工具变量与两阶段最小二乘法ppt课件.ppt

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1、采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen1Instrumental Variables & 2SLS工具变量与两阶段最小二乘法w y = b0 + b1x1 + b2x2 + . . . bkxk + uw x1 = p0 + p1z + p2x2 + . . . pkxk + v采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermedia

2、te Econometrics, Yan Shen2Chapter Outline本章提要nOmitted Variables in a simple regression model简单回归中的遗漏变量nIV estimation of the Multiple Regression多方程回归中的工具变量估计nTwo Stage Least Squares两阶段最小二乘法nIV solutions to errors-in-variables problem用工具变量解决测量误差问题nTesting for endogeneity检验内生性采用PP管及配件：根据给水设计图配置好PP管及配件，

3、用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen3Lecture 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

4、regression model多方程回归的IV估计采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen4Problem 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？采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物

5、Intermediate Econometrics, Yan Shen5The 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 presence of the omitted variable使用一种对遗漏变量稳健的估计方法。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保

6、证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen6Why 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 采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接

7、部位干净无污物Intermediate Econometrics, Yan Shen7Why 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采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂

8、直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen8Instrumental Variable: Who qualifies?什么样的变量可以作为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

9、(z,u) = 0 (15.4)即Cov(z,u) = 0采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen9nThe instrument must be correlated with the endogenous variable 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？采用PP管及配件：根

10、据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen10About 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。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管

11、材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen11About 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。采用PP管

12、及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen12Example: wage determination例子：工资决定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

13、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采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen13Example: wage determination例子：工资决定nProblem: since the error term contains IQ, and education correlates with I

14、Q, 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。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen14Exa

15、mple: 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。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen15Example: wage determination例子：工资决定nIV used in the litera

16、ture:在文献中使用的IVnMothers education母亲教育水平nNumber of siblings. 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

17、能力不相关的。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen16When an IV is 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 thi

18、s 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，并且这些矩可以通过样本估计。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen17When an IV is Available: Estimation当IV存在

19、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xxzzyyzziiii1b采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen18When an IV is Available: Estimation当IV存在时：估计nWhen z=x we obtain th

20、e OLS 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。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen1

21、9When an IV is Available: Estimation当IV存在时：估计nWhen assumptions (15.4) and (15.5) hold, 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。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Inte

22、rmediate Econometrics, Yan Shen20When 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 assumptions (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

23、时，给定渐近方差， 可以估计b1的标准差采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen21When an IV is Available: Inference当IV存在时：推断 112122,222,The asymptotic variance of is : the population variance of x. x: the population variance of u. u: the square of the popul

24、ation correlatixx zxx zVarnbbsbs ss的渐近方差是的总体方差的总体方差on between x and z. xz， 总体相关系数的平方采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen22When an IV is Available: Inference当IV存在时：推断 2212,222,22The estimated asymptotic variance is,: the total sum of s

25、quares of .: R from the regressing x on z.xz R : the consistent estimator of inxx zxiix zseSST RSSTxxR sbss估计的渐近方差为（ 的平方和）（将 对 回归的）2troduced before. s前面提到的对的一致估计采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen23IV versus OLS estimationIV与OLS估计n S

26、tandard 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。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen24IV versus OLS

27、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。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切

28、断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen25The 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的渐近偏差采用P

29、P管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen26The Effect of Poor Instruments不好的工具变量导致的问题nPrefer IV if Corr(z,u)/Corr(z,x) Corr(x,u)xuxuuxCorrxzCorruzCorrssbbssbb),( plim :OLS),(),(plim :IV1111采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的

30、圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen27Reporting R-Squared after IV Estimation在IV估计后报告R方nSince the variance of y cannot be decomposed nicely when x and u are correlated, the R-squared is not very meaningful after the IV estimation.由于当x与u相关时我们不能对y进行简单的分解，在IV估计之后的R方没有太多意义。nIt cannot be use

31、d in the usual way to compute F tests of joint restrictions.也不能用它来计算检验线性约束的 F 统计量。 采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen28IV Estimation in the Multiple Regression Case多元回归的工具变量估计n IV estimation can be extended to the multiple regressio

32、n caseIV估计可以扩展到多元回归情形n Call the model we are interested in estimating the structural model我们将感兴趣的模型称为结构方程。n Our problem is that one or more of the variables are endogenous问题在于一个或多个变量是内生的。n We need an instrument for each endogenous variable对每个内生变量都我们都需要一个工具变量。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边

33、剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen29The Structural Equation结构方程i012211121111Consider the model(1) y.Now, the endogenous variables are:y and y . Both are correlated with u .The exogenous variable is still x , hence, cov(x ,)0.Equation (1) is an example ofyxuubbbi01221112111

34、1 structural equation.(1) y.y yuxcov(x ,)0.yxuubbb考虑模型此时，内生变量为 和。他们都与 相关。外生变量仍然是 ，所以方程（1）是结构方程的一个例子。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen30The Structural Equation: Example结构方程：例子D111S222Structural equation has explicit meaning. (e.g.)A

