北大計量經(jīng)濟(jì)學(xué)講義-第二講課件

1、The Simple Regression Model (1)簡單二元回歸y = b0 + b1x + u1Intermediate Econometrics Yan ShenChapter Outline 本章大綱Definition of the Simple Regression Model 簡單回歸模型的定義Deriving the Ordinary Least Squares Estimates 普通最小二乘法的推導(dǎo)Mechanics of OLS OLS的操作技巧Units of Measurement and Functional Form測量單位和函數(shù)形式Expected Va

2、lues and Variances of the OLS estimators OLS估計量的期望值和方差Regression through the Origin 過原點回歸2Intermediate Econometrics Yan ShenLecture Outline 講義大綱Some Terminology 一些術(shù)語的注解A Simple Assumption 一個簡單假定Zero Conditional Mean Assumption 條件期望零值假定 What is Ordinary Least Squares 何為普通最小二乘法Deriving OLS Estimates 普

3、通最小二乘法的推導(dǎo)3Intermediate Econometrics Yan ShenSome Terminology 術(shù)語注解 In the simple linear regression model, where y = b0 + b1x + u, we typically refer to y as theDependent Variable, orLeft-Hand Side Variable, orExplained Variable, orRegressand在簡單二元回歸模型y = b0 + b1x + u中， y通常被稱為因變量，左邊變量，被解釋變量，或回歸子。4Inter

4、mediate Econometrics Yan ShenSome Terminology術(shù)語注解 In the simple linear regression of y on x, we typically refer to x as theIndependent Variable, orRight-Hand Side Variable, orExplanatory Variable, orRegressor, orCovariate, orControl Variables在y 對 x進(jìn)行回歸的簡單二元回歸模型中， x通常被稱為自變量，右邊變量，解釋變量，回歸元，協(xié)變量，或控制變量。5I

5、ntermediate Econometrics Yan ShenSome Terminology術(shù)語注解Equation y = b0 + b1x + u has only one nonconstant regressor x, it is called a simple linear regression model, or two-variables regression model, or bivariate linear regression model. 等式y(tǒng) = b0 + b1x + u只有一個非常數(shù)回歸元。我們稱之為簡單回歸模型， 兩變量回歸模型或雙變量回歸模型.6Inte

6、rmediate Econometrics Yan ShenSome Terminology術(shù)語注解The coefficients b0 , b1 are called the regression coefficients. b0 is also called the constant term or the intercept term, or intercept parameter. b1 represents the marginal effects of the regressor, x. It is also called the slope parameter.b0 , b1被

7、稱為回歸系數(shù)。 b0也被稱為常數(shù)項或截矩項，或截矩參數(shù)。 b1代表了回歸元x的邊際效果，也被成為斜率參數(shù)。7Intermediate Econometrics Yan ShenSome Terminology術(shù)語注解 The variable u is called the error term or disturbance in the relationship. It represents factors other than x that can affect y. u 為誤差項或擾動項，它代表了除了x之外可以影響y的因素。8Intermediate Econometrics Yan S

8、henSome Terminology術(shù)語注解Meaning of linear: linear means linear in parameters, not necessarily mean that y and x must have a linear relationship.There are many cases that y and x have nonlinear relationship, but after some transformation, they are linear in parameters.For example, y=eb0+b1x+u .線性的含義：

9、y 和x 之間并不一定存在線性關(guān)系，但是，只要通過轉(zhuǎn)換可以使y的轉(zhuǎn)換形式和x的轉(zhuǎn)換形式存在相對于參數(shù)的線性關(guān)系，該模型即稱為線性模型。9Intermediate Econometrics Yan ShenExamples 簡單二元回歸模型例子A simple wage equationwage= b0 + b1(years of education) + ub1 : if education increase by one year, how much more wage will one gain.上述簡單工資函數(shù)描述了受教育年限和工資之間的關(guān)系, b1 衡量了多接受一年教育工資可以增加多少

10、.10Intermediate Econometrics Yan ShenA Simple Assumption關(guān)于u的假定 The average value of u, the error term, in the population is 0. That is, E(u) = 0(2.5) It it restrictive?我們假定總體中誤差項u的平均值為零. 該假定是否具有很大的限制性呢?11Intermediate Econometrics Yan ShenA Simple Assumption關(guān)于u的假定If for example, E(u)=5. Then y = (b0

