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Unconditional Quantile Regressions

Author(s): Sergio Firpo, Nicole M. Fortin and Thomas Lemieux

Source:

Econometrica,

Vol. 77, No. 3 (May, 2009), pp. 953-973

Published by: The Econometric Society

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Econometnca,

Vol.

77,

No. 3

(May,

2009),

953-973

UNCONDITIONAL

QUANTILE

REGRESSIONS

Sergio

Firpo,

Nicole M.

Fortin,

and

Thomas Lemieux1

propose

new

regression

method to evaluate the

impact

changes

in the distri-

bution of the

explanatory

variables on

quantiles

the unconditional

(marginal)

distrib-

ution of

outcome variable.

The

proposed

method

consists of

running

regression

the

(recentered)

influence function

(RIF)

the

unconditional

quantile

the

explana-

tory

variables. The

influence

function,

widely

used tool in

robust

estimation,

easily

computed

for

quantiles,

well as

for other

distributional statistics. Our

approach,

thus,

can

readily generalized

other

distributional

statistics.

Keywords: Influence

functions,

unconditional

quantile,

RIF

regressions,

quantile

regressions.

INTRODUCTION

this

paper,

propose

new

computationally

simple regression

method

estimate

the

impact

changing

the distribution

explanatory

variables,

on the

marginal

quantiles

the

outcome

variable, Y,

other

functional

of the

marginal

distribution

Y. The method

consists of

running

regres-

sion

of a

transformation

the

(recentered)

influence function

defined below

of the

outcome

variable

on the

explanatory

variables. To

distinguish

our

ap-

proach

from

commonly

used

conditional

quantile regressions

(Koenker

and

Bassett

(1978),

Koenker

(2005)),

we call our

regression

method an

uncondi-

tional

quantile regression.2

Empirical

researchers are

often interested

changes

the

quantiles,

de-

noted

qT,

of the

marginal

(unconditional)

distribution,

FY(y).

For

example,

may

want

to estimate the direct effect

dqT(p)/dp

increasing

the

proportion

of unionized

workers,

Pr[X

1],

the Tth

quantile

of the

distribu-

tion

wages,

where

1 if

the

workers

unionized

and X

otherwise.

In the

case

the

mean

/a,

the

coefficient

standard

regression

of Y

is a

measure

the

impact

increasing

the

proportion

unionized

*We thank the co-editor and three

referees for

helpful suggestions.

are also indebted

to Joe

Altonji,

Richard

Blundell,

David

Card,

Vinicius

Carrasco,

Marcelo

Fernandes,

Chuan

Goh,

Jinyong

Hahn,

Joel

Horowitz,

Guido

Imbens,

Shakeeb

Khan,

Roger

Koenker,

Thierry

Magnac,

Ulrich

Millier,

Geert

Ridder,

Jean-Marc

Robin,

Hal

White,

and

seminar

participants

CESG2005, UCL,

CAEN-UFC, UFMG,

Econometrics

Rio

2006, PUC-Rio,

IPEA-RJ,

SBE

Meetings

2006,

Tilburg University, Tinbergen

Institute,

Leuven,

ESTE-2007,

Harvard-MIT

Econometrics

Seminar, Yale, Princeton, Vanderbilt,

and

Boston

University

for useful

comments

on earlier versions of

the

manuscript.

Fortin

and Lemieux thank

SSHRC for

financial

support.

Firpo

thanks

CNPq

for financial

support.

Usual disclaimers

apply.

2The "unconditional

quantiles"

are

the

quantiles

of the

marginal

distribution of

the

outcome

variable Y.

Using "marginal"

instead of

"unconditional" would

confusing,

however,

since

also

use

the word

"marginal"

refer to the

impact

small

changes

covariates

(marginal

effects).

Econometric

Society

DOI:

10.3982/ECTA6822

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954

FIRPO,

FORTIN,

AND T.

LEMIEUX

workers on

the mean

wage, dji(p)/dp.

