03.20.2013 - Flavio Cunha

Eliciting Maternal Beliefs about the Technology
of Skill Formation

Jennifer Culhane, Flávio Cunha, and Irma Elo
CHOP and University of Pennsylvania

March 2013

Inequality in Skills Arise Early in Life of Individuals
Figure
Dynamics of Cognitive Skills
For Different Groups of Permanent Income
0.8

0.6

0.4

0.2

0.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
‐0.2

‐0.4

‐0.6

‐0.8
Bottom Quartile Second Quartile Third Quartile Top Quartile

Inequality in Skills and Inequality in Investments
Investments in Human Capital of Children
by Quartiles of Permanent Income
.8
.6
.4
.2
0

0 1 2 3 4 5
Investments (hours per day)

kdensity Bottom kdensity Second
kdensity Third kdensity Top

Figure
Choice Set and Preferences
120

100 Preferences

80
Other Expenditures

60

40

Choice Set

20

0
0 5 10 15 20 25 30 35 40

Child Development

Parental Information about the Technology of Skill
Formation

1. Iodine De¢ ciency:Field, Robles, and Torero (2009); Roy
(2009).
2. The 1964 Surgeon General Report on Smoking: Aizer and
Stroud (2010).
3. Time spent and appropriateness of activities: Kalil, Ryan, and
Corey (2012).

Figure
Choice Set and Preferences
120

Red: Mother underestimates returns
100
Blue: Mother has unbiased returns

80
Other Expenditures

60

40

20

0
0 5 10 15 20 25 30 35 40

Child Development

The Technology of Skill Formation

I Let xi denote investment.
I Let qi ,0 denote the human capital of the child at birth.
I Let qi ,1 denote the human capital of the child at age 24
months.
I Let θ i and νi denote unobservable components.

ln qi ,1 = ln A + ρ ln qi ,0 + γ ln xi + θ i + νi

Maternal Beliefs about the Technology Parameters

I Mother has beliefs about γ.


I For mother i, γ is a random variable:

γ N µγ,i , σ2
γ,i

and µγ,i , σ2 are heterogeneous across mothers.
γ,i



γ N µγ,i , σ2
γ,i

γ,i
I Is this empirically important?



γ N µγ,i , σ2
γ,i

γ,i
I Is this empirically important?
I Unfortunately, it is not possible to answer this question with
data on xi , qi ,0 , and qi ,1 unless one is willing to make strong
assumptions.

Today

I Objective estimation of the technology of skill formation
(based on the CNLSY/79 data).
I Elicit subjective beliefs about the technology of skill formation
(based on the MKIDS data).
I Compare objective estimates with subjective beliefs.

Objective Estimation of the Technology of Skill Formation:

I Consider a Cobb-Douglas speci…cation (CHS, 2010).
I Our goal is to estimate the parameters (especially γ) of the
following equation:

ln qi ,1 = ln A + ρ ln qi ,0 + γ ln xi + θ i + νi

1. Need to …nd a metric for q and x (time)
2. Account for measurement error in q and x (IRT analysis of
MSD, factor analysis of HOME-SF).
3. Address endogeneity of x (FE, IVFE).

Table 4
Objective Estimation of the Technology of Skill Formation
Dependent variable: Natural log of skills around age 24 months1
FE Procedure
16 to 32 19 to 29
13 to 35 Months
Months Months
(1) (2) (5) (8)
Overall Overall Overall Overall
2
Natural Logarithm of health conditions at birth 0.588*** 0.597*** 0.517** 0.608*
(0.17) (0.17) (0.21) (0.34)
3
Natural logarithm of investments 0.206*** 0.204*** 0.235*** 0.363***
(0.03) (0.03) (0.04) (0.06)
Constant 2.222*** ‐1.516*** ‐1.624*** ‐1.943**
(0.38) (0.39) (0.50) (0.91)
Dummy for the child's age at the time of the
Yes No No No
interview
Natural logarithm of child's age at the time of the
No Yes Yes Yes
interview
Observations 2,984 2,984 2,218 1,400
R‐squared 0.775 0.769 0.676 0.502
Number of Mothers 2,168 2,168 1,757 1,202
Robust standard errors in parentheses. All regressions have dummy variables for the child's gender,
birthorder, and year of birth to capture cohort effects. All the regressions also have dummy variables for
maternal age at the time of the child's birth. In column (1), we add dummy variables for the age of the
child at the time of the interview. In columns (2)‐(12), we replace the dummies with one continuous
variable (the natural log of the child's age at the time of the interview).
*** p<0.01, ** p<0.05, * p<0.1

