计量经济学课件03

MultipleRegressionAnalysisy=b0+b1x1+b2x2+...bkxk+u1.Estimation1Economics20-Prof.AndersonParallelswithSimpleRegression

b0isstilltheintercept

b1tobkallcalledslopeparameters

uisstilltheerrorterm(ordisturbance)Stillneedtomakeazeroconditionalmeanassumption,sonowassumethatE(u|x1,x2,…,xk)=0Stillminimizingthesumofsquaredresiduals,sohavek+1firstorderconditions2Economics20-Prof.AndersonInterpretingMultipleRegression3Economics20-Prof.AndersonA“PartiallingOut”Interpretation4Economics20-Prof.Anderson“PartiallingOut”continued

Previousequationimpliesthatregressingyonx1

andx2givessameeffectofx1asregressingyonresidualsfromaregressionofx1onx2Thismeansonlythepartofxi1thatisuncorrelatedwithxi2arebeingrelatedtoyisowe’reestimatingtheeffectofx1onyafterx2hasbeen“partialledout”5Economics20-Prof.AndersonSimplevsMultipleRegEstimate6Economics20-Prof.AndersonGoodness-of-Fit7Economics20-Prof.AndersonGoodness-of-Fit(continued)

Howdowethinkabouthowwelloursampleregressionlinefitsoursampledata?Cancomputethefractionofthetotalsumofsquares(SST)thatisexplainedbythemodel,callthistheR-squaredofregressionR2=SSE/SST=1–SSR/SST8Economics20-Prof.AndersonGoodness-of-Fit(continued)9Economics20-Prof.AndersonMoreaboutR-squared

R2canneverdecreasewhenanotherindependentvariableisaddedtoaregression,andusuallywillincreaseBecauseR2

willusuallyincreasewiththenumberofindependentvariables,itisnotagoodwaytocomparemodels10Economics20-Prof.AndersonAssumptionsforUnbiasedness

Populationmodelislinearinparameters:y=b0+b1x1+b2x2+…+bkxk

+uWecanusearandomsampleofsizen,{(xi1,xi2,…,xik,

yi):i=1,2,…,n},fromthepopulationmodel,sothatthesamplemodelisyi=b0+b1xi1+b2xi2+…+bkxik

+ui

E(u|x1,x2,…xk)=0,implyingthatalloftheexplanatoryvariablesareexogenousNoneofthex’sisconstant,andtherearenoexactlinearrelationshipsamongthem11Economics20-Prof.AndersonTooManyorTooFewVariables

Whathappensifweincludevariablesinourspecificationthatdon’tbelong?Thereisnoeffectonourparameterestimate,andOLSremainsunbiasedWhatifweexcludeavariablefromourspecificationthatdoesbelong?OLSwillusuallybebiased12Economics20-Prof.AndersonOmittedVariableBias13Economics20-Prof.AndersonOmittedVariableBias(cont)14Economics20-Prof.AndersonOmittedVariableBias(cont)15Economics20-Prof.AndersonOmittedVariableBias(cont)16Economics20-Prof.AndersonSummaryofDirectionofBiasCorr(x1,x2)>0Corr(x1,x2)<0b2>0PositivebiasNegativebiasb2<0NegativebiasPositivebias17Economics20-Prof.AndersonOmittedVariableBiasSummary

Twocaseswherebiasisequaltozerob2=0,thatisx2doesn’treallybelonginmodelx1andx2areuncorrelatedinthesampleIfcorrelationbetweenx2,x1andx2,yisthesamedirection,biaswillbepositiveIfcorrelationbetweenx2,x1andx2,yistheoppositedirection,biaswillbenegative18Economics20-Prof.AndersonTheMoreGeneralCase

Technically,canonlysignthebiasforthemoregeneralcaseifalloftheincludedx’sareuncorrelatedTypically,then,weworkthroughthebiasassumingthex’sareuncorrelated,asausefulguideevenifthisassumptionisnotstrictlytrue19Economics20-Prof.AndersonVarianceoftheOLSEstimators

NowweknowthatthesamplingdistributionofourestimateiscenteredaroundthetrueparameterWanttothinkabouthowspreadoutthisdistributionisMucheasiertothinkaboutthisvarianceunderanadditionalassumption,soAssumeVar(u|x1,x2,…,xk)=s2(Homoskedasticity)20Economics20-Prof.AndersonVarianceofOLS(cont)

Letxstandfor(x1,x2,…xk)AssumingthatVar(u|x)=s2alsoimpliesthatVar(y|x)=s2

The4assumptionsforunbiasedness,plusthishomoskedasticityassumptionareknownastheGauss-Markovassumptions21Economics20-Prof.AndersonVarianceofOLS(cont)22Economics20-Prof.AndersonComponentsofOLSVariances

Theerrorvariance:alargers2impliesalargervariancefortheOLSestimatorsThetotalsamplevariation:alargerSSTjimpliesasmallervariancefortheestimatorsLinearrelationshipsamongtheindependentvariables:alargerRj2impliesalargervariancefortheestimators23Economics20-Prof.AndersonMisspecifiedModels24Economics20-Prof.AndersonMisspecifiedModels(cont)

Whilethevarianceoftheestimatorissmallerforthemisspecifiedmodel,unlessb2=0themisspecifiedmodelisbiasedAsthesamplesizegrows,thevarianceofeachestimatorshrinkstozero,makingthevariancedifferencelessimportant25Economics20-Prof.AndersonEstimatingtheErrorVariance

Wedon’tknowwhattheerrorvariance,s2,is,becausewedon’tobservetheerrors,uiWhatweobservearetheresiduals,ûiWecanusetheresidualstoformanestimateoftheerrorvariance26Economics20-Prof.AndersonErrorVarianceEstimate(cont)

df=n–(k+1),ordf=n–k–1

df(i.e.degreesoffreedom)isthe(num