第六讲极大似然估计Word格式.docx
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在具有n个观察值的随机样本中,每个观察值的密度函数为。
由于n个随机观察值是独立的,其联合密度函数为
函数被称为似然函数,通常记为,或者。
与Greene书中定义的区别
Theprobabilitydensityfunction,orpdfforarandomvariabley,conditionedonasetofparameters,,isdenoted.Thisfunctionidentifiesthedatageneratingprocessthatunderliesanobservedsampleofdataand,atthesametime,providesamathematicaldescriptionofthedatathattheprocesswillproduce.Thejointdensityofnindependentandidenticallydistributed(iid)observationsfromthisprocessistheproductoftheindividualdensities;
(17-1)
Thisjointdensityisthelikelihoodfunction,definedasafunctionoftheunknownparametervector,,whereisusedtoindicatethecollectionofsampledata.
Notethatwewritethejointdensityasafunctionofthedataconditionedontheparameterswhereaswhenweformthelikelihoodfunction,wewritethisfunctioninreverse,asafunctionoftheparameters,conditionedonthedata.
Thoughthetwofunctionsarethesame,itistobeemphasizedthatthelikelihoodfunctioniswritteninthisfashiontohighlightourinterestintheparametersandtheinformationaboutthemthatiscontainedintheobserveddata.
However,itisunderstoodthatthelikelihoodfunctionisnotmeanttorepresentaprobabilitydensityfortheparametersasitisinSection16.2.2.Inthisclassicalestimationframework,theparametersareassumedtobefixedconstantswhichwehopetolearnaboutfromthedata.
Itisusuallysimplertoworkwiththelogofthelikelihoodfunction:
.(17-2)
Again,toemphasizeourinterestintheparameters,giventheobserveddata,wedenotethisfunction.Thelikelihoodfunctionanditslogarithm,evaluatedat,aresometimesdenotedsimplyand,respectivelyor,wherenoambiguitycanarise,justor.
Itwillusuallybenecessarytogeneralizetheconceptofthelikelihoodfunctiontoallowthedensitytodependonotherconditioningvariables.Tojumpimmediatelytooneofourcentralapplications,supposethedisturbanceintheclassicallinearregressionmodelisnormallydistributed.Then,conditionedonit’sspecificisnormallydistributedwithmeanandvariance.Thatmeansthattheobservedrandomvariablesarenotiid;
theyhavedifferentmeans.Nonetheless,theobservationsareindependent,andaswewillexamineincloserdetail,
(17-3)
whereisthematrixofdatawithrowequalto.
2、识别问题
Therestofthischapterwillbeconcernedwithobtainingestimatesoftheparameters,andintestinghypothesesaboutthemandaboutthedatageneratingprocess.
Beforewebeginthatstudy,weconsiderthequestionofwhetherestimationoftheparametersispossibleatall—thequestionofidentification.Identificationisanissuerelatedtotheformulationofthemodel.
Theissueofidentificationmustberesolvedbeforeestimationcanevenbeconsidered.
Thequestionposedisessentiallythis:
Supposewehadaninfinitelylargesample—thatis,forcurrentpurposes,alltheinformationthereistobehadabouttheparameters.
Couldweuniquelydeterminethevaluesoffromsuchasample?
Aswillbeclearshortly,theanswerissometimesno.
注意:
希望大家能够熟练地写出不同分布的密度函数,以及对应的似然函数。
这是微观计量经济学的基本功。
特别是正态分布、Logistic分布。
更一般地讲,指数类分布的密度函数。
17.3Efficientestimation:
thePrincipleofMaximumLikelihood
Theprincipleofmaximumlikelihoodprovidesameansofchoosinganasymptoticallyefficientestimatorforaparameterorasetofparameters.
Thelogicofthetechniqueiseasilyillustratedinthesettingofadiscretedistribution.
Considerarandomsampleofthefollowing10observationsfromaPoissondistribution:
5,0,1,1,0,3,2,3,4,and1.
Thedensityforeachobservationis
Sincetheobservationsareindependent,theirjointdensity,whichisthelikelihoodforthissample,is
.
Thelastresultgivestheprobabilityofobservingthisparticularsample,assumingthataPoissondistributionwithasyetunknownparametergeneratedthedata.Whatvalueofwouldmakethissamplemostprobable?
Figure17.1plotsthisfunctionforvariousvaluesof.Ithasasinglemodeat,whichwouldbethemaximumlikelihoodestimate,orMLE,of.
Considermaximizingwithrespectto.Sincethelogfunctionismonotonicallyincreasingandeasiertoworkwith,weusuallymaximizeinstead;
insamplingfromaPoissonpopulation,
Fortheassumedsampleofobservations,
and
Thesolutionisthesameasbefore.Figure17.1alsoplotsthelogoftoillustratetheresult.
Thereferencetotheprobabilityofobservingthegivensampleisnotexactinacontinuousdistribution,sinceaparticularsamplehasprobabilityzero.Nonetheless,theprincipleisthesame.
Thevaluesoftheparametersthatmaximizeoritslogarethemaximumlikelihoodestimates,denoted.Sincethelogarithmisamon