Chapter 6
Series correlation and Heteroscedasticity
Check the classical linear regression model whether holding classical assume or not!
→ E (ei )  0 ??
 2
0


2



 ?
→Var(e)=  2 I  
.


.


2
0
 

( no series correction and homoscedasticity (constant error variance))
＊ ∵The central-limit theorem, ∴we can use standard statistical tests for large sample
series( without normally assumption)
A.
Heteroscedasticity: unequal error variance, ie. var(ei) =  2
1. ˆ LSE is unbiased, inefficient, consistent
2. corrections for heteroscedasticity:
We can apply GLES (generalized least squares estimate)
Case 1: known Var (ei )   i2
yi   xi  ei ……………..the model
yi
Apply  i
V a (rei ) 
i

x
i

ei
i
= yi   xi  ei
*
*
*
1
v a re(i )  1,
 i2
ie. the transformed error term is homoscedasticity
form the transformed model to get ˆ G L S ,E ie. → ˆ GLSE is BLUE.
e
Ie. weighted least squares estimate ( to minimize  ( i ) 2 )
i
2 GLSE
2 LSE
→ R
<R
(∵ GLSE uses a transformed dep. Var.)
*
＊ In econometric modeling, the assumption of homoscedasticity will be unreasonable in
corss-setion studies (ex. a cross-section study of firms in one industry, or of family income
and expenditure). Heteroscedasticity does not usually occur in time series studies.
1
Case 2: if unknown Var (ei )
(1) ex. the housing expenditure model, first, using OLS method to estimate model, and
testing whether the heteroscedasticity is present or not.
(2) To find the sample variance of regression  i2
(3) follow the case 1, get the transformed model → ˆ OLS
case 3: if Var (ei )  c X 2 K i c≠0, constant.
……error variance vary directly with an independent variable.
(1) finding transformed model:
(
1
relative weights)
X Ki
X KI
Yi

X
X
e
 0  1 1i   2 2i   K
 i
X Ki
X Ki
X Ki
X Ki
X Ki X Ki
Y *   K   0 X 0*  1 X 1*   2 X 2*  e *
→ Var (ei ) 
*
1
var( ei )  c , transform data into a constant variance of error
X K2 i
(2) 可由 the transformed model to get ˆ LSE
(= ˆ GLSE )
→ this case is likely to occur quite frequently in both macroeconomics and microeconomic
empirical studies.
2
3. test for heteroscedasticity:
H0:  12   22  ....   g2
H1:  i2  cX K2 i
The Goldfeld-Quandt test:
找出 與  ê2 遞增有關的 independent variable,
(1) 將所有觀察值按 X K 值(遞增) to rearrange
(2) deleted the middle d obs. (approximately 1/5 )
ex total 50 obs. delete 10 obs , obs 1-20, 31-50 two part
(3) Run 2 separate regressions; regression 1 associated with low values of X K , and
regression 2 associated with high values of X K .
Each regression involves (N-d)/2 obs.
and (N-d-2K)/2=[(N-d)/2]-K degrees of freedom
can get ESS1 and ESS2 (the error sum of square associated with each regression)
(4) Assuming that ê is normally distributed (and no serial correlation)
ESS2/ESS1 ~ F (N-d-2k)/2, (N-d-2K)/2, α
→if F < critical value , accept HO (homoskcedasticity)
If F> critical value, reject HO
3
B.
serial correlation (or autocorrelation)
if
E( e t , es )≠0 as t≠s,
ie. correlation between disturbances≠0
var( e t )=  i2 t= 1, 2, …, T
→ ˆ LSE is unbiased, inefficient, consistent ( if explanation var. without lagged dep. Var.)
→ ˆ GLSE is BLUE
㎝
＊   0 : ˆ is biased downward (smaller), ∴ ˆ ˆ is smaller the value of t is getting large,
easy to reject H0 , ( type I error ),
get the error conclusion
＊ êt with pattern, each êt not independent
4
1.
1st order autoregressive disturbance, AR(1)
0   1
et   et 1   t ,
 t ~ (0,  2 I )
obey assumption of SLRM
E( et )= 0, var(e)=  2 
 1

