北大计量经济学讲义-第五讲课件.ppt

1、Intermediate Econometrics,Yan Shen1Multiple Regression Analysis:OLS Asymptotics(1)多元回归分析：OLS的渐近性（1）y=b0 +b1x1+b2x2 +.+bkxk+uIntermediate Econometrics,Yan Shen2Chapter Outline 本章提纲nConsistency 一致性一致性nAsymptotic Normality and Large Sample Inference 渐近正态和大样本推断渐近正态和大样本推断nAsymptotic Efficiency of OLS OLS

2、的渐近有效性 Intermediate Econometrics,Yan Shen3Lecture Outline 本课提纲nWhat do we mean by saying consistency 一致性的含义是什么nConsistency of OLS estimators OLS估计量的一致性nThe Inconsistency of OLS when the zero conditional mean assumption fails 当零条件均值假设不成立时OLS没有一致性。nWhat do we mean by asymptotic normality and large sam

3、ple inference 渐近正态性和大样本推断的含义是什么nThe asymptotic normality of OLS OLS的渐近正态Intermediate Econometrics,Yan Shen4Why considering consistency?为什么考虑一致性nWe have discussed the following finite sample(small sample)properties of the OLS estimators and test statistics:我们已经讨论了有限样本(小样本）中OLS估计量和检验统计量具有的如下性质：nUnbias

4、edness of OLS estimators(MLR.1-4)在MLR.1-4下 OLS估计量具有无偏性nBLUE of OLS estimators(MLR.1-5)在MLR.1-5下 OLS估计量是最优线性无偏估计量nMVUE of OLS estimators(MLR.1-6)在MLR.1-6 下OLS估计量是最小方差无偏估计量nThe distribution of t(F)statistic is t(F)distributiont(F)统计量的分布为t(F)分布。nThese properties hold for any sample size n.样本容量为任意n时，这些性

5、质都成立。Intermediate Econometrics,Yan Shen5Why consider consistency?为什么考虑一致性nSince in many situations the error term is not normally distributed,it is important to know the asymptotic properties(large sample properties),i.e.,the properties of OLS estimator and test statistics when the sample size grows

6、 without bound.由于在很多情形下误差项可能呈现非正态分布，了解OLS 估计量和检验统计量的渐近性，即当样本容量任意大时的特性就是重要的问题。Intermediate Econometrics,Yan Shen612nLet be an estimator of based on a sample,.,.is a consistent estimator of if for every 0,Pr(|W|)0 as n.When is consistent,we also say that is the probability limit nnnnWyyyWW of,written

7、as lim().nnWpWWhat is Consistency什么是一致性令 是基于样本 的关于 的估计量。如果对于任何 ，当 时 便是 的一个一致估计量。当 具有一致性时，我们也称 为 的概率极限，写作是nPr(|W|)0 n 0 lim().npWnWnW12,.,ny yynWnWIntermediate Econometrics,Yan Shen7Consistency v.s.unbiasedness一致性与无偏性nIs it possible for an estimator to be biased in finite sample but consistent in lar

8、ge sample?一个估计量是否有可能在有限样本中是有偏的但又具有一致性？nSuppose true value of z=0,a random variable x=z with probability(n-1)/n,and x=n with probability 1/n.假设Z的真值为0，一个随机变量X以(n-1)/n的概率取值为Z，而以1/n的概率取值为n。nE(x)=z*(n-1)/n+n*1/n=1 X的期望为1nplim(x)is the value of x as n goes to infinity.Therefore plim(x)=z=0.记plim(x)为n趋向无穷大

9、时x的取值。因此 plim(x)=z=0.Intermediate Econometrics,Yan Shen8Consistency v.s.unbiasedness一致性与无偏性nIs it possible for an estimator to be unbiased but inconsistent?是否有可能(一个估计量)是无偏却不一致的？nSuppose true value of z=0,a random variable x=0.5 with probability 0.5,and x=-0.5 with probability 0.5.Then E(x)=0.But as

10、x always fluctuates around the line x=0,its variance does not vanish as n goes to infinity.Therefore,it is an inconsistent estimator of z.n假设Z的真值为0，一个随机变量X以0.5的概率取0.5，而以0.5的概率取-0.5，那么X的期望为0。但是 X总是在X=0这条线上下摆动，当n趋向无穷大时，它的方差并不会趋于0。因此，它是Z的不一致的估计量。Intermediate Econometrics,Yan Shen9Consistency v.s.unbias

