Efficient Estimation with Panel Data When Instruments Are

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Efficient Estimation with Panel Data When Instruments Are Predetermined: An Empirical
Comparison of Moment-Condition Estimators
Author(s): James P. Ziliak
Reviewed work(s):
Source: Journal of Business & Economic Statistics, Vol. 15, No. 4 (Oct., 1997), pp. 419-431
Published by: American Statistical Association
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Efficient
Estimation
With
Panel
Data
When
Instruments Are
An
Predetermined:
Empirical
Estimators
Comparisonof Moment-Condition
James P. ZILIAK
Departmentof Economics,Universityof Oregon, Eugene, OR 97403-1285
I examinethe empiricalperformanceof instrumental
variablesestimatorswith predetermined
instrumentsin an applicationto life-cycle labor supplyunderuncertainty.The estimatorsstudied
aretwo-stageleastsquares,generalizedmethod-of-moments
(GMM),forwardfilter,independently
weightedGMM,and split-sampleinstrumentalvariables.I comparethe bias/efficiencytrade-off
for the estimatorsusing bootstrapalgorithmssuggestedby Freedmanandby BrownandNewey.
Resultsindicatethatthe downwardbias in GMMis quitesevereas the numberof momentconditions expands,outweighingthe gainsin efficiency.Theforward-filter
estimator,however,has lower
bias andis moreefficientthantwo-stageleast squares.
KEY WORDS: Bootstrap;Life-cyclelaborsupply;Overidentifying
restrictions;Split samples.
The panel-dataliteratureofferslittle guidanceon the rel- stratedvia MonteCarlosimulationthatthe bias in OMD is
ative empiricalperformanceof instrumentalvariables(IV) quite severe,the sourceof which,like feasiblegeneralleast
estimatorswhen appliedto samples of the size typically squares,is dueto a correlationbetweenthe samplemoments
encounteredin practice.For example,when the numberof and the estimatedweight matrixused in optimallyminiobservationsis large (say,greaterthan500), are the small- mizing the distance between populationand sample mosample concernsof a bias/efficiencytrade-offin general- ments. They attemptedto correctfor the bias by developized method-of-moments
(GMM)raisedby Tauchen(1986) ing a split-sampleestimator,called independentlyweighted
and Altonji and Segal (1994) binding?Moreover,if bias OMD (IW-OMD)but foundthatboth OMD and IW-OMD
is present,is it due to a correlationbetween the sample aredominatedin termsof lowerbias androotmean
squared
moments and the sample weight matrix (Altonji and Se- error(RMSE)by equallyweightedminimumdistance.Algal 1994) or to instrumentsweakly correlatedwith the en- thoughthe findingsof bias in GMMby Tauchenandby Aldogenousregressoras recentlydiscussedby Bound,Jaeger, tonjiandSegal seem conclusive,they arefor smallsamples,
and Baker (1995) and Angrist and Krueger(1995)? The andthereis no a
prioriexpectationthat comparableresults
latterissue is crucialfor efficiencyconsiderationsbecause exist in a
large panel-datasetting. If comparablenegative
the optimalnumberof momentsmay containinstruments results
againstGMM do exist, then a case could be made
datedfar into the past, possibly weakeningthe correlation for less efficientestimatorssuch as
two-stageleast squares
between the instrumentsand the endogenousregressor(s).
(2SLS) or Keane and Runkle's (1992) forward-filter(FF)
In this article,I examinethe samplepropertiesof several estimator.
panel-dataIV estimatorsfirst by applyingthem to a wellParallelto the time series researchon IV estimatorsof
known life-cycle labor-supplymodel and then comparing
and Segal (1994), Nelson and Startz (1990), and
the estimatorsin terms of the bias/efficiencytrade-offvia Altonji
Tauchen(1986) is cross-sectionalresearchon the propera bootstrapMonteCarlo.
ties of IV estimatorswhen the correlationbetweenthe inFor many panel-dataapplications,GMM is the obvious
strumentsand the endogenousregressoris weak (Bekker
estimatorof choice:It does not requirea full specification
and Stock 1994; Angristand Krueger1995;
of the stochasticprocess (Hansenand Singleton 1982), it 1994; Staiger
Boundet al. 1995).As pointedout by Boundet al., a weak
is consistent asymptoticallyunder a variety of situations
correlationbetweenthe instrumentsandthe endogenousreincludingwhen the only instrumentsavailableare predetermined ratherthan strictlyexogenous(Andersonand Hsiao gressorcan lead to (1) a large standarderror,(2) a bias in
IV even if a weak correlationexists between the instru1982; Arellano and Bond 1991), and it attains the effiments and the structuralerror,and (3) a bias in IV toward
ciency bound in the class of IV estimators(Chamberlain
least squares(OLS) as the explanatorypower of
ordinary
1987; Ahn and Schmidt 1995; Arellanoand Bover 1995).
the
instruments
approaches0. They demonstratedthat the
Tauchen(1986)demonstrated,
however,thatin samplestypresultsfrom the overidentifiedmodels
ically encounteredin time series applications(N = 50 or returns-to-schooling
and
reported
by Angrist
Krueger(1991) are biased toward
75) GMM is biased as the numberof momentconditions
OLS
due
to
correlated
instrumentsand,worse still,
weakly
expands,leadingto a bias/efficiencytrade-off,and thus he
recommendedthe use of "suboptimal"
instrumentsets.
AltonjiandSegal (1994)extendedthe small-sampleanal? 1997 American Statistical Association
ysis of Tauchento least squaresoptimalminimum-distance
Journal of Business & Economic Statistics
(OMD) estimationof covariancestructures.They demonOctober 1997, Vol. 15, No. 4
419
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All use subject to JSTOR Terms and Conditions
420
Journalof Business & EconomicStatistics,October1997
thatsimilarIV resultsare foundby using instrumentsfrom tions test. Because each algorithmis asymptoticallyvalid
a uniformrandom-number
for the estimator,I comparebootstrapresultsfrom the apgenerator.
the
criticisms
et
of
Bound
and
al.,
Addressing
Angrist
proachesof both FreedmanandBrownandNewey.
a
convenient
From a 10-yearbalancedpanel of men, I find the fol(1995)
Krueger
developed computationally
IV
as
an
alternative
to
estimator
2SLS.
(SSIV)
split-sample
lowing results of note. The downwardbias in GMM is
SSIV is not biased towardOLS;however,the estimatoris quite severeas the numberof momentconditionsexpands,
biasedtoward0, so they proposedto "inflate"the SSIV es- outweighingthe gains in efficiency.The bias is due to a
timateswith a bias-correctionfactor,giving unbiasedSSIV correlationbetween the sample momentsused in estima(USSIV).TheyconcludedthatUSSIV gives resultscompa- tion and the estimatedweight matrix.The IW-GMMesrableto 2SLS but at a substantialloss of efficiencydue to timatoris generally successful at eliminatingthe bias in
the smallersamplesused in estimation.Becausepaneldata GMMparameterestimatesusing Freedman'sbootstrapalwith a long time series presentthe opportunityfor many gorithm;however,the standarderrorsfrom asymptoticthe(possiblyweaklycorrelated)instruments,SSIV andUSSIV ory seem to understatethe truesamplingvariation,andthe
test tendsto overreject.Thelevoveridentifying-restrictions
may be viable alternativesto standard2SLS or GMM.
In this article,I focus on panel-dataIV estimatorsthat els distortionin the overidentifying-restrictions
test persists
are consistent asymptoticallywhen only predetermined in models with many moments,even after recenteringthe
instrumentsare available. Applications that fall within distributionusingBrownandNewey's (1995)algorithm.Fithis class of estimatorsincludedynamicmodels, rational- nally,the bias in FF parameterestimatesis less thanGMM
models. and 2SLS, and it is more efficientthan2SLS.
expectationsmodels, and simultaneous-equations
Predeterminedinstrumentscomplicate the estimation of
AND PREDETERMINED
1. ESTIMATION
thateliminate
suchmodelsbecausecertaintransformations
INSTRUMENTS
the model'stemporallypersistentlatentheterogeneity,such
as deviationsfromtime-means(the "within"estimator),are
I begin with a brief overviewof the IV estimatorsused
inconsistentwhen instrumentsare predetermined(Keane in the empiricalapplicationandbootstrapsimulations.Conand Runkle 1992). The estimatorsconsideredhere include siderthe linearregressionfor individuali (i= 1,..., N) in
2SLS, GMM,FF, SSIV,USSIV, and an IV analogto IW- time t (t = 1,...,T)
OMD thatI refer to as IW-GMM.
I study the sample propertiesof the estimatorsin a
realistic setting using data from the Panel Study of InYit= +i Xit3 Eit,
(1)
come Dynamics (PSID). The empiricalmodel employed
is MaCurdy's(1985) life-cycle labor-supplymodel under uncertainty.MaCurdy'smodel is pertinentto the is- wherea2 representsfixed latentheterogeneity,xit is a (1 x
sues studiedhere becauseuncertaintysuggests a rational- K) vector of predeterminedexplanatoryvariables,/ is a
expectationssolution to the consumer'sproblem,thereby (K x 1) vectorof parametersto estimate,andEitis a random
makingthe instrumentset predetermined.In addition,the errorthatvariesover i and t andis assumeddistributediid
model producesestimatesof the intertemporalsubstitution (0, cr). Under the assumptionof fixed effects, the latent
elasticity,a key parameterused in understandingthe co- heterogeneityis correlatedwith the explanatoryvariables
movementsin earningsand hoursover the businesscycle. for all periods;thatis, E[ailxit] 4 0 for all t.