1  

1  2  
 T 1
 
 2   T 1 

   T 2 

1

 T 2

1




 2
Prove: var(et)=
;
1  2
 var( et )  E (et2 )  E[(  et 1   t ) 2 ]  E (  2 et21   t2  2  et 1 t )
=  2 E (et 1 ) 2  E ( t2 )   2 var( et )   2
Cov( et , et 1 )= E (et  et 1 )  E[(  et 1   t )  et `1 ]  E (  2 et21  et 1 t )
 2

=  E (et 1 )   var( et 1 )   var( et ) 
1  2
cov( et , et 1 )
cov( et , et 1 )


var( et )
[var( et )]1 / 2  [var( et 1 )]1 / 2
2
ie.  is the correlation coefficient between et and et 1 .
＊ using GLSE:
a. if small sample
PP'  I ,
P' P   1
 1  2

 
→ P  0
T T

 

 0
0  
1
0 
 1 0






0
0

0
0



1
→ The transformed model
 1  2 Y 
1


 Y2  Y1 
P   Y3  Y2   Y *
T 1







 YT  YT 1 



Pe  
T 1




5
1   2 e1 

e2   e1 
e3   e2 




eT   eT 1 
 e*
 1  2

1 
PX  
T K


 1  
1   2 X 12
X 22  X 12

X T 2  X T 1, 2

1   2 X 1K

X 2 K  X 1K


 X TK  X T 1, K


  X*



 var( e * )  var( P e)  E ( P e e' P' )  PE (ee' )  P'   2 PP'   2 I
→ ˆ
∵ Can using OLSE run transformed model
b. if large sample;
P0
(T 1)T
 
 0

 0

 
 0
1

0
0 
1 
 1






0
0
0
0


1
P0 : loosing the 1st obs. of P. (drop the 1st obs. of PX)
P 0Y  P0 X 0  P0 e
→ the transformed model
# obs.= T-1
＊ ie. Yt  Yt 1  ( X t   X t 1 )  0  (et   et 1 ) , t=2,3,….T
＊
if know the ρ , OLSE the transformed model = an approximate GLSE → ̂ 0
＊
̂ 0
is not efficient as ˆ in finite sample
(b.)
(a)
2.
AR(2):
 A22


P  0
T T



et  1 et 1   2 et 2   t
 2
0

0
 1
 2

0
1
 1
0
1



 2
 1

1st


0


 2nd


0 ,  PY 


other obs.




Yt  1Yt 1   2Yt  2 


1
6
→ 1st, 2nd obs. Forms more complex, so drop two obs. and apply GLSE method estimate.
IF the model : Yt    X t  et …(1) we can get the transformed model:
＊ Yt   (1  1   2 )  ( X t  1 X t 1  2 X t 2  1Yt 1   2Yt 2   t . (2) t = 3, 4,….T
(a) apply OLSE estimate the equation (1) → get êt
(b) regress êt on eˆt 1 , eˆt  2 → get ̂1 , ̂ 2
(c) Take ̂1 , ̂ 2 into the Equation (2), and apply OLSE estimate equation (2) → get ˆ , ˆ
3. test for series correlation
a. Durbin –Watson test:
, i.e.   0
, i.e.   0, or   0
H0: AR(0)
H1: AR (1)
T
D.W . 
 (eˆ
t 2
for AR(1) only
t
 eˆt 1 ) 2
=
T
 eˆ
t 1
2
t
cov( et , et 1 )
var( e~t )
T: #obs
Table; d l and dU
(1) dU  D. W .  2
d l  D. W .  dU
0  D. W .  d l
(2) 4  d l  D. W .  4
4  d l  D. W .  4  dU
2  D. W .  4  dU
accept H0
result indeterminate
reject H0,   0
(positive serial correlated present)
reject H0,   0 (negative serial correlated present)
result indeterminate
accept
H0 ,
7
＊ (1) It’s a test for AR(1).
(2) The regression must include a constant.
(3) If outliners are present in the residuals, the statistic may be misleading.
(4) In appropriate when the independent variables include lagged dependent variable.
In this case, the appropriate test is the Durbin h test.
h  (1 
1
T
D. W .)
~ N (0, 1)
2
1  T  var( ˆ )
4. the generation of forecasting in AR case:
a. AR(1) case: ex Yt    X t  et ,
et   et 1  vt , t =1,2,…, T
→ must to use et   et 1  vt to forecast eˆt 1
eT 1   eT  vT 1
 eT 1  ˆ (YT  ˆ  ˆX T )  ˆ eˆT ,
ˆ , ˆ : GLSE
( ∵the systematic component of disturbance , ∴ can estimate it.)
 YˆT 1  ˆ  ˆX T 1  ˆ eˆT
Systematic component of regression model
eT 2   eT 1  vT 2
eˆT  2  ˆ eˆT 1  ˆ 2 eˆT
 YˆT  2  ˆ  ˆX T  2  ˆ 2 eˆT
YˆT l  ˆ  ˆX T l  ˆ l eˆT
is the best linear unbiased forecasting.
b. AR(2) case: Yt    X t  et ,
→ eT 1  1 eT   2 eT 1  vT 1
YˆT 1  ˆ  ˆX T 1  ˆ 1 eˆT  ˆ 2 eˆT 1
et  1 et 1   2 et 2  vt
t =1,2,…, T
 eˆT 1  ˆ1 eˆT  ˆ 2 eˆT 1
is the best linear unbiased forecasting.
.
8
＊ cov( et , et  s )   S  e2
S: arbitrary integer if AR(1)
 et 1  et 2  vt 1
<Prove> et   et 1  vt   2 et  2   vt 1
=  3 et 3   2 vt  2   vt 1  vt  ....
=  S et S   S 1vt ( S 1)   S 2 vt ( S 2)  ....  vt
et  et S   S et2S   S 1et S vt ( S 1)   S 2 et s vt ( S 2)  .
E( )=0
 E (et , vt )  0
∴ Cov(et , et  s )  E (et , et  S )   S E (et2 S )   S  e2
＊ perform transformation, i.e. generalized difference ;
Yt   0  1 X 1t   2 X 2t  et
if model:
∴ Yt 1   0  1 X 1t 1   2 X 2t 1  et 1 → Yt 1   0  1 X 1t 1   2 X 2t 1   et 1
→ (Yt  Yt 1 )   0 (1   )  1 ( X 1t  X 1t 1 )   2 ( X 2t  X 2t 1 )  (et   et 1 )
Yt*
 0*

