Why
does
DFL
only
do
Step
1?
Because
the
DFL
procedure
simply
reweights
the
function
n
mapping
X
n
s
to
w s
(given
by
β
t
b
, β
t
w
in
our
QR
model)
using
the
change
in
the
density
of
X
n
s
between
t
and
τ
(
g
t
(
X
)
to
g
τ
(
X
)).
Since
DFL
do
not
explicitly
model
’prices’
(
β
t
b
, β
t
w
in
the
quantile
model),
they
do
not
take
Steps
2
and
3
where
these
prices
are
varied.
Similarly,
the
Lemieux
(2004)
procedure
for
reweighting
residual
densities
can
be
written
as
Δ
Q
L
θ
=
Q
θ
f
g
τ
(
X
)
, β
t
b
= 0
, β
t
w
−
Q
θ
f
g
t
(
X
)
, β
t
b
= 0
, β
t
w
= Δ
Q
θ
X
.
Unlike
the
quantile
approach,
Lemieux
estimates
β
t
w
using
OLS.
But
this
difference
is
unlikely
to
be
important.
5.6.3
Advantages
and
disadvantages
of
the
quantile
decomposition
relative
to
other
approaches
An
advantage
of
the
QR
approach
is
that
it
nests
JMP,
DFL,
and
all
extensions
to
DFL
that
have
been
recently
proposed.
I
would
also
argue
that
it
handles
each
of
these
cases
somewhat
more
transparently
than
the
competing
techniques.
A
second
virtue
of
QR
is
that
procedure
explicitly
models
the
separate
roles
of
quantities,
and
between and
withingroup
prices
to
overall
inequality.
That
is,
DFL
and
extensions
never
explicitly
estimate
prices,
although
these
prices
are
implicit
in
the
tool.
By
contrast,
JMP
do
estimate
prices
(both
observed
and
unobserved).
But
in
practice,
their
residual
pricing
function
does
not
quite
work
as
advertised
unless
one
conditions
the
residual
distribution
(
F
t
(
θ

X
it
))
very
finely
on
all
combinations
of
X
n
s
.
A
third
virtue
of
QR
is
that
it
satisfies
the
addingup
property.
That
is,
if
the
QR
model
fits
the
data
well,
the
sum
of
the
components
of
the
decomposition
will
add
up
to
the
total.
(Of
course,
it
is
still
a
sequential
decomposition;
the
order
of
operations
matters.)
Finally,
unlike
JMP
and
extensions,
QR
provides
a
consistent
treatment
of
between and
withingroup
prices
(there
is
no
’hybridization’
of
OLS
and
logit/probit
models).
The
QR
decomposition
has
two
notable
disadvantages.
First,
it
is
parametric.
The
precision
of
the
simulation
will
depend
on
the
fit
of
the
QR
model
which
in
turn
depends
on
the
characteristics
of
the
data
and
the
richness
of
the
QR
model.
By
contrast,
the
DFL
procedure
and
its
extensions
never
actually
parameterize
the
conditional
distribution
of
wages,
F
(
w

X
).
Hence,
the
treatment
of
F
(
w

X
)
in
DFL
is
fully
nonparametric.
Notably,
DFL
must
parameterize
the
reweighting
function
(through
the
probit/logit).
I’ve
never
seen
any
work
documenting
the