隔壁計(jì)量-2015s普通最小二乘法操作和解釋

中級計(jì)量經(jīng)濟(jì)INTERMEDIATEChapterOutlineMotivationforMultiple使用多元回歸的MechanicsandInterpretationofOrdinaryLeast普通最小二乘法的操作和TheExpectedValuesoftheOLSEstimatorsTheVarianceoftheOLSEstimatorsEfficiencyofOLS:TheGauss-MarkovTheorem LectureOutlineTheMLR.1–MLR.4假定MLR.1TheUnbiasednessoftheOLSestimatesOverorUnderspecificationof模型設(shè)定不足或過度OmittedVariable遺漏變量的SamplingVarianceoftheOLSslopeestimates TheexpectedvalueoftheOLS目標(biāo)：利用觀測到的信息（數(shù)據(jù)）揭示變量之經(jīng)濟(jì)但是，數(shù)據(jù)與建立計(jì)量經(jīng)濟(jì)模型的假設(shè)之間往缺乏精確性是歸納推理系統(tǒng)化的最 Rao(2004):不確定的知識+所含不確定性度量的知報(bào)告新的不確定的知識時(shí)， 告存在錯(cuò)誤的 TheexpectedvalueoftheOLSWenowturntothestatisticalpropertiesofOLSforestimatingtheparametersinanunderlyingpopulationmodel.Statisticalpropertiesarethepropertiesofestimatorswhenrandomsamplingisdonerepeatedly.Notmeaningfultotalkaboutthestatisticalpropertiesofasetofestimatesobtainedfromasinglesample. 假定MLR.1（對參數(shù)而言為線性Inthepopulationmodel,thedependentvariableyisrelatedtotheindependentvariablexandtheerroruasy=0+1x1+2x2+1,2…,k:unknownparametersofuisanrandomerrororrandomdisturbanceThepopulationmodelisalsocalledthetruemodel,toallowforthepossibilitythatwemightestimateamodelthatdiffersfromit. AssumptionMLR.2(Random假定MLR.2（隨機(jī)抽樣性Wecanusearandomsampleofsizenfromthepopulation,{(xi1,xi2,…,xik;yi):i=1,…,n},whereidenotesobservation,andj=1,…,kdenotesthejthregressor.xi2,…,xik;yi):i=1,…,n其中i代表第i個(gè)觀察值，Sometimeswewrite有 AssumptionsMLR.3假定E(u|xi1,xi2…,Whenthisassumptionholds,wesayalloftheexplanatoryvariablesareexogenous;whenitfails,wesaythattheexplanatoryvariablesare。Wewillpayparticularattentiontothecasethatassumption3failsduetoomittedvariables.特別注意當(dāng)重要變量缺省導(dǎo)致假定3不成立的情 AssumptionMLR.4假定MLR.4Noperfectcollinearity（不存在完全共線性）Inthesample,noneoftheindependentvariablesisconstant,andtherearenoexactlinearrelationshipsamongtheindependent樣本自變量無常數(shù)，自變量之間也不存在嚴(yán)格的線性關(guān)Whenoneregressorisanexactlinearcombinationoftheotherregressor(s),wesaythemodelsuffersfromperfectcollinearity. AssumptionMLR.4假定Examplesofperfectcollinearity：y=0+1x1+2x2+3x3+u, x2=，y=0+1log(inc)+2log(inc2)+uy=0+1x1+2x2+3x3+ x1+x2+x3+x4 2x2+…+kxk+u,當(dāng)y0+1x1+2x2+…+kxk+u,n<(k+1)也發(fā)生完全共線性的情況Whydoesperfectcollinearity AssumptionMLR.4假定 ysis.首先，你可以從ceterisparibus分析的角度考慮。Second,fromthe ofpartiallingout.你可以從Third,fromtheimplementationofOLSThedenominatoroftheOLSestimatoris0whenthereisperfectcollinearity,hencetheOLSestimatorcannotbeperformed.在完全共線性情況下，OLS估計(jì)量的分母為零，因此OLS估計(jì)量不能估算具體估 AssumptionMLR.4假定考慮人口受教育程度對人均工 同期，高中以上人數(shù)占比edu值和最大值分別是：5.1%,21.6%, AssumptionMLR.4假定上海青 江東重 東四 福 海南北

高中以上人數(shù)占比年平均工資（元年平均工資（元 擬合 AssumptionMLR.4假定.gen.regwageedunote:eduomittedbecauseof1=F(=2ProbF=R-=AdjR-Root==Std.t[95% Theorem3.1(UnbiasednessofUnderassumptionsMLR.1throughMLR.4,theOLSestimatorsareunbiasedestimatorofthepopulationparameters,thatisjj

)

j Theorem3.1(UnbiasednessofUnbiasednessisthepropertyofanestimator,thatis,theprocedurethatcanproduceanestimateforaspecificsample,notanestimate.無偏性是估計(jì)量的特性，而不是估計(jì)值的特性。估計(jì)量 法（過程，該方法使得給定一個(gè)樣本，我們可以得到一組們評價(jià)的是方法的優(yōu)Notcorrecttosay“5percentisanestimateofthereturnofeducation”.“5%是教育匯 TooManyorTooFew變量太多還是太少 Whathappensifweincludevariablesinourspecificationthatdonotbelong?如果我們在設(shè)定中包含了不屬于真實(shí)模型的變量會Amodelisoverspecifedwhenoneormoreoftheindependentvariablesisincludedinthemodeleventhoughithasnopartialeffectonyinthepopulation.。 ThereisnoeffectontheunbiasednessoftheOLSestimators.ButitcanhaveundesirableeffectsonthevariancesoftheOLS TooManyorTooFewWhatifweexcludeavariablefromourspecificationthatdoesbelong?如果我們在設(shè)定中排除了一個(gè)本屬于真實(shí)模型的變量會如Ifavariablethatactuallybelongsinthetruemodelisomitted,wesaythemodelisunderspecified.如果一個(gè)實(shí)際上屬于真實(shí)模型的變量被遺漏，我們說此模OLSwillusuallybe 此時(shí)OLS通常有Derivingthebiascausedbyomittinganimportantvariableisanexampleof OmittedVariableSupposethey01x12x2andsatisfieMLR.-MLR.4,butdueourignoranceordataunavailaity, y