11、edness一致性与无偏性nUnbiased estimators are not necessarily consistent,but those whose variances shrink to zero as the sample size grows are consistent.无偏估计量未必是一致的，但是那些当样本容量增大时方差会收缩到零的无偏估计量是一致的。Intermediate Econometrics,Yan Shen10Consistency一致性n Under the Gauss-Markov assumptions OLS is BLUE,but in other

12、cases it wont always be possible to find unbiased estimators 在高斯马尔可夫假定下OLS 是最优线性无偏估计量，但在别的情形下不一定能找到无偏估计量。n In those cases,we may settle for estimators that are consistent,meaning as n ,the distribution of the estimator collapses to the true parameter valuen在那些情形下，我们只要找到一致的估计量，即当n 时,这些估计量的分布退化为参数的真值。

13、Intermediate Econometrics,Yan Shen11Sampling Distributions as n increases当n增加时样本的分布b1n1n2n3n1 n2 n3Sampling distri-bution of 1例：n1:每次从班上抽取10人，抽若干次后，平均身高的分布；n2:每次从班上抽取100人，抽若干次后，平均身高的分布；n3:每次从班上抽取200人，抽若干次后，平均身高的分布。Intermediate Econometrics,Yan Shen12Consistency of OLSOLS的一致性n Theorem 5.1:Under Assum

14、ptions MLR.1 through MLR.4,the OLS estimator is consistent for both the intercept and slope parameters.定理5.1:在假设MLR.1到MLR.4下，OLS截距估计量和斜率估计量都是一致的估计量。n Consistency can be proved for the simple regression case in a manner similar to the proof of unbiasednessn对简单回归而言，证明估计量的一致性和证明无偏性的方法是类似的。Intermediate

15、Econometrics,Yan Shen13Proving Consistency证明一致性11121111121111112111The OLS estimated slope parameter from simple regression isiiiiiiiiixx yxxxx uxxnxx unxxbbb简单回归中的斜率估计量即Intermediate Econometrics,Yan Shen14Proving Consistency证明一致性1112111111111211111Because as,0 does not converge to zero,plim.niiiiii

16、nnxx unxxnxx unxxbbbbb 由于当 趋于无穷时分子趋于零而分母不趋于零，故 的概率极限即。Intermediate Econometrics,Yan Shen15Proof of OLS Consistency证明OLS的一致性nA general proof of consistency of the OLS estimators from the multivariate regression case can be shown through matrix manipulations.多元回归中OLS估计量的一致性的证明可以通过矩阵运算得到。Intermediate E

17、conometrics,Yan Shen16A Weaker Assumption一个更弱的假定n For unbiasedness,we assumed a zero conditional mean E(u|x1,x2,xk)=0 要获得估计量的无偏性，我们假定零条件期望 E(u|x1,x2,xk)=0n For consistency,we can have the weaker assumption of zero mean and zero correlation(MLR.3)E(u)=0 and Cov(xj,u)=0,for j=1,2,k 而要获得估计量的一致性，我们可以使用更

18、弱的假定：零期望和零相关性假定。n Without this assumption,OLS will be biased and inconsistent!如果这个较弱的假定也不成立，OLS将是有偏而且不一致的。Intermediate Econometrics,Yan Shen1701 12 201 12 2112121True model:You think:,so that and then,plimwhere,yxxvyxuuxvCov x xVar xbbbbbbbbb Deriving the Inconsistency推导不一致性n Define the asymptotic b

19、ias as,then consider the following true model and estimated model:定义渐近偏差为：，并考虑下面的真实模型和待估计 模型。11plimbb11plimbbIntermediate Econometrics,Yan Shen18Asymptotic Bias(cont)渐近偏差(续)n So,thinking about the direction of the asymptotic bias is just like thinking about the direction of bias for an omitted varia

20、ble 所以，考虑渐近偏差的方向就像是考虑存在一个 遗漏变量时偏差的方向。n Main difference is that asymptotic bias uses the population variance and covariance,while bias uses the sample counterparts 主要的区别在于渐近偏差用总体方差和总体协方差表示，而偏差则是基于它们在样本中的对应量。n Remember,inconsistency is a large sample problem it doesnt go away as add data 记住，不一致性是一个大样本

