Dec

ompo

sitio

n m

etho

ds in

ec

onom

ics

Nic

ole

Forti

n, U

BC

Thom

as L

emie

ux, U

BC

Ser

gio

Firp

o, E

ES

P-F

GV

This

chap

ter

!U

ses

the

clas

sic

wor

k of

Oax

aca

(197

3) a

nd

Blin

der (

1973

) for

the

mea

nas

its

poin

t of

depa

rture

!Fo

cuse

s on

rece

nt d

evel

opm

ents

(las

t 15

year

s) o

n ho

w to

go

beyo

nd th

e m

ean

!P

rovi

des

empi

rical

illu

stra

tions

and

dis

cuss

es

appl

icat

ions

thro

ugho

ut!

Sug

gest

s a

�use

r gui

de� o

f bes

t pra

ctic

es

Wha

t is n

ew si

nce

Oax

aca?

!M

etho

ds fo

r goi

ng b

eyon

d th

e m

ean

mot

ivat

ed b

y:"

Ineq

ualit

y lit

erat

ure

(JM

P, D

FL, e

tc.)

"In

tere

st fo

r �w

hat h

appe

ns w

here

�, e.

g. g

ende

r gap

and

gl

ass

ceilin

g!

Con

nect

ion

with

trea

tmen

t effe

ct li

tera

ture

"H

elps

form

aliz

e an

d cl

arify

som

e as

pect

s of

de

com

posi

tions

"S

truct

ural

vs.

uns

truct

ural

mod

ellin

g!

Ong

oing

issu

es"

Bas

e gr

oup

prob

lem

(Oax

aca

and

Ran

som

, 199

9)"

Sel

ectio

n (N

eal a

nd J

ohns

on, P

etro

ngol

oan

d O

livet

ti, e

tc.)

Plan

of t

he p

rese

ntat

ion

!S

tatu

s of

the

pape

r�!

Qui

ck re

fresh

er o

n th

e O

axac

a de

com

posi

tion

!C

over

the

mai

n co

ntrib

utio

n of

the

chap

ter a

s a

set o

f six

mai

n �ta

ke a

way

� poi

nts

!E

xpla

in w

here

we

are

in te

rms

of w

ritin

g an

d w

hat i

s th

e �to

do�

list

Refr

eshe

r on

Oax

aca

deco

mpo

sitio

n

!W

ant t

o de

com

pose

the

diffe

renc

e in

the

mea

n of

an

outc

ome

varia

ble Y

betw

een

two

grou

ps A

and B

!G

roup

s co

uld

also

be

perio

ds, r

egio

ns, e

tc.

!P

ostu

late

line

ar m

odel

for Y

, with

con

ditio

nally

inde

pend

ent

erro

rs:

!Th

e di

ffere

nce ∆

=E(Y

B)-E

(YA) c

an b

e de

com

pose

d as

A fe

w re

mar

ks!

We

focu

s on

this

par

ticul

ar d

ecom

posi

tion,

but

we

coul

d al

so c

hang

e th

e or

der,

show

the

inte

ract

ion

term

, etc

. D

oes

not a

ffect

the

subs

tanc

e of

the

argu

men

t in

mos

t ca

ses.

!In

the

�agg

rega

te� d

ecom

posi

tion,

we

only

div

ide ∆

into

its

two

com

pone

nts ∆

S(w

age

stru

ctur

e ef

fect

) and

∆X

(com

posi

tion

effe

ct).

!In

the

�det

aile

d�de

com

posi

tion

we

also

look

at t

he

cont

ribut

ion

of e

ach

indi

vidu

al c

ovar

iate

(or

corr

espo

ndin

g β)

!Th

e �in

terc

ept�

com

pone

nt o

f ∆S,

βB

0-β A

0, is

the

wag

e st

ruct

ure

effe

ct fo

r the

bas

e gr

oup.

