Alexandru M. Lefter

Brian P. McCall

English

Research Papers in Economics

Decomposing Wage Distributions with Self-Selection
by
Alexandru M. Lefter
Brian P. McCall
University of Minnesota
May 20052
Introduction
There is ample empirical evidence that the wage structure in the United States changed markedly
during the 1980s and early 1990s (for a comprehensive survey of the literature, see Katz and
Autor, 1999). Despite unanimous agreement that an important dimension of this change is
related to the sharp increase in wage dispersion for both men and women, there has been some
debate in the labor economics field as to the roles played by various supply-side, demand-side,
and institutional factors in this rampant expansion of wage inequality (detailed discussions of
these factors can be found in Topel, 1997; Johnson, 1997; and Fortin and Lemieux, 1997). The
empirical strategies used to assess the relative contributions of these factors are typically based
on generalizations of the decomposition methodology developed by Blinder (1973) and Oaxaca
(1973). One shortcoming of most studies that investigated the determinants of the changes in
wage inequality is that they ignored possible changes in the selection process that led some
individuals to become part of the labor force and others to stay out of it. In this paper we propose
semi-parametric methods that allow assessing the effect of the changes in these self-selection
mechanisms on the changes in the entire distribution of wages.
Preliminaries
Since we will be modeling the wage distribution in terms of its hazard function, it is useful to
begin our discussion with a brief overview of the properties of hazard functions. The main
objective of this outline is to show that the probability density function of a random variable that
meets certain general conditions is nothing but a simple transformation of its hazard function. It
is worth noting here that although the hazard function is generally used for response variables
that come in the form of a duration, wage, earnings and income variables have similar properties
(approximately continuous distributions over positive values), and therefore can be analyzed
using all techniques available in the duration analysis literature. The benefits of such an
empirical strategy are beginning to be acknowledged (see, for example, Donald, Green, and
Paarsch, 2000).
Let 0 T  represent a random duration variable, and t a particular value of T. The cumulative
distribution function (c.d.f.) of T is given by ( ) ( ), F t T t    where 0 t  . The survivor function is
defined as ( ) 1 ( ) ( ). S t F t T t      In other words, S(t) represents the probability that an event
has not occurred by time t, o r   t h a t   t h e   i n d i v i d u a l   h a s   “ s u r v i v e d ”   p a s t t. Throughout this section,
we assume that T is continuous and denote the probability density function (p.d.f.) of T
by
( )
( ) .
dF t
f t
dt
 For 0, t  ( | ) t T t t T t       is the probability of leaving the initial state
in the interval [t, t +  t) given survival until time t. The hazard function for T is defined as:
0
( | )
( ) lim
t
t T t t T t
t
t


     


(1)
Thus, the hazard function is the instantaneous rate of leaving per unit of  t i m e   ( t h e   “ e s c a p e ”   r a t e ) .  
From equation ( 1 )   i t   f o l l o w s   t h a t ,   f o r   “ s m a l l ”    t, ( | ) ( ) . t T t t T t t t          The hazard can3
then be used to approximate a conditional probability in much the same way that the height of
the p.d.f. of T can be used to approximate an unconditional probability. We can express the
hazard function in terms of the p.d.f. and c.d.f. of T very simply. First, note that:
( ) ( ) ( )
( | )
( ) 1 ( )
t T t t F t t F t
t T t t T t
T t F t
       
       
  
(2)
When the c.d.f. is differentiable, we can take the limit of the right hand side of equation (2),
divided by  t, as  t approaches zero from above:
0
( ) ( ) 1 ( ) ( )
( ) lim
1 ( ) 1 ( ) ( ) t
F t t F t f t f t
t
t F t F t S t


  
   
  
Because the derivative of S(t) is – f(t), we have:
ln ( )
( )
d S t
t
dt
   (3)
Integrating (3) and using the fact that F(0) = 0, we can get the c.d.f. of T in terms of the hazard
function:
0 ( ) 1 exp ( )
t
F t s ds            (4)
Straightforward differentiation of (4) gives the p.d.f. of T in terms of the hazard function:
0 ( ) ( )exp ( )
t
f t t s ds            (5)
Therefore, any probability can be computed using the hazard function. Equation (5) is
particularly important for the empirical methods proposed in this paper since it shows that the
probability density function of a positive random variable can be easily recovered from its hazard
function. This suggests that once selectivity-corrected estimates of the parameters of a hazard
function are obtained, it is possible to look at the effects of self-selection on the entire
distribution of the variable of interest.
Methods for Decomposing Wage Distribution Changes
In this section we will use the results from hazard models to develop semi-parametric methods
that can be used to decompose changes in wage distributions while accounting for self-selection.
We assume that the distribution of wages w depends on a J-dimensional vector of observed
regressors x and an unobserved variable v. Rewriting the conditional distribution of wages in
terms of (5), we have:4
0 ( | , ) ( | , ) ( | , ) ( | , )exp ( | , )
w
f w v w v S w v w v s v ds              x x x x x (6)
We will parameterize the hazard function according to the mixed proportional hazard model with
time-constant covariates and time-varying coefficients (for details, see McCall, 1996):
  ( | , ) exp ( ) ( ) w v v w w      x β x (7)
where ()  is a continuous function and exp[ (w)] is the baseline hazard, ()  β is a J-dimensional
vector of continuous functions measuring the effects of the observed regressors x, and v is a
nonnegative, unobserved, random variable distributed independently of x.
Substituting (7) into (6) yields:
      0 ( | , ) exp ( ) ( ) exp exp ( ) ( )
w
f w v v w w v s s ds          x β x β x (8)
To see how the unobserved variable v affects the wage distribution, we can use integration by
parts to obtain the following equation:
0 ( | , ) ( | , ) w v S w v dw

