10.1
Heteroskedasticity
Objectives
•
•
•
•
What is heteroskedasticity?
What are the consequences?
How is heteroskedasticity identified?
How is heteroskedasticity corrected?
ECON 7710, 2010
10.2
Main empirical model for Unit 10:
foodexpi = 0 + 1incomei + i.
foodexp: Family food expenditure
income : Family income
Least squares estimates, US data (UE_Tab0301)
ˆf oodexp
se
i
 40 . 77  0 . 128
 22 . 14 
 0 . 031 
***
Income
R  0 . 3171 , N  40.
2
Is this the best estimated equation?
ECON 7710, 2010
i
10.3
1. The Nature of Heteroskedasticity
In a regression about firms, for the same
mistake,
million
billion
ECON 7710, 2010
10.4
Heteroskedasticity is a problem that
occurs when the error term does not
have a constant variance.
CLRM: Each error term comes from the same
probability distribution.
Assumption CLRM.5 is violated!
ECON 7710, 2010
10.5
Regression Model
Yi = 0 + 1X1i + 2X2i + i
zero mean:
E(i|X1i,X2i) = 0
homoskedasticity:
var(i|X1i,X2i) = s 2
no autocorrelation: cov(i, j|X1i,X2i,X1j,X2j) = 0
i= j
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10.6
Identical distributions for
observations i and j
Distribution for i
Distribution for j
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10.7
Homoskedasticity
Yi = 0 + 1Xi + i
var(i|Xi) = s2 for all i
f(Y)
.
.
0
X1
X2
X3
.
X4
ECON 7710,
2010
Conditional
Distribution
.
X
10.8
Heteroskedasticity
Yi = 0 + 1Xi + i
var(i|Xi) = si2 for all i
ECON 7710, 2010
Conditional Distribution
10.9
ECON 7710, 2010
10.10
ECON 7710, 2010
10.11
Pure heteroskedasticity
Different variances of the error term.
Correctly specified PRF.
Impure heteroskedasticity
Different variances of the error term.
Specification error.
ECON 7710, 2010
10.12
2. Detecting Heteroscedasticity
2.1 Graphical Method
Plotting foodexp against income
(for one regressor)
Scatter Diagram of Regressing foodexp on income
280
200
foodexp
Example 1:
Food expenditure,
US Data
(UE_Tab0301)
240
160
120
80
40
200
400
600
800
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income
1,000
1,200
10.13
Example 1: Food expenditure, US Data,
UE_Tab0301
Plotting e2 against
income.
Plotting e
against income.
7,000
120
6,000
80
squared residual
5,000
residual
40
0
4,000
3,000
2,000
-40
-80
200
1,000
400
600
800
1,000
1,200
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income
0
200
400
600
800
income
1,000
1,200
10.14
Example 2: textbook data, (Woody3)
***
***
***
**
ˆ
Y  102,192  9, 075 N  0.35 P  1.29 I
se
R  0.6182, N = 33.
2
40,000
30,000
residual
20,000
10,000
0
-10,000
-20,000
-30,000
0
50,000 100,000 150,000 200,000 250,000
ECON 7710, 2010
Population
10.15
3.2 Park Test
Model
Yi = 0 + 1X1i + … + KXKi + t i = 1,…,N (*)
Suppose it is suspected that var(i) depends on Zi
in the form of
var( i) = si2 = s2Zi1evi
lnsi2 = lns2 + 1lnZki + vi
Ho: 1 = 0 (Homoskedastic errors);
HA: 1  0 (Heteroskedastic
errors).
ECON 7710, 2010
10.16
Step 1: Estimate the equation (*) with OLS and
obtain the residuals.
e i  Yi  Yˆi  Yi  ˆ 0  ˆ1 X 1 i 
 ˆ K X K i
Step 2: Regress the natural log of squared
residuals on the natural log of a possible
proportionality factor
ln(ei2) = 0 + 1lnZi + vi
where vi is an error term satisfying all classical
assumptions.
ECON 7710, 2010
10.17
Step 3
If the coefficient of lnZ is significantly
different from zero, then it would suggest that
there is heteroscedastic pattern in the residuals
with respect to Z. Otherwise, homoscedastic errors
cannot be rejected.
Example 3: Park Test: US data (UE_Tab0301)
^
ln(e2) = -7.46 + 2.07** ln(income)
t
(2.28)
p-value
(0.0284)
ECON 7710, 2010
10.18
Advantages of the Park test:
a. The test is simple.
b. It provides information about the variance structure.
Limitations of the Park test:
a. The distribution of the dependent variable is
problematic.
b. It assumes a specific functional form.
c. It does not work when the variance depends on two or
more variables.
d. The correct variable with which to order the
observations must be identified first.
e. It cannot handle partitioned
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2010
10.19
3.3 White’s Test
Model
Yi = 0 + 1X1i + 2X2i + i i = 1,…,N (*)
Suppose it is suspected there may be
heteroskedasticity but we are not sure of its
functional form.
Ho: The conditional variance of i is constant.
HA: The conditional variance of i is not constant.
ECON 7710, 2010
10.20
Step 1: Estimate the equation (*) with OLS and
obtain the residuals.
e  Y  Yˆ  Y  ˆ  ˆ X  ˆ X
i
i
i
i
0
1
1i
2
2i
Step 2: Regress the squared residuals on all
explanatory variables, all cross product terms and
the square of each explanatory variable.
ei2 = 0 + 1X1i + 2X2i
+ 3X1i2 + 4X2i2
+ 5X1iX2i + vi
ECON 7710, 2010
10.21
Step 3: Test the overall significance of the
equation in Step 2. (df = number of regressors)
Statistic = NR2white ~ 2df
Critical value (cv) = 2df,
Reject the hypothesis of homoskedasticity if
NR2err > cv.
Example 4: White test: US data (UE_Tab0301)
^
e2 = 1924 – 7.4 income + 0.0088income2*
R2 = 0.3646, N = 40, NR2 = 14.58
cv = 2(2, 0.01) =
9.21.
ECON 7710, 2010
10.22
Advantages of the White test:
a. It does not assume a specific functional form.
b. It is applicable when the variance depends on two
or more variables.
Limitations of the White test:
a. It is an large-sample test.
b. It provides no information about the variance
structure.
c. It loses many degrees of freedom when there are
many regressors.
d. It cannot handle partitioned data.
e. It also captures specification
errors.
ECON 7710, 2010
10.23
3. Consequences of Heteroskedasticity
If heteroskedasticity appears but OLS is
used for estimation, how are the OLS
estimates affected?
Unaffected: OLS estimators are still linear and
unbiased because, on average, overestimates
are as likely as underestimates.
 
