03.20.2013 - Flavio Cunha

Eliciting Maternal Beliefs about the Technologyof Skill Formation

Jennifer Culhane, Flávio Cunha, and Irma EloCHOP and University of Pennsylvania

March 2013

Inequality in Skills Arise Early in Life of Individuals

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Figure Dynamics of Cognitive Skills

For Different Groups of Permanent Income

Bottom Quartile Second Quartile Third Quartile Top Quartile

Inequality in Skills and Inequality in Investments

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kdensity Bottom kdensity Second

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by Quartiles of Permanent IncomeInvestments in Human Capital of Children

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Child Development

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FigureChoice Set and Preferences

Choice Set

Preferences

Parental Information about the Technology of SkillFormation

1. Iodine Deffi ciency:Field, Robles, and Torero (2009); Roy(2009).

2. The 1964 Surgeon General Report on Smoking: Aizer andStroud (2010).

3. Time spent and appropriateness of activities: Kalil, Ryan, andCorey (2012).

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Child Development

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FigureChoice Set and Preferences

Red: Mother underestimates returnsBlue: Mother has unbiased returns

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 24months.

I Let θi and νi denote unobservable components.

ln qi ,1 = lnA+ ρ ln qi ,0 + γ ln xi + θi + 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 , σ2γ,i

)and µγ,i , σ

2γ,i are heterogeneous across mothers.

I Is this empirically important?I Unfortunately, it is not possible to answer this question withdata on xi , qi ,0, and qi ,1 unless one is willing to make strongassumptions.


I Mother has beliefs about γ.I For mother i , γ is a random variable:

γ ∼ N(

µγ,i , σ2γ,i

)and µγ,i , σ





γ ∼ N(

µγ,i , σ2γ,i

)and µγ,i , σ


I Is this empirically important?

I Unfortunately, it is not possible to answer this question withdata on xi , qi ,0, and qi ,1 unless one is willing to make strongassumptions.



γ ∼ N(

µγ,i , σ2γ,i

)and µγ,i , σ



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 specification (CHS, 2010).I Our goal is to estimate the parameters (especially γ) of thefollowing equation:

ln qi ,1 = lnA+ ρ ln qi ,0 + γ ln xi + θi + νi

1. Need to find a metric for q and x (time)

2. Account for measurement error in q and x (IRT analysis ofMSD, factor analysis of HOME-SF).

3. Address endogeneity of x (FE, IVFE).

16 to 32 Months

19 to 29 Months

(1) (2) (5) (8)Overall Overall Overall Overall

Natural Logarithm of health conditions at birth2 0.588*** 0.597*** 0.517** 0.608*(0.17) (0.17) (0.21) (0.34)

Natural logarithm of investments3 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 interview

Yes No No No

Natural logarithm of child's age at the time of the interview

No Yes Yes Yes

Observations 2,984 2,984 2,218 1,400R‐squared 0.775 0.769 0.676 0.502Number of Mothers 2,168 2,168 1,757 1,202

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

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).

Table 4Objective Estimation of the Technology of Skill Formation

Dependent variable: Natural log of skills around age 24 months1

FE Procedure

13 to 35 Months

(3) (4) (6) (7) (9) (10)Black White Black White Black White

Natural Logarithm of health conditions at birth2 0.550** 0.621*** 0.782** 0.377 ‐0.405 0.533(0.24) (0.22) (0.35) (0.28) (0.53) (0.55)

Natural logarithm of investments3 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


Yes Yes Yes Yes Yes Yes

Observations 867 2,117 658 1,560 403 997R‐squared 0.827 0.771 0.812 0.681 0.900 0.504Number of Mothers 641 1,527 519 1,238 350 852

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



FE Procedure


13 to 35 Months 16 to 32 Months 19 to 29 Months


I The fixed-effect procedure requires strict exogeneity ofparental investments.

I This rules out any form of feedback from ν to x .I Consider two different instruments: permanent incomebetween 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.

(3) (4) (6) (7) (9) (10)Black White Black White Black White

Natural Logarithm of health conditions at birth2 0.550** 0.621*** 0.782** 0.377 ‐0.405 0.533(0.24) (0.22) (0.35) (0.28) (0.53) (0.55)

Natural logarithm of investments3 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


Yes Yes Yes Yes Yes Yes

Observations 867 2,117 658 1,560 403 997R‐squared 0.827 0.771 0.812 0.681 0.900 0.504Number of Mothers 641 1,527 519 1,238 350 852

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



FE Procedure


13 to 35 Months 16 to 32 Months 19 to 29 Months


I We consider a second instrument. To see motivation, considerthe difference between the third and second child in thehousehold:

ln q3,1− ln q2,1 = ρ (ln q3,0 − ln q2,0)+γ (ln x3,1 − ln x2,1)+ (ν3,1 − ν2,1)

I Now, suppose that

1. ν3,1 − ν2,1 = η3,1.2. η3 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 werechosen before the birth of the third child, the differenceln x2,1 − ln x1,1 is not correlated with η3,1 if (1) and (2) aretrue.

