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   e2k
corr(et , et  k ) 
e2k
e2e2
 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
xt1  1  
xt2  xt  xt 1
yt  xt11  xt22  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 x12  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  x02  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