计量经济学课件英文版伍德里奇

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伍德里奇计量经济学课件 (11)

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动因：优点
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如果影响y的其它因素与x不相关，则改变x 可以保证u不变，从而x对y的影响可以被识别出来。多元回归分析更适合于其它条件不变情况下的分析，因为多元回归分析允许我们明确地控制许多其它也同时影响因变量的因素。
Introductory Econometrics 6 of 89
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Introductory Econometrics
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动因：一个例子
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考虑一个简单版本的解释教育对小时工 wage b 0 b1educ b 2 exper u 资影响的工资方程。
• exper：在劳动力市场上的经历，用年衡量
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在这个例子中，“在劳动力市场上的经历”被明确地从误差项中提出。
Introductory Econometrics
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“排除其它变量影响”（续）
上述方程意味着：将y同时对x1和x2回归得出的x1的影响与先将x1对x2回归得到残差，再将y对此残差回归得到的x1 的影响相同。 n 这意味着只有x1中与x2不相关的部分与 y有关，所以在x2被“排除影响”之后，我们再估计x1对y的影响。
过原点的回归
y b1 x1 b 2 x2 b k x k x b x b x b y
1 1 2 2 k k
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x b x b x ) min y ( b i 1 i 1 2 i 2 k ik
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ˆ b ˆ x b ˆ x b ˆ x u ˆ yi b 0 1 i1 2 i2 k ik i
Introductory Econometrics
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计量经济学课件英文版伍德里奇

Two methods to estimate
2016/9/22 Department of Statistics-Zhaoliqin 17
Some Terminology, cont.
β0 :intercept parameter β1 :slope parameter means that a one-unit increase in x changes the expected value of y by the amount β1,holding the other factors in u fixed.
Department of Statistics-Zhaoliqin 16
2016/9/22
Some Terminology, cont.
y = β0 + β1x + u, E [u|x] = 0. β0 + β1x is the systematic part of y. u is the unsystematic part of y. u is denoted the error term. Other terms for u : error shock disturbance residual (sometimes in reference to ﬁtted error term)
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2016/9/22
Department of Statistics-Zhaoliqin
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Hence, we are interested in studying is the mean of wages given the years of Education that will be denoted as E [Wages|Education] Following economic theory, we assume a speciﬁc model for E[Wages|Education]

伍德里奇计量经济学课件 (1)

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计量经济学
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若贝尔经济学奖获奖名单
2004 Finn Kydland , Edward Prescott 2003 Robert F. Engle, Clive W. J. Granger 2002 Daniel Kahneman, Vernon L. Smith 2001 George A. Akerlof, A. Michael Spence, Joseph E. Stiglitz 2000 James J Heckman, Daniel L McFadden 1999 Robert A. Mundell 1998 Amartya Sen 1997 Robert C. Merton, Myron S. Scholes 1996 James A. Mirrlees, William Vickrey
INTERMEDIATE ECONOMETRICS
计量经济学导论
Fall, 2012
1
Outline
有关信息 n 什么是计量经济学 n 计量经济学的作用 n 数据: 输入数据 n 经验分析的步骤 n 本课程涵盖的内容
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信息：课程——计量经济学
金融计量学课号：01663 学分：4 课程性质：教育部规定核心课程
△诺贝尔经济学奖与计量经济学
77位获奖者中10位直接因为对计量经济学发展的贡献而获奖 1969 R. Frish J. Tinbergen 1973 W. Leotief 1980 L. R. Klein 1984 R. Stone 1989 T. Haavelmo 2000 J. J. Heckman D. L. McFadden 2003 R. F. Engle C. W. J. Granger

计量经济学课件英文版伍德里奇

atistics by Zhaoliqin
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1.3 The Structure of Economic Data
Cross Sectional Time Series Panel
Department of Statistics by Zhaoliqin
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Types of Data – Cross Sectional
Department of Statistics by Zhaoliqin 10
Why study Econometrics?
An empirical analysis uses data to test a theory or to estimate a relationship
A formal economic model can be tested
Theory may be ambiguous as to the effect of some policy change – can use econometrics to evaluate the program
Department of Statistics by Zhaoliqin 11
1.2 steps in empirical economic analysis
Welcome to Econometrics
Department of Statistic：赵丽琴 liqinzhao_618@
Department of Statistics by Zhaoliqin
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About this Course
Textbook: Jeffrey M. Wooldridge, Introductory Econometrics—A Modern Approach. Main Software: Eviews. Sample data can be acquired from internet. If time permitted, R commands will be introduced.

