2003_Levinsohn_Estimating Production Functions Using Inputs to Control for Unobservables

7/27/2019 2003_Levinsohn_Estimating Production Functions Using Inputs to Control for Unobservables

1/26

The Review of Economic Studies Ltd.

Estimating Production Functions Using Inputs to Control for UnobservablesAuthor(s): James Levinsohn and Amil PetrinSource: The Review of Economic Studies, Vol. 70, No. 2 (Apr., 2003), pp. 317-341Published by: Oxford University Press

Stable URL: http://www.jstor.org/stable/3648636.Accessed: 25/03/2014 04:53

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2/26

Review

fEconomic

tudies

2003)

0,

317-341

0034-6527/03/00120317$02.00

?

2003TheReview fEconomictudiesimited

Estimatingroductionunctions

Using nputs

o Control or

Unobservables

JAMES

LEVINSOHN

University

fMichigan

and

AMIL PETRIN

UniversityfChicago

First ersion

eceivedune

000;

inal

ersion

ccepted

ctober

002

Eds.)

We dd o hemethods

or

onditioning

ut

erially

orrelatednobservedhockso he

roduction

technology.

e build n deas

first

eveloped

n

Olley

ndPakes

1996).

They

how

how

ouse

nvestment

to control

or orrelationetween

nput

evels ndthe nobserved

irm-specificroductivity

rocess.

We

show hat ntermediate

nputs

those

nputs

hich re

ypically

ubtractedut

n a

value-added

roduction

function)

an also solve his

imultaneity

roblem.

We discuss ome

heoreticalenefits

f

xtending

he

proxy

hoice

et n this irection

ndour

mpirical

esults

uggest

hese enefitsan be

important.

1. INTRODUCTION

Economists

egan elating

utput

o

nputs

n

the

arly

800's.

A

large

iteraturen

estimating

production

unctions as

followed,

n

part

because much

of

economic

heory ields

estable

implications

hat rerelated

o

the

echnology

nd

optimizing

ehaviour.1

Since at east s

early

s Marschak

ndAndrews

1944),

applied

esearchers

ave

worried

about he

otential

orrelation

etween

nput

evels nd he nobserved

irm-specificroductivity

shocks

n

the estimation

f

production

unction

arameters.

he economics

nderlying

his

concern re

intuitive.

irms hathave

a

large

positive roductivity

hock

may respond y

using

more

nputs.

o the

extent hat his

s

true,

rdinary

east

squares

OLS)

estimates f

production

unctions

ill

yield

iased

parameter

stimates,nd,

y mplication,

iased stimates

of

productivity.

Many lternativesoOLS havebeenproposed,nd we add to this etbyextendinglley

and Pakes

1996).

They

show

the conditions nderwhich n

investment

roxy

ontrols or

correlation

etween

nput

evels and the unobserved

roductivity

hock.Their

pproach

as

the

advantage

hat,

or

many

uestions,

t

is no

moredifficulto

implement

hanOLS. We

showwhen ntermediate

nputs

those

nputs

hich

re

typically

ubtractedut

n

a value-added

production

unction)

an alsosolve his

imultaneity

roblem.

We discuss ome

potential

enefits

of

expanding

he hoice et

of

proxies

o nclude

hese

nputs.

1. Much

of he

arly pplied

work

xploring

his

elationship

as

pioneered y

gricultural

conomists

ike

Von

Thuenen

a

colleague

f

Cournot),

ho ollected ata t

his

farm

n the

1820's omeasure he

marginal roduct

f

nputs

and the

substitutability

etween

nputs.

lux

1913),

using

ne of the first vailable

manufacturing

ensuses,

etails

relationships

etween

nputs

nd

output

or

manufacturing

irmsn

England.

hambers

1997)

provides

brief

istory

ofproductionunctionstimation.

317

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3/26

318

REVIEW OF ECONOMIC

STUDIES

One

benefits

strictly

ata-driven.

t turns ut that he nvestment

roxy

s

only

valid

for

plants

eporting

on-zero

nvestment.

This

is due to

an

invertibility

ondition

escribed

below.)

Pronounced

djustment

osts,

which

o not

necessarily

nvalidate

he

use of

nvestment

as

a

proxy,

re the

ikely

eason

hat verone-half f our

sample

reports

ero nvestment. e

are

concerned

bout

runcating

ll

of these

lants.Using

ntermediate

nput

roxies

nstead f

investmentvoids his roblem. his s becausefirmsinourdata, t east) lmost lways eport

positive

se

of

ntermediate

nputs

ike

materials

r

electricity.2

To

the

xtent

hat on-convex

djustment

osts re an

important

ssue,

ntermediate

nputs

may

confer

nother

enefit.

f

adjustment

osts

ead

to

kink

points

n

the nvestmentemand

function,

lantsmay

not

ntirelyespond

o some

productivity

hocks,

nd

correlation

etween

the

egressors

ndthe rror

erm an

remain.f

t

s

less

costly

o

adjust

he

ntermediate

nput,

t

may

espond

more

ully

o

the ntire

roductivity

erm.

Another

ice

feature

f the

ntermediate

nput

s

that

t

provides simple

inkbetween

the estimation

trategy

nd

the economic

heory, rimarily

ecause

intermediate

nputs

re

not

typically

tatevariables.

We

develop

this

ink,

deriving

he conditions hat

musthold

if

intermediate

nputs

re to

be a valid

proxy

or he

productivity

hock.

We

suggest

hree

specificationests or valuatingny proxy's erformancendfor omparingmongproxies

when

more

han ne is available.

We

also derive he

expected

irections

f

bias

on

the

OLS

estimates

elative o our

ntermediate

nput

pproach

when

simultaneity

xists.We

take

the

framework

o four

Chilean

manufacturing

ndustries,

indingignificant

ifferences

etween

OLS and

our

pproach

hat

re also consistent

ith

imultaneity.3

Many

estimators ave

been

developed

to

address

simultaneity

nder different

ata

generating

rocesses

o

whichOLS

is notrobust.We

compare

stimates

etween

OLS,

fixed

effects,

he

Olley-Pakes

nvestment

roxy

stimator,

ur ntermediate

nput

roxy

stimator

nd

a Blundell-Bond

GMM estimator

a

lagged-input

nstrumental

ariables

IV)

estimator

ith

fixed

ffects,

ime

ffects,

R(1)

and

MA(O)

shocks,

ll of which aise

potential

imultaneity

problems).

While

hesemodels

o not

enerally

est ne

nother,

ny

est

ejecting

o

differences

between stimates ellsus that he wo

processes

annot othbe

compatible

ith he

ndustry

under onsideration.

hese

results dd

to the vidence

f

simultaneity

roblem

nd

shed ome

light

n ts

underlying

ature.

The

remainder

fthe

paper

s

organized

s

follows.

ection

provides very

rief eview

of

the

simultaneityroblem.

n

Section

3,

we

introduce ur

intermediate

nputproxy,

nd

develop

he

conditions nder

which

t

will be

a

validestimator.

ection

describes

ur

data,

andSection

includes

hedetails

f

he

stimation

pproach.

n Section

we

present

ur

esults,

while ection

concludes.

ppendices

nclude

monotonicity

roof,

short-cutor

implifying

estimation,

nd

a

recipe

or

ur stimationoutine.

2. ESTIMATION

N

THE

PRESENCE OF SIMULTANEITY

We

write

irm's

production

t time as

yit

=

f(xit, Eit;

P)

with

P

parameters.it

includes

inputs

hat re

easily

djusted

nd those hat

volve

ver ime

n

response

o

beliefs.

he errors

{Eit

}I1

are

often

hought

f

as Hicks

neutral

roductivity

hocks.