Table 4
FE Procedure
13 to 35 Months 16 to 32 Months 19 to 29 Months
(3) (4) (6) (7) (9) (10)
Black White Black White Black White
2
Natural Logarithm of health conditions at birth 0.550** 0.621*** 0.782** 0.377 ‐0.405 0.533
(0.24) (0.22) (0.35) (0.28) (0.53) (0.55)
3
Natural logarithm of investments 0.202*** 0.198*** 0.216*** 0.230*** 0.459*** 0.436***
(0.04) (0.04) (0.06) (0.06) (0.11) (0.10)
No No No No No No
interview
Yes Yes Yes Yes Yes Yes
interview
Observations 867 2,117 658 1,560 403 997
R‐squared 0.827 0.771 0.812 0.681 0.900 0.504
Number of Mothers 641 1,527 519 1,238 350 852

Robust standard errors in parentheses. All regressions have dummy variables for the child's gender, birthorder, and year of birth to
capture cohort effects. All the regressions also have dummy variables for maternal age at the time of the child's birth. In column (1),
we add dummy variables for the age of the child at the time of the interview. In columns (2)‐(12), we replace the dummies with one
continuous variable (the natural log of the child's age at the time of the interview).
*** p<0.01, ** p<0.05, * p<0.1


I The …xed-e¤ect procedure requires strict exogeneity of
parental investments.
I This rules out any form of feedback from ν to x.
I Consider two di¤erent instruments: permanent income
between ages 0-36 months of the child.
I Advantage: allows for some form of feedback from ν to x.
I However, imposes no feedback from ν to permanent income.

Table 4
FE Procedure
13 to 35 Months 16 to 32 Months 19 to 29 Months
(3) (4) (6) (7) (9) (10)
Black White Black White Black White
2
Natural Logarithm of health conditions at birth 0.550** 0.621*** 0.782** 0.377 ‐0.405 0.533
(0.24) (0.22) (0.35) (0.28) (0.53) (0.55)
3
Natural logarithm of investments 0.202*** 0.198*** 0.216*** 0.230*** 0.459*** 0.436***
(0.04) (0.04) (0.06) (0.06) (0.11) (0.10)
No No No No No No
interview
Yes Yes Yes Yes Yes Yes
interview
Observations 867 2,117 658 1,560 403 997
R‐squared 0.827 0.771 0.812 0.681 0.900 0.504
Number of Mothers 641 1,527 519 1,238 350 852

Robust standard errors in parentheses. All regressions have dummy variables for the child's gender, birthorder, and year of birth to
capture cohort effects. All the regressions also have dummy variables for maternal age at the time of the child's birth. In column (1),
we add dummy variables for the age of the child at the time of the interview. In columns (2)‐(12), we replace the dummies with one
continuous variable (the natural log of the child's age at the time of the interview).

*** p<0.01, ** p<0.05, * p<0.1


I We consider a second instrument. To see motivation, consider
the di¤erence between the third and second child in the
household:

ln q3,1 ln q2,1 = ρ (ln q3,0 ln q2,0 ) + γ (ln x3,1 ln x2,1 ) + (ν3,1 ν

I Now, suppose that
1. ν3,1 ν2,1 = η 3,1 .
2. η 3 is not revealed to the parent until the third child is born.
I Then, if we look at families in which ln x2,1 and ln x1,1 were
chosen before the birth of the third child, the di¤erence
ln x2,1 ln x1,1 is not correlated with η 3,1 if (1) and (2) are
true.
I Allows feedback from η 3,1 to ln x3,1 , but imposes assumption
on the serial correlation of ν.