伍德里奇计量经济学国外课件ch01

Economics 20 - Prof. Anderson 3
Types of Data – Cross Sectional
Cross-sectional data is a random sample Each observation is a new individual, firm, etc. with information at a point in time If the data is not a random sample, we have a sample-selection problem
Ecoቤተ መጻሕፍቲ ባይዱomics 20 - Prof. Anderson 7
Example: Returns to Education
A model of human capital investment implies getting more education should lead to higher earnings In the simplest case, this implies an equation like
Welcome to Economics 20
What is Econometrics?
Economics 20 - Prof. Anderson
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Why study Econometrics?
Rare in economics (and many other areas without labs!) to have experimental data Need to use nonexperimental, or observational, data to make inferences Important to be able to apply economic theory to real world data

伍德里奇《计量经济学导论--现代观点》1

T his appendix derives various results for ordinary least squares estimation of themultiple linear regression model using matrix notation and matrix algebra (see Appendix D for a summary). The material presented here is much more ad-vanced than that in the text.E.1THE MODEL AND ORDINARY LEAST SQUARES ESTIMATIONThroughout this appendix,we use the t subscript to index observations and an n to denote the sample size. It is useful to write the multiple linear regression model with k parameters as follows:y t ϭ␤1ϩ␤2x t 2ϩ␤3x t 3ϩ… ϩ␤k x tk ϩu t ,t ϭ 1,2,…,n ,(E.1)where y t is the dependent variable for observation t ,and x tj ,j ϭ 2,3,…,k ,are the inde-pendent variables. Notice how our labeling convention here differs from the text:we call the intercept ␤1and let ␤2,…,␤k denote the slope parameters. This relabeling is not important,but it simplifies the matrix approach to multiple regression.For each t ,define a 1 ϫk vector,x t ϭ(1,x t 2,…,x tk ),and let ␤ϭ(␤1,␤2,…,␤k )Јbe the k ϫ1 vector of all parameters. Then,we can write (E.1) asy t ϭx t ␤ϩu t ,t ϭ 1,2,…,n .(E.2)[Some authors prefer to define x t as a column vector,in which case,x t is replaced with x t Јin (E.2). Mathematically,it makes more sense to define it as a row vector.] We can write (E.2) in full matrix notation by appropriately defining data vectors and matrices. Let y denote the n ϫ1 vector of observations on y :the t th element of y is y t .Let X be the n ϫk vector of observations on the explanatory variables. In other words,the t th row of X consists of the vector x t . Equivalently,the (t ,j )th element of X is simply x tj :755A p p e n d i x EThe Linear Regression Model inMatrix Formn X ϫ k ϵϭ .Finally,let u be the n ϫ 1 vector of unobservable disturbances. Then,we can write (E.2)for all n observations in matrix notation :y ϭX ␤ϩu .(E.3)Remember,because X is n ϫ k and ␤is k ϫ 1,X ␤is n ϫ 1.Estimation of ␤proceeds by minimizing the sum of squared residuals,as in Section3.2. Define the sum of squared residuals function for any possible k ϫ 1 parameter vec-tor b asSSR(b ) ϵ͚nt ϭ1(y t Ϫx t b )2.The k ϫ 1 vector of ordinary least squares estimates,␤ˆϭ(␤ˆ1,␤ˆ2,…,␤ˆk )؅,minimizes SSR(b ) over all possible k ϫ 1 vectors b . This is a problem in multivariable calculus.For ␤ˆto minimize the sum of squared residuals,it must solve the first order conditionѨSSR(␤ˆ)/Ѩb ϵ0.(E.4)Using the fact that the derivative of (y t Ϫx t b )2with respect to b is the 1ϫ k vector Ϫ2(y t Ϫx t b )x t ,(E.4) is equivalent to͚nt ϭ1xt Ј(y t Ϫx t ␤ˆ) ϵ0.(E.5)(We have divided by Ϫ2 and taken the transpose.) We can write this first order condi-tion as͚nt ϭ1(y t Ϫ␤ˆ1Ϫ␤ˆ2x t 2Ϫ… Ϫ␤ˆk x tk ) ϭ0͚nt ϭ1x t 2(y t Ϫ␤ˆ1Ϫ␤ˆ2x t 2Ϫ… Ϫ␤ˆk x tk ) ϭ0...͚nt ϭ1x tk (y t Ϫ␤ˆ1Ϫ␤ˆ2x t 2Ϫ… Ϫ␤ˆk x tk ) ϭ0,which,apart from the different labeling convention,is identical to the first order condi-tions in equation (3.13). We want to write these in matrix form to make them more use-ful. Using the formula for partitioned multiplication in Appendix D,we see that (E.5)is equivalent to΅1x 12x 13...x 1k1x 22x 23...x 2k...1x n 2x n 3...x nk ΄΅x 1x 2...x n ΄Appendix E The Linear Regression Model in Matrix Form756Appendix E The Linear Regression Model in Matrix FormXЈ(yϪX␤ˆ) ϭ0(E.6) or(XЈX)␤ˆϭXЈy.