課件計(jì)量經(jīng)濟(jì)學(xué)19章7虛擬變量

1、Multiple Regression Analysis y = b0 + b1x1 + b2x2 + . . . bkxk + u 5. Dummy VariablesChapter Outline1. 描述定性信息描述定性信息Describing Qualitative Information2. 一個(gè)虛擬變量作解釋變量一個(gè)虛擬變量作解釋變量A Single Dummy Independent Variable3. 用多個(gè)虛擬變量表示多種分類數(shù)據(jù)用多個(gè)虛擬變量表示多種分類數(shù)據(jù)Using Dummy Variables For Multiple Categories4. 與虛擬變量有關(guān)的交互

2、項(xiàng)與虛擬變量有關(guān)的交互項(xiàng)Interactions Involving Dummy Variables5. 虛擬變量作因變量：線性概率模型虛擬變量作因變量：線性概率模型A Binary Dependent Variable: The Linear Probability Model6. 關(guān)于政策分析與項(xiàng)目評價(jià)的進(jìn)一步討論關(guān)于政策分析與項(xiàng)目評價(jià)的進(jìn)一步討論More On Policy Analysis And Program EvaluationLecture Outline 1. 定性信息與虛擬變量定性信息與虛擬變量Qualitative information & Dummy Variable

3、s 2. 虛擬變量作為解釋變量虛擬變量作為解釋變量(截距項(xiàng)截距項(xiàng))Dummy Independent Variables 3. 與虛擬變量有關(guān)的交互項(xiàng)與虛擬變量有關(guān)的交互項(xiàng)Interactions Involving Dummy Variables：Chow test1.定性信息與虛擬變量定性信息與虛擬變量Qualitative information & Dummy Variables數(shù)量信息與定性信息數(shù)量信息與定性信息Quantitative & Qualitative information 連續(xù)變量(Continuous Variables): Quantitative informa

4、tion: wage, years of education, experience, weight, sales, price, pop 離散變量(Discrete Variables): Qualitative information: gender (1: male 2: female); race(1.black; 2. white; 3. others); marital status (1: single; 2: married); region(1. eastern; 2. central ; 3. western); education attainment (1: prima

5、ry; 2: junior 3. high; 4: college) training (1. trainees; 2. nontrainees); insurance(1. participating; 2. not participating); industry (1.agriculture; 2: manufacture; 3: service;4. others) Income groups (1. 5000); age group (1. 60); Ordinal variables: Credit rating (low to high: 1 2 3 4 5);Dummy Var

6、iables & Qualitative information A dummy variable 是一種只取1或0兩個(gè)數(shù)值的變量. Examples: (1) sex: 1: male 2: female male (= 1 if male, 0 otherwise); female (= 1 if female, 0 otherwise)(2) region: 1. eastern; 2. central ; 3. western) eastern (=1 if eastern, 0 otherwise); central (=1 if central, 0 otherwise) west

7、ern (=1 if western, 0 otherwise) Dummy variables are also called: 二值變量(binary variables), 0-1變量(zero-one variables)2. 虛擬變量作為解釋變量虛擬變量作為解釋變量(截距項(xiàng)截距項(xiàng))Dummy Independent Variables2. 虛擬變量作為解釋變量虛擬變量作為解釋變量 Case 1: y = b0 + d0d + b1x + u Case 2: y = b0 + d1d1 + d1d2 + b1x + u Case 3: y = b0 + d1d1 + d1d2 + d1

8、d1d2 + b1x + u Case 4: y = b0 + d0d + d1dx+b1x + u d: dummy variable虛擬變量Case 1: y = b0 + d0d + b1x + u 考慮一個(gè)簡單工資方程： wage = b0 + d0 female + b1 educ + u If female =0, then wage = b0 + b1educ + u If female =1, then wage = (b0 + d0) + b1educ + u d0 can be interpreted as an intercept shift (截距項(xiàng)變動(dòng))wage =