Table 4
IV
13 to 35 Months
(11) (12)
Overall Overall
2
Natural Logarithm of health conditions at birth 0.602*** 0.56
(0.21) (0.45)
Natural logarithm of investments3 0.428* 0.417**
(0.25) (0.21)
Constant ‐1.635*** ‐2.648***
(0.50) (0.98)
No No
interview
Yes Yes
interview
Observations 2,798 1,254
R‐squared
Number of Mothers 2,134 1,048
3
First Stage ‐ Dependent variable: Natural logarithm of investments
Natural logarithm of family income (average between 0.142***
ages 0 and 36 months) (0.044)
3
Lagged Natural logarithm of investments ‐0.298***
(0.077)

Eliciting Beliefs about the Technology of Skill Formation

I Let E [ ln q1 j q0 , x, θ ] denote the maternal expectation of child
development at age 24 months when the unobserved
component is θ, investment is x, and the child’ health
s
condition at birth is q0 .
I To estimate µγ,i , we need to measure E [ ln q1 j q0 , x, θ ] for
two di¤erent levels of x. To see why:

E [ ln q1 j q0 , x, θ ] = ln A + ρ ln q0 + µγ,i ln x + θ i
¯ ¯

E [ ln q1 j q0 , x, θ ] = ln A + ρ ln q0 + µγ,i ln x + θ i
¯ ¯
I Subtracting and reorganizing terms:

E [ ln q1 j q0 , x, θ ]
¯ E [ ln q1 j q0 , x, θ ]
µγ,i = ¯
ln x
¯ ln x
¯

I In order to measure returns to investments, we need to come
up with a way to measure maternal expectation of child
development, E [ ln q1 j q0 , x, θ ] .
I Take the Motor-Social Development Instrument, but instead
of asking:

“Has your child ever spoken a partial sentence with three words or
more?”

I Suppose we ask:

“What do you think is the youngest age and the oldest age a baby
learns to speak a partial sentence with three words or more?”

I We need to do it for two di¤erent levels of x.


I Transform the information of age range into probability.
I Assumption: The mother believes that development is
logistically distributed within the age range she supplies.

Figure 3
Transforming age range into probability
1
.75
Probability
.5
.25
0

0 4 8 12 16 20 24 28 32 36 40 44 48
Age (in months)

Figure 3
1
.75
Probability
.5
.25
0

0 4 8 12 16 20 24 28 32 36 40 44 48
Age (in months)

Logistic prediction, high High investment

Figure 3
1 .75
Probability
.5 .25
0

0 4 8 12 16 20 24 28 32 36 40 44 48
Age (in months)

Logistic prediction, high High investment
Logistic prediction, low Low investment

Transform the Probability into Expected Human Capital

I Remember that the NHANES contains data on a
representative set of children from ages 2 to 47 months.
I The NHANES dataset allows us to estimate the probability
that a child age A has already spoken a partial sentence with
three words or more.

Figure 4
1
.75
Probability
.5
.25
0 Probability as a Function of Child's Age

0 4 8 12 16 20 24 28 32 36 40 44 48
Child Age (Months)

Speak partial sentence, data Speak partial sentence, predicted

Ignoring Measurement Error

I If we are willing to ignore measurement error (i.e., assume that
the response to “speak partial sentence” exactly measures
maternal expectation), we can obtain µγ,i by comparing the
response for the two di¤erent levels of investments.
I Suppose that x = 3 months per year (i.e., x = 6 hours per
¯ ¯
day) and x = 1 month per year (i.e., x = 2 hours per day).
¯ ¯
Then:
ln(22) ln(16)
µγ,i = ' 29%.
ln(3) ln(1)

Accounting for Measurement Error

I One way to allow for measurement error is to include other
(repeated) measurements.
I Added bonus: Because MSD items vary in di¢ culty, they
provide a way to check whether answers are consistent.

