北大計(jì)量經(jīng)濟(jì)學(xué)講義-第三講.ppt

1、Intermediate Econometrics, Yan Shen,1,Multiple Regression Analysis: Estimation(1)多元回歸分析：估計(jì)（1）,y = b0 + b1x1 + b2x2 + . . . bkxk + u,Intermediate Econometrics, Yan Shen,2,Chapter Outline 本章大綱,Motivation for Multiple Regression 使用多元回歸的動(dòng)因 Mechanics and Interpretation of Ordinary Least Squares 普通最小二乘法的操

2、作和解釋 The Expected Values of the OLS Estimators OLS估計(jì)量的期望值 The Variance of the OLS Estimators OLS估計(jì)量的方差 Efficiency of OLS: The Gauss-Markov Theorem OLS的有效性：高斯馬爾科夫定理,Intermediate Econometrics, Yan Shen,3,Lecture Outline 課堂大綱,Motivation for multivariate Analysis 使用多元回歸的動(dòng)因 The Model 模型 The Estimation 估計(jì)

3、 Properties of the OLS estimates OLS估計(jì)的性質(zhì) The Partialling out Interpretation 對(duì)“排除其它變量影響”的解釋 Simple versus multiple regressions 比較簡(jiǎn)單回歸模型與多元回歸模型 Goodness of Fit 擬合優(yōu)度,Intermediate Econometrics, Yan Shen,4,Motivation: Advantage 動(dòng)因：優(yōu)點(diǎn),The primary drawback of the simple regression analysis for empirical w

4、ork is that it is very difficult to draw ceteris paribus conclusions about how x affects y. 在實(shí)證工作中使用簡(jiǎn)單回歸模型的主要缺陷是：要得到在其它條件不變的情況下， x對(duì)y的影響非常困難。 Whether the ceteris paribus effects are reliable or not depends on whether the conditional mean assumption is realistic. 在其它條件不變情況假定下我們估計(jì)出的x對(duì)y的影響值是否可信依賴，完全取決于條

5、件均值零值假設(shè)是否現(xiàn)實(shí)。 If other factors that affecting y are not correlated with x, changing x can ensure that u is not changed, and the effect of x on y can be identified. 如果影響y的其它因素與x不相關(guān)，則改變x可以保證u不變，從而x對(duì)y的影響可以被識(shí)別出來(lái)。,Intermediate Econometrics, Yan Shen,5,Motivation : Advantage動(dòng)因：優(yōu)點(diǎn),Multiple regression analys

6、is is more amenable to ceteris paribus analysis because it allows us to explicitly control for many other factors that simultaneously affect the dependent variable. 多元回歸分析更適合于其它條件不變情況下的分析，因?yàn)槎嘣貧w分析允許我們明確地控制許多其它也同時(shí)影響因變量的因素。 Multiple regression models can accommodate many explanatory variables that may

7、 be correlated. 多元回歸模型能容許很多解釋變量，而這些變量可以是相關(guān)的。 Important for drawing inference about causal relations between y and explanatory variables when using non-experimental data. 在使用非實(shí)驗(yàn)數(shù)據(jù)時(shí)，多元回歸模型對(duì)推斷y與解釋變量間的因果關(guān)系很重要。,Intermediate Econometrics, Yan Shen,6,Motivation : Advantage動(dòng)因：優(yōu)點(diǎn),It can explain more of the v

8、ariation in the dependent variable. 它可以解釋更多的因變量變動(dòng)。 It can incorporate more general functional form. 它可以表現(xiàn)更一般的函數(shù)形式。 The multiple regression model is the most widely used vehicle for empirical analysis. 多元回歸模型是實(shí)證分析中最廣泛使用的工具。,Intermediate Econometrics, Yan Shen,7,Motivation: An Example動(dòng)因：一個(gè)例子,Consider