A common practice in panel data is to eliminate the
For each estimator,I sequentiallybuildup the momentmatrix by adding extra years as instruments,and I test the fixed effect by takingdeviationsfrom the individual'stime
specificationwith Sargan'stest of the overidentifyingre- series means, known as the within transformation.In IV
estimationwith predeterminedinstruments,however,the
strictions(Godfrey1988).
is inconsistent.In particular,for the
with
a
the
I complement empiricalinvestigation
bootstrap withintransformation
MonteCarloas developedby Efron(1979) andextendedto within transformationto be consistentit is necessaryfor
IV by Freedman(1984) and Freedmanand Peters (1984). the instrumentset to be strictlyexogenousto the model's
The bootstrapis used to comparethe estimatorsin terms error term for all periods, E[eis Wit] = 0 for all s, t;
of bias, efficiency,RMSE, medianabsoluteerror(MAE), however,predeterminedinstrumentsonly guaranteeweak
and asymptoticcoveragerates. The strengthof the boot- exogeneity, E[Eis]Wit] = 0 for all s > t. This inconsisstrapover standardMonte Carlo analysis lies in the fact tency carries over into the class of endogenousrandomthat,like the IV estimatorsthemselves,the researchersim- effects estimatorsstudiedby Hausmanand Taylor(1981).
ply approximatesthe empiricaldistributionof the estimator The first-difference(Andersonand Hsiao 1982;Keaneand
estimateof the underlyingerrordis- Runkle 1992; Schmidt, Ahn, and Wyhowski 1992) and
with a nonparametric
tribution.AlthoughFreedman'smethod is asymptotically orthogonal-deviations
(Arellanoand Bover 1995) transforestimatorsfromoveridentifiedmod- mationsare consistentwhen appliedwith lagged levels of
validfor bootstrapping
els (Hahnin press),BrownandNewey (1995) andHall and predetermined(or endogenous)regressorsas instruments.
Horowitz(in press) showed that the Freedmanalgorithm Orthogonaldeviationsmay offer efficiencygains over first
does not yield an improvementin termsof coveragerates differencesbecausedifferencingexacerbatesmeasurement
(Maeshiroand Vali 1988). Each transover asymptotictheory,and more importantly,it gives the errors-in-variables
will
to the estimators.
be
formation
restricfor
the
size
applied
overidentifying
wrong
asymptotically
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Ziliak:EfficientEstimationWithPanel Data
421
Estimators
1.1 Method-of-Moments
The first estimatorI consideris 2SLS, which minimizes
the distancebetweenthe samplemomentsand the population moments,givingequalweightto each observation.The
2SLS estimatorproducesconsistentparameterestimatesfor
eitherthe first-differenceor orthogonal-deviations
transformationsand is given as
=
(2)
/32SLS (X'P(W)X)- (X'P(W)y),
where W is an (N(T - 1) x L) matrix of instruments,
P(W) = W(W'W)-1W' is the projectionmatrixof instruments,E is the stacked(N(T - 1) x 1) vectorof residuals,
X is the stacked(N(T-1) x K) matrixof regressors,andy
is the stacked(N(T - 1) x 1) dependentvariable.Underthe
assumptionof conditionalhomoscedasticity,E[e21W]= 0,
inferencefor the estimated2SLS parametersis conducted
with the variance-covariance
matrix
var(32SLS)
-=
S=
2(X'P(W)X)-1
NT N-K
(3)
(y- X32SLS)2.
As notedby White (1982), 2SLS standarderrorsareinconsistent when the conditionalhomoscedasticityassumption
is violated;thus, he proposeda robustcovariancematrix
estimatedas
var(02SLS) = (X'P(W)X)-1X'P(W)
x QP(W)X(X'P(W)X)-1,
(4)
where Q is a diagonalmatrixof squaredresiduals.
Because the number of instrumentstypically exceeds
the numberof parametersestimated (L > K), one can
test the overidentifyingrestrictionswith the Sargantest
as e(32SLS)P(W)E(G32SLS))/-E
, whichis asymptoticallydistributedX2 with L - K df (Godfrey1988). In the case of
conditionalheteroscedasticity,
Hansen's(1982)robustvariantof the overidentifying-restrictions
test givenlateris necessary for consistentinference.
1.1.1 GeneralizedMethod-of-Moments.Hansen(1982)
and White (1982) showedthat improvementsin efficiency
over 2SLS are possible by optimally weighting the distance between the sample and populationmoments,with
the weight being the inverse of the covariancematrixof
sample moments.The ensuing GMM estimatortypically
relies on residualsfrom the 2SLS estimatorfor an initial
consistent estimate of the covariancematrix. The GMM
estimatoris
andHansen'sversionof the overidentifying-restrictions
test
is e(Ogmm)'P(W(S))e(igmm),
which is distributedasymptoticallyX2with (L - K) df.
1.1.2 ForwardFiltering. As an alternativeto 2SLSand
GMM, Keane and Runkle (1992) proposedan estimator
called the forward-filter(FF) estimator.The FF estimator
eliminatesall forms of serial correlationwhile still maintaining orthogonalitybetween the initial instrumentset,
whichcontainslaggedvaluesof predetermined/endogenous
variables,and the structuralerror.Thereis a similaritybetweenforward-filtering
andthe orthogonal-deviations
transformationdevelopedby Arellanoand Bover (1995) in that
both methodsdemeanthe variableswith only currentand
futurevalues;however,filteringis likely to be superiorbecauseit eliminatesall formsof serialcorrelation.Although
Schmidtet al. (1992) arguedthat filteringis irrelevantif
one exploits all samplemomentsduringestimation,filtering maybe a desirablealternativeto GMMin practice.First,
becausethe dimensionof the GMMmomentmatrixgrows
exponentiallyas the numberof time periodsandregressors
intractableandthe overexpands,it can be computationally
identifyingrestrictionsare less likely to be satisfied,possibly due to a weak correlationbetweeninstrumentsandendogenousregressors.Second,if the small-sampleevidence
from Tauchen(1986) and Altonji and Segal (1994) carries
over to the panel-datasetting, a bias/efficiency trade-off
will arise with the optimalGMMestimator.
Similarto GMMthe FF estimatoris a two-stepestimator.
In the first step, the first-differenced
equationis estimated
residby 2SLS. The ((T - 1) x 1) vectorof first-difference
uals for individuali from the 2SLS regression,Ei,2SLS, are
used in constructinga (T - 1) x (T - 1) covariancematrix,
N=1 ,2SLSI,2SLS.The inverseof the covariFD
ance matrixis then filteredby a Choleskydecomposition,
CFF - Chol(E1), thateliminatesserialcorrelationin the
differencederrors.The second step involves transforming
the originalstackedN(T - 1) first-differenceobservations
by QFF = IN 0 CFF, leadingto the FF estimator
3FF= (X'QFFP(W)QFFX)-1(X'QFFP(W)QFFY), (7)
=
with variance-covariance matrix
var(3FF)
2(X'QFF
As
noted
infer(1992),
P(W)QFFX)-1.
by Hayashi
ence with the FF estimatoris inconsistentif the conditional homoscedasticity assumption is violated; however,
heteroscedasticity-robust variants of the variance and the
overidentifying restrictions test are easy to compute.
1.1.3 Independently Weighted Generalized Method-of-
Moments. In an attemptto mitigatethe finite-samplebias
in OMD,AltonjiandSegal (1994)developeda new estimator
calledindependentlyweightedOMD.I extendtheiridea
=
(5)
(X'P(W(S))X)-1(X'P(W(S))y),
/3gmm
to the case of IV with panel data. The motivationbehind
where P(W(S)) = W(S)-1W' is the projectionmatrixof IW-GMMis to breakthe correlationbetween the sample
instrumentsand S = W'2W is the optimal weight ma- momentsusedin estimation,(1/N)[W'E],andthe estimated
trix that permits both conditionalheteroscedasticityand weight matrix,S = W'2W, that is constructedwith the
autocorrelationin the covariancematrix Q. The variance- same data. The procedureis to randomlysplit the sample
into independentgroups(g), say two (g = 1,2), with group
covariancematrixfor the GMMestimatoris
1 used in constructingthe weight matrix,S1 = W2i W1,
(6) and group 2 used in constructingthe sample momentsto
var( gmm)= (X'P(W(S))X)-1,
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422
Journalof Business & EconomicStatistics,October1997
estimate,(1/N)[W2e2]. The samplesplit mustoccuron the
cross-sectionaldimensionof the data becauseeach crosssectionalunit retainsits own time series for instruments.
