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!
Classical Multiple linear regression
model (CMLRM):
1,model
2,random sample0 1 1 kkY b b X b X
1{ ;,}i i k iY X X
0 1 1 ( 1,,)i i k k i iY b b X b X i n
Matrix form:
1 1 1 0 1
1
1 2 2 1 2
1
1
1
,,,
1
k
k
n
n k n k n
X X b
Y
X X b
Y
X X b
Y X b ε
Y X b ε
3,Model assumption:
1,
2.
3,is non-random,
4,
5,Normality assumption
E( )?ε 0
2E( ) ( i s a u n i t m a t r i x )
nnε ε II
X
r a n k ( ) = 1knX
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
1? (b X X ) X Y
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? (b X X ) X Y
E ( )?bb
21v a r ( ) E [ ( ) ( ) ] ( )b b b b b X X
6,Residual and Estimation of
the population variance
1,Residual
1) P is idempotent (幂等的 )
2)
3)
4)
2?
[]
()
-1
-1
e Y Y Y X b I X ( X X ) X Y
P Y P I X ( X X ) X
E( )?e0
2v a r ( ) E ( )e e e P
2
1
t r ( )
n
i
i
e
Y Y b X Y e e
2,Estimator for
2?
2
2?
( 1 ) ( 1 )
ie
n k n k
ee
22?( ) ( )
1
ee
EE
nk
7,Goodness-of-fit testing
1,1) Total sum of squares:
2) Explained sum of squares:
2,Coefficient of determination:
2 2T S S ( )
iY Y n Y YY
2 2 2?E S S ( )
iiY Y e n Y b X Y
2
2
22
E S S
1
TS S ( )i
nY
R
Y Y n Y
e e b X Y
YY
3,Adjusted R-squared:
2
2
2
( 1 ) ( 1 )( 1 )
11
() 1
1
i
nRnk
R
YY nk
n
ee
8,Hypothesis testing
1,Significance test for the population
regression equation
1) Hypothesis:
2) F-statistic:
01,0 v skH b b
010
,t he r e e xs i t s su c h t ha t 0kH k b?
E S S
(,1 )
R S S
1
kF F k n k
nk
3) Testing
Given a significance level,pick up
the critical value,
if,reject,
otherwise,accept it.
2,Significance test for a single parameter
1) Hypothesis:
2) t-statistic,
F?
FF
0H
01,0 v s,0jjH b H b
() v a r ( )
jj
j j
bb
t
Sb b
9,Forecasting
1,Point forecast:
Given,
then a predictor
2,Interval forecast:
Let
then
1 1,1 2,1,1( 1,,,)n n n k nX X XX
11nnY Xb
0 1 1?nne Y Y
21
1 1 1v a r ( ) [ 1 ( ) ]n n ne?
X X X X
Let
Then
Homework:
1
1 1 1( ) 1 ( )n n ne
X X X X
1
1
( 1 )
()
n
n
e
t t n k
e?
对以下数据进行建模 分析 并做必要的假设检验（显著性水平 =5% ）
Y,国家财政收入（不包括债务收入）（ 单位亿元）
X1,国内生产总值（ 单位亿元）
X2,进出口贸易总额（ 单位亿元）
X3,国家外汇（ 单位亿美元）
资料如下表,
y x1 x2 x3
1978 1132.3 3624.1 355,0 1.67
1979 1146.4 4038.2 454,0 8.4
1980 1159.9 4517.8 570,0 - 12.96
1981 1175.8 4862.4 735.3 27.08
1982 12 12.3 5294.7 771.3 69.86
1983 1367,0 5934.5 860.1 89.01
1984 1642.9 7171,0 1201,0 82.2
1985 2004.8 8964.4 2066.7 26.44
1986 2122,0 10202.2 2580.4 20.72
1987 2199.4 11962.5 3084.2 29.23
1988 2357.2 14928.3 3821.8 33.72
1989 2664.9 16909.2 4155.9 55.5
1990 2937.1 18547.9 5560.1 110.93
1991 3149.5 21617.8 7225.8 217.12
1992 3483.4 26638.1 9119.6 194.43
1993 4349,0 34634.4 11271,0 211.99
1994 5218.1 46759.4 20381.9 516.2
1995 6242.2 58478.1 23499.9 735.97
1996 7408,0 67884.6 24133.8 1050.29
1997 8651.1 74462.6 26967.2 1398.9
1998 9876,0 78345.2 26896.3 1449.6
1999 11444.1 82067.5 29896.3 1546.75
2000 13395.2 89468.1 39274.2 1655.7
2001 16386,0 97314.8 42183.6 2121.65
2002 18913.9 102397.9 51378.3 2864.07
(Multiple linear regression model)
You are required to get familiar with matrix
algebra for mastering this chapter!