11、+5)+ b1x + (u-5),therefore, E(u)=E(u-5)=0.This is not a restrictive assumption, since we can always use b0 to normalize E(u) to 0.上述推導(dǎo)說明我們總可以通過調(diào)整常數(shù)項來實現(xiàn)誤差項的均值為零, 因此該假定的限制性不大.12Intermediate Econometrics Yan ShenZero Conditional Mean Assumption 條件期望零值假定 We need to make a crucial assumption about how u

12、and x are related We want it to be the case that knowing something about x does not give us any information about u, so that they are completely unrelated. That isE(u|x) = E(u)。我們需要對u和 x之間的關(guān)系做一個關(guān)鍵假定。理想狀況是對x的了解并不增加對u的任何信息。換句話說，我們需要u和 x完全不相關(guān)。13Intermediate Econometrics Yan ShenZero Conditional Mean As

13、sumption 條件期望零值假定 Since we have assumed E(u) = 0, therefore, E(u|x) = E(u) = 0. (2.6)What does it mean?由于我們已經(jīng)假定了E(u) = 0，因此有E(u|x) = E(u) = 0。該假定是何含義？14Intermediate Econometrics Yan ShenZero Conditional Mean Assumption 條件期望零值假定 In the example of education, suppose u represents innate ability, zero c

14、onditional mean assumption meansE(ability|edu=6)=E(ability|edu=18)=0.The average level of ability is the same regardless of years of education.在教育一例中，假定u 代表內(nèi)在能力，條件期望零值假定說明不管解釋教育的年限如何，該能力的平均值相同。 15Intermediate Econometrics Yan ShenZero Conditional Mean Assumption 條件期望零值假定 Question: Suppose that a sco

15、re on a final exam, score, depends on classes attended (attend) and unobserved factors that affect exam performance (such as student ability). Then consider model score =b0 + b1attend +uWhen would you expect it satisfy (2.6)?假設(shè)期末成績分?jǐn)?shù)取決于出勤次數(shù)和影響學(xué)生現(xiàn)場發(fā)揮的因素，如學(xué)生個人素質(zhì)。那么上述模型中假設(shè)（2.6）何時能夠成立？16Intermediate Eco

16、nometrics Yan ShenZero Conditional Mean Assumption 條件期望零值假定 (2.6) implies the population regression function, E(y|x) , satisfies E(y|x) = b0 + b1x.E(y|x) as a linear function of x, where for any x the distribution of y is centered about E(y|x).(2.6)說明總體回歸函數(shù)應(yīng)滿足E(y|x) = b0 + b1x。該函數(shù)是x的線性函數(shù)，y的分布以它為中心。1

17、7Intermediate Econometrics Yan Shen.x1=5x2 =10E(y|x) = b0 + b1xyf(y)給定x時y的條件分布18Intermediate Econometrics Yan ShenDeriving the Ordinary Least Squares Estimates 普通最小二乘法的推導(dǎo) Basic idea of regression is to estimate the population parameters from a sample Let (xi,yi): i=1, ,n denote a random sample of si

18、ze n from the population For each observation in this sample, it will be the case that yi = b0 + b1xi + ui回歸的基本思想是從樣本去估計總體參數(shù)。 我們用(xi,yi): i=1, ,n 來表示一個隨機樣本，并假定每一觀測值滿足yi = b0 + b1xi + ui。19Intermediate Econometrics Yan Shen.y4y1y2y3x1x2x3x4u1u2u3u4xyPopulation regression line, sample data pointsand t

19、he associated error terms總體回歸線，樣本觀察點和相應(yīng)誤差E(y|x) = b0 + b1x20Intermediate Econometrics Yan ShenDeriving OLS Estimates普通最小二乘法的推導(dǎo) To derive the OLS estimator we need to realize that our main assumption of E(u|x) = E(u) = 0 also implies that Cov(x,u) = E(xu) = 0 Why? Remember from basic probability that

