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 nonunion 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 nonunion 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 socalled threefold 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.