2. For

many

irm-levelata

ets his runcation

s not rivial.

lthough

esearchers

orking

ith he

ongitudinal

research atabase

rom heU.S.

(as

did

Olley

nd

Pakes)

or

comparable

anufacturing

ensuses rom

heU.K.

or

France

may

not

ind

alf

f heir

ample eporting

ero

nvestment,

uch f

he

lant-level

esearch

eing

onducted

oday

s

on

easier-to-obtain

ata

from

ountries

ike

Turkey,

olumbia,

Mexico nd ndonesia.

n

these

ountries,

s

well

as

Chile,

the zero

nvestment

roblem

s more

ikely

o

oom

arge.

3. In

this

aper,

we

only eport

he

esults

rom he our

argest

manufacturing

ndustries.n earlier ersions f

the

paper,

we

report

esults or

ight

ndustries.

his s an

effort

o

keep

the

number

f

reported

esults

manageable.

Readers nterested

n

seeing

esults

or ll of the ndustries

an

access the

NBER

Working

aper

893

(Levinsohn

nd

Petrin,999).



4/26

LEVINSOHN

& PETRIN ESTIMATING

PRODUCTION

FUNCTIONS 319

A

simultaneityroblem

riseswhen heres

contemporaneous

orrelation

etween

it

and

Eit.

This

imultaneity

iolates

heOLS conditions

or

nbiased

nd consistent

stimation.

t can

arise

with

irm-levelata

when

nput

hoices

espond

o

shocks.

Marschak nd

Andrews

1944)

suggest

his

roblem

may

be

most

ronounced

or

nputs

hat

djust apidly.

pplied

esearchers

have

pent

much

ffort

ddressing

he

conometric

roblem

hese orrelations

onfer.

Ina multivariateontextt sgenerallympossibleosign he iasesoftheOLS coefficients

when

imultaneity

xists nd here

re

many nputs.

ome ntuitionbout

he

ias can be derived

from n

analysis

f

the OLS

estimates or

two-inputroduction

unction,

ith ne

freely

variable

nput

it

call

t

abour)

nd one

quasi-fixed

nput it

call

t

capital):

Yit

=

80

+

Ilit

+

fkkit Eit.

OLS

estimates or he

nputs

re

A=

+

^2

0,101k,k

l,k

and, ymmetrically,

+k

k 2

1,1rk,k

-

7/l,k

where

a,b

denotes he

ample

ovariance etween

and

b.

We

consider

ias

n

three

ifferentases. Since

the

denominator

s

always ositive,

he

ign

of he

ias s

determined

y

henumerator.

f

only

abour

esponds

othe

hock

say

more abour

is

hired

n

response

o

productivity

hock,

o

al,,

>

0),

and

apital

s

not

orrelated ith

abour,

then

fl

will end o

be biased

up

but

k

willremain nbiased.

f

only

abour

esponds

othe hock

and

capital

nd

abour re

positively

orrelated

negative

ias on the

apital

oefficientan also

result.

inally,

f

capital

nd

abour re

positively

orrelatednd abour's

orrelation

ith he

productivityhockshigherhan apital's orrelation,1will end ooverestimate1andAkwill

usually

nderestimate

k.

For hort

anels

we thinkhese ast wo ases

may

e most

elevant,

s

between-firmariation

ften

lays

dominant

ole

n

dentification,

nd

capital

nd abour end

to

be

highly

orrelated

n

this imension.

Within

stimatorsre

a common lternative

o

OLS,

using

nly

he

variation

ithin-firm

to

protect gainst

potential

orrelation

etween nobserved

irm-specific

ixed ffects

like

managerialuality)

nd

nput

hoices.

ometimes,

he

between-firmariation

an be

important

for

obtaining recise

stimates

f

output

lasticities ssociatedwith tate

variables

in

short

panels,

firms

may

not

adjust

capital

much).

Thus

within

stimators ffer

more

protection

against

irm-specific

ffects

han

OLS,

but

hey

an exacerbate ther

roblems

yreducing

he

signal .

An instrumentalariable IV) estimatorchievesconsistency y instrumentinghe

explanatory

ariables

with

egressors

hat re

correlated ith he

nputs

utuncorrelated ith

Eit.

The

IV

approach

an also alleviate

measurementrror

roblems,

hich

end o be

most

pronounced

n

capital.4

otentialnstrumentst thefirm-levelnclude

nput rices

nd

agged

values

f

nput

se.

Firm-level

nput rices

re

rarely

bserved.

agged

values f

nputs

revalid

instruments

f

he

ag

time

s

ong

nough

obreak he

dependence

etween he

nput

hoices

nd

the

erially

orrelatedhock.

lundell ndBond

2000)

develop sophisticated

ousin fthe V

4.

Measurementrror

n

capital

asthe

ame

mplications

or he

arameter

stimates

s the

imultaneityroblem

described

bove.

n

particular,

f

apital

nd abour re

positively

orrelatednd

apital

s

measured ith

rror,

henoise

willtend o

attenuate

apital's

oefficientowards

ero nd ts ssociated

utput

hange

willbe

incorrectly

ttributed

o

labour.


giambot


5/26

320

REVIEW OF

ECONOMIC STUDIES

approach

hat s robust

o

firm-specific

ixed

ffects,

erially

orrelated

roductivity

hocks nd

measurementrror.5

The

nvestment

roxy

Olleyand Pakes (1996) suggest novelapproach o addressinghis imultaneityroblem.

They

nclude

n

the

estimation

quation proxy

which

hey

derive

rom

structural

odel

of the

optimizing

irm.

he

proxy

ontrols or he

part

f

theerror orrelated

ith

nputs y

annihilating

ny

variation

hat

s

possibly

elated o the

roductivity

erm.

We

simplify

slightly)

heir

model,

writing

he

production

unction

n

ogs

as

Yt

=

f0

+

Oflt

+

fkkt

wt?

rt.

(1)

Inputs

re

divided

nto

freely

ariable ne

(lt)

and

the tate ariable

apital kt).6

t

s assumed

to be

additivelyeparable

n

a transmitted

omponent

tot)

nd

an

.i.d.

component

?rt).

he

key

differenceetween

t

and

rt

s that heformers a

state

ariable,

nd

hence

mpacts

hefirm's

decision

ules,

while he

atter as no

mpact

n thefirm's

ecisions.

Olley-Pakeswritenvestments usta functionfthe wo tate ariablesntheirmodel, t

and

ot,

or

it

=

it(Wt,

kt).

When

wt

s

stochastically

ncreasing

n

past

values,

Pakes

1996)

proves

hat

ptimizing

irms

choosing

o invest ave

investmentunctionshat re

strictly

ncreasing

n

the unobserved

productivity

hock.

Basically,

etter

roductivity

hocks

oday

meanbetterhocks

n

the

future,

and

this

eads

to

capital

ccumulation.

The

monotonicity

llows

t wt,

t)

to

be

nvertedo

yield

t

as a functionf nvestment

nd

capital,

or

wt

=

ot

(it,

kt).

One can thenrewrite

1)

as

Yt

=

fPlt

t

(it,

kt)+

nt,

(2)

where

ot

it,

kt)

=

Fo

+

Wkktot it,kt).

A

first-stage

stimatorhat

s linear n

It

and

non-parametric

n

'kt

can be

used

to

obtain

a consistentstimate

f

01.7

Olley

and Pakes

use

a

fourth-order

olynomial

n

it

and

kt

to

approximate

-),

estimating

2)

using

OLS,

with

utput

egressed

n

abour

nd he

polynomial

terms.