Unl

ess

the

othe

r β�s

ar

e th

e sa

me

in g

roup

Aan

d B

, βB

0-β A

0w

ill a

rbitr

arily

de

pend

on

the

base

gro

up c

hose

n.

The

six ta

ke-a

way

poi

nts

1.Th

e w

age

stru

ctur

e ef

fect

(∆S)

can

be

inte

rpre

ted

as a

tre

atm

ent e

ffect

2.G

oing

bey

ond

the

mea

n is

a �s

olve

d� p

robl

em fo

r the

ag

greg

ate

deco

mpo

sitio

n3.

Goi

ng b

eyon

d th

e m

ean

is m

ore

diffi

cult

for t

he d

etai

led

deco

mpo

sitio

n4.

The

anal

ogy

betw

een

quan

tile

and

stan

dard

(mea

n)

regr

essi

ons

is n

ot h

elpf

ul5.

Dec

ompo

sing

pro

porti

ons

is e

asie

r tha

n de

com

posi

ng

quan

tiles

6.Th

ere

is n

o ec

onom

etric

sol

utio

n to

the

base

gro

up p

robl

em

1. T

he w

age

stru

ctur

e ef

fect

(∆S)

can

be

inte

rpre

ted

as a

trea

tmen

t eff

ect

!Th

e co

nditi

onal

inde

pend

ence

ass

umpt

ion

(E(ε

|X)=

0) u

sual

ly

invo

ked

in O

axac

a de

com

posi

tions

can

be

repl

aced

by

the

wea

ker i

gnor

abilit

yas

sum

ptio

n to

com

pute

the

aggr

egat

e de

com

posi

tion

!Fo

r exa

mpl

e, a

bilit

y (ε

) can

be

corre

late

d w

ith e

duca

tion

(X) a

s lo

ng a

s th

e co

rrela

tion

is th

e sa

me

in g

roup

s A

and

B.

!A

lso

help

s pr

ovid

e a

slig

htly

mor

e st

ruct

ural

foun

datio

n to

the

deco

mpo

sitio

n.!

If w

e ha

ve Y

G=m

G(X

, ε) a

nd ig

nora

bilit

y, th

en:

"∆

Sso

lely

refle

cts

chan

ges

in th

e m

(.) fu

nctio

ns (A

TET)

"∆

Xso

lely

refle

cts

chan

ges

in th

e di

strib

utio

n of

X a

nd ε

(igno

rabi

lity

key

for t

his

last

resu

lt).

1. T

he w

age

stru

ctur

e ef

fect

(∆S)

can

be

inte

rpre

ted

as a

trea

tmen

t eff

ect

!A

num

ber o

f est

imat

ors

for A

TET=

∆S

have

be

en p

ropo

sed

in th

e tre

atm

ent e

ffect

lit

erat

ure

"In

vers

e pr

obab

ility

wei

ghtin

g (IP

W),

mat

chin

g, e

tc.

!Fo

rmal

resu

lts e

xist

, e.g

. IP

W is

effi

cien

t for

"A

TET

(Hira

no, I

mbe

ns, a

nd R

idde

r, 20

03)

"Q

uant

iletre

atm

ent e

ffect

s (F

irpo,

200

7)!

We

like

this

sin

ce it

pro

vide

s a

theo

retic

al

just

ifica

tion

for D

FL�s

appr

oach

�

1. T

he w

age

stru

ctur

e ef

fect

(∆S)

can

be

inte

rpre

ted

as a

trea

tmen

t eff

ect

!W

hen

the

treat

men

t effe

ct Y

iB-Y

iAis

het

erog

enou

s, th

e A

TET

depe

nds

on th

e ch

arac

teris

tics

of g

roup

B.

!Th

e di

ffere

nce

in in

terc

epts

βB

0-β A

0ca

n be

inte

rpre

ted

as th

e A

TE in

the

base

gro

up!

