows the p-values of a test for whether the 15 coefficient estimates of Inputi for each alternative specification
are statistically the same as the corresponding ones from Specification (7) in Table 7. Alternative specifications: (1’) the full list
of added controls can be seen in footnote 41; (2’) lagged time inputs of all 15 activities; (3’) lagged skill measures of other types
and interactions of any two skills, which is not implemented for this cohort because of lack of data; (4’) full list of added controls
can be seen in footnote 44; (5’) 16 time inputs (15 original time inputs plus school activities), whereby the p-value refers to a
test of whether the 15 coefficients of the original time inputs are statistically unchanged with respect to specification (7); (6’)
proportions of each time input spent in educational activities (e.g. reading): 7 additional covariates; (7’) proportions of each
time input spent in general care (i.e having meals): 7 additional covariates; (8’) proportions of each time input spent watching
TV: 7 additional covariates; (9’) proportions of each time input spent at home: 14 additional covariates; (10’) proportions of
each time input that partner actually participates in the activity: 12 additional covariates; (11’) proportions of each time input
spent during weekends: 15 additional covariates; (12’) the age range of the older cohort is changed to be 5-12 years old.
53
Table 19: Alternative Specifications: Older Children, B-spline
(1’)
(2’)
(3’)
(4’)
(5’)
(6’)
(7’)
(8’)
(9’)
(10’)
(11’)
(12’)
Alternative Specifications
(7) + more controls
(7) + lagged inputs
(7) + lagged skills
(7) + measurement error controls
(7), school time as a separate input
(7) + prop. educational activities
(7) + prop. general care
(7) + prop. watching TV
(7) + prop. at home
(7) + prop. participation time
(7) + prop. weekend time
(7), change age of cohorts
Math
0.940
1.000
0.946
0.887
0.999
0.999
1.000
0.923
0.915
0.996
1.000
0.908
Vocabulary
0.997
1.000
0.871
0.869
1.000
0.990
0.969
0.970
0.999
0.909
0.997
0.224
Comprehension
0.991
1.000
0.830
0.978
0.995
0.937
1.000
0.990
1.000
0.998
0.974
0.795
Non-cognitive
0.999
1.000
0.994
0.999
1.000
0.989
1.000
1.000
0.861
0.997
0.847
0.313
Note: This table shows the p-values of a test for whether the coefficient estimates of Inputi for each alternative specification
are statistically the same as the corresponding ones from Specification (7) in Table 9. All specifications in this table are in the
form of a linear B-Spline with 2 knots placed at 33rd and 67th percentiles of each time input, whenever possible. Alternative
specifications: (1’) the full list of added controls can be seen in footnote 41; (2’) lagged time inputs of all 15 activities; (3’)
lagged skill measures of other types and interactions of any two skills; (4’) full list of added controls can be seen in footnote
44; (5’) 30 time inputs (27 original time inputs plus 3 school inputs), whereby the p-value refers to a test of whether the 27
coefficients of the original time inputs are statistically unchanged with respect to specification (7); (6’) proportions of each time
input spent in educational activities (e.g. reading): 7 additional covariates; (7’) proportions of each time input spent in general
care (i.e having meals): 7 additional covariates; (8’) proportions of each time input spent watching TV: 7 additional covariates;
(9’) proportions of each time input spent at home: 14 additional covariates; (10’) proportions of each time input that partner
actually participates in the activity: 12 additional covariates; (11’) proportions of each time input spent during weekends: 15
additional covariates; (12’) the age range of the older cohort is changed to be 13-17 years old.