We follow he

xposition

n

Robinson

1988)

to

llustratehe dea

further.t s

suggestive

f

howone

might

mplement

lternative

on-parametric

stimators

which

we do

in

what

ollows).

Robinson

1988)

takes he

xpectation

f

equation

2)

conditional

n

t

and

kt.

This s

given y

E[yt I it,kt]= ?1E[lt I

it,

kt] qt(it,kt) (3)

5. It s also

possible

o

directly

pecify

he

arametricrocess

hat he

roductivity

erm ollows.

owever,

ven

f

we are

willing

o

characterizehe

dynamic equence

Xit,

Eit

-1

as a

parametricrocess

nd

want

nly

o estimatehe

parameters

f

his

rocess,

e still

ave

significantroblem.

y

tself,

nowledge

f

he

rocess

up

tothe

parameters)

is

not

nough

o

control or he

imultaneity

etween

it

nd

Xit

over ime

ecause he

process

Xit,

it,

~i'

follows

path

hat

epends pon

ts

tarting

alues

Xi

1,

Ei ).

This s an nitial

onditions

roblem

see

Heckman

1981)

and

Pakes

(1996)),

where stimationf

parameters

or stochastic

rocess

hat

epends

pon

ime-ordered

utcomess

impossible

unless he

rocess

s initialized .

ne solutions to nitialize he bserved

rocess y

ssuming

he

istory

s

exogenous,

i.e. that

{Xit,

1it}T-1

is

independent

f

Xit,

Eit

T,

where

T

is thefirst ate

a firm

s observed. second olution

splits

he

ample

nto wo

parts,

he

irst

art

fwhich s usedto

estimate

tarting

alues

see

Robertsnd

Tybout,

997).

6. For

implicity,

e assume

as

they

o)

that

apital

s the

nly

tate

ariable

ver

which he irm as

control.

7. We will

lways

se

t

-)

when

iscussing

he

on-parametric

art

f

this

irst

tage;

ts

rguments

ill

hange,

but t

will

always

nclude

apital

nd the

roxy

ariable.More

generally,

t

(.)

will

lways

have s

arguments

ll of he

endogenous

tate ariables

nd he

roxy

ariable.



6/26

LEVINSOHN

&

PETRIN ESTIMATING

PRODUCTION FUNCTIONS

321

because:

(i)

irt

s mean

independent

f

it

and

kt;

and

(ii)

E[0tt(it,

kt)

it,

kt]

=

t(it,

kt).

Subtractingquation

3)

from

2)

yields

Yt

E[yt

I

t,kt]

=

lI(lt

E[lt

Iit,

kt])?+

-r.

(4)

By assumptiontrhs mean ndependentf t andthus fthe ransformedegressort

-

E[ltI

it,

kt]),

o

no-intercept

LS can be used to obtain onsistentstimates

f

fil.

Since

capital

nters

ql(.)

twice,

more

omplete

model

s

needed o

dentify

k.

Olley

nd

Pakes ssume

hat

wt

follows first-orderarkov

rocess

ndthat

apital

oes not

mmediately

respond

o

?t,

the

nnovations

n

productivity

ver ast

period's

xpectation,iven y

=t

ct

-

E[wot

ot-I].

Defining

1*

s

output

et

f

abour's

ontribution,

hey

write

Yt

=Y

- Alt

=

fo

+

-kkt

+

E[wOt

wt-11

t,

(5)

where

t7 =-

t+

rt.

Under

hese

ssumptionsegressing

t

on

kt

nd a

consistentstimate

f

E[Ct

I

at-

produces

consistentstimate f

fk

(because

both

t

and

rit

reuncorrelated

ith

kt).8

When his

pproach

works,

t

can

have

dvantages

elative o

OLS,

within,

nd traditional

instrumentalariable

stimators

see

Griliches

nd

Mairesse,

998).9

When he nvestment

roxy

mightail

Investment

s a control

n

a

state

variable,

omething

hich

y

definition

s

costly

o

adjust.

Costs

of

adjustment

an

cause

problems

or

stimation

n

different

ays.

Firms

hat

make

only

ntermittent

nvestmentsill have their

ero-investmentbservations

runcated

rom he

estimationoutinethemonotonicityondition oes nothold fortheseobservations).or

manufacturing

ensuses

his

an be

a

large ortion

f

thedata.

While

his

runcation

ssuerelates

nly

o

efficiency,

on-convex

djustment

osts

may

ead

tokinks

n the nvestmentunction

hat ffecthe

esponsiveness

f nvestment

o

the

ransmitted

shock ven

when nvestments

undertaken.10

or

xample,

uppose

t

ct,

kt)

has

some

maximal

level

of

investment or all

possible

outcomes of

Ot.

Then

it

wt,

kt)

=

it

ot,

kt)

when

Ot

>

t

kt),

for thekink

point

Jit

kt).

The

error erm n

(4)

becomes

ot

+

(Ot

-

0)t

kt)),

which

is

correlated

ith

t.

Alternatively,

uppose

Wt

s

nstead he xtentowhich

t

s

known tthe

ime

of

the

nvestment

ecision,

and that

t

=

it

6t,

kt)

s monotonic

n

&1t.

Again,

wt

-

t)

remains n

the rror

erm.

f

course

n

both ases

the nvestment

roxy

s

helpful

ecause

tcontrols

or

o)t.

8.

Note

hat

P0

is not

eparately

dentified

rom hemean

f

E[wt

I

wt-1]

without

omefurtherestriction.

9.

FromGriliches

nd

Mairesse

1998)

with he ariable

eferences

hanged

o

be consistent

ith ur

notation:

The

major

nnovation

f

Olley

nd Pakes s to

bring

n

a new

quation,

he nvestment

quation,

s a

proxy

or

o,

theunobservedransmitted

omponent

f

E.

Trying

o

proxy

or

he

unobserved

w

if

t

can be done

correctly)

as

several

dvantages

ver

he

usual

within

stimators

or

themore

eneral

Chamberlain

nd GMM

type

stimators):

t does

not ssume

hat

cw

educes

o a

fixed

over

ime)

firm

ffect;

t

eaves

more

dentifying

ariance

n

1

and

k,

and hence

s

a less

costly

olution o the

omitted

ariable

nd/or

imultaneityroblem;

nd t hould lso be

substantively

ore nformative.

10.

Doms and

Dunne

1998)

and

Attanasio,

acelli

and Dos Reis

(2000)

report

umpy

nertial

ehaviour

n

investmentatafrom .S.

andU.K.

plant-levelurveys,uggesting

on-convex

djustment

osts

xist.



7/26

322

REVIEW OF ECONOMIC STUDIES

3.

INTERMEDIATE NPUTS

AS

PROXIES

We

now add a second

freely

ariable

nput,

,

whichwe

call

the ntermediate

nput

perhaps

materials

r

energy).11

riting

he

og

of

output

s a functionf he

og

of

nputs

nd he

hocks

we have

Yt= fO+ llt kkt + ftit+ wt+trt. (6)

The ntermediate

nput's

emand unctions

given

s

tt

=

tt(Wt, kt),

and t must

e monotonic

n

ct

for

ll

(relevant)

t

to

qualify

s

a

valid

proxy.

ote hat

nput

and

output

rices

reassumed

o

be

common cross irms

they

re

suppressed),

ndtheres no

error

n

the

nput

emand

unction.12

n

the

data

ection

we use these

onditions

o

help

hoose

between andidate

roxies.

Assuming

monotonicity

olds

one can

invert he

nput

emand

unction

o obtain

ot

=

ot

tt,kt).