21、问题。因此，当数据增加时候这个问题并不会消失。Intermediate Econometrics,Yan Shen19Consistency with endogeneity有内生性时的一致性nConsider the true model to be y=b0+b1x1+b2x2+u but cov(u,x1)=0.考虑真实模型为y=y=b0+b1x1+b2x2+u，但而u和x1相关。nWhen cov(x1,x2)=0 but cov(u,x2)=0,then OLS estimators for b1 and b2 are inconsistent.若x1 和x2相关，而u和x2不相关，

22、则对b1和b2的OLS估计量都是不一致的。nWhen cov(x1,x2)=0 and cov(u,x2)=0,only b1 is inconsistent.若x1 和x2不相关，且u和x2不相关，则只有对b1的OLS估计量是不一致的Intermediate Econometrics,Yan Shen20Asymptotic Normality and Large Sample Inference渐近正态和大样本推断nConsistency of an estimator is an important property,but it alone does not allow us to p

23、erform statistical inference.估计量的一致性是一条重要性质，但我们并不能只靠它来进行统计推断。nRecall that under the Classical Linear Model assumptions,the sampling distributions are normal,so we could derive t and F distributions for testing 在经典线性模型假设下，样本的分布是正态分布，因而我们能够推出t分布和F分布用于检验。Intermediate Econometrics,Yan Shen21Asymptotic N

24、ormality and Large Sample Inference渐近正态和大样本推断nThis exact normality was due to assuming the population error distribution was normal 这种准确的正态分布来自于总体误差的分布是正态分布的假定。n This assumption of normal errors implied that the distribution of y,given the xs,was normal as welln这个正态误差的假定意味着当x给定时，y的分布也是正态分布。Intermedi

25、ate Econometrics,Yan Shen22Asymptotic Normality and Large Sample Inference渐近正态和大样本推断nWhy normality assumption is needed?为什么需要正态性假定？nFor proving unbiasedness?No.为了证明无偏性？-不是nFor proving BLUE?No.为了证明最优线性估计量？不是nFor making exact inference when using t or F statistics?Yes.为了能够用t统计量和F统计量作精确的推断？是的Intermedia

26、te Econometrics,Yan Shen23Asymptotic Normality and Large Sample Inference渐近正态和大样本推断n Easy to come up with examples for which this exact normality assumption will fail 很容易碰到一些例子，其中严格的正态性假定并不能成立n Any clearly skewed variable,like wages,arrests,savings,etc.cant be normal,since a normal distribution is s

27、ymmetric 任何一个明显不对称的变量，像工资，拘捕次数，储蓄量，等等都不可能服从正态分布，因为正态分布是对称的。Intermediate Econometrics,Yan Shen24Asymptotic Normality and Large Sample Inference渐近正态和大样本推断nWhat to do?我们要做些什么？nWill estimators become approximately normally distributed when sample size gets large?当样本容量变大时是否估计量会渐近地趋向于正态分布？nWe are discussi

28、ng whether OLS estimator satisfy asymptotic normality.我们讨论是否OLS估计量满足渐近正态性。Intermediate Econometrics,Yan Shen25Central Limit Theorem中心极限定理w Based on the central limit theorem,we can show that OLS estimators are asymptotically normalw 基于中心极限定理，我们能够证明OLS估计量是渐近正态。w Asymptotic Normality implies that P(Zz

29、)F(z)as n,or P(Zz)F(z)渐近正态意味着当n 时，P(Zz)F(z)或者 P(Zz)(z)w The central limit theorem states that the standardized average of any population with mean m and variance s2 is asymptotically N(0,1),or 中心极限定理指出任何一个均值为而方差为2 的总体的标准化平均值的分布渐近趋向于N(0,1),或记作1,0 NnYZaYsm1,0 NnYZaYsmIntermediate Econometrics,Yan Shen

30、26Theorem 5.2:Asymptotic Normality of OLS定理5.2:OLS的渐近正态性2222Under the Gauss-Markov assumptions MLR.1-MLR.5MLR.1-MLR.5(i)is asymptotically normally distributed.That is,Normal 0,where is the asymptotic varianjjajjjjnaabbbbss在高斯马尔可夫假定下是渐近正态分布的，即为22212ij212ijce of,and plim,where r are the residuals from

