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 1 / 24

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. 2 / 24

Objectives Notation Model Results Notation Y ij : potential lifetime earnings of person i if schooling level j is chosen. X ij : vector of observed talent or ability indicators of person i . τ i : unobserved talent component relevant for person i (unobserved by the econometrician). Z i : family background (proxies of financial constraints). ω i : taste effects. V ij : value of choosing school level j for person i . 3 / 24

Objectives Notation Model Results Thus = y j ( X i , τ i ) j = 1 , .., n Y ij = g ( y j , Z i , ω i ) , V ij and i belongs to j if V ij = max ( V i 1 , ..., V in ) . Finally, ( τ, ω ) ∼ F ( τ, ω ) . 4 / 24

Objectives Notation Model Results The Model If person i chooses a , y ai ( t ) = 0, 0 < t ≤ S y ai ( t ) = y ai exp [ g ai ( t − S )] , S ≤ t < ∞ , where S is the incremental schooling period associated with a over b . t − S is market experience. 5 / 24

Objectives Notation Model Results If alternative b is chosen, y bi ( t ) = y bi exp [ g bi t ] , 0 ≤ t < ∞ . Therefore, � � � ∞ y ai V ai = y ai ( t ) exp ( − r i t ) dt = exp ( − r i S ) ( r i − g ai ) S � ∞ y ai V bi = y bi ( t ) exp ( − r i t ) dt = ( r i − g bi ) . 0 6 / 24

Objectives Notation Model Results Assumptions (1) Infinite horizon. (2) Constant rate of discount for each person, such that r i > g ai , g bi . (3) Ignore direct costs of school. 7 / 24

Objectives Notation Model Results Selection Selection Rule � V ai � I i = ln V bi = ln y ai − ln y bi − r i S − ln ( r i − g ai ) + ln ( r i − g bi ) A Taylor approximation to the nonlinear terms around their population mean values ( g a , g b , r ) yields I i = α 0 + α 1 ( ln y ai − ln y bi ) + α 2 g ai + α 3 g bi + α 4 r i . (1) 8 / 24

Objectives Notation Model Results Selection In Equation (1): α 1 = 1 ∂ I 1 α 2 = = ( r − g a ) > 0 ∂ g a ∂ I 1 α 3 = = − ( r − g b ) < 0 ∂ g b � � ( g a − g b ) α 4 = − S + ( r − g a )( r − g b ) The selection criterion is Pr ( choose a ) = Pr ( V ai > V bi ) = Pr ( I > 0 ) Pr ( choose b ) = Pr ( V ai ≤ V bi ) = Pr ( I ≤ 0 ) . 9 / 24

Objectives Notation Model Results Earnings/Discount Earnings and Discount Functions ln y ai = X i β a + u 1 i g ai = X i γ a + u 2 i ln y bi = X i β b + u 3 i g bi = X i γ b + u 4 i r i = Z i δ + u 5 i ( u 1 i , u 2 i , u 3 i , u 4 i , u 5 i ) ∼ N ( 0 , Σ) 10 / 24

Objectives Notation Model Results Reduced Reduced Form I i = α 0 + X [ α 1 ( β a − β b ) + α 2 γ a + α 3 γ b ] + α 4 Z δ + α 1 ( u 1 − u 3 ) + α 2 u 2 + α 3 u 4 + α 5 u 5 = W π − ǫ , where W = [ X , Z ] and − ǫ = α 1 ( u 1 − u 3 ) + α 2 u 2 + α 3 u 4 + α 5 u 5 . 11 / 24

Objectives Notation Model Results Reduced Thus Pr ( a is observed ) = Pr ( W π > ǫ ) � W π � = F , σ ǫ where F ( . ) is the standard normal c.d.f. ⇒ Probit Model. 12 / 24

Objectives Notation Model Results Bias Selection Bias and Earnings Functions Pr [ observing y a ( t )] = Pr ( I > 0 ) = Pr ( W π > ǫ ) . Thus E ( ln y a | I > 0 ) = X β a + E ( u 1 | W π > ǫ ) = X β a + σ 1 ρ 1 λ a , σ 1 ǫ where σ 2 1 = Var ( u 1 ) , ρ 1 = σ 1 σ ǫ , and � W π � − f σ ǫ λ a = � W π � . F σ ǫ 13 / 24