(E.7)It can be shown that (E.7) always has at least one solution. Multiple solutions do not help us,as we are looking for a unique set of OLS estimates given our data set. Assuming that the kϫ k symmetric matrix XЈX is nonsingular,we can premultiply both sides of (E.7) by (XЈX)Ϫ1to solve for the OLS estimator ␤ˆ:␤ˆϭ(XЈX)Ϫ1XЈy.(E.8)This is the critical formula for matrix analysis of the multiple linear regression model. The assumption that XЈX is invertible is equivalent to the assumption that rank(X) ϭk, which means that the columns of X must be linearly independent. This is the matrix ver-sion of MLR.4 in Chapter 3.Before we continue,(E.8) warrants a word of warning. It is tempting to simplify the formula for ␤ˆas follows:␤ˆϭ(XЈX)Ϫ1XЈyϭXϪ1(XЈ)Ϫ1XЈyϭXϪ1y.The flaw in this reasoning is that X is usually not a square matrix,and so it cannot be inverted. In other words,we cannot write (XЈX)Ϫ1ϭXϪ1(XЈ)Ϫ1unless nϭk,a case that virtually never arises in practice.The nϫ 1 vectors of OLS fitted values and residuals are given byyˆϭX␤ˆ,uˆϭyϪyˆϭyϪX␤ˆ.From (E.6) and the definition of uˆ,we can see that the first order condition for ␤ˆis the same asXЈuˆϭ0.(E.9) Because the first column of X consists entirely of ones,(E.9) implies that the OLS residuals always sum to zero when an intercept is included in the equation and that the sample covariance between each independent variable and the OLS residuals is zero. (We discussed both of these properties in Chapter 3.)The sum of squared residuals can be written asSSR ϭ͚n tϭ1uˆt2ϭuˆЈuˆϭ(yϪX␤ˆ)Ј(yϪX␤ˆ).(E.10)All of the algebraic properties from Chapter 3 can be derived using matrix algebra. For example,we can show that the total sum of squares is equal to the explained sum of squares plus the sum of squared residuals [see (3.27)]. The use of matrices does not pro-vide a simpler proof than summation notation,so we do not provide another derivation.757The matrix approach to multiple regression can be used as the basis for a geometri-cal interpretation of regression. This involves mathematical concepts that are even more advanced than those we covered in Appendix D. [See Goldberger (1991) or Greene (1997).]E.2FINITE SAMPLE PROPERTIES OF OLSDeriving the expected value and variance of the OLS estimator ␤ˆis facilitated by matrix algebra,but we must show some care in stating the assumptions.A S S U M P T I O N E.1(L I N E A R I N P A R A M E T E R S)The model can be written as in (E.3), where y is an observed nϫ 1 vector, X is an nϫ k observed matrix, and u is an nϫ 1 vector of unobserved errors or disturbances.A S S U M P T I O N E.2(Z E R O C O N D I T I O N A L M E A N)Conditional on the entire matrix X, each error ut has zero mean: E(ut͉X) ϭ0, tϭ1,2,…,n.In vector form,E(u͉X) ϭ0.(E.11) This assumption is implied by MLR.3 under the random sampling assumption,MLR.2.In time series applications,Assumption E.2 imposes strict exogeneity on the explana-tory variables,something discussed at length in Chapter 10. This rules out explanatory variables whose future values are correlated with ut; in particular,it eliminates laggeddependent variables. Under Assumption E.2,we can condition on the xtjwhen we com-pute the expected value of ␤ˆ.A S S U M P T I O N E.3(N O P E R F E C T C O L L I N E A R I T Y) The matrix X has rank k.This is a careful statement of the assumption that rules out linear dependencies among the explanatory variables. Under Assumption E.3,XЈX is nonsingular,and so ␤ˆis unique and can be written as in (E.8).T H E O R E M E.1(U N B I A S E D N E S S O F O L S)Under Assumptions E.1, E.2, and E.3, the OLS estimator ␤ˆis unbiased for ␤.P R O O F:Use Assumptions E.1 and E.3 and simple algebra to write␤ˆϭ(XЈX)Ϫ1XЈyϭ(XЈX)Ϫ1XЈ(X␤ϩu)ϭ(XЈX)Ϫ1(XЈX)␤ϩ(XЈX)Ϫ1XЈuϭ␤ϩ(XЈX)Ϫ1XЈu,(E.12)where we use the fact that (XЈX)Ϫ1(XЈX) ϭIk . Taking the expectation conditional on X givesAppendix E The Linear Regression Model in Matrix Form 