工具变量与两阶段最小二乘法

IntermediateEconometrics,YanShen1InstrumentalVariables&2SLS工具变量与两阶段最小二乘法y=b0+b1x1+b2x2+...bkxk+ux1=p0+p1z+p2x2+...pkxk+vIntermediateEconometrics,YanShen2ChapterOutline本章提要OmittedVariablesinasimpleregressionmodel简单回归中的遗漏变量IVestimationoftheMultipleRegression多方程回归中的工具变量估计TwoStageLeastSquares两阶段最小二乘法IVsolutionstoerrors-in-variablesproblem用工具变量解决测量误差问题Testingforendogeneity…检验内生性IntermediateEconometrics,YanShen3LectureOutline本课提要Motivation:WhyusingIV?出发点：为何用工具变量？StatisticalInferencewiththeIVestimatorIV估计中的统计推断PropertiesofIVwithapoorIV“坏”工具变量的性质ComputingRsquaresafterIV计算IV估计的R方IVestimationofthemultipleregressionmodel多方程回归的IV估计IntermediateEconometrics,YanShen4Problemtostartwith从这个问题出发…Ifimportantvariablesareomitted,whatshouldwedo?如果一些重要的变量被遗漏，我们应当怎么办？IntermediateEconometrics,YanShen5Thewaysout一些办法Ignoretheproblem,pretendthatitdoesnotexist忽略这个问题，假装这个问题并不存在Findanduseasuitableproxy使用代理变量Usesanestimationmethodthatrecognizesthepresenceoftheomittedvariable使用一种对遗漏变量稳健的估计方法。IntermediateEconometrics,YanShen6WhyUseInstrumentalVariables?为何使用工具变量？InstrumentalVariables(IV)estimationisusedwhenyourmodelhasendogenousx’s当模型解释变量具有内生性时，使用工具变量估计Thatis,whenCov(x,u)≠0即，Cov(x,u)≠0时IntermediateEconometrics,YanShen7WhyUseInstrumentalVariables?为何使用工具变量？Thus,IVcanbeusedtoaddresstheproblemofomittedvariablebias所以，IV可以用来解决遗漏变量偏差Additionally,IVcanbeusedtosolvetheclassicerrors-in-variablesproblem而且，IV可用来解决经典的测量误差问题IntermediateEconometrics,YanShen8InstrumentalVariable:Whoqualifies?什么样的变量可以作为IV？Inorderforavariable,z,toserveasavalidinstrumentforx,thefollowingmustbetrue针对内生变量x的一个有效的工具变量z应当满足如下条件Theinstrumentmustbeexogenous工具变量应为外生Thatis,Cov(z,u)=0(15.4)即Cov(z,u)=0IntermediateEconometrics,YanShen9Theinstrumentmustbecorrelatedwiththeendogenousvariablex工具变量应与内生变量x相关Thatis,Cov(z,x)≠0(15.5)InstrumentalVariable:Whoqualifies?什么样的变量可以作为IV？IntermediateEconometrics,YanShen10AboutCov(z,u)关于Cov(z,u)WehavetousecommonsenseandeconomictheorytodecideifitmakessensetoassumeCov(z,u)=0为了判断Cov(z,u)=0这一假定是否合理，我们不得不依赖于常识和经济理论。IntermediateEconometrics,YanShen11AboutCov(z,x)WecantestifCov(z,x)≠0我们可以检验是否Cov(z,x)≠0JusttestingH0:p1=0inx=p0+p1z+v只需检验H0:p1=0inx=p0+p1z+vSometimeswerefertothisregressionasthefirst-stageregression.有时我们将这个回归称为第一阶段回归。IntermediateEconometrics,YanShen12Example:wagedetermination例子：工资决定Supposethetruemodelregresseslog(wage)oneducation(educ)andability(abil).假定真实模型将对数工资对教育和能力回归Nowabilityisunobserved,andtheproxy,IQ,isnotavailable.现在能力不可观测，而且没有代理变量IQTheactualregression:regresslog(wage)oneduc.