As is well

known,

the same

coeffi-

cient

can

also be

interpreted

as an

impact

on the conditional mean.3 Un-

fortunately,

the

coefficient

/3T

from a

single

conditional

quantile

regression,

j8T

Fy\r\X

0),

generally

different

from

dqT(p)/dp

(Pr[Y

qT\X

1]-

Pr[Y

qT\X

0])/fY(qT),

the

effect of

increasing

the

proportion

of unionized workers on the rth

quantile

the unconditional dis-

tribution of Y.4

new

approach

is therefore needed to

provide practitioners

with

easy way

compute

dqT(p)/dp,

especially

when

not univariate

and

binary

as in

the above

example.

Our

approach

builds

upon

the

concept

of the

influence function

(IF),

widely

used tool

the robust estimation

statistical

or econometric

mod-

els. As its name

suggests,

the influence function

IF(

v, FY)

of a distributional

statistic

v(FY )

represents

the influence of

an individual observation on

that dis-

tributional statistic.

Adding

back the statistic

v(FY)

to the influence

function

yields

what we call the recentered

influence function

(RIF).

One convenient

fea-

ture of the

RIF

is that its

expectation

equal

v(FY).5

Because

influence

functions can be

computed

for most distributional

statistics,

our

method

easily

extends to other choices of v

beyond

quantiles,

such

as the

variance,

the Gini

coefficient,

and

other

commonly

used

inequality

measures.6

For

the Tth

quantile,

the influence function

IF(Y;

qT,FY)

known to be

equal

qT})/fy{qr)'

As a

result, RIF(

qT,

FY)

simply

equal

IF(

qT,

FY).

We call the conditional

expectation

of the

RIF(

v, FY)

modeled as a function of the

explanatory

variables,

£[RIF(Y;

FY)\X]

mv(X),

the

RIF

regression

model.1

the case

quantiles,

£[RIF(Y;

qT,

FY)\X]

mT(X)

can be viewed as an unconditional

quantile

regression.

show that the

average

derivative of the unconditional

quantile

regression,

E[mfT(X)],

corresponds

to the

marginal

effect on the

unconditional

quantile

of a

small location shift

the distribution

covariates,

holding

everything

else

constant.

Our

proposed

approach

can be

easily

implemented

as an

ordinary

least

squares (OLS) regression.

the case of

quantiles,

the

dependent

variable

the

regression

RIF(Y;

qr,FY)

1{Y

ÇtD/MÇt).

It is

easily

3The

conditional

mean

interpretation

is the

wage change

that a worker

would

expect

when

her union

status

changes

from

non-unionized

unionized,

E(Y\X

0).

Since the unconditional mean is

fi(p)

pE(Y\X

p)E{Y\X

0),

follows that

drtpydp

E{Y\X

E(Y\X

4The

expression

for

dqT(p)/dp

is obtained

by implicit

differentiation

applied

FY(qT)

•

(Pi[Y

<qT\X

l]- Pr[7

qT\X

0])

Pr[Y

qT\X

0].

Such

property

important

some

situations,

although

for

the

marginal

effects in

which we

are interested in this

paper

the

recentering

not

fundamental.

Firpo,

Fortin,

and

Lemieux

(2007b),

the

recentering

is useful because it allows

us to

identify

the

intercept

and

perform

Oaxaca-type decompositions

at various

quantiles.

6See

Firpo,

Fortin,

and Lemieux

(2007b)

for such

regressions

on the variance

and Gini.

7In the case of the

mean,

since the

RIF is

simply

the outcome

variable

regression

RIF(

fi)

on X is the same as an OLS

regression

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UNCONDITIONAL

QUANTILE

REGRESSIONS

955

computed

estimating

the

sample quantile

qT,

estimating

the

density

fY(qT)

at that

point

qT using

kernel

(or

other)

methods,

and

forming

dummy

variable

qT], indicating

whether the value of

the outcome variable is below

qT.

Then we can

simply

run

OLS

regression

of this new

dependent

variable on

the

covariates,

although

suggest

sophisticated

estimation methods

Section

We view our

approach

as an

important complement

to the literature

con-

cerned with the

estimation of

quantile

functions.