20、 Cov(X,Y) = E(XY) E(X)E(Y)由E(u|x) = E(u) = 0 可得Cov(x,u) = E(xu) = 0 。21Intermediate Econometrics Yan ShenDeriving OLS continued普通最小二乘法的推導(dǎo) We can write our 2 restrictions just in terms of x, y, b0 and b1 , since u = y b0 b1x E(y b0 b1x) = 0 Ex(y b0 b1x) = 0These are called moment restrictions可將u = y

21、b0 b1x代入以得上述兩個矩條件。22Intermediate Econometrics Yan ShenDeriving OLS using M.O.M.使用矩方法推導(dǎo)普通最小二乘法 The method of moments approach to estimation implies imposing the population moment restrictions on the sample moments。矩方法是將總體的矩限制應(yīng)用于樣本中。23Intermediate Econometrics Yan ShenDerivation of OLS普通最小二乘法的推導(dǎo) We wa

22、nt to choose values of the parameters that will ensure that the sample versions of our moment restrictions are true目標(biāo)是通過選擇參數(shù)值，使得在樣本中矩條件也可以成立。 The sample versions are as follows:24Intermediate Econometrics Yan ShenDerivation of OLS普通最小二乘法的推導(dǎo)Given the definition of a sample mean, and properties of sum

23、mation, we can rewrite the first condition as follows根據(jù)樣本均值的定義以及加總的性質(zhì)，可將第一個條件寫為25Intermediate Econometrics Yan ShenDerivation of OLS普通最小二乘法的推導(dǎo)26Intermediate Econometrics Yan ShenSo the OLS estimated slope is因此OLS估計出的斜率為27Intermediate Econometrics Yan ShenSummary of OLS slope estimateOLS斜率估計法總結(jié) The s

24、lope estimate is the sample covariance between x and y divided by the sample variance of x. If x and y are positively correlated, the slope will be positive. If x and y are negatively correlated, the slope will be negative. Only need x to vary in our sample.斜率估計量等于樣本中x 和 y 的協(xié)方差除以x的方差。若x 和 y 正相關(guān)則斜率為正

25、，反之為負(fù)。28Intermediate Econometrics Yan ShenMore OLS 關(guān)于OLS的更多信息 Intuitively, OLS is fitting a line through the sample points such that the sum of squared residuals is as small as possible, hence the term least squares。 The residual, , is an estimate of the error term, u, and is the difference between

26、the fitted line (sample regression function) and the sample point。OLS法是要找到一條直線，使殘差平方和最小。殘差是對誤差項的估計，因此，它是擬合直線（樣本回歸函數(shù)）和樣本點之間的距離。29Intermediate Econometrics Yan Shen.y4y1y2y3x1x2x3x41234xySample regression line, sample data pointsand the associated estimated error terms 樣本回歸線，樣本數(shù)據(jù)點和相關(guān)的誤差估計項30Intermedia

27、te Econometrics Yan ShenAlternate approach to derivation推導(dǎo)方法二 Given the intuitive idea of fitting a line, we can set up a formal minimization problem That is, we want to choose our parameters such that we minimize the following:正式解一個最小化問題，即通過選取參數(shù)而使下列值最?。?1Intermediate Econometrics Yan ShenAlternate

28、approach, continued推導(dǎo)方法二 If one uses calculus to solve the minimization problem for the two parameters you obtain the following first order conditions, which are the same as we obtained before, multiplied by n如果直接解上述方程我們得到下面兩式，這兩個式子等于前面兩式乘以n32Intermediate Econometrics Yan ShenLecture Summary 講義總結(jié)Int

29、roduce the simple linear regression model.Introduce the method of ordinary least squares to estimate the slope and intercept parameters using data from a random sample.介紹簡單線性回歸模型介紹通過隨機樣本的數(shù)據(jù)運用普通最小二乘法估計斜率和截距的參數(shù)值33Intermediate Econometrics Yan ShenThe Simple Regression Model (2)簡單二元回歸y = b0 + b1x + u34

30、Intermediate Econometrics Yan ShenChapter Outline 本章大綱Definition of the Simple Regression Model 簡單回歸模型的定義Deriving the Ordinary Least Squares Estimates 推導(dǎo)普通最小二乘法的估計量Mechanics of OLS OLS的操作技巧Unites of Measurement and Functional Form 測量單位和回歸方程形式Expected Values and Variances of the OLS estimators OLS估計量