事实上使用的回归：将对数工资对教育回归IntermediateEconometrics,YanShen13Example:wagedetermination例子：工资决定Problem:sincetheerrortermcontainsIQ,andeducationcorrelateswithIQ,endogeneityproblemappears.问题：由于误差项包含IQ，并且教育水平与能力相关，此时会出现教育的内生性问题。AgoodIVneedtobehighlycorrelatedwitheducation,butnotcorrelatedwiththeerrorterm.一个好的IV应当与教育水平高度相关，并且与误差项不相关。IntermediateEconometrics,YanShen14Example:wagedetermination例子：工资决定IsIQagoodinstrument?IQ是好的工具变量吗？No.Itcorrelateswithbotheducationandtheerrorterm.不。它同时与教育和误差项相关。IntermediateEconometrics,YanShen15Example:wagedetermination例子：工资决定IVusedintheliterature:在文献中使用的IVMother’seducation母亲教育水平Numberofsiblings.Hypothesis:moresiblingsisassociatedwithloweraveragelevelsofeducation.兄弟姐妹数目。假说：兄弟姐妹越多，平均受教育水平越低IfwewithtouseeitherofthemasIV,weneedtobeconfidentthattheyarenotcorrelatedwithability.无论我们使用其中的哪一个作为IV，我们都需要肯定它们是与能力不相关的。IntermediateEconometrics,YanShen16WhenanIVisAvailable:Estimation当IV存在时：估计Fory=b0+b1x+u,andgivenourassumptions(15.4)and(15.5),b1canbeidentified.对于y=b0+b1x+u，且给定假定(15.4)及（15.5），b1可以被识别Inthiscontext,identificationmeansthatwecanwriteb1intermsofpopulationmomentsthatcanbeestimatedinsamples.这里，识别是指我们可以将b1表示为总体矩的函数，并且这些矩可以通过样本估计。IntermediateEconometrics,YanShen17WhenanIVisAvailable:Estimation当IV存在时：估计SinceCov(z,y)=b1Cov(z,x)+Cov(z,u),sob1=Cov(z,y)/Cov(z,x)ThentheIVestimatorforb1is则b1的工具变量估计为xxzzyyzziiii1ˆbIntermediateEconometrics,YanShen18WhenanIVisAvailable:Estimation当IV存在时：估计Whenz=xweobtaintheOLSestimatorofb1.当z=x时，我们得到b1的OLS估计Thismeanswhenxisexogenous,itcanbeusedasitsownIV,andtheIVestimatorisidenticaltoOLSinthiscase.这意味着当x是外生时，可以用它作自己的IV，这时的IV估计与OLS估计恒等。IntermediateEconometrics,YanShen19WhenanIVisAvailable:Estimation当IV存在时：估计Whenassumptions(15.4)and(15.5)hold,onecanshowthattheIVestimatorisconsistentforb1,afterapplyingthelawoflargenumbers.当假定(15.4)和(15.5)成立时，可以应用大数定律证明IV估计是b1的一致估计。IntermediateEconometrics,YanShen20WhenanIVisAvailable:Inference当IV存在时：推断Thehomoskedasticityassumptionis同方差假定：E(u2|z)=s2=Var(u)(15.11)Whenassumptions(15.4),(15.5),(15.11)hold,giventheasymptoticvariance,thestandarderrorforb1canbeestimated.当假定(15.4),(15.5),(15.11)成立时，给定渐近方差，可以估计b1的标准差IntermediateEconometrics,YanShen21WhenanIVisAvailable:Inference当IV存在时：推断112122,222,ˆTheasymptoticvarianceofisˆˆ:thepopulationvarianceofx.x:thepopulationvarianceofu.u:thesquareofthepopulationcorrelatixxzxxzVarnbbsbsss的渐近方差是的总体方差的总体方差onbetweenxandz.xz，总体相关系数的平方IntermediateEconometrics,YanShen22WhenanIVisAvailable:Inference当IV存在时：推断2212,222,22Th