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CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 1
1. What is the nature of heteroscedasticity?
2. What are its consequences?
3. How does one detect it?
4. What are the remedial measures?
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 2
The Nature of Heteroscedasticity
• One of the important assumptions of the classical linear regression model is that the variance of each disturbance term ui, conditional on the chosen values of the explanatory variables, is some constant number equal to s2.
• This is the assumption of homoscedasticity, or equal (homo) spread (scedasticity), that is, equal variance.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 3
The Nature of Heteroscedasticity
• Symbolically,
• When the variance is nonconstant then we have heteroscedasticity:
niuE i
, ... ,2 ,1 )( 22 s (11.1.1)
22 )( ii
uE s (11.1.2)
2
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 4
FIGURE 11.1: Homoscedastic disturbances
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 5
FIGURE 11.2: Heteroscedastic disturbances
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 6
The Nature of Heteroscedasticity
• There are several reasons why the variances of ui may be variable:
1. Following the error–learning models, as people learn, their errors of behavior become smaller over time.
In this case, is expected to decrease (Fig. 11.3).2 i
s
3
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 7
FIGURE 11.3: Illustration of heteroscedasticity
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 8
The Nature of Heteroscedasticity
2. As incomes grow, people have more discretionary income and hence more scope for choice about the disposition of their income. Hence, is likely to increase with income.
2
i s
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 9
The Nature of Heteroscedasticity
3. As data collecting techniques improve, is likely to decrease.
2
i s
Thus, banks that have sophisticated data processing equipment are likely to commit fewer errors in the monthly or quarterly statements of their customers than banks without such facilities.
4
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 10
The Nature of Heteroscedasticity
4. Heteroscedasticity can also arise as a result of the presence of outliers.
An outlier is an observation that is much different (either very small or very large) in relation to the observations in the sample (Fig. 11.4).
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 11
FIGURE 11.4: The relationship between stock prices and consumer prices
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 12
The Nature of Heteroscedasticity
5. Another source of heteroscedasticity arises from violating Assumption 9 of the CLRM, namely, that the regression model is correctly specified.
Very often, what looks like heteroscedasticity may be due to the fact that some important variables are omitted from the model.
5
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 13
The Nature of Heteroscedasticity
Recall our study of advertising impressions retained (Y) in relation to advertising expenditure (X).
If you regress Y on X only and observe the residuals from this regression, you will see one pattern.
But if you regress Y on X and X2, you will see another pattern (Fig. 11.5).
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 14
FIGURE 11.5: Residuals from the regression of (a) impressions on advertising expenditure and (b) impressions on Adexp and Adexp2
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 15
The Nature of Heteroscedasticity
6. Another source of heteroscedasticity is skewness in the distribution of one or more regressors included in the model.
Examples are economic variables such as income, wealth, and education.
It is well known that the distribution of income and wealth in most societies is uneven.
6
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 16
The Nature of Heteroscedasticity
7. Other sources of heteroscedasticity: As David Hendry notes, heteroscedasticity can also arise because of
(1) incorrect data transformation (e.g., ratio or first difference transformations).
(2) incorrect functional form (e.g., linear vs. log- linear models).
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 17
The Nature of Heteroscedasticity
• As an illustration of heteroscedasticity, consider Table 11.1.
• This table gives data on compensation per employee in 10 nondurable goods manufacturing industries, classified by the employment size of the firm or establishment for the year 1958.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 18
The Nature of Heteroscedasticity
TABLE 11.1: COMPENSATION PER EMPLOYEE ($) IN NONDURABLE MFG. INDUSTRIES ACCORDING TO EMPLOYMENT SIZE OF ESTABLISHMENT, 1958
7
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 19
FIGURE 11.6: Standard deviation of compensation and mean compensation
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 20
OLS Estimation in the Presence of Heteroscedasticity
• What happens to OLS estimators and their variances if we introduce heteroscedasticity by letting
but retain all other assumptions of the classical model?