9、a simple version of the wage equation for obtaining the effect of education on hourly wage: 考慮一個(gè)簡(jiǎn)單版本的解釋教育對(duì)小時(shí)工資影響的工資方程。 exper: years of labor market experience exper：在勞動(dòng)力市場(chǎng)上的經(jīng)歷，用年衡量 In this example experience is explicitly taken out of the error term. 在這個(gè)例子中，“在勞動(dòng)力市場(chǎng)上的經(jīng)歷”被明確地從誤差項(xiàng)中提出。,Intermediate Econ

10、ometrics, Yan Shen,8,Motivation: An Example動(dòng)因：一個(gè)例子,Consider a model that says family consumption is a quadratic function of family income: 考慮一個(gè)模型：家庭消費(fèi)是家庭收入的二次方程。 Cons = b0 + b1 inc+b2 inc2 +u Now the marginal propensity to consume is approximated by 現(xiàn)在，邊際消費(fèi)傾向可以近似為 MPC= b1 +2b2,Intermediate Econometr

11、ics, Yan Shen,9,The Model with k Independent Variables含有k個(gè)自變量的模型,The general multiple linear regression model can be written as 一般的多元線性回歸模型可以寫(xiě)為,Intermediate Econometrics, Yan Shen,10,Parallels with Simple Regression類似于簡(jiǎn)單回歸模型,b0 is still the intercept b0仍是截距 b1 to bk all called slope parameters b1到bk

12、都稱為斜率參數(shù) u is still the error term (or disturbance) u仍是誤差項(xiàng)（或干擾項(xiàng)） Still need to make a zero conditional mean assumption, so now assume that 仍需作零條件期望的假設(shè)，所以現(xiàn)在假設(shè) E(u|x1,x2, ,xk) = 0 Still minimizing the sum of squared residuals, so have k+1 first order conditions 仍然最小化殘差平方和，所以得到k+1個(gè)一階條件,Intermediate Econ

13、ometrics, Yan Shen,11,Obtaining the OLS Estimates如何得到OLS估計(jì)值,The method of ordinary least squares chooses the estimates to minimize the sum of squared residuals, 普通最小二乘法選擇能最小化殘差平方和的估計(jì)值，,Intermediate Econometrics, Yan Shen,12,Obtaining the OLS Estimates如何得到OLS估計(jì)值,Intermediate Econometrics, Yan Shen,13

14、,Obtaining the OLS Estimates如何得到OLS估計(jì)值,The first order conditions are also the sample counterparts of the related population moments. 一階條件也是相關(guān)的總體矩在樣本中的對(duì)應(yīng)。 After estimation we obtain the OLS regression line, or the sample regression function (SRF) 在估計(jì)之后，我們得到OLS回歸線，或稱為樣本回歸方程（SRF）,Intermediate Economet

15、rics, Yan Shen,14,Interpreting Multiple Regression對(duì)多元回歸的解釋,Intermediate Econometrics, Yan Shen,15,Example: Determinants of College GPA例子：大學(xué)GPA的決定因素,Two-independent-variable regression 兩個(gè)解釋變量的回歸 pcolGPA: predicted values of college grade point average pcolGPA:大學(xué)績(jī)點(diǎn)預(yù)測(cè)值 hsGPA : high school GPA hsGPA : 高

16、中績(jī)點(diǎn) ACT : achievement test score ACT :成績(jī)測(cè)驗(yàn)分?jǐn)?shù) pcolGPA = 1.29 + 0.453hsGPA+0.0094ACT,Intermediate Econometrics, Yan Shen,16,Example: Determinants of College GPA例子：大學(xué)GPA的決定因素,One-independent-variable regression 一個(gè)解釋變量的回歸 pcolGPA = 2.4 +0.0271ACT The coefficients on ACT is three times larger. ACT的系數(shù)大三倍。

17、 If these two regressions were both true, they can be considered as the results of two different experiments. 如果這兩個(gè)回歸都是對(duì)的，它們可以被認(rèn)為是兩個(gè)不同實(shí)驗(yàn)的結(jié)果。,Intermediate Econometrics, Yan Shen,17,Holding other factors fixed“保持其它因素不變”的含義,The power of multiple regression analysis is that it allows us to do in non-exp

