2009-7-24 1
Chapter 5
How to use the multiple linear regression
model
22009-7-24
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(,,; )kY F X X
0 1 1 1 1( ) (,,) (,,)k k k kf Y g X X g X X
0 1 1 2 2e x p ( )Y X X
32009-7-24
2,Reciprocal model:
3,Double-log(constant elasticity) model:
0 1 1 2 2
1Y
XX
l n l nYX
ln
E l a stic ity ( c o n st,)
ln
d Y X d Y
d X Y d X
N o t e,
42009-7-24
But the following model is not inherently
linear:
For others,see page72-73
12Y X X
52009-7-24
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,
62009-7-24
A dummy variable is a binary variable,that
is,
Example,
1 i f A h a p p e n s
0 i f A h a p p e n s
D
1,
0,
D
解 放 后解 放 前
1,s t u d e n t
0,t e a c h e r
D
*
*
*
1,
,1 9 7 9
0,t
tt
Dt
tt
72009-7-24
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,t t t t t
C b b X b D D?
反 常 年 份正 常 年 份
0 1 1 1 2 2
1,
( ),
0,t t t t t
C b b b D X D?
正 常 年 份反 常 年 份
0 0 1 1 1 1
1,
( ),
0,t t t t t t
C b b D b b D X D?
正 常 年 份反 常 年 份
82009-7-24
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,
,1 9 7 9
0,
t
tt
Dt
tt
*0 1 2 ()t t t t ttY b b X b X X D
92009-7-24
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 1k q k q k q k q k kY X X X X
01,0k q kH
102009-7-24
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)
22
R U R U R R
,2
U R U R
( E S S E S S ) / ( ) /
E S S / ( ) ( 1 ) / ( )q n k
q R R qF
n k R n k?
112009-7-24
Regression model with random
explanatory variables:
1)若 X与误差项独立,则前面所得结论仍成立,
2)若 X与误差项不相关,则 OLSE是一致估计也是渐进无偏的,
3)若 X与误差项相关,则 OLSE不是一致估计也不是渐进无偏的,OLS失效,
Homework,P87,ex5.1 and 5.2
Chapter 5
How to use the multiple linear regression
model
22009-7-24
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(,,; )kY F X X
0 1 1 1 1( ) (,,) (,,)k k k kf Y g X X g X X
0 1 1 2 2e x p ( )Y X X
32009-7-24
2,Reciprocal model:
3,Double-log(constant elasticity) model:
0 1 1 2 2
1Y
XX
l n l nYX
ln
E l a stic ity ( c o n st,)
ln
d Y X d Y
d X Y d X
N o t e,
42009-7-24
But the following model is not inherently
linear:
For others,see page72-73
12Y X X
52009-7-24
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,
62009-7-24
A dummy variable is a binary variable,that
is,
Example,
1 i f A h a p p e n s
0 i f A h a p p e n s
D
1,
0,
D
解 放 后解 放 前
1,s t u d e n t
0,t e a c h e r
D
*
*
*
1,
,1 9 7 9
0,t
tt
Dt
tt
72009-7-24
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,t t t t t
C b b X b D D?
反 常 年 份正 常 年 份
0 1 1 1 2 2
1,
( ),
0,t t t t t
C b b b D X D?
正 常 年 份反 常 年 份
0 0 1 1 1 1
1,
( ),
0,t t t t t t
C b b D b b D X D?
正 常 年 份反 常 年 份
82009-7-24
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,
,1 9 7 9
0,
t
tt
Dt
tt
*0 1 2 ()t t t t ttY b b X b X X D
92009-7-24
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 1k q k q k q k q k kY X X X X
01,0k q kH
102009-7-24
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)
22
R U R U R R
,2
U R U R
( E S S E S S ) / ( ) /
E S S / ( ) ( 1 ) / ( )q n k
q R R qF
n k R n k?
112009-7-24
Regression model with random
explanatory variables:
1)若 X与误差项独立,则前面所得结论仍成立,
2)若 X与误差项不相关,则 OLSE是一致估计也是渐进无偏的,
3)若 X与误差项相关,则 OLSE不是一致估计也不是渐进无偏的,OLS失效,
Homework,P87,ex5.1 and 5.2
