厦门大学：《高级经济计量学》讲义 第四章 多元线性回归模型

Chapter3多元线性回归模型 (Muitiple linear regression model) You are required to get familiar with matrix algebra for mastering this chapter

Chapter 3 多元线性回归模型 (Multiple linear regression model) You are required to get familiar with matrix algebra for mastering this chapter!

a Classical Multiple linear regression model (CMLRM) 1. model X=b+bX1+…+ 6, X+E 2. random sample M, XI k y=b+bx1+…+b如+1(=1,…,n)

◼ Classical Multiple linear regression model (CMLRM): 1. model 2. random sample Y b b X b X 0 1 1 k k = + + + + 1 { ; , } Y X X i i ki 0 1 1 ( 1, , ) i i k ki i Y b b X b X i n = + + + + = 

Matrix form Y=Xb+8 k1 8 1Ⅹ 2,b Ⅹ

Matrix form: 11 1 0 1 1 12 2 1 2 1 1 1 , , , 1 k k n n kn k n X X b Y X X b Y X X b    = +                       = = = =                         Y Xb ε Y X b ε

3. Model assumption: 1.E()=0 2. E(Ge=oI,(I, is a unit matrix) 3. Xis non-random 4. rank(X)=k+l< n 5. Normality assumption E~N(0,o2)(i=1,…,n)

3. Model assumption: 1. 2. 3. is non-random. 4. 5. Normality assumption E( ) ε = 0 2 E( ) ( is a unit matrix)  n n εε = I I X rank( )= 1 X k n +  2 (0, ) ( 1, , )   i N i n =

assumptions 1 and 5 imply that the errors are Independent As in the case of the univariate linear regression models, we can estimate the regression coefficients of the multiple linear regression models by using the ordinary least squares procedure. In matrix form the olse is b=(XX XY

assumptions 1 and 5 imply that the errors are Independent. As in the case of the univariate linear regression models, we can estimate the regression coefficients of the multiple linear regression models by using the ordinary least squares procedure. In matrix form, the OLSE is 1 ˆ ( − b X X) X Y =  

4. OLSE for the CMlrM b=(XXXY 5. Properties of the olse for the cMlRM E(b)=b 2. var(b)=El(b-b)(b-b=O(XX) 3. The gauss-Markoy theorem is still true The ol se for the cmlrm is the blue

4. OLSE for the CMLRM 5. Properties of the OLSE for the CMLRM 1. 2. 3. The Gauss-Markov theorem is still true: The OLSE for the CMLRM is the BLUE. 1 ˆ ( − b X X) X Y =   E( ) b b ˆ = 2 1 ˆ ˆ ˆ var( ) E[( )( ) ] ( )  − b b b b b X X = − − =  

6. Residual and estimation of 2 the population variance o 1. Residual e=Y-Y=Y-Xb=[-X(XXXY PY(P=I-X(XX) X 1) P is idempotent(幂等的) 2)E(e)=0 3)var(e=e(ee)=oP 4) >2=rY-bX'Y=tr(ee)

6. Residual and Estimation of the population variance 1. Residual 1) P is idempotent (幂等的) 2) 3) 4) 2  ˆ ˆ [ ] ( ) = − = − = −   = = −   -1 -1 e Y Y Y Xb I X(X X) X Y PY P I X(X X) X E( ) e 0 = 2 var( ) E( ) e ee P = =   2 1 ˆ tr( ) n i i e =  = − = Y Y bX Y ee   

2. Estimator for O ee n-(k+1)n-(k+1) E(a2)=E( 2 O k-1

2. Estimator for 2  2 2 ˆ ( 1) ( 1) i e n k n k   = = − + − +  e e 2 2 ( ) ( ) ˆ 1 e e E E n k    = = − −

7. Goodness-of-fit testing 1)Total sum of squares TSS=∑(x-y)=YY-ny2 2)Explained sum of squares ESS=∑(y-y)2-∑e2=bXY-ny2 2 Coefficient of determination 2 ESS ee b'XY-ny R TSs∑(x-Y2)YY-ny

7. Goodness-of-fit testing 1. 1) Total sum of squares: 2) Explained sum of squares: 2. Coefficient of determination: 2 2 TSS ( ) Y Y nY i = − = −  Y Y 2 2 2 ESS ( ) ˆ Y Y e nY i i = − − = −   b X Y   2 2 2 2 ESS ˆ 1 TSS ( ) i nY R Y Y nY    − = = − =  − −  e e b X Y Y Y

3. Adjusted R-squared ee 2 R2=1 n-(k+1)_1(m-1)(1-R) n-k-1

3. Adjusted R-squared: 2 2 2 ( 1) ( 1)(1 ) 1 1 ( ) 1 1 i n k n R R Y Y n k n  − + − − = − = − − − − −  e e