Education and Self-Selection Robert Willis and Sherwin Rosen Journal - - PowerPoint PPT Presentation

Objectives Notation Model Results

Education and Self-Selection

Robert Willis and Sherwin Rosen Journal of Political Economy, 1979, vol. 87 James Heckman University of Chicago ECON 345 April 16, 2007

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Objectives Notation Model Results

Objectives of the Paper Specification and estimation of a model of demand for college education. To estimate life earnings conditioned on actual school choices taking into account selection bias. To determine the extent to which alternative earnings prospects influence the decision to attend college.

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Objectives Notation Model Results

Notation

Yij: potential lifetime earnings of person i if schooling level j is chosen. Xij : vector of observed talent or ability indicators of person i. τi : unobserved talent component relevant for person i (unobserved by the econometrician). Zi : family background (proxies of financial constraints). ωi : taste effects. Vij : value of choosing school level j for person i.

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Objectives Notation Model Results

Thus Yij = yj(Xi, τi) j = 1, .., n Vij = g(yj, Zi, ωi) , and i belongs to j if Vij = max(Vi1, ..., Vin) . Finally, (τ, ω) ∼ F(τ, ω) .

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Objectives Notation Model Results

The Model If person i chooses a, yai(t) = 0, 0 < t ≤ S yai(t) = yai exp[gai(t − S)], S ≤ t < ∞ , where S is the incremental schooling period associated with a over b. t − S is market experience.

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Objectives Notation Model Results

If alternative b is chosen, ybi(t) = ybi exp[gbit], 0 ≤ t < ∞ . Therefore, Vai = ∞

S

yai(t) exp(−rit)dt =

• yai

(ri − gai)

• exp(−riS)

Vbi = ∞ ybi(t) exp(−rit)dt = yai (ri − gbi) .

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Objectives Notation Model Results

Assumptions

(1) Infinite horizon. (2) Constant rate of discount for each person, such that ri > gai, gbi. (3) Ignore direct costs of school.

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Objectives Notation Model Results Selection

Selection Rule Ii = ln Vai Vbi

• =

ln yai − ln ybi − riS − ln(ri − gai) + ln(ri − gbi) A Taylor approximation to the nonlinear terms around their population mean values (ga, gb, r) yields Ii = α0 + α1 (ln yai − ln ybi) + α2gai + α3gbi + α4ri . (1)

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Objectives Notation Model Results Selection

In Equation (1):

α1 = 1 α2 = ∂I ∂ga = 1 (r − ga) > 0 α3 = ∂I ∂gb = − 1 (r − gb) < 0 α4 = −

• S +

(ga − gb) (r − ga)(r − gb)

• The selection criterion is

Pr(choose a) = Pr(Vai > Vbi) = Pr(I > 0) Pr(choose b) = Pr(Vai ≤ Vbi) = Pr(I ≤ 0) .

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Objectives Notation Model Results Earnings/Discount

Earnings and Discount Functions ln yai = Xiβa + u1i gai = Xiγa + u2i ln ybi = Xiβb + u3i gbi = Xiγb + u4i ri = Ziδ + u5i (u1i, u2i, u3i, u4i, u5i) ∼ N(0, Σ)

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Objectives Notation Model Results Reduced

Reduced Form Ii = α0 + X [α1(βa − βb) + α2γa + α3γb] + α4Zδ + α1(u1 − u3) + α2u2 + α3u4 + α5u5 = Wπ − ǫ , where W = [X, Z] and −ǫ = α1(u1 − u3) + α2u2 + α3u4 + α5u5 .

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Objectives Notation Model Results Reduced

Thus Pr(a is observed) = Pr(Wπ > ǫ) = F Wπ σǫ

• ,

where F(.) is the standard normal c.d.f. ⇒ Probit Model.

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Objectives Notation Model Results Bias

Selection Bias and Earnings Functions Pr [observing ya(t)] = Pr(I > 0) = Pr(Wπ > ǫ). Thus E(ln ya|I > 0) = Xβa + E(u1|Wπ > ǫ) = Xβa + σ1ρ1λa , where σ2

1 = Var(u1), ρ1 = σ1ǫ σ1σǫ, and

λa = −f Wπ

σǫ

• F

Wπ

σǫ

.