9、b0 + d0 female + b1 educ + u 在零值條件期望假定( zero conditional mean)下： E(wage| female, educ) = b0 + d0 female + b1 educ (1) E(wage| female=1, educ) = (b0 + d0 ) + b1 educ (2) E(wage| female=0, educ) = b0 + b1 educ d0 = E(wage| female=1, educ) - E(wage| female=0, educ) d0 (an intercept shift): 給定教育年限educ，女

10、性平均工資比男性平均工資高d0元。 Example of d0 0E(wage|female,educ) = b0 + d0 female + b1 educ Example of d0 0 and d1 0wage = b0 + d0female+ d1female*educ+ b1educ+u = (b0 + d0female)+ (d1female+ b1)educ+uFemale: female = 1Male: female = 0b0 + d0b0 檢驗(yàn)不同群體之間工資方程的差異F test & Chow test 簡單工資方程： wage = b0 + b1educ+u (p)

11、不同性別之間的工資方程可能是不同的： Female: wage = b0f + b1f educ + u (f ) Female: wage = b0m + b1m educ + u (m) H0: b0f = b0m, b1f = b1m; H1：H0 is not true.F test 方法1：F test wage =b0 + d0female+ b1educ + d1female*educ +u (UR) H0: d0=0, d1=0; H1: H0 is not true wage = b0 + b1educ+u (R) F test： (SSRr-SSRur)/(k+1) / S

12、SRur/n-2(k+1)一種簡便F test: Chow test (1) 用女性(female=1)子樣本估計(jì)女性工資方程(f)，用男性 (female=0)子樣本估計(jì)男性工資方程(m), 分別得到SSRf和SSRm, 則有無約束工資方程的SSRur可以寫作：SSRur= SSRf+SSRm (2) 用所有群體(男、女)混合(pooling)在一起的總樣本, 估計(jì)受約束工資方程(R), 得到SSRr; (由于受約束模型是用所有群體混合(pooling)在一起的總樣本，故作者記作：SSRp = SSRr ) (3) F test: (k工資方程(p)中解釋變量的個(gè)數(shù))1(2/)() 1/()

13、()1(2/) 1/()(knSSRSSRkSSRSSRSSRknSSRkSSRSSRFmfmfpururr更多解釋變量的回歸方程 工資方程： lwage = b0 + b1educ+ b2exper+ u (p) 不同性別之間的工資方程可能是不同的： Female: lwage = b0f + b1f educ+b2fexper + u (f ) male: lwage = b0m + b1m educ+b2mexper + u (m) H0: b0f = b0m, b1f = b1m , b2f = b2m; H1：H0 is not true. lwage = b0 + d0female

14、+b1educ+ d1female*educ +b2exper +d2female*exper +u (UR) H0: d0=0, d1=0 , d2=0; H1: H0 is not true lwage = b0 +b1educ+b2exper +u (R)不同時(shí)期的結(jié)構(gòu)方程差異 lwage = b0 + b1educ+ b2exper+ u (p) 不同時(shí)期之間的工資方程可能是不同的： 1995: lwage = b095 + b195 educ+b295exper + u (95) 2001: lwage = b001 + b101 educ+b201exper + u (01) H0

15、: b095 = b001, b195 = b101 , b295 = b201; H1：H0 is not true.2 year samples: Define: s01 (=1 if year=2001, 0 if year=1995) lwage = b0 + d0s01+b1educ+ d1s01*educ +b2exper +d2s01*exper +u (UR) H0: d0=0, d1=0 , d2=0; H1: H0 is not true lwage = b0 +b1educ+b2exper +u (R)關(guān)于Chow test Chow test是F test的一種特例,因

