计量经济第三章.ppt

1、Intermediate Econometrics, Yan Shen,1,Multiple Regression Analysis: Estimation (2)多元回归分析：估计（2）,y = b0 + b1x1 + b2x2 + . . . bkxk + u,Intermediate Econometrics, Yan Shen,2,Chapter Outline 本章大纲,Motivation for Multiple Regression 使用多元回归的动因 Mechanics and Interpretation of Ordinary Least Squares 普通最小二乘法的

2、操作和解释 The Expected Values of the OLS Estimators OLS估计量的期望值 The Variance of the OLS Estimators OLS估计量的方差 Efficiency of OLS: The Gauss-Markov Theorem OLS的有效性：高斯马尔科夫定理,Intermediate Econometrics, Yan Shen,3,Lecture Outline 课堂大纲,The MLR.1 MLR.4 Assumptions 假定MLR.1 MLR.4 The Unbiasedness of the OLS estima

3、tes OLS估计值的无偏性 Over or Under specification of models 模型设定不足或过度设定 Omitted Variable Bias 遗漏变量的偏误 Sampling Variance of the OLS slope estimates OLS斜率估计量的抽样方差,Intermediate Econometrics, Yan Shen,4,The expected value of the OLS estimatorsOLS估计量的期望值,We now turn to the statistical properties of OLS for esti

4、mating the parameters in an underlying population model. 我们现在转向OLS的统计特性，而我们知道OLS是估计潜在的总体模型参数的。 Statistical properties are the properties of estimators when random sampling is done repeatedly. We do not care about how an estimator does in a specific sample. 统计特性是估计量在随机抽样不断重复时的性质。我们并不关心在某一特定样本中估计量如何。,

5、Intermediate Econometrics, Yan Shen,5,Assumption MLR.1 (Linear in Parameters)假定 MLR.1（对参数而言为线性）,In the population model (or the true model), the dependent variable y is related to the independent variable x and the error u as 在总体模型(或称真实模型）中，因变量y与自变量x和误差项u关系如下 y= b0+ b1x1+ b2x2+ +bkxk+u (3.31) where

6、b1, b2 , bk are the unknown parameters of interest, and u is an unobservable random error or random disturbance term. 其中， b1, b2 , bk 为所关心的未知参数，u为不可观测的随机误差项或随机干扰项。,Intermediate Econometrics, Yan Shen,6,Assumption MLR.2 (Random Sampling)假定 MLR.2（随机抽样性）,We can use a random sample of size n from the po

7、pulation, 我们可以使用总体的一个容量为n的随机样本 (xi1, xi2, xik; yi): i=1,n， where i denotes observation, and j= 1,k denotes the jth regressor. 其中i 代表观察，j=1,k代表第j个回归元 Sometimes we write 有时我们将模型写为 yi= b0+ b1xi1+ b2xi2+ +bkxik+ui (3.32),Intermediate Econometrics, Yan Shen,7,Assumption MLR.3 假定MLR.3,MLR.3 (No perfect co

8、llinearity) （不存在完全共线性） : In the sample, none of the independent variables is constant, and there are no exact linear relationships among the independent variables. 在样本中，没有一个自变量是常数，自变量之间也不存在严格的线性关系。 When one regressor is an exact linear combination of the other regressor(s), we say the model suffers

9、from perfect collinearity. 当一个自变量是其它解释变量的严格线性组合时，我们说此模型有严格共线性。 Examples of perfect collinearity：完全共线性的例子： y= b0+ b1x1+ b2x2+ b3x3+u， x2 = 3x3， y= b0+ b1log(inc)+ b2log(inc2 )+u y= b0+ b1x1+ b2x2+ b3x3+ b4x4 u，x1 +x2 +x3+ x4 =1.,Intermediate Econometrics, Yan Shen,8,Perfect collinearity also happens

