On Asymptotic Normality when the Number of Regress

OnAsymptoticNormalitywhentheNumberofRegressorsIncreasesandtheMinimumEigenvalueofX0X=nDecreases1A.RonaldGallantDepartmentofStatisticsNorthCarolinaStateUniversityRaleigh,NC27695-8203USAJuly1989RevisedAugust1990CorrectionsNovember19901ThispaperisaspecializationtotheunivariatecaseoftheresultsinGallant,A.Ronald,andGeraldoSouza(1991),\OntheAsymptoticNormalityofFourierFlexibleFormEstimates,JournalofEconometrics50,329{353.AbstractConditionsunderwhicharegressionestimatebasedonpregressorsisasymptoticallynor-mallydistributedwhentheminimumeigenvalueofX0X=ndecreaseswithpareobtained.Theresultsarerelevanttotheregressionsontruncatedseriesexpansionsthatariseinneuralnetworks,demandanalysis,andassetpricingapplications.01IntroductionThepleadingtermsofaseriesexpansionfjg1j=1areoftenusedinregressionanalysistoeitherrepresentorapproximatetheconditionalexpectationwithrespecttoxofadependentvariabley.Eitherexplicitlyorimplicitly,pusuallygrowswiththesamplesizenintheseapplications.Thegrowthmayfollowsomedeterministicrulefpngoritmaybeadaptivewithpincreasedwhenat-testrejectsorsomemodelselectionrulesuchasSchwarz’s(1978)criterion,Mallow’s(1973)Cp;orcrossvalidationsuggestsanincrease.Themostfamiliarexamplesofexpansionsviewedasrepresentationsareexperimentaldesigns,whichareHadamardexpansionsinthelevelsoffactors.Inthiscase,whichwerefertoastherstparadigm,thedataarepresumedtohavebeengeneratedaccordingtotheregressionyt=pXj=1ojj(xt)+ett=1;2;:::;nPerhapsthemostfamiliarexamplesofexpansionsviewedasapproximationsoccurinre-sponsesurfaceanalysiswhentheobservedresponseisregressedonapolynomialinthecontrolvariables.Inthiscase,whichwerefertoasthesecondparadigm,thedataarepresumedtohavebeengeneratedaccordingtotheregressionyt=go(xt)+ett=1;2;:::;nandgp(xj)=Ppj=1jj(x)isregardedasanapproximationtogo.Whileouranalysiscoversbothparadigms,thesecondprovidestheprimarymotivation.Infact,itisthreespecicapplicationsthatmotivatethiswork,althoughourresultsobvi-ouslyhavewiderapplicability.Thesethreeapplicationsare:(i)thestatisticalinterpretationofneuralnetworks(White,1989)asusedinrobotics,navigationaids,speechinterpretation,andotherarticialintelligenceapplications[feedforwardnetworkscanbeviewedasseriesexpansionregressions(GallantandWhite,1988)],(ii)exiblefunctionalformsasusedinconsumerandfactordemandanalysis(Barnett,GewekeandYue,1989),and(iii)Hermiteexpansionsasusedinassetpricingandsampleselectionapplications(GallantandTauchen,1989;GallantandNychka,1987).Foravarietyofreasons{mimickingbiologicalstructure,avoidingunrealisticboundaryconditions,orappendingaleadingspecialcasetotheexpan-1sioneithertoimprovetheapproximationortesthypotheses{thesethreeapplicationsshareacommonfeature:theeigenvaluesoftheppmatrixX0X=n=(1=n)nXt=1[1(xt);;p(xt)]0[1(xt);;p(xt)](oritsrstordercounterpartiftheanalysisisnonlinear)declineasnandpincreasetogether.Intheseapplications,theindependentvariablesfxtgarealmostinvariablyobtainedbysamplingfromsomecommondistribution(x)sothatresultsaremoreusefullystatedintermsoftheppmatrixGp=Z[1(x);;p(x)]0[1(x);;p(x)]d(x)IfonetriedtostateresultsintermsofconditionsontheeigenvaluesofX0X=n(orothercharacteristicssuchasthediagonalelementsofthehatmatrix)resultswouldbeconditionalupontheparticularsequencefxtgthatobtainedofwhich,presumably,onlytherstntermsareknown.Moreover,evenifonewerepresumedtohavetheentiresequencefxtg1t=1availableforinspection,onewouldstillbeinvolvedintheconceptualcircularityofhavingtoproposearulepn;checktheeigenvaluesofX0X=nasntendstoinnity,andifthatruledidn’twork,totryagain.Withouttheresultsofthispaper,whicharerenementsofAndrews’(1988)results,onecouldnotstatearulefpngaprioriwhichwouldsatisfytheconditionsonX0X=nforeverysequencefxtgencounteredinapplications.ProvidingamoreelegantproofthatgivesfasterratesthanAndrews(1988)insomecasesisthemaincontributionofthepaper.AnextensiontoadaptiverulesusingresultsduetoEastwood(1987)ispossible;seeEastwoodandGallant(1987)orAndrews(1988)forexamples.(WeciteoneoftheearlyversionsofAndrews’workwhichisrichinhistoryandexamples;laterversionsimprovetheresultsbuthavedeletedsomeofthisinterestingmaterial.)Tohelpxideas,weillustratetheratesofdeclineonemightencounterinapplicationswithanexampletakenfromGallant(1984)regardingthelogcostfunctionofarmwhentherm’soutputistheonlyfreevariable.Thelogcostfunctionofarmgivesthelogarithmofthecosttoarmoverayear,say,toproducexunitsoflogoutputatspeciedlogpricesofthefactorsofproduction.Sinceweshallholdpricesxed,theirargumentsaresuppressedandwewritealogcostfunctionasg(x).Unitsofmeasurementareirrelevant,andthecapital2stockofthermisxed(thusboundingfeasibleoutputfromaboveandbelow),sowecanassumethat0x2withoutlossofgenerality.Dataisgeneratedaccordingtoyt=go(xt)+ett=1;2;:::;nIfoneapproximatedgo(x)bytheFourierseriesgp(xj)=u0+2KXj=1[ujcos(jxt)vjsin(jxt)]p=2K+1andwereuniformover[0,2]thentheeigenvaluesofGpwouldbeboundedfromaboveandbelowforallp(Tolstov,1962).Butthiswouldimplythatthelogcostfunctionofarmisperiodicsothatconditionsatthelowestfeasibleoutputarethesameasatthehighes