16、此，需要同方差假定(MRL5): 不同群體(如男女)之間，誤差方差相同：Var(uf|x)=Var(um|x)=s2 Chow test中，須注意估計(jì)不同方程，樣本也不同：子群體方程需用子樣本，受約束方程需用所有群體混合在一起的總樣本。其他應(yīng)用：允許截距不同，檢驗(yàn)斜率是否相同 不同性別, 工資方程的截距參數(shù)不同： lwage = b0 + d0female + b1educ+ b2exper+ u (p) 不同性別之間, 工資方程的斜率參數(shù)也可能不同： Female: lwage = b0f + b1f educ+b2fexper + u (f ) male: lwage = b0m + b1

17、m educ+b2mexper + u (m) H0: b1f = b1m , b2f = b2m; H1：H0 is not true. lwage = b0 + d0female +b1educ+ d1female*educ +b2exper +d2female*exper +u (UR) H0: d1=0 , d2=0; H1: H0 is not true lwage = b0 + d0female + b1educ+ b2exper+ u (R) 方法1： F test 方差2：類似Chow test 用女性樣本估計(jì)方程(f),得SSRf;用女性樣本估計(jì)方程(m),得SSRm: SS

18、Rur=SSRf+SSRm 估計(jì)受約束方程(R), SSRr . 注:(R)中含有dummy:female; 在(R)中，則不含 計(jì)算F統(tǒng)計(jì)量總結(jié) 1. 定性信息與虛擬變量定性信息與虛擬變量Qualitative information & Dummy Variables 2. 虛擬變量作為解釋變量虛擬變量作為解釋變量(截距項(xiàng)截距項(xiàng))Dummy Independent Variables 3. 與虛擬變量有關(guān)的交互項(xiàng)與虛擬變量有關(guān)的交互項(xiàng)Interactions Involving Dummy Variables：Chow test作業(yè)與思考題 思考題：7.1, 7.2, 7.6 計(jì)算機(jī)練習(xí)：

19、7.10，7.15Multiple Regression Analysis y = b0 + b1x1 + b2x2 + . . . bkxk + u 5. Dummy VariablesChapter Outline1. 描述定性信息描述定性信息Describing Qualitative Information2. 一個(gè)虛擬變量作解釋變量一個(gè)虛擬變量作解釋變量A Single Dummy Independent Variable3. 用多個(gè)虛擬變量表示多種分類數(shù)據(jù)用多個(gè)虛擬變量表示多種分類數(shù)據(jù)Using Dummy Variables For Multiple Categories4. 與

20、虛擬變量有關(guān)的交互項(xiàng)與虛擬變量有關(guān)的交互項(xiàng)Interactions Involving Dummy Variables5. 虛擬變量作因變量：線性概率模型虛擬變量作因變量：線性概率模型A Binary Dependent Variable: The Linear Probability Model6. 關(guān)于政策分析與項(xiàng)目評價(jià)的進(jìn)一步討論關(guān)于政策分析與項(xiàng)目評價(jià)的進(jìn)一步討論More On Policy Analysis And Program EvaluationLecture Outline 1. 定性信息與虛擬變量定性信息與虛擬變量Qualitative information & Dummy

21、 Variables 2. 虛擬變量作為解釋變量虛擬變量作為解釋變量Dummy Independent Variables 3. 虛擬變量作因變量虛擬變量作因變量Dummy Dependent Variable 4. 政策分析中的選擇性與內(nèi)生性政策分析中的選擇性與內(nèi)生性Selection And Endogeneity In Policy Analysis1.定性信息與虛擬變量定性信息與虛擬變量Qualitative information & Dummy Variables數(shù)量信息與定性信息數(shù)量信息與定性信息Quantitative & Qualitative information 連續(xù)變

22、量(Continuous Variables): Quantitative information: wage, years of education, experience, weight, sales, price, pop 離散變量(Discrete Variables): Qualitative information: gender (1: male 2: female); race(1.black; 2. white; 3. others); marital status (1: single; 2: married); region(1. eastern; 2. central

