厦门大学：《高级经济计量学》讲义 第五章 How to use the multiple linear regression model

hapter5 How to use the multiple linear regression moce 2021/2/20

2021/2/20 1 Chapter 5 How to use the multiple linear regression model

General linear models Suppose that we have a model X=F(X1…,Xk2E) It is inherently linear for the parameters if it can be transformed into f()=A+B81(X12…,X)+…+B8(X1,…,X)+E EXamples 1.Exponential model Y=exp(B+B, X,+B,x2)e 2021/2/20

2021/2/20 2 General linear models Suppose that we have a model: It is inherently linear for the parameters if it can be transformed into Examples: 1. Exponential model: 1 ( , , ; ) Y F X Xk =  0 1 1 1 1 ( ) ( , , ) ( , , ) k k k k f Y g X X g X X = + + + +     0 1 1 2 2 Y X X = + + exp( )    

2. Reciprocal model Y Bo+BX+B,X,+8 3. Double-log constant elasticity)model Iny=a+BinX+& ote dyx dIny Elasticity dx y dInx B(const) 2021/2/20

2021/2/20 3 2. Reciprocal model: 3. Double-log(constant elasticity) model: 0 1 1 2 2 1 Y     X X = + + + ln ln Y X = + +    ln Elasticity (const.) ln dY X d Y dX Y d X = = =  Note :

But the following model is not inherently linear Y=XXP +8 For others, see page72-73 2021/2/20

2021/2/20 4 But the following model is not inherently linear: For others, see page72-73 Y X X1 2   = +

a What is the dummy variable 属性因素: Man and woman City and country, developed country and developing country 经济环境的变化: Peace and war 淡季和旺季 风调雨顺和洪灾旱灾 All these have a good or bad influence on economy 2021/2/20

2021/2/20 5 ◼ What is the dummy variable 属性因素: Man and Woman, City and country, developed country and developing country 经济环境的变化: Peace and war 淡季和旺季 风调雨顺和洪灾旱灾 All these have a good or bad influence on economy

a dummy variable is a binary variable that IS I if a happens D lO if a happens EXample: D ∫1,解放后 「1, student D 0解放前0, teacher 2t≥t t=1979 o. t< t 2021/2/20

2021/2/20 6 A dummy variable is a binary variable, that is, Example: 1 if A happens 0 if A happens D  =   1, 0, D  =   解放后 解放前 1, student 0, teacher D  =   * * * 1, , 1979 0, t t t D t t t   = =   

加法方式:消费函数 C=b+bhX1+bD+E,D1=常年份 0,正常年份 乘法方式:消费函数 1,正常年份 1=b+(b+b C D2)X1+E,D2 0,反常年份 同时考虑两方式 C=b+bD+(+D)x+,D=/正常年份 0,反常年份 2021/2/20

2021/2/20 7 加法方式: 消费函数 乘法方式: 消费函数 同时考虑两方式: 0 1 2 1 1 1, , 0, C b b X b D D t t t t t   = + + + =   反常年份 正常年份 0 1 11 2 2 1, ( ) , 0, C b b b D X D t t t t t   = + + + =   正常年份 反常年份 0 01 1 11 1, ( ) , 0, C b b D b b D X D t t t t t t   = + + + + =   正常年份 反常年份

临界值虚拟变量(Note:p76,总消费函数例 子是引入临界值虚拟变量) 进口消费品额Y,国民收入X 「1,t≥t D t=1979 * 0.t<t 进口消费品回归模型 y=b。+bx,+b2(X-x,)D+6 2021/2/20

2021/2/20 8 临界值虚拟变量(Note: p76, 总消费函数例 子是引入临界值虚拟变量） 进口消费品额 Y, 国民收入 X 进口消费品回归模型 * * * 1, , 1979 0, t t t D t t t   = =    0 1 2 ( )* t t t t t t Y b b X b X X D = + + − +

Lcase studies(P77-78) Use of t and F-tests to test hypotheses nvolving multiple parameters Joint tests on several regression parameters Suppose that we have a model Y=B+B2X2+…+B k-qk-qPk-q+lk-a+y+. +BX,+8 And the null hypothesis is Bbo:B-=…=k=0 2021/2/20

2021/2/20 9 case studies(P77-78) Use of t and F-tests to test hypotheses Involving multiple parameters 1. Joint tests on several regression parameters: Suppose that we have a model And the null hypothesis is 1 2 2 1 1 k q k q k q k q k k Y X X X X       − − − + − + = + + + + + + + 0 1 : 0 H k q k   − + = = =

The test statistic is (ESS -ESSUR) / q (RUR -RR)/q ESSUR /(n-k)(1-Ri UR丿(n-k) 2. Tests involving linear functions of the regression parameters 3. Tests involving the equality of coefficients of different regression equations Chow- test (a special F-test) 2021/2/20 10

2021/2/20 10 The test statistic is 2. Tests involving linear functions of the regression parameters 3. Tests involving the equality of coefficients of different regression equations: Chow- test (a special F-test) 2 2 R UR UR R , 2 UR UR (ESS ESS ) / ( ) / ESS /( ) (1 ) /( ) q n k q R R q F n k R n k − − − = = − − −