Transcript Slide 1
Autocorrelation
Hill et al Chapter 12
The nature of the problem
• Time series data:
– Observations follow a natural ordering
through time.
• Autocorrelation
– The error term contains a carryover from
previous shocks.
– Related to, or correlated with, the effects of
the earlier shocks.
Violation of assumption MR4
yt 1 2 xt 2 3 xt 3 et
E (et ) 0
var(et ) 2
cov et , es 0 for t s
Area response for sugar cane
ln At 1 2 ln Pt et
yt 1 2 xt et
yt 1 2 xt et
yt ln At
xt ln Pt
Least squares estimation
yˆ t = 6.111+ 0.971 xt
(0.169) (0.111)
R2 = 0.706
(std. errors)
First order autoregressive AR(1)
errors
yt 1 2 xt et
et et 1 vt
cov(vt , vs ) 0
var(vt ) v2
E (vt ) 0
Properties of an AR(1) error
Assumption:
1 1
Properties
E (et ) 0
2
var(et ) e2 v 2
1
cov et , et k e2k
corr(et , et k )
e2k
e2e2
k
Consequences of Autocorrelation
• The least squares estimator is still a linear
unbiased estimator, but it is no longer
best.
• The formulas for the standard errors
usually computed for the least squares
estimator are no longer correct
• Hence confidence intervals and
hypothesis tests that use these standard
errors may be misleading.
Transforming the model
yt 1 2 xt et
et et 1 vt
yt 1 2 xt et 1 vt
et 1 yt 1 1 2 xt 1
et 1 yt 1 1 2 xt 1
yt 1 2 xt yt 1 1 2 xt 1 vt
yt yt 1 1 1 2 xt xt 1 vt
y yt yt 1
xt1 1
xt2 xt xt 1
yt xt11 xt22 vt
t
Transforming the first observation
y1 1 2 x1 e1
2
var(et ) e2 v 2
1
1 2 y1 1 2 1 1 2 x12 1 2 e1
y1 x11 1 x12 2 e1
11
2
1
2
x 1
e 1 e1
y1 1 2 y1
x12 1 2 x1
v2
2
var(e ) (1 ) var(e1 ) (1 )
v
1 2
1
2
2
Implementing GLS
eˆt yt b1 b2 xt
eˆt eˆt 1 vˆt
T
ˆ
eˆ eˆ
t 2
T
t t 1
2
ˆ
e
t 1
t 2
Sugar cane area continued
x32
x32 ˆ x22
T
ˆ
eˆ eˆ
t 2
T
t t 1
eˆ
t 2
2.2919 (0.342)(2.1637)
1.5519
0.342
2
t 1
1
ln( Aˆt ) =6.164+1.007ln(Pt)
y 1 ˆ y1
2
1 0.342 2 3.3673
(0.213) (0.137)
3.1642
xt*1
xt
xt*2
yt
yt*1
1
0.93970
-2.5868
-2.4308
3.3673
3.1642
2
0.65799
-2.1637
-1.2790
4.2627
3.1110
3
0.65799
-2.2919
-1.5519
3.7677
2.2798
4
0.65799
-2.2045
-1.4206
4.4998
3.2215
The Durbin-Watson test
T
H0 : 0
H1 : 0
H1 : 0
H1 : 0
d
eˆ eˆ
t 2
t
T
2
ˆ
e
t
t 1
2 1 ˆ
t 1
2
Critical values for D-W
• The distribution of d under the null depends on
the values of the explanatory variables.
• Critical values cannot be tabulated.
• Use software to compute appropriate p-value.
• Use the bounds test.
– Define two new stats which do not have a distribution
dependent on the data.
– dL < d < du
Critical values for the bounds test
If d d Lc , reject H 0 : 0 and accept H1 : 0 ;
if d dUc , do not reject H 0 : 0 ;
if d Lc d dUc , the test is inconclusive.
Sugar Cane Example
d Lc 1.393
dUc 1.514
d 1.291 d Lc ,
A lagrange multiplier test
yt 1 2 xt et 1 vt
eˆt yt b1 b2 xt
t = 2.006 F = 4.022 p-value = 0.054
Points to note in testing for
autocorrelation
• In the LM test, the estimated residual for
t=1 is missing. Either omit the first
observation or make e0=0.
• D-W test is exact in finite samples, LM is a
large sample test.
• D-W is not valid when one of the variables
is a lagged dependent variable.
• The LM test can be used for higher order
forms of autocorrelation.
Prediction with AR(1) errors
y0 1 x02 e0
yˆ 0 b1 b2 x0
yT 1 1 2 xT 1 eT 1
1 2 xT 1 eT vT 1
eT yT ˆ 1 ˆ 2 xT
yˆT 1 ˆ 1 ˆ 2 xT 1 ˆ eT
yˆT h ˆ 1 ˆ 2 xT h ˆ heT