54
Table 20: Alternative Specifications: Younger Children, B-spline
(1’)
(2’)
(4’)
(5’)
(6’)
(7’)
(8’)
(9’)
(10’)
(11’)
(12’)
Alternative Specifications
(7) + more controls
(7) + lagged inputs
(7) + measurement error controls
(7), school time as a separate input
(7) + prop. educational activities
(7) + prop. general care
(7) + prop. watching TV
(7) + prop. at home
(7) + prop. participation time
(7) + prop. weekend time
(7), change age of cohorts
Math
0.951
1.000
0.993
0.999
1.000
1.000
1.000
0.860
1.000
1.000
0.659
Vocabulary
0.979
0.993
0.987
0.998
1.000
1.000
1.000
0.269
1.000
0.999
0.690
Comprehension
0.821
1.000
0.908
1.000
0.999
1.000
1.000
0.896
0.999
0.999
0.384
Non-cognitive
0.973
0.998
1.000
0.999
0.997
1.000
1.000
0.984
0.998
1.000
0.573
Note: This table shows the p-values of a test for whether the coefficient estimates of Inputi for each alternative specification
are statistically the same as the corresponding ones from Specification (7) in Table 11. All specifications in this table are in the
form of a linear B-Spline with 2 knots placed at 33rd and 67th percentiles of each time input, whenever possible. Alternative
specifications: (1’) the full list of added controls can be seen in footnote 41; (2’) lagged time inputs of all 15 activities; (3’)
lagged skill measures of other types and interactions of any two skills, which is not implemented for this cohort because of lack
of data; (4’) full list of added controls can be seen in footnote 44; (5’) 30 time inputs (27 original time inputs plus 3 school
inputs), whereby the p-value refers to a test of whether the 27 coefficients of the original time inputs are statistically unchanged
with respect to specification (7); (6’) proportions of each time input spent in educational activities (e.g. reading): 7 additional
covariates; (7’) proportions of each time input spent in general care (i.e having meals): 7 additional covariates; (8’) proportions
of each time input spent watching TV: 7 additional covariates; (9’) proportions of each time input spent at home: 14 additional
covariates; (10’) proportions of each time input that partner actually participates in the activity: 12 additional covariates;
(11’) proportions of each time input spent during weekends: 15 additional covariates; (12’) the age range of the older cohort is
changed to be 5-12 years old.
55
Figure 1: Intuition for the Test of Exogeneity
E[Skilli |Inputji = x]
E[Skilli |Inputji = x, Covariatesji ]
E[Skilli |Inputji =0, Covariatesji ]
E[Skilli |Inputji =0]
Inputji
Inputji
(a) Correlation between Time Input and
Child Skill, Unconditional
(b) Correlation between Time Input and
Child Skill, Conditional on Covariates
Figure 2: Why are Unobservables Discontinuous at Inputji = 0?
Mother’s Typei
E[Mother’s Typei |Inputji = 0]
Inputji
56
Figure 3: Types of Confounders
Confounderi
Confounderi
E[Confounderi |Inputji = 0]
E[Confounderi |Inputji = 0]
Inputji , Inputj?
i
(a) Type (a1) vs. Type (a2)
Inputji , Inputj?
i
(b) Type (b)
Note: Inputj?
i represents the optimal choice of input j by individual i. Red range: Support of confounder
among all observations of sample. Blue range: Support of confounder among all observations of sample for
which Inputji = 0. The confounder is of type (a1) if some of its correlation with Skilli happens for values of
the confounder in the blue range, otherwise it is of type (a2).
57
Figure 4: Evidence of Bunching
.9
.8
Cumulative Density
.4
.5
.6
.7
.3
.2
.1
0
0
.1
.2
.3
Cumulative Density
.4
.5
.6
.7
.8
.9
1
(b) Passive Time with Friends, Older Cohort
1
(a) Self Active Time, Older Cohort
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Self Active time
0
10
15
20
25
30
35
40
Inactive time with friends
45
50
55
60
.9
.8
Cumulative Density
.4
.5
.6
.7
.3
.2
.1
0
0
.1
.2
.3
Cumulative Density
.4
.5
.6
.7
.8
.9
1
(d) Active Time with Father, Younger Cohort
1
(c) Active Time with Mother, Older Cohort
5
0
5
10
15
20
25
30
35
40
Active time with mother
45
50
55
60
0
5
10
15
20
25
30
Active time with father
35
40
45
Note: Each plot shows the cumulative density function of the time spent in the corresponding activity for
the corresponding cohort. The fact that these plots cross the vertical axis not at the origin is direct evidence
of bunching, as it implies the probability density function is discontinuously larger at zero. Time described
in the horizontal axis is reported in hours per week, but continuously (in minutes).
