八高研一下2课件chapter4the least squares estimator part

1、CHAPTER 4 The Least Squares EstimatorTABLE 4.1 Assumptions of the Classical Linear Regression Model4.2 Motivating Least Squares4.3 Finite Sample Properties of Least Squares4.4 Large Sample Properties of the Least Squares Estimator4.5 Interval Estimation4.6 Prediction and Forecasting4.7 Data Problems

2、Assignments: Exercises 6-10, 12, 13, 15, 16, Applications 1, 2TABLE 4.1 Assumptions of the Classical Linear Regression ModelChapterA1. Linearity:A2. Full rank: The nK sample data matrix, X, has full column rank.A3. Exogeneity of the independent variables:There is no correlation between the disturban

3、ces and the independent variables.A4. Homoscedasticity and nonautocorrelation: Each disturbance, i , has the same finite variance, 2, and is uncorrelated with every other disturbance, j , conditioned on X.A5. Stochastic or nonstochastic data:A6. Normal distribution: The disturbances are normally dis

4、tributed.4.2 Motivating Least Squares (1/3)ChapterThe method of least squares mimics the sample relationship in the population.4.2.1 The Population Orthogonality Conditions4.2 Motivating Least Squares (2/3)ChapterDrop Assumption A1 that Ey|x is linear, i.e. assume Ey|x=f(x). We use the MSE rule as a

5、 criterion to seek an optimal linear predictor of y, which we denote xg.4.2.2 Minimum Mean Squared Error Predictor 4.2 Motivating Least Squares (3/3)ChapterConsider the problem of finding a linear unbiased estimator. If we seek the one that has smallest variance, we will be led once again to least s

6、quares. This proposition (GaussMarkov Theorem) will be proved in Section .2.3 Minimum Variance Linear Unbiased Estimation4.3 Finite Sample Properties of Least SquaresChapter 4.3.1 Unbiased Estimation4.3.2 Bias Caused by Omission of Relevant Variables4.3.3 Inclusion of Irrelevant Variables4.3.

7、4 The Variance of the Least Squares Estimator4.3.5 The GaussMarkov Theorem4.3.6 The Implications of Stochastic Regressors4.3.7 Estimating the Variance of the Least Squares Estimator4.3.8 The Normality Assumption 4.3 Finite Sample Properties of Least Squares (1/10)ChapterSectionb are linear, unbiased

8、 estimators of b.Linearity means being linear in y. 4.3.1 Unbiased Estimation4.3 Finite Sample Properties of Least Squares (2/10)ChapterSectionSuppose a correctly specified model would beBut we regress y on only X1, then the estimator isThe omitted variable formula is4.3.2 Bias Caused by Omission of

9、 Relevant Variables (1/2)Example 4.2 Omitted Variable (tablef2-2.wf1)有2种方法计算 。直接用formula,不过要注意b1是2维的。另一种就是用模型的离差形式。4.3 Finite Sample Properties of Least Squares (3/10)ChapterSectiontablef2-2.wf1ls log(gasexp/gasp/pop) c log(gasp) log( e)ls log(gasexp/gasp/pop) c log(gasp)ls log( e) c log(gasp)注意工作文件

10、中序列名与模型中名字不完全对应4.3.2 Bias Caused by Omission of Relevant Variables (2/2)4.3 Finite Sample Properties of Least Squares (4/10)ChapterSectionSuppose that a correctly specified regression model would beIf we regress then the estimator isThe estimators are still unbiased. The cost is that the precision o

11、f estimators is reduced.4.3.3 Inclusion of Irrelevant Variables4.3 Finite Sample Properties of Least Squares (5/10)ChapterSectionExample 4.3 Sampling Variance in a simple Regression Model4.3.4 The Variance of the Least Squares Estimator4.3 Finite Sample Properties of Least Squares (6/10)ChapterSecti

12、onTHEOREM 4.2 GaussMarkov TheoremIn the linear regression model with regressor matrix X, the least squares estimator b is the minimum variance linear unbiased estimator of . For any vector of constants w, the minimum variance linear unbiased estimator of w in the regression model is wb, where b is t

13、he least squares estimator.PROOF: Only A1 to A4 are necessary.Denote a linear, unbiased estimator b0=Cy such that Eb0|X=.4.3.5 The GaussMarkov Theorem (1/2)先用具体例子柯布道格拉斯 函数说明w含义。4.3 Finite Sample Properties of Least Squares (7/10)ChapterSection4.3.5 The GaussMarkov Theorem (2/2)4.3 Finite Sample Prop

14、erties of Least Squares (8/10)ChapterSectionTHEOREM 4.3 GaussMarkov Theorem (Concluded)In the linear regression model, the least squares estimator b is the minimum variance linear unbiased estimator of whether X is stochastic or nonstochastic, so long as the other assumptions of the model continue t

15、o hold.4.3.6 The Implications of Stochastic Regressors4.3 Finite Sample Properties of Least Squares (9/10)ChapterSectionWe can not obtain the data of the disturbances, and use the residuals as proxies. But the residuals are imperfect estimates of their population counterparts, as4.3.7 Estimating the

16、 Variance of the Least Squares Estimator和书上略有些不同。4.3 Finite Sample Properties of Least Squares (10/10)ChapterSectionUnder Assumption A6,4.3.8 The Normality Assumption4.4 Large Sample Properties of the Least Squares EstimatorChapter4.4.1 Consistency of the Least Squares Estimator of 4.4.2 Asymptotic