E ˆ k   k
k
 0 ,1,  , K 
ECON 7710, 2010
10.24
3.1 OLS estimators are inefficient.
Some fluctuations of the error term are
attributed to the variation in independent
variables.
There are other linear and unbiased
estimators that have smaller variances
than the OLS estimator.
ECON 7710, 2010
10.25
3.2 Unreliable Hypothesis Testing
var
ols
 biased
 
ˆ  var
k
hetero
 
ˆ
k
 
se ˆ k
 unreliable testing conclusion
ECON 7710, 2010
10.26
4. Remedies
4.1 Heteroskedasticity-Corrected
Standard Errors
Yi = 0 + 1X1i + 2X2i + i
heteroskedasticity:
var(i) = si 2
OLS estimators are unbiased.
The standard errorsECON
of7710,
OLS
2010 are biased.
10.27
A heteroskedasticity-consistent (HC) standard
error of an estimated coefficient is a standard
error of an estimated coefficient adjusted for
heteroskedasticity.
a. HC standard errors are consistent for any
type of heteroskedasticity.
b. Hypothesis tests are valid with HC standard
errors in large samples.
c. Typically, HC se > OLS se
ECON 7710, 2010
10.28
Example 5:
Yi = 0 + 1Xi + i, var(i|Xi) = si.
incorrect
variance formula:
correct
variance formula:
 
var ˆ 1 
s
2

Xi  X

2



var ˆ  


X  X  



2
2
si X i  X
1
2
i
ECON 7710, 2010
2
10.29
HC estimator of the variance of the slope
coefficient in a simple regression model

 