I Allows feedback from η3,1 to ln x3,1, but imposes assumptionon the serial correlation of ν.

(11) (12)Overall Overall

Natural Logarithm of health conditions at birth2 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


Yes Yes

Observations 2,798 1,254R‐squaredNumber of Mothers 2,134 1,048

0.142***(0.044)

Lagged Natural logarithm of investments3 ‐0.298***(0.077)

First Stage ‐ Dependent variable: Natural logarithm of investments3

Natural logarithm of family income (average between ages 0 and 36 months)



IV13 to 35 Months

Eliciting Beliefs about the Technology of Skill Formation

I Let E [ ln q1| q0, x , θ] denote the maternal expectation of childdevelopment at age 24 months when the unobservedcomponent is θ, investment is x , and the child’s healthcondition at birth is q0.

I To estimate µγ,i , we need to measure E [ ln q1| q0, x , θ] fortwo different levels of x . To see why:

E [ ln q1| q0, x̄ , θ] = lnA+ ρ ln q0 + µγ,i ln x̄ + θi

E [ ln q1| q0, x¯ , θ] = lnA+ ρ ln q0 + µγ,i ln x¯+ θi

I Subtracting and reorganizing terms:

µγ,i =E [ ln q1| q0, x̄ , θ]− E [ ln q1| q0, x¯ , θ]

ln x̄ − ln x¯


I In order to measure returns to investments, we need to comeup with a way to measure maternal expectation of childdevelopment, E [ ln q1| q0, x , θ] .

I Take the Motor-Social Development Instrument, but insteadof asking:

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

I Suppose we ask:

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

I We need to do it for two different levels of x .


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

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Transforming age range into probabilityFigure 3

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roba

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y

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Logistic prediction, high High investment


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roba

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y

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roba

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Logistic prediction, low Low investment


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y

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Logistic prediction, low Low investment


Transform the Probability into Expected Human Capital

I Remember that the NHANES contains data on arepresentative set of children from ages 2 to 47 months.

I The NHANES dataset allows us to estimate the probabilitythat a child age A has already spoken a partial sentence withthree words or more.

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

Speak partial sentence, data Speak partial sentence, predicted

Probability as a Function of Child's AgeFigure 4

Ignoring Measurement Error

I If we are willing to ignore measurement error (i.e., assume thatthe response to “speak partial sentence” exactly measuresmaternal expectation), we can obtain µγ,i by comparing theresponse for the two different levels of investments.

I Suppose that x̄ = 3 months per year (i.e., x̄ = 6 hours perday) and x

¯= 1 month per year (i.e., x

¯= 2 hours per day).

Then:

µγ,i =ln(22)− ln(16)ln(3)− ln(1) ' 29%.

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 diffi culty, theyprovide a way to check whether answers are consistent.

Speaks partial sentence

Knows own age and sex

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Age range into probabilitySpeaks partial sentence


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Probability into expected development



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Comparing answers across scenariosFigure 5


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, nothealthy, high, low.

I Why create more scenarios? We can test the Cobb-Douglasspecification.


I Taking differences between scenarios (qh, xh) and (qh, xl )yields:

µhγ,i =E [ ln q1| qh, xh, θi ]− E [ ln q1| qh, xl , θi ]

ln xh − ln xl.

I Analogously, take differences between scenarios (qu , xh) and(qu , xl ):

µuγ,i =E [ ln q1| qu , xh, θi ]− E [ ln q1| qu , xl , θi ]

ln xh − ln xl.

I If we find that µhγ,i 6= µuγ,i , then parents do not believeproduction function is Cobb-Douglas.

Lowest age Highest age Lowest age Highest age Lowest age Highest age Lowest age Highest ageMSD 30: Let someone know, without crying, that wearing wet or soiled pants or diapers bothers him or her?

14.58 28.39 18.52 32.74 16.81 30.32 21.11 34.16

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 someone's help?

27.30 39.37 30.97 41.81 28.64 40.42 32.88 43.00

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 2 parts of the body besides a head?

33.42 43.35 35.85 44.59 34.16 43.78 37.13 45.34

Average lowest and highest age by MSD item and scenario1Table 6

Scenario 1 Scenario 2Scenario 3 Scenario 4

Health condition at birth is "high"2

Investment is "high"3 Investment in "low"3 Investment is "high"3 Investment in "low"3Health condition at birth is "low"2

0.5

11

.5

2 2.5 3 3.5 4x

High investment Low investment

Health condition at birth is high

0.5

11

.5

2 2.5 3 3.5 4x

Health is high Health is low

Investment is high

0.5

11

.5

2 2.5 3 3.5 4x

High investment Low investment

Health condition at birth is low

0.5

11

.5

2 2.5 3 3.5 4x

Health is high Health is low

Investment is low

Kernel density of expected human capital at age 24 monthsFigure 7

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

Figure 8Importance of measurement error in responses

Signal Item Noise Item‐Scenario Noise

Mean Median25th

Percentile75%

PercentileStd

Deviation

Overall items and Scenarios1 8.8% 4.5% ‐4.5% 23.0% 32.9%

High versus low heath conditions at birthOverall 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 itemsCognitive 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 birthCognitive 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%