Objectives Notation Model Results Bias λ a represents the truncated mean (with truncation point W π σ ǫ ) . It is commonly called the Mills Ratio. Furthermore, 0 ) = X γ a + σ 2 ǫ E ( g a | I > λ a σ ǫ 0 ) = X β b + σ 3 ǫ E ( ln y b | I ≤ λ b σ ǫ 0 ) = X γ b + σ 4 ǫ E ( g b | I ≤ λ b , σ ǫ where � W π � f σ ǫ � W π � . λ b = 1 − F σ ǫ 14 / 24

Objectives Notation Model Results Estimation Estimation X β a + β ∗ ln y a = a λ a + η 1 X γ a + γ ∗ g a = a λ a + η 2 X β b + β ∗ ln y b = b λ b + η 3 X γ b + γ ∗ g b = b λ b + η 4 λ a and λ b are not known. Apply a standard two-step procedure. 15 / 24

Objectives Notation Model Results Estimation From the probit model we compute � � W i � − f π/σ ǫ � λ ai = � � W i � F π/σ ǫ � � W i � f π/σ ǫ � λ bi = � � . W i � 1 − F π/σ ǫ These expression can be used in the estimation of the earning equations. This produces consistent estimates for β a , β b , γ a , and γ b . 16 / 24

Objectives Notation Model Results Estimation Finally, using the consistent estimates � y a i � � X i ( � β a − � ln = β b ) y b i � X i � g ia = γ a � X i � g ib = γ b to estimate the model Pr ( choose a ) � �   α 1 ( � β a − �   α 0 + X β b ) + α 2 � γ a + α 3 � γ b + α 4 Z δ   > ǫ     = Pr   .     σ ǫ σ ǫ (2) 17 / 24

Objectives Notation Model Results Estimation Model (2) is used to test the economic restrictions: α 1 = 1 ∂ I 1 α 2 = = ( r − g a ) > 0 ∂ g a ∂ I 1 α 3 = = − ( r − g b ) < 0 ∂ g b � � ( g a − g b ) α 4 = − S + ( r − g a )( r − g b ) 18 / 24

Objectives Notation Model Results Other Other Tests From ln y ai = X i β a + u 1 i g ai = X i γ a + u 2 i we obtain ln y ai ( t ) = X i ( β a + t γ a ) + u 1 i + tu 2 i . Similarly, ln y bi ( t ) = X i ( β b + t γ b ) + u 3 i + tu 4 i . 19 / 24

Objectives Notation Model Results Other Therefore Pr ( choose a ) � θ 0 + θ 1 [ ln y a ( t ) − ln y b ( t )] + θ 2 g a + θ 3 g b + θ 4 r � > ǫ = Pr . σ ǫ σ ǫ [ ln y a ( t ) − ln y b ( t )] = ln y a − ln y b + ( g a − g b ) t − g a S This implies the restrictions: θ 1 = α 1 ( t − S ) θ 1 + θ 2 = α 2 − t θ 1 + θ 3 = α 3 These represent a check of the validity of the model. 20 / 24

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. 21 / 24

Objectives Notation Model Results Identification β a , β b , γ a and γ b are estimated from the selectivity-bias-corrected earnings equations. α 1 /σ ǫ , α 2 /σ ǫ , α 3 /σ ǫ , α 4 δ/σ ǫ are obtained from the structural probit. There are 15 additional parameters in the variance-covariance matrix Σ . However, � W i π � σ j + σ j ǫ λ ai − λ 2 var ( η ij ) = if j = 1 , 2 ai σ ǫ σ ǫ � W i π � σ j + σ j ǫ λ bi − λ 2 var ( η ij ) if j = 3 , 4 . = bi σ ǫ σ ǫ 22 / 24

Objectives Notation Model Results Identification Similar expressions hold for Cov ( η i 1 , η i 2 ) and Cov ( η i 3 , η i 4 ) . Since, it is possible to estimate σ j , two within-group covariances, and four covariances σ j ǫ for j = 1 , .., 4, we have 11 statistics (including σ ǫ ) to estimate 15 parameters. Consequently, we cannot estimated all the covariances without further assumptions. 23 / 24

Objectives Notation Model Results Results 24 / 24