X


ei
est . var ˆ 1 
2
X i  X 
2
i X

2
2


Example 6: HC Standard Errors, US
data (UE_Tab0301)
ˆfo o d ex p = 4 0 .7 7  0 .1 3 *** in co m e
i
i
o ls se
h c se
 2 2 .1 4 
 2 4 .3 2 
 0 .0 3 1 
 0 .0 3 9 
R  0 .3 1 7 1, ECON
N 7710,
= 42010
0.
2
10.30
4.2 Weighted Least Squares
Yi = 0 + 1X1i + 2X2i + i
E(i) = 0
var(i) = si 2
si 2 = c Zi 2
ECON 7710, 2010
cov(t, s) = 0
t =s
The variance is
assumed to be
proportional to
the value of Zi2
10.31
Step 1: Decide which variable is proportional to
the heteroskedasticity.
Step 2: Divide all terms in the original model by
that variable (divide by Zi ).
ECON 7710, 2010
10.32
Step 3: Run least squares on the transformed
model which has new variables. Note that the
transformed model have an intercept only if Z is
one of the explanatory variables.
For example, if Zi = X2i, then
ECON 7710, 2010
10.33
Example 7: WLS: US data (UE_Tab0301)
ˆ
 foodexp

 incom e

1
***
  0.1577  21.2858
 0.02342 
14.0380  incom e

se
R  0.0570, N = 40.
2
What are values of the estimated coefficients
of the original model?
Has the problem of heteroskedasticity solved?
ECON 7710, 2010
10.34
Comparing different estimates: US data
(UE_Tab0301)
0
1
OLS estimate
40.77
0.128***
OLS se
22.14
0.031
HC se
24.32
0.039
WLS estimate
21.28
0.158***
WLS se
14.03
0.023
The WLS estimates have improved upon
ECON 7710, 2010
those of OLS.
10.35
Other possibilities
• var(i) = cZi
• var(i) = cZi
• var(i) = c(a1X1i + a2X2i)
ECON 7710, 2010
10.36
In large samples HC standard errors
are consistent measures for any type of
heteroscedasticity. CI & t-test are valid.
ECON 7710, 2010
10.37
4.3 Re-specifying the Regression Model
The heteroskedasticity may be impure.
4.3.1 Use another functional form
E.g., Double-log: Less variation
Example 8: US data (UE_Tab0301)


***
ˆ
ln foodexp  0.30  0.69 ln  incom e 
 0.90 
 0.14 
se
R  0.4014, N = 40.
2
The hypothesis of constant
variance can be rejected.
ECON 7710, 2010
10.38
Example 9: India data (Food_India55)
Empirical model:
foodexpi = 0 + 1totexpi + i.
ˆfoodexp  94.21**  0.44 *** totexp
i
se
 50.86 
 0.078 
R  0.3698, N = 55.
2
The hypothesis of homoskedasticity can
be rejected by the Park and White tests.
ECON 7710, 2010
10.39
Which model is the best?
Double-log


