Human Capital:Theory

Lent TermLecture 2

Dr. Radha Iyengar

What is Human Capital? Part of original conception of inputs in

production. Adam Smith said that there were 4 inputs in which we might invest:

1. Machines or mechanical inputs2. Building/infrastructure3. land 4. human capital

Education and “General” Human CapitalWe’re going to study first education

(schooling including college/graduate education)

This is important because it is: Expandable and maybe doesn’t depreciate

(like physical capital) Transportable and shareable (not true with

“specific capital”)

What are we going to study? Theory

Static Model (Card) Dynamic Model (Heckman)

The goal of theory is to motivate the large body of empirical work

Empirics Some talk of methods (identification, diff-in-

diff, IV) Reconciling different estimates Economics of Education (briefly!)

A Static Model of Human Capital

Acquisition(for details see: David Card, “Causal Effect of Education

on Earnings” Handbook of Labor Economics)

Basis of Empirical Estimates Common form of estimation:log( y ) = a + bS + cX + dX2 + e (1)

Usually called a Mincer Regression

Some Empirical Facts1. A simple regression model with a linear

schooling term and a low-order polynomial in potential experience explains 20-35% of the variation in observed earnings data, with predictable and precisely-estimated coefficients in almost all applications.

2. Returns to education vary across the

population with observables, such as school quality or parent’s education

OLS Estimates1. 10 percent upward bias on OLS estimates

of the return to education (based on the most recent, “best” twins studies)

2. Estimates of the return to schooling based on brothers or fraternal twins contain positive ability bias, but less than the corresponding OLS estimate.

Does IV Fix the problem? IV estimates of the return to education

based on family background systematically higher than corresponding OLS

estimates probably have a bigger ability bias than OLS

estimates

IV estimates of the return to education based on intervention in the school system about 20 percent more than the OLS estimates. return to schooling for these subgroups are

especially high, and cannot be generalized to the population.

A Static Model of Education and Earnings Because of its tractability, Card uses a

static model that abstracts away from the relationship between completed schooling and earnings over the lifecycle. (we’ll do a dynamic model next).

Two assumptions: that most people finish schooling and only then

enter the labor force (smooth transition). the effect of schooling independent of

experience (Separability above)

The basics Simple Linear regression first introduced by

Mincer Takes the general form of linearity in Schooling,

quadratic in experience.

Assumptions:1. separability of experience and education. 2. log-earnings are linear in education.

correct measure of schooling is years of education each year of schooling is the same. (more on this later)

(1) )log( 2 edXcXbSay

Wages or earnings? Earnings conflates hours and wages Card reports that about two-thirds of the

returns to education are due to the effect of education on earnings—the rest attibutable to the effects on hours/week and week/year.

The specification in (1) explains about 20-30 percent of the variation in earnings data.

Why use Semi-Log Specification? log earnings are approximately normally

distributed. Heckman and Polachek show that the

semi-log form is the best in the the Box-Cox class of transformations. (we can talk about this more later in the empirical part)

Defining some Terms Let our utility function U(S, y) = log(y) –

h(s) where y is earnings, S is years of schooling, and h(s) is an increasing, convex function. Then, define our discounted present value (DPV) function:

rrsSydtrtSy /)exp()()exp()(

Simple relationship between returns and costs

So that we have h(S) = r*S more generally we could have a convex

h(.) function if the marginal cost of each year of schooling increases faster than the foregone earnings for that year—maybe because of credit constraints)

ResultsOptimal schooling is implicitly defined by

That is there are two sources of heterogeneity:

1. Differences in costs (represented by h(S))2. Difference in marginal returns (represented

by y’(S)/y(S))

(2) )(

)(')('

Sy

SySh

Optimal Schooling a simple specification of these two

components

(define E(b) = b and E(r)= r and k1, k2> 0)

This gives us the optimal schooling expression:

(3a) )()('

1SkbSySy

i

(3b) )(' 2SkrSh i

(4) )/()( 21* kkrbS iii

Interpretation of Equilibrium Individuals do not necessarily know the parameters of their

earnings functions when they make their schooling choices.

bi interpretation: individual's best estimate of his/her earnings gain per year of education, as of early adulthood.