23、; 3. western); education attainment (1: primary; 2: junior 3. senior; 4: college) training (1. trainees; 2. nontrainees); insurance(1. participant; 2. nonparticipants); industry (1.agriculture; 2: manufacture; 3: service;4. others) Income groups (1. 5000); age group (1. 60); Ordinal variables: Credi

24、t rating (low to high: 1 2 3 4 5);虛擬變量與定性信息虛擬變量與定性信息Dummy Variables & Qualitative information 虛擬變量虛擬變量(A dummy variable) 是一種只取1或0兩個(gè)數(shù)值的變量. 虛擬變量可以用來表示定性信息：Examples: (1) sex: 1: male 2: female male (= 1 if male, 0 otherwise); female (= 1 if female, 0 otherwise)(2) region: 1. eastern; 2. central ; 3. we

25、stern) eastern (=1 if eastern, 0 otherwise); central (=1 if central, 0 otherwise) western (=1 if western, 0 otherwise) Dummy variables are also called: 二值變量(binary variables), 0-1變量(zero-one variables)Dummy Varibles2. 虛擬變量作為解釋變量虛擬變量作為解釋變量 Dummy Independent Variables Case 1: y = b0 + d0d + b1x + u Ca

26、se 2: y = b0 + d1d1 + d1d2 + b1x + u Case 3: y = b0 + d1d1 + d1d2 + d1d1d2 + b1x + u Case 4: y = b0 + d0d + d1dx+b1x + u d: dummy variable虛擬變量Case 1: y = b0 + d0d + b1x + u 考慮一個(gè)簡單工資方程： wage = b0 + d0 female + b1 educ + u If female =0, then wage = b0 + b1educ + u If female =1, then wage = (b0 + d0) +

27、 b1educ + u d0 can be interpreted as an intercept shift (截距項(xiàng)變動(dòng))wage = b0 + d0 female + b1 educ + u 在零值條件期望假定( zero conditional mean)下： (1) E(wage|female,educ)= b0 + d0 female + b1 educ (2) E(wage| female=1, educ) = (b0 + d0 ) + b1 educ (3) E(wage| female=0, educ) = b0 + b1 educ d0 = E(wage| female=1

28、, educ) - E(wage| female=0, educ) d0 (an intercept shift): 給定教育年限educ，女性平均工資比男性平均工資高d0元。 Example of d0 0E(wage|female,educ) = b0 + d0 female + b1 educ Example of d0 0 E(wage|female,educ) = b0 + d0 female + b1 educ回歸結(jié)果wage = b0 + d0 female + b1 educ + uIf female =0, then lwage = b0 + b1educ If female

29、 =1, then lwage = (b0 + d0) + b1educ d0 0 and d1 0wage = b0 + d0female+ d1female*educ+ b1educ+u = (b0 + d0female)+ (d1female+ b1)educ+uFemale: female = 1Male: female = 0b0 + d0b0 檢驗(yàn)不同群體之間工資方程的差異F test & Chow test 簡單工資方程： wage = b0 + b1educ+u (p) 不同性別之間的工資方程可能是不同的： Female: wage = b0f + b1f educ + u (

30、f ) Female: wage = b0m + b1m educ + u (m) H0: b0f = b0m, b1f = b1m; H1：H0 is not true.F test 方法1：F test wage =b0 + d0female+ b1educ + d1female*educ +u (UR) H0: d0=0, d1=0; H1: H0 is not true wage = b0 + b1educ+u (R) F test： (SSRr-SSRur)/(k+1) / SSRur/n-2(k+1)一種簡便F test: Chow test (1) 用女性(female=1)子樣