Thus,

he ntermediate

nput

eplaces

nvestment,

ith

Pt

-)

now

given

s

a function

ofthe

ntermediate

nput

nd

capital,

r

t tt, kt)

=

Po

+

Pkkt

+

lttt (t

(tt, kt).

(7)

The

equation

or

he econd

tage

hanges

o

Yt

=

fB

+

fkkt

+

p1tt

E[o[wtot-11]

t.

(8)

Similar

o

nvestment,

or

ny

alue

f

(/k,

,)

we can estimate

E[owt

cot-1].

While

E[kt

4]

=

0

is assumed o

stillhold for

8),

E[tti

f]

=

0 does

not

generally

oldbecause

the

ntermediate

input

s correlated

ith

7

(it

responds

o

at).

Since

firmshoose

t-1

before ither

omponent

of

tf7

s

realized,

t shouldbe uncorrelated

ith

*.

It

should

lso be

correlated

ith

tt

(via,

for

xample,

ize correlationver

timedue to

irreversibility

n

capital

nvestment

nd/or he

persistence

n

cot),

o

we

use

E[tt-lq*i]

=

0 to

obtain dentification.

The

monotonicity

ondition

The

monotonicity

onditionor ntermediate

nputs

s

dentical

o hat or

nvestment;

onditional

on

capital, profit

maximizing

ehaviour

must ead

more

productive

irms o use more

intermediate

nputs.

We

magine

story

here ncreases

n

productivity

ncrease he ntermediate

input'smarginal

roductivity.

his

n

turn eads firmso increase

utput,

hich eads to more

input

se.

Aside

from ome

regularity

onditions

n

the

production

unction

(.),

conditional

n

k,

the

ign

f

the

hange

n

ntermediate

nput

se for

small

hange

n

w is

given

y

sign

=

sign(

ftlfo

-

filfa))

(9)

(where

il

s

the

econd erivativef

f

-)

with

espect

o

1,

tc.).13

ptimizing

ehaviour

mplies

the

marginal roduct

f abour eclines

s

labour

ncreases,

o

fil

0

(and

fic

>

0),

so

-fil

fto

>

0. If

fl

=

0

or

fi,

=

0

and

f,,

>

0

the

monotonicity

ondition

11. The contributionf these

nputs

s

usually

ubtracted

rom

utput

efore

stimation

implicitly) sing

a

separability

ssumption.

hus,

heir

se adds ittle

urden o

necessary

ata

requirements

lready

acedwhen

stimating

production

unctions.

12.

Loosening

he atter

ssumption

s the

ubject

f

ongoing

ork.

13.

Appendix

contains he ull erivation.



8/26

LEVINSOHN

& PETRIN

ESTIMATING

PRODUCTION FUNCTIONS 323

holds.

f

fi

>

0 the esult

s

driven

y

he

ross-partial

f

output

ith

espect

o he ntermediate

input

nd

abour.

f

the

marginal roduct

f

the ntermediate

nputweakly

ncreases s labour

use

ncreases

fi

>

0)

then

he

esult olds. ven

fthe

marginal roduct

alls

with

ncreases

n

labour,

he ondition

ay

till old

it

will

depend

n

relative

magnitudes

f he

wo

products

n

theR.H.S.

of

9)).

One advantage f thisestimators that t is easy to verifywhether hemonotonicity

condition

s consistent

ith

ome

ommon

echnologies

sed

by

conomists

e.g.

Cobb-Douglas

or constant

lasticity

f

substitution).

his result ontrasts

ith he

proof

hat

nvestment

s

monotonic

n

productivity

see

Pakes,

1996).

If

one wishes o use a modelthat iffersven

slightly

rom

hat

f

Olley

nd

Pakes,

t

becomes

necessary

o

re-investigate

he

ppropriateness

of the

nvestment

roxy

sing

he

firm's

ynamic roblem,

nd this an be difficult

as

Pakes

demonstrates).

notheronvenient

roperty

fthis stimators that he

monotonicity

ondition

is

not

mposed,

o we

can

check

o see

if

t s

empiricallyustified.

We show

how

to do

this

n

Section .

Commonnput rices andother ommon nobservedactors)

As

with

nvestment,

he ntermediate

nput

emand

quation

t

=

tt

Wt,kt)

s not ndexed

y

other actors

like

nput

rices).

f

nput

rices

re observed nd

not

ommon cross

firms,

hey

can be

included

irectly

n the

demand

unction,

oosening

he ommon actor

rice

estriction.

If

they

re

notobserved

ut

one

suspects nput rice

atios

ary

ver

ime,

r

by

region,

r

by

urban/rural

ocation,

r

by ype

f

firm,

necanestimate ifferentunctions

or hese

ime

eriods

or

regions

if

hey

re

observed),

making

stimationobust o

these ifferencest the

xpense

f

placing reater

emands

n thedata.

4.

DATA

Inorder o mplementhe ntermediatenput roxy, eneeddata.Weusean8-year anelfrom

Chile

hat as also been

usedelsewhere.14hisChilean ata

s

representative

f

many

irm-level

panels

n

the ense

hat thas

many

irm-level

ariables

including any

ntermediate

nputs),

t

is not ensored

or

ntry

nd

exit,

nd

thas a reasonable ime-seriesimension

o

t.

The data

et

s

comprised

f

plant-level

ata

f6665

plants

n

Chile

from 979

o 1986.

The

data are a

manufacturing

ensus

covering

ll

plants

with t least 10

employees

nd collected

by

Chile's Instituto

acional de Estadistica

INE).

A

very

detailed

description

f how the

longitudinal

amples

were ombined

nto

panel

s found

n

Lui

(1991).15

In

an

attempt

o

keep

the

analysismanageable,

we focus

on

the

four

argest

ndustries

(excluding etroleum

nd

refining).

he

3-digit

evel

ndustries

along

with heir SIC

codes)

are

Metals

381),

Textiles

321),

Food

Products

311)

and

Wood Products

331).16

The data

are observed

nnually

nd

they

nclude

ross

evenue

our

output

ndex),

ndices f abour nd

capital

nputs,

nd

measure

f

he ntermediate

nputs lectricity,

aterials,

ndfuels.17

abour

14.

See,

for

xample,

ybout,

e

Melo and Corbo

1991),

Lui

(1993),

Lui and

Tybout

1996),

Levinsohn

1999)

and

Pavcnik

1999).

15.

Due to

the

way

hat

he ata

re

reported,

e

treat

lants

s

firms,

lthough

here re

ertainly

ulti-plant

irms

in

the

ample.

We will

not

apture

he xtento which

multi-plant

irms

xperience

cale or

scope

conomies ue

totheir

multi-plant

ature.

either

rewe able to

nvestigate

hether

entry

s a new

firm,

new

plant

rom n

existing

irm,

r

simply

iversificationf

n

existing lant

r

firms

discussed

n

Dunne,

Roberts nd Samuelson

1988).

16. Results

or henext our

argest

ndustries,

ther

hemicals

352),

Beverages

313),

Printing

nd

Publishing

(342)

and

Apparel

322)

are

reported

n NBER

Workingaper

819 and lso available t thewebsites

f

both

uthors.

17.

Revenue

s

ourmeasure

f

plant utput

ecause

as

in

most irm-level

ata)

we do

not bserve

plant-level

measure

f

physical utput.

Many xamples

f

output eing irectly

bserved

re

found

n thevoluminousiterature

n

agricultural

conomics.



9/26

324

REVIEW OF

ECONOMIC STUDIES

is the

number

f

man-years

ired or

roduction,

nd firms

istinguish

etween

heir

lue-

nd

white-collar

orkers. ross

evenue,

apital,

materials,

lectricity,

nd

fuels achhave

heir

wn

annual

rice

eflator

most

f hem

rovided y

heBanco Central e

Chile)

nd re achdeflated

to

real 1980 Chilean

esos.