31、 regressing on the other independent variables.plim rjjjjjjijjjijjnananrxanrxbbsbb其中是的渐近方差，而，其中是对其它自变量进行回归的残差。Intermediate Econometrics,Yan Shen27Theorem 5.2:Asymptotic Normality of OLS定理5.2:OLS的渐近正态性2222(ii)is a consistent estimator of()(iii)For each j,Normal 0,1ajjjiisessssbbb是的一个一致估计量Intermediate

32、 Econometrics,Yan Shen28What is assumed and not assumed in Theorem 5.2在定理5.2中什么是我们的假定而什么不是nThe normality assumption MLR.6 is dropped.去掉了正态性假定MLR.6nStill assumed:仍然假定：nThe distribution of the error has finite variance.误差的分布具有有限的方差nZero conditional mean零条件期望nHomoskedasticity同方差性nLinear structure and r

33、andom sample线性结构和随机样本Intermediate Econometrics,Yan Shen29Understanding Theorem 5.2理解定理5.2222222jjWhy considering,not just in(i)?Because()1(/)(),.Notice the sample variance of x is SST/,jjjjjjjjjjjjjjjjijjjijnnVSSTRSST SSRSSTSSRSSTxxSSRrnbbbbbbbbsbss为什么在(i)中考虑，而不是注jjjjxSST/and the sample variance of

34、is SSR/.SSR/ijijnrnrn意到 的样本方差为，而 的样本方差为。Intermediate Econometrics,Yan Shen30Understanding Theorem 5.2理解定理5.22222ij2ij221(),where is the population variance of r.r1Let,then().As n,()shrinks to zero at the speed of1/.Therefore,only when we scale jjrrrjrjjVSSRncVcnVnssbssssbsbb其中是 的总体方差。up by ncan we d

35、iscuss the asymptotic distribution.n()njjVbb当时，以1/n的速度减小到零。因此，我们只有按的比例增大，才能讨论渐近分布。Intermediate Econometrics,Yan Shen31Asymptotic Normality(cont)渐近正态（续）n Because the t distribution approaches the normal distribution for large df,we can also say thatn因为自由度df很大的 t分布接近于正态分布，我们也 可以得到 1knajjjtsebbbw Note

36、that while we no longer need to assume normality with a large sample,we do still need homoskedasticityw 注意到尽管我们在大样本中不再需要正态性假定，我们仍然需要同方差性Intermediate Econometrics,Yan Shen32Asymptotic Standard Errors渐近标准误差w If u is not normally distributed,we sometimes will refer to the standard error as an asymptoti

37、c standard error,sincew 如果u不是正态分布，我们有时把标准误差称作是渐近标准误差，因为ncseRSSTsejjjjjbsb,122w So,we can expect standard errors to shrink at a rate proportional to the inverse of n 所以，我们预计标准误差减小的速度与n成正比Intermediate Econometrics,Yan Shen33Multiple Regression Analysis:Asymptotics多元回归分析：渐近性(2)ny=b0+b1x1+b2x2+.bkxk+uIn

38、termediate Econometrics,Yan Shen34Lecture Outline 本课提纲nThe asymptotic normality of OLS OLS的渐近正态性nLarge sample tests 大样本检验nThe Asymptotic t statistic t统计量的渐近性nThe LM statistic LM统计量nThe Asymptotic Efficiency of OLS OLS的渐近有效Intermediate Econometrics,Yan Shen35Lagrange Multiplier statistic拉格朗日乘子统计量n On

39、ce we are using large samples and relying on asymptotic normality for inference,we can use more than t and F stats 当我们使用大样本并且依靠渐近正态性进行推断时，除了t和F统计量，我们还可以使用别的统计量。n The Lagrange multiplier or LM statistic is an alternative for testing multiple exclusion restrictions 拉格朗日乘子或LM统计量是检验多重限定性约束的另一种选择。n Becau