10、when y= b0+ b1x1+ b2x2+ b3x3+u , n(k+1). 当y= b0+ b1x1+ b2x2+ b3x3+u , n(k+1) 也发生完全共线性的情况。 The denominator of the OLS estimator is 0 when there is perfect collinearity, hence the OLS estimator cannot be performed. You can check this by looking at the formula of the estimator for b2 in the session dis

11、cussing the partialling-out effect. 在完全共线性情况下，OLS估计量的分母为零，因此OLS估计量不能得到。你可以回顾讨论“排除其它变量影响”部分中的b2估计量的式子，来检验这一点。,Intermediate Econometrics, Yan Shen,9,Assumptions MLR.4 假定 MLR.4,MLR.4 (Zero Conditional Mean) （零条件均值） : E(u| xi1, xi2, xik)=0. (3.36) When this assumption holds, we say all of the explanator

12、y variables are exogenous; when it fails, we say that the explanatory variables are endogenous. 当该假定成立时，我们称所有解释变量均为外生的；否则，我们则称解释变量为内生的。 We will pay particular attention to the case that assumption 3 fails because of omitted variables. 我们将特别注意当重要变量缺省时导致假定3不成立的情况。,Intermediate Econometrics, Yan Shen,1

13、0,Theorem 3.1 (Unbiasedness of OLS)定理 3.1（OLS的无偏性）,Under assumptions MLR.1 through MLR.4, the OLS estimators are unbiased estimator of the population parameters, that is (3.37) 在假定MLR.1MLR.4下，OLS估计量是总体参数的无偏估计量，即,Intermediate Econometrics, Yan Shen,11,Including irrelevant variables or Omitted Variabl

14、e : 包涵了不相关变量或者忽略了变量,What happens if we include variables in our specification that dont belong? 如果我们在设定中包含了不属于真实模型的变量会怎样？ A model is overspecifed when one or more of the independent variables is included in the model even though it has no partial effect on y in the population 尽管一个（或多个）自变量在总体中对y没有局部效

15、应，但却被放到了模型中，则此模型被过度设定。 There is no effect on our parameter estimate, and OLS remains unbiased. But it can have undesirable effects on the variances of the OLS estimators. 过度设定对我们的参数估计没有影响，OLS仍然是无偏的。但它对OLS估计量的方差有不利影响。,Intermediate Econometrics, Yan Shen,12,Including irrelevant variables or Omitted Va

16、riable : 包涵了不相关变量或者忽略了变量,What if we exclude a variable from our specification that does belong? 如果我们在设定中排除了一个本属于真实模型的变量会如何？ If a variable that actually belongs in the true model is omitted, we say the model is underspecified. 如果一个实际上属于真实模型的变量被遗漏，我们说此模型设定不足。 OLS will usually be biased. 此时OLS通常有偏。,Int

17、ermediate Econometrics, Yan Shen,13,Omitted Variable Bias遗漏变量的偏误,Intermediate Econometrics, Yan Shen,14,Omitted Variable Bias (cont)遗漏变量的偏误（续）,Intermediate Econometrics, Yan Shen,15,Omitted Variable Bias Summary遗漏变量的偏误总结，Table 3.2,Two cases where bias is equal to zero 两种偏误为零的情形 b2 = 0, that is x2 do

18、esnt really belong in model b2 = 0，也就是，x2实际上不属于模型 x1 and x2 are uncorrelated in the sample 样本中x1与x2不相关 If correlation between x2 , x1 and x2 , y is the same direction, bias will be positive 如果x2与 x1间相关性和x2与y间相关性同方向，偏误为正。 If correlation between x2 , x1 and x2 , y is the opposite direction, bias will

19、be negative 如果x2与 x1间相关性和x2与y间相关性反方向，偏误为负。,Intermediate Econometrics, Yan Shen,16,Omitted Variable Bias Summary遗漏变量的偏误 总结,Intermediate Econometrics, Yan Shen,17,Summary of Direction of Bias偏误方向总结,Intermediate Econometrics, Yan Shen,18,The More General Case更一般的情形,Technically, it is more difficult to