→ assume   1

1 X 1t *
 2 X 2t *


*
et 1
perform transformation = first difference
Yt *  Yt  Yt 1 ； X t*  X t  X t 1 ；  0 (1   )  0,  no cons tan t term
＊ Durbin –Watson test:
for AR(1):
when the independent variables include lagged dependent variable, the appropriate test is the
Durbin h test.
→ if T× vâr( ˆ ) >1, the Durbin h test is not valid.
Other testing method:
(the model Yt    Yt 1  X t  et )
(1) run OLS → get êt
(2) To drop the 1st obs. for simplicity
(i.e. # obs. =T-1)
(3) OLS : eˆt   *  * eˆt 1   *Yt 1   * X t   t
→ H 0 :  *  0,
＊ ˆ 
Cov(eˆt , eˆt 1 )
,
Var (eˆt )
H1 :  *  0
if AR(1)
9
(using t test)
3. Alternative procedures for ̂ :
a. the Cochrance-Orcutt procedure:
(1) OLS the original model
(2) OLS et   et 1  vt
→ get êt
→ get ̂
(3) perform the transformed regression: (i.e. transform Y and X’s by generalized difference)
Yt*  Yt   Yt 1 ； X t*  X t   X t 1
ˆ
ˆ
→ get ˆ 0 , ˆ1, .......
ˆ
ˆ
(4) take the ˆ 0 , ˆ1, ....... into the original model → get êˆt
(5) OLS eˆˆt   eˆˆt 1   t → get ̂ˆ
Repeater above steps.
(6) When
→ get new ̂
the new ˆ  the old ˆ <0.01 or 0.05, or after 10 or 20 times estimates of ρ have
been obtained, then stop the iterations.
[ there is no guarantee that the final ̂ will be the optimal estimate (i.e. min.  eˆi2 ), because
it may lead to a local rather than a global minimum .]
b. the Hildreth-Lu procedure:
try difference ρ ,
-1< ρ < 1, ρ≠0
→ choose ρ , whth min. ESS,
→ OLS the transformed model
this method  MLE .
c. the Durbin produre:
OLS
Yt  Yt 1   0 (1   )  1 ( X 1t  X 1t 1 )  vt → get ̂
[∵ if the transformed model: Yt  Yt 1   0 (1   )  1 ( X 1t  X 1t 1 )  vt ]
[＊ take ̂ into the transformed model, and OLS re-estimated ˆ →GLSE ]
10