58
11
3.8
4
11.5
Conditional Mean
12.5
12
13
Conditional Mean
4.2
4.6
4.4
4.8
13.5
5
14
Figure 5: Evidence of Power to Detect Endogeneity from Omitted Variables
0
2
4
6
8 10 12 14 16 18 20 22 24
Active time with mother (hours per week)
26
28
30
0
4
6
8 10 12 14 16 18 20 22 24
Active time with father (hours per week)
26
28
30
(b) Number of Books Child Has, Younger Children
40
30
60
35
Conditional Mean
40
45
50
55
Conditional Mean
80 100 120 140 160 180 200 220 240
(a) Mother’s Level of Education (Years),
Younger Children
2
0
2
4
6
8 10 12 14 16 18 20 22 24 26
Don't know or refuse to answer (hours per week)
28
30
0
(c) Hours Mother Works Per Week, Older Children
2
4
6
8 10 12 14 16 18 20 22 24
Passive time with father (hours per week)
26
28
30
(d) Household Income ($1,000s Per Year),
Older Children
Note: In each plot, the vertical axis shows the mean of a potential confounder conditional on a given level of time input (i.e.
horizontal axis variable). The scatter plot represents the observed conditional mean of the confounder (aggregated to the next
hour of the time input). At zero time input, we show the 95% confidence interval. The solid curve represents a third order
local polynomial regression of the confounder on the time input, using time input data at the minute level. The shaded region
represents the 95% confidence interval for this regression with an out-of-sample prediction at zero minutes. See footnote 20 for
more details on the regression and confidence interval.
59
Figure 6: Evidence of Power to Detect Endogeneity from Simultaneity
(a) Child’s Birth Weight (Pounds), Older Children
0
5
.1
5.5
.2
6
.3
Conditional Mean
6.5 7 7.5 8
Conditional Mean
.4 .5 .6 .7
8.5
.8
9
.9
9.5
1
(b) Child is White, Older Children
0
2
4
6
8 10 12 14 16 18 20 22 24
Passive time with father (hours per week)
26
28
30
0
4
6
8 10 12 14 16 18 20 22 24
Passive time with father (hours per week)
26
28
30
(d) Child’s Height (Inches), Younger Children
40
50
42
60
44
70
Conditional Mean
48
46
50
Conditional Mean
100
80
90
52
110
54
56
120
(c) Child’s Age (Months), Younger Children
2
0
2
4
6
8 10 12 14 16 18 20 22 24
Active time with friends (hours per week)
26
28
30
0
2
4
6
8 10 12 14 16 18 20 22 24
Active time with friends (hours per week)
26
28
30
Note: In each plot, the vertical axis shows the mean of a potential confounder conditional on a given level of time input (i.e.
horizontal axis variable). The scatter plot represents the observed conditional mean of the confounder (aggregated to the next
hour of the time input). At zero time input, we show the 95% confidence interval. The solid curve represents a third order
local polynomial regression of the confounder on the time input, using time input data at the minute level. The shaded region
represents the 95% confidence interval for this regression with an out-of-sample prediction at zero minutes. See footnote 20 for
more details on the regression and confidence interval.
60
Figure 7: Evidence of Power to Detect Endogeneity from Measurement Error
(b) Child Completed Weekday Diary (With or
Without Help), Older Children
0
.35
.1
.4
.2
.45
.3
Conditional Mean
.4 .5 .6 .7
Conditional Mean
.5
.55
.6
.8
.65
.9
1
.7
(a) Weekend Diary was Completed Without
Help, Younger Children
0
2
4
6
8 10 12 14 16 18 20 22 24
Active time with mother (hours per week)
26
28
30
0
2
4
6
8 10 12 14 16 18 20 22 24
Passive time with others (hours per week)
26
28
30
Note: In each plot, the vertical axis shows the mean of a potential confounder conditional on a given level of time input (i.e.
horizontal axis variable). The scatter plot represents the observed conditional mean of the confounder (aggregated to the next
hour of the time input). At zero time input, we show the 95% confidence interval. The solid curve represents a third order
local polynomial regression of the confounder on the time input, using time input data at the minute level. The shaded region
represents the 95% confidence interval for this regression with an out-of-sample prediction at zero minutes. See footnote 20 for
more details on the regression and confidence interval.