Each groupis used alternatelyin constructingthe weight
matrixand the samplemomentsso that the resultingIWGMM estimatoris the averageof the independentestimations:
OIW-GMM =
S
(X1,Wg(W?9gQgW g<W9Xg)1
g=1
x
Wf9g- gW,)(X;Wg(W'9
1Wyg),
(8)
where -g refers to the excludedgroup.Because the sampling errorsof the populationmoments and weight matrix are independent,the averageis a consistent estimator.The varianceof the estimatedaverageis constructedas
var(CIW-GMM)
=
(1/G2)[var(l31) + -- + var(CG)],where
the covariancebetweenith andjth samplesplitis 0 by construction.The IW-GMMestimatoris expectedto be less efficientthanGMMbecauseof the loss in degreesof freedom.
In the applicationto follow, I fix G = 2 for the IW-GMM
estimator.
1.1.4 Split-Sample Instrumental Variables. Angrist
where W21 = W1(W2W2)-1W2X2. The asymptoticcovariancematrixtakesthe usualformas 2SLS underconditional
homoscedasticityas0, (X' W21(W1 W21)-1W1X1i)1 and
the robustcovariancematrixis the sameas in Equation(4)
with W21replacingW. For obviousreasons,thereis no test
of the overidentifyingrestrictionsin this case.
1.2 Choice of Instrument Set
To this point,little attentionhas been givento the specificationof the instrumentset otherthanthe overidentifyingrestrictionstest. When the matrix of explanatoryvariables is predetermined,
the first-differenceand orthogonaldeviationstransformationsmake instrumentsdated t - 1
and earliervalid for estimation.In the standardIV estimator, choosing one-period lagged instruments, W
=
X(t_1),
leaves the systemjust identified(L = K), whereaschoosing instruments dated t - 1 and t - 2 imposes p = 2K - K
overidentifyingrestrictionsbut sacrificesan extraperiodof
datafor each observation.
Schmidtet al. (1992)arguedthatefficiencygainsarepossible if, insteadof the usualinstrumentset, one exploitsall
of the linearmomentsavailableas impliedby the orthogonality conditionsE[WiEci,d] = 0, where Wi is the matrixof
instrumentsfor individuali and Ei,d is the vector of firstdifferenceor orthogonaldeviationsresiduals.The Schmidt
et al. approachuses levels of instrumentsfrom different
time periodsfor differentobservations.The instrumentset
for individuali when all the regressorsare predetermined
is constructedas
and Krueger(1995) confrontedthe problemof a weak correlation between the instrumentsand regressorsin their
earlierwork by developingthe split-sampleIV (SSIV) estimator.Recall that if the instrumentsand the endogenous regressor(s)are weakly correlatedand there exists a
0 ? 0
0
X, 0 0 0 ? - 0
(weak)correlationbetweenthe errorfromthe first-stagefit0
0
0
0
0
0
?
? ?
X2
Xl
ted value and the structuralerror,then IV is biased.SSIV
breaksthe correlationbetweenthe two errorsby randomly wi =
0o0 o o
splittingthe samplein half and using one-halfto estimate
.
"
"
S. . .
T-1
x1Xl
2
the first-stageequationand the other half to estimatethe
structuralparameters.Let sample2 estimatethe first-stage
(11)
equationandcombinethatwith W1to formthe fittedvalue
for X1, which is then regressedon yl, yielding the SSIV which has dimension (T - 1) x (T)(T - 1)(K)/2 and where
estimator
zt(t = 1,. .., T- 1) are the lagged levels of the explanatory
variables.For example,in period2 variablesfromperiod1
- W2X2)-1
= (X' W2
W'
)
are valid instruments,in period 3 variablesfrom both peWI(W2W2
(W2W2)-1
issIv
riods 1 and 2 are valid, and so on until period T, where
x XW2(W
2)1 Wjy, (9) variablesfrom periods 1 to T - 1 are valid instruments.
= 15 and K = 10,
which underconditionalhomoscedasticityhas an asymp- When T and K are both large, say T
there
are
restrictions,
1,040 overidentifying
highlighting the
totic covariance matrix of var(3ssiv) = &2(XW2(W
computational burden of the Schmidt et al. approach. In the
W2)-1W•X2)-1. A loss of efficiency relative to 2SLS is
to follow, I compare the efficiency of the stanexpected with SSIV because only one-half of the observa- application
set to the stacked instrument set in Equadard
instrument
tions are used in estimation. This divergence in efficiency
tion
(11).
may be exacerbated when heteroscedasticity-robust covariance matrices of Equation (4) are employed.
2. AN APPLICATIONTO LIFE-CYCLELABOR
Because SSIV is biased toward 0, Angrist and Krueger
SUPPLY UNDER UNCERTAINTY
(1995) inflated the SSIV estimator with a bias-correction
A focal point of interest in the labor-supply literature is
factor, resulting in a just-identified 2SLS estimator in samand efficient estimation of the intertemporalsubconsistent
ple 1. USSIV is consistent under group asymptotics and is
stitution elasticity (ISE). The ISE measures intertemporal
given by the formula
changes in hours of work due to an anticipated change in
the
real wage and aids in understanding comovements in
/uSSIV
= (X W 21(W
W2)W X)-1
earnings and hours over the business cycle. Consequently,
estimates of the ISE are of import to public policy
reliable
x XW;1(VINW21w)-1w21y
(10)
,
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Ziliak:EfficientEstimationWithPanel Data
in labormarkets.The life-cycle labor-supplymodel developedby MaCurdy(1985) andAltonji(1986)formsthe basis
of the empiricalexercise.
Based on a utility functionthat is additivelyseparable
betweenconsumptionandleisure,the double-loglife-cycle
labor-supplyfunctionfor individuali (i = 1,..., N) in time
t (t = 1,..., T) is
423
based on the first-differencetransformationand 46 based
on the orthogonal-deviations
transformation.The FF estimatoris not includedunderorthogonaldeviationsbecause
filteringwould be redundantin this case. I begin with the
standardinstrumentset with zit's fromt- 1 andt- 2, along
with lagged wages from t - 2. I then sequentiallybuildup
the stackedinstrumentset fromEquation(11). Becausethe
ISE
is the parameterof primaryinterest,I only presentreInhit = Aio+ 6 Inwit + zitly + Eit,
(12)
sults for the ISE in Table 1. The efficiencypatternin the
where In is the naturallog operator;hit is annualhours
variableswith changesin momentconditions
of work; A0ois the marginalutility of initial wealth that demographic
is similarto the patternin the ISE. The weightingmatrix
is a functionof all futurewages, assets, prices, and tastes for the first-difference
GMMestimatorthatcorrectedfor a
and is correlatedwith the explanatoryvariables;zit is a first-order
moving averageerrorfailed to be positive defvectorof time-varyingdemographics;and6 is the ISE painite in the stacked-momentmodels, even with modified
rameter.The assumptionsbetweenthe regressorsandstruc- Bartlett
weights; hence, the results reportedonly correct
tural randomerrorare E[cisln wit] = 0 for all s > t and for conditional
As a point of departure,
heteroscedasticity.
E[Eis zit] = 0 for all s > t. Wages are assumedto be en- I presentOLS estimatesat the bottomof Table 1. The OLS
to accountfor the pos- first-differenceISE is .111 with a
dogenousratherthanpredetermined
heteroscedasticity-robust
sible presenceof nonlinearincometaxesand/orhumancap- standarderrorof
.079, andthe orthogonal-deviations
ISE is
ital considerations,or possiblymeasurementerror.
.176 with a heteroscedasticity-robust
standarderrorof .074.
Estimationproceedsby takingfirstdifferencesor orthog2.2.1 Base-CaseResults. In the base case with demoonal deviationson Equation(12) to eliminate the unobgraphicvariablesfromt - 1 and t - 2 andwages fromt - 2
served marginalutility of wealth and then using the es- as instrumentsfor the
the firsttimatorsoutlinedin Section 1. Because of the endogene- difference2SLS and contemporaneousperiod,
GMM estimatesof the ISE are .21
ity of wages and the desire to use lagged values of wages and .52, respectively;however,neitherestimateis
signifias instruments,an extrayear of data is lost; consequently, cant even at the 10%level. As a benchmark
to judge the
there are a maximum of (T - 2) x (T - 2)[(K - 1)(T first-difference
base-caseresults,Altonji(1986)estimateda
+ 1) + T]/2 linear momentconditionsto exploit in estiof
values
from .01 to .45, most of which are imprerange
mation.The stackedinstrumentmatrixin Equation(11) for
estimated.
