Public-PRIVATE Pay Differentials
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standard practice.6 Maximum likelihood of the bivariate probit leads to four sample
selection correction terms
where P = Puv. Again, φ and Φ are the univariate standard normal density and
distribution functions respectively, and F the bivariate standard normal distribution
function. Equations (5) to (8) collapse to the usual univariate Heckman expressions
for Puv = 0.
λ i,pi
= Φ( Zγ)Φ
B∖μ - PZ'iγ
.(1 - P 2)1/2 ,
× F ( Bμ, z'â, ρ ) 1
λ i,s 1
=Φ( B',μ^)φ
Ziγ - PBiμ
.(1 - P 2)1/2 .
× F ( Biμ^, Z,iγ, P )
-1
λi , p 2
= Φ( ziγ)Φ
Biμ - pZ'â
(1 - P 2)1/2.
×F(—B',μ, Z'iγ-P) 1
(5)
(6)
(7)
λi,s2
= -φ( B'1μι)Φ
Z'iγ'- PB]∣μ
.(1 - P 2)1/2.
× F(-B '1μ, ZY-P) 1
(8)
Now, the wage equations (1) and (2) can be re-written as
''
E(ln w1,i | Xi,Pi = 1,Si = 1) = X1,iβ1 +σ11P1uλi, p1 +σ11P1vλi,s1 (9)
E(ln w2,i | Xi ,Pi = 1,Si = 0) = X2,iβ2 +σ22P2uλi, p2 +σ22P2vλi,s2 (10)
where the correction terms are according to (5) to (8).
6 Since the covariance of ε1 and ε2 is not identifiable in this model, the covariance matrix has
been split into two matrices (see Co et al. 1999 for details; see Koop and Poirier 1997 for a
discussion on these identification issues).