758E(␤ˆ͉X)ϭ␤ϩ(XЈX)Ϫ1XЈE(u͉X)ϭ␤ϩ(XЈX)Ϫ1XЈ0ϭ␤,because E(u͉X) ϭ0under Assumption E.2. This argument clearly does not depend on the value of ␤, so we have shown that ␤ˆis unbiased.To obtain the simplest form of the variance-covariance matrix of ␤ˆ,we impose the assumptions of homoskedasticity and no serial correlation.A S S U M P T I O N E.4(H O M O S K E D A S T I C I T Y A N DN O S E R I A L C O R R E L A T I O N)(i) Var(ut͉X) ϭ␴2, t ϭ 1,2,…,n. (ii) Cov(u t,u s͉X) ϭ0, for all t s. In matrix form, we canwrite these two assumptions asVar(u͉X) ϭ␴2I n,(E.13)where Inis the nϫ n identity matrix.Part (i) of Assumption E.4 is the homoskedasticity assumption:the variance of utcan-not depend on any element of X,and the variance must be constant across observations, t. Part (ii) is the no serial correlation assumption:the errors cannot be correlated across observations. Under random sampling,and in any other cross-sectional sampling schemes with independent observations,part (ii) of Assumption E.4 automatically holds. For time series applications,part (ii) rules out correlation in the errors over time (both conditional on X and unconditionally).Because of (E.13),we often say that u has scalar variance-covariance matrix when Assumption E.4 holds. We can now derive the variance-covariance matrix of the OLS estimator.T H E O R E M E.2(V A R I A N C E-C O V A R I A N C EM A T R I X O F T H E O L S E S T I M A T O R)Under Assumptions E.1 through E.4,Var(␤ˆ͉X) ϭ␴2(XЈX)Ϫ1.(E.14)P R O O F:From the last formula in equation (E.12), we haveVar(␤ˆ͉X) ϭVar[(XЈX)Ϫ1XЈu͉X] ϭ(XЈX)Ϫ1XЈ[Var(u͉X)]X(XЈX)Ϫ1.Now, we use Assumption E.4 to getVar(␤ˆ͉X)ϭ(XЈX)Ϫ1XЈ(␴2I n)X(XЈX)Ϫ1ϭ␴2(XЈX)Ϫ1XЈX(XЈX)Ϫ1ϭ␴2(XЈX)Ϫ1.Appendix E The Linear Regression Model in Matrix Form759Formula (E.14) means that the variance of ␤ˆj (conditional on X ) is obtained by multi-plying ␴2by the j th diagonal element of (X ЈX )Ϫ1. For the slope coefficients,we gave an interpretable formula in equation (3.51). Equation (E.14) also tells us how to obtain the covariance between any two OLS estimates:multiply ␴2by the appropriate off diago-nal element of (X ЈX )Ϫ1. In Chapter 4,we showed how to avoid explicitly finding covariances for obtaining confidence intervals and hypotheses tests by appropriately rewriting the model.The Gauss-Markov Theorem,in its full generality,can be proven.T H E O R E M E .3 (G A U S S -M A R K O V T H E O R E M )Under Assumptions E.1 through E.4, ␤ˆis the best linear unbiased estimator.P R O O F :Any other linear estimator of ␤can be written as␤˜ ϭA Јy ,(E.15)where A is an n ϫ k matrix. In order for ␤˜to be unbiased conditional on X , A can consist of nonrandom numbers and functions of X . (For example, A cannot be a function of y .) To see what further restrictions on A are needed, write␤˜ϭA Ј(X ␤ϩu ) ϭ(A ЈX )␤ϩA Јu .(E.16)Then,E(␤˜͉X )ϭA ЈX ␤ϩE(A Јu ͉X )ϭA ЈX ␤ϩA ЈE(u ͉X ) since A is a function of XϭA ЈX ␤since E(u ͉X ) ϭ0.For ␤˜to be an unbiased estimator of ␤, it must be true that E(␤˜͉X ) ϭ␤for all k ϫ 1 vec-tors ␤, that is,A ЈX ␤ϭ␤for all k ϫ 1 vectors ␤.(E.17)Because A ЈX is a k ϫ k matrix, (E.17) holds if and only if A ЈX ϭI k . Equations (E.15) and (E.17) characterize the class of linear, unbiased estimators for ␤.Next, from (E.16), we haveVar(␤˜͉X ) ϭA Ј[Var(u ͉X )]A ϭ␴2A ЈA ,by Assumption E.4. Therefore,Var(␤˜͉X ) ϪVar(␤ˆ͉X )ϭ␴2[A ЈA Ϫ(X ЈX )Ϫ1]ϭ␴2[A ЈA ϪA ЈX (X ЈX )Ϫ1X ЈA ] because A ЈX ϭI kϭ␴2A Ј[I n ϪX (X ЈX )Ϫ1X Ј]Aϵ␴2A ЈMA ,where M ϵI n ϪX (X ЈX )Ϫ1X Ј. Because M is symmetric and idempotent, A ЈMA is positive semi-definite for any n ϫ k matrix A . This establishes that the OLS estimator ␤ˆis BLUE. How Appendix E The Linear Regression Model in Matrix Form 760Appendix E The Linear Regression Model in Matrix Formis this significant? Let c be any kϫ 1 vector and consider the linear combination cЈ␤ϭc1␤1ϩc2␤2ϩ… ϩc k␤k, which is a scalar. The unbiased estimators of cЈ␤are cЈ␤ˆand cЈ␤˜. ButVar(c␤˜͉X) ϪVar(cЈ␤ˆ͉X) ϭcЈ[Var(␤˜͉X) ϪVar(␤ˆ͉X)]cՆ0,because [Var(␤˜͉X) ϪVar(␤ˆ͉X)] is p.s.d. Therefore, when it is used for