The orthogonal-deviations
2SLS estimates
cisely
the labor-supplymodel contains,in period 3, wages from are more efficient than
their first-differencecounterparts,
period 1 along with demographicsin periods 1 and 2 as possibly due to reducedmeasurementerrorrelativeto difinstruments,wages fromperiods1 and 2 along with demo- ferencing;however,thereis no
generalpatternof efficiency
graphicsfrom periods 1 to 3 are instrumentsin period4, gainsacrossthe otherorthogonal-deviations
estimators.The
and so on until period T when there are (T - 2) lagged IW-GMMestimates
across
the
first-difference
and
vary
wages and (K - 1)(T - 1) lagged demographic variables as orthogonal-deviations
transformations
in thatthe formerininstruments.
flate the correspondingGMM estimateby about40% and
the
latterdeflateit by about40%. Of course,the objective
2.1 Data
of IW-GMMis to correctfor bias in GMM,whetherit be
The dataused to estimatethe life-cycle labor-supplypa- positive or
negativebias. On the contrary,SSIV and USrameterscome fromWavesXII-XXI (calendaryears 1978- SIV
uniformlyinflatetheir2SLS counterpartsacrossspec1987) of the PSID.The sampleis selectedon manydimen- ifications,althoughthere is a substantialloss in
efficiency
sions and is similar to other researchstudyinglife-cycle due to smaller
samplesizes. Interestingly,underconditional
modelsof laborsupply.The sampleis restrictedto continu- homoscedasticity,the FF estimator
performsbest on effiously married,continuouslyworking,prime-agemen aged ciency groundsin the base case, even surpassingGMM;
22-51 in 1978 from the Survey Research Center random however,once one accountsfor
heteroscedasticity,GMM
subsample of the PSID. In addition the individual must ei- and IW-GMMimproveon the FF estimates.
ther be paid an hourly wage rate or must be salaried, and
The resultsfrom the Sarganoveridentifying-restrictions
he cannot be a piece-rate worker or self-employed. This tests aremixedacrossthe models.The
conditionallyhomoselection process resulted in a balanced panel of 532 men scedasticp values of .02 and .01 from the first-difference
over 10 years or 5,320 observations. The real wage rate, wit, 2SLS and FF estimatorsdo not lend much
supportto the
is the hourly wage reported by the panel participant rather choice of instruments;
however,the GMMand orthogonalthan the average wage (annual earnings over annual hours) deviations versions do not reject the
overidentifying
to minimize division bias (Borjas 1981). The predetermined restrictions.The test rejectsthe IW-GMMspecificationsuntime-varying taste shifters, zit, include a quadratic in age, der first differencesbut does not reject SSIV models.Two
the number of children in the household, and a dummy vari- importantfindings that emerge in the base case and are
able for bad health.
magnifiedin the stacked-momentresultsto follow are that
(1) once the Sargantest is modifiedfor the presenceof het2.2 Results
eroscedasticity,the overidentifyingrestrictionsare not reIn Table 1 I report the results from 101 regressions, 55 jected in any of the 2SLS or FF specificationsand (2) as
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424
Journal of Business & Economic Statistics, October 1997
Table1. Intertemporal
SubstitutionElasticitiesUnderAlternativeMoment-Condition
Estimators
Firstdifference
Moments
2SLS
Base case:
T - 1, T - 2
GMM
FF
IW-GMM SSIV
USSIV
2SLS
Orthogonaldeviations
GMM IW-GMM
SSIV
USSIV
.2091
(.4155)
(.4231)
[.0234]
[.3147]
.5192
(.3638)
[.4482]
.1350
(.3163)
(.3857)
[.0138]
[.1925]
.7161
(.3498)
[.0059]
.7158
(.6183)
(.7109)
[.3447]
[.4356]
1.7491
(2.0709)
(1.8272)
.2552
(.3773)
(.4488)
[.1361]
[.5744]
.4506
(.4011)
[.6431]
.2603
(.4044)
[.0392]
.8931
(.6427)
(.8237)
[.5862]
[.8326]
.5428
(.1808)
(.2259)
[.0700]
[.3837]
.3942
(.1504)
[.4248]
.5422
(.1525)
(.2007)
[.0284]
[.2778]
.3971
(.0846)
[.0000]
.6752
(.2120)
(.4878)
[.0000]
[.0002]
6.0887
(7.3626)
(7.1342)
.7130
(.1719)
(.2119)
[.0697]
[.1185]
.6158
(.1586)
[.2040]
.5069
(.0908)
[.0000]
.4289
(.1764)
(.4880)
[.8188]
[.9315]
8.2390
(19.205)
(18.444)
3
.5620
(.1436)
(.1817)
[.0632]
[.3932]
.3916
(.1158)
[.4560]
.2868
(.0584)
[.0000]
.3768
(.1609)
(.3564)
[.0017]
[.0065]
-3.4139
(3.6721)
(4.2839)
.3724
(.0626)
[.0000]
.1933
(.1013)
[.3192]
.1832
(.0462)
[.0000]
.6702
(.8119)
(1.2464)
.2778
(.1031)
[.2363]
.2819
(.0497)
[.0000]
.1714
(.0898)
[.0480]
.2838
(.0412)
[.0000]
.3502
(.1084)
(.1772)
[.0000]
[.1927]
.2763
(.1028)
(.1758)
[.0000]
[.2051]
.1524
(.0802)
[.3509]
.1448
(.0799)
[.1303]
.4031
(.0346)
[.0000]
.1413
(.0743)
[.1849]
.3082
(.0287)
[.0000]
-.0654
(.0850)
(.2331)
[.0000]
[.0000]
.7629
(1.1336)
(1.6145)
.2653
(.1003)
(.1686)
[.0000]
[.1663]
.0659
(.0691)
[.3562]
.1051
(.0679)
[.1709]
.2781
(.0262)
[.0000]
.0931
(.0674)
[.3787]
.1227
(.0661)
[.1926]
.2544
(.0255)
[.0000]
-.0800
(.0825)
(.2270)
[.0000]
[.0000]
-.0824
(.0822)
(.2207)
[.0000]
[.0000]
.8239
(.9977)
(1.3779)
.2814
(.0987)
(.1623)
[.0000]
[.1875]
.1115
(.0247)
(.0791)
.0881
(.1168)
(.2886)
[.0000]
[.0002]
.0218
(.1044)
(.2389)
[.0000]
[.0000]
.0299
(.0940)
(.2242)
[.0000]
[.0000]
.0158
(.0849)
(.2109)
[.0000]
[.0000]
.0017
(.0834)
(.2035)
[.0000]
[.0000]
.0046
(.0827)
(.1949)
[.0000]
[.0000]
.1070
(.1342)
(.3406)
[.3946]
[.4759]
-.0489
(.1064)
(.3007)
[.0000]
[.0000]
-.0898
(.0972)
(.2779)
[.0000]
[.0000]
-.0754
(.0910)
(.2550)
[.0000]
[.0000]
-2.4665
(6.1924)
(7.2017)
.3774
(.1279)
(.1780)
[.0000]
[.2211]
.3123
(.1179)
(.1742)
[.0000]
[.0705]
.5885
(.1353)
(.1792)
[.0109]
[.0959]
.5432
(.1242)
(.1733)
[.0002]
[.0772]
.4006
(.1133)
(.2000)
[.0000]
[.0125]
.4051
(.1044)
(.1860)
[.0000]
[.0253]
.3650
(.1005)
(.1799)
[.0000]
[.0505]
.3408
(.0984)
(.1738)
[.0000]
[.0364]
.3512
(.0971)
(.1694)
[.0000]
[.0431]
.3768
(.1156)
[.2209]
T - 1 to T - 4
.5093
(.1225)
(.1779)
[.0089]
[.2096]
.4568
(.1118)
(.1711)
[.0003]
[.2715]
.3108
(.1008)
(.1829)
[.0927]
[.0824]
{9}
Stacked cases:
T - 1 to T - 2
{72}
T-
1 to T-
{107}
{137}
T - 1 to T - 5
{162}
T - 1 to T - 6
{182}
T - 1 to T - 7
{197}
T - 1 to T- 8
{207}
T-
1to T-9
{212}
OLS
.1186
(.0878)
[.1393]
.1017
(.0741)
[.3510]
.3517
(.0927)
(.1778)
[.0000]
[.1383]
.3069
(.0893)
(.1677)
[.0000]
[.2126]
.2781
(.0869)
(.1592)
[.0000]
[.1593]
.2961
(.0856)
(.1540)
[.0000]
[.1793]
.3170
(.0346)
[.0000]
.3615
(.0311)
[.0000]
.3444
(.0247)
[.0000]
.3479
(.0237)
[.0000]
.3173
(.0228)
[.0000]
-.1169
(.5247)
(.7319)
-.2473
(.6665)
(.9609)
-.2577
(.7653)
(1.0709)
-.1355
(.7492)
(.9927)
-.1808
(.7238)
(.9267)
1.1224
(.7987)
(.8270)
.3272
(.9251)
(1.4420)
.5920
(.7368)
(1.1203)
.7449
(.9966)
(1.4588)
.7825
(.9308)
(1.2813)
.1755
(.0224)
(.0743)
are in squarebrackets.The second valuesin parenthesesandsquarebrackets
restrictions
NOTE:Standarderrorsare in parenthesesandp valuesforthe nullhypothesisof correctoveridentifying
standarderrorsand p values. N = 532 and T = 8. The numbersbetween( } are the numberof instruments
forthe 2SLS, FF,SSIV,and USSIVestimatorsreferto heteroscedasticity-robust
used in estimation.
the ratio of instruments to cross-sectional sample size rises,
Hansen's test seems to overreject. This will become evident
in Subsection 2.2.2 when one compares GMM to IW-GMM
and 2SLS to SSIV.