Constructionf

the

apital

ariable s documentedn

Lui

(1991).

It is

the

um

of

the

real

pesovalueofdepreciateduildings, achineryndvehicles, achof which s assumed o have

a

depreciation

ate

8)

of

5,

10 and 20%

respectively.18

hus each

type

f

capital

Kj

evolves

according

o

Kjt

=

(1

-

Sj)Kj,t-1

+

ijt,

and

the

otal

apital

ndex

t time

is

Kt

=

Kjt.

Our

capital

ariable

s

constructed

n a

slightly

ifferent

anner rom

lley-Pakes

s

they

assume

nvestment

eported

ast

period

nters

he

production

unction

s

capital

n

this

period.

We assume nvestment

ccurring

n

this

eriod

nters

apital

n

this

eriod.

bviously,

he

etails

on the imingfdatacollection,hetimingf the actual nvestmentecision, nd thecapital

adjustmentrocess

will determine hich

if either)

f these

ssumptions

s

appropriate.

e

do

not

know hesedetailsfor

our

data,

o for

us

the

choice

s

somewhat

rbitrary,lthough

the

decision ffects he

proxy's

mplementation.19

nderour accumulation

rocess, oday's

investmentecision

must

e made

knowing

nly

he

utcome

f

wt-1,

r

capital

via nvestment)

will

respond

o

wt,

violating

he

consistency

ondition.20 nder his

scenario,

ext

period's

investment

s

the

proxy

or his

period's

hock

it

responds

ully

o

cot).

n

Olley-Pakes

his

period's

nvestments the

proxy

or his

eriod's

ot,

nd ast

period's

nvestmentnters

apital

this

eriod.

he subtletiesf

iming

re

learly

mportant

or hese ariantsf he

roxy pproach.

Table

1

provides

omemacroeconomic

ackground

s well

as

some

ummary

tatistics

or

the

ndustries

e

examine.

y

1979,

most f

Pinochet's conomic

olicies

were

lready

n

place.

The LatinAmericanebt risisedto recessionn 1982and1983during hichndustrialutput

and

employment

ell.

ndustrial

utput

ose

gain

n

1984,

talled

n

1985,

nd then ontinued

to

rise

throughout

hedecade.

These

macroeconomic

ycles

re

apparent

n

thefirst

olumn

f

Table

1

where eal GDP is

reported

or

1979-1986.We

will take hese

macroeconomic

ycles

into ccount

y allowing

he

wt

tt,

kt)

function

o

be

different

or ach of

these hree ifferent

time

eriods

so

t

=

1, 2,

3).

It s also

evident

rom

able

1

that

his

eriod

s

characterized

y

major

onsolidationnd

exit;

henumber f

plants

alls

n

every ndustry

rom

he

beginning

o the

nd

of

the

ample

(although

here s also

entry

n our

sample).

The

original

work

by Olley

and

Pakes devoted

significant

fforto

highlighting

he

mportance

fnot

using

n

artificially

alanced

ample

and

the

electionssues hat risewith hebalanced

ample).They

lso show

nce

they

move o the

unbalanced

anel,

heir election orrectionoes not

hange

heir esults.We

simply

ote hat

our

ample

s

unbalanced,

ndwe

do

not

ocus

n selectionssues.21

18. No initial

apital

tock s

reported

or ome

plants, lthough

nvestment

s

recorded.

When

ossible,

we used

a

capital

eries hatwas

reported

or

subsequent

ase

year.

or a

small

numberf

plants, apital

tock

s not

eported

in

any

year.

or hese

lants,

e

estimated

projected

nitial

apital

tock ased

on other

eported

lant

bservables. e

then

sedthe nvestmentata o fill

ut

he

apital

tock

ata.

19. Not

knowing

he

iming

f the ctual

nvestment

ecisions nd

capital djustmentrocess

s

the urrentorm

for

conometricesearch ecause he

details

f

nvestmentehaviourre

lmost

ever

eported.

20. This

story

s

consistent

f,

for

xample,

nvestmentecorded

oday

was ordered efore his

eriod's

hock s

known

say

at the nd of ast

period)

nd

tcomeson-line his

eriod.

21. Wehave

xperimented

ith

ual

ndex election orrectionsf

Olley

ndPakes nd

found

s

they

ndGriliches

and

Mairesse

1998)

did that he election riterion

ade

ittle ifferencencethe

imultaneity

orrection

as

n

place.

In order

o

focus

n the

ntermediate

nputsssue,

we do not

nclude hosemethods

rresultsn

this

aper.



10/26

LEVINSOHN

& PETRIN ESTIMATING

PRODUCTION FUNCTIONS 325

TABLE

1

Some

descriptive

tatisticsn

Chilean

manufacturing

Industry

ISIC)

Food

products

311)

Metals

381)

Textiles

321)

Wood

products

331)

Value Value

Value

Value

Year GDP Plants added Plants added Plants added Plants added

1979

997.6

1537

39-0

459 10.0

503

12-4

524 10-4

1980 1075.3

1439

43.4

447

11.0 445

12.9

449

8.7

1981 1134-7

1351 42.7 413

11.5 403

11.3

406 6.8

1982 974.9

1319

47.0 365

8-1

350

8.7

358

6-5

1983 968.0

1297

42.9

322

8.3 327

9-7

335

8.1

1984

1029.4

1340 46.8 358

11.4

336

10.4

339

10.3

1985

1054-6

1338 49.1 351 9.6 337

10-8

342 10-1

1986 1114.3

1288

61.4 347 9.6 331

12.9

313 5-3

Note: GDP

figures

rom

he

nternationalinancial tatistics earbook. DP

andvalue

dded

n

millions

of

1980

pesos.

TABLE 2

Per cent

f

non-zero

bservations

Industry

ISIC)

Investment Fuels Materials

Electricity

Food

products

311)

42.7 78.0

99.8

88-3

Metals

381)

44-8 63-1 99-9

96.5

Textiles

321)

41.2

51.2 99.9

97-0

Wood

products

331)

35.9 59.3 99.7

93.8

Choosing

mong

ntermediate

nputs

The

discussion

hus

far

has focused

on

using

ntermediate

nputs

s

the

proxy

ariable.

n

practice,

here re

ypically

everalntermediate

nputs

nd he

uestion

f

choosing

mong

hem

naturally

rises. etails

f he

ndustry

nd he

requency

nd

ype

f

uestions

sked ffirmsan

play

n

mportant

ole

n

choosing

mong

nputs.

ne

natural

ay

o

start

valuating

he

otential

usefulness

f

a

proxy

s to

count ts zero values.

n

general,

he

number

f zeros

bounds

rom

below

the

number f observations

hatmust e

truncated

rom he

estimationoutine.

able

2

lists

the

percentage

f

firm-level

bservations

eporting

on-zero

evels of

investment,

uels,

materials,

nd

electricity.

t

suggests

hat

heres

significant

ariability

n

zero

vs.

non-zero

se

across

nputs.

s

described

arlier,

hese ero

observations

ay

lso

reflect inks

n

thefactor

demand urves

rising

rom

for

xample)

djustment

osts,

which

an

violate

he

monotonicity

condition.

Table

2

indicates

hat,

n

our

data,

many

irms

o notundertakenvestment

very eriod

(year).

Forthese

bservations,

o

Olley-Pakes

roxy

s available. his eadstoa ratherevere

efficiency

oss

as we wouldhave to

truncatever

50% of

the

observations

n

each

industry.

