Notice that this decomposition sums to the total

Notice that this decomposition sums to the total observed change: Δ Q X
θ
+Δ Q b
θ
+Δ Q w
θ
=
Δ Q
θ
. This is an important advantage over the JMP procedure, in which the ’residual price
and quantity component’ must be estimated as a remainder term after the other two com-
ponents are calculated.
5.6.2 Quantile implementation of DFL
Interestingly, the notation above makes it apparent that the DFL procedure is simply the
first step of the JMP decomposition above. In particular,
Δ Q
DF L
θ
= Q
θ
f g
τ
( X ) , β
t
b
, β
t
w
− Q
θ
f g
t
( X ) , β
t
b
, β
t
w
= Δ Q
θ
X
.
24
( ( )) ( ))
( ( )) ( ))
( ( )) ( ))
( ( )) ( ( ))
(
(
(

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 within-group 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 adding-up 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
within-group 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
non-parametric. Notably, DFL must parameterize the reweighting function (through the
probit/logit). I’ve never seen any work documenting the