2.2.2 Stacked Moment Results. Turning now to the
stacked moment-condition results reveals a striking change
in the efficiency of the ISE. With 72 moment conditions
in the stacked t - 1 and t - 2 case, significant efficiency
gains are achieved for all the estimators, with the average
reduction in standard errors around 50%. The FF estimator outperforms 2SLS in efficiency in all cases, but GMM
is more efficient than FF in all of the stacked-moment
specifications. No clear efficiency pattern emerges between
orthogonal deviations and first-difference GMM, nor with
heteroscedasticity-robust 2SLS across the transformations.
Quite remarkably,IW-GMM outperforms GMM across the
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Ziliak: Efficient Estimation With Panel Data
425
bias-correctionfactormay be undesirablewith manyoveridentifyingrestrictions.The FF estimatorfrom Keaneand
Orthogonal-deviations Runkle
(1992)does not appearto sufferfromthe samebias
F test
WaldTest
as in GMMandyet is more efficientthan2SLS.
Table2. First-StageF and WaldTests for Wage Changes
First-difference
F test
Moments
Base case:
T - 1, T -
2.2.3 Testsfor WeakInstruments. An obvious question
2
2.541
20.095
3.979
30.293
[.007]
[.017]
[.000]
2
1.182
[.141]
92.589
[.052]
1.382
[.019]
104.331
[.008]
3
1.197
[.084]
129.862
[.066]
1.303
[.021]
150.497
[.004]
4
1.098
[.208]
160.540
[.083]
1.154
[.109]
187.569
[.003]
5
1.074
[.251]
210.043
[.007]
1.092
[.205]
245.218
[.000]
6
1.138
[.104]
241.216
[.002]
1.120
[.134]
286.980
[.000]
7
1.149
[.080]
271.128
[.000]
1.097
[.175]
309.148
[.000]
8
1.146
[.079]
291.226
[.000]
1.079
[.214]
326.667
[.000]
9
1.160
[.061]
325.604
[.000]
1.080
[.210]
362.342
[.000]
{9}
Stacked cases:
T - 1 to T {72}
T- 1 to T{107}
T- 1 to T {137}
T- 1 to T {162}
T- 1 to T {182}
T- 1 to T {197}
T- 1 to T {207}
T - 1 to T {212}
WaldTest
[.000]
NOTE: The nullhypothesisis thatthe instrumentsjointlyexplainnone of the variationin the
wagechanges. P valuesare givenin brackets.Forthe F test, the numerator
degrees of freedom
are (# of moments- 1) and the denominator
degrees of freedomare NT - N - #moments,
=
=
whereN 532 and T 8. The degrees of freedomforthe Waldtest arethe numberof moment
conditions.
boardon efficiencygrounds.This resultis surprisinggiven
that IW-GMMonly uses half of the sample for each estimationand may point to some small-sampleadvantages
of IW-GMMover GMM.Likewise, SSIV standarderrors
underhomoscedasticitydominate2SLS and FF; however,
2SLS and FF are far superioronce one controls for heteroscedasticity.Proceedingdown the columns, note that,
as additionalmomentsare addedto the instrumentset, the
ISE is estimatedmorepreciselyas predicted.For example,
exploitingall moments(212)reducesthe 2SLS andFF standarderrorsby at least 45% and the GMM standarderrors
by at least 55%comparedto the suboptimalmatrixwith 72
moments.
The efficiency gains of extra instruments,nonetheless,
come at a cost. A clear patternof downwardbias in the
first-differenceand orthogonal-deviations
GMM and SSIV
estimators emerges in the stacked-moment results of Table
1. With 72 moments imposed, the first-difference 2SLS and
GMM ISE parameters differ by 27%, but the difference is
67% with 212 moments. The bias is even more severe under
orthogonal deviations. Unlike GMM, the finding that the
SSIV parameter estimates converge toward 0 as the number of overidentifying restrictions expands is an expected
property of the estimator (Angrist and Krueger 1995). What
is not expected is the erratic and puzzling behavior of the
USSIV estimator. The estimated ISE ranges from 8.239 in
the 72-moment orthogonal-deviations case to -3.414 in the
107-moment first-difference case. No obvious explanation
emerges for the disappointing USSIV results, but the findings indicate that "forcing" exact identification through the
arises:Is the bias in GMMdue to instrumentsweakly correlatedwith the wage or due to a correlationbetweenthe
samplemomentsandthe estimatedweight matrix?Several
corroboratingfacts suggest that a correlationbetweenthe
sample moments and the estimatedweight matrix is the
sourceof the bias. In Table2 I reportfirst-stageF tests and
Waldtests withtheirassociatedp valuesfor the endogenous
wage regressor.I presentthe Waldtest becausethe F test
is inconsistentwhen thereis heteroscedasticity.
As seen in
Table 2, the F test frequentlyleads to the incorrectconclusion that there are weakly correlatedinstrumentswhen
the Waldtest does not, suggestingthatwhen heteroscedasticity is presentthe Waldtest is the more appropriatetest
for the correlationbetweeninstrumentsandregressors.The
conclusionthat the instrumentsare not weak is strengthenedby notingthatthe overidentifying-restrictions
test does
not rejectthe modelunderheteroscedasticity-robust
GMM,
2SLS, or FE
Potentiallythe strongestpiece of evidence that the bias
comes from the estimatedweight matrixis the IW-GMM
parameterestimates.Beginningwith 162 moments,the IWGMM estimates are at least twice as large as GMM on
averageand are comparableto 2SLS and FF estimates,a
resultthat would not ariseif therewere no correlationbetweenthe GMMsamplemomentsandestimatedweightmatrix or if the instrumentsareweak.Unfortunately,
Hansen's
test
all
the
of
IW-GMM
overidentifying-restrictions rejects
specifications(andall SSIV modelsbeginningwith 137 moments). This "overrejection"as the ratio of moments to
cross-sectionalsamplesize increases,whichreachesa maximum of 212/266 underIW-GMMcomparedto 212/532
underGMM,may indicatea weaknessin the test.
2.2.4 Summary. The resultsof Tables 1 and 2 suggest
that GMM is biased downwardrelativeto 2SLS and FF
as the numberof moment conditionsexpandsbecause of
a correlationbetweenthe estimatedweight matrixand the
samplemoments.Moreover,IW-GMMis successfulin correctingthe bias in GMM;however,the IW-GMMparameter estimatesare more variablethan 2SLS and FF and the
overidentifying restrictions are rejected in each specification. SSIV is biased toward 0 as the number of instruments
expands and USSIV performs poorly. Finally, inference,
whether it be standard errors, overidentifying restrictions,
or first-stage F tests, can be quite misleading when the assumption of conditional homoscedasticity is incorrect.
3.
BOOTSTRAPPING OVERIDENTIFIEDMODELS
To investigate the sample properties of the estimators
such as the potential bias/efficiency trade-off in GMM,
more deeply, I now turn to the bootstrap. The bootstrap, recently surveyed by Efron and Tibshirani (1993) and Jeong
and Maddala (1993), is a powerful statistical technique for
the computation of measures of variability,confidence inter-
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426
Journalof Business & EconomicStatistics,October1997
vals, andbias of an estimator.In the currentapplication,the
bootstrapis a naturalalternativeto standardMonte Carlo
estimateof
analysisbecauseit is basedon a nonparametric
the underlyingerrordistribution.Recall that a strengthof
each of the IV estimatorsconsideredin Section2 emanates
fromtheirlackof dependenceon an a priorispecifiedsmallsampledistribution.Moreover,the parametricMonteCarlo
is difficultto implementbecausethe labor-supplyliterature
offerslittle help in the way of thejoint small-sampledistributionof wages andhours,makingthe choiceof a distribution a dubiousexerciseat best. Consequently,the bootstrap
maintainsthe semiparametric,
empirical-basedspiritof the
article.
The typicalregression-based
bootstrapis a multistepprocedurewherebythe researcherresampleswith replacement
the estimatedresiduals,constructsa new dependentvariable
as the sum of the fittedvalue from the regressionplus the
bootstrappedresidual,reestimatesthe model, and repeats
the exercise B times (b = 1,..., B). There are then B ob-
samplingalgorithmas an alternativeto Freedman'smethod.
The idea is to resamplefrom the multinomialdistribution
in which each observationreceives a differentprobability
weight,iA, such thatthe samplemoments,9i = g(?, X, W),
are satisfiedby construction,EZ=1 ig = 0. In general,
jiA
is solved numericallyfrom a linearprogrammingproblem;
however,Brown and Newey suggesteda closed-formsolution for the probabilitygiven as Ai = (1 g'V-l-i)/
and V =
[N (1 - g-V-1g)], where
= 1/N -Igi
1/N EN_, gi. The estimated probabilities are expected to
satisfy the usual regularityconditionssuch as •i > 0 and
Si= li = 1.
