Studenmund (2006): Chapter 10

1
Lecture 10
Studenmund (2006) Chapter 10
Heteroscedasticity
Heteroskedasticity

• Objectives
• What is heteroscedasticity?
• What are the consequences?
• How is heteroscedasticity be identified?
• How is heteroscedasticity be corrected?

2
The probability density function for Yi at three
different levels of family income, X1 i , are
identical.
3
Homoscedastic pattern of errors
The scattered points spread out quite equally
4
Heteroscedasticity Case
The variance of Yi increases as family income,
X1i, increases.
5
Heteroscedastic pattern of errors
The scattered points spread out quite unequally
6
Var(?i) E(?i2) ?i2 ? ?2
Definition of Heteroscedasticity
Refer to lecture notes Supplement 03A
7
Consequences of heteroscedasticity
1. OLS estimators are still linear and unbiased
8
Detection of heteroscedasticity
9
Detection of heteroscedasticity Graphical method
10
(No Transcript)
11
Statistical test (i) Park test
12
Example Studenmund (2006), Equation 10.21 (Table
10.1), pp.370-1
Park Test Practical In EVIEWS Procedure 1
PCON petroleum consumption in the
ith state REG motor vehicle registration TAX
the gasoline tax rate
13
Graphical detection
14
Procedure 2 Obtain the residuals, take square
and take log
15
Horizontal variable
16
(No Transcript)
17
(No Transcript)
18
Procedure 3 4
19
(ii)Breusch-Pagan test, or LM test
(3) Compute LMW n?R2
(4) Compare the W and ?2df (where the df is (q)
of regressors in (2)) if W gt ?2df gt reject
the Ho
20
Yi ?0 ?1X1i ?2X2i ?3X3i ?i
21
BPG test for a linear model PCON?0?1REG?2Tax?
The W-statistic indicates that the
heteroscedasticity is existed.
22
(No Transcript)
23
(iiia) Whites general heteroscedasticity test
(no cross-term) (The White Test)
(3) Compute W (or LM) n?R2
(4) Compare the W and ?2df (where the df is of
regressors in (2)) if W gt ?2df gt reject the Ho
24
(iiib) Whites general heteroscedasticity test
(with cross-terms) (The White Test)
(3) Compute W (or LM) n?R2
(4) Compare the W and ?2df (where the df is of
regressors in (2)) if W gt ?2df gt reject the Ho
25
With cross-term
No cross-term
26
White test for a linear model PCON?0?1REG?2Tax
? The W-statistic indicates that the
heteroscedasticity is existed.
27
(No Transcript)
28
Another example 8.4 (Wooldridge(2003), pp.258)
The White test for a linear model PCON?0?1REG?2
Tax? The test statistic indicates
heteroscedasticity is existed.
29
Testing the log-log model
W
30
Remedy Weighted Least Squares (WLS)
Suppose Yi ?0 ?1 X1i ?2 X2i
?i
E(?i) 0, E(?i ?j ) 0 i ? j
Vqr (?i2) ?i2 ?2 f (ZX2i) ?2Zi2
If all Zi 1 (or any constant), homoscedasticity
returns. But Zi can be any value, and it is the
proportionality factor.
In the case of ?2 was known To correct the
heteroscedasticity Transform the regression
gt Y ?0 X0 ?1 X1 ?2 X2 ?i
31
Why the WLS transformation can remove the
heteroscedasticity?
These three results satisfy the assumptions of
classical OLS.
32
These plots suggest variance is increasing
proportional to X2i2. The scattered plots
spreading out as nonlinear pattern.
gt Yi ?1 X0 ?2 X1 ?3 ?
Where ?i satisfies the assumptions of classical
OLS
33
Example Studenmund (2006), Eq. 10.24, pp.374
C.V.0.3392
34
(No Transcript)
35
OLS result
Refers to Studenmund (2006), Eq.(10.28), pp.376
36
(No Transcript)
37
Alternative remedy of heteroscedasticity
reformulate with per capita
38
Now, after the reformulation The test statistic
value indicates that the heteroscedasticity is
not existed.
39
W lt ?2df gt not reject Ho
40
This plot suggests a variance is increasing
proportional to X2i. The scattered plots
spreading out as a linear pattern
41
Transformation divided by the squared root term
--_at_sqrt(X)
42
Compare to the transformation divided by the REG,
the CV of That one is smaller.
43
Example Gujarati (1995), Table 11.5, pp.388
Simple OLS result RD 192.99 0.0319 Sales
SEE 2759 (0.194) (3.830)
C.V. 0.9026
44
?2(0.05, 2) 5.9914 ?2(0.10, 2) 4.60517
45
Observe the shape pattern of residuals linear or
nonlinear?
46
Transformation equations
47
Transformation divided by the squared root term
--_at_sqrt(X)
48
calculate the C.V. 0.8195
49
After transformation by _at_sqrt(x), the
W-statistic indicates there is no
heteroscedasticity
?2(0.05, 2) 5.9914 ?2(0.10, 2) 4.60517
W lt ?2df gt not reject Ho
50
After transformation by _at_sqrt(Xi), residuals
still spread out
51
Transformation divided by the suspected variable
(Xi)
52
Calculate the C.V. 0.7467
53
After transformation divided by the suspected X,
the W-statistic indicates there is no
heterose\cedasticity
?2(0.05, 2) 5.9914 ?2(0.10, 2) 4.60517
W lt ?2df gt not reject Ho
54
After transformation divided by Xi, residuals
spread out more stable