This is very useful since we can then use results


This is very useful since we can then use results from the treatment
effect literature.

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7
The wage structure effect ( Δ
S
) can be
interpreted as a treatment effect

The conditional independence assumption (E( ε |X)=0) usually
invoked in Oaxaca decompositions can be replaced by the weaker
ignorability assumption (D
g ╨
ε |X) to compute the aggregate
decomposition

For example, ability ( ε ) can be correlated with education (X) as long
as the correlation is the same in groups A and B.

This is the standard assumption used in “selection on observables”
models where matching methods are typically used to estimate the
treatment effect.

Main result: If we have Y
G
=m
G
(X, ε ) and ignorability, then:

Δ
S
solely reflects changes in the m(.) functions (ATET)

Δ
X
solely reflects changes in the distribution of X and ε
(ignorability key for this last result).
The wage structure effect ( Δ
S
) can be
interpreted as a treatment effect

A number of estimators for ATET= Δ
S
have been proposed in the
treatment effect literature

Inverse probability weighting (IPW), matching, etc.

Formal results exist, e.g. IPW is efficient for

ATET (Hirano, Imbens, and Ridder, 2003)

Quantile treatment effects (Firpo, 2007)

This has been widely used in the decomposition literature since
DiNardo, Fortin, and Lemieux (1996).
Formal derivation of the identification
result (Handbook chapter)
Formal derivation (2)

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8
Formal derivation (3) Some intuition about Proposition 1

The wage setting model we use, Y
g
=m
g
(X, ε ) is very general

Includes the linear model y
g
= x β
g
+ ε as a special case

There are three reasons why wages can be different between
groups g=A and g=B:

Differences in the wage setting equations m
a
(.) and m
b
(.)

Differences in the distribution of X for the two groups

Differences in the distribution of ε for the two groups

The ignorability assumption states that the distribution of ε given X is
the same for the two groups, though this does not mean that
E( ε |X)=0.

So once we control for differences between the X’s in the two
groups, we also implicitly control for differences in the ε ’s.

Only source of difference left is, thus, differences in the wage
structures m
a
(.) and m
b
(.).
A few caveats discussed in the chapter

This general result (Proposition 1) only works for the aggregate
decomposition. More assumptions have to be imposed to get at the
detailed decomposition.

We are implicitly ruling out general equilibrium effects under
assumption (simple counterfactual treatment).

For instance, in the absence of unions, wages in the nonunion
sector may change as firms are no longer confronted with the threat
of unionization

The wage structure observed among nonunion workers, m
a
(.), is no
longer a valid counterfactual for union workers.