For each experiment,I computethe bias as (1/B) Eb=1
6b- 6, where6bis the bthbootstrapestimateof the ISE and
6 is the pseudotruevalueof the ISE definedlater,the averstandarderrorconstructed
age heteroscedasticity-consistent
fromthe asymptoticformulasin Section1 (SE-A),the bootstrap standard error as [(Bl(b
-
B)2)
-
servationsfrom which to computemeasuresof bias, vari- 1)11/2/vB (SE-B), the RMSE definedas the squareroot
of the sum of bias squaredand the averagevarianceconability,or confidenceintervals.This approachis consistent structedfrom
formulas(RMSE-A),the RMSE
only underthe assumptionsof conditionalhomoscedastic- definedas the asymptotic
root
of the sum of bias squaredand
square
ity, no serialdependence,andnonstochasticregressors.
and the MAE. In addition,I
SE
boot
(RMSE-B),
squared
Whenthe regressorsare stochasticor thereis conditional
a
measure
of
of asymptoticinferenceby
reliability
report
heteroscedasticityas is typicalin IV estimation,Freedman
the
rates
on a 95%two-tailedconficonstructing coverage
(1984) and Freedmanand Peters (1984) suggested an al- dence
intervalas the fractionof times the pseudotruevalue
ternativeprocedure.Instead of resamplingthe residuals,
of the ISE falls within the intervalbased on asymptotic
one resamplessimultaneouslythe estimatedresidualsalong
formulasand asymptoticcriticalvalues.
standard-error
with the regressorsandinstruments.More specifically,one
Because the t distributiondoes not adjust confidence
resampleswith replacementfrom (E,X, W), whereE is the intervalsfor the presenceof skewness in the underlying
or orthogonal-deviations
vectorof estimatedfirst-difference
population,I also constructbootstrap-tcriticalvalues that
residuals,X is the matrixof first-differenceor orthogonal- have the
to accountfor skewness (Efronand Tibdeviationsregressors,and W is the matrixof instruments. shirani ability
1993, pp. 159-162). The idea is to approximate
Call the constructed"pseudodata"(g*,X*, W*). The new the
asymptoticpivotal t statistic from the bootstrapas
dependentvariableis Y* - X*3 + ^*, which is regressed t(b) = (6b - 6)/se(b), where se(b) is the bth estimated
on X* with W* as instrumentsto generatea new 3*, and
asymptoticstandarderror.The a and 1 - a critical valthe procedureis repeatedB times.This approach,in which ues are foundby orderingthe t(b)'s for the entirebootstrap
each observationhas equal probabilityweight 1/N of be- sampleand are then used in constructingbootstrap-tconing drawnfromthe discreteempiricaldistributionfunction, fidenceintervals.The bootstrap-tmay offer asymptoticreis an asymptoticallyvalid methodof bootstrappingan IV finementsover critical values from first-orderasymptotic
estimator,even when the model is overidentified,and pro- theory (Brown and Newey 1995; Hall and Horowitzin
vides asymptoticcoverageratesequalto theirnominalrates press).Finally,to gaugethe potentialgainsin the Brownand
(Hahnin press).
Newey algorithmoverFreedman'sfor correctingthe size of
Brown and Newey (1995) and Hall and Horowitz (in
press), however, showed that the Freedman method does
not yield an improvement in terms of coverage rates over
first-orderasymptotic theory, and more importantly,it gives
the wrong size, even asymptotically, for the overidentifyingrestrictions test. The problem lies in the fact that when the
model is overidentified the sample moments are typically
not 0, and thus one bootstraps from an empirical distribution that is a poor approximation to the true underlying
distribution. Hall and Horowitz attempted to mitigate this
problem by recentering the moments at their sample values,
and Brown and Newey suggested recentering the empirical
distributionby imposing the moment conditions on the data.
Because Brown and Newey showed that their approach
is efficient in the class of bootstrap methods, I use their re-
the overidentifying-restrictions test, I report the fraction of
times the bootstrap value of the overidentifying-restrictions
test exceeds the asymptotic critical X2 value at the .05 level
(Boot-J).
3.1 Results
To make relative comparisons of bias, RMSE, and MAE
across the estimators I need to "prime" the bootstrap (i.e.,
construct?) with a common vector of parameters.This approach of choosing a common set of starting values mimics the parametric Monte Carlo and has been applied to
the bootstrap by Freedman and Peters (1984). Because the
focus is on the ISE, I use the estimated ISE most often
encountered in male labor-supply studies as the pseudotrue
value of 5. Pencavel (1986) noted that the ISE has a cen-
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Ziliak: Efficient Estimation With Panel Data
427
Table3. BootstrapComparisonsof First-Difference
Moment-Condition
EstimatorsWhereObservationsAre DrawnWithProbability1/N
Estimators
Bias
SE-A
SE-B
2SLS
GMM
FF
IW-GMM
SSIV
USSIV
-.108
.174
-.097
.189
.208
.828
.485
.360
.351
.459
.854
15.621
.474
.330
.309
.565
.909
6.082
.497
.400
.365
.497
.879
15.643
2SLS
GMM
FF
IW-GMM
SSIV
USSIV
.128
-.093
.055
.013
.137
-1.107
.169
.097
.142
.070
.310
3.4E2
.199
.108
.156
.337
.273
42.527
.212
.134
.152
.071
.339
3.4E2
RMSE-A
RMSE-B
MAE
95%
asym-t
coverage
95%boot-t
critical
values
.263
.280
.183
.336
.451
1.683
.99
.95
.98
.88
1.00
.99
-1.59,
-1.35,
-1.93,
-1.48,
-1.63,
-1.04,
1.71
2.15
1.29
3.99
1.67
1.53
.29
.13
.44
.66
.29
.145
.108
.136
.149
.160
4.965
.85
.81
.87
.47
.95
1.00
-1.53,
-3.34,
-1.79,
-7.28,
-2.13,
-1.19,
3.27
.98
2.63
8.06
1.95
1.69
.99
.99
.99
1.00
.63
.147
.179
.090
.156
.211
.927
.74
.21
.87
.19
.84
.96
-3.73,
-5.62,
-3.74,
-15.1,
-3.07,
-2.26,
4.05
-.2
1.49
19.1
.31
1.79
.99
1.00
1.00
1.00
.90
.084
.172
.069
.144
.192
1.099
.79
.04
.85
.14
.83
.99
-4.50,
-6.0,
-3.69,
-27.4,
-2.86,
-1.58,
3.12
-1.74
1.84
32.5
-.31
1.41
1.00
1.00
1.00
1.00
.95
Boot-J
rejection
rate:5%
9 moments
.486
.373
.324
.596
.932
6.138
72 moments
.236
.142
.168
.337
.306
42.541
162 moments
2SLS
GMM
FF
IW-GMM
SSIV
USSIV
.022
-.169
-.046
--.020
-.216
-.932
.131
.055
.119
.029
.188
11.598
.219
.077
.128
.229
.124
5.707
.220
.186
.136
.230
.249
5.782
.132
.178
.128
.035
.286
11.636
212 moments
2SLS
GMM
FF
IW-GMM
SSIV
USSIV
-.014
-.175
-.037
.044
-.209
-.372
.124
.043
.100
.018
.175
13.467
.166
.047
.106
.219
.096
5.783
.125
.180
.107
.047
.273
13.472
.166
.181
.112
.224
.231
5.794
The pseudotruevalueof the ISEis fixedat .21 forallcalculations.Biasis computedas the differencebetweenthe averagebootstrap
NOTE:Allcalculationsare based on 100 bootstrapreplications.
estimateand the pseudotrueISE,SE-Ais the averagestandarderrorcomputedfromasymptotictheory,SE-B is the averagestandarderrorconstructedfromthe bootstrap,RMSE-Ais the root
meansquarederrorcomputedwiththe variancefromasymptotictheory,RMSE-Bis the rootmeansquarederrorcomputedwiththe variancefromthe bootstrap,MAEis the medianabsoluteerror,
95%asym-t coverageis the fractionof times the pseudotruevalue of the ISEfallswithinthe 95%confidenceintervalbased on standarderrorsand criticalvalues fromasymptotictheory,95%
restrictionstest exceeds
boot-tcriticalvalues are the .025 and .975 valuesfromthe bootstrapt statistic,and boot-Jrejectionrateis the fractionof timesthe bootstrapvalueof the overidentifying
the criticalX2 valueat the .05 level.
tral tendency of .20, which, coincidentally,is consistent
with the 2SLS 9-momentmodel of Table 1; thus,the 2SLS
9-momentparametersform the startingvalues. I present
100 bootstrapreplicationsfor the 9-, 72-, 162-, and 212momentcases for both the first-differenceand orthogonalThe small numberof bootstrap
deviationstransformations.
Monte Carlo drawsis sufficientfor measuresof bias and
variabilityfor most samplesizes but is sufficientfor confidenceintervalsonly in largersamples(Hall 1986;Efronand
Tibshirani1993, p. 161). All experimentswere conducted
in Gauss with a Pentium-90,where the models with 212
momentstook at least 48 hours to compute.I presentthe
resultsfrom the Freedmanalgorithmin Tables3 and4 and
fromBrownandNewey's algorithmin Tables5 and6. I focus on the first-differenceresultsbecausethe implications
from orthogonaldeviationsare largelythe same.