  x x (9)
Thus,
          0 0 0 0 ( | , ) exp exp ( ) ( ) exp exp ( ) ( )
v
w w
w v v s s ds dw s s ds dw  
 
             x β x β x (10)
Differentiating (10) with respect to v gives:
     
         
   
0
0
0 0 0
0
exp exp ( ) ( )
( | , )
exp exp ( ) ( ) exp ( ) ( )
( | , 1) ln ( | , 1)
v
w
v
w w
v
d s s ds
d w v
dw
dv dv
s s ds s s ds dw
S w v S w v dw

 



  

 
      
  


  

β x
x
β x β x
x x
(11)
Evaluating (11) at v = 1 and using integration by parts yields:
 
0
1
( | , ) 1
( | , 1)ln ( | , 1)
4 v
d w v
S w v S w v dw
dv



     
x
x x5
In other words, at v = 1, a small change in v reduces the mean wage independently of x.
As is common in the selection literature, we model an individual’ s labor force participation
decision by an index function c. More specifically, we assume that the wage is observed when
c = 1 and not observed when c = 0, where:
  ( 1| , ) 1 exp exp( ) c         z π z (12)
and z is an L-dimensional vector of regressors,  is an L-dimensional vector of parameters
measuring the effect of the regressors on the probability of working, and is an unobserved
random variable. The possibility of selection bias arises when the unobserved variables and v
are correlated. We denote the joint cumulative distribution function of the unobservables and v
by G( , v).
The log-likelihood function for the resulting selectivity-corrected model can be written as:
   
             
1
1
0
ln( ) ln exp exp( )
1 exp exp( ) exp ( ) ( ) exp exp ( ) ( ) ( , )
i
i
N c
i
i v
c
w
i i i
L
v w w v s s ds dG v


   


   
        
  

π z
π z β x β x
Maximizing this likelihood function gives estimates of  (w),  (w),  , and G( , v), which can be
used to estimate selectivity-corrected probability density functions. McCall (1996) presented
conditions under which the parameters of the mixed proportional hazard model with time-
varying coefficients can be identified. The only additional condition required for identifiablility
when self-selection is incorporated in this model is that the vector z contains at least one variable
that is not included in the vector x.
To illustrate how the empirical strategy outlined above can be used to decompose changes in
wage distributions, consider the following selectivity-corrected probability density functions,
where the subscripts 1 and 2 indicate time periods:
        1 1 1 1 1 1 1 1 1 1 1 1 1 0 ( | , , 1) exp ( ) ( ) exp exp ( ) ( ) | , 1
w
v
f w c v w w v s s ds dG v c             z x β x β x z
        2 2 2 2 2 2 2 2 2 2 2 2 2 0 ( | , , 1) exp ( ) ( ) exp exp ( ) ( ) | , 1
w
v
f w c v w w v s s ds dG c              z x β x β x z
Denote the joint cumulative distribution function of the observables z and x at time 1 and time 2
by 1( , ) Q z x and 2( , ), Q z x respectively. Then, the unconditional distributions of wages are:
        1 1 1 1 1 1 1 1 1 0
,
( | 1) exp ( ) ( ) exp exp ( ) ( ) | , 1 ( , )
w
v
f w c v w w v s s ds dG v c dQ  
 
         
   
z x
β x β x z zx6
        2 2 2 2 2 2 2 2 2 0
,
( | 1) exp ( ) ( ) exp exp ( ) ( ) | , 1 ( , )
w
v
f w c v w w v s s ds dG v c dQ  
 
         
   
z x
β x β x z zx
The differences between these two unconditional distributions can be investigated using a
generalized version of the Blinder/Oaxaca decomposition methodology (for details, see DiNardo,
Fortin, and Lemieux, 1996):
1 1 1 1
1 1 1 1 1 1 1 1
,
2 2 1 1 2 2 2 2 2 2 2 2
, , , , ,
2 2 2 2 2 2 1 2
( | 1) ( | 1) ( | 1) ( | 1) ( | 1) ( | 1)
( | 1) ( | 1) ( | 1) ( | 1)
Q Q Q G
Q G Q G Q G
f w c f w c f w c f w c f w c f w c
f w c f w c f w c f w c
                  
               
β β
where, for example,
1 1 ,
2 2 ( | 1)
Q G f w c  is a counterfactual probability density function showing what
the distribution of wage would have looked like had the joint distributions of the observables and
unobservables remained unchanged between time 1 and time 2. More specifically:
       