estimating any linear combination of ␤, OLS yields the smallest variance. In particular, Var(␤ˆj͉X) ՅVar(␤˜j͉X) for any other linear, unbiased estimator of ␤j.The unbiased estimator of the error variance ␴2can be written as␴ˆ2ϭuˆЈuˆ/(n Ϫk),where we have labeled the explanatory variables so that there are k total parameters, including the intercept.T H E O R E M E.4(U N B I A S E D N E S S O F␴ˆ2)Under Assumptions E.1 through E.4, ␴ˆ2is unbiased for ␴2: E(␴ˆ2͉X) ϭ␴2for all ␴2Ͼ0. P R O O F:Write uˆϭyϪX␤ˆϭyϪX(XЈX)Ϫ1XЈyϭM yϭM u, where MϭI nϪX(XЈX)Ϫ1XЈ,and the last equality follows because MXϭ0. Because M is symmetric and idempotent,uˆЈuˆϭuЈMЈM uϭuЈM u.Because uЈM u is a scalar, it equals its trace. Therefore,ϭE(uЈM u͉X)ϭE[tr(uЈM u)͉X] ϭE[tr(M uuЈ)͉X]ϭtr[E(M uuЈ|X)] ϭtr[M E(uuЈ|X)]ϭtr(M␴2I n) ϭ␴2tr(M) ϭ␴2(nϪ k).The last equality follows from tr(M) ϭtr(I) Ϫtr[X(XЈX)Ϫ1XЈ] ϭnϪtr[(XЈX)Ϫ1XЈX] ϭnϪn) ϭnϪk. Therefore,tr(IkE(␴ˆ2͉X) ϭE(uЈM u͉X)/(nϪ k) ϭ␴2.E.3STATISTICAL INFERENCEWhen we add the final classical linear model assumption,␤ˆhas a multivariate normal distribution,which leads to the t and F distributions for the standard test statistics cov-ered in Chapter 4.A S S U M P T I O N E.5(N O R M A L I T Y O F E R R O R S)are independent and identically distributed as Normal(0,␴2). Conditional on X, the utEquivalently, u given X is distributed as multivariate normal with mean zero and variance-covariance matrix ␴2I n: u~ Normal(0,␴2I n).761Appendix E The Linear Regression Model in Matrix Form Under Assumption E.5,each uis independent of the explanatory variables for all t. Inta time series setting,this is essentially the strict exogeneity assumption.T H E O R E M E.5(N O R M A L I T Y O F␤ˆ)Under the classical linear model Assumptions E.1 through E.5, ␤ˆconditional on X is dis-tributed as multivariate normal with mean ␤and variance-covariance matrix ␴2(XЈX)Ϫ1.Theorem E.5 is the basis for statistical inference involving ␤. In fact,along with the properties of the chi-square,t,and F distributions that we summarized in Appendix D, we can use Theorem E.5 to establish that t statistics have a t distribution under Assumptions E.1 through E.5 (under the null hypothesis) and likewise for F statistics. We illustrate with a proof for the t statistics.T H E O R E M E.6Under Assumptions E.1 through E.5,(␤ˆjϪ␤j)/se(␤ˆj) ~ t nϪk,j ϭ 1,2,…,k.P R O O F:The proof requires several steps; the following statements are initially conditional on X. First, by Theorem E.5, (␤ˆjϪ␤j)/sd(␤ˆ) ~ Normal(0,1), where sd(␤ˆj) ϭ␴͙ෆc jj, and c jj is the j th diagonal element of (XЈX)Ϫ1. Next, under Assumptions E.1 through E.5, conditional on X,(n Ϫ k)␴ˆ2/␴2~ ␹2nϪk.(E.18)This follows because (nϪk)␴ˆ2/␴2ϭ(u/␴)ЈM(u/␴), where M is the nϫn symmetric, idem-potent matrix defined in Theorem E.4. But u/␴~ Normal(0,I n) by Assumption E.5. It follows from Property 1 for the chi-square distribution in Appendix D that (u/␴)ЈM(u/␴) ~ ␹2nϪk (because M has rank nϪk).We also need to show that ␤ˆand ␴ˆ2are independent. But ␤ˆϭ␤ϩ(XЈX)Ϫ1XЈu, and ␴ˆ2ϭuЈM u/(nϪk). Now, [(XЈX)Ϫ1XЈ]Mϭ0because XЈMϭ0. It follows, from Property 5 of the multivariate normal distribution in Appendix D, that ␤ˆand M u are independent. Since ␴ˆ2is a function of M u, ␤ˆand ␴ˆ2are also independent.Finally, we can write(␤ˆjϪ␤j)/se(␤ˆj) ϭ[(␤ˆjϪ␤j)/sd(␤ˆj)]/(␴ˆ2/␴2)1/2,which is the ratio of a standard normal random variable and the square root of a ␹2nϪk/(nϪk) random variable. We just showed that these are independent, and so, by def-inition of a t random variable, (␤ˆjϪ␤j)/se(␤ˆj) has the t nϪk distribution. Because this distri-bution does not depend on X, it is the unconditional distribution of (␤ˆjϪ␤j)/se(␤ˆj) as well.From this theorem,we can plug in any hypothesized value for ␤j and use the t statistic for testing hypotheses,as usual.Under Assumptions E.1 through E.5,we can compute what is known as the Cramer-Rao lower bound for the variance-covariance matrix of unbiased estimators of ␤(again762conditional on X ) [see Greene (1997,Chapter 4)]. This can be shown