Figure 8: Evidence of Power to Detect Endogeneity from Over-Aggregation of Inputs
(b) Proportion of Passive time with Friends
Watching TV, Older Children
0
0
.1
.05
.2
.1
.3
Conditional Mean
.4 .5 .6 .7
Conditional Mean
.15
.2
.25
.8
.3
.9
1
.35
(a) Proportion of Active Time with Father
Spent at Home, Younger Children
0
2
4
6
8 10 12 14 16 18 20 22 24
Passive time with father (hours per week)
26
28
30
0
2
4
6
8
10 12 14 16 18 20
Active time with friends
22
24
26
28
30
Note: In each plot, the vertical axis shows the mean of a potential confounder conditional on a given level of time input (i.e.
horizontal axis variable). The scatter plot represents the observed conditional mean of the confounder (aggregated to the next
hour of the time input). At zero time input, we show the 95% confidence interval. The solid curve represents a third order
local polynomial regression of the confounder on the time input, using time input data at the minute level. The shaded region
represents the 95% confidence interval for this regression with an out-of-sample prediction at zero minutes. See footnote 20 for
more details on the regression and confidence interval.
61
Figure 9: Evidence of Power to Detect Endogeneity from Non-Linear Effects
(b) ActiveTime with Grandparents (Distribution), Younger Children
6
.95
Cumulative Probability
.75
.8
.85
.9
.7
2
Conditional Mean
3
4
5
1
7
(a) Active Time with Grandparents (1st Moment), Younger Children
0
.6
1
.65
When active time with mother is 0
When active time with mother is 1
When active time with mother is 2
When active time with mother is 3
0
2
4
6
8 10 12 14 16 18 20 22 24
Active time with mother (hours per week)
26
28
30
0
5
10
15
20
25
30
35
Active time with grandparents (hours per week)
40
45
Note: In the plots of the right, we show the cumulative density function of the confounder for selected values of the time input
(in hours), for the confounder and time input shown in the corresponding plot of the left. In the plots of the left, the vertical
axis shows the mean of a potential confounder conditional on a given level of time input (i.e. horizontal axis variable). The
scatter plot represents the observed conditional mean of the confounder (aggregated to the next hour of the time input). At
zero time input, we show the 95% confidence interval. The solid curve represents a third order local polynomial regression of
the confounder on the time input, using time input data at the minute level. The shaded region represents the 95% confidence
interval for this regression with an out-of-sample prediction at zero minutes. See footnote 20 for more details on the regression
and confidence interval.
62
Appendix
Table 21: Non-cognitive Skills Loading Factors
Cheats or tells lies
Bullies or mean to others
Feels no sorry after misbehaving
Breaks things on purpose
Has sudden changes in mood
Feels no love
Too fearful or anxious
Feels worthless or inferior
Sad or depressed
Cries too much
Easily confused
Has obsessions
Rather high strung, tense and nervous
Argues too much
Disobedient
Stubborn, sullen, or irritable
Has a very strong temper
Has difficulty concentrating
Impulsive, or acts without thinking
Restless or overly active
Has trouble getting along with other children
Not liked by other children
Withdrawn, does not get involved with others
Clings to adults
Demands a lot of attention
Too dependent on others
Thinks before acting, not impulsive
Generally well behaved, does what adults request
Can get over being upset quickly
Waits turn in games and other activities
Gets along well with other children
Admired by other children
Cheerful, happy
Tries things for himself/herself
Does neat, careful work
Curious and exploring, likes new experiences
Younger Cohort
0.4988
0.5519
0.4300
0.4874
0.5532
0.4679
0.4455
0.4894
0.5354
0.4139
0.5152
0.4735
0.5248
0.5822
0.5439
0.6125
0.6117
0.5895
0.5933
0.5370
0.5942
0.4502
0.4028
0.3216
0.5431
0.4666
0.4883
0.5719
0.4433
0.5142
0.6195
0.5779
0.4721
0.3619
0.4185
0.1932
Older Cohort
0.5272
0.5513
0.4729
0.4803
0.5647
0.5351
0.4776
0.5751
0.6158
0.3418
0.5314
0.5828
0.5353
0.5771
0.5872
0.6310
0.6362
0.5874
0.6395
0.5324
0.6106
0.4747
0.4533
0.2812
0.5203
0.4593
0.5418
0.5788
0.4623
0.4801
0.6168
0.5424
0.5283
0.4122
0.4201
0.2397
Note: The larger is the factor loading, the larger is the conditional correlation between the variables and the factor (i.e. the
measure of non-cognitive skills).
