Chapter 7 自相关
Autocorrelation or
Serial correlation
1,Suppose the linear regression model is
如果 不恒成立,则称误差项是序列相关的 (自相关的 )。 (serial correlation)
2,产生序列相关的原因及序列相关的影响
1)原因
( 1)经济行为的惯性或冲击的惯性( SARS)
( 2)模型误设( Model misspecification)
2)序列相关对估计与检验的影响与异方差性类似。
0 1 1t t k k t tY = b b X b X +
c o v (,) 0 ( )ij ij
3,如误差项的序列相关具有形式则称为一阶序列相关,其中 。
:正序列相关
:负序列相关
:不序列相关
Assumption:
1t t t
1
01
10
0
22E ( ) 0,E ( ) c on st,
tt
E ( ) 0 ( )ts ts
4,有关推论
1) 2)
3)
4)
0
s
t t s
s

( ) 0 ( )stE s t
2 2 2v a r ( ) ( 1 ) 1,,
t tn
1
2
2
12
1
1
()
1
n
n
nn
E










ε ε
5)
11
112
22
1
c ov (,) c ov (,)
[ v a r ( ) ] [ v a r ( ) ]
t t t t
tt




Tests for Autocorrelation
1,Durbin-Watson test
1) Suppose the linear regression model is
一阶自相关系数,。 将 用残差代替并运用 OLS于以上 AR(1)可得估计
0 1 1 0 ( 0 )t t k k t tY = b b X b X + b
1 ( A R ( 1) )t t t
1
2
c ov (,)tt


2
1122?
nn
t t ttte e e
t?
2) Hypothesis,
Construct the D-W statistic01
,0 v s,0HH
2
1
12
2
2 1
1
()
2 1 2( 1 )
n
tt
ttt
n
t
t
t
ee
ee
d
e
e






04d
0 ( 2
0 ( 0 2
0 ( 4 2
d
d
d



不 相 关 )
正 相 关 )
负 相 关 )
3) 检验判断:
ULdd上 临 界 值,,下 临 界 值,
0
0
0
( 1 ) r e je c t
( 2) 4 - 4 r e je c t
( 3 ) 4 no t r e je c t
( 4) O the r w ise,n o c on c l usion s
L
L
UU
o d d H
d d H
d d d H



2,Test for autocorrelation when a lagged
dependent variable serves as an
independent variable
Suppose that
Durbin h statistic (T is the observation No.)
1t t t tY Y X
(0,1 )?
1 v a r ( )
ThN
T
( 1 )?
2 1 v a r ( )
dTh
T?

自相关修正
1,Generalized Differencing(广义差分法)
It is necessary that is known.
2,Cochrane-Orcutt Approach(P101)?
Generalized Least
Squares Method
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
Assumptions:
1)
2) 否则,存在异方差性或自相关,也可能两者同时存在。
E( )?ε 022E ( )
ij nnε ε
r a n k ( ) = 1knX
I 同 方 差 且 无 自 相 关
Generalized least squares approach
Since is a positive definite matrix,there
exists an invertible matrix such that
Thus,
Let
Then (A)
P
P Ω P = I
P Y = P X b + P ε
* * *Y = P Y,X = P X,ε =P ε
* * *Y = X b + ε
It is easy to get
And the rest of the assumptions still holds.
The OLS applies to (A),Hence
* * * 2E( ),E( )ε 0 ε ε I



* * - 1 * *
-1
- 1 - 1 - 1
b = ( X X ) X Y
[ ( PX ) PX ] ( PX ) PY
(X Ω X ) X Ω Y
Homework:分别用广义差分法和 Cochrane-
Orcutt法为以下数据建模:
Y X
1967 4010.000 22418.00
1968 3711.000 22308.00
1969 4004.000 23319.00
1970 4151.000 24180.00
1971 4569.000 24893.00
1972 4582.000 25310.00
1973 4697.000 25799.00
1974 4753.000 25886.00
1975 5062.000 26868.00
1976 5669.000 28134.00
1977 5628.000 29091.00
1978 5736.000 29450.00
1979 5946.000 30705.00
1980 6501.000 32372.00
1981 6549.000 33152.00
1982 6705.000 33764.00
1983 7104.000 34411.00
1984 7609.000 35429.00
1985 8100.000 36200.00