For

Olley-Pakes,

ho

use

dataon

arger

irms

n

the

apital-intensive

.S.

telecommunications

industry,

nly

% of

firm/year

bservations

re ero.

Positive

se of materialss

reported

or ver

99% of the

ample'sfirm-year

bservations

for ll

four

ndustries.

or

lectricity

he ractionf

non-zero bservationss

only

lightly

ower.

Fuels

are

non-zero

ormost

bservations,

utmaterials

nd/or

lectricity

re

preferred

s a

proxy

by

this

non-zeros

riterion. e will focus

principally

n these

wo

candidates,

lthough

he

ideas

we

discuss

pply

roadly

o

any

nput

nder

onsideration.

Except

for

lectricity,

e

have

very

ittle nformationn

input

rices.

Electricity

rices

were

fully egulatedy1982,

with he aw

requiring

hat

ll

plants onsuming

ess than MW



11/26

326 REVIEW OF ECONOMIC STUDIES

of

electricity

e

able

to

purchase lectricity

t a fixed nd

common

nit

rice.

his covers ver

90%

ofour

ample.22

Two

further

onsiderations

rising

rom

he

estimation

ssumptions

an

help provide

guidance

n

choosing mong

nputs.

irst,

heestimation

pproach

ssumes

here

s no

error

in the

nput

emand

quation,

o

for

ny

apital

evel nd

productivity

hock,

firms

assumed

toreadily e ableto obtain

tt

wt,kt).Thismaybeproblematicor lectricity,hichnChile t

the

imewas not

very eliably enerated

rdelivered. lant lowdownsnd

hutdownsaused

by

unreliable

upplymay

ead to

observed

lectricity

sage

that

s

different

rom

rue

emand. o

some

extent,

similar

tory

or

materialsnd/or

uels

may

hold,

specially

or

irmsocated n

areaswhere vents

ike

bad weather

an ead to

disruptions

n

delivery.

Second,

while measurement

rror s

always

a

concern,

conometric

heory

ells us

that

t

takes

on

a

heightenedmportance

hen

using non-parameteric

stimators. ence

inputs

measuredwith

ess

error

re

generally

referred

s

proxies.

On this

matter,

otential

measurement

roblems

rise

f

nputs

re

stored

eriod

o

period

nd

changes

n

inventories

of

inputs

re not

directly

bserved

for

xample,

irms

nlyreport

ew

nput urchases).23

n

our data firms ecord he amount

f

electricity

hey

urchase, enerate,

nd

sell,

so we can

compute onsumptionirectly.he inabilityostore lectricityor ongperiodsmeans hat ts

use should

e

highly

orrelated

ith

he

year-to-yearroductivity

erms.

Materials nd

fuels,

n

the

other

and,

may

be

easy

to store ver

ime,

ndhencenew

purchases

f these

nputs

which

we

observe)

may

not

xactly

rack

nputs

sed

n

production.

Three

pecification

ests

Because

the choice of a

proxy

has an

arbitrary

albeit

nformed)

lement,

we

suggest

hree

specification

ests.

irst,

n informal

ut

mportantpecification

heck

s

to

plot

he

proxy

s

a function

f its

two

explanatory

ariables. o be

empirically

onsistent

ith

he

model,

he

productivityhock should ncrease n theuse of the ntermediatenput, olding hecapital

level constant.

f

thefunctions

monotonic ut

decreasing,

r

f

the

functionoes not

satisfy

monotonicity,

ne

might

eed o

group

irms

ccording

o

some

ther

bservable(s)

o

oosen he

common actor

rice

estriction.

ne

could

till

se the

proxy

n

principle

hen he

function

s

monotone

ecreasing

onditional

n

capital,

ut he

nterpretation

f he

productivity

erm hat

it's

proxying

or

must

e

modified

n

a

way

hatmakes he

heory

onsistent ith

his esult

i.e.

why

oes

t

decrease

s

input

se

ncreases,

onditionaln

capital?).

A second est

sks whether

e

get

he ameestimates

sing

ither

lectricity

r materials.

Not

rejecting

hat he

stimatesre

the

ame

uggests

ither

nput

nd

the

ingle

actor

w)

may

be sufficient

or

modelling roduction. nfortunately,

his est oes not

provide

lear

guidance

as to the

problem

f

one

rejects; ejection

oes not

mean hat oth

model

pecifications

ail

one

may

becorrect).

Finally,

s

suggested

n

Olley-Pakes,

he

reely

ariable

nput,

abour,

hosen

n

this

eriod

should

ot e

correlated

ith

he

nnovation

n

productivity

ext

eriod

i.e. Corr(lt,

+l

=

0)).

We extend

histest o

include

ll six

inputs,

nd this

provides

s with ix

over-identifying

conditionshatwe use to test he ramework.

22. The aw

requires

he

egulated rice

be

within 10% band

round he

verage

rice

n

the

freely egotiated

contracts

everyone

ver2

MW).

For a discussion

f these

nd other ssues

relating

o the

regulation

f

electricity

n

Chile,

ee Bitran

nd

Saez

(1994).

23.

Input

nventories

ay

e

likely

o

occur

when he

torage

osts

re ow

nd

delivery

s not

ust-in-time

r

nput

prices ary ignificantly

ver ime.



12/26

LEVINSOHN

&

PETRIN

ESTIMATING PRODUCTION FUNCTIONS

327

5. ESTIMATION

In

this

section,

we cover

only

the

specifics

f how our estimation outine s

implemented.

Consistencyroofs

or urestimators oulduse results rom akes nd

Olley

1995).

Readers

interested

n

implementing

ur estimation outine re also directed o the estimation

ecipe

inAppendix ,which rovides detailed uide o the pproach.

The

irst

tage

To

keep

the

xposition

traightforward,

e

approximate

he

production

unction

ith

Cobb-

Douglas technology.24

e

write

Yt

=

fo

+

3kkt

fslt

+

fultu

eet

+

ff

ft

+

fmmt

+

tot

+

7t,

(10)

where

yt

s

the

og

of

gross utput

n

year

,

kt

s the

og

of the

plant's apital

tock,

l

is the

log

of skilled abour

nput,

'

is

the

og

of the

unskilledabour

nput,

nd

mt,

ft

and

et

denote

log-levels

f

materials,

uels

nd

electricity.

Weproceed s ifmaterials ere heproxy, ewriting10) as

Yt

=

fslt

+

ultu

+

feet

+

ff

ft +

t

(mt,

kt) +

1t,

(11)

where

Pt mt,

kt)

=

fo

+

fmmt fkkt

+

wt

mt, kt).

As with

Olley-Pakes,

11)

can be

estimated

sing

OLS

(and

a

polynomial

xpansion

n

mt

nd

kt

o

approximate

t -)).

We

take n

alternative

pproach

o

explore

different

on-parametric

stimator. e first

estimate

he

onditional oments

(yt

I

kt,

mt),

E(Ilt

I

kt,

mt),

E(ls

I

kt,

mt),E(et

I

kt,

mt),

and

E

(ft

I kt,

mt)

by

regressing

t

for

xample)

n

kt

and

mt.25

We use

a

locallyweighted

quadratic east squares approximation,lthoughn principle ne could use anyconsistent

parametric

r

non-parametric

stimatoror ach

ofthese onditional

eans.26We then

ubtract

the

xpectation

f

11)

conditionaln

kt,mt)

from

11)

to obtain

Yt

-

E(yt I kt,

mt)

=

fi

(lt

-

E(lt

I kt,

mt))

+

(ltu

-

E(ltu

I

kt,

mt))

+

fe(et

-

E(et

I

kt,

mt))

+

Of(ft

-

E(ft

I

kt,

mt))

+

rt.

