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1、Introductory Econometrics,Lijun Jia,1,Multiple Regression Analysis多元回归分析,y=b0+b1x1+b2x2+.bkxk+u 6.Heteroskedasticity(HSK)异方差,Introductory Econometrics,Lijun Jia,2,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 estimationOLS估计后的“对异方差稳健”统计推断,Introductory
2、Econometrics,Lijun Jia,3,Testing for HSK 检验异方差The Breusch-Pagen TestB-P 检验The White TestWhite检验,Introductory Econometrics,Lijun Jia,4,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
3、最小二乘法：可行GLS,Introductory Econometrics,Lijun Jia,5,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 v
4、ariance of u is different for different values of the xs,then the errors are heteroskedastic如果u 的方差随x变化，那么误差是异方差的。,Introductory Econometrics,Lijun Jia,6,.,Education level,primary,secondary,f(y|x),Illustration of Heteroskedasticity异方差图示,college,.,.,E(y|x)=b0+b1x,wage,Introductory Econometrics,Lijun J
5、ia,7,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。,Introdu
6、ctory Econometrics,Lijun Jia,8,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
7、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。,Introductory Econometrics,Lijun Jia,9,The Breusch-Pagen Test for HSK用B-P检验异方差,Essentially we want to test H0:Var(u|x1,x2,xk)=s2,（8.11）which is equivalent to H0:E(u2|x1,x2,xk)=E(u2)=s2
8、本质上，我们想检验H0:Var(u|x1,x2,xk)=s2 这等价于检验H0:E(u2|x1,x2,xk)=E(u2)=s2（因为 zero conditional expectation）,Introductory Econometrics,Lijun Jia,10,If we assume the relationship between u2 and xj will be linear,can test it as a set of linear restrictions如果我们假设u2 和xj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 u2=d0+d1x1+dk
9、 xk+v（8.12）this means testing H0:d1=d2=dk=0（8.13）所以,对于 u2=d0+d1x1+dk xk+v 这意味着检验 H0:d1=d2=dk=0,Introductory Econometrics,Lijun Jia,11,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 x1,xk.在零假设下，通常可以假定误差v与x1,xk独立
10、Then either F or LM statistics for overall significance of the independent variables in explaining u2 can be used to test HSK.那么，如果将u2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 u2 is not normally distributed in the sample.由于u2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。,Introductory
11、 Econometrics,Lijun Jia,12,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 xs,can use the R2 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R2构造F 检验。(8.15),Introductory
12、 Econometrics,Lijun Jia,13,The Breusch-Pagen Test for HSK用B-P检验异方差,Introductory Econometrics,Lijun Jia,14,The Breusch-Pagen Test for HSK用B-P检验检验异方差,Introductory Econometrics,Lijun Jia,15,The Breusch-Pagen Test for HSK用B-P检验检验异方差,Introductory Econometrics,Lijun Jia,16,5.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
13、者p值很小，则可以拒绝零假设！,Introductory Econometrics,Lijun Jia,17,The White Test for HSK用White检验检验异方差,The Breusch-Pagan test will detect any linear forms of heteroskedasticityB-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 xsWhite检验通过加入 x 平方项和交叉项引入了一定
14、的非线性。Still just using an F or LM to test whether all the xj,xj2,and xjxh are jointly significant仍然是用F和LM检验来检验xj,xj2,xjxh是否联合显著,Introductory Econometrics,Lijun Jia,18,The White Test for HSK用White检验检验异方差,This can get to be unwieldy pretty quickly.这个办法很快就会显出其笨重之处。For example,if we have three explanator
15、y variables,x1,x2,and x3then the White test will have 9 restrictions:3 on levels,3 on squares,and 3 on cross-products.例如，如果我们有三个解释变量x1,x2,x3那么White检验有9个约束，三个对线性项，三个对平方项，三个对交叉项。（如8.19）With small samples,degrees of freedom will soon be run out with more regressors.在小样本情形，自由度将会随着解释变量数目增加而迅速减少。,Introduc
