,Click to edit Master title style,Click to edit Master text styles,G,Third level,Georgia,Fifth level,*,Introductory Econometrics,Lijun Jia,*,Multiple Regression Analysis,多元回归分析,y=,b,0,+,b,1,x,1,+,b,2,x,2,+.,b,k,x,k,+u,6.Heteroskedasticity(HSK),异方差,1,Introductory Econometrics,Lijun Jia,y=b0+b1x1+b2x2+.,Lecture Outline,本课提要,What is HSK,什么是异方差,Consequences of HSK,异方差的影响,HSK-Robust Inference after OLS estimation,OLS,估计后的“对异方差稳健”统计推断,2,Introductory Econometrics,Lijun Jia,Lecture Outline 本课提要What is HS,Testing for HSK,检验异方差,The Breusch-Pagen Test,B-P,检验,The White Test,White,检验,3,Introductory Econometrics,Lijun Jia,Testing for HSK 检验异方差3Introduc,Weighted Least squares,加权最小二乘法,WLS when HSK is known up to a multiplicative constant,当在比例意义上已知异方差时的加权最小二乘法,WLS when HSK is of unknown form:the feasible GLS,当异方差具有未知形式时的加权最小二乘法：可行,GLS,4,Introductory Econometrics,Lijun Jia,Weighted Least squares加权最小二乘法,What is Heteroskedasticity(HSK),什么是异方差,Recall the assumption of homoskedasticity implied that conditional on the explanatory variables,the variance of the unobserved error,u,was constant,同方差假定意味着条件于解释变量，不可观测误差的方差为常数,If this is not true,that is if the variance of,u,is different for different values of the,x,s,then the errors are heteroskedastic,如果,u,的方差随,x,变化，那么误差是异方差的。,5,Introductory Econometrics,Lijun Jia,What is Heteroskedasticity(HS,.,Education level,primary,secondary,f(,y|x,),Illustration of Heteroskedasticity,异方差图示,college,.,.,E(,y,|,x,)=,b,0,+,b,1,x,wage,6,Introductory Econometrics,Lijun Jia,.Education level primarysecond,Why do we care?,为何关心,异方差,？,The standard errors of the estimates,are,biased if we have heteroskedasticity.,如果存在异方差，那么估计值的标准差是有偏的。,If the standard errors are biased,we can not use the usual,t,statistics or,F,statistics or,LM,statistics for drawing inferences.,如果标准差有偏，我们就不能应用通常的,t,统计量或,F,统计量来进行统计推断。,7,Introductory Econometrics,Lijun Jia,Why do we care?为何关心异方差？The st,Testing for HSK,检验异方差,Reason No.1:We may prefer to see the usual OLS standard errors and test statistics reported unless there is evidence of heteroskedasticity.,理由,1,：除非有证据显示异方差存在，我们仍会偏好于常规,OLS,的标准差及检验统计量。,Reason No.2:If heteroskedasticity is present,the OLS estimator is no longer the BLUE,then it is possible to obtain a better estimator than OLS.,理由,2,：如果异方差存在，,OLS,不再是,BLUE,，那么就有可能得到比,OLS,更好的估计量。,8,Introductory Econometrics,Lijun Jia,Testing for HSK检验异方差Reason No,The Breusch-Pagen Test for HSK,用,B-P,检验异方差,Essentially we want to test,H,0,:Var(,u|x,1,x,2,x,k,)=,s,2,（,8.11,）,which is equivalent to H,0,:E(,u,2,|x,1,x,2,x,k,)=E(,u,2,)=,s,2,本质上，我们想检验,H,0,:Var(,u|x,1,x,2,x,k,)=,s,2,这等价于检验,H,0,:E(,u,2,|x,1,x,2,x,k,)=E(,u,2,)=,s,2,（因为,zero conditional expectation,）,9,Introductory Econometrics,Lijun Jia,The Breusch-Pagen Test for HSK,If we assume the relationship between,u,2,and,x,j,will be linear,can test it as a set of linear restrictions,如果我们假设,u,2,和,x,j,之间具有线性关系，则可以通过一组线性约束来完成检验。,So,for,u,2,=,d,0,+,d,1,x,1,+,d,k,x,k,+v,（,8.12,）,this means testing,H,0,:,d,1,=,d,2,=,d,k,=0,（,8.13,）,所以,对于,u,2,=,d,0,+,d,1,x,1,+,d,k,x,k,+v,这意味着检验,H,0,:,d,1,=,d,2,=,d,k,=0,10,Introductory Econometrics,Lijun Jia,If we assume the relationship,The Breusch-Pagen Test for HSK,用,B-P,检验检验异方差,Under the null hypothesis,it is often reasonable to assume that the error,v,is independent of,x,1,x,k,.,在零假设下，通常可以假定误差,v,与,x,1,x,k,独立,Then either,F,or,LM,statistics for overall significance of the independent variables in explaining,u,2,can be used to test HSK.,那么，如果将,u,2,视为被解释变量，检验全部解释变量显著性的,F,统计量就可以用来检验异方差。,They are asymptotically valid test since,u,2,is not normally distributed in the sample.,由于,u,2,在样本中不是正态分布，这些统计量只在渐近的意义下适用。,11,Introductory Econometrics,Lijun Jia,The Breusch-Pagen Test for HSK,The Breusch-Pagen Test for HSK,用,B-P,检验异方差,The error cannot be observed by can be estimated from OLS residuals.,不可观测的误差可以通过,OLS,残差进行估计。,After regressing the residuals squared on all of the,x,s,can use the,R,2,to form an,F,or,LM,test.,将残差平方对所有的,x,回归之后，可以通过,R,2,构造,F,检验,。,(8.15),12,Introductory Econometrics,Lijun Jia,The Breusch-Pagen Test for HSK,The Breusch-Pagen Test for HSK,用,B-P,检验异方差,13,Introductory Econometrics,Lijun Jia,The Breusch-Pagen Test for HSK,The Breusch-Pagen Test for HSK,用,B-P,检验检验异方差,14,Introductory Econometrics,Lijun Jia,The Breusch-Pagen Test for HSK,The Breusch-Pagen Test for HSK,用,B-P,检验检验异方差,15,Introductory Econometrics,Lijun Jia,The Breusch-Pagen Test for HSK,5.,然后看,F,和,LM,值的大小，或者对应的,p,值。如果,F,和,LM,值很大或者,p,值很小，则可以拒绝零假设！,16,Introductory Econometrics,Lijun Jia,5.然后看F和LM值的大小，或者对应的p值。如果F和LM值很,The White Test for HSK,用,White,检验检验异方差,The Breusch-Pagan test will detect any linear forms of heteroskedasticity,B-P,检验可以识别任意线性形式的异方差,The White test allows for nonlinearities by using squares and cross products of all the,x,s,White,检验通过加入,x,平方项和交叉项引入了一定的非线性。,Still just using an F or LM to test whether all the,x,j,x,j,2,and,x,j,x,h,are jointly significant,仍然是用,F,和,LM,检验来检验,x,j,x,j,2,x,j,x,h,是否联合显著,17,Introductory Econometrics,Lijun Jia,The White Test for HSK用White检,The White Test for HSK,用,Whi