18、erimental environments what natural scientists are able to do in a controlled laboratory setting: keep other factors fixed. 多元回歸分析的優(yōu)勢(shì)在于它使我們能在非實(shí)驗(yàn)環(huán)境中去做自然科學(xué)家在受控實(shí)驗(yàn)中所能做的事情：保持其它因素不變。,Intermediate Econometrics, Yan Shen,18,Properties 性質(zhì),The sample average of the residuals is zero. 殘差項(xiàng)的樣本平均值為零 The sample co

19、variance between each independent variable and the OSL residuals is zero. 每個(gè)自變量和OLS協(xié)殘差之間的樣本協(xié)方差為零。 The point is always on the OLS regression line. 點(diǎn) 總位于OLS回歸線上。,Intermediate Econometrics, Yan Shen,19,A “Partialling Out” Interpretation對(duì)“排除其它變量影響”的解釋,Consider regression line of 考慮回歸線 One way to express

20、 is 的一種表達(dá)是 is obtained in the following way: 由以下方式得出：,Intermediate Econometrics, Yan Shen,20,A “Partialling Out” Interpretation 對(duì)“排除其它變量影響”的解釋,Regress our first independent variable x1 on our second independent variable x2 , and then obtain the residual . 將第一個(gè)自變量對(duì)第二個(gè)自變量進(jìn)行回歸，然后得到殘差 。 In other words,

21、is the residual from the regression 換句話說(shuō)， 是由回歸 得到的殘差。 Then, do a simple regression of y on to obtain . 然后，將y向 進(jìn)行簡(jiǎn)單回歸得到 。,Intermediate Econometrics, Yan Shen,21,“Partialling Out” continued“排除其它變量影響”（續(xù)）,Previous equation implies that regressing y on x1 and x2 gives same effect of x1 as regressing y on

22、 residuals from a regression of x1 on x2 上述方程意味著：將y同時(shí)對(duì)x1和x2回歸得出的x1的影響與先將x1對(duì)x2回歸得到殘差，再將y對(duì)此殘差回歸得到的x1的影響相同。 This means only the part of x1 that is uncorrelated with x2 are being related to y , so were estimating the effect of x1 on y after x2 has been “partialled out” 這意味著只有x1中與x2不相關(guān)的部分與y有關(guān)，所以在x2被“排除影響

23、”之后，我們?cè)俟烙?jì)x1對(duì)y的影響。,Intermediate Econometrics, Yan Shen,22,“Partialling Out” continued“排除其它變量影響”（續(xù)）,In the general model with k explanatory variables, can still be written as in equation , but the residual comes from the regression of x1 on x2 , xk. 在一個(gè)含有k個(gè)解釋變量的一般模型中， 仍然可以寫(xiě)成 ，但殘差 來(lái)自x1對(duì)x2 , xk的回歸。 Thus

24、 measures the effect of x1 on y after x2, , xk.has been partialled out. 于是 度量的是，在排除x2 , xk等變量的影響之后， x1對(duì)y的影響。,Intermediate Econometrics, Yan Shen,23,Simple vs Multiple Regression Estimates比較簡(jiǎn)單回歸和多元回歸估計(jì)值,Intermediate Econometrics, Yan Shen,24,Simple vs Multiple Regression Estimates比較簡(jiǎn)單回歸和多元回歸估計(jì)值,This

25、is because there exists a simple relationship 這是因?yàn)榇嬖谝粋€(gè)簡(jiǎn)單的關(guān)系 where is the slope coefficient from the simple regression of x2 on x1 . The proof. 這里， 是x2對(duì)x1的簡(jiǎn)單回歸得到的斜率系數(shù)。證明如下。,Intermediate Econometrics, Yan Shen,25,Intermediate Econometrics, Yan Shen,26,Simple vs Multiple Regression Estimates簡(jiǎn)單回歸和多元回歸估計(jì)