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Objectives Notation Model Results Bias

λa represents the truncated mean (with truncation point Wπ

σǫ ).

It is commonly called the Mills Ratio. Furthermore,

E(ga|I > 0) = Xγa + σ2ǫ σǫ λa E(ln yb|I ≤ 0) = Xβb + σ3ǫ σǫ λb E(gb|I ≤ 0) = Xγb + σ4ǫ σǫ λb ,

where

λb = f Wπ

σǫ

• 1 − F

Wπ

σǫ

.

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Objectives Notation Model Results Estimation

Estimation ln ya = Xβa + β∗

aλa + η1

ga = Xγa + γ∗

aλa + η2

ln yb = Xβb + β∗

bλb + η3

gb = Xγb + γ∗

bλb + η4

λa and λb are not known. Apply a standard two-step procedure.

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Objectives Notation Model Results Estimation

From the probit model we compute

• λai

= −f

• Wi

π/σǫ

• F
• Wi

π/σǫ

• λbi

= f

• Wi

π/σǫ

• 1 − F
• Wi

π/σǫ . These expression can be used in the estimation of the earning equations. This produces consistent estimates for βa, βb, γa, and γb.

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Objectives Notation Model Results Estimation

Finally, using the consistent estimates ln

• yai

ybi

• =

Xi( βa − βb)

• gia

= Xi γa

• gib

= Xi γb to estimate the model Pr(choose a) = Pr         α0 + X

• α1(

βa − βb) + α2 γa + α3 γb

• + α4Zδ

σǫ > ǫ σǫ         . (2)

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Objectives Notation Model Results Estimation

Model (2) is used to test the economic restrictions: α1 = 1 α2 = ∂I ∂ga = 1 (r − ga) > 0 α3 = ∂I ∂gb = − 1 (r − gb) < 0 α4 = −

• S +

(ga − gb) (r − ga)(r − gb)

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Objectives Notation Model Results Other

Other Tests From ln yai = Xiβa + u1i gai = Xiγa + u2i we obtain ln yai(t) = Xi (βa + tγa) + u1i + tu2i . Similarly, ln ybi(t) = Xi (βb + tγb) + u3i + tu4i .

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Objectives Notation Model Results Other

Therefore

Pr(choose a) = Pr θ0 + θ1 [ln ya(t) − ln yb(t)] + θ2ga + θ3gb + θ4r σǫ > ǫ σǫ

• .

[ln ya(t) − ln yb(t)] = ln ya − ln yb + (ga − gb) t − gaS This implies the restrictions: θ1 = α1 (t − S)θ1 + θ2 = α2 −tθ1 + θ3 = α3 These represent a check of the validity of the model.

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Objectives Notation Model Results Identification

Identification Exclusion restriction:X and Z must have elements that are not in common. Even if Z = X, identification could be reached from nonlinearity. Heckman (1979): The effect of specification error in selection models.

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Objectives Notation Model Results Identification

βa, βb, γa and γb are estimated from the selectivity-bias-corrected earnings equations. α1/σǫ, α2/σǫ, α3/σǫ, α4δ/σǫ are obtained from the structural probit. There are 15 additional parameters in the variance-covariance matrix Σ. However, var(ηij) = σj + σjǫ σǫ Wiπ σǫ λai − λ2

ai

• if j = 1, 2

var(ηij) = σj + σjǫ σǫ Wiπ σǫ λbi − λ2

bi

• if j = 3, 4 .

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Objectives Notation Model Results Identification

Similar expressions hold for Cov(ηi1, ηi2) and Cov(ηi3, ηi4). Since, it is possible to estimate σj, two within-group covariances, and four covariances σjǫ for j = 1, .., 4, we have 11 statistics (including σǫ) to estimate 15 parameters. Consequently, we cannot estimated all the covariances without further assumptions.

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Objectives Notation Model Results

Results

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