20、derive the sign of omitted variable bias with multiple regressors. 从技术上讲，要推出多元回归下缺省一个变量时各个变量的偏误方向更加困难。 But remember that if an omitted variable has partial effects on y and it is correlated with at least one of the regressors, then the OLS estimators of all coefficients will be biased. 我们需要记住，若有一个对y

21、有局部效应的变量被缺省，且该变量至少和一个解释变量相关，那么所有系数的OLS估计量都有偏。,Intermediate Econometrics, Yan Shen,19,The More General Case更一般的情形(3.49-3.50),Intermediate Econometrics, Yan Shen,20,The More General Case更一般的情形,Intermediate Econometrics, Yan Shen,21,Variance of the OLS Estimators OLS估计量的方差,Now we know that the sampling

22、 distribution of our estimate is centered around the true parameter。现在我们知道估计值的样本分布是以真实参数为中心的。 Want to think about how spread out this distribution is 我们还想知道这一分布的分散状况。 Much easier to think about this variance under an additional assumption, so 在一个新增假设下，度量这个方差就容易多了，有：,Intermediate Econometrics, Yan Sh

23、en,22,Assumption MLR.5 (Homoskedasticity)假定MLR.5（同方差性）,Assume Homoskedasticity: 同方差性假定： Var(u|x1, x2, xk) = s2 . Means that the variance in the error term, u, conditional on the explanatory variables, is the same for all combinations of outcomes of explanatory variables. 意思是，不管解释变量出现怎样的组合，误差项u的条件方差都

24、是一样的。 If the assumption fails, we say the model exhibits heteroskedasticity. 如果这个假定不成立，我们说模型存在异方差性。,Intermediate Econometrics, Yan Shen,23,Variance of OLS (cont)OLS估计量的方差（续）,Let x stand for (x1, x2,xk) 用x表示(x1, x2,xk) Assuming that Var(u|x) = s2 also implies that Var(y| x) = s2 假定Var(u|x) = s2，也就意味着

25、Var(y| x) = s2 Assumption MLR.1-5 are collectively known as the Gauss-Markov assumptions. 假定MLR.1-5共同被称为高斯马尔科夫假定,Intermediate Econometrics, Yan Shen,24,Theorem 3.2 (Sampling Variances of the OLS Slope Estimators)定理 3.2（OLS斜率估计量的抽样方差）,Intermediate Econometrics, Yan Shen,25,Interpreting Theorem 3.2对定理

26、3.2的解释,Theorem 3.2 shows that the variances of the estimated slope coefficients are influenced by three factors: 定理3.2显示：估计斜率系数的方差受到三个因素的影响： The error variance 误差项的方差 The total sample variation 总的样本变异 Linear relationships among the independent variables 解释变量之间的线性相关关系,Intermediate Econometrics, Yan S

27、hen,26,Interpreting Theorem 3.2: The Error Variance对定理3.2的解释（1）：误差项方差,A larger s2 implies a larger variance for the OLS estimators. 更大的s2意味着更大的OLS估计量方差。 A larger s2 means more noises in the equation. 更大的s2意味着方程中的“噪音”越多。 This makes it more difficult to extract the exact partial effect of the regresso

28、r on the regressand. 这使得得到自变量对因变量的准确局部效应变得更加困难。 Introducing more regressors can reduce the variance. But often this is not possible, neither is it desirable. 引入更多的解释变量可以减小方差。但这样做不仅不一定可能，而且也不一定总令人满意。 s2 does not depends on sample size. s2 不依赖于样本大小,Intermediate Econometrics, Yan Shen,27,Interpreting T

29、heorem 3.2: The total sample variation对定理3.2的解释（2）：总的样本变异,A larger SSTj implies a smaller variance for the estimators, and vice versa. 更大的SSTj意味着更小的估计量方差，反之亦然。 Everything else being equal, more sample variation in x is always preferred. 其它条件不变情况下， x的样本方差越大越好。 One way to gain more sample variation is