Figure 5
Comparing answers across scenarios
Age range into probability Probability into expected development
1

1
Speaks partial sentence Speaks partial sentence
.25 .5 .75

.25 .5 .75
Knows own age and sex Knows own age and sex
0

0
0 4 8 12 16 20 24 28 32 36 40 44 48 0 4 8 12 16 20 24 28 32 36 40 44 48
Age (in months) Child Age (Months)
1

1
Speaks partial sentence Speaks partial sentence
Knows own age and sex
.75

.75
.5

.5
.25

.25

Knows own age and sex
0

0

0 4 8 12 16 20 24 28 32 36 40 44 48 0 4 8 12 16 20 24 28 32 36 40 44 48
Age (in months) Child Age (Months)


I We create four hypothetical scenarios:
1. (qh , xh ) = child is healthy at birth and investment is high.
2. (qu , xh ) = child is not healthy at birth and investment is high.
3. (qh , xl ) = child is healthy at birth and investment is low.
4. (qu , xl ) = child is not healthy at birth and investment is low.
I A video explains to participants what we mean by healthy, not
healthy, high, low.
I Why create more scenarios? We can test the Cobb-Douglas
speci…cation.


I Taking di¤erences between scenarios (qh , xh ) and (qh , xl )
yields:

E [ ln q1 j qh , xh , θ i ] E [ ln q1 j qh , xl , θ i ]
µh =
γ,i .
ln xh ln xl
I Analogously, take di¤erences between scenarios (qu , xh ) and
(qu , xl ):

E [ ln q1 j qu , xh , θ i ] E [ ln q1 j qu , xl , θ i ]
µu =
γ,i .
ln xh ln xl

I If we …nd that µh 6= µu , then parents do not believe
γ,i γ,i
production function is Cobb-Douglas.

Table 6
1
Average lowest and highest age by MSD item and scenario
2 2
Health condition at birth is "high" Health condition at birth is "low"
Investment is "high"3 Investment in "low"3 Investment is "high"3 Investment in "low"3
Scenario 1 Scenario 3 Scenario 2 Scenario 4
Lowest age Highest age Lowest age Highest age Lowest age Highest age Lowest age Highest age
MSD 30: Let someone know, without crying, that
wearing wet or soiled pants or diapers bothers him or 14.58 28.39 18.52 32.74 16.81 30.32 21.11 34.16
her?

MSD 35: Speak a partial sentence of 3 words or more? 22.26 35.69 26.63 39.02 24.66 37.88 29.09 40.53

MSD 38: Count 3 objects correctly? 25.08 37.72 28.70 40.84 26.54 38.63 30.50 42.26

MSD 40: Know his/her age and sex? 25.35 38.70 28.97 40.91 27.16 39.99 31.10 42.35

MSD 41: Say the names of at least 4 colors? 26.06 38.85 30.23 41.84 28.24 39.86 32.08 43.30

MSD 36: Say his/her first and last name together without
27.30 39.37 30.97 41.81 28.64 40.42 32.88 43.00
someone's help?

MSD 47: Count out loud up to 10? 27.96 40.58 31.42 42.96 29.11 41.28 33.03 43.78

MSD 48: Draw a picture of a man or woman with at least
33.42 43.35 35.85 44.59 34.16 43.78 37.13 45.34
2 parts of the body besides a head?

Figure 7
Kernel density of expected human capital at age 24 months
Health condition at birth is high Investment is high
1 1.5

1 1.5
.5

.5
0

0
2 2.5 3 3.5 4 2 2.5 3 3.5 4
x x

High investment Low investment Health is high Health is low

Health condition at birth is low Investment is low
1 1.5

1 1.5
.5

.5
0

0

2 2.5 3 3.5 4 2 2.5 3 3.5 4
x x

High investment Low investment Health is high Health is low

Figure 8
Importance of measurement error in responses

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Signal Item Noise Item‐Scenario Noise

Table 9
Subjective expectations about the technology of skill formation
Not accounting for measurement error
25th 75% Std
Mean Median
Percentile Percentile Deviation
Overall items and Scenarios1 8.8% 4.5% ‐4.5% 23.0% 32.9%

High versus low heath conditions at birth
Overall items, only scenarios with high health conditions at birth 12.7% 7.2% 0.0% 29.1% 36.5%
Overall items, only scenarios with low health conditions at birth 4.9% 2.9% ‐7.2% 21.1% 33.6%