17、Normality of the Least Squares Estimator4.4.3 Consistency of s2 and the Estimator of Asy. Varb4.4.4 Asymptotic Distribution of a Function of b: The Delta Method4.4.5 Asymptotic Efficiency4.4.6 Maximum Likelihood Estimation4.4 Large Sample Properties of the Least Squares Estimator (1/12)ChapterSectio

18、n4.4.1 Consistency of the Least Squares Estimator of (1/4)THEOREM D.1 Convergence in Quadratic Mean (convergence in mean square)4.4 Large Sample Properties of the Least Squares Estimator (2/12)ChapterSectionA5a. (xi, i ) i = 1, . . . , n is a sequence of independent observations.A5b. 4.4.1 Consisten

19、cy of the Least Squares Estimator of (2/4)4.4 Large Sample Properties of the Least Squares Estimator (3/12)ChapterSection4.4.1 Consistency of the Least Squares Estimator of (3/4)4.4 Large Sample Properties of the Least Squares Estimator (4/12)ChapterSectionTABLE 4.2 Grenander Conditions for Well-Beh

20、aved DataG1. For each column of X, xk, ifHence, xk does not degenerate to a sequence of zeros. Sums of squares will continue to grow as the sample size increases. No variable will degenerate to a sequence of zeros.G2. This condition implies that no single observation will ever dominate and as n, ind

21、ividual observations will e less important.G3. Let Rn be the sample correlation matrix of the columns of X, excluding the constant term if there is one. Then a positive definite matrix. This condition implies that the full rank condition will always be met. We have already assumed that X has full ra

22、nk in a finite sample, so this assumption ensures that the condition will never be violated.4.4.1 Consistency of the Least Squares Estimator of (4/4)4.4 Large Sample Properties of the Least Squares Estimator (5/12)ChapterSectionIn this section, we drop the assumption of normality of the disturbances

23、, and assume that observations are independent.THEOREM 4.4 Asymptotic Distribution of b with Independent ObservationsIf i are independently distributed with mean zero and finite variance 2 and xik is such that the Grenander conditions are met, thenPROOF:4.4.2 Asymptotic Normality of the Least Square

24、s Estimator (1/2)4.4 Large Sample Properties of the Least Squares Estimator (6/12)ChapterSection4.4.2 Asymptotic Normality of the Least Squares Estimator (2/2)4.4 Large Sample Properties of the Least Squares Estimator (7/12)ChapterSectionTHEOREM D.5 Khinchines Weak Law of Large NumbersIf xi, i = 1,

25、. . . , n is a random (i.i.d.) sample from a distribution with finite mean Exi = , then4.4.3 Consistency of s2 and the Estimator of Asy. Varb (1/2)4.4 Large Sample Properties of the Least Squares Estimator (8/12)ChapterSection4.4.3 Consistency of s2 and the Estimator of Asy. Varb (2/2)4.4 Large Samp

26、le Properties of the Least Squares Estimator (9/12)ChapterSectionTHEOREM 4.5 Asymptotic Distribution of a Function of bIf f(b) is a set of continuous and continuously differentiable functions of b such that and if Theorem 4.4 holds, thenIn practice, the estimator of the asymptotic covariance matrix

27、would beNote that4.4.4 Asymptotic Distribution of a Function of b: The Delta Method (1/3)4.4 Large Sample Properties of the Least Squares Estimator (10/12)ChapterSectionExample 4.4 Nonlinear Functions of Parameters: The Delta Methodthe short-run price and e elasticities are 2 and 3. The long-run ela

28、sticities are 2/(1 ) and 3/(1 ), respectively. How to estimate the long-run elasticities and their standard errors?4.4.4 Asymptotic Distribution of a Function of b: The Delta Method (2/3)4.4 Large Sample Properties of the Least Squares Estimator (11/12)ChapterSectiontablef2-2.wf1equation eq1.ls log(

29、gasexp/gasp/pop) c log(gasp) log( e) log(pnc) log(puc) log(gasexp(-1)/gasp(-1)/pop(-1)scalar f2=c(2)/(1-c(6)scalar f3=c(3)/(1-c(6) matrix eqcov=eq1.coefcovmatrix(2,6) cc=0cc(1,2)=1/(1-c(6)cc(1,6)=c(2)/(1-c(6)2cc(2,3)=cc(1,2)cc(2,6)=c(3)/(1-c(6)2matrix(2,2) csxxc=cc*eqcov*transpose(cc)scalar se2=sqrt

30、(csxxc(1,1)scalar se3=sqrt(csxxc(2,2)4.4.4 Asymptotic Distribution of a Function of b: The Delta Method (3/3)4.4 Large Sample Properties of the Least Squares Estimator (12/12)ChapterSectionDEFINITION 4.1 Asymptotic EfficiencyAn estimator is asymptotically efficient if it is consistent, asymptoticall

31、y normally distributed, and has an asymptotic covariance matrix that is not larger than the asymptotic covariance matrix of any other consistent, asymptotically normally distributed estimator.4.4.5 Asymptotic Efficiency4.5 Interval EstimationChapter4.5.1 Forming a Confidence Interval for a Coefficient 4.5.2 Confidence Intervals Based on Large Samples 4.5.3 Confidence Interval for a Linear Combination of Coefficients4.5 Interval Estimation (1/8)ChapterSection4.5.1 For