工具变量与两阶段最小二乘法演示教学.ppt
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1、工具变量与两阶段最小二乘法Lecture Outline本课提要nMotivation:Why using IV?出发点:为何用工具变量?nStatistical Inference with the IV estimatorIV 估计中的统计推断nProperties of IV with a poor IV“坏”工具变量的性质nComputing R squares after IV计算IV估计的R方nIV estimation of the multiple regression model多方程回归的IV估计2Intermediate Econometrics,Yan ShenProb
2、lem to start with从这个问题出发nIf important variables are omitted,what should we do?如果一些重要的变量被遗漏,我们应当怎么办?3Intermediate Econometrics,Yan ShenThe ways out一些办法nIgnore the problem,pretend that it does not exist忽略这个问题,假装这个问题并不存在nFind and use a suitable proxy使用代理变量nUses an estimation method that recognizes the
3、presence of the omitted variable使用一种对遗漏变量稳健的估计方法。4Intermediate Econometrics,Yan ShenWhy Use Instrumental Variables?为何使用工具变量?n Instrumental Variables(IV)estimation is used when your model has endogenous xs当模型解释变量具有内生性时,使用工具变量估计n That is,when Cov(x,u)0 即,Cov(x,u)0时 5Intermediate Econometrics,Yan ShenW
4、hy Use Instrumental Variables?为何使用工具变量?nThus,IV can be used to address the problem of omitted variable bias所以,IV可以用来解决遗漏变量偏差n Additionally,IV can be used to solve the classic errors-in-variables problem而且,IV可用来解决经典的测量误差问题6Intermediate Econometrics,Yan ShenInstrumental Variable:Who qualifies?什么样的变量可以
5、作为IV?n In order for a variable,z,to serve as a valid instrument for x,the following must be true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n The instrument must be exogenous工具变量应为外生n That is,Cov(z,u)=0(15.4)即Cov(z,u)=07Intermediate Econometrics,Yan ShennThe instrument must be correlated with the endogenous varia
6、ble x工具变量应与内生变量 x 相关n That is,Cov(z,x)0(15.5)Instrumental Variable:Who qualifies?什么样的变量可以作为IV?8Intermediate Econometrics,Yan ShenAbout Cov(z,u)关于Cov(z,u)n We have to use common sense and economic theory to decide if it makes sense to assume Cov(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。9Intermed
7、iate Econometrics,Yan ShenAbout Cov(z,x)nWe can test if Cov(z,x)0 我们可以检验是否Cov(z,x)0 n Just testing H0:p1=0 in x=p0+p1z+v只需检验 H0:p1=0 in x=p0+p1z+vn Sometimes we refer to this regression as the first-stage regression.有时我们将这个回归称为第一阶段回归。10Intermediate Econometrics,Yan ShenExample:wage determination例子:工
8、资决定nSuppose the true model regresses log(wage)on education(educ)and ability(abil).假定真实模型将对数工资对教育和能力回归nNow ability is unobserved,and the proxy,IQ,is not available.现在能力不可观测,而且没有代理变量IQnThe actual regression:regress log(wage)on educ.事实上使用的回归:将对数工资对教育回归11Intermediate Econometrics,Yan ShenExample:wage det
9、ermination例子:工资决定nProblem:since the error term contains IQ,and education correlates with IQ,endogeneity problem appears.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。nA good IV need to be highly correlated with education,but not correlated with the error term.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。12Intermediate Econ
10、ometrics,Yan ShenExample:wage determination例子:工资决定nIs IQ a good instrument?IQ是好的工具变量吗?nNo.It correlates with both education and the error term.不。它同时与教育和误差项相关。13Intermediate Econometrics,Yan ShenExample:wage determination例子:工资决定nIV used in the literature:在文献中使用的IVnMothers education母亲教育水平nNumber of si
11、blings.Hypothesis:more siblings is associated with lower average levels of education.兄弟姐妹数目。假说:兄弟姐妹越多,平均受教育水平越低nIf we with to use either of them as IV,we need to be confident that they are not correlated with ability.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。14Intermediate Econometrics,Yan ShenWhen an IV i
12、s Available:Estimation当IV存在时:估计n For y=b0+b1x+u,and given our assumptions(15.4)and(15.5),b1 can be identified.对于y=b0+b1x+u,且给定假定(15.4)及(15.5),b1可以被识别nIn this context,identification means that we can write b1 in terms of population moments that can be estimated in samples.这里,识别 是指我们可以将b1表示为总体矩的函数,并且这
13、些矩可以通过样本估计。15Intermediate Econometrics,Yan ShenWhen an IV is Available:Estimation当IV存在时:估计nSince Cov(z,y)=b1Cov(z,x)+Cov(z,u),sonb1=Cov(z,y)/Cov(z,x)n Then the IV estimator for b1 is则b1的工具变量估计为16Intermediate Econometrics,Yan ShenWhen an IV is Available:Estimation当IV存在时:估计nWhen z=x we obtain the OLS
14、estimator of b1.当z=x时,我们得到b1的OLS估计n This means when x is exogenous,it can be used as its own IV,and the IV estimator is identical to OLS in this case.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。17Intermediate Econometrics,Yan ShenWhen an IV is Available:Estimation当IV存在时:估计nWhen assumptions(15.4)and(15.5)h
15、old,one can show that the IV estimator is consistent for b1,after applying the law of large numbers.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。18Intermediate Econometrics,Yan ShenWhen an IV is Available:Inference当IV存在时:推断n The homoskedasticity assumption is 同方差假定:E(u2|z)=s2=Var(u)(15.11)n When assum
16、ptions(15.4),(15.5),(15.11)hold,given the asymptotic variance,the standard error for b1 can be estimated.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19Intermediate Econometrics,Yan ShenWhen an IV is Available:Inference当IV存在时:推断20Intermediate Econometrics,Yan ShenWhen an IV is Available:Inference当IV
17、存在时:推断21Intermediate Econometrics,Yan ShenIV versus OLS estimationIV与OLS估计n Standard error in IV case differs from OLS only in the Rx,z2 from regressing x on zIV与OLS的标准差的不同之处仅在于将x对z回归得到的Rx,z2 n Since Rx,z2 1,IV standard errors are larger由于Rx,z2 1,IV的标准差会比较大。22Intermediate Econometrics,Yan ShenIV ver
18、sus OLS estimationIV与OLS估计nWhen Cov(x,u)0,OLS is inconsistent.IV is consistent when assumptions(15.4),(15.5)holds.当Cov(x,u)0,OLS不是一致估计,当(15.4),(15.5)成立时,IV是一致估计。n The stronger the correlation between z and x,the smaller the IV standard errorsx和z的相关性越强,IV的标准差越小。23Intermediate Econometrics,Yan ShenThe
19、 Effect of Poor Instruments不好的工具变量导致的问题nWhat if our assumption that Cov(z,u)=0 is false?如果Cov(z,u)=0 不成立会怎样?n The IV estimator will be inconsistent,tooIV估计将不再为一致估计n Can compare asymptotic bias in OLS and IV可以对比OLS和IV的渐近偏差24Intermediate Econometrics,Yan ShenThe Effect of Poor Instruments不好的工具变量导致的问题n
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