31、的期望值和方差Regression through the Origin 過原點的回歸35Intermediate Econometrics Yan ShenLecture Outline 講義大綱Algebraic Properties of OLS OLS的代數(shù)特性Goodness of fit 擬合優(yōu)度Using Stata for OLS regression使用stata做OLS 回歸Effects of Changing Units in Measurement on OLS Statistics改變測量單位對OLS統(tǒng)計量的效果36Intermediate Econometrics

32、 Yan Shen Mechanics of OLS OLS的操作技巧Example: CEO Salary and Return on Equity 例：CEO的薪水和資本權(quán)益報酬率37Intermediate Econometrics Yan ShenExample: CEO Salary and Return on Equity 例：CEO的薪水和資本權(quán)益報酬率Salary: annual salary measured in $1000. In the 1990 data above, (min, mean, max)=(223, 1281, 14822).變量salary衡量了已10

33、00美元為單位的年薪，其最小值，均值和最大值分別如上。Roe: net income/common equity, three-year average，（0.5, 17.18,56.3)Roe凈收入/所有者權(quán)益，為三年平均值。N=209. The estimated relation(estimated salary)=963.191 + 18.501roe.38Intermediate Econometrics Yan ShenExample: CEO Salary and Return on Equity 例：CEO的薪水和資本權(quán)益報酬率Interpretation:對估計量的解釋：96

34、3.19: The salary that the CEO will get when roe=0.常數(shù)項的估計值衡量了當(dāng)roe為零時CEO的薪水。18.5: If ROE increases by one percentage point, then salary is going to increase by 18.5, i.e., $18,500.b1 的估計值反應(yīng)了ROE若增加一個百分點工資將增加18500美元。If roe=30, what is the estimated salary?39Intermediate Econometrics Yan ShenAlgebraic Pr

35、operties of OLS OLS的代數(shù)性質(zhì) The sum of the OLS residuals is zero OLS 殘差和為零 （p24) Thus, the sample average of the OLS residuals is zero as well 因此 OLS 的樣本殘差平均值也為零.40Intermediate Econometrics Yan ShenAlgebraic Properties of OLS OLS的代數(shù)性質(zhì)The sample covariance between the regressors and the OLS residuals is

36、 zero回歸元（解釋變量）和OLS殘差之間的樣本協(xié)方差為零 (p25)41Intermediate Econometrics Yan ShenAlgebraic Properties of OLS OLS的代數(shù)性質(zhì)The OLS regression line always goes through the mean of the sample.OLS回歸線總是通過樣本的均值。42Intermediate Econometrics Yan ShenAlgebraic Properties of OLS OLS的代數(shù)性質(zhì)We can think of each observation as b

37、eing made up of an explained part, and an unexplained part, 我們可把每一次觀測看作由被解釋部分和未解釋部分構(gòu)成.Then the fitted values and residuals are uncorrelated in the sample. 預(yù)測值和殘差在樣本中是不相關(guān)的43Intermediate Econometrics Yan ShenAlgebraic Properties of OLS OLS的代數(shù)性質(zhì)44Intermediate Econometrics Yan ShenMore Terminology更多術(shù)語De

38、fine the total sum of square as 定義總平方和為45Intermediate Econometrics Yan ShenMore Terminology更多術(shù)語SST is a measure of the total sample variation in the ys; that is, it measures how spread out the ys are in the sample. 總平方和是對y在樣本中所有變動的度量，即它度量了y在樣本中的分散程度If we divide SST by n-1, we obtain the sample varia

39、nce of y.將總平方和除以n-1,我們得到y(tǒng)的樣本方差。46Intermediate Econometrics Yan ShenMore Terminology更多術(shù)語Explained Sum of Squares (SSE)is defined as 解釋平方和定義為It measures the sample variation in the predicted value of ys. 它度量了y的預(yù)測值的在樣本中的變動47Intermediate Econometrics Yan ShenMore Terminology更多術(shù)語Residual Sum of Squares i

40、s defined as 殘差平方和定義為SSR measures the sample variation in the residuals.殘差平方和度量了殘差的樣本變異48Intermediate Econometrics Yan ShenSST, SSR and SSEThe total variation in y can always be expressed as the sum of the explained variation SSE and the unexplained variation SSR, i.e.y 的總變動可以表示為已解釋的變動SSE和 未解釋的變動SSR

