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1、IIIGENERALAPPROACHESTONONLINEARESTIMATIONInthispartwebeginourstudyofnonlineareconometricmethods.Whatwemeanbynonlinearneedssomeexplanationbecauseitdoesnotnecessarilymeanthattheunderlyingmodeliswhatwewouldthinkofasnonlinear.Forexample,supposethepopulationm
2、odelofinterestcanbewrittenasy¼xbþu,but,ratherthanassumingEðujxÞ¼0,weassumethatthemedianofugivenxiszeroforallx.ThisassumptionimpliesMedðyjxÞ¼xb,whichisalinearmodelfortheconditionalmedianofygivenx.[Theconditionalmean,EðyjxÞ,mayormaynotbelinearinx.]Thestan-
3、dardestimatorforaconditionalmedianturnsouttobeleastabsolutedeviations(LAD),notordinaryleastsquares.LikeOLS,theLADestimatorsolvesaminimi-zationproblem:itminimizesthesumofabsoluteresiduals.However,thereisakeydi¤erencebetweenLADandOLS:theLADestimatorcannotb
4、eobtainedinclosedform.Thelackofaclosed-formexpressionforLADhasimplicationsnotonlyforobtainingtheLADestimatesfromasampleofdata,butalsofortheasymptotictheoryofLAD.AlltheestimatorswestudiedinPartIIwereobtainedinclosedform,afactwhichgreatlyfacilitatesasympto
5、ticanalysis:weneedednothingmorethantheweaklawoflargenumbers,thecentrallimittheorem,andthebasicalgebraofprobabilitylimits.Whenanestimationmethoddoesnotdeliverclosed-formsolutions,weneedtousemoreadvancedasymptotictheory.Inwhatfollows,‘‘nonlinear’’describes
6、anyprob-leminwhichtheestimatorscannotbeobtainedinclosedform.Thethreechaptersinthispartprovidethefoundationforasymptoticanalysisofmostnonlinearmodelsencounteredinapplicationswithcrosssectionorpaneldata.Wewillmakecertainassumptionsconcerningcontinuityanddi
7、¤erentiability,andsoproblemsviolatingtheseconditionswillnotbecovered.InthegeneraldevelopmentofM-estimatorsinChapter12,wewillmentionsomeoftheapplicationsthatareruledoutandprovidereferences.Thispartofthebookisbyfarthemosttechnical.Wewillnotdwellonthesome-t
8、imesintricateargumentsusedtoestablishconsistencyandasymptoticnormalityinnonlinearcontexts.Forcompleteness,wedoprovidesomegeneralresultsonconsis-tencyandasymptoticnormalityforgeneralclassesofestimators.However,forspecificest