26、值的比較,Intermediate Econometrics, Yan Shen,27,Simple vs Multiple Regression Estimates簡(jiǎn)單回歸和多元回歸估計(jì)值的比較,In the case with k independent variables, the simple regression and the multiple regression produce identical estimate for x1 only if 在k個(gè)自變量的情況下，簡(jiǎn)單回歸和多元回歸只有在以下條件下才能得到對(duì)x1相同的估計(jì) (1) the OLS coefficients o

27、n x2 through xk are all zero, or （1）對(duì)從x2到xk的OLS系數(shù)都為零，或 (2) x1 is uncorrelated with each of x2 , xk. （2） x1與x2 , xk中的每一個(gè)都不相關(guān)。,Intermediate Econometrics, Yan Shen,28,Summary 總結(jié),In this lecture we introduce the multiple regression. 在本次課中，我們介紹了多元回歸。 Important concepts: 重要概念： Interpreting the meaning of

28、OLS estimates in multiple regression 解釋多元回歸中OLS估計(jì)值的意義 Partialling effect 局部效應(yīng)（其它情況不變效應(yīng)） Properties of OLS OLS的性質(zhì) When will the estimates from simple and multiple regression to be identical 什么時(shí)候簡(jiǎn)單回歸和多元回歸的估計(jì)值相同,Intermediate Econometrics, Yan Shen,29,Multiple Regression Analysis: Estimation (2)多元回歸分析：估

29、計(jì)（2）,y = b0 + b1x1 + b2x2 + . . . bkxk + u,Intermediate Econometrics, Yan Shen,30,Chapter Outline 本章大綱,Motivation for Multiple Regression 使用多元回歸的動(dòng)因 Mechanics and Interpretation of Ordinary Least Squares 普通最小二乘法的操作和解釋 The Expected Values of the OLS Estimators OLS估計(jì)量的期望值 The Variance of the OLS Estima

30、tors OLS估計(jì)量的方差 Efficiency of OLS: The Gauss-Markov Theorem OLS的有效性：高斯馬爾科夫定理,Intermediate Econometrics, Yan Shen,31,Lecture Outline 課堂大綱,The MLR.1 MLR.4 Assumptions 假定MLR.1 MLR.4 The Unbiasedness of the OLS estimates OLS估計(jì)值的無(wú)偏性 Over or Under specification of models 模型設(shè)定不足或過(guò)度設(shè)定 Omitted Variable Bias 遺

31、漏變量的偏誤 Sampling Variance of the OLS slope estimates OLS斜率估計(jì)量的抽樣方差,Intermediate Econometrics, Yan Shen,32,The expected value of the OLS estimatorsOLS估計(jì)量的期望值,We now turn to the statistical properties of OLS for estimating the parameters in an underlying population model. 我們現(xiàn)在轉(zhuǎn)向OLS的統(tǒng)計(jì)特性，而我們知道OLS是估計(jì)潛在的總

32、體模型參數(shù)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. 統(tǒng)計(jì)性質(zhì)是估計(jì)量在隨機(jī)抽樣不斷重復(fù)時(shí)的性質(zhì)。我們并不關(guān)心在某一特定樣本中估計(jì)量如何。,Intermediate Econometrics, Yan Shen,33,Assumption MLR.1 (Linear in Parameters)假定 MLR.1（

33、對(duì)參數(shù)而言為線性）,In the population model (or the true model), the dependent variable y is related to the independent variable x and the error u as 在總體模型(或稱真實(shí)模型）中，因變量y與自變量x和誤差項(xiàng)u關(guān)系如下 y= b0+ b1x1+ b2x2+ +bkxk+u where b1, b2 , bk are the unknown parameters of interest, and u is an unobservable random error or