63
Table 22: Exogeneity Test Results: Older Children, Not Value-Added
Controls
(1)
(2)
(3)
(4)
(5)
(6)
(7)
No Controls
Child Chrs.
Mother Demographic Chrs.
Family Demographic Chrs.
Family Environmental Chrs.
Other Environmental Chrs.
School Experience
Math
F-stat
p-Value
3.406
0.000
1.214
0.254
1.077
0.373
0.919
0.543
0.943
0.515
0.834
0.640
0.838
0.635
Vocabulary
F-stat
p-Value
2.468
0.001
1.719
0.042
1.644
0.056
1.490
0.101
1.333
0.174
1.351
0.164
1.349
0.165
Comprehension
F-stat
p-Value
2.009
0.012
1.681
0.049
1.466
0.110
1.385
0.146
1.283
0.205
1.205
0.261
1.300
0.194
Non-cognitive
F-stat
p-Value
1.722
0.041
1.281
0.206
1.208
0.258
1.326
0.178
1.220
0.249
1.175
0.284
1.169
0.289
Note: Entries in bold are “surviving specifications” for which we cannot reject exogeneity at 10% of significance. Each specification contains different control variables: (1) no controls; (2) child characteristics; (3) mother demographic characteristics;
(4) family demographic characteristics; (5) family environmental characteristics; (6) school characteristics; (7) child’s school
experience. All standard errors are corrected for heteroskedasticity. See footnote 27 for a full description of the control variables.
All standard errors are corrected for heteroskedasticity.
64
Table 23: Effects of Children’s Time Allocation: Older Children, Not Value-Added
Active time with mother
Passive time with mother
Active time with father
Passive time with father
Active time with grandparents
Passive time with grandparents
Active time with siblings
Passive time with siblings
Active time with friends
Passive time with friends
Self Active time
Self Passive time
Active time with others
Passive time with others
Don’t know or refuse to answer
R-Square
Observations
Exogeneity test F-statistic
Exogeneity test p-value
Math
Vocabulary
Comprehension
Non-cognitive
0.009**
(0.004)
0.005*
(0.003)
0.018**
(0.007)
0.002
(0.005)
0.022*
(0.012)
-0.005
(0.005)
0.003
(0.007)
0.008*
(0.004)
0.010**
(0.004)
0.003
(0.004)
0.009**
(0.003)
0.004
(0.003)
0.011
(0.007)
0.002
(0.004)
0.005
(0.004)
0.478
1453
0.838
0.635
0.004
(0.004)
0.002
(0.003)
0.013
(0.008)
0.001
(0.005)
0.028*
(0.016)
-0.004
(0.006)
-0.003
(0.007)
0.004
(0.004)
0.008*
(0.004)
-0.002
(0.004)
0.001
(0.003)
0.002
(0.003)
-0.003
(0.006)
-0.005
(0.005)
0.003
(0.004)
0.422
1455
1.349
0.165
0.003
(0.004)
0.000
(0.003)
0.010
(0.009)
0.010**
(0.005)
0.036**
(0.013)
-0.002
(0.006)
-0.004
(0.007)
0.004
(0.004)
0.002
(0.004)
0.001
(0.004)
0.004
(0.003)
0.000
(0.003)
0.002
(0.008)
-0.006
(0.004)
0.003
(0.004)
0.436
1453
1.300
0.194
0.001
(0.004)
0.004
(0.003)
0.009
(0.008)
0.005
(0.006)
0.018
(0.019)
-0.009
(0.008)
0.011
(0.008)
0.000
(0.006)
-0.002
(0.005)
-0.004
(0.005)
0.004
(0.004)
0.003
(0.004)
0.005
(0.009)
0.003
(0.006)
-0.001
(0.006)
0.134
1454
1.169
0.289
Note: Standard errors corrected for heteroskedasticity are in parentheses. All estimates are for specification (7). * Significant
at the 10% level. ** Significant at the 5% level. See footnote 27 for a full description of the control variables.