(12)

No-intercept

LS

is then sedon this

quation

o estimate

irst-stage

arameters.

This

completes

he

irst

tage.

Although

here

re everal

stimation

teps

n

a

more

eneral

non-parametricpproach

ike

ours,

o

single

tep

s more

omplicated

han

(locallyweighted)

least

squares egression.27

f

we were

only

concerned ith he

marginal

roductivities

f the

variable

nputs

except

he

oefficient

n the

proxy

ariable),

we

could

top

here.

To obtain he

capital oefficient,plant-level easure fproductivity,nd/orn estimate freturnso scale

we

need more

omplete

model or

t

-)

because

both

lectricity

nd

capital

nter

ttwice.

24. As

Olley-Pakes

ote,

he

pproach

pplies

o

quite eneral

roduction

echnologies.

25. See

Section

.2,

and

specially

he irstew

ages

ofSection .2.1 of

Pagan

nd

Ullah

1999),

for

(relatively)

understandable

iscussion fkernel-basedstimates fthe oefficients

n the inear

erms f he

stimatingquation.

26. Readers

notfamiliar ith

ocal

quadratic egression

moothing ight

ind

t

helpful

o think

f this

tep

s

using

weighted

east

quares

o

construct

redictions

f

Yt

given

kt,mt)

using

s

regressors

hebasis for second-order

polynomial

pproximation

n

kt,mt).

For

nyparticularoint

k

,

m*)

forwhich

n

estimate f

the

xpected

alueof

Yt

s

necessary,

he

egression

eights

he bservations

losest othe

oint

k*,m*)

most

eavily.

consistent

stimator

of

E(yt

I kt

=

k7,

mt

=

m*)

s

the

ntercept

rom his ocal

quadratic egression.

27. We haveused

the

OLS-with-a-polynomial-approximationpproachallowing

or

ifferent

ub-periods

f the

sample

ccording

omacroeconomic

ycles)

ndwe find

n

most ases that

third-order

olynomial

pproximationives

very

imilar stimates

fthe

arameters.



13/26

328

REVIEW OF

ECONOMIC STUDIES

The

econd

tage

We use

two

moment

onditions

o

dentify

m

nd

fk.

As

with

Olley-Pakes,

urfirstmoment

condition

dentifies

k

by

ssuming

hat

apital

oes

not

espond

othe

nnovation

n

productivity

?t.

The secondmomentdentifies

im

y using

he act hat ast

period's

materialshoice hould

be uncorrelatedith he nnovationnproductivityhisperiod. hesepopulationmomentsre

given

y

E[(?t

+

it)kt]

=

E[?tkt]

=

0,

(13)

and

E[(?t

+

rlt)mt-1]

E[ltmt-i]

=

0.

(14)

We obtain n estimate

f

the esidual

rom he

ollowing

elationship:

t

rlt(*)

=

Yt

-

fisl

-

ult'

-

feet

-

Of

ft

-

O*mt

-

O*kt

E[cot

I

ot-1],

wherewe

explicitly

eference

heresidual s a functionf the wo

parameters

*

=

(P*,

Pf).

To estimate

[wt

ot-1]

we use

the stimatesf

t

obtained rom

hefirst

tage

esults

and

he

candidatealues

(m*

,

l9)).28

We also include he

six

over-identifying

onditions,

ielding

n

total

eight

population

moment

onditions

iven

y

the

vector f

expectations

E[(?t

+

tlt)Zt],

where

Zt

is thevector

iven y

Zt

=

{kt,

mt-1,

_,

It

_,

et-1,

ft-1,

kt-1,

mt-2}.

Finally,

we

obtain

stimates

Ok,

m)

by

minimizing

heGMM

criterionunction

Q(P*)

min

~8h=1

i

(tTio(+i,t

it(*))Ziht

,

(15)

ii

mni3

.sh=i

t=Ti0

where

indexing

irmss

explicit,

indexes he

eight

nstruments,

nd

Tio

and

Til

index he

second nd astperiod irm is observed.

Inferencesing

he

bootstrap

Measuring

he

recision

f

our

stimates

equires

s

to

ccount

or

he ariancen

every

stimator

that

nters urroutine

and

all of their

ovariances).

here re 11

estimatingquations

n

total,

and

many

stimates

et

used

more han

nce.

Pakes

and

Olley

1995)

provide

he

heoretical

framework

or

omputing

symptotic

tandardrrors.

Instead

of

undertaking

his

difficult

ask,

we

use

the

bootstrap

or

nference.29his

technique

re)samples

he

empirical

distribution

f

the

observed

data to construct ew

bootstrapped

amples.

he valueofthe tatistics

computed

or

ach

of

these

amples,

nd he

distributionfestimatesogeneratedrovideshebootstrappproximationo the rue ampling

distribution

fthe tatistic.

Our

resampling

ule reats ach

set

of

firm-level

bservations

ogether

s an

independent,

identical raw

rom he

verall

opulation

f

firms. e

sample

with

eplacement

nd

with

qual

probability

rom he ets ffirm-levelbservationsn the

riginal ample.

A

bootstrap

ample

s

considered

omplete

hen thas a number

f

firm-year

bservations

hat

quals

or

ust

xceeds)

thenumber f

firm-year

bservations

n the

riginal

ata.

The

bootstrap

s

easy

to

implement.

n

addition,

t also

provides symptotic

efinements

for

many

statistics,

ncluding

he

asymptotically

ivotal

ones

in

our

analysis.Finally,

he

28. See

Appendix

.

29.

See

Horowitz

2001)

for

n

overview

f

the

ootstrap.



14/26

LEVINSOHN

& PETRIN

ESTIMATING

PRODUCTION

FUNCTIONS 329

bootstrap

makes nference

n

differences

etween stimators

emarkably

traightforward.

he

usual

difficulty

hen

constructing

n

estimate

f the

variance

f

differencess the need for

the

covariance erm etween he two estimators

they

re

estimated

n

the same

sample).

A

distribution

f

differencesbtains cross he

ootstrapped

amples

y

ubtracting

ne estimate

from he

ther

for

ach of

these

amples).

he

sampling

istributiono

obtained

utomatically

accounts or he ovariance etween he stimators.

The

bootstrap pproach

must be

slightly

modifiedwhen

using

more moments han

parameters

o

obtain stimates

as

we

do

with he est f

over-identifying

onditions).

he

ogic

of the

bootstrap

equires

hat stimates btained

singbootstrappedamples

must

mplement

moments

hat

qual

zero n the

population

rom

hich he

bootstrapamples

re drawn

thats,

theobserved

ata).

Since this

population

s the

original

ata,

he

bootstrapped

oments ave

to be recentred

y

the stimated

alues

of themoments

sing

he

riginal

ata

at

the

bjective

function

inimum).

The

Bond nd Blundell V

approach

An alternativeV estimationtrategyhat lso deals with he core issue of simultaneitys

proposed

y

Blundell ndBond

2000).

They

tart ith he

ollowing

odel

in

their

otation):

Yit

=

i'xit

+

Ytrl+

i

+

vit

+

mit),

where

y

and x are

log)

output

nd

nputs

same

as

above),

y,

s

a

time-specific

ffect,

7i

s a

firm-specific

ixed

ffect,

it

s

AR(1),

and

mit

s

MA(0)

(say) arising

rom

measurement

rror.

n

this

model

ii,

vit

and

mit

can all

potentially

esult

n

estimateshat rebiased.Their V estimator

is robust

o correlation

etween

ach

of

these rrors

nd

potentially

ismeasured

nput

hoices

(at

the

xpense

f

placing ignificant

emands

n the

data).

