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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
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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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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).
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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.
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giambot
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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.
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http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp7/27/2019 2003_Levinsohn_Estimating Production Functions Using Inputs to Control for Unobservables
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
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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.
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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