41、之和，即SST=SSE+SSR49Intermediate Econometrics Yan ShenProof that SST = SSE + SSR證明 SST = SSE + SSR50Intermediate Econometrics Yan ShenProof that SST = SSE + SSRTherefore, SST = SSE + SSR.We have used the fact that the fitted value and residuals are uncorrelated in the sample.該證明中我們使用了一個事實, 即樣本中因變量的擬合值和

42、殘差不相關(guān).51Intermediate Econometrics Yan ShenGoodness-of-Fit擬合優(yōu)度 How do we think about how well our sample regression line fits our sample data?我們?nèi)绾魏饬繕颖净貧w線是否很好地擬合了樣本數(shù)據(jù)呢? Can compute the fraction of the total sum of squares (SST) that is explained by the model, call this the R-squared of regression可以計算模

43、型解釋的總平方和的比例，并把它定義為回歸的R-平方 R2 = SSE/SST = 1 SSR/SST52Intermediate Econometrics Yan ShenGoodness-of-Fit擬合優(yōu)度R-squared is the ratio of the explained variation compared to the total variation. R-平方是已解釋的變動占所有變動的比例It is thus interpreted as the fraction of the sample variation in y that is explained by x. 它

44、因此可被看作是y的樣本變動中被可以被x解釋的部分The value of R-squared is always between zero and one.R-平方的值總是在0和1之間53Intermediate Econometrics Yan ShenGoodness-of-Fit擬合優(yōu)度In the social sciences, low R-squareds in regression equations are not uncommon, especially for cross-sectional analysis. 在社會科學(xué)中，特別是在截面數(shù)據(jù)分析中, 回歸方程得到低的R-平

45、方值并不罕見。It is worth emphasizing that a seemingly low R-squared does not necessarily mean that an OLS regression equation is useless.值得強調(diào)的是表面上低的R-平方值不一定說明OLS回歸方程是沒有價值的54Intermediate Econometrics Yan ShenGoodness-of-Fit擬合優(yōu)度Example 2.8CEO Salary and Return on EquityCEO薪水和凈資產(chǎn)回報Example 2.9Voting outcomes

46、and Campaign Expenditures競選結(jié)果和選舉活動開支55Intermediate Econometrics Yan ShenUsing Stata for OLS regressions使用 Stata 進(jìn)行OLS回歸 Now that weve derived the formula for calculating the OLS estimates of our parameters, youll be happy to know you dont have to compute them by hand我們已經(jīng)推導(dǎo)出公式計算參數(shù)的OLS估計值，所幸的是我們不必親手去計

47、算它們。 Regressions in Stata are very simple, to run the regression of y on x, just type在Stata中進(jìn)行回歸非常簡單，要讓y對x進(jìn)行回歸，只需要輸入 reg y x56Intermediate Econometrics Yan ShenUnits of Measurement 測量單位Suppose the salary is measured in hundreds of dollars, rather than in thousands of dollars, say salarhun. 假定薪水的單位是美

48、元，而不是千美元，salarys.What will be the OLS intercept and slope estimates in the regression of salarhun on roe? 在Salarys對roe進(jìn)行回歸時OLS截距和斜率的估計值是多少？57Intermediate Econometrics Yan ShenUnits of Measurement 測量單位The estimated sample regression is changed from (estimated salarys)=963.191 + 18.501roeto (estimated

49、 salarys)=963191 + 18501roeIn general, when the dependent variable is multiplied by the constant c, but nothing has changed for the independent variable, the OLS intercept and slope estimates are also multiplied by c. 一般而言，當(dāng)因變量乘上常數(shù)c，而自變量不改變時，OLS的截距和斜率估計量也要乘上c。58Intermediate Econometrics Yan ShenUnit

50、s of Measurement 測量單位If redefine roedec = roe/100, then the sample regression line will change from 如果定義 roedec = roe/100，那么樣本回歸線將會從(estimated salary)=963.191 + 18.501roe to 改變到 (estimated salary)=963.191 + 1850.1roedecIn general, if the independent variable is divided or multiplied by some nonzero