One might expect this estimate to vary less across individuals than their realized values of schooling

the distribution of bi may change over time with shifts in labor market conditions, technology, etc. (Skill Premium)

Some Assumptions treat bi as known at the beginning of the lifecycle

and fixed over time:

assumption probably leads to some overstatement of the role of heterogeneity of bi in the determination of schooling and earnings outcomes.

for simplicity, assume jointly symmetric distribution of b and r.

Returns to schooling From our equilibrium expression (4) can

get expression for returns to schooling

Even in this simple model there is a distribution of returns unless Linear indifferent curves with uniform slope

Linear opportunity curves, with uniform slope0 2 kandirri

0 1 kandibbi

Within vs. Between Variation Within: Eq. (4) as a partial equilibrium description

of the relative education choices of a cohort of young adults, given their family backgrounds and the institutional environment and economic conditions that prevailed during their late teens and early 20s.

Differences across cohorts in these background factors will lead to further variation in the distribution of marginal returns to education in the population as a whole.

Earnings and Schooling Eqn From equation 3A (FOC), we get

Note that individual heterogeneity affects both the intercept and the slope

Defining αi = ai + a0

Use this with eqn (4), to define schooling choice in terms of a, b, and r

(5)

Linear Estimating Function Define λ0 and ψ0 as the parameters from

the linear projection of ai and bi on where is E(Si )

(6a)(6b)

That is:

SS i S

OLS estimates of b Using this notation, we can write the

probability limit of the OLS estimate:

(7)where the avg. marginal return to schooling

in the population is:

SSSkbp OLS 00010blim

Homogeneous Returns Let bi = b and k1 = 0

Then (5) implies the OLS estimate is not consistent, with upward bias of l0 %.

The bias comes from the correlation of ability to the marginal cost of schooling.

0lim OLSbp

Heterogeneous Schooling Reintroducing a heterogeneous b

we get additional bias terms in due to the self-selection of years of schooling.

The size of this bias depends on the importance of the variation in b in determining the overall variance of schooling outcomes.

Sbp OLS 00lim

What did we learn The linear model appears to fit so well because

there is a bias introduced by heterogeneity which is convex and independent of the concavity of the opportunity curve.

More simply put, the concavity from quadratic term in (5) is offset by the convexity from y0 giving an approximately linear relationship.

Understanding Observed Linearity-1 Case 1: gets the standard ability “omitted

variable bias” return to schooling Let ai vary by individual (heterogeneous) b be fixed across individuals. Bias comes from the correlation between

ability and marginal cost of schooling so σra < 0 which implies that λ0 > 0.

Understanding Observed Linearity-2 Case 2:

ai and bi both vary across individuals. cross-sectional upward bias because of self

selection. So depending on the relative variance of these

components will determine the convexity and concavity.

Understanding Observed Linearity-3 Rewrite (5) and reorganize terms:

This is linear if ψ0≈ 2k1

The bigger the contribution of bi to the overall variance of schooling, ψ0 is bigger and the more the convexity

iiii vSuSkSSbc 212

1000i )()()log(y

What about Measurement Error? The downward bias of measurement error is often

thought to offset some if not all of the upward bias in a,b from ability, only be true if the error is not correlated with level of

schooling Unlikely because individuals with high levels of schooling

cannot report positive errors in schooling whereas individuals with very low levels of schooling cannot report negative errors in schooling.

Given this correlation, the measurement error may actually exacerbates the attenuation bias.

IV in a Heterogeneous World

even minor difference in mean earnings between the two groups will be exaggerated by the IV procedure.

For example, natural experiments inference are based on small differences between groups of individuals who attended schools at different times, places, etc. However, the uses of these differences might be difficult to generalize.