30、 to increase the sample size. 增加样本方差的一种方法是增加样本容量。 This components of parameter variance depends on the sample size.参数方差的这一组成部分依赖于样本容量。,Intermediate Econometrics, Yan Shen,28,Interpreting Theorem 3.2: multicollinearity对定理3.1的解释（3）：多重共线性,A larger Rj2 implies a larger variance for the estimators 更大的Rj2

31、意味着更大的估计量方差。 A large Rj2 means other regressors can explain much of the variations in xj. 如果Rj2较大，就说明其它解释变量解释可以解释较大部分的该变量。 When Rj2 is very close to 1, xj is highly correlated with other regressors, this is called multicollinearity. 当Rj2非常接近1时， xj与其它解释变量高度相关，被称为多重共线性。 Severe multicollinearity means

32、the variance of the estimated parameter will be very large. 严重的多重共线性意味着被估计参数的方差将非常大。,Intermediate Econometrics, Yan Shen,29,Interpreting Theorem 3.2: multicollinearity对定理3.2的解释（3）：多重共线性,Multicollinearity is a data problem. 多重共线性是一个数据问题 Could be reduced by appropriately dropping certain variables, or

33、 collecting more data, etc. 可以通过适当的地舍弃某些变量，或收集更多数据等方法来降低。 Notice that a high degree of correlation between certain independent variables can be irrelevant as to how well we can estimate other parameters in the model. 注意：虽然某些自变量之间可能高度相关，但与模型中其它参数的估计程度无关。,Intermediate Econometrics, Yan Shen,30,Varianc

34、es in Misspecified Models误设模型中的方差,The tradeoff between bias and variance is important for considering whether to include an additional variable in the regression. 在考虑一个回归模型中是否该包括一个特定变量的决策中，偏误和方差之间的消长关系是重要的。 Suppose the true model is y = b0 + b1x1 + b2x2 +u then we have 假定真实模型是 y = b0 + b1x1 + b2x2 +

35、u， 我们有,Intermediate Econometrics, Yan Shen,31,Variances in Misspecified Models误设模型中的方差,Consider the misspecified model 考虑误设模型是 the estimated variance is 估计的方差是 When x1 and x2 has zero correlation, 当x1和x2不相关时 otherwise 否则,Intermediate Econometrics, Yan Shen,32,Consequences of Dropping x2 舍弃x2的后果,Inte

36、rmediate Econometrics, Yan Shen,33,Estimating the Error Variance 估计误差项方差,We wish to form an unbiased estimator of s2. 我们希望构造一个s2 的无偏估计量 If we knew u, an unbiased estimator of s2 can be formed by calculate the sample average of the u 2 如果我们知道 u，通过计算 u 2的样本平均可以构造一个s2的无偏估计量 We dont know what the error

37、variance, s2, is, because we dont observe the errors, ui. 我们观察不到误差项 ui ，所以我们不知道误差项方差s2。,Intermediate Econometrics, Yan Shen,34,Estimating the Error Variance估计误差项方差,What we observe are the residuals, i 我们能观察到的是残差项i 。 We can use the residuals to form an estimate of the error variance 我们可以用残差项构造一个误差项方差

38、的估计 df = n (k + 1), or df = n k 1 df (i.e. degrees of freedom) is the (number of observations) (number of estimated parameters) df（自由度）,是观察点个数被估参数个数,Intermediate Econometrics, Yan Shen,35,Estimating the Error Variance估计误差项方差,The division of n-k-1 comes from E(Sum of squared residuals)=(n-k-1) s2. 上式中除以n-k-1是因为残差平方和的期望值是(n-k-1)s2. Why degree of freedom is n-k-1 ? 为什么自由度是n-k-1 Because k+1 restrictions are imposed when deriving the OLS