Primarily motor vs. primarily cognitive items
Cognitive items, all scenarios 7.6% 0.0% ‐0.9% 20.8% 33.4%
Motor items, all scenarios 9.9% 4.6% ‐8.0% 23.7% 38.7%

Primarily motor vs. primarily cognitive items by health condition at birth
Cognitive items, high health conditions at birth 10.3% 0.0% 0.0% 26.1% 35.6%
Cognitive items, low health conditions at birth 4.9% 0.0% ‐1.8% 17.0% 34.5%
Motor items, high health conditions at birth 14.8% 9.0% 0.0% 35.7% 43.8%
Motor items, low health conditions at birth 4.9% 0.0% ‐6.1% 22.3% 41.2%

Accounting for measurement error
25th 75% Std
Mean Median
Percentile Percentile Deviation
Overall items and Scenarios1 7.4% 3.9% ‐3.4% 21.0% 32.9%

High versus low heath conditions at birth
Overall items, only scenarios with high health conditions at birth 11.8% 6.7% 1.3% 29.4% 36.1%
Overall items, only scenarios with low health conditions at birth 3.0% 0.2% ‐8.2% 15.8% 32.8%

Primarily motor vs. primarily cognitive items
Cognitive items, all scenarios 5.1% 2.1% ‐4.0% 16.3% 36.4%
Motor items, all scenarios 6.9% 3.1% ‐5.0% 19.0% 35.4%

Primarily motor vs. primarily cognitive items by health condition at birth
Cognitive items, high health conditions at birth 10.2% 4.2% ‐0.4% 24.7% 39.2%
Cognitive items, low health conditions at birth 0.0% ‐4.4% ‐7.4% 13.3% 36.9%
Motor items, high health conditions at birth 16.6% 10.4% 3.4% 36.1% 39.9%
Motor items, low health conditions at birth ‐2.8% ‐4.1% ‐13.9% 7.3% 36.6%

1
Health conditions at birth are "high" if (1) the baby's weight at birth is 8 pounds, the baby's length at birth is 20 inches, and the
gestational age is 9 months. Health conditions at birth are low if the baby's weight at birth is 5 pounds, the baby's length at birth is
18 inches, and the gestational age is 7 months. When investments are "high", the mother spends 6 hours/day interacting with the
baby. In contrast, when investment is "low", the mother spends only 2 hours/day interacting with the baby.

Table 11
Subjective beliefs about the technology of skill formation
How do maternal mean beliefs change with definitions of health conditions at birth and investments?
Gestation: 9 months Gestation: 9 months Gestation lasts 9 months
Healthy Birth weight: 7 pounds Healthy Birth weight: 7 pounds Normal Birth weight: 8 pounds
Birth length: 20 inches Hospital time: 3 days Birth length: 20 inches
Health conditions at birth
Gestation: 7 months Gestation: 7 months Gestation: 8.5 months
Not Healthy Birth weight: 5 pounds Not Healthy Birth weight: 5 pounds Small Birth weight: 6 pounds
Birth length: 19 inches Hospital time: 7 days Birth length: 19 inches

High 6 hours/day High 4 hours/day High 4 hours/day
Investments
Low 2 hours/day Low 3 hours/day Low 3 hours/day
Number of observations = 42 Number of observations = 71 Number of observations
Mean Median Std Dev Mean Median Std Dev Mean Median Std Dev

Overall items and Scenarios 7.2% 6.9% 28.1% 18.7% 7.6% 48.4% 22.6% 1.8% 41.1%

Overall items, only scenarios with high
8.3% 13.8% 37.9% 29.8% 11.9% 59.0% 28.9% 8.4% 49.5%
health conditions at birth
Overall items, only scenarios with low
6.1% 1.1% 29.2% 7.7% ‐3.2% 49.2% 16.3% 8.2% 42.4%
health conditions at birth

Cognitive items, all scenarios 2.7% 1.0% 27.9% 9.7% ‐3.4% 65.1% 14.8% ‐3.7% 49.6%

Motor items, all scenarios 9.3% 8.6% 32.5% 24.9% 16.9% 41.6% 24.2% 13.6% 52.1%

Figure 9
Checking the robustness of logit assumption
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1