40、se the LM statistic uses an auxiliary regression its sometimes called an nR2 stat LM统计量使用一个辅助性的回归，因此它有时被叫做nR2 统计量。Intermediate Econometrics,Yan Shen36LM Statistic(cont)LM统计量(续)n Suppose we have a standard model,y=b0+b1x1+b2x2+.bkxk+u and our null hypothesis is H0:bk-q+1=0,.,bk=0 假设我们有一个标准模型y=b0+b1x1

41、+b2x2+.bkxk+u 而我们的零假设为:H0:bk-q+1=0,.,bk=0n First,we just run regression on the restricted model 首先，我们对被约束的模型进行回归01 1122212.Second,take the residuals,and regress on,.,(i.e.the variables),where is from the second regression,.,kqkqkuukyxxuuuxxxallLMnRRuuxxxbbb其 次，记 录 残 差并 将对进 行 回 归（即 所 有 的22,uuLMnRR自 变

42、 量)。那 么其 中来 自 于第 二 个 回 归Intermediate Econometrics,Yan Shen37LM StatisticLM统计量nIF the H null is true,then R-squared should be close to zero,since should be approximately uncorrelated with all the independent variables.如果H0为真，那么R-平方应该接近零，因为 应该近似地与所有自变量都不相关。nWe need to decide how close is close to zero

43、.我们需要判断接近零的程度。u u Intermediate Econometrics,Yan Shen38LM StatisticLM统计量222222,so can choose a criticalvalue,from a distribution,orjust calculate a p-value for,aqqqaqqqLMcLMp所以可以选择一个分布的临界值c，或计算的 值。n The following steps can be used for testing the joint significance of a set of q independent variables

44、.接下去的步骤可以用来检验一组q个自变量的联合显著性。Intermediate Econometrics,Yan Shen39Steps involved in LM testLM检验中的步骤nRegress y on restricted set of independent variables and save the residuals,.将y对被约束的自变量进行回归，保存残差 。nRegress on all of the independent variables and obtain the R-squared,将 对所有自变量进行回归，得到相应的R-平方。nCompute LM=

45、n 计算 LM=nnCompare LM to the appropriate critical value in a chi-square distribution.n将LM 值与卡方分布中相应的临界值进行比较。u u u 2uR2uRu Intermediate Econometrics,Yan Shen40Characteristics of LM testLM检验的特性nLM statistics is sometimes referred to as n-R-squared statistic,or score statistic.LM统计量有时被称作是n-R-平方统计量，或者得分统

46、计量nAll that matters are 相关的因素只有nNumber of restrictions,q 约束q的个数nThe size of the auxiliary R-squared 辅助R-平方的大小nThe sample size 样本容量nIrrelevant:不相关的因素:nDegree of freedom of the unrestricted model 未约束模型中自由度的个数。nR squared from the unrestricted model and restricted model(the first-step regression model)未

47、约束模型和被约束模型(第一步的回归模型)的R-平方Intermediate Econometrics,Yan Shen41LM test PK F test&t testLM检验与F检验和t检验的优劣对比nWith a large sample,the result from an F test and from an LM test should be similar 在大样本中，F检验和LM检验得到的结果相似。n Unlike the F test and t test for one exclusion,the LM test and F test will not be identic

48、al 只有一个约束时，F检验和t检验是等价的，然而LM检验和F检验并不等价。nThe main regression and the auxiliary regression should use the same set of observations.主回归和辅助回归必须使用相同的一组观测值。Intermediate Econometrics,Yan Shen42Asymptotic Efficiency渐近有效n Under the Gauss-Markov assumptions,estimators besides OLS can be consistent 在高斯-马尔可夫假定下，

49、OLS估计量以外的估计量可以具有一致性。n However,under the Gauss-Markov assumptions,the OLS estimators will have the smallest asymptotic variances 但是，在高斯-马尔可夫假定下，OLS估计量具有最小的渐近方差。n We say that OLS estimators are asymptotically efficient among a certain class of estimators under the Gauss-Markov assumptions.我们说在高斯-马尔可夫假

50、定下OLS估计量是渐近有效的估计量。Intermediate Econometrics,Yan Shen43Asymptotic Efficiency渐近有效nImportant to remember our assumptions though,if not homoskedastic,then the above conclusion is not true.重要的一点是如果同方差的假定不成立，上述结论也不能成立。nTo prove that OLS estimators are asymptotically efficient,one needs to(1)present an est