1 1 ,
2 2 2 2 2 2 1 1 1 0 ( | 1) exp ( ) ( ) exp exp ( ) ( ) | , 1 ( , )
w Q G
v
f w c v w w v s s ds dG v c dQ  
 
         
     β x β x z zx
A clear advantage of this semi-parametric decomposition technique is to be found in its focus on
the entire density of wages, which allows examining the effects of self-selection at any point in
the wage distribution. Its main limitation is that the order of the decomposition can influence the
magnitude of effects of interest, although this problem can be addressed by reversing the order of
the decomposition. Another limitation is related to the absence of exact methods for computing
standard errors, but this can be dealt with using a bootstrap procedure.
Conclusions
The purpose of this paper was to discuss a semi-parametric methodology that can be used to
investigate changes in wage distributions that are due to changes in labor force participation
decisions over time. This methodology borrows techniques from the duration analysis literature
and has the advantage of modeling the effects of self-selection on the entire distribution of
wages, and not just the mean or some pre-specified percentiles thereof, as is common in
regression analysis or quintile regression frameworks. Although the discussion was limited to
differences within the same wage distribution between two point in time, with an application to
changes in wage inequality, it is worth mentioning that it can be easily extended to explore
differences between two wage distributions at the same point in time, with an application to
male-female wage differentials. Furthermore, it can be used to analyze changes over time in the
differences between two wage distributions, which would allow examining the effects of the
changing patterns of self-selection on the narrowing of the gender gap, another important
dimension of the change that characterized the U.S. wage structure during the 1980s and early
1990s (Blau, 1998).7
References
B l a u ,   F .   ( 1 9 9 8 ) ,   “ T r e nds in the Well-Being of American Women, 1970-1 9 9 5 , ”   Journal of
Economic Literature, 36(1), 112-165.
Blinder, A.S. (1973) "Wage Discrimination: Reduced Form and Structural Estimates,"
Journal of Human Resources, 8, 436-455.
Dinardo, J., N. Fortin, and T.   L e m i e u x   ( 1 9 9 6 ) ,   “ L a b o r   Ma r k e t   I n s t i t u t i o n s   a n d   t h e  
Distribution of Wages, 1973-1992: A Semi-P a r a m e t r i c   A p p r o a c h , ”   Econometrica, 64(5),
1001-1044.
D o n a l d   S . J . ,   G r e e n ,   D . A . ,   a n d   H . J .   P a a r s c h   ( 2 0 0 0 )   ,   “ D i f f e r e n c e s   i n   Wa g e   D i s t r i b u t i o n s  
between Canada and the United States: An Application of a Flexible Estimator of
D i s t r i b u t i o n   F u n c t i o n s   i n   t h e   P r e s e n c e   o f   C o v a r i a t e s , ”   Review of Economic Studies, 67,
609-633.
J o h n s o n ,   G .   ( 1 9 9 7 ) ,   “ C h a n g e s   i n   E a r n i n g s   I n e q u a l i t y :   T h e   R o l e   o f   D e m a n d   S h i f t s , ”  
Journal of Economic Perspectives, 11(2), 41-54.
Katz, L. and D. Autor (1999),  “ C h a n g e s   i n   t h e   Wa g e   S t r u c t u r e   a n d   E a r n i n g s   I n e q u a l i t y , ”  
in Handbook of Labor Economics Vol. 3, O. Ashenfelter and D. Card Eds. (North
Holland: Amsterdam), 1463-1555.
Fortin, N., and T. Lemieux  ( 1 9 9 7 ) ,   “ I n s t i t u t i o n a l   C h a n g e   a n d   R i s i n g   Wa g e   I n e q u a l i t y :   I s  
t h e r e   a   L i n k a g e ? , ”   Journal of Economic Perspectives, 11(2), 75-96.
Mc C a l l ,   B . P .   ( 1 9 9 6 ) ,   “ T h e   I d e n t i f i a b i l i t y   o f   t h e   Mi x e d   P r o p o r t i o n a l   H a z a r d s   Mo d e l   w i t h  
Time-V a r y i n g   C o e f f i c i e n t s , ”   Econometric Theory, 12, 733-738.
O a x a c a ,   R .   ( 1 9 7 3 ) ,   “ Ma l e -F e m a l e   Wa g e   D i f f e r e n t i a l s   i n   U r b a n   L a b o r   Ma r k e t s , ”  
International Economic Review, 14, 693-709.
T o p e l ,   R . H .     ( 1 9 9 7 ) ,   “ F a c t o r   P r o p o r t i o n s   a n d   R e l a t i v e   Wa g e s :   T h e   S u p p l y -Side
D e t e r m i n a n t s   o f   Wa g e   I n e q u a l i t y , ”   Journal of Economic Perspectives, 11(2), 54-74.

C h a n g e s i n t h e Wa g e S t r u c t u r e a n d E a r n i n g s I n e q u a l i t y , ”

e m i e u x

Wage Discrimination: Reduced Form and Structural Estimates,&quot;

Decomposing Wage Distributions with Self-Selection

http://www.legacy-irc.csom.umn.edu/RePEC/hrr/papers/0705.pdf

Decomposing Wage Distributions with Self-Selection

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics