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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 Stable URL: http://www.jstor.org/stable/1392488 . Accessed: 15/11/2012 19:20 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected] . American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal of Business &Economic Statistics. http://www.jstor.org This content downloaded by the authorized user from 192.168.72.231 on Thu, 15 Nov 2012 19:20:25 PM All use subject to JSTOR Terms and Conditions 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 This content downloaded by the authorized user from 192.168.72.231 on Thu, 15 Nov 2012 19:20:25 PM 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 This content downloaded by the authorized user from 192.168.72.231 on Thu, 15 Nov 2012 19:20:25 PM All use subject to JSTOR Terms and Conditions 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, This content downloaded by the authorized user from 192.168.72.231 on Thu, 15 Nov 2012 19:20:25 PM All use subject to JSTOR Terms and Conditions 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) , This content downloaded by the authorized user from 192.168.72.231 on Thu, 15 Nov 2012 19:20:25 PM All use subject to JSTOR Terms and Conditions 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 This content downloaded by the authorized user from 192.168.72.231 on Thu, 15 Nov 2012 19:20:25 PM All use subject to JSTOR Terms and Conditions 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 This content downloaded by the authorized user from 192.168.72.231 on Thu, 15 Nov 2012 19:20:25 PM All use subject to JSTOR Terms and Conditions 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- This content downloaded by the authorized user from 192.168.72.231 on Thu, 15 Nov 2012 19:20:25 PM All use subject to JSTOR Terms and Conditions 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- This content downloaded by the authorized user from 192.168.72.231 on Thu, 15 Nov 2012 19:20:25 PM All use subject to JSTOR Terms and Conditions 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- This content downloaded by the authorized user from 192.168.72.231 on Thu, 15 Nov 2012 19:20:25 PM All use subject to JSTOR Terms and Conditions 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 This content downloaded by the authorized user from 192.168.72.231 on Thu, 15 Nov 2012 19:20:25 PM All use subject to JSTOR Terms and Conditions 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. This content downloaded by the authorized user from 192.168.72.231 on Thu, 15 Nov 2012 19:20:25 PM All use subject to JSTOR Terms and Conditions - .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 This content downloaded by the authorized user from 192.168.72.231 on Thu, 15 Nov 2012 19:20:25 PM All use subject to JSTOR Terms and Conditions 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.] REFERENCES Ahn, S. C., and Schmidt, P. (1995), "Efficient Estimation of Models for Dynamic Panel Data," Journal of Econometrics, 68, 5-27. Altonji, J. G. (1986), "Intertemporal Substitution in Labor Supply: Evidence From Micro Data,"Journal of Political Economy, 94, S176-S215. Altonji, J. G., and Segal, L. M. 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