Is conditionally independent of x y gi g k x k 1 x ik

is conditionally independent of X :
Y
gi
= ±
g 0
+
K X
k =1
X
ik
±
gk
+ °
gi
; g = A; B; (1)
where E ( °
gi
j X
i
) = 0 , and X is the vector of covariates ( X
i
= [ X
i 1
; ::; X
iK
] ). As is well
known, the overall di/erence in average outcomes between group B and A ,
b
°
°
O
= Y
B
° Y
A
;
3

can be written as: 3
b
°
°
O
=
(
b
±
B 0
°
b
±
A 0
) +
K
X
k =1
X
Bk
°
b
±
Bk
°
b
±
Ak
±
| {z }
b
°
°
S
(Unexplained)
+
K
X
k =1
²
X
Bk
° X
Ak
³
b
±
Ak
| {z }
b
°
°
X
(Explained)
where
b
±
g 0
and
b
±
gk
( k = 1 ; ::; K ) are the estimated intercept and slope coe¢ cients, re-
spectively, of the regression models for groups g = A; B . The °rst term in the equation
is what is usually called the ±unexplained² e/ect in Oaxaca decompositions. Since we
mostly focus on wage decompositions in this chapter, we typically refer to this °rst ele-
ment as the ±wage structure²e/ect ( °
°
S
). The second component, °
°
X
, is a composition
e/ect, which is also called the ±explained² e/ect (by di/erences in covariates) in OB
decompositions.
In the above decomposition, it is straightforward to compute both the overall composi-
tion and wage structure e/ects, and the contribution of each covariate to these two e/ects.
Following the existing literature on decompositions, we refer to the overall decomposition
(separating °
°
O
in its two components °
°
S
and °
°
X
) as an aggregate decomposition . The
detailed decomposition involves subdividing both °
°
S
, the wage structure e/ect, and °
°
X
,
the composition e/ect, into the respective contributions of each covariate, °
°
S;k
and °
°
X;k
,
for k = 1 ; ::; K .
The chapter is organized around the following ±take away²messages:
A. The wage structure e/ect can be interpreted as a treatment e/ect
This point is easily seen in the case where group B consists of union workers, and
group A consists of non-union workers. The raw wage gap ° ° can be decomposed as
the sum of the ±e/ect² of unions on union workers, °
°
S
, and the composition e/ect
linked to di/erences in covariates between union and non-union workers, °
°
X
. We can
3
The decomposition can also be written by exchanging the reference group used for the wage structure
and composition e/ects as follows:
b
°
°
O
=
´
(
b
°
B 0
°
b
°
A 0
) +
K
P
k =1
X
Ak
°
b
°
Bk
°
b
°
Ak
±
µ
+
´
K
P
k =1
²
X
Bk
° X
Ak
³
b
°
Bk
µ
.
Alternatively, the so-called three-fold decomposition uses the same reference group for both ef-
fects, but introduces a third interaction term:
b
°
°
O
=
´
(
b
°
B 0
°
b
°
A 0
) +
K
P
k =1
X
Ak
°
b
°
Bk
°
b
°
Ak
±
µ
+
´
K
P
k =1
²
X
Bk
° X
Ak
³
b
°
Ak
µ
+
´
K
P
k =1
°
X
Bk
° X
Ak
)
°
b
°
Bk
°
b
°
Ak
±±
µ
. While these various versions of the
basic decomposition are used in the literature, using one or the other does not involve any speci°c esti-
mation issues. For the sake of simplicity, we thus focus on the one decomposition introduced in the text
for most of the chapter.