3.1.1 Results From Freedman Algorithm. Beginning
with the base case of 9 moments in Table 3, 2SLS and
FF tend to be biased toward OLS (6oLs = .11), which is
consistentwith the presenceof a finite-samplebias in IV
estimators.The absolute value of bias in GMM exceeds
both 2SLS and FF, whereasGMM dominatesall three of
the split-sampleestimatorsin terms of lower bias. Com-
paringthe averageasymptoticstandarderror(SE-A)to the
bootstrapstandarderror(SE-B) suggeststhat thereis little
differencefor 2SLS, FF, andGMMstandarderrors,butthe
asymptoticand bootstrapstandarderrorsdivergefor IWGMM,SSIV,andUSSIV.FF dominatesall of the estimators
in termsof lower RMSEand MAE, but GMMdoes better
than2SLS in termsof RMSEbut not for MAE. Moreover,
GMMdominatesall of the split-sampleestimatorsfor lower
RMSEandMAE.In termsof asym-tcoveragerates,GMM
performsbest, 2SLS, FF, SSIV,and USSIV tend to underreject, and IW-GMMoverrejects.The bootstrap-tcritical
values suggest that there is a slight departurefrom symmetry for each of the estimatorsand that asymptoticcritical values are too large.The boot-J rejectionrates,however, reveal that there is a serious level distortionin the
test computedwith Freedman's
overidentifying-restrictions
algorithm.Overall,Keaneand Runkle'sFF estimatorperforms best in the base case.
The most obvious trend arising in the stacked-moment
cases is the increasingbias in GMMas the numberof moment conditionsexpands.The bias in the GMM ISE rises
(i.e., becomes more negative)from -.093 under 72 moments to -.175 with 212 moments.This finding,reenforc-
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Journal of Business & Economic Statistics, October 1997
428
Moment-Condition
EstimatorsWhereObservationsAre DrawnWithProbability1/N
Table4. BootstrapComparisonsof Orthogonal-Deviations
Estimators
Bias
SE-A
SE-B
RMSE-A
RMSE-B
MAE
95%
asym-t
coverage
95%boot-t
critical
values
Boot-J
rejection
rate:5%
9 moments
2SLS
GMM
IW-GMM
SSIV
USSIV
.099
.303
.068
.419
.312
.432
.370
.444
.753
5.992
.440
.366
.451
.647
3.793
.443
.478
.450
.862
5.999
2SLS
GMM
IW-GMM
SSIV
USSIV
.280
.011
.124
.058
.355
.165
.096
.069
.292
31.512
.219
.113
.264
.168
12.082
.325
.097
.142
.298
31.512
.452
.475
.457
.771
3.806
.259
.317
.285
.494
1.079
.96
.87
.90
.99
.97
-1.85,
-1.68,
-1.72,
-1.61,
-1.05,
2.16
2.67
2.63
1.69
2.11
.30
.16
.73
.12
.270
.080
.160
.130
3.825
.62
.88
.45
.99
1.00
-.80,
-2.36,
-8.95,
-1.78,
-1.17,
4.40
2.62
11.5
1.39
1.28
1.00
1.00
1.00
.42
.160
.140
.184
.255
1.220
.77
.27
.20
.83
.91
-2.24,
-5.21,
-17.3,
-3.11,
-2.23,
7.36
.12
17.3
.07
2.32
1.00
1.00
1.00
.85
.121
.150
.155
.243
1.274
.84
.07
.13
.71
.98
-2.49,
-6.62,
-23.7,
-2.81,
-1.49,
7.62
-1.21
40.5
-.01
2.32
1.00
1.00
1.00
.90
72 moments
.356
.114
.292
.178
12.087
-
162 moments
2SLS
GMM
IW-GMM
SSIV
USSIV
.112
-.141
.042
-.252
-1.458
.141
.053
.031
.222
83.064
.209
.074
.248
.134
10.147
.180
.150
.052
.336
83.077
2SLS
GMM
IW-GMM
SSIV
USSIV
.087
-.154
.009
-.239
-.039
.123
.041
.019
.186
17.758
.175
.056
.251
.101
6.229
.151
.159
.021
.302
17.759
.238
.159
.252
.286
10.252
-
212 moments
.196
.164
.251
.259
6.230
-
NOTE: See note to Table3.
ing the results of Table 1, extends Tauchen's(1986) and
AltonjiandSegal's(1994) findingof bias in GMMin small
samplesto the large-samplecase of panel data.The 2SLS
andFF estimatorsare not biasedtowardOLS as in the base
case, while IW-GMMhas negligible bias in each of the
stacked-moment
simulations.Consistentwith the resultsof
Table 1 is the bias toward0 in SSIV and USSIV as the
numberof momentconditionsincreases.The USSIV estimatordoes not succeedin its primaryfunctionto "inflate"
the SSIV estimates.
In terms of statisticalinference,there appearsto be little differencebetweenthe bootstrapstandarderrorandthe
averageasymptoticstandarderrorfor the FF andGMMestimators.Onthe contrary,the standarderrorsdo divergefor
the otherestimators,especiallyIW-GMMandUSSIV.This
suggests that the small IW-GMMstandarderrorsin Table
1 may understatethe true samplingvariabilityin the estimator,makingthe bootstrapuseful for correctlyestimating
IW-GMMstandarderrors.Examiningthe 95%asym-tcoverageratesindicatesthatall of the estimatorsexceptUSSIV
tend to overrejectthe event thatthe pseudotrueISE lies in
the confidenceregion,with the overrejectionbeing particularly acute for both GMM and IW-GMM.In GMM, the
asymptoticconfidenceregion is distortedbecause of the
bias in the parameterestimatestoward0, coupled with a
tight standarderror.IW-GMM,on the otherhand,has little
bias in the estimatedISE'srelativeto GMM,butbecausethe
asymptoticstandarderrorsare estimatedtightly,the confidence intervalis short, excludingthe pseudotrueISE too
frequently.Moreover,inspectingthe boot-t criticalvalues
revealsthatGMMdoes not even include0 in the criticalregion in the 162- and 212-momentcases. This suggeststhat
the bootstrap-tconfidenceintervalis useful for GMMand
IW-GMMto capturethe skewnessin the underlyingdistribution.The FF estimatorcontinuesto outperform2SLS in
coveragerates;meanwhile,given the bias in the estimated
ISE, the SSIV and USSIV estimatorsgive relativelygood
coverage.
Comparingthe omnibusmeasuresof estimatorperformance(RMSEandMAE)for the stacked-moment
bootstrap
simulationsrevealsthatFF dominates2SLS in all cases and
both FF and 2SLS alwaysdominateGMMexcept with 72
moments.Highlightingthe problemsnoted by Brownand
Newey (1995) and Hall and Horowitz(in press),however,
the overidentifying-restrictions
test is distortedseverely,resultingin 100%rejectionrateswhen the nominalrejection
rate is 5%.It mustbe stressedthat,as establishedby Hahn
(in press),overrejectionof the overidentifyingrestrictions
does not invalidatethe otherbias/efficiencyresultsin Tables 3 and 4--the estimatorsand overidentifyingrestrictions are separate.
3.1.2 ResultsFromBrownand NeweyAlgorithm. Tables 5 and 6 contain the bootstrapresults for the firstdifferenceandorthogonal-deviations
transformations
using
Brown and Newey's (1995) algorithm.The USSIV estimatoris not presentedbecause the motivationfor Brown
and Newey's methodis to get a correct overidentifyingrestrictionstest, which by definitiondoes not exist for the
USSIV. Comparedto Table 3, the most strikingresult in
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Ziliak: Efficient Estimation With Panel Data
429
Table5. BootstrapComparisonsof First-Difference
Moment-Condition
EstimatorsWhereObservationsAre DrawnWithProbability
pi
Estimators
Bias
SE-A
SE-B
RMSE-A
RMSE-B
MAE
95%
asym-t
coverage
95%boot-t
critical
values
Boot-J
rejection
rate:5%
9 moments
2SLS
GMM
FF
IW-GMM
SSIV
.383
.316
.316
.251
.460
-.045
-.034
-.034
-.041
-.043
.306
.268
.323
.264
.351
.309
.270
.325
.267
.353
.385
.317
.316
.254
.462
.197
.161
.210
.143
.220
.99
.96
.95
.96
.97
-1.86,
-2.07,
-2.06,
-1.99,
-2.35,
1.09
1.42
2.03
2.17
.99
.03
.00
.12
.03
.00
.121
.130
.118
.116
.088
.84
.57
.73
.24
.81
-2.96,
-3.65,
-3.83,
-22.4,
-3.37,
1.61
1.08
.84
2.95
1.12
.45
.31
.58
.96
.00
72 moments
2SLS
GMM
FF
IW-GMM
SSIV
.117
.074
.101
.034
.078
-.086
-.120
-.115
-.141
-.081
.123
.093
.105
.188
.082
.150
.151
.156
.235
.115
.145
.141
.153
.145
.112
162 moments
2SLS
GMM
FF
IW-GMM
SSIV
-.099
-.108
-.103
.061
.039
.051
.082
.049
.061
.116
.115
.115
.128
.119
.120
.099
.111
.111
.58
.25
.43
-4.99, .84
-5.55, -.4
-4.91, .50
.45
.71
.84
.056
.054
.138
.138
.120
.33
-4.79, .29
.00
.142
.136
.129
.30
.03
.23
.08
-5.39, .36
-6.6, -1.83
-5.45, -.16
-5.35, -1.33
.93
.74
.96
.03
-
-.127
212 moments
2SLS
GMM
FF
IW-GMM
SSIV
.051
.031
.045
-.134
-.131
-.125
-
-
-.144
.046
.068
.040
.049
-
.035
.143
.135
.133
.150
.137
.134
-
.152
-
-
.149
.144
NOTE: See note to Table3.