12 16 20 24 28 32 36
Age

Logit distribution Upper triangular distribution
Lower triangular distribution

Table 12
Subjective beliefs about the technology of skill formation
Checking sensitivity of the logit assumption
Not accounting for measurement error
Logit Lower Triangular Upper Triangular
Mean Median Std Dev Mean Median Std Dev Mean Median Std Dev

Overall items and Scenarios 8.8% 4.5% 32.9% 15.0% 6.7% 24.7% 16.7% 8.2% 27.5%

Overall items, only scenarios with
12.7% 7.2% 36.5% 17.3% 8.0% 29.7% 18.6% 11.0% 32.4%
high health conditions at birth
Overall items, only scenarios with
4.9% 2.9% 33.6% 12.6% 4.7% 21.5% 14.8% 5.9% 24.9%
low health conditions at birth

Cognitive items, all scenarios 7.6% 0.0% 33.4% 13.0% 0.7% 24.1% 15.2% 0.4% 28.1%

Motor items, all scenarios 9.9% 4.6% 38.7% 16.7% 7.7% 29.6% 18.1% 9.6% 31.8%

Parental preferences

I Suppose that parental preferences are described by the
following utility function:

(c c )1 σ
¯ 1 1
q1 σ
1
u (c, q1 ) = +α
1 σ 1 σ
I The parameters:
1. σ: How much parents are willing to substitute child
development for other goods and services.
2. α: How much parents “value” child development relative to
other goods and services.
3. c : Minimum expenditure on goods and services.
¯

Parental preferences

I The problem of the (representative) parent is:
" #
(c c )1 σ 1
¯ 1
q1 σ 1
max E +α I
c ,x 1 σ 1 σ

I Parents face the constraints:

Budget: c + πx = y

Belief: ln q1 = ln A + ρ ln q0 + µγ ln x + θ + ν

Information set: I = fπ, y , θ, µγ , σ2 g
γ

I And child development follows the “true” production function:

ln q1 = ln A + ρ ln q0 + γ ln x + θ + ν

Estimating parental preferences

I In order to estimate the parameters σ, α, and c we collect
¯
data on hypothetical choices.
I We tell the parent that q0 is high.
I We present the parent with a scenario of y and π.
I We ask them to choose how to allocate resources between c
and x.
I Then we vary y and π, one at a time.
I We use the choices to estimate the parameters σ, α, and c .
¯

Figure 10
Demand for investments
2.1
2
1.9
1.8
1.7
1.6

25 30 35 40 45 50
Price

low income medium income
high income

Model simulation

I Suppose that γ = 0.2, µγ = 0.15, and σ2 = 0.05.
γ
I Question: How much higher would investments be if we were
able to convince the parent that γ = 0.2?
I Investments would increase by 6%.
I Lower bound?

Conclusion
I In general, economic models of human development assume
that mothers know the parameters of the technology of skill
formation.
I In this paper, we formulate a model of human development
that allows for subjective beliefs about these parameters.
I Policy relevance: Allows us to understand the channels
through which information acquisition, di¤usion, and learning
about parenting may a¤ect the developmental outcoumes of
young children.
I When such models are structurally estimated, the variation in
observed investments across families is attributed to shocks,
heterogeneity in the characteristics of the children or families,
but not to the heterogeneity in the knowledge base of the
mother.
I Important for policy and methodological reasons.

Conclusion
I Methodological relevance: When models that don’ account
t
for beliefs are structurally estimated, the variation in observed
investments across families is attributed to shocks,
heterogeneity in the characteristics of the children or families,
but not to the heterogeneity in the knowledge base of the
mother.
I Unfortunately, this arises because heterogeneity in information
cannot be separately identi…ed from other unobserved
heterogeneity.
I To solve this problem, we develop and implement a survey
methodology that elicits these subjective beliefs.
I Future work:
1. Re…ne elicitation of beliefs;
2. Con…rm that elicited beliefs predict investment (if so, go to 3,
if not back to 1)
3. Experimentally manipulate beliefs to estimate its causal e¤ect
on investments.

03.20.2013 - Flavio Cunha

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