Chapter 3 一元线性回归模型第一节 回归分析与回归方程回归分析,
1.根据经济理论或考察样本数据去设定回归方程
Y,dependent variable;
:independent
:random error or disturbance term
(,,)Y f X Z
,XZ
A special and simple case (univariate linear
regression model),
这是本章研究的重点。
2,参数估计 (Estimation of parameter)
3,Testing
4,Predicting
设有样本为,则
01Y b b X
{,}iiYX
01 ( 1,,)i i iY b b X i n
模型的假设,
1.
2,(同方差 )
3.
4,
满足这四条件的 LRM称为经典线性回归模型 (CLRM)。
( ) 0iE
( ) 0ijE i j
2v ar ( )
i
( ) 0iiEX
由假设得
Population regression equation (function)
The pity is the parameters are unknown.
我们要利用样本来估计参数,如得参数估计值
,则 称为
sample regression equation (function).
How to estimate them? The OLS method.
01()E Y b b X
01bb和
01Y b b X
普通最小二乘法 (Ordinary least squares
procedure):
求 使残差平方和最小,
Let
Then (OLSE)
01bb和
2 2 2
01
( ) [ ( ) ]
i i i i ie Y Y Y b b X
a n di i i ix X X y Y Y
2
1
01

i i ib x y x
b Y b X



The properties of the OLSE:
1,无偏性 (unbiased):
2.
1 1 0 0
( ),( )E b b E b b
22
2
10 22
v a r ( ),v a r ( ) i
ii
X
bb
x n x


2
01 2
c o v (,)
i
X
bb
x

3,关于样本 的线性性,
4,Gauss-Markov theorem,如果是经典线性回归模型 (CLRM),则其参数的 OLSE
为 BLUE。即,在所有线性无偏估计中,OLSE的方差最小。
{}iY
10 2
1,( ) ( )i
i i i i i
j
x
b k Y b Xk Y k
nx

01 ( 1,,)i i iY b b X i n
Estimation of the variance of the random
disturbance term,:
We know and it is unknown,
Thus,
and so on are also unknown,To estimate
them,we have to first evaluate,It is not
difficult to show that
is an unbiased estimator for,
2?
2 v a r( )
i
22
2
10 22
v a r ( ),v a r ( ) i
ii
Xbb
x n x


2?
22? ( 2 )
ien
2?
Where
are the residuals,
Example3.1(P39)(how to use Eviews)
(iieY
01
()
i i ie Y b b X
模型的假设：
5,Normality assumption:
The properties of the OLSE:
5,
2( 0,)
i N
22
2
1 1 0 022
(,),(,)i
ii
X
b N b b N b
xx

Model testing(模型的检验 ):
总离差分解公式,
即,
TSS = ESS + RSS
TSS,Total sum of squares
ESS,Error (residual) sum of squares
RSS,Regression (explained) sum of
squares
222( ) ( ) ( )
i i i iY Y Y Y Y Y
2 2 2?
i i iy e y
1.Goodness-of-fit testing(R2检验 ):
Coefficient of determination(判定系数 ):
In general,the larger R2,the better.
2,Sample coefficient of correlation:
2
2
2
E S S R S S
1
TS S TS S
i
i
y
R
y

2rR
3,Hypothesis testing
We have known
Let (standard error)
22
2
1 1 0 022
(,),(,)i
ii
X
b N b b N b
xx

22? ( 2 )
ien
2
1
()
iS b x
Then
And we can test the following hypothesis:
Moreover,interval estimator for
is
11
1
( 2 )
()
bb
t t n
Sb

0 1 1 1,v e r s u s,H b c H b c
(1 )
1b
11
2
()b t S b

Forecasting(预测 ) 1,Point forecasting
Since we know
and the sample regression equation
then given,what about and?
As
(an unbiased estimator for )
01Y b b X
01
Y b b X
1nX?
1nY? 1()nEY?
1 0 1 1 1 0 1 1 1
( ) ( ) ( )
n n n n nE Y E b b X b b X E Y
1()nEY?
and (误差均匀 )
Naturally,we use as a point predictor for both
and,
2,Interval forecasting
(1) Forecast interval for
Forecast error:
1nY?
11?( ) 0nnE Y Y
1?nY?
1()nEY?
1nY?
1 1 1
n n ne Y Y
The variance of the forecast error:
Therefore
2
11
2
1 0 1 1 1 0 1
v a r ( ) ( )
v a r ( ) v a r ( ) v a r ( ) 2 c o v (,)
nn
n n n
e E e
b X b X b b?



2
2 1
2
()1
1 n
i
XX
nx




2
2 1
1 2
()1
( 0,1 )nn
i
XX
eN
nx




It is a pity that is unknown,Fortunately,
we have
Thus,
Hence,a Forecast interval for is
22? ( 2 )
ien
2?
1 1 1
22
22 11
22
( 2 )
( ) ( )11
11
n n n
nn
ii
e Y Y
tn
X X X X
n x n x







1nY?(1 )
(2) Forecast interval for
Similarly,we can obtain a forecast
interval for,
2
1
1 2
2
()1?
1 nn
i
XX
Yt
nx?


1()nEY?
(1 )
1()nEY?
2
1
1 2
2
()1?
nn
i
XX
Yt
nx?