34、random disturbance term. 其中， b1, b2 , bk 為所關(guān)心的未知參數(shù)，u為不可觀測(cè)的隨機(jī)誤差項(xiàng)或隨機(jī)干擾項(xiàng)。,Intermediate Econometrics, Yan Shen,34,Assumption MLR.2 (Random Sampling)假定 MLR.2（隨機(jī)抽樣性）,We can use a random sample of size n from the population, 我們可以使用總體的一個(gè)容量為n的隨機(jī)樣本 (xi1, xi2, xik; yi): i=1,n， where i denotes observation, and

35、j= 1,k denotes the jth regressor. 其中i 代表觀察，j=1,k代表第j個(gè)回歸元 Sometimes we write 有時(shí)我們將模型寫(xiě)為 yi= b0+ b1xi1+ b2xi2+ +bkxik+ui,Intermediate Econometrics, Yan Shen,35,Assumptions MLR.3 假定 MLR.3,MLR.3 (Zero Conditional Mean) （零條件均值） : E(u| xi1, xi2, xik)=0. When this assumption holds, we say all of the explana

36、tory variables are exogenous; when it fails, we say that the explanatory variables are endogenous. 當(dāng)該假定成立時(shí)，我們稱所有解釋變量均為外生的；否則，我們則稱解釋變量為內(nèi)生的。 We will pay particular attention to the case that assumption 3 fails because of omitted variables. 我們將特別注意當(dāng)重要變量缺省時(shí)導(dǎo)致假定3不成立的情況。,Intermediate Econometrics, Yan She

37、n,36,Assumption MLR.4 假定MLR.4,MLR.4 (No perfect collinearity) （不存在完全共線性） : In the sample, none of the independent variables is constant, and there are no exact linear relationships among the independent variables. 在樣本中，沒(méi)有一個(gè)自變量是常數(shù)，自變量之間也不存在嚴(yán)格的線性關(guān)系。 When one regressor is an exact linear combination of

38、 the other regressor(s), we say the model suffers from perfect collinearity. 當(dāng)一個(gè)自變量是其它解釋變量的嚴(yán)格線性組合時(shí)，我們說(shuō)此模型有嚴(yán)格共線性。 Examples of perfect collinearity：完全共線性的例子： 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. Perfect collinearity a

39、lso happens when y= b0+ b1x1+ b2x2+ b3x3+u , n(k+1). 當(dāng)y= b0+ b1x1+ b2x2+ b3x3+u , n(k+1) 也發(fā)生完全共線性的情況。 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

40、 session discussing the partialling-out effect. 在完全共線性情況下，OLS估計(jì)量的分母為零，因此OLS估計(jì)量不能得到。你可以回顧討論“排除其它變量影響”部分中的b2估計(jì)量的式子，來(lái)檢驗(yàn)這一點(diǎn)。,Intermediate Econometrics, Yan Shen,37,Theorem 3.1 (Unbiasedness of OLS)定理 3.1（OLS的無(wú)偏性）,Under assumptions MLR.1 through MLR.4, the OLS estimators are unbiased estimator of the pop

41、ulation parameters, that is 在假定MLR.1MLR.4下，OLS估計(jì)量是總體參數(shù)的無(wú)偏估計(jì)量，即,Intermediate Econometrics, Yan Shen,38,Theorem 3.1 (Unbiasedness of OLS)定理 3.1（OLS的無(wú)偏性）,Unbiasedness is the property of an estimator, that is, the procedure that can produce an estimate for a specific sample, not an estimate. 無(wú)偏性是估計(jì)量的特性，

42、而不是估計(jì)值的特性。估計(jì)量是一種方法（過(guò)程），該方法使得給定一個(gè)樣本，我們可以得到一組估計(jì)值。我們?cè)u(píng)價(jià)的是方法的優(yōu)劣。 Not correct to say “5 percent is an unbiased estimate of the return of education”. 不正確的說(shuō)法:“5%是教育匯報(bào)率的無(wú)偏估計(jì)值?！?Intermediate Econometrics, Yan Shen,39,Too Many or Too Few Variables變量太多還是太少了？,What happens if we include variables in our specifica