However,

unlike Imbens

and

Newey

(2009),

Chesher

(2003),

and

Florens,

Heckman,

Meghir,

and

Vyt-

lacil

(2008),

who considered

the identification of structural

functions defined

from conditional

quantile

restrictions

the

presence

endogenous

regres-

sors,

our

approach

concerned

solely

with

parameters

that

capture changes

unconditional

quantiles

the

presence

exogenous regressors.

The structure

of the

paper

as follows.

the next

section,

define the

key object

interest,

the "unconditional

quantile partial

effect"

(UQPE)

and

show

how RIF

regressions

for the

quantile

can

used to estimate the

UQPE.

also

link

this

parameter

the

structural

parameters

of a

general

model and

the conditional

quantile

partial

effects

(CQPE).

The estimation issues

are

ad-

dressed

Section

3. Section

presents

empirical application

our method

that

illustrates well the difference between our method and conditional

quan-

tiles

regressions.

We conclude

Section 5.

2. UNCONDITIONAL

PARTIAL

EFFECTS

2.1.

General

Concepts

assume that

is observed

the

presence

of covariates

that

and X have

joint

distribution,

FY,x(;

•)

'•

-►

[0,

1],

and

the

support

of X.

analogy

with

standard

regression

coefficient,

our

object

interest

is the effect of a small increase

the location of

the distribution of the

explanatory

variable

on the rth

quantile

of the

unconditional distribution

. We

represent

this small location shift

the distribution

X in

terms of

the counterfactual

distribution

Gx(x).

definition,

the unconditional

(mar-

ginal)

distribution function of

can be written

(1)

FY(y)

FYlx(y\X

•

dFx(x).

Under the

assumption

that the conditional

distribution

FY\x(-)

is unaffected

this small

manipulation

the distribution

counterfactual

distribution

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956

FIRPO,

N. M.

FORTIN,

AND T.

LEMIEUX

G*Y,

can be

obtained

replacing

Fx(x)

with

Gx(x)8:

(2)

(y)

FYlx(y\X

•

dGx(x).

Our

regression

method builds on

some

elementary properties

the

influ-

ence

function,

measure introduced

Hampel (1968,

1974)

study

the in-

finitesimal

behavior of

real-valued

functional

v(FY),

where

v:Fv-+R,

and

where

a class of

distribution functions

such that

\v(F)\

+oo.

Let

another distribution in the

same class. Let

FY,t.GY

represent

the

mixing

distribution,

which is t

away

from

the direction of the

proba-

bility

distribution

FY,t.Gy

•

(GY

FY)

FY,

where

The directional

derivative of v

the

direction of the distribution

can be

written

(3)

HFy,,gy)-HFy)

dv{FYtt.GY)

t=Q

JlF(y;p,FY).d(GY-FY)(y),

where

IF(};;

v, FY)

dv(FY,t.Ay)/dt\t=0,

with

denoting

the

probability

mea-

sure that

puts

mass

1 at

the value

The von

Mises

(1947)

linear

approximation

the functional

v(FYjt.GY)

v(FY,t.GY)

v(FY)

t.j

IF(y;

v, FY)

•

d(GY

FY)(y)

r(t;v;GY,FY),

where

r(t\

GY, FY)

is a

remainder term. We

define

the

recentered

influence

function

(RIF)

formally

as the

leading

terms of the above

expansion

for

the

particular

case where

and

1. Since

IF(y;

)

•

dFY(y)

definition,

it follows that

RIFO;

v, FY)

v(FY)

IF(s; v, FY)

•

dAy{s)

KFy)

IF(y;i/,Fy).

Finally,

note that the last

equality

equation (3)

also holds

for

RIF(>>;

v,FY).

the

presence

covariates

we can use the law of iterated

expecta-

tions

express

v(FY )

terms of the

conditional

expectation

RIF(_y;

v, FY)

Instead of

assuming

a constant conditional

distribution

FY\x(-\-),

we could allow the condi-

tional

distributions to

vary

long

they converge

as the

marginal

distributions

of X

converge

to one another.

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