to be ␴2(X ЈX )Ϫ1,which is exactly the variance-covariance matrix of the OLS estimator. This implies that ␤ˆis the minimum variance unbiased estimator of ␤(conditional on X ):Var(␤˜͉X ) ϪVar(␤ˆ͉X ) is positive semi-definite for any other unbiased estimator ␤˜; we no longer have to restrict our attention to estimators linear in y .It is easy to show that the OLS estimator is in fact the maximum likelihood estima-tor of ␤under Assumption E.5. For each t ,the distribution of y t given X is Normal(x t ␤,␴2). Because the y t are independent conditional on X ,the likelihood func-tion for the sample is obtained from the product of the densities:͟nt ϭ1(2␲␴2)Ϫ1/2exp[Ϫ(y t Ϫx t ␤)2/(2␴2)].Maximizing this function with respect to ␤and ␴2is the same as maximizing its nat-ural logarithm:͚nt ϭ1[Ϫ(1/2)log(2␲␴2) Ϫ(yt Ϫx t ␤)2/(2␴2)].For obtaining ␤ˆ,this is the same as minimizing͚nt ϭ1(y t Ϫx t ␤)2—the division by 2␴2does not affect the optimization—which is just the problem that OLS solves. The esti-mator of ␴2that we have used,SSR/(n Ϫk ),turns out not to be the MLE of ␴2; the MLE is SSR/n ,which is a biased estimator. Because the unbiased estimator of ␴2results in t and F statistics with exact t and F distributions under the null,it is always used instead of the MLE.SUMMARYThis appendix has provided a brief discussion of the linear regression model using matrix notation. This material is included for more advanced classes that use matrix algebra,but it is not needed to read the text. In effect,this appendix proves some of the results that we either stated without proof,proved only in special cases,or proved through a more cumbersome method of proof. Other topics—such as asymptotic prop-erties,instrumental variables estimation,and panel data models—can be given concise treatments using matrices. Advanced texts in econometrics,including Davidson and MacKinnon (1993),Greene (1997),and Wooldridge (1999),can be consulted for details.KEY TERMSAppendix E The Linear Regression Model in Matrix Form 763First Order Condition Matrix Notation Minimum Variance Unbiased Scalar Variance-Covariance MatrixVariance-Covariance Matrix of the OLS EstimatorPROBLEMSE.1Let x t be the 1ϫ k vector of explanatory variables for observation t . Show that the OLS estimator ␤ˆcan be written as␤ˆϭΘ͚n tϭ1xt Јx t ΙϪ1Θ͚nt ϭ1xt Јy t Ι.Dividing each summation by n shows that ␤ˆis a function of sample averages.E.2Let ␤ˆbe the k ϫ 1 vector of OLS estimates.(i)Show that for any k ϫ 1 vector b ,we can write the sum of squaredresiduals asSSR(b ) ϭu ˆЈu ˆϩ(␤ˆϪb )ЈX ЈX (␤ˆϪb ).[Hint :Write (y Ϫ X b )Ј(y ϪX b ) ϭ[u ˆϩX (␤ˆϪb )]Ј[u ˆϩX (␤ˆϪb )]and use the fact that X Јu ˆϭ0.](ii)Explain how the expression for SSR(b ) in part (i) proves that ␤ˆuniquely minimizes SSR(b ) over all possible values of b ,assuming Xhas rank k .E.3Let ␤ˆbe the OLS estimate from the regression of y on X . Let A be a k ϫ k non-singular matrix and define z t ϵx t A ,t ϭ 1,…,n . Therefore,z t is 1ϫ k and is a non-singular linear combination of x t . Let Z be the n ϫ k matrix with rows z t . Let ␤˜denote the OLS estimate from a regression ofy on Z .(i)Show that ␤˜ϭA Ϫ1␤ˆ.(ii)Let y ˆt be the fitted values from the original regression and let y ˜t be thefitted values from regressing y on Z . Show that y ˜t ϭy ˆt ,for all t ϭ1,2,…,n . How do the residuals from the two regressions compare?(iii)Show that the estimated variance matrix for ␤˜is ␴ˆ2A Ϫ1(X ЈX )Ϫ1A Ϫ1؅,where ␴ˆ2is the usual variance estimate from regressing y on X .(iv)Let the ␤ˆj be the OLS estimates from regressing y t on 1,x t 2,…,x tk ,andlet the ␤˜j be the OLS estimates from the regression of yt on 1,a 2x t 2,…,a k x tk ,where a j 0,j ϭ 2,…,k . Use the results from part (i)to find the relationship between the ␤˜j and the ␤ˆj .(v)Assuming the setup of part (iv),use part (iii) to show that se(␤˜j ) ϭse(␤ˆj )/͉a j ͉.(vi)Assuming the setup of part (iv),show that the absolute values of the tstatistics for ␤˜j and ␤ˆj are identical.Appendix E The Linear Regression Model in Matrix Form 764。