31、本估計(jì)女性工資方程(f)，用男性 (female=0)子樣本估計(jì)男性工資方程(m), 分別得到SSRf和SSRm, 則有無約束工資方程的SSRur可以寫作：SSRur= SSRf+SSRm (2) 用所有群體(男、女)混合(pooling)在一起的總樣本, 估計(jì)受約束工資方程(R), 得到SSRr; (由于受約束模型是用所有群體混合(pooling)在一起的總樣本，故作者記作：SSRp = SSRr ) (3) F test: (k工資方程(p)中解釋變量的個(gè)數(shù))1(2/)() 1/()()1(2/) 1/()(knSSRSSRkSSRSSRSSRknSSRkSSRSSRFmfmfpururr

32、工資方程的性別差異：更多解釋變量的回歸方程 工資方程： lwage = b0 + b1educ+ b2exper+ u (p) 不同性別之間的工資方程可能是不同的： Female: lwage = b0f + b1f educ+b2fexper + u (f ) male: lwage = b0m + b1m educ+b2mexper + u (m) H0: b0f = b0m, b1f = b1m , b2f = b2m; H1：H0 is not true. lwage = b0 + d0female+b1educ+ d1female*educ +b2exper +d2female*ex

33、per +u (UR) H0: d0=0, d1=0 , d2=0; H1: H0 is not true lwage = b0 +b1educ+b2exper +u (R)F testChow test不同時(shí)期的結(jié)構(gòu)方程差異 lwage = b0 + b1educ+ b2exper+ u (p) 不同時(shí)期之間的工資方程可能是不同的： 1995: lwage = b095 + b195 educ+b295exper + u (95) 2001: lwage = b001 + b101 educ+b201exper + u (01) H0: b095 = b001, b195 = b101 ,

34、b295 = b201; H1：H0 is not true.2 year samples: Define: s01 (=1 if year=2001, 0 if year=1995) lwage = b0 + d0s01+b1educ+ d1s01*educ +b2exper +d2s01*exper +u (UR) H0: d0=0, d1=0 , d2=0; H1: H0 is not true lwage = b0 +b1educ+b2exper +u (R)關(guān)于Chow test Chow test是F test的一種特例,因此，需要同方差假定(MRL5): 不同群體(如男女)之間，

35、誤差方差相同：Var(uf|x)=Var(um|x)=s2 Chow test中，須注意估計(jì)不同方程，樣本也不同：子群體方程需用子樣本，受約束方程需用所有群體混合在一起的總樣本。其他應(yīng)用：允許截距不同，檢驗(yàn)斜率是否相同 不同性別, 工資方程的截距參數(shù)不同： lwage = b0 + d0female + b1educ+ b2exper+ u (p) 不同性別之間, 工資方程的斜率參數(shù)也可能不同： Female: lwage = b0f + b1f educ+b2fexper + u (f ) male: lwage = b0m + b1m educ+b2mexper + u (m) H0: b

36、1f = b1m , b2f = b2m; H1：H0 is not true. lwage = b0 + d0female +b1educ+ d1female*educ +b2exper +d2female*exper +u (UR) H0: d1=0 , d2=0; H1: H0 is not true lwage = b0 + d0female + b1educ+ b2exper+ u (R) 方法1： F test 方差2：類似Chow test 用女性樣本估計(jì)方程(f),得SSRf;用女性樣本估計(jì)方程(m),得SSRm: SSRur=SSRf+SSRm 估計(jì)受約束方程(R), SSR

37、r . 注:(R)中含有dummy:female; 在(R)中，則不含 計(jì)算F統(tǒng)計(jì)量3. 虛擬變量作因變量虛擬變量作因變量Dummy Dependent VariableA Binary Dependent Variable: The Linear Probability Model 以前均是連續(xù)變量作因變量： 工資，體重 虛擬變量也可以做因變量： 某個(gè)事件是否發(fā)生：行為選擇結(jié)果： 參與勞動(dòng)力市場：parti: 1: in ; 0: out 參與培訓(xùn)項(xiàng)目：training: 1:trainees; 0: no 是否上大學(xué)：college: 1: in; 0: out 虛擬變量所代表的事件: 1