35、 prototype micro model for an agricultural good is(quantity)()(),(quantity)()( ),DSpriceincomepricerain fallbb结构方程具有明显的含义对农产品的微观模型DS(quantity)(quantity) .Each of the above equation is a structural equation.The parameters are called structural parameters.上面的每一个方程都是结构方程其中的参数称为结构参数。采用PP管及配件：根据给水设计图配置好P

36、P管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen31The Reduced-Form Equation方程的简约式nThe reduced-form equation expresses one of the endogenous variables as a function of all exogenous variables and stochastic disturbances.简约式的方程将内生变量表示为所有外生变量和随机扰动的函数。nThe corresponding

37、 parameters are reduced-form parameters.对应的参数称为简约式的参数。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen32The Reduced-Form Equation:Example简约式的方程：例子121221212121211221122121212121Eliminating the equilibrium condition we get,where I,r are the exogeno

38、us variables.IrDSDSpIrqIrbb b b 消去均衡条件得这里， ， 为外生变量采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen33Multiple Regression IV多元回归的IVn Now, we write the structural model as y1 = b0 + b1y2 + b2 x1 + u1, where y2 is endogenous and x1 is exogenous现在，将结构

39、模型写为y1 = b0 + b1y2 + b2 x1 + u1，其中y2为内生，为x1外生n Let z1 be the instrument for y2, we need the following assumptions to get a valid IV estimates. 令z1为y2的工具变量，为了得到一个有效的IV，我们需要做如下假设采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen34Assumptions for Mult

40、iple Regression IV多个回归中关于IV的假定nE(u1) = 0 nCov(x1,u1) = 0 and nCov(z1,u1) = 0 nTherefore, E(z1u1 )= E(x1u1 )=0.nWe now have 3 moment conditions for 3 unknowns.现在我们有三个未知量，三个矩条件采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen35Obtaining the IV estim

41、ators得到IV估计量10122111101221111012211j0,0,0,Solving this set of linear equations we canget the instrumental variables estimators for ,0,1,2.niiiiniiiiiniiiiiyyxxyyxzyyxjbbbbbbbbbb通过解这一组线j,0,1,2jb性方程我们可以得到对的工具变量估计采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometri

42、cs, Yan Shen36The Reduced form for y2 y2的简约式nWe still need z2 to be correlated with y2我们仍然需要 z2与 y2相关ny2 = p0 + p1x1 + p2z1 + v2, where p2 0n This reduced form equation regresses the endogenous variable on all exogenous ones.这个简约式方程将内生变量对所有的外生变量回归。nThe above procedure can be extended to more explana

43、tory variables case.上述过程可以扩展为更多个解释变量的情形。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen37Example: Return to Education例子：教育回报nCard (1995) use wage and education data for a sample of men in 1976 to estimate the return to education.Card(1995)使用1976

44、年关于men的工资和教育数据估计教育回报nInstrumental variable: whether one grew up near a four-year college (nearc4).工具变量：是否在本科大学附近长大。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen38Example: Return to Education例子：教育回报nFor distance to be a valid instrument如果距离是一个有

45、效的IV，应有n(1) It must be uncorrelated with the error term that determines wage.（1）它必须和影响工资的误差项不相关采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen39Example: Return to Education例子：教育回报nFor distance to be a valid instrument如果距离是一个有效的IV，应有n(2) It must

46、be partially correlated with educ.(2)它必须部分地与教育变量相关nThis can be checked by regress educ on nearc4 and all of the exogenous variables appearing in the equation.可以通过将教育对距离和其他外生变量回归来查看这一点。采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen40Example: Ret

47、urn to Education例子：教育回报nThis means we estimate the reduced form for educ.这意味着我们估计了教育的简约式的方程nThis estimation gives a coefficient of 0.32 with a standard error of 0.088.估计的结果：系数为0.32,标准差为0.088nImplies that in 1976, other things being fixed, on average, people who lived near a college had about one-thi

48、rd of a year more education than those who didnt. 意味着在1976年，控制住其它因素后，平均来讲，居住在大学附近的人比其他人多受约4个月的教育采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen41Example: Return to Education例子：教育回报ExplanatoryOLS (sd)IV (sd)Educ0.075 (0.003)0.132 (0.055)Exper0.08

49、5 (0.007)0.108 (0.024)Exper2-0.0023 (0.003) -0.0023(-0.003)Black-0.199 (0.018)-0.147 (0.032)South-0.148 (0.026)-0.145 (0.027)Observations30103010R squared0.30.238Other controls采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen42Example: Return to E

50、ducation例子：教育回报nThe IV estimates of the return to education is almost twice as large as the OLS estimate.对教育回报的IV估计几乎是OLS估计的两倍nThe standard error is over 18 times larger.标准差比OLS估计大18倍采用PP管及配件：根据给水设计图配置好PP管及配件，用管件在管材垂直角切断管材，边剪边旋转，以保证切口面的圆度，保持熔接部位干净无污物Intermediate Econometrics, Yan Shen43Example: Retu