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OnAsymptoticNormalitywhentheNumberofRegressorsIncreasesandtheMinimumEigenvalueofX0X=nDecreases1A.RonaldGallantDepartmentofStatisticsNorthCarolinaStateUniversityRaleigh,NC27695-8203USAJuly1989RevisedAugust1990CorrectionsNovember19901ThispaperisaspecializationtotheunivariatecaseoftheresultsinGallant,A.Ronald,andGeraldoSouza(1991),\OntheAsymptoticNormalityofFourierFlexibleFormEstimates,JournalofEconometrics50,329{353.AbstractConditionsunderwhicharegressionestimatebasedonpregressorsisasymptoticallynor-mallydistributedwhentheminimumeigenvalueofX0X=ndecreaseswithpareobtained.Theresultsarerelevanttotheregressionsontruncatedseriesexpansionsthatariseinneuralnetworks,demandanalysis,andassetpricingapplications.01IntroductionThepleadingtermsofaseriesexpansionf jg1j=1areoftenusedinregressionanalysistoeitherrepresentorapproximatetheconditionalexpectationwithrespecttoxofadependentvariabley.Eitherexplicitlyorimplicitly,pusuallygrowswiththesamplesizenintheseapplications.Thegrowthmayfollowsomedeterministicrulefpngoritmaybeadaptivewithpincreasedwhenat-testrejectsorsomemodelselectionrulesuchasSchwarz’s(1978)criterion,Mallow’s(1973)Cp;orcrossvalidationsuggestsanincrease.Themostfamiliarexamplesofexpansionsviewedasrepresentationsareexperimentaldesigns,whichareHadamardexpansionsinthelevelsoffactors.Inthiscase,whichwerefertoasthe rstparadigm,thedataarepresumedtohavebeengeneratedaccordingtotheregressionyt=pXj=1 oj j(xt)+ett=1;2;:::;nPerhapsthemostfamiliarexamplesofexpansionsviewedasapproximationsoccurinre-sponsesurfaceanalysiswhentheobservedresponseisregressedonapolynomialinthecontrolvariables.Inthiscase,whichwerefertoasthesecondparadigm,thedataarepresumedtohavebeengeneratedaccordingtotheregressionyt=go(xt)+ett=1;2;:::;nandgp(xj )=Ppj=1 j j(x)isregardedasanapproximationtogo.Whileouranalysiscoversbothparadigms,thesecondprovidestheprimarymotivation.Infact,itisthreespeci capplicationsthatmotivatethiswork,althoughourresultsobvi-ouslyhavewiderapplicability.Thesethreeapplicationsare:(i)thestatisticalinterpretationofneuralnetworks(White,1989)asusedinrobotics,navigationaids,speechinterpretation,andotherarti cialintelligenceapplications[feedforwardnetworkscanbeviewedasseriesexpansionregressions(GallantandWhite,1988)],(ii) exiblefunctionalformsasusedinconsumerandfactordemandanalysis(Barnett,GewekeandYue,1989),and(iii)Hermiteexpansionsasusedinassetpricingandsampleselectionapplications(GallantandTauchen,1989;GallantandNychka,1987).Foravarietyofreasons{mimickingbiologicalstructure,avoidingunrealisticboundaryconditions,orappendingaleadingspecialcasetotheexpan-1sioneithertoimprovetheapproximationortesthypotheses{thesethreeapplicationsshareacommonfeature:theeigenvaluesofthep pmatrixX0X=n=(1=n)nXt=1[ 1(xt); ; p(xt)]0[ 1(xt); ; p(xt)](orits rstordercounterpartiftheanalysisisnonlinear)declineasnandpincreasetogether.Intheseapplications,theindependentvariablesfxtgarealmostinvariablyobtainedbysamplingfromsomecommondistribution (x)sothatresultsaremoreusefullystatedintermsofthep pmatrixGp=Z[ 1(x); ; p(x)]0[ 1(x); ; p(x)]d (x)IfonetriedtostateresultsintermsofconditionsontheeigenvaluesofX0X=n(orothercharacteristicssuchasthediagonalelementsofthehatmatrix)resultswouldbeconditionalupontheparticularsequencefxtgthatobtainedofwhich,presumably,onlythe rstntermsareknown.Moreover,evenifonewerepresumedtohavetheentiresequencefxtg1t=1availableforinspection,onewouldstillbeinvolvedintheconceptualcircularityofhavingtoproposearulepn;checktheeigenvaluesofX0X=nasntendstoin nity,andifthatruledidn’twork,totryagain.Withouttheresultsofthispaper,whicharere nementsofAndrews’(1988)results,onecouldnotstatearulefpngaprioriwhichwouldsatisfytheconditionsonX0X=nforeverysequencefxtgencounteredinapplications.ProvidingamoreelegantproofthatgivesfasterratesthanAndrews(1988)insomecasesisthemaincontributionofthepaper.AnextensiontoadaptiverulesusingresultsduetoEastwood(1987)ispossible;seeEastwoodandGallant(1987)orAndrews(1988)forexamples.(WeciteoneoftheearlyversionsofAndrews’workwhichisrichinhistoryandexamples;laterversionsimprovetheresultsbuthavedeletedsomeofthisinterestingmaterial.)Tohelp xideas,weillustratetheratesofdeclineonemightencounterinapplicationswithanexampletakenfromGallant(1984)regardingthelogcostfunctionofa rmwhenthe rm’soutputistheonlyfreevariable.Thelogcostfunctionofa rmgivesthelogarithmofthecosttoa rmoverayear,say,toproducexunitsoflogoutputatspeci edlogpricesofthefactorsofproduction.Sinceweshallholdprices xed,theirargumentsaresuppressedandwewritealogcostfunctionasg(x).Unitsofmeasurementareirrelevant,andthecapital2stockofthe rmis xed(thusboundingfeasibleoutputfromaboveandbelow),sowecanassumethat0 x 2 withoutlossofgenerality.Dataisgeneratedaccordingtoyt=go(xt)+ett=1;2;:::;nIfoneapproximatedgo(x)bytheFourierseriesgp(xj )=u0+2KXj=1[ujcos(jxt) vjsin(jxt)]p=2K+1and wereuniformover[0,2 ]thentheeigenvaluesofGpwouldbeboundedfromaboveandbelowforallp(Tolstov,1962).Butthiswouldimplythatthelogcostfunctionofa rmisperiodicsothatconditionsatthelowestfeasibleoutputarethesameasatthehighes
本文标题:On Asymptotic Normality when the Number of Regress
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