伍德里奇计量经济学讲义8PPT课件

• What happens if we include variables in our specification that don’t belong? • There is no effect on our parameter estimate, and OLS remains unbiased • What if we exclude a variable from our specification that does belong? • OLS will usually be biased
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第18页/共28页
Variance of the at the sampling distribution of our estimate is centered around the true parameter
Want to think about how spread out this distribution is
第12页/共28页
Omitted Variable Bias (cont)
Recall thetruemodel,so that
yi b0 b1xi1 b2 xi2 ui , so t he
n um erat o rbeco m es
xi1 x1 b0 b1xi1 b2xi2 ui b1 xi1 x1 2 b2 xi1 x1 xi2 xi1 x1 ui
1
第1页/共28页
Interpreting Multiple Regression
yˆ bˆ0 bˆ1x1 bˆ2x2 ... bˆk xk , so yˆ bˆ1x1 bˆ2 x2 ... bˆk xk ,
so holdingx2,...,xk fixedimpliesthat

伍德里奇计量经济学课件 (1)

Ragnar Frisch Norway
Jan Tinbergen the Etherlands
The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel 1973 "for the development of the input-output method and for its application to important economic problems"
n n
近20位担任过世界计量经济学会会长 30余位左右在获奖成果中应用了计量经济学
17
计量经济学
若贝尔经济学奖获奖名单 2010彼得·戴蒙德和戴尔·莫滕森、克里斯托弗·皮萨里季斯失业 2009奥利弗·威廉森、艾利诺-奥斯特罗姆公共资源管理 2008 保罗-克鲁格曼国际贸易模式 2007赫维奇马斯金迈尔森机制设计 2006埃德蒙·费尔普斯通货膨胀与失业 2005罗伯特·奥曼和托马斯·谢林博弈论
n
8
计算机及软件
Eviews n Stata n S-plus n SAS n ☆R
n
9
教材及参考书
★Introductory Econometrics》(4E),Jeffrey M. Woodldridge, 2009（英文改编版《计量经济学导论》，已经由中国人民大学出版社2010年6月出版《Basic Econometrics》(fourth edition),Damodar N. Gujarrati，2003 《金融计量经济学》， Chris Brooks，西南财经大学出版社，2005 《经济计量分析》，William H.Greene，中国人民大学出版社 2007年《计量经济学（第3版）》，李子奈、潘文卿，高等教育出版社，2010年

伍德里奇计量课件 (19)

Economics 20 - Prof. Anderson 3
Using the Data
Create variables appropriate for analysis For example, create dummy variables from categorical variables, create hourly wages, etc. Check the data for missing values, errors, outliers, etc. Recode as necessary, be sure to report what you did
Economics 20 - Prof. Anderson 9
Interpreting Your Results
Keep theory in mind when interpreting results Be careful to keep ceteris paribus in mind Keep in mind potential problems with your estimates – be cautious drawing conclusions Can get an idea of the direction of bias due to omitted variables, measurement error or simultaneity
Economics 20 - Prof. Anderson 10
Further Issues
Some problems are just too hard to easily solve with available data May be able to approach the problem in several ways, but something wrong with each one Provide enough information for a reader to decide whether they find your results convincing or not

伍德里奇计量经济学导论ppt课件

l 等式右边的变量被称为解释变量（Explanaiory Variable）或自变量（Independent Variable）、右边变量、回归元，协变量，或控制变量。
l 等式y = b0 + b1x + u只有一个非常数回归元。我们称之为简单回归模型，两
变量回归模型或双变量回归模型.
ppt课件.
A simple wage equation
wage= 0 + 1 (years of education) + u 1 : if education increase by one year, how much more wage
will one gain.
上述简单工资函数描述了受教育年限和工资之间的关系, 1衡量
66
一、回归的含义
Ø 回归的历史含义 l F.加尔顿最先使用“回归（regression）”。 l 父母高，子女也高；父母矮，子女也矮。 l 给定父母的身高，子女平均身高趋向于“回归”到全体人口的平均身高。
ppt课件.

7
Ø 回归的现代释义
回归分析用于研究一个变量关于另一个（些）变量的具体依赖关系的计算方法和理论。
110 115 120 130 135 140
－ 6 750
200 220 240 260
120 136 140 144 145
－－ 5 685
135 137 140 152 157 160 162
7 1043
137 145 155 165 175 189
－ 6 966
150 152 175 178 180 185 191
60 — — 93 107 115 — — — —
65 74 — 95 110 120 — 140 — 175

伍德里奇计量课件 (16)

Economics 20 - Prof. Anderson 5
Identification of Demand Equation
w
D
S (z=z1) S (z=z2) S (z=z3)
h
Economics 20 - Prof. Anderson 6
Using IV to Estimate Demand
So, we can estimate the structural demand equation, using z as an instrument for w First stage equation is w = p0 + p1z + v2 Second stage equation is h = a2ŵ + u2 Thus, 2SLS provides a consistent estimator of a2, the slope of the demand curve We cannot estimate a1, the slope of the supply curve
Economics 20 - Prof. Anderson 10
Rank and Order Conditions
We refer to this as the rank condition Note that the exogenous variable excluded from the first equation must have a non-zero coefficient in the second equation for the rank condition to hold Note that the order condition clearly holds if the rank condition does – there will be an exogenous variable for the endogenous one

伍德里奇计量经济学课件 (17)

male | .0344839 .0107014 3.22 0.001 .0134881 .0554797
white | .0463804 .0150704 3.08 0.002 .0168127 .0759482
cigs | -.0052704 .001026 -5.14 0.000 -.0072834 -.0032573
n 无论Var(u|x) = Var(y|x)是否依赖于x，它们都可以一致地估计总体R平方。
Introductory Econometrics
10 of 75
为何关心异方差？
n 如果存在异方差，那么估计值的标准误差是有偏的。
n 如果标准误差有偏，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。
motheduc | -.0008691 .0024551 -0.35 0.723 -.0056859 .0039478
lfaminc | .0131714 .0083708 1.57 0.116 -.0032519 .0295947
_cons | 4.659946 .0377218 123.53 0.000 4.585937 4.733955
异方差存在时的方差
n V ar(bˆ j) 开平方被称为
n 异方差稳健的标准误差，或 n White标准误差，或 n Huber标准误差，或 n Eicker 标准误差
Introductory Econometrics
16 of 75
稳健标准误差
n 稳健标准误差可以用来进行推断。 n 有时可以将估计的方差乘以n/(n – k – 1)来
V ar(bˆ j )
(
rˆi2juˆi2 SSR j )2
,

伍德里奇计量经济学课件Chapte (6)

Only change will happen to the intercept. log(cyi)=log(c)+log(yi), the new intercept will be β0-log(c) log(dxi)=log(d)+log(xi), the new intercept will be β0+βjlog(d)
bwght =child birth weight, in ounces cigs =number of cigarette smoked by the mother while pregnant, per day faminc =annual family income, in thousands of dollars.
3

4
[Example]: Infant birth weight on cigarette smoking and family income (cont.)