IV-2 Define a linear relationship between

returns to schooling and a set of characteristics, Z, i.e.

So the earnings function can be rewritten as:

iiii Zr

iiiiiii ZkbZS 0/)(

kbb iii /)( k/1

IV-3 The big news: In the presence of heterogeneous

returns to education the conditions to get an interpretable IV estimator of very strong. The requirements are that we have individual specific

heterogeneity components that are mean independent of the instrument.

The second moment of the return to education is also independent of the instrument

The conditional expectation of the unobserved component of optimal school choice is linear in b.

Family Background IV-1 The strategy: use variables such as

parents education, characteristics of parents to control for unobserved ability.

The key idea: if a and S uncorrelated then we get an unbiased estimate, otherwise, we get an upward bias

Family Background IV-3 To illustrate this, consider a linear log

earnings function:

linear projection of unobserved ability component on individual schooling and a measure of family background (Fi):

iii aSbay 0log

iiii uFFSSa ')()( 21

Comparing Regressions: Homogeneous Case In order to compare this to the regression of a on

S alone, define:

Using these, we could compare three potential estimators: OLS from univariate regression of earnings on schooling

—bOLS

OLS from bivariate regression of earnings on schooling and family background—bbiv

IV estimator using Fi as an instrument for Si (bIV)

s 210 2SFFs

IVOLSbiv bpbpbpb limlimlim

Comparing Regressions: Heterogeneous case introducing heterogeneity across

individuals in b, so that

we can relate ψ0 as follows

Assuming , ,

iiii vFFSSb )()( 21

)/( 221210 FSFs

011 S 022 S 0F

IVOLSbiv bpbpbpb limlimlim

Siblings/Twins Models The key idea behind this strategy: some of

the unobserved differences that bias a cross-sectional comparison of education and earnings are based on family characteristics

Key Assumption: within families, these differences should be fixed.

Differencing between schooling levels of individuals will yield consistent results.

Defining “Family Effect” Define “pure family effects” model as the

aij=aj and bij=bj

linear projection of a and bi – b on the observed schooling outcomes of the two family members:

iiii uFFSSa )()( 21

iiii vFFSSbb )()( 21

Estimating with “Family Effects” Assuming that bi, S1i, S2i have a jointly

symmetric distribution Earning functions are then:

Taking differences, a within family difference in log earnings model:

12122111111 )()(log iiii eSSSScy

22122121122 )()(log iiii eSSSScy

iiii eSSy 2211log

When Family Effects Models Work With identical twins, it is natural to impose

the symmetry conditions so that λ1=λ2=λ, ψ1=ψ2=ψ and

With these assumptions and the pure family effects specification, all biases from ability and schooling are sucked up by the family average schooling component which differences out.

SSS 21

When Family Effects Don’t Work In the case of siblings, or father-son pairs

it seems less plausible.

Relax the family effects model as follows:

1221211111 )()( iiii uSSSSa

1222211212 )()( iiii uSSSSa

1221211111 )()( iiii vSSSSbb

2222211212 )()( iiii vSSSSbb

Why doesn’t it work For a randomly-ordered siblings or

fraternal twins, it is natural to assume that the projection coefficients satisfy the symmetry restrictions so that λ11=λ22, λ12=λ21, ψ12=ψ21, ψ11=ψ21

From this, the earnings model eqn’s are:

From this system, is not identifiable.

121211111 )log( iiii eSScy

222212122 )log( iiii eSScy

“Family Effect” or OLS Models? Without a “pure family effect” and

symmetric it is only possible to estimate an upper bound measure of the marginal returns to schooling.

there is no guarantee that this bound is tighter than the bound implied by the cross-sectional OLS estimator.

It is possible that the OLS estimator has a smaller upward bias than the within family estimator.