the base case of 9 momentsis the dramaticimprovement
in boot-J rejectionrates. This is exemplifiedby 2SLS in
which the boot-J rejectionrate falls from .29 to .03. The
test still overrejectswith the FF estimatorbut 70% fewer
times than under the Freedmanmethod.In addition,the
Brown and Newey algorithmimproveson the efficiency
of the bootstrapestimates,especiallyfor 2SLS, IW-GMM,
and SSIV.In termsof bias, FF andGMMare identical,but
GMMdominates2SLS andFF in termsof lowerMAE and
RMSE-B.IW-GMM,however,performsbest overallin the
EstimatorsWhereObservationsAre DrawnWithProbability
Moment-Condition
Table6. BootstrapComparisonsof Orthogonal-Deviations
Ai
Estimators
Bias
SE-A
SE-B
RMSE-A
RMSE-B
MAE
95%
asym-t
coverage
95%boot-t
critical
values
Boot-J
rejection
rate:5%
9 moments
2SLS
GMM
IW-GMM
SSIV
-.009
-.044
-.081
.011
.396
.341
.291
.438
.454
.338
.390
.424
.396
.344
.302
.438
.454
.341
.399
.425
.319
.240
.230
.254
.92
.96
.86
.98
-2.11,
-2.27,
-2.74,
-1.94,
2.21
1.68
2.45
1.25
.10
.02
.03
.00
.109
.109
.089
.142
.76
.62
.33
.56
-3.74,
-3.72,
-9.86,
-4.16,
1.46
.38
3.71
1.33
.32
.20
.77
.03
.126
.119
.41
.13
-5.82, -.11
-6.01, -.99
.32
.75
72 moments
2SLS
GMM
IW-GMM
SSIV
-.101
-.108
-.090
-.119
.096
.071
.032
.098
.112
.072
.138
.101
.140
.129
.095
.155
.151
.130
.138
.156
162 moments
2SLS
GMM
IW-GMM
SSIV
-.130
-.122
.053
.037
.068
.042
.140
.127
-
-
-
-
-.142
.046
.041
.149
.147
.129
-
.147
-
-
-
.140
.12
-6.75, -.67
.00
.136
.139
.30
.05
-5.52, -.37
-7.15, -1.56
.94
.85
-
212 moments
2SLS
GMM
IW-GMM
SSIV
-.138
-.136
-.167
.049
.030
.048
.058
.037
.029
.147
.140
-
.174
.149
.141
-
.170
-
.171
-
-
.04
-6.38, -1.86
NOTE: See note to Table3.
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-
.03
Journalof Business & EconomicStatistics,October1997
430
base case in termsof lower RMSE and MAE. Because of
the gains in efficiencyand in levels of the boot-J test, the
Brown and Newey method seems to be preferableto the
Freedmanmethodwith few moments.
On the other hand,problemsarise with the Brown and
Newey algorithm,especiallyin the many-momentmodels.
First, some of the bootstrap probabilities (Pji), although
summingto one overall, are negative. This problem increased with the numberof moment conditions,ranging
from a low of .5% of the observationswith 9 moments
in 2SLS and GMMto over 40% with IW-GMMwith 212
moments.I consideredseveralmethodsto deal with this,
andthe methodI chose was to redistributea fractionof the
negativeprobabilitiesto each observationandthen assigna
small positiveprobability(1/100th of the smallestpositive
probability)to those observationswith negativeprobabilities, making sure the probabilitiessum to 1. The results
are not sensitive to the methodused. Second, because so
much weight (i.e., a "high"probability)is given to a subset of the observationsfor the IW-GMMestimatorswith
162 and 212 moments,the estimatorsare singular.Recall
that in samplingwith replacementan observationcan be
drawnmorethanonce, which can occurwith greaterprobabilityunderthe BrownandNewey algorithm.This singularityproblempersistedwith every bootstrapsampleover
a four-dayperiodand with differentrandomsamplesplits
of the originaldata as well. Consequently,I am unableto
reporton the IW-GMMestimatorfor these cases. Brown
and Newey also reporteddifficultiesdue to a singularity
problemin theirempiricalapplication.
Withthe exceptionof SSIV,the boot-J test overrejectsin
casesjust as in Tables3 and4. The disthe stacked-moment
tortionin levels, althoughless thanthe Freedmanmethod,
becomesmore severeas additionalmomentsare appended
to the instrumentset. It is importantto note thatboth Hall
and Horowitz, with their Monte Carlo, and Brown and
Newey, with their empiricalapplication,found that level
distortionspersistin the test even afterrecenteringthe distribution.In termsof bias, the estimatorsdo not distinguish
themselvesas in the Freedmanmethod,and each tends to
centeron the OLS estimate.However,2SLS and FF continueto dominateGMMin termsof lower bias. Moreover,
asym-tcoverageis worsefor eachestimator,suggestingthat
boot-t critical values may improve on coverage over firstorder asymptotics. The central conclusion from the previous
bootstrap results remains the same though; namely, in comparing the estimators across all criteria, the FF estimator
continues to be preferred in this application.
4.
SUMMARYAND CONCLUSIONS
In this article I compared several IV estimators for paneldata models with predeterminedinstruments. The empirical
results from the life-cycle labor-supply model, in conjunction with results from two separate bootstrap Monte Carlo
experiments, are summarized as follows. First, and most
important,the GMM estimator is biased downward relative
to the 2SLS and FF estimators as the number of moment
conditions approaches the optimal number of moments be-
causeof a correlationbetweenthe estimatedweightmatrix
andthe samplemoments.This leads to poorcoveragerates
for confidenceintervalsbased on asymptoticcritical values but providesa clearrole for the bootstrap-tconfidence
intervalas a basis of inferenceunderGMM. GMM performs reasonablywell with suboptimalinstrumentsbut is
not recommendedfor panel-dataapplicationswhen all of
the momentsare exploitedfor estimation.
Second, the IW-GMMestimatoris generallysuccessful
at eliminatingthe bias in GMM parameterestimatesusing Freedman'sbootstrapalgorithm;however,the standard
errorsfrom asymptotictheoryseem to understatethe true
test
samplingvariationandthe overidentifying-restrictions
is biasedtowardrejection,possiblydue to the high ratioof
momentsto cross-sectionsamplesize. Furtherresearchon
boththe asymptoticand small-samplepropertiesof this estimatoris needed.In the meantime,the bootstrapis likelyto
be usefulin IW-GMMfor the constructionof bothstandard
errorsandconfidenceintervals.Third,the USSIVestimator
tendsto be highlyunstableandinefficient,in contrastto the
SSIV estimatorwhichworksfairlywell in modelswithfew
overidentifyingrestrictions.
Fourth,cautionis warrantedwhen conductinginference
underthe assumptionof conditionalhomoscedasticity.On
several occasions, incorrectlyassuminghomoscedasticity
led to the rejectionof the overidentifyingrestrictionsand
to the conclusionof weakly correlatedinstruments.When
is present,one shoulduse robustvariants
heteroscedasticity
of the test of overidentifyingrestrictionsandthe Waldtest
for first-stagetests of correlationbetweeninstrumentsand
regressors.In addition,after adjustingfor conditionalheteroscedasticity,thereis no obviousefficiencygain of using
orthogonaldeviationsratherthanfirst-differences.
Fifth,furtherresearchis neededon improvingmethodsof
bootstrappingestimatorsfrom overidentifiedmodels. The
test perlevels distortionin the overidentifying-restrictions
sisted in models with many moments,even afterrecentering the distributionusing BrownandNewey's (1995) algorithm.Moreover,complicationsarosein the implementation
of theiralgorithm,notablynegativeprobabilitiesandsingularityproblems.Hence,if the modelhas numerousmoment
restrictionsandthe focus is solely on the estimatorandconsistentconfidenceintervalsandnot on the overidentifyingrestrictions test, then Freedman's (1984) method of giving
equal weight to each observation is recommended.
Finally, although 2SLS is still a reliable method of estimation, the FF estimator performs best in terms of the
bias/efficiency trade-off. It offers more efficient estimation
than 2SLS but does not have the problem of excess bias
found in GMM parameter estimates. Consequently, the FF
estimator is attractive relative to the other estimators considered here.
ACKNOWLEDGMENTS
I thank the editor, an associate editor, and an anonymous
referee for many helpful comments that have improved this
article substantially. I also have benefited from discussions
with George Evans, Tom Kniesner, Larry Singell, and Wes
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Ziliak:EfficientEstimationWithPanel Data
Wilson. Most especially,I am indebtedto CarlosMartinsFilho and MarkThomafor their insightfuladvice. I alone
am responsiblefor any remainingerrors.
[Received December 1994. Revised July 1996.]
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