[Example]: Infant birth weight on cigarette smoking and family income (cont.)
taking the natural log ― the most popular device in econometrics ― of x, y or both;
― ― ―

using quadratic forms of x; use interactions of x variables.
Occasional you’ll see reference to a “standardized coefficient” or “beta coefficient” which has a specific meaning. Idea is to replace y and each x variable with a standardized version – i.e. subtract mean and divide by standard deviation. Coefficient reflects how many y standard deviation of y change with respect to an one standard deviation change in x, holding other factors fixed. So we are measuring effects not in terms of the original units of y and x, but in standard deviation units.

伍德里奇计量经济学导论第六版英文课件

伍德里奇计量经济学导论第六版英文课件Woodridge's Introduction to Econometrics, 6th Edition, is a comprehensive textbook that covers the fundamentals of econometrics in a clear and concise manner. The accompanying courseware is designed to help students further understand and apply the concepts discussed in the book.The courseware includes PowerPoint slides, practice quizzes, and interactive exercises to enhance students' learning experience. The slides cover the key topics in each chapter, providing visual aids to help students grasp complex concepts. The quizzes allow students to test their understanding of the material and receive immediate feedback on their performance. The interactive exercises provide hands-on practice withreal-world data sets, helping students develop their econometric skills.In addition to the courseware, students have access to online resources such as supplementary readings, video tutorials, and self-assessment tools. These resources are designed to support students in their learning journey and provide additional assistance when needed.Overall, Woodridge's Introduction to Econometrics, 6th Edition, is a valuable resource for students studying econometrics. The comprehensive courseware offers a range of tools to support students in their learning, making it easier for them to understand and apply the concepts discussed in the textbook. With its clear explanations and practical exercises, this courseware is an essential companion for students looking to excel in econometrics.。

伍德里奇计量课件 (9)

b1s Cov x1 , u b1e1 ˆ plim b1 b1 b1 2 Var x1 s x* s e2
Economics 20 - Prof. Anderson 8
Proxy Variables (continued)
What do we need for for this solution to give us consistent estimates of b1 and b2? E(x3* | x1, x2, x3) = E(x3* | x3) = d0 + d3x3 That is, u is uncorrelated with x1, x2 and x3* and v3 is uncorrelated with x1, x2 and x3 So really running y = (b0 + b3d0) + b1x1+ b2x2 + b3d3x3 + (u + b3v3) and have just redefined intercept, error term x3 coefficient
Economics 20 - Prof. Anderson 13
Measurement Error in an Explanatory Variable
Define measurement error as e1 = x1 – x1* Assume E(e1) = 0 , E(y| x1*, x1) = E(y| x1*) Really estimating y = b0 + b1x1 + (u – b1e1) The effect of measurement error on OLS estimates depends on our assumption about the correlation between e1 and x1 Suppose Cov(x1, e1) = 0 OLS remains unbiased, variances larger

伍德里奇计量课件 (8)

Economics 20 - Prof. Anderson 10
The Breusch-Pagan Test
Don’t observe the error, but can estimate it with the residuals from the OLS regression After regressing the residuals squared on all of the x’s, can use the R2 to form an F or LM test The F statistic is just the reported F statistic for overall significance of the regression, F = [R2/k]/[(1 – R2)/(n – k – 1)], which is distributed Fk,
Economics 20 - Prof. Anderson 8
A Robust LM Statistic
Run OLS on the restricted model and save the residuals ŭ Regress each of the excluded variables on all of the included variables (q different regressions) and save each set of residuals ř1, ř2, …, řq Regress a variable defined to be = 1 on ř1 ŭ, ř2 ŭ, …, řq ŭ, with no intercept The LM statistic is n – SSR1, where SSR1 is the sum of squared residuals from this final regression

伍德里奇计量经济学课件 (14)

ˆ b 1
Introductory Econometrics 15 of 54
证明一致性
Because as n , n 1 xi1 x1 ui 0 n
1
xi1 x1
2
does not converge to zero,
2
ˆ b plimb 1 1
Introductory Econometrics
17 of 54
一个更弱的假定
n n n
要获得估计量的无偏性，我们假定零条件期望 – E(u|x1, x2,…,xk) = 0 而要获得估计量的一致性，我们可以使用更弱的假定：零期望和零相关性假定。如果这个较弱的假定也不成立，OLS将是有偏而且不一致的。
Introductory Econometrics
18 of 54
推导不一致性
n
b ，并考虑下面定义渐近偏差为：plimb 1 1 的真实模型和待估计模型。
True model: y b 0 b1 x1 b 2 x2 v u b 2 x2 v and then, b b plimb
n
Introductory Econometrics
5 of 54
为什么考虑一致性
n
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。
Introductory Econometrics
6 of 54
什么是一致性
令 W n 是基于样本 y1 , y2 ,..., yn 的关于的估计量。如果对于任何 >0 ，当 n 时 Pr(|Wn | ) 0

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