第一章-古典一元线性回归2011-川大-计量经济学-课件.ppt

第一篇古典线性回归模型,CHAPTERONECLASSICALTWO-VARIABLELINEARREGRESSIONMODEL,Twovariable:onedependentvariableonlyoneexplanatoryvarialbeLinear:Yislinearfunctionofparameters.,IMPORTANTdifferentiation1、correlation：determinatestochastic（no-determinate）相关关系：确定性非确定性2、Correlationandcausality相关关系与因果关系3、Linearity:linearityinthevariablelinearityintheparameters线性：变量线性参数线性,1.1Two-variablelinearregressionmodel,Population:asetofallpossibleoutcomesofarandomvariableSubpopulation：1.1.1populationregressionfunctionandpopulationregressionlinePRF：Yt=0+1Xt+utstochasticE(YtXt)=0+1XtdeterminateYt=E(YtXt)+utSEET.F2.1,X:explanatory,independentvariable,fixed-valuevariableY:dependentvariable,randomvariable，：regressioncoefficientsU:disturbanceorerrorterm,isarandomvariableThetaskofregressionistoestimatethePDF，thatis，toestimatethevalueofunknown，onthebasisofobservationsonYandX。,errortermustandsfortheaggregateeffectofallfactorswhichareexcludedfrommodelbutindeedaffectY:1、variablesexcludedfrommodelwitheffectonYvaguenessoftheory,negligibleeffect,noavailablityofdata2、intrinsicrandomofhumanbeingsbehavoir3、measurementerror4、errorofmodelforms,1.1.2sampleregressionfunctionandsampleregressionline(f2.4,t2.4,t2.5)SRFSeef2.5,ThetaskofregressionistoestimatethePDFonthesampleinformationofYandX,PrimaryobjectiveinregressionanalysisistoestimatethePRFonthebasisoftheSRF，oronthesampleinformationofYandX,buttheestimationofthePRFbasedontheSRFisatbestanapproximation。ThenextquestionishowtheSRFshouldbeconstitutedascloseaspossibletothePRFeventhoughweneverknowwhatisthetruePRF。Methodofestimation：Ordinaryleastsquare：最小二乘法Maximumlikelyhood：最大似然法,1.1.2themeaningofthetermlinear,LinearityinvariablesLinearityinparametersconditionalmeansofYislinearfunctionofparameters.,1.2estimationmethodordinaryleastsquare（OLS）,Now,theandPRFisunknown,ourtaskinregressionanalysisistoestimatethePRFonthebasisoftheSRF，oronthesampleinformationofYandX,buttheestimationofthePRFbasedontheSRFisatbestanapproximation。WeassumethattheclosertheSRFistoasetofsampledataonYandX,thebettertheSRFfitsthePRF.,Scatter-graph,Y0X,PRF:E(Yi)=0+1Xi,Enlargedlocalarea,actual,estimate,residual,fundamentalthoughtofOLS：tomakeSRFascloseaspossibletoPRF,theresidualofeverypointshouldbeassmallaspossibleHowtochosetomakethesumofsquaredresidualsassmallaspossible,?,Criterionofminimizingthesumofsquaredresiduals(Thesumofsquaredresidualsisafunctionofestimatorofparameters.),1.3CLRMsAssumption,CLRM:classicallinearregressionmodel1、Modelislinearinparameters，Xisfixed-value2、ZeromeanvalueofdisturbanceuiE（ui|Xi）=0ItmeansthatthesefactorexcludedfromthemodeldontsystematicallyaffectthemeanvalueofY.,3、Homoscedasticityorequalvarianceofui(thevarianceofuiisthesameforallobservation.)Var（ui|Xi）=Eui-E(ui|Xi)2=E（ui）2=2Heteroscedasticity:differentvarianceofui,varianceofuivarieswithX.,Heteroscedasticity:differentvarianceofui,varianceofuivarieswithX.Var（ui|Xi）=Eui-E(ui|Xi)2=E（ui）2=i2,ConditionaldistributionofDisturbanceui,4、Noautocorrelationbetweenthedisturbances，namely，noserialcorrelation。Cov(ui,uj|Xi,Xj)=Eui-E(ui)|Xiuj-E(uj)|Xj=E(ui|Xi)(uj|Xj)=0ijYonlydependsonXandcurrentuwithoutrelationtootherus.,Positiveserialcorrelation,+ui,+ui,-ui,-ui,Negativeserialcorrelation,+ui,+ui,-ui,-ui,Zerocorrelation,+ui,+ui,-ui,-ui,5、ZerocovariancebetweenuiandXiBecauseXisfixedvalueanduisrandomvariable,thisassumptionissatisfiedautomatically.ThisassumptionguaranteethattheeffectofXanduonYcanbeseparatedeasily.,6、thenumberofobservationsnisgreaterthanthenumberofparameterstobeestimated。,1.4statisticalpropertiesofOLSestimators,Giventheassumptionsoftheclassicallinearregressionmodel,theestimatorsofOLSpossessidealoroptimumproperties:Best,Linear,unbiasedestimators(blue)Guass-Markovtheorem,linearity,arelinearfunctionsofYiorui.Becausethelaterarenormalrandomvariable,theestimatorofparametersarenormalrandomvariables,UNBIASEDtheexpectedvalueofestimatorofparametersareequaltoitstruevalue。,Minimumvarianceorbest,theestimatorofOLShasminimumvarianceintheclassofallsuchlinearunbiasedestimator,namely,efficientestimatorInstatistic,theprecisionorreliabilityofestimateismeasuredbyitsvarianceorstandarderror.ThesmallertheSE,thebettertheestimate,最小二乘估计量和的方差,Becausefollownormaldistribution,then,isvarianceoferrorterm,附：OLS估计量的其他性质,1.5THECOEFFICIENTOFDETERMINATION,Thecoefficientofdeterminationisameasureof“goodnessoffit”,indicateshow“well”thesampleregressionlinefitsthedata.1.5.1variationanalysis,istotalvariationaboutobservationfromthesamplemeans，isthepartthatcanbeexplainedbyexplanatoryvariable,istheresidualpartthatcantbeexplainedbymodel.,Totalsumofsquare,explainedsumofsquare,residualsumofsquare,ESSstandsforthepartofTSSthatcanbeexplainedbyRegression,so:ThegreaterESS,thesmallerRSS,thebetterSRFfitsthedata,1.5.2goodnessoffit(coefficientofdetermination）,WedefinethegoodnessoffitR2asfollowing:,R2iscalledgoodnessoffitorcoefficientofdetermination。,将RSSTSSESS代入上式可得：,Wehavetheconclusion:,TheGreaterR2，thecloserR2to1,thebetterSRFfitsampledata,1.5.3correlationcoefficient(相关系数）Correlationcoefficientquantitivelymeasuresthelinearcorrelationbetweentwovariables,denotedasr,Property:（1）randhassamesign（2）r:-1r1r=1，variableXandYareper