课程高级计量经济学英文课件胡ch03multiple regression analysis1estimation

Economics20-Prof.Anderson1MultipleRegressionAnalysis1

（多元回归分析）y=b0+b1x1+b2x2+...+bkxk+u1.Estimation（估计）Economics20-Prof.Anderson23.1Motivationformultipleregression3.2MechanicsandinterpretationofOLS3.3TheexpectedvalueoftheOLSestimators3.4ThevarianceoftheOLSestimators3.5EfficiencyofOLS:theGauss-MarkovtheoremEconomics20-Prof.Anderson3ParallelswithSimpleRegression

与简单回归的对比

b0isstilltheintercept

b1tobkallcalledslopeparameters

uisstilltheerrorterm(ordisturbance)Stillneedtomakeazeroconditionalmeanassumption,sonowassumethatE(u|x1,x2,…,xk)=0Stillminimizingthesumofsquaredresiduals,sohavek+1firstorderconditionsEconomics20-Prof.Anderson4InterpretingMultipleRegressionEconomics20-Prof.Anderson5A“PartiallingOut”Interpretation

(“排除其他变量影响”的解释)Economics20-Prof.Anderson6“PartiallingOut”continued

Previousequationimpliesthatregressingyonx1

andx2givessameeffectofx1asregressingyonresidualsfromaregressionofx1onx2Thismeansonlythepartofxi1thatisuncorrelatedwithxi2arebeingrelatedtoyisowe’reestimatingtheeffectofx1onyafterx2hasbeen“partialledout”Economics20-Prof.Anderson7Economics20-Prof.Anderson8SimplevsMultipleRegEstimate

（简单回归与多元回归估计值的比较）Economics20-Prof.Anderson9Goodness-of-Fit（拟合优度）Economics20-Prof.Anderson10Goodness-of-Fit(continued)

Howdowethinkabouthowwelloursampleregressionlinefitsoursampledata?Cancomputethefractionofthetotalsumofsquares(SST)thatisexplainedbythemodel,callthistheR-squaredofregressionR2=SSE/SST=1–SSR/SSTGoodness-of-Fit(continued)Economics20-Prof.Anderson11Goodness-of-Fit(continued)Economics20-Prof.Anderson12MoreaboutR-squared

R2canneverdecreasewhenanotherindependentvariableisaddedtoaregression,andusuallywillincreaseBecauseR2

willusuallyincreasewiththenumberofindependentvariables,itisnotagoodwaytocomparemodels，即使用R2作为判断是否应在模型中增加一个或几个解释变量的工具很不适当。通常使用：校正R2、AIC信息准则、SC准则，假设检验（ch4）Economics20-Prof.Anderson13AssumptionsforUnbiasedness（4个）

参数线性：Populationmodelislinearinparameters:y=b0+b1x1+b2x2+…+bkxk

+u

随机抽样：Wecanusearandomsampleofsizen,{(xi1,xi2,…,xik,yi):i=1,2,…,n},fromthepopulationmodel,sothatthesamplemodelisyi=b0+b1xi1+b2xi2+…+bkxik

+ui

零条件均值：E(u|x1,x2,…xk)=0,implyingthatalloftheexplanatoryvariablesareexogenous

不存在完全共线性：Noneofthex’sisconstant,andtherearenoexactlinearrelationshipsamongthemEconomics20-Prof.Anderson14TooManyorTooFewVariables

过度设定或设定不足

Whathappensifweincludevariablesinourspecificationthatdon’tbelong?Thereisnoeffectonourparameterestimate,andOLSremainsunbiasedWhatifweexclude（遗漏）

avariablefromourspecificationthatdoesbelong?OLSwillusuallybebiasedEconomics20-Prof.Anderson15OmittedVariableBias

（遗漏变量的偏误）Economics20-Prof.Anderson16OmittedVariableBias(cont)Economics20-Prof.Anderson17OmittedVariableBias(cont)Economics20-Prof.Anderson18OmittedVariableBias(cont)Economics20-Prof.Anderson19SummaryofDirectionofBiasCorr(x1,x2)>0Corr(x1,x2)<0b2>0PositivebiasNegativebiasb2<0NegativebiasPositivebiasEconomics20-Prof.Anderson20OmittedVariableBiasSummary

Twocaseswherebiasisequaltozerob2=0,thatisx2doesn’treallybelonginmodelx1andx2areuncorrelatedinthesampleIfcorrelationbetweenx2,x1andx2,yisthesamedirection,biaswillbepositiveIfcorrelationbetweenx2,x1andx2,yistheoppositedirection,biaswillbenegativeEconomics20-Prof.Anderson21TheMoreGeneralCase

Technically,canonlysignthebiasforthemoregeneralcaseifalloftheincludedx’sareuncorrelatedTypically,then,weworkthroughthebiasassumingthex’sareuncorrelated,asausefulguideevenifthisassumptionisnotstrictlytrueEconomics20-Prof.Anderson22VarianceoftheOLSEstimators（OLS估计量的方差）

NowweknowthatthesamplingdistributionofourestimateiscenteredaroundthetrueparameterWanttothinkabouthowspreadoutthisdistributionisMucheasiertothinkaboutthisvarianceunderanadditionalassumption,so同方差性：AssumeVar(u|x1,x2,…,xk)=s2(Homoskedasticity)Economics20-Prof.Anderson23VarianceofOLS(cont)

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

The4assumptionsforunbiasedness,plusthishomoskedasticityassumptionareknownastheGauss-MarkovassumptionsEconomics20-Prof.Anderson24VarianceofOLS(cont)Economics20-Prof.Anderson25ComponentsofOLSVariances

Theerrorvariance:alargers2impliesalargervariancefortheOLSestimatorsThetotalsamplevariation:alargerSSTjimpliesasmallervariancefortheestimatorsLinearrelationshipsamongtheindependentvariables:alargerRj2impliesalargervariancefortheestimatorsEconomics20-Prof.Anderson26MisspecifiedModels

（误设模型中的方差）Economics20-Prof.Anderson27MisspecifiedModels(cont)

Whilethevarianceoftheestimatorissmallerforthemisspecifiedmodel,unlessb2=0themisspecifiedmodelisbiasedAsthesamplesizegrows,thevarianceofeachestimatorshrinkstozero,makingthevariancedifferencelessimportantEconomics20-Prof.Anderson28Estimatingth