Eac

h co

mpo

nent

E(X

Bk)(β

Bk-β A

k) in

dica

tes

by h

ow m

uch

the

ATE

cha

nges

whe

n w

e sw

itch

from

XBk

=0 (b

ase

grou

p) to

XBk

=E(X

Bk).

!N

ot c

lear

this

is, i

n ge

nera

l, a

sens

ible

way

of d

escr

ibin

g he

tero

gene

ity in

the

treat

men

t effe

ct.

!N

eeds

som

e ec

onom

ics

to h

elp

here

, for

inst

ance

A

=bla

cks,

B=w

hite

s, a

nd X

kis

uni

on s

tatu

s du

mm

y (A

shen

felte

r)

Goi

ng b

eyon

d th

e m

ean

is a

�sol

ved�

pr

oble

m fo

r the

agg

rega

te d

ecom

posit

ion

!C

an d

irect

ly a

pply

non

-par

amet

ric m

etho

ds

(IPW

, mat

chin

g, e

tc.)

from

the

treat

men

t effe

ct

liter

atur

e.!

Igno

rabi

lity

is c

ruci

al, b

ut m

G(X

, ε) d

oes

not n

eed

to b

e lin

ear

!In

fere

nce

by b

oots

trap

or a

naly

tical

sta

ndar

d er

rors

in th

e ca

se o

f IP

W (�

gene

rate

d re

gres

sor�

corr

ectio

n re

quire

d)

!IP

W/D

FL v

ery

easy

to u

se w

ith la

rge

and

wel

l be

have

d (n

o su

ppor

t pro

blem

) dat

a se

ts.

Goi

ng b

eyon

d th

e m

ean

is m

ore

diff

icul

t fo

r the

det

ailed

dec

ompo

sitio

n!

Unt

il re

cent

ly, t

here

wer

e on

ly a

few

par

tial (

and

not

alw

ays

satis

fact

ory)

way

s of

per

form

ing

a de

taile

d de

com

posi

tion

for g

ener

al d

istri

butio

nal m

easu

res

(qua

ntile

sin

par

ticul

ar):

"D

FL c

ondi

tiona

l rew

eigh

ting

for t

he c

ompo

nent

s of

∆X

linke

d to

dum

my

cova

riate

s (e

.g. u

nion

s)"

Mac

hado

-Mat

a qu

antil

ere

gres

sion

s fo

r com

pone

nts

of ∆

S.

"S

eque

ntia

l DFL

-type

rew

eigh

ting,

add

ing

one

cova

riate

at a

tim

e. S

ensi

tive

to o

rder

use

d as

in a

sim

ple

regr

essi

on.

!A

mor

e pr

omis

ing

appr

oach

is to

est

imat

e fo

r pr

opor

tions

, and

inve

rt ba

ck to

qua

ntile

s. R

IF

regr

essi

on o

f Firp

o, F

ortin

and

Lem

ieux

(200

9) is

on

e po

ssib

le w

ay o

f doi

ng s

o (m

ore

on th

is s

oon)

Goi

ng b

eyon

d th

e m

ean

is m

ore

diff

icul

t fo

r the

det

ailed

dec

ompo

sitio

n!

We

also

pro

pose

a m

ore

gene

ral c

ondi

tiona

l rew

eigh

ting

appr

oach

!In

tuiti

on fo

r com

pone

nts

of ∆

X:

"W

hen

sequ

entia

lly a

ddin

g re

gres

sors

, the

effe

ct fo

r the

last

one

is

cons

iste

nt s

ince

all

othe

r cov

aria

tes

have

bee

n co

ntro

lled

for.

"S

imila

rly, c

ompa

ring

the

effe

ct o

btai

ned

by re

wei

ghtin

gon

all

X�s

vs.

al

l X�s

exc

ept X

kgi

ves

the

corr

ect e

ffect

of X

k."