Classical Multiple linear regression
model (CMLRM):
1,model
2,random sample0 1 1 kkY b b X b X
1{ ;,}i i k iY X X
0 1 1 ( 1,,)i i k k i iY b b X b X i n
Matrix form:
1 1 1 0 1
1
1 2 2 1 2
1
1
1
,,,
1
k
k
n
n k n k n
X X b
Y
X X b
Y
X X b
Y X b ε
Y X b ε
3,Model assumption:
1,
2.
3,is non-random,
4,
5,Normality assumption
E( )?ε 0
2E( ) ( i s a u n i t m a t r i x )
nnε ε II
X
r a n k ( ) = 1knX
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
1? (b X X ) X Y
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? (b X X ) X Y
E ( )?bb
21v a r ( ) E [ ( ) ( ) ] ( )b b b b b X X
6,Residual and Estimation of
the population variance
1,Residual
1) P is idempotent (幂等的 )
2)
3)
4)
2?
[]
()
-1
-1
e Y Y Y X b I X ( X X ) X Y
P Y P I X ( X X ) X
E( )?e0
2v a r ( ) E ( )e e e P
2
1
t r ( )
n
i
i
e
Y Y b X Y e e
2,Estimator for
2?
2
2?
( 1 ) ( 1 )
ie
n k n k
ee
22?( ) ( )
1
ee
EE
nk
7,Goodness-of-fit testing
1,1) Total sum of squares:
2) Explained sum of squares:
2,Coefficient of determination:
2 2T S S ( )
iY Y n Y YY
2 2 2?E S S ( )
iiY Y e n Y b X Y
2
2
22
E S S
1
TS S ( )i
nY
R
Y Y n Y
e e b X Y
YY
3,Adjusted R-squared:
2
2
2
( 1 ) ( 1 )( 1 )
11
() 1
1
i
nRnk
R
YY nk
n
ee
8,Hypothesis testing
1,Significance test for the population
regression equation
1) Hypothesis:
2) F-statistic:
01,0 v skH b b
010
,t he r e e xs i t s su c h t ha t 0kH k b?
E S S
(,1 )
R S S
1
kF F k n k
nk
3) Testing
Given a significance level,pick up
the critical value,
if,reject,
otherwise,accept it.
2,Significance test for a single parameter
1) Hypothesis:
2) t-statistic,
F?
FF
0H
01,0 v s,0jjH b H b
() v a r ( )
jj
j j
bb
t
Sb b
9,Forecasting
1,Point forecast:
Given,
then a predictor
2,Interval forecast:
Let
then
1 1,1 2,1,1( 1,,,)n n n k nX X XX
11nnY Xb
0 1 1?nne Y Y
21
1 1 1v a r ( ) [ 1 ( ) ]n n ne?
X X X X
Let
Then
Homework:
1
1 1 1( ) 1 ( )n n ne
X X X X
1
1
( 1 )
()
n
n
e
t t n k
e?
对以下数据进行建模 分析 并做必要的假设检验（显著性水平 =5% ）
Y,国家财政收入（不包括债务收入）（ 单位亿元）
X1,国内生产总值（ 单位亿元）
X2,进出口贸易总额（ 单位亿元）
X3,国家外汇（ 单位亿美元）
资料如下表,
y x1 x2 x3
1978 1132.3 3624.1 355,0 1.67
1979 1146.4 4038.2 454,0 8.4
1980 1159.9 4517.8 570,0 - 12.96
1981 1175.8 4862.4 735.3 27.08
1982 12 12.3 5294.7 771.3 69.86
1983 1367,0 5934.5 860.1 89.01
1984 1642.9 7171,0 1201,0 82.2
1985 2004.8 8964.4 2066.7 26.44
1986 2122,0 10202.2 2580.4 20.72
1987 2199.4 11962.5 3084.2 29.23
1988 2357.2 14928.3 3821.8 33.72
1989 2664.9 16909.2 4155.9 55.5
1990 2937.1 18547.9 5560.1 110.93
1991 3149.5 21617.8 7225.8 217.12
1992 3483.4 26638.1 9119.6 194.43
1993 4349,0 34634.4 11271,0 211.99
1994 5218.1 46759.4 20381.9 516.2
1995 6242.2 58478.1 23499.9 735.97
1996 7408,0 67884.6 24133.8 1050.29
1997 8651.1 74462.6 26967.2 1398.9
1998 9876,0 78345.2 26896.3 1449.6
1999 11444.1 82067.5 29896.3 1546.75
2000 13395.2 89468.1 39274.2 1655.7
2001 16386,0 97314.8 42183.6 2121.65
2002 18913.9 102397.9 51378.3 2864.07