65
Figure 10: Further Evidence of Power of Test (1 of 2)
(b) Mother’s Age at Child Birth, Younger Children
30
29
Conditional Mean
26
28
27
25
24
30
32
Conditional Mean
34
38
36
40
42
(a) Mother’s Age, Younger Children
0
2
4
6
8 10 12 14 16 18 20 22 24
Active time with father (hours per week)
26
28
30
0
2
4
6
8 10 12 14 16 18 20 22 24
Active time with mother (hours per week)
26
28
30
(d) Child Lives with Biological Parents, Older
Children
1
0
.2
.1
.3
.2
.4
.3
Conditional Mean
.7
.5
.6
Conditional Mean
.4 .5 .6 .7
.8
.8
.9
.9
1
(c) Mother is Married, Older Children
0
2
4
6
8 10 12 14 16 18 20 22 24
Passive time with father (hours per week)
26
28
30
0
4
6
8 10 12 14 16 18 20 22 24
Passive time with father (hours per week)
26
28
30
(f) Caregiver Spent on School Supplies Last
Year, Older Children
0
.8
.1
.2
.3
Conditional Mean
.4 .5 .6 .7
.8
Conditional Mean
.82 .84 .86 .88 .9 .92 .94 .96 .98
.9
1
1
(e) Child Has Musical Instrument at Home,
Older Children
2
0
2
4
6
8 10 12 14 16 18 20 22 24
Passive time with father (hours per week)
26
28
30
0
Notes: See the note for Figure 5 in the text.
66
2
4
6
8 10 12 14 16 18 20 22 24
Active time with mother (hours per week)
26
28
30
Figure 11: Further Evidence of Power of Test (2 of 2)
(b) Neighborhood is Safe at Night (Rating 1-5),
Older Children
.4
10.5
.6
11
.8
11.5
Conditional Mean
1
1.2 1.4 1.6
Conditional Mean
12 12.5 13 13.5
1.8
14
2
2.2
14.5
(a) Father’s Level of Education (Years),
Younger Children
0
2
4
6
8 10 12 14 16 18 20 22 24
Active time with mother (hours per week)
26
28
30
0
4
6
8 10 12 14 16 18 20 22 24 26
Don't know or refuse to answer (hours per week)
28
30
(d) Primary Caregiver Completed Weekday Diary, Older Children
0
0
.1
.1
.2
.2
.3
Conditional Mean
.5
.3
.4
Conditional Mean
.4 .5 .6 .7
.6
.8
.7
.9
1
.8
(c) Child Completed Weekend Diary (With or
Without Help), Older Children
2
0
2
4
6
8 10 12 14 16 18 20 22 24 26
Don't know or refuse to answer (hours per week)
28
30
0
4
6
8 10 12 14 16 18 20 22 24 26
Don't know or refuse to answer (hours per week)
28
30
(f) Proportion of Passive Time with Mother
Watching TV, Younger Children
0
.2
.05
.3
.1
Conditional Mean
.4
.5
Conditional Mean
.15 .2 .25 .3 .35
.6
.4
.45
.7
.5
(e) Proportion of Active time with Friends Engaging in Arts and Crafts, Older Children
2
0
2
4
6
8
10 12 14 16 18 20
Passive time with friends
22
24
26
28
30
0
Notes: See the note for Figure 5 in the text.
67
2
4
6
8
10 12 14 16 18 20
Passive time with father
22
24
26
28
30