Rep

eatin

g th

e pr

oced

ure

for e

ach

Xk

give

s th

e rig

ht m

argi

nal

cont

ribut

ion

of e

ach

Xk,

thou

gh th

e k

effe

cts

do n

ot s

um u

p to

the

tota

l (in

tera

ctio

n ef

fect

s).

!A

rew

eigh

ting

appr

oach

can

als

o be

use

d to

com

pute

the

com

pone

nts

of ∆

S(a

s in

DiN

ardo

and

Lem

ieux

, 199

7):

"R

estri

ct s

ampl

e to

Xk=

0 (o

r oth

er b

ase

grou

p va

lue)

"R

ewei

ghto

n th

e X

-kot

her c

ovar

iate

s to

hav

e th

e sa

me

dist

ribut

ion

as in

the

full

sam

ple.

"G

ives

the

dist

ribut

ion

whe

n th

e w

age

stru

ctur

e ef

fect

of X

kha

s be

en

set t

o ze

ro.

Qua

ntile

regr

essio

ns d

o no

t help

!Te

mpt

ing

to ru

n qu

antil

ere

gres

sion

s (s

ay fo

r the

med

ian)

and

pe

rform

a d

ecom

posi

tion

as in

the

case

of t

he m

ean

(Oax

aca)

!D

oes

not w

ork

beca

use

ther

e ar

e tw

o in

terp

reta

tions

to β

for t

he

mea

n"

Con

ditio

nal m

ean:

E(

Y|X

) = Xβ

"U

ncon

d. m

ean

(LIE

): E(

Y) =

EX(E

(Y|X

)) =

EX(X

)β!

But

the

LIE

doe

s no

t wor

k fo

r qua

ntile

s"

Con

ditio

nal q

uant

ile:

Qτ(

Y|X

) = Xβτ

"U

ncon

d. q

uant

ile:

Qτ≠

E X(Qτ(

Y|X

)) =

EX(X

)βτ

!O

nly

the

first

inte

rpre

tatio

n w

orks

for βτ,

whi

ch is

not

use

ful f

or

deco

mpo

sing

unc

ondi

tiona

l qua

ntile

s

Dec

ompo

sing

prop

ortio

ns is

eas

ier t

han

deco

mpo

sing

quan

tiles

!E

xam

ple:

10

perc

ent o

f men

ear

n m

ore

than

80K

a y

ear,

but o

nly

5 pe

rcen

t of w

omen

do

so.

!E

asy

to d

o a

deco

mpo

sitio

n by

runn

ing

LP m

odel

s fo

r the

pr

obab

ility

of e

arni

ng le

ss (o

r mor

e) th

an 8

0K, a

nd p

erfo

rm a

O

axac

a de

com

posi

tion

on th

e pr

opor

tions

.!

By

cont

rast

, muc

h le

ss o

bvio

us h

ow to

dec

ompo

se th

e di

ffere

nce

betw

een

the

90th

quan

tile

for m

en (8

0K) a

nd w

omen

(say

65K

)!

But

func

tion

linki

ng p

ropo

rtion

s an

d qu

antil

esis

the

cum

ulat

ive

dist

ribut

ion.

!

Cou

nter

fact

ual p

ropo

rtion

s →

Cou

nter

fact

ual c

umul

ativ

e →

Cou

nter

fact

ual q

uant

iles

!C

an b

e illu

stra

ted

grap

hica

lly

Figu

re 1

: Rel

atio

nshi

p B

etw

een

Prop

ortio

ns a

nd Q

uant

iles

01

04

Y (q

uant

iles)

Proportion (cumul. prob. (F(Y))

Men

(A)

Wom

en (B

)

Q.5

.5 .1

Q.9

Q.1

.9

Cou

nter

fact

ual

prop

ortio

ns c

ompu

ted

usin

g LP

mod

el

Figu

re 2

: RIF

Reg

ress

ions

: Inv

ertin

g Lo

cally

01

04

Y (q

uant

iles)