***
ˆ
ln foodexp  1.15  0.74 ln  totexp 
 0.78 
 0.12 
se
R  0.4125, N = 55.
2
HC
ˆfoodexp  94.21**  0.44 *** totexp.
i
o ls se
h c se
ˆ
foodexp
WLS
totexp
 5 0 .8 6 
 4 3 .2 6 
 76.5439
 37.9435 
se
ECON 7710, 2010
**
 0 .0 7 8 
 0 .0 7 4 
1
totexp
 0.4650
 0.0632 
***
.
10.40
4.3.2 Other reformulations
E.g., take average of variables related to
the size of observed units, adding more
variables
Example 10: Data set “Concert”
The concert tour of a singer in the US
revenue = 0 + 1adv + 2stad + 3cd
+ 4radio + 5weekend + .
ECON 7710, 2010
10.41
(1)
rˆ evenue  73  3.15adv  34.66stad  8.30cd
 300 radio  356 w eekend
rˆ evenue
 81
stad
(2)
 2.10
stad
adv
 50.20 stad  7.53
stad
cd
stad
se
 176
radio
stad
rˆ evenue
(3)
1
pop
 293
w eekend
stad
  22  2.21
adv
pop
2.53radio  4.28 w eekend
ECON 7710, 2010
 109
stad
pop
 7.93
cd
pop
10.42
Remarks:
•The variable Z is difficult to identify. The
functional relationship between the error and Z is
not known. Use WLS at last.
•With correct WLS, we expect the standard
errors of the regression coefficients will be
smaller than the OLS counterparts.
•A log transformation usually reduces the
degree of heteroskedasticity.
•The hypothesis of homoskedasticity should not
7710, 2010
be rejected in the newECON
model.
10.43
5. A Complete Example
Sources: Section 8.2.2 (pp. 255 – 256)
Section 10.5 (pp. 369 – 376)
Empirical regression model
pconi = 0 + 1regi + 2taxi + 3uhmi + i.
pconi1:
regi :
taxi :
uhm :
petroleum consumption in the ith state
motor vehicle registrations in the ith state (‘000)
the gasoline tax rate in the ith state(cents per gallon)
ECON 7710,
2010
urban highway miles
wihtin
the ith state
10.44
Equation 1
^ = 389.57*** – 0.061reg – 36.47***tax + 60.76***uhm
pcon
se, vif
(0.04, 24.3) (13.15, 1.1) (10.26, 24.9)
Adj. R2 = 0.9192, N = 50.
Equation 2
^ = 551.69*** + 0.19***reg – 53.59***tax
pcon
se
(0.012)
Adj. R2 = 0.8607, N = 50.
ECON 7710, 2010
(16.86)
10.45
Graphical investigation
1,200
residual
800
400
0
-400
-800
0
5,000
10,000
REG
ECON 7710, 2010
15,000
20,000
10.46
Park test
^
ln(e2) = 1.65 + 0.95***ln(REG)
se
R2 = 0.1657, N = 50
(0.3083)
White test
^e2 = 11,098,291 + 140REG – 0.0005REG2 –
12.84REGTAX – 237,873TAX + 12347TAX2.
R2 = 0.6645, N = 50, NR2 = 33.22.
Checking for other specifications:
Double log, quadraticECON 7710, 2010
10.47
(4)
^ = 551.69*** + 0.19***reg – 53.59***tax
pcon
hc se
(0.022)
(23.90)
R2 = 0.8664, N = 50.
p cˆ on
(5)
reg
 0 . 01367 
 218 . 539
 48 . 1033 
***
1
 17 . 3890
 4 . 6822 
reg
se
R
2
 0 . 3600 , N  50
p cˆ on
(6)
 0 . 1678
***
pop
 0 . 1684
**
 0 . 1082
 0 . 07159 
reg
pop
se
R
2
 0 . 1989 , N  50
ECON 7710, 2010
 0 . 0103
 0 . 00349 
***
tax
***
tax
reg
10.48
Selected Exercises
Ch. 10: Q. 1, 3, 4, 5, 8, 10, 12, 14
ECON 7710, 2010
10.49
Regression Model
Yi = 0 + 1X1i + 2X2i + i
zero mean:
E(i|X1i,X2i) = 0
homoskedasticity:
var(i|X1i,X2i) = s 2
no autocorrelation: cov(i, j|X1i,X2i,X1j,X2j) = 0
i= j
2
heteroskedasticity: ECON 7710,var(
|X
,X
)
=
s
i 1i 2i
i
2010
10.50
Heteroskedasticity
Yi = 0 + 1Xi + i
var(i|Xi) = si2 for all i
f(Y)
.
.
0
X1
X2
X3
ECON 7710, 2010
Conditional Distribution
.
X
10.51
Step 3: Test the overall significance of the
equation in Step 2. (df = number of regressors)
Statistic = NR2err ~ 2df
Critical value (cv) = 2df,
Reject the hypothesis of homoskedasticity if
NR2err > cv.
ECON 7710, 2010
10.52
Step 1: Decide which variable is proportional to
the heteroskedasticity.
Step 2: Divide all terms in the original model by
that variable (divide by Zi ).
Yi
Zi
β 0
1
Zi
β1
X 1i
Zi
 2
X 2i

Zi
*
*
*
*
*
Yi  β 0 X 0i  β 1 X 1i  β 2 X 2i   i
ECON 7710, 2010
i
Zi
10.53
Step 3: Run least squares on the transformed
model which has new variables. Note that the
transformed model have an intercept only if Z is
one of the explanatory variables.
For example, if Zi = X2i, then
Yi
Zi
 β0
1
Zi
Y  β 0X
*
i
*
0i
 β1
X 1i
Zi
β X
*
1
1i
ECON 7710, 2010
 2 
 β2  
i
Zi
*
i
10.54
In large samples HC standard errors
are consistent measures for any type of
heteroscedasticity. CI & t-test are valid.
OLS estimator
WLS
HC
se’s
Improve
No
Standard errors Improve Improve
Specific form
Yes
No
Large sample
No
Yes
ECON 7710, 2010