They

use two

kinds f moments

or dentification.he first et uses

input

evels

agged

at east

wo

periods

s

instruments

n thefirst-differenced

quations

where

utput

s also

first-

differencedo condition n theAR(1) productivityerm). heyreport ifficultyn obtaining

precise

stimates

sing ust

thesemoment onditions.

hey

dd

an

additional et of moments

that

ses

suitably

agged

irstifferencesfvariables

s

instruments

or

he

quations

n

evels .

The additionalmomentsower he tandardrrors

nd

pass

an

over-identifying

est.

We

mplement

heBond ndBlundell stimator

s an alternative

o our

pproach

nd

report

the esults

n

the

next

ection.

6. RESULTS

In this

ection,

e

present

everal ets fresults ith everal

bjectives

n

mind. ur

over-riding

goal

is to llustrateow one can most

sefullymplement

he

ntermediate

nputs pproach.

o

one

approach

ill

be

appropriate

or

ll industries

n

all timeframes.

nstead,

he

ight

pproach

willdepend nthe etails f hendustryeing tudied. owards his nd,we show he easoning

used

to select

mong

roxies.

We

find

using

either

materials

r

electricity

s a

proxy

yields

statistically

ignificant

estimates

fthe

parameters

f

production

unctionsn the

Chilean

ase.

The

estimates

ighlight

howestimators

sing

ntermediate

nputs

o

control

or

nobservables

iffern

predictable

nd

informative

ays

rom ther

xisting

nd

ommonly

sedestimators.ur

results

re

fairly

obust

across

he

ndustries e

examine.

Thebase case

We

begin by presentingroduction

unction

stimates

orthe four

ndustriesiscussed

n

Section -Food Products311),Metals 381),Textiles321) andWoodProducts331). Table3



15/26

330 REVIEW OF ECONOMIC STUDIES

TABLE 3

Base

case

parameter

stimates

or our

ndustries

(bootstrapped

tandard rrorsn

parentheses)

Industry

ISIC

code)

Input

311 381 321

331

Unskilled

abour

0.139

0-172

0.130

0-193

(0-010) (0-033) (0-024) (0-034)

Skilled

abour

0.051 0.188 0.155

0.133

(0-009) (0-025) (0-026) (0-030)

Electricity

0-085 0-081 0-005

0-047

(0-007) (0-015)

(0-019) (0-021)

Fuels

0.023 0.020 0-038

0.021

(0-004) (0-011) (0-010) (0-014)

Materials

0.500 0.420

0-500

0.550

(0-078) (0.091) (0-118)

(0-086)

Capital

0.240

0.290

0-180

0.190

(0-053) (0-094) (0-095) (0-090)

Returns

o scale

1-037

1.172

1.007

1.133

(0-059) (0-075) (0-113)

(0-157)

No. obs.

6115 1394

1129

1032

presents

heresults

sing

materials

s the ntermediate

nput

roxy.30

e

find hat oefficients

arepreciselystimatedt standardevelsof statisticalignificance.The soleexceptions the

coefficientn

electricity

n

SIC

321.)

Especially

n

the

argest

ndustry

ISIC 311),

estimatesre

quite

precise.

There re

significant

ifferences

n

the

production

unctions

cross hese

four

ndustries,

but

none

of the ndustries

eally

tands ut as

having

radically

ifferent

echnology.

n

all

industries,

he oefficientn materials

s

the

argest

nd

consistently

overs round .50.

Capital

is

usually

he actor ith henext

ighest

oefficient.eturns

o

scale

range

rom

.04

for

SIC

311)

to

1.172

for

SIC

381),

although

stimatesre

generally

ot

ignificantly

ifferentrom

constant

eturns. e will

ndirectly

eturno these

ase

case

results

ince

many

f our oncerns

are focused

ot n the oefficients

n

Table 3

per

se,

but ather ow hese oefficientsifferrom

those btained ith raditionalstimators.

Alternative

roxies

The results

n

thebase case use

materials s the ntermediate

nput roxy.

here

re,

though,

other andidate

ntermediate

nputs,

nd these

nclude

fuels

nd

electricity.

n

Section

,

we

discussed

why

we

prefer

aterials

nd

electricity

o fuels s our

proxy.

he main eason

s both

arenon-zero or

irtually

ll firms

or

ll

time

eriods

basically

liminating

heneed o

truncate

observations).

ection also

suggested

hree

ther

pecification

ests or he hoice f he

roxy.

30.

See

Levinsohn nd Petrin

2000)

for esults rom hese our ndustriesnd four

thers

ith

lectricity

s

the

proxy.

evinsohn nd

Petrin

1999)

provides

stimates

f

value-added

roduction

unctionsnd

mplied roductivity

numberslso

using

he

lectricityroxy.



16/26

LEVINSOHN

& PETRIN ESTIMATING PRODUCTION

FUNCTIONS

331

The results f each of these hree ests

re

reported

ere.

n

each

test,

we takematerials

s the

principle

roxy

nd

electricity

s

the

ther

andidate.

The

first

pecification

est nvolves

isually xamining

he unction

t

=

ot

mt,

kt).

Recall

that he

monotonicity

ondition

equires

hat his unctione

increasing

n materials

conditional

on

capital).

he first

pecification

est

imply raphs

he moothed

o-function

nd ooks

for his

monotonicity.ecausewe believe hat hisfunction aydifferver he hreemacroeconomic

cycles

n

our

data,

we

havethree uchfunctionso

graph.

he first

pans

he

years

1979-1981,

the econd

1982-1983,

nd

the

hird 984-1986.

A

good example

f

the results hat ome out of thisexercise

s

provided y

ISIC 321

(textiles).

he three

anels

of

Figure

show he moothed

lots

for

materials

n

this

ndustry.

The verticalxismeasures

he stimated

roductivity

hock,

while he

xis

running

eft

measures

materials

sage

and

the axis

running ight

measures

apital.

f

the

monotonicity

ondition

always

held,

onditional

n

any

observedevel of

capital

materials

sage

would ncrease

when

productivity

ncreased.

n

thefirst

anel,productivity

s indeed

ncreasing

n

materials

or ll

levels of

capital.

This also

appears

o be the case

in

the second

time

period.

n the third

period,

t low levels

of materials

sage

and

high

evels

of

capital,

he

monotonicity

ondition

is sometimesiolated. verall,we find hat

monotonicityppears

o

argely

oldfor he

iggest

threendustries

or ll three

eriods.31

In

anyparticular

ndustry

hese unctions

ften iffercross

he

periods

as

in ISIC

321).

Additionally,

ithinn

ndustry-timeeriod,

he

ates f ncrease or

roductivity

ppear

o

vary

widely

cross

apital

evels

also

as

in

ISIC

321).

These results

mply

hat he

non-parametric

approaches

re

mportant;hey rovide

flexibility

obustothese

ifferences.f

course,

othing

in

our

methodologyrecludes

hese

non-parametriclots

from

ooking

iketheAndes-full

of

(smoothed)

eaks

nd

valleys-and

we find his o be the ase for

ur mallest

ndustry

or he

lasttwotime

eriods.

or this

ndustry

ne couldundertake

ome

of the

pecification

hanges

suggested

n

Section

to see

if

hey

estore

monotonicity.

Our

second

specification

est

ompares

he

parameter

stimates btained

withdifferent

proxies.

We

simply

sk fthe hoiceof

proxy

mattersor he

resulting

stimates.f themodel

is

correct,

ny proxy

atisfying

he

monotonicity

ondition

hould

heoreticallyive

similar

parameter

stimates

or he

reely

ariable

nputs

other

han he

Documents

2003_Levinsohn_Estimating Production Functions Using Inputs to Control for Unobservables