Proportion (cumul. prob. (F(Y))

Q.5

.5C

ount

erfa

ctua

l qu

antil

e (m

edia

n)

.1

Q.9

Q.1

.9

Cou

nter

fact

ual

prop

ortio

n

Slo

pe o

f the

cu

mul

ativ

e(d

ensi

ty)

Figu

re 3

: Inv

ertin

g G

loba

lly

01

04

Y (q

uant

iles)

Proportion (cumul. Prob. (F(Y))

Men

(A)

Wom

en (B

)

Q.5

.5 .1

Q.9

Q.1

.9

Cou

nter

fact

ual

prop

ortio

ns c

ompu

ted

usin

g LP

mod

el

Dec

ompo

sing

prop

ortio

ns is

eas

ier t

han

deco

mpo

sing

quan

tiles

!FF

L re

cent

ered

influ

ence

func

tion

(RIF

) reg

ress

ions

"R

un L

P m

odel

s (o

r log

it/pr

obit)

for b

eing

bel

ow a

giv

en

quan

tiles

, and

div

ide

by d

ensi

ty (s

lope

of c

umul

ativ

e) to

lo

cally

inve

rt."

Dep

ende

nt v

aria

ble

is d

umm

y 1(

Y<Q

τ) di

vide

d by

den

sity

→

influ

ence

func

tion

for t

he q

uant

ile.

"R

IF a

ppro

ach

wor

ks fo

r oth

er d

istri

butio

nal m

easu

res

(Gin

i, va

rianc

e, e

tc.)

!C

hern

ozhu

kov

et a

l. (2

009)

"E

stim

ate

�dis

tribu

tiona

l reg

ress

ions

� (LP

, log

itor

pro

bit)

for

each

val

ue o

f Y (s

ay a

t eac

h pe

rcen

tile)

"In

vert

back

glo

bally

to re

cove

r cou

nter

fact

ual q

uant

iles

Ther

e is

no e

cono

met

ric so

lutio

n to

he

base

gro

up p

robl

em!

Ele

men

ts o

f the

det

aile

d de

com

posi

tion

are

wel

l def

ined

for ∆

X.!

Effe

ct o

f cha

ngin

g th

e di

strib

utio

n of

Xk

(gro

up A

to g

roup

B) h

oldi

ng th

e di

strib

utio

n of

the

othe

r cov

aria

tes

cons

tant

!N

o ba

se g

roup

pro

blem

for e

lem

ents

∆X.

!Fo

r ∆S,

how

ever

, the

re a

re a

s m

any

deta

iled

deco

mpo

sitio

ns a

s th

ere

are

base

gro

ups

!O

K w

hen

the

base

gro

up is

of p

artic

ular

eco

nom

ic in

tere

st. F

or

exam

ple,

if b

ase

grou

p =

unsk

illed

(0 e

xper

ienc

e, p

rimar

y ed

ucat

ion)

!O

ther

wis

e th

e w

hole

exe

rcis

e is

not

ver

y us

eful

!B

ette

r to

find

inte

rest

ing

way

s of

cha

ract

eriz

ing

the

hete

roge

neity

in th

e tre

atm

ent e

ffect

to g

ive

som

e gu

idan

ce o

n w

hat a

re th

e in

tere

stin

g ec

onom

ic fa

ctor

s at

wor

k.!

(The

oret

ical

) exa

mpl

e: g

ende

r gap

sm

all i

n m

ost o

ccup

atio

ns, b

utla

rge

in a

few

�top

-end

� occ

upat

ions

. Can

then

com

pute

cou

nter

fact

ual

gend

er g

ap if

nob

ody

was

in th

ese

few

top-

end

occu

patio

ns.

To-d

o lis

t

!Fi

nish

the

writ

e-up

!A

dd e

mpi

rical

app

licat

ion(

s)!

Dis

cuss

em

piric

al a

pplic

atio

ns