16、tory Econometrics,Lijun Jia,19,Alternate form of the White testWhite检验的变形,Consider that the fitted values from OLS,are a function of all the xs考虑到OLS的预测值是所有x的函数。Thus,2 will be a function of the squares and cross products.Therefore,and 2 can proxy for all of the xj,xj2,and xjxh.因此，2是平方项和交叉项的函数。和 2可以用
17、来替代所有的xj,xj2,xjxh,Introductory Econometrics,Lijun Jia,20,Alternate form of the White testWhite检验的变形,Regress the residuals squared on and 2 and use the R2 to form an F or LM statistic,将残差平方对 和 2回归(8.20)，用R2来构建F或LM统计量Now we only need to test 2 restrictions now.现在只需要检验两个约束Page 283 检验过程！,Introductory Ec
18、onometrics,Lijun Jia,21,White test,Keep the R-squared from this regression，,Introductory Econometrics,Lijun Jia,22,Weighted Least Squares加权最小二乘法,While its always possible to estimate robust standard errors for OLS estimates,if we know something about the specific form of the heteroskedasticity,we ca
19、n transform the model into one that has homoskedastic errors called weighted least squares.对OLS估计稳健标准差总是可能办到的，但是，如果我们知道一些关于异方差结构的信息，我们可以将原模型转化为具有同方差的新模型，这称为加权最小二乘法。,Introductory Econometrics,Lijun Jia,23,Weighted Least Squares加权最小二乘法,In such cases weighted Least squares is more efficient estimates t
20、han OLS,and it produces t and F statistics that have t and F distributions.在这些情况中，加权最小二乘法比OLS更为有效。对应的t 和 F 统计量具有t 和 F 分布。,Introductory Econometrics,Lijun Jia,24,Generalized Least Squares广义最小二乘法,Estimating the transformed equation by OLS is an example of generalized least squares(GLS)通过OLS估计变换后的方程可以作
21、为广义最小二乘法（GLS）的一个例子 GLS will be BLUE in this caseGLS在这种情形下为BLUE GLS is a weighted least squares(WLS)procedure where each squared residual is weighted by the inverse of Var(ui|xi)GLS是加权最小二乘法（WLS）在权重为Var(ui|xi)倒数时的特例。,Introductory Econometrics,Lijun Jia,25,Weighted Least Squares加权最小二乘法,While it is intu
22、itive to see why performing OLS on a transformed equation is appropriate,it can be tedious to do the transformation尽管对变换后的模型做OLS是直观的，但是变换本身可能很繁琐。Weighted least squares is a way of getting the same thing,without the transformation 加权最小二乘法可以完成相同的目的，但是不需要进行变换。Idea is to minimize the weighted sum of squ
23、ares(weighted by 1/hi)想法是最小化加权平方和（权重为1/hi）,Introductory Econometrics,Lijun Jia,26,Weighted Least Squares加权最小二乘法,Introductory Econometrics,Lijun Jia,27,Feasible GLS可行GLS,More typical is the case where you dont know the form of the heteroskedasticity更典型的情形是你并不知道异方差的形式 In this case,you need to estimate
24、 h(xi)此时，你需要估计h(xi)Typically,we start with the assumption of a fairly flexible model,such as我们可以从一个非常灵活的方程形式入手 Var(u|x)=s2exp(d0+d1x1+dkxk)Since we dont know the d,must be estimated 由于d未知，我们必须对它进行估计。,Introductory Econometrics,Lijun Jia,28,Feasible GLS(continued)可行GLS,Our assumption implies that我们的假定
25、意味着 u2=s2exp(d0+d1x1+dkxk)v,where E(v|x)=1.ln(u2)=a0+d1x1+dkxk+e Where E(e)=1 and e is independent of x其中E(e)=1 且 e 独立于xNow,we know that is an estimate of u,so we can estimate this by OLS现在，我们知道 是u 的一个估计，所以我们可以通过OLS对其进行估计。,Introductory Econometrics,Lijun Jia,29,Feasible GLS(continued)可行GLS,Now,an es
26、timate of h is obtained as=exp(),and the inverse of this is our weight对h的估计可以通过=exp()得到，其倒数为我们的权重 So,what did we do?那么，我们做了什么呢？Run the original OLS model,save the residuals,square them and take the log对原方程做OLS回归，保存残差，平方之，并取自然对数 Regress ln(2)on all of the independent variables and get the fitted values,将ln(2)对全部解释变量回归，得到预测值 Do WLS using 1/exp()as the weight将1/exp()作为权重，做WLS,