x1)1 x1 OmittedVariableBias遺漏變量的偏誤（續(xù)yi01xi12xi2ui,sothenumerator

2

xx

x OmittedVariableBias遺漏變量的偏誤（續(xù)

x1xi2xi1x1 2xx2xx2 由于E(ui)0， E

x

OmittedVariableBias遺漏變量的偏誤（續(xù)

xthen

x soE

OmittedVariableBias遺漏變量的偏誤總結(jié) Twocaseswherebiasisequaltozero兩種偏誤為零的情2=0,thatisx2doesnotreallybelongin2=0，也就是，x2實(shí)際上不屬于模x1andx2areuncorrelatedinthe樣本中x1與x2不相 OmittedVariableBias遺漏變量的偏 總Ifcorrelationbetweenx2,x1andx2,yissamedirection,biaswillbepositive如果x2x1間相關(guān)性和x2與y間相關(guān)性同方向 Ifcorrelationbetweenx2x1andx2yistheoppositedirection,biaswillbenegative如果 SummaryofDirectionofCorr(x1,x2)>Corr(x1,x2)<2>2<NegativePositive Example:Aglobalmrg_dat"D:\data\CFPS\cfps2010"use$mrg_dat\2010adult.dta*keeptheindependentvariableskeepqk601qa1age Example:A*gendependentvariable(DV)withsetrelationshiptoIVs+randomgeny=5+ e+.3*qa1age+4**runthedesiredregy e reg e2of=F(=ProbF=R-=AdjR-=Root=yt[95%e........ reg 1of=F(=ProbF==AdjR-=Root=yt[95%e.... regy1of F(ProbF=R-=AdjR-=Root=yt[95%..... e1of=F(=ProbF==AdjR-=Root=et[95%.. Omitted-VariableBiasThereality-2isNowaytoknowthesignofCorr(x1,x2)astheomittedvariableisoftenunobserved.Whattodo?們無法準(zhǔn)確知道Corr(x1x2)的符號。怎么辦呢？Relyoneconomictheoriesandintuitionmakeaneducatedguessofthesign. TheMoreGeneral Technically,itismoredifficulttoderivethesignofomittedvariablebiaswithmultipleregressors. Butrememberthatifanomittedvariablehaspartialeffectsonyanditiscorrelatedwithatleastoneoftheregressors,thentheOLSestimatorsofallcoefficientswillbebiased. VarianceoftheOLS假定MLR.5（同方差性Var(u|x1,x2,…,xk)=s2Meansthatthevarianceintheerrorterm,u,conditionalontheexplanatoryvariables,isthesameforallcombinationsofesofexplanatoryIftheassumptionfails,wesaythemodelexhibits VarianceofOLS(cont.)Letxstandsfor(x1,用x表示(x1 AssumingthatVar(u|x)=2alsoimpliesthatVar(y|x)=2假定Var(u|x)=2，也就意味著Var(y|x)= AssumptionsMLR.1-5arecollectivelyknownastheGauss-Markovassumptions.假定MLR.1-5共同被稱 假 Theorem3.2(SamplingVariancesoftheOLSSlopeGiventheGauss-Markov給 假 R2R2

2SST2

,jSSTj

xj

andR2isthejfromregressingxjonallotherj2其中 x x，2 R2是x向所有其它x回歸所得到的 InterpretingTheorem Theorem3.2showsthatthevariancesoftheestimatedslopecoefficientsareinfluencedbythree定理3.2顯示：估計(jì)斜率系數(shù)的方差受到三個(gè)因素的影Theerror誤差項(xiàng)的方Thetotalsample總的樣本變Linearrelationshipsamongtheindependent解釋變量之間的線性相關(guān)關(guān) InterpretingTheorem3.2:TheError對定理3.2的解釋（1）：誤差項(xiàng)方Alarger2impliesalargervariancefortheOLS更大的2意味著更大的OLS估計(jì)量方差A(yù)larger2meansmorenoisesinthe更大的2意味著方程中的“噪音”越多Thismakesitmoredifficulttoextracttheexactpartialeffectoftheregressorontheregressand. Introducingmoreregressorscanreducethevariance.Butoftenthisisnotpossible,neitherisitdesirable. 2doesnotdependonsample2不依賴于樣本大小 InterpretingTheorem3.2:ThetotalsampleAlargerSSTjimpliesasmallervariancefortheestimators,andviceversa.更大的SSTj意味著更小的估計(jì)量方差，反之亦然Everythingelsebeingequal,moresamplevariationinxisalwayspreferred.其它條件不變情況下，x的樣本方差越大越好Onewaytogainmoresamplevariationistoincreasethesamplesize.增加樣本方差 法是增加樣本容量Thiscomponentofparametervariancedependsonthesamplesize.參數(shù)方差的這一組成部分依賴于樣本容 InterpretingTheorem3.2:對定理3.1的解釋