复旦管理学院考博资料——计量经济学Lecture 7
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The OLS estimators are still linear, unbiased, consistent, and asymptotically normal
However, they are no longer efficient Normally underestimating the error variance
4
First-order Autocorrelation: AR(1)
Specification
ut ut1 t
E
(t
)
0,Var(t
)
2
,
Cov(t
,
ts
)
0
To ensure that AR(1) is a stationary stochastic process, 1
Properties
Inertia in most economic time series: GDP, Price indexes, etc
Specification bias
Omitted variables Incorrect functional forms Lagged dependent variables
0
0
M
0
0L
1
1
2
0
0
L 1 2
1
MA (q)
ut t 1t1 2t2 L qtq
8
Relations between AR and MA
AR (1) and MA (∞)
ut ut1 t Lut t (1 L)ut t
ut (1 L)1t (I L 2L2 L )t
Specification
Properties
ut t t1, 1
E(ut ) 0
Var(ut )
0
E(ut2 )
(1
2
)
2
2
Cov(ut , ut1)
E (ut ut 1 )
1
2
Cov(ut , uts ) 0, s 2
1
1 0
1 2
, s
0, s
2
7
First-order Moving Average: MA(1)
autocorrelation coefficient converges to zero.
Var-Cov matrix
AR (p)
1 L T1
Var(u) 2
M
1
L
T
2
M O M
T 1 T 2 L
1
ut 1ut1 2ut2 L put p t
6
First-order Moving Average: MA(1)
MA(1) suggests that ut has a rather short memory: The errors are only interrelated in two successive
periods
Var-Cov matrix
1
1 2 L
0
1 2
1L
0
Var(u)
2
M
MO M
Corr(ut ,uts )
Cov(ut ,uts X) Var(ut X)Var(uts
X)
s 0
s
2
Var(u
X)
E(uu'
X)
1
1 L 2 L
T T
1 2
2
1 1
1 L T 1
1
L
T
2
M M O M M M O M
T 1 T 2 L
2
T
1
T 2
L
1
2
Sources of Serial Correlation
Data transformation: Quarterly data, first differencing, etc
Nonstationarity: Mean, variance, and covariance of a time series do not change over time
Econometrics (I)
Lecture 7 Autocorrelation and Regression with Time Series Data
Dr. Sun Pei (孙霈) Associate Professor in Industrial Economics School of Management, Fudan University
E(ˆ 2 X) E( uˆ 'uˆ X) 2
T k
Underestimating Var(βˆ) and inflating t-statistics The problems are not resolved by using large
t t1 2t2 L qtq q0
MA(1) and AR (∞)
ut t t1 (I L)t
t (I L)1ut (I L 2L2 L )ut
ut ut1 2ut2 L
ut
ut 1
u2 t2
L
t
(1) p1 put p t
p 1
9
Consequences of Autocorrelation
Serially Correlated Disturbances
For time series regression model
Yt 1 2 Xt2 L k Xtk ut ,t 1, 2,L ,T Var(ut X) E(ut2 X) 2 0 ,Cov(ut ,uts X) E(ututs X) s 0
E(ut ) 0
Var(ut ) 0
1
2
2
2
Cov(ut , uts ) s
2 s , s
Leabharlann Baidu
s 0
s
5
First-order Autocorrelation: AR(1)
AR(1) suggests that ut has a long memory, but when s, the time interval, goes to infinity, the
3
The Lag Operator
LYt = Yt-1, L2Yt = L(LYt) = Yt-2 Lq Yt = Yt-q , L0 = I, L0Yt = Yt
The lag operator can be treated as a scalar Polynomials
p (L) I 1L 2L2 L p Lp p (L)Yt Yt 1Yt1 2 Lt2 L p Lt p (I L)1 I L 2L2 L
However, they are no longer efficient Normally underestimating the error variance
4
First-order Autocorrelation: AR(1)
Specification
ut ut1 t
E
(t
)
0,Var(t
)
2
,
Cov(t
,
ts
)
0
To ensure that AR(1) is a stationary stochastic process, 1
Properties
Inertia in most economic time series: GDP, Price indexes, etc
Specification bias
Omitted variables Incorrect functional forms Lagged dependent variables
0
0
M
0
0L
1
1
2
0
0
L 1 2
1
MA (q)
ut t 1t1 2t2 L qtq
8
Relations between AR and MA
AR (1) and MA (∞)
ut ut1 t Lut t (1 L)ut t
ut (1 L)1t (I L 2L2 L )t
Specification
Properties
ut t t1, 1
E(ut ) 0
Var(ut )
0
E(ut2 )
(1
2
)
2
2
Cov(ut , ut1)
E (ut ut 1 )
1
2
Cov(ut , uts ) 0, s 2
1
1 0
1 2
, s
0, s
2
7
First-order Moving Average: MA(1)
autocorrelation coefficient converges to zero.
Var-Cov matrix
AR (p)
1 L T1
Var(u) 2
M
1
L
T
2
M O M
T 1 T 2 L
1
ut 1ut1 2ut2 L put p t
6
First-order Moving Average: MA(1)
MA(1) suggests that ut has a rather short memory: The errors are only interrelated in two successive
periods
Var-Cov matrix
1
1 2 L
0
1 2
1L
0
Var(u)
2
M
MO M
Corr(ut ,uts )
Cov(ut ,uts X) Var(ut X)Var(uts
X)
s 0
s
2
Var(u
X)
E(uu'
X)
1
1 L 2 L
T T
1 2
2
1 1
1 L T 1
1
L
T
2
M M O M M M O M
T 1 T 2 L
2
T
1
T 2
L
1
2
Sources of Serial Correlation
Data transformation: Quarterly data, first differencing, etc
Nonstationarity: Mean, variance, and covariance of a time series do not change over time
Econometrics (I)
Lecture 7 Autocorrelation and Regression with Time Series Data
Dr. Sun Pei (孙霈) Associate Professor in Industrial Economics School of Management, Fudan University
E(ˆ 2 X) E( uˆ 'uˆ X) 2
T k
Underestimating Var(βˆ) and inflating t-statistics The problems are not resolved by using large
t t1 2t2 L qtq q0
MA(1) and AR (∞)
ut t t1 (I L)t
t (I L)1ut (I L 2L2 L )ut
ut ut1 2ut2 L
ut
ut 1
u2 t2
L
t
(1) p1 put p t
p 1
9
Consequences of Autocorrelation
Serially Correlated Disturbances
For time series regression model
Yt 1 2 Xt2 L k Xtk ut ,t 1, 2,L ,T Var(ut X) E(ut2 X) 2 0 ,Cov(ut ,uts X) E(ututs X) s 0
E(ut ) 0
Var(ut ) 0
1
2
2
2
Cov(ut , uts ) s
2 s , s
Leabharlann Baidu
s 0
s
5
First-order Autocorrelation: AR(1)
AR(1) suggests that ut has a long memory, but when s, the time interval, goes to infinity, the
3
The Lag Operator
LYt = Yt-1, L2Yt = L(LYt) = Yt-2 Lq Yt = Yt-q , L0 = I, L0Yt = Yt
The lag operator can be treated as a scalar Polynomials
p (L) I 1L 2L2 L p Lp p (L)Yt Yt 1Yt1 2 Lt2 L p Lt p (I L)1 I L 2L2 L