MultipleRegressionAnalysisy=b0+b1x1+b2x2+...bkxk+u1.Estimation1多元線性回歸模型估計共31頁，您現在瀏覽的是第1頁!ParallelswithSimpleRegression

b0isstilltheintercept

b1tobkallcalledslopeparameters

uisstilltheerrorterm(ordisturbance)Stillneedtomakeazeroconditionalmeanassumption,sonowassumethatE(u|x1,x2,…,xk)=0Stillminimizingthesumofsquaredresiduals,sohavek+1firstorderconditions2多元線性回歸模型估計共31頁，您現在瀏覽的是第2頁!theOLSregressionlineorthesample

regression

function(SRF).istheOLS

intercept

estimateandaretheOLS

slope

estimates.Stilluseordinary

least

squarestogettheestimates:3多元線性回歸模型估計共31頁，您現在瀏覽的是第3頁!OLSFirstOrderConditions

Thisminimizationproblemcanbesolvedusingmultivariablecalculas.Thisleadstok+1linearequationink+1unknown:……4多元線性回歸模型估計共31頁，您現在瀏覽的是第4頁!InterpretingMultipleRegression5多元線性回歸模型估計共31頁，您現在瀏覽的是第5頁!“PartiallingOut”continued

Previousequationimpliesthatregressingyonx1

andx2givessameeffectofx1asregressingyonresidualsfromaregressionofx1onx2Thismeansonlythepartofxi1thatisuncorrelatedwithxi2arebeingrelatedtoyisowe’reestimatingtheeffectofx1onyafterx2hasbeen“partialledout”6多元線性回歸模型估計共31頁，您現在瀏覽的是第6頁!Goodness-of-Fit7多元線性回歸模型估計共31頁，您現在瀏覽的是第7頁!Goodness-of-Fit(continued)8多元線性回歸模型估計共31頁，您現在瀏覽的是第8頁!AssumptionsforUnbiasedness

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,andtherearenoexactlinearrelationshipsamongthem9多元線性回歸模型估計共31頁，您現在瀏覽的是第9頁!OmittedVariableBias10多元線性回歸模型估計共31頁，您現在瀏覽的是第10頁!OmittedVariableBias(cont)11多元線性回歸模型估計共31頁，您現在瀏覽的是第11頁!SummaryofDirectionofBiasCorr(x1,x2)>0Corr(x1,x2)<0b2>0PositivebiasNegativebiasb2<0NegativebiasPositivebias12多元線性回歸模型估計共31頁，您現在瀏覽的是第12頁!TheMoreGeneralCase

Technically,canonlysignthebiasforthemoregeneralcaseifalloftheincludedx’sareuncorrelatedTypically,then,weworkthroughthebiasassumingthex’sareuncorrelated,asausefulguideevenifthisassumptionisnotstrictlytrue13多元線性回歸模型估計共31頁，您現在瀏覽的是第13頁!VarianceofOLS(cont)

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

The4assumptionsforunbiasedness,plusthishomoskedasticityassumptionareknownastheGauss-Markovassumptions14多元線性回歸模型估計共31頁，您現在瀏覽的是第14頁!ComponentsofOLSVariances

Theerrorvariance:alargers2impliesalargervariancefortheOLSestimatorsThetotalsamplevariation:alargerSSTjimpliesasmallervariancefortheestimatorsLinearrelationshipsamongtheindependentvariables:alargerRj2impliesalargervariancefortheestimators15多元線性回歸模型估計共31頁，您現在瀏覽的是第15頁!MisspecifiedModels(cont)

Whilethevarianceoftheestimatorissmallerforthemisspecifiedmodel,unlessb2=0themisspecifiedmodelisbiasedAsthesamplesizegrows,thevarianceofeachestimatorshrinkstozero,makingthevariancedifferencelessimportant16多元線性回歸模型估計共31頁，您現在瀏覽的是第16頁!ErrorVarianceEstimate(cont)

df=n–(k+1),ordf=n–k–1df(i.e.degreesoffreedom)isthe(numberofobservations)–(numberofestimatedparameters)17多元線性回歸模型估計共31頁，您現在瀏覽的是第17頁!AFittedorPredictedValueForobservationi,thefittedvalueisTheresidualforobservationiisdefinedasinthesimpleregressioncase,Theproperties123Thepoint()isalwaysontheOLSregressionline:18多元線性回歸模型估計共31頁，您現在瀏覽的是第18頁!A“PartiallingOut”Interpretation19多元線性回歸模型估計共31頁，您現在瀏覽的是第19頁!SimplevsMultipleRegEstimate20多元線性回歸模型估計共31頁，您現在瀏覽的是第20頁!Goodness-of-Fit(continued)

Howdowethinkabouthowwelloursampleregressionlinefitsoursampledata?Canputethefractionofthetotalsumofsquares(SST)thatisexplainedbythemodel,callthistheR-squaredofregressionR2=SSE/SST=1–SSR/SST21多元線性回歸模型估計共31頁，您現在瀏覽的是第21頁!MoreaboutR-squared

R2canneverdecreasewhenanotherindependentvariableisaddedtoaregression,andusuallywillincreaseBecauseR2

willusuallyincreasewiththenumberofindependentvariables,itisnotagoodwaytoparemodels22多元線性回歸模型估計共31頁，您現在瀏覽的是第22頁!TooManyorTooFewVariables

Whathappensifweincludevariablesinourspecificationthatdon’tbelong?Thereisnoeffectonourparameterestimate,andOLSremainsunbiasedWhatifweexcludeavariablefromourspecificationthatdoesbelong?OLSwillusuallybebiased23多元線性回歸模型估計共31頁，您現在瀏覽的是第23頁!OmittedVariableBias(cont)24多元線性回歸模型估計共31頁，您現在瀏覽的是第24頁!OmittedVariableBias(cont)25多元線性回歸模型估計共31頁，您現在瀏覽的是第25頁!OmittedVariableBiasSummary

Twocaseswherebiasisequaltozerob2=0,thatisx2doesn’treallybelonginmodelx1andx2areuncorrelatedinthesampleIfcorrelationbetweenx2,x1andx2,yisthesamedirection,biaswillbepositiveIfcorrelationbetweenx2,x1andx2,yistheoppositedirection,biaswillbenegative26多元線性回歸模型估計共31頁，您現在瀏覽的是第26頁!VarianceoftheOLSEstimators

NowweknowthatthesamplingdistributionofourestimateiscenteredaroundthetrueparameterWanttothinkabouthowspreadoutthisdistributionisMucheasiertothinkaboutthisvarianceunderanadditionalassumption,soAssumeVar