43、tion that dont belong? 如果我們?cè)谠O(shè)定中包含了不屬于真實(shí)模型的變量會(huì)怎樣？ 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 盡管一個(gè)（或多個(gè)）自變量在總體中對(duì)y沒(méi)有局部效應(yīng)，但卻被放到了模型中，則此模型被過(guò)度設(shè)定。 There is no effect on our parameter estimate, and

44、OLS remains unbiased. But it can have undesirable effects on the variances of the OLS estimators. 過(guò)度設(shè)定對(duì)我們的參數(shù)估計(jì)沒(méi)有影響，OLS仍然是無(wú)偏的。但它對(duì)OLS估計(jì)量的方差有不利影響。,Intermediate Econometrics, Yan Shen,40,Too Many or Too Few Variables變量太多還是太少了？,What if we exclude a variable from our specification that does belong? 如果我們?cè)谠O(shè)

45、定中排除了一個(gè)本屬于真實(shí)模型的變量會(huì)如何？ If a variable that actually belongs in the true model is omitted, we say the model is underspecified. 如果一個(gè)實(shí)際上屬于真實(shí)模型的變量被遺漏，我們說(shuō)此模型設(shè)定不足。 OLS will usually be biased. 此時(shí)OLS通常有偏。 Deriving the bias caused by omitting an important variable is an example of misspecification analysis. 推導(dǎo)

46、由遺漏重要變量所造成的偏誤，是模型設(shè)定分析的一個(gè)例子。,Intermediate Econometrics, Yan Shen,41,Omitted Variable Bias遺漏變量的偏誤,Intermediate Econometrics, Yan Shen,42,Omitted Variable Bias (cont)遺漏變量的偏誤（續(xù)）,Intermediate Econometrics, Yan Shen,43,Omitted Variable Bias (cont)遺漏變量的偏誤（續(xù)）,Intermediate Econometrics, Yan Shen,44,Omitted V

47、ariable Bias (cont)遺漏變量的偏誤（續(xù)）,Intermediate Econometrics, Yan Shen,45,Omitted Variable Bias Summary遺漏變量的偏誤 總結(jié),Two cases where bias is equal to zero 兩種偏誤為零的情形 b2 = 0, that is x2 doesnt really belong in model b2 = 0，也就是，x2實(shí)際上不屬于模型 x1 and x2 are uncorrelated in the sample 樣本中x1與x2不相關(guān) If correlation betw

48、een x2 , x1 and x2 , y is the same direction, bias will be positive 如果x2與 x1間相關(guān)性和x2與y間相關(guān)性同方向，偏誤為正。 If correlation between x2 , x1 and x2 , y is the opposite direction, bias will be negative 如果x2與 x1間相關(guān)性和x2與y間相關(guān)性反方向，偏誤為負(fù)。,Intermediate Econometrics, Yan Shen,46,Omitted Variable Bias Summary遺漏變量的偏誤 總結(jié),

49、Intermediate Econometrics, Yan Shen,47,Summary of Direction of Bias偏誤方向總結(jié),Intermediate Econometrics, Yan Shen,48,Omitted-Variable Bias 遺漏變量偏誤,In general , b2 is unknown; and when a variable is omitted, it is mainly because of this variable is unobserved. In other words, we do not know the sign of Co

50、rr(x1, x2). What to do? 但是，通常我們不能觀測(cè)到b2 ，而且，當(dāng)一個(gè)重要變量被缺省時(shí)，主要原因也是因?yàn)樵撟兞繜o(wú)法觀測(cè)，換句話說(shuō)，我們無(wú)法準(zhǔn)確知道Corr(x1, x2)的符號(hào)。怎么辦呢？ We rely on economic theories and intuition to make a educated guess of the sign. 我們將依靠經(jīng)濟(jì)理論和直覺(jué)來(lái)幫助我們對(duì)相應(yīng)符號(hào)做出較好的估計(jì)。,Intermediate Econometrics, Yan Shen,49,Example: hourly wage equation例子：小時(shí)工資方程,Supp