51、constant, c, then the OLS slope coefficient is multiplied or divided by c, but the intercept will not change. 一般而言，如果自變量除以或乘上某個非零常數(shù)，c，那么OLS斜率將乘以或除以c，而截距則不改變。59Intermediate Econometrics Yan ShenIncorporating Nonlinearrities in Simple Regression在簡單回歸中加入非線性Linear relationships are not general enough fo

52、r all economic applications.線性關(guān)系并不適合所有的經(jīng)濟(jì)學(xué)運用 However, it is rather easy to incorporate many nonlinearities into simple regression analysis by appropriately defining the dependent and independent variables. 然而，通過對因變量和自變量進(jìn)行恰當(dāng)?shù)亩x, 我們可以在簡單回歸分析中非常容易地處理許多y和x之間的非線性關(guān)系.60Intermediate Econometrics Yan ShenThe

53、 Natural Logarithm自然對數(shù) 61Intermediate Econometrics Yan ShenIn the wage-education example, now suppose the percentage increase in wage is the same, given one more year of education. 在工資-教育的例子中，假定每增加一年的教育，工資的百分比增長都是相同的A model that gives a constant percentage effect is 能夠給出不變的百分比效果的模型是If , we have62Int

54、ermediate Econometrics Yan ShenExample 2.10A log Wage Equation將對數(shù)工資方程Compared to 和該方程相比63Intermediate Econometrics Yan ShenAnother important use of the natural log is in obtaining a constant elasticity model自然對數(shù)的另一個重要用途是用于獲得彈性為常數(shù)的模型In the example of CEO Salary and Firm Sales, a constant elasticity m

55、odel is在CEO的薪水和企業(yè)銷售額的例子中，常數(shù)彈性模型是64Intermediate Econometrics Yan ShenCombination of functional forms available from using either the original variable or its natural log 變量的原始形式和其自然對數(shù)的不同組合 65Intermediate Econometrics Yan ShenThe Simple Regression Model 簡單二元回歸 (3)y = b0 + b1x + u66Intermediate Econome

56、trics Yan ShenChapter Outline本章大綱 Definition of the Simple Regression Model 二元回歸模型的定義Deriving the Ordinary Least Squares Estimates 推導(dǎo)普通最小二乘法的估計量Mechanics of OLS OLS的操作技巧Unites of Measurement and Functional Form 測量單位和函數(shù)形式Expected Values and Variances of the OLS estimators OLS估計量的期望值和方差Regression thro

57、ugh the Origin 過原點回歸67Intermediate Econometrics Yan ShenExpected Values and Variances of the OLS EstimatorsOLS估計量的期望值和方差We will study the properties of the distributions of OLS estimators over different random samples from the population. 從總體中抽取的不同的隨機樣本可得到不同的OLS估計量，我們將研究這些OLS估計量的分布。We begin by estab

58、lishing the unbiasedness of OLS under a set of assumptions.首先，我們在一些假定下證明OLS的無偏性。68Intermediate Econometrics Yan ShenAssumption SLR.1 (Linear in Parameters):假定SLR.1 （關(guān)于參數(shù)是線性的）In the population model, the dependent variable y is related to the independent variable x and the error u as 在總體模型中，因變量 y 和自變

59、量 x 和殘差 u 的關(guān)系可寫作y = b0 + b1x + u , where b0 and b1 are the population intercept and slope parameters respectively. 其中 b0 和 b1 分別是總體的截距參數(shù)和斜率參數(shù)69Intermediate Econometrics Yan ShenAssumption SLR.2 (Random Sampling):假定SLR.2 (隨機抽樣):Assume we can use a random sample of size n, (xi, yi): i=1, 2, , n, from

60、the population model. Thus we can write the sample model yi = b0 + b1xi + ui 假定我們從總體模型隨機抽取容量為n的樣本, (xi, yi): i=1, 2, , n, 那么可以寫出樣本模型為 yi = b0 + b1xi + ui70Intermediate Econometrics Yan ShenAssumptions SLR.3 and SLR.4假定 SLR.3 和 SLR.4 SLR.3 , Zero Conditional Mean: SLR.3, 零條件期望：Assume E(u|x) = 0 and t