Take-Homes from the Static Model 1 The OLS estimator has two ability biases,

the intercept the slope. The bias in the slope may be relatively small if there is not

much heterogeneity. The necessary conditions for IV estimators to be

consistent is strict many plausible instruments recover only the weighted

average of marginal returns of the affected subgroups. . If the OLS estimator is upward biased, then the IV

estimator is likely even more so

If twins or siblings have identical abilities, then a within-family estimator will recover an asymptotically unbiased estimator

otherwise a within-family estimator will be biased the extent to which depends on the relative

importance the variance in schooling attributable to ability differences in families versus the population.

Take-Homes from the Static Model 2

Measurement errors biases are potentially important in interpreting the estimates from different procedures. OLS estimates are probably downward biased

by about 10% OLS estimates that control for family

background may be downward biased by about 15% or more

within-family differenced estimates may be downward-biased by 20-30% with the upper range more likely for identical twins.

Take-Homes from the Static Model 3

Empirical EstimatesAuthor Instrument OLS IV

Angrist and Krueger Quarter of birth .070(.000)

0.101(0.033)

Staiger and Stock (Quarter of birth)*(state)*(year) .063(.000)

.060(.030)

Kane and Rouse Tuition at 2 and 4 year state colleges and distance to nearest college

.080(.005)

.091(.033)

Card Distance to nearby 4-year college Distance*parent education

-- .097(.048)

Conneely and Uusitalo Indicator for living in university town in 1980

.085(.001)

.110(.024)

Malluccio Distance to local private school or high school

.073/.063(.011)/(.006)

.145/.113(.041)/(.033)

Harmon and Walker Changes in minimum schooling leaving age

.061(.001)

.153(.015)

Does this Explain Differences in Empirical Estimates? -1 Appears that IV-based studies estimate a

return to schooling that’s about 30% more than OLS estimates: Why?

1. Bound and Jaeger: • IV estimate are even further upward biased

than the corresponding OLS estimates by unobserved differences between the characteristics of treatment and comparison groups implicit in the IV scheme..

2. Ability bias is that OLS estimates of the return to schooling are relatively small

• the gaps between IV and OLS estimates reflect the downward bias in OLS estimates attributable to measurement error.

• Most likely: measurement error bias itself seems like it could only explain about 10% of the bias.

Does this Explain Differences in Empirical Estimates? -2

3. Publication bias: only want to publish papers with large and significant point estimates

• Ashenfelter and Harmon cite a positive correlation across studies between IV-OLS gap in estimated returns and the sampling error of the IV estimates.

Does this Explain Differences in Empirical Estimates? -3

4. Underlying heterogeneity: • Factors like compulsory schooling or accessibility

of schools are more likely to affect the schooling choices of individuals who would otherwise have relatively low schooling levels.

• If these individuals have higher than average marginal returns to schooling, then IV estimators based on compulsory schooling or school proximity should yield higher than average marginal returns.

Does this Explain Differences in Empirical Estimates? -4

General Conclusions1. Consistent with summary of the literature from the 60s and 70s

by Grilliches, • the average return to education in a given population is not

much below the estimate that emerges from a simple cross-sectional regression of earnings on education.

• The “best available” evidence from the latest studies of identical twins suggests a small upward bias of about 10% in the simple OLS estimates

2. Estimate of the return to schooling based on comparisons of brother or fraternal twins contain some positive ability bias • less than the corresponding OLS• ability differences appear to exert relatively less influence

on within-family schooling difference

General Conclusions 23. IV estimates of the return to education based on family

background are systematically higher than corresponding OLS estimates and may contain a bigger upward bias

4. Returns to education vary across the population with such observable factors as school quality and parental education

5. IV estimates of the return to education based on interventions in the school system tend to be 20% or more above the corresponding OLS estimates. There is some evidence that this is due to the higher than average marginal returns of the individuals targeted by these programs.

Next Week… Empirical Education Papers

Twins IV estimates

Some Education Production Function What are returns to various Education inputs Will post extra reading on course web-page