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Human Capital: Theory Lent Term Lecture 2 Dr. Radha Iyengar

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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 inputs 2. Building/infrastructure 3. land 4. human capital

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Education and “General” Human Capital We’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”)

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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!)

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A Static Model of Human Capital Acquisition (for details see: David Card, “Causal Effect of Education on Earnings” Handbook of Labor Economics)

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Basis of Empirical Estimates Common form of estimation: log( y ) = a + bS + cX + dX 2 + e(1) Usually called a Mincer Regression

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Some Empirical Facts 1. 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

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OLS Estimates 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.

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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.

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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)

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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)

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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 percent of the variation in earnings data.

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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)

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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:

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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)

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Results Optimal 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))

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Optimal Schooling a simple specification of these two components (define E(b) = b and E(r)= r and k 1, k 2 > 0) This gives us the optimal schooling expression:

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Interpretation of Equilibrium Individuals do not necessarily know the parameters of their earnings functions when they make their schooling choices. b i 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)

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Some Assumptions treat b i as known at the beginning of the lifecycle and fixed over time: assumption probably leads to some overstatement of the role of heterogeneity of b i in the determination of schooling and earnings outcomes. for simplicity, assume jointly symmetric distribution of b and r.

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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 slope

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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.

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Earnings and Schooling Eqn From equation 3A (FOC), we get Note that individual heterogeneity affects both the intercept and the slope Defining α i = a i + a 0 Use this with eqn (4), to define schooling choice in terms of a, b, and r (5)

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Linear Estimating Function Define λ 0 and ψ 0 as the parameters from the linear projection of a i and b i on where is E(S i ) (6a) (6b) That is:

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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:

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Homogeneous Returns Let b i = b and k 1 = 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.

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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.

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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 y 0 giving an approximately linear relationship.

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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 σ ra 0.

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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.

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Understanding Observed Linearity-3 Rewrite (5) and reorganize terms: This is linear if ψ 0 ≈ 2k 1 The bigger the contribution of b i to the overall variance of schooling, ψ 0 is bigger and the more the convexity

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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.

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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.

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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:

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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.

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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

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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 (F i ):

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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—b OLS OLS from bivariate regression of earnings on schooling and family background—b biv IV estimator using F i as an instrument for Si (b IV )

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Comparing Regressions: Heterogeneous case introducing heterogeneity across individuals in b, so that we can relate ψ 0 as follows Assuming,,

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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.

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Defining “Family Effect” Define “pure family effects” model as the a ij =a j and b ij =b j linear projection of a and b i – b on the observed schooling outcomes of the two family members:

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Estimating with “Family Effects” Assuming that b i, S 1i, S 2i have a jointly symmetric distribution Earning functions are then: Taking differences, a within family difference in log earnings model:

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When Family Effects Models Work With identical twins, it is natural to impose the symmetry conditions so that λ 1 =λ 2 =λ, ψ 1 =ψ 2 =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.

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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:

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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 λ 11 =λ 22, λ 12 =λ 21, ψ 12 =ψ 21, ψ 11 =ψ 21 From this, the earnings model eqn’s are: From this system, is not identifiable.

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“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.

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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

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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

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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

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Empirical Estimates AuthorInstrumentOLSIV Angrist and KruegerQuarter of birth.070 (.000) (0.033) Staiger and Stock(Quarter of birth)*(state)*(year).063 (.000).060 (.030) Kane and RouseTuition at 2 and 4 year state colleges and distance to nearest college.080 (.005).091 (.033) CardDistance to nearby 4-year college Distance*parent education (.048) Conneely and UusitaloIndicator for living in university town in (.001).110 (.024) MalluccioDistance to local private school or high school.073/.063 (.011)/(.006).145/.113 (.041)/(.033) Harmon and WalkerChanges in minimum schooling leaving age.061 (.001).153 (.015)

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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..

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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

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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

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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

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General Conclusions 1. 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

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General Conclusions 2 3. 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.

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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

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