38、: success; 0: failure 將一個(gè)虛擬變量y作為因變量: y = b0 + b1x1 + b2x2 + . . . bkxk + u 如何解釋參數(shù)？ MLR.3成立，則: E(y|x) = b0 + b1x1 + b2x2 + . . . Bkxk 參數(shù)b1衡量：在給定x2-xk不變的情況下，x變化1單位， y的期望E(y|x)變化多少。 當(dāng)y 是一個(gè)虛擬變量(0-1)時(shí)，可以有： E(y|x) = P(y=1|x) 即：給定x, 事件(y=1)發(fā)生的概率P(y=1|x), 與y的均值E(y=1|x)是相等的. 故： P(y=1|x) = b0 + b1x1 + b2x2 +

39、. . . bkxk 參數(shù)b1解釋為：在給定x2-xk不變情況下，x變化1單位， 事件y=1發(fā)生的概率P(y=1|x)變化多少。 P(y=1|x): response probability(響應(yīng)概率) 虛擬變量為因變量的線性回歸模型，稱線性概率模型(linear probability model, LPM)例：遷移決定模型 教育對勞動(dòng)力遷移的影響：mig = b0 + b1age + b2agesq+ b3 junior +b4senior+ b5college+ b6married+ b7child + b8citynet +b9distance+ b10discrim+others+

40、uMig:1:migrants; 0:nonmigrantsEduc: 1: primary; 2: junior; 3. senior;4.collegeLPM的不足之處 1. 預(yù)測值可能在0,1之外。 這與預(yù)測值的概率解釋不一致。 2. 與(1)相關(guān)的問題是：事件發(fā)生概率P(y=1|x), 與x可能取值之間可能是非線性關(guān)系。 LPM假定，在x的不同取值點(diǎn)上，x變化1單位對于P(y=1)的影響(marginal effects, 邊際效應(yīng))相同。 然而，有時(shí)，在x的不同取值點(diǎn)上，x變化1單位對于P(y=1)的影響是不同的：在x的邊界值上，x變化1單位對于P(y=1)的影響，與在x均值上，x變

41、化1單位對于P(y=1)的影響，并不相同。 LPM所估計(jì)的邊際效應(yīng)，在x均值附近比較準(zhǔn)確。 3. LPM 不符合同方差假定(MLR.5). 無法正確OLS估計(jì)量的方差(漸近方差)，從而影響正確推斷。 Var(y|x)= Ey-E(y|x)2=E(y2|x) - E(y|x)2 = E(y|x) - E(y|x)2 = p(x)-p(x)2 = p(x)1-p(x) E(y|x)=P(y=1|x)=p(x), Var(y)=E(y2)-E(y)2 方法：校正異方差LPM的優(yōu)點(diǎn) LPM容易估計(jì)，容易理解(估計(jì)系數(shù)即為邊際效應(yīng))。 在均值附近，對于邊際效應(yīng)估計(jì)也比較準(zhǔn)確。 關(guān)鍵是，不必對誤差項(xiàng)u的分布進(jìn)行假定。在大樣本下，MLR1-4保證OLS估計(jì)量具有一致性。4. 政策分析中的選擇性與內(nèi)生性政策分析中的選擇性與內(nèi)生性Selection And Endogeneity In Policy Analysis 虛擬變量的一個(gè)經(jīng)典的應(yīng)用：評估社會(huì)政策/項(xiàng)目效果 例如：性別/種族/戶籍歧視，失業(yè)保險(xiǎn)/醫(yī)療保險(xiǎn)/低保政策，培訓(xùn)/扶貧項(xiàng)目. 項(xiàng)目評估(program evaluation): 評估某社會(huì)政策或項(xiàng)目(如培訓(xùn))對個(gè)人、家庭、企業(yè)、社區(qū)、地區(qū)的影響 通常, 將參加項(xiàng)目者，稱為治療組/試驗(yàn)組(tre