Mean Median25th

Percentile75%

PercentileStd

DeviationOverall items and Scenarios1 7.4% 3.9% ‐3.4% 21.0% 32.9%

High versus low heath conditions at birthOverall 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 itemsCognitive 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 birthCognitive 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%

Not accounting for measurement errorSubjective expectations about the technology of skill formation

Table 9

1Health 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.

Accounting for measurement error

High High High

Low Low Low

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 health conditions at birth

8.3% 13.8% 37.9% 29.8% 11.9% 59.0% 28.9% 8.4% 49.5%

Overall items, only scenarios with low health conditions at birth

6.1% 1.1% 29.2% 7.7% ‐3.2% 49.2% 16.3% 8.2% 42.4%

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%

4 hours/day

3 hours/day

4 hours/day

3 hours/day

6 hours/day

2 hours/day

Health conditions at birth

Investments

How do maternal mean beliefs change with definitions of health conditions at birth and investments?

Table 11Subjective beliefs about the technology of skill formation

Number of observations = 71 Number of observationsNumber of observations = 42

Healthy Birth weight: 7 poundsHospital time: 3 days

Normal

Not HealthyGestation: 7 months

Not Healthy Birth weight: 5 poundsHospital time: 7 days

Small Birth weight: 6 poundsBirth length: 19 inches

Gestation: 8.5 months

Gestation lasts 9 monthsBirth weight: 8 poundsBirth length: 20 inches

HealthyGestation: 9 months

Gestation: 7 monthsBirth weight: 5 poundsBirth length: 19 inches

Gestation: 9 monthsBirth weight: 7 poundsBirth length: 20 inches

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Logit distribution Upper triangular distribution

Lower triangular distribution

Checking the robustness of logit assumptionFigure 9

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 high health conditions at birth

12.7% 7.2% 36.5% 17.3% 8.0% 29.7% 18.6% 11.0% 32.4%

Overall items, only scenarios with low health conditions at birth

4.9% 2.9% 33.6% 12.6% 4.7% 21.5% 14.8% 5.9% 24.9%

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%

Not accounting for measurement errorLower Triangular Upper TriangularLogit

Checking sensitivity of the logit assumption

Table 12Subjective beliefs about the technology of skill formation

Parental preferences

I Suppose that parental preferences are described by thefollowing utility function:

u (c, q1) =(c − c̄)1−σ − 1

1− σ+ α

q1−σ1 − 11− σ

I The parameters:

1. σ: How much parents are willing to substitute childdevelopment for other goods and services.

2. α: How much parents “value” child development relative toother goods and services.

3. c̄ : Minimum expenditure on goods and services.

Parental preferences

I The problem of the (representative) parent is:

maxc ,x

E

[(c − c̄)1−σ − 1

1− σ+ α

q1−σ1 − 11− σ

∣∣∣∣∣ I]

I Parents face the constraints:

Budget: c + πx = y

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

Information set: I = {π, y , θ, µγ, σ2γ}

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

ln q1 = lnA+ ρ ln q0 + γ ln x + θ + ν

Estimating parental preferences

I In order to estimate the parameters σ, α, and c̄ we collectdata 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 cand x .

I Then we vary y and π, one at a time.I We use the choices to estimate the parameters σ, α, and c̄ .

1.6

1.7

1.8

1.9

22.

1

25 30 35 40 45 50Price

low income medium income

high income

Demand for investmentsFigure 10

Model simulation

I Suppose that γ = 0.2, µγ = 0.15, and σ2γ = 0.05.I Question: How much higher would investments be if we wereable 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 assumethat mothers know the parameters of the technology of skillformation.

I In this paper, we formulate a model of human developmentthat allows for subjective beliefs about these parameters.

I Policy relevance: Allows us to understand the channelsthrough which information acquisition, diffusion, and learningabout parenting may affect the developmental outcoumes ofyoung children.

I When such models are structurally estimated, the variation inobserved 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 themother.

I Important for policy and methodological reasons.

ConclusionI Methodological relevance: When models that don’t accountfor beliefs are structurally estimated, the variation in observedinvestments 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 themother.

I Unfortunately, this arises because heterogeneity in informationcannot be separately identified from other unobservedheterogeneity.

I To solve this problem, we develop and implement a surveymethodology that elicits these subjective beliefs.

I Future work:

1. Refine elicitation of beliefs;2. Confirm that elicited beliefs predict investment (if so, go to 3,if not back to 1)

3. Experimentally manipulate beliefs to estimate its causal effecton investments.

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03.20.2013 - Flavio Cunha