第六讲极大似然估计.docx

第六讲极大似然估计.docx

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第六讲极大似然估计

第六讲极大似然估计

TheLikelihoodFunctionandIdentificationoftheParameters（极大似然函数及参数识别）

1、似然函数的表示

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