51、ose the model log(wage) = b0+b1educ + b2abil +u is estimated with abil omitted. What is the direction of bias for b1? 假定模型 log(wage) = b0+b1educ + b2abil +u，在估計(jì)時(shí)遺漏了abil。 b1的偏誤方向如何？ Since in general ability has positive partial effect on y and ability and education years is positive corrected, we exp

52、ect b1 to have a upward bias. 因?yàn)橐话銇?lái)說(shuō)ability對(duì)y有正的局部效應(yīng)，并且ability和education years正相關(guān)，所以我們預(yù)期b1上偏。,Intermediate Econometrics, Yan Shen,50,The More General Case更一般的情形,Technically, it is more difficult to derive the sign of omitted variable bias with multiple regressors. 從技術(shù)上講，要推出多元回歸下缺省一個(gè)變量時(shí)各個(gè)變量的偏誤方向更加困難。

53、 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. 我們需要記住，若有一個(gè)對(duì)y有局部效應(yīng)的變量被缺省，且該變量至少和一個(gè)解釋變量相關(guān)，那么所有系數(shù)的OLS估計(jì)量都有偏。,Intermediate Econometrics, Yan Shen,51,The More General

54、Case更一般的情形,Intermediate Econometrics, Yan Shen,52,The More General Case更一般的情形,Intermediate Econometrics, Yan Shen,53,Variance of the OLS Estimators OLS估計(jì)量的方差,Now we know that the sampling distribution of our estimate is centered around the true parameter?，F(xiàn)在我們知道估計(jì)值的樣本分布是以真實(shí)參數(shù)為中心的。 Want to think about

55、 how spread out this distribution is 我們還想知道這一分布的分散狀況。 Much easier to think about this variance under an additional assumption, so 在一個(gè)新增假設(shè)下，度量這個(gè)方差就容易多了，有：,Intermediate Econometrics, Yan Shen,54,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 .

56、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，不管解釋變量出現(xiàn)怎樣的組合，誤差項(xiàng)u的條件方差都是一樣的。 If the assumption fails, we say the model exhibits heteroskedasticity. 如果這個(gè)假定不成立，我們說(shuō)模型存在異方差性。,Intermediate

57、Econometrics, Yan Shen,55,Variance of OLS (cont)OLS估計(jì)量的方差（續(xù)）,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，也就意味著Var(y| x) = s2 Assumption MLR.1-5 are collectively known as the Gauss-Markov assumptions. 假定MLR.1-5共同被稱為高斯馬爾科夫假定,

58、Intermediate Econometrics, Yan Shen,56,Theorem 3.2 (Sampling Variances of the OLS Slope Estimators)定理 3.2（OLS斜率估計(jì)量的抽樣方差）,Intermediate Econometrics, Yan Shen,57,Interpreting Theorem 3.2對(duì)定理3.2的解釋,Theorem 3.2 shows that the variances of the estimated slope coefficients are influenced by three factors:

59、定理3.2顯示：估計(jì)斜率系數(shù)的方差受到三個(gè)因素的影響： The error variance 誤差項(xiàng)的方差 The total sample variation 總的樣本變異 Linear relationships among the independent variables 解釋變量之間的線性相關(guān)關(guān)系,Intermediate Econometrics, Yan Shen,58,Interpreting Theorem 3.2: The Error Variance對(duì)定理3.2的解釋（1）：誤差項(xiàng)方差,A larger s2 implies a larger variance for the OLS estimators. 更大的s2意味著更大的OLS估計(jì)量方差。 A larger s2 means more noises in the equation. 更大的s2意味著方程中的“噪音”越多。 This makes it more difficult to extract the exact partial effect of the regressor on the regressand. 這使得得到自變量對(duì)因變量的準(zhǔn)確局部效應(yīng)變得更加困難。 Introduc