多元线性回归模型估计

MultipleRegressionAnalysisy=b0+b1x1+b2x2+...bkxk+u1.Estimation1多元线性回归模型估计共31页，您现在浏览的是第1页!ParallelswithSimpleRegression

b0isstilltheintercept

b1tobkallcalledslopeparameters

uisstilltheerrorterm(ordisturbance)Stillneedtomakeazeroconditionalmeanassumption,sonowassumethatE(u|x1,x2,…,xk)=0Stillminimizingthesumofsquaredresiduals,sohavek+1firstorderconditions2多元线性回归模型估计共31页，您现在浏览的是第2页!theOLSregressionlineorthesample

regression

function(SRF).istheOLS

intercept

estimateandaretheOLS

slope

estimates.Stilluseordinary

least

squarestogettheestimates:3多元线性回归模型估计共31页，您现在浏览的是第3页!OLSFirstOrderConditions

Thisminimizationproblemcanbesolvedusingmultivariablecalculas.Thisleadstok+1linearequationink+1unknown:……4多元线性回归模型估计共31页，您现在浏览的是第4页!InterpretingMultipleRegression5多元线性回归模型估计共31页，您现在浏览的是第5页!“PartiallingOut”continued

Previousequationimpliesthatregressingyonx1

andx2givessameeffectofx1asregressingyonresidualsfromaregressionofx1onx2Thismeansonlythepartofxi1thatisuncorrelatedwithxi2arebeingrelatedtoyisowe’reestimatingtheeffectofx1onyafterx2hasbeen“partialledout”6多元线性回归模型估计共31页，您现在浏览的是第6页!Goodness-of-Fit7多元线性回归模型估计共31页，您现在浏览的是第7页!Goodness-of-Fit(continued)8多元线性回归模型估计共31页，您现在浏览的是第8页!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多元线性回归模型估计共31页，您现在浏览的是第9页!OmittedVariableBias10多元线性回归模型估计共31页，您现在浏览的是第10页!OmittedVariableBias(cont)11多元线性回归模型估计共31页，您现在浏览的是第11页!SummaryofDirectionofBiasCorr(x1,x2)>0Corr(x1,x2)<0b2>0PositivebiasNegativebiasb2<0NegativebiasPositivebias12多元线性回归模型估计共31页，您现在浏览的是第12页!TheMoreGeneralCase

Technically,canonlysignthebiasforthemoregeneralcaseifalloftheincludedx’sareuncorrelatedTypically,then,weworkthroughthebiasassumingthex’sareuncorrelated,asausefulguideevenifthisassumptionisnotstrictlytrue13多元线性回归模型估计共31页，您现在浏览的是第13页!VarianceofOLS(cont)

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

The4assumptionsforunbiasedness,plusthishomoskedasticityassumptionareknownastheGauss-Markovassumptions14多元线性回归模型估计共31页，您现在浏览的是第14页!ComponentsofOLSVariances

Theerrorvariance:alargers2impliesalargervariancefortheOLSestimatorsThetotalsamplevariation:alargerSSTjimpliesasmallervariancefortheestimatorsLinearrelationshipsamongtheindependentvariables:alargerRj2impliesalargervariancefortheestimators15多元线性回归模型估计共31页，您现在浏览的是第15页!MisspecifiedModels(cont)

Whilethevarianceoftheestimatorissmallerforthemisspecifiedmodel,unlessb2=0themisspecifiedmodelisbiasedAsthesamplesizegrows,thevarianceofeachestimatorshrinkstozero,makingthevariancedifferencelessimportant16多元线性回归模型估计共31页，您现在浏览的是第16页!ErrorVarianceEstimate(cont)

df=n–(k+1),ordf=n–k–1df(i.e.degreesoffreedom)isthe(numberofobservations)–(numberofestimatedparameters)17多元线性回归模型估计共31页，您现在浏览的是第17页!AFittedorPredictedValueForobservationi,thefittedvalueisTheresidualforobservationiisdefinedasinthesimpleregressioncase,Theproperties123Thepoint()isalwaysontheOLSregressionline:18多元线性回归模型估计共31页，您现在浏览的是第18页!A“PartiallingOut”Interpretation19多元线性回归模型估计共31页，您现在浏览的是第19页!SimplevsMultipleRegEstimate20多元线性回归模型估计共31页，您现在浏览的是第20页!Goodness-of-Fit(continued)

Howdowethinkabouthowwelloursampleregressionlinefitsoursampledata?Canputethefractionofthetotalsumofsquares(SST)thatisexplainedbythemodel,callthistheR-squaredofregressionR2=SSE/SST=1–SSR/SST21多元线性回归模型估计共31页，您现在浏览的是第21页!MoreaboutR-squared

R2canneverdecreasewhenanotherindependentvariableisaddedtoaregression,andusuallywillincreaseBecauseR2

willusuallyincreasewiththenumberofindependentvariables,itisnotagoodwaytoparemodels22多元线性回归模型估计共31页，您现在浏览的是第22页!TooManyorTooFewVariables

Whathappensifweincludevariablesinourspecificationthatdon’tbelong?Thereisnoeffectonourparameterestimate,andOLSremainsunbiasedWhatifweexcludeavariablefromourspecificationthatdoesbelong?OLSwillusuallybebiased23多元线性回归模型估计共31页，您现在浏览的是第23页!OmittedVariableBias(cont)24多元线性回归模型估计共31页，您现在浏览的是第24页!OmittedVariableBias(cont)25多元线性回归模型估计共31页，您现在浏览的是第25页!OmittedVariableBiasSummary

Twocaseswherebiasisequaltozerob2=0,thatisx2doesn’treallybelonginmodelx1andx2areuncorrelatedinthesampleIfcorrelationbetweenx2,x1andx2,yisthesamedirection,biaswillbepositiveIfcorrelationbetweenx2,x1andx2,yistheoppositedirection,biaswillbenegative26多元线性回归模型估计共31页，您现在浏览的是第26页!VarianceoftheOLSEstimators

NowweknowthatthesamplingdistributionofourestimateiscenteredaroundthetrueparameterWanttothinkabouthowspreadoutthisdistributionisMucheasiertothinkaboutthisvarianceunderanadditionalassumption,soAssumeVar