22 )( ii
uE s
• To answer this question, we shall revert to the two – variable model.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 21
OLS Estimation in the Presence of Heteroscedasticity
• Applying the usual formula, the OLS estimator of b2 is
iii uXY
21 bb
22
22
)(
ˆ
ii
iiii
i
ii
XXn
YXYXn
x
yx b
(11.2.1)
8
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 22
OLS Estimation in the Presence of Heteroscedasticity
• But its variance is now given by the following expression:
22
22
2 )( )ˆvar(
i
ii
x
x s b (11.2.2)
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 23
OLS Estimation in the Presence of Heteroscedasticity
• The usual variance formula obtained under the assumption of homoscedasticity is:
(11.2.3)
2
2
2 )ˆvar(
i x
s b
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 24
OLS Estimation in the Presence of Heteroscedasticity
• Heteroscedasticity Homoscedasticity
22
22
2 )( )ˆvar(
i
ii
x
x s b
(11.2.2)
2
2
2 )ˆvar(
i x
s b
(11.2.3)
• Of course, if for each i, the two formulas will be identical.
22 ss i
9
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 25
The Method of Generalized Least Squares (GLS)
• Why is the usual OLS estimator of b2 not best, although it is still unbiased?
• Intuitively, we can see the reason from Table 11.1.
• There is considerable variability in earnings between employment classes.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 26
The Method of Generalized Least Squares (GLS)
• If we were to regress per–employee compensation on the size of employment, we would like to use the knowledge that there is considerable interclass variability in earnings.
• We would like to devise the estimating scheme so that observations coming from populations with greater variability are given less weight than those coming from populations with smaller variability.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 27
The Method of Generalized Least Squares (GLS)
• OLS does not follow this strategy: It assigns equal weight to each observation.
• But a method of estimation, known as generalized least squares (GLS), takes such information into account explicitly and is therefore capable of producing estimators that are BLUE.
10
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 28
The Method of Generalized Least Squares (GLS)
• To see how this is done, let us continue with the familiar two–variable model:
iii uXY
21 bb (11.3.1)
which for ease of algebraic manipulation we write as
iiii uXXY
201 bb (11.3.2)
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 29
The Method of Generalized Least Squares (GLS)
iiii uXXY
201 bb (11.3.2)
where X0i = 1 for each i. The reader can see that these two formulations are identical.
• Now assume that the heteroscedastic variances are known.
2
i s
• Divide (11.3.2) through by si to obtain ….
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 30
The Method of Generalized Least Squares (GLS)
(11.3.3)
which for ease of exposition we write as
i
i
i
i
i
i
i
i uXXY
ss b
s b
s 2 0
1
iii
uXXY 2011
bb (11.3.4)
• What is purpose of transforming the original model?
11
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 31
The Method of Generalized Least Squares (GLS)
(11.3.5)
1
)( cesin )( 1
known is since )( 1
)()var(
222
2
22
2
2
2
iii
i
ii
i
i
i
ii
uE
uE
u EuEu
ss s
s s
s
which is a constant.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 32
The Method of Generalized Least Squares (GLS)
• GLS is OLS on the transformed variables that satisfy the standard least –squares assumptions.
• The actual mechanics of estimating and are as follows. First, we write down the SRF of (11.3.3)
i
i
i
i
i
i
i
i uXXY
ss b
s b
s ˆˆˆ
2
0
1
1 b̂
2 b̂
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 33
The Method of Generalized Least Squares (GLS)
i
i
i
i
i
i
i
i uXXY
ss b
s b
s ˆˆˆ
2
0
1
or
iii
uXXY ˆˆˆ 2011
bb (11.3.6)
12
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 34
The Method of Generalized Least Squares (GLS)
• Now, to obtain the GLS estimators, we minimize
that is,
22011 2 )ˆˆ(ˆ
iii XXYu bb
2
2
0
1
2
ˆˆˆ
i
i
i
i
i
i
i
i XXYu
s b
s b
ss (11.3.7)
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 35
The Method of Generalized Least Squares (GLS)
• The GLS estimator of is
and its variance is given by
(11.3.9)
2 b
222 )())((
))(())((ˆ iiiii
iiiiiiii
XwXww
YwXwYXww b (11.3.8)
222 )())((
)( )ˆvar(
iiiii
i
XwXww
w b
where .2/1 ii
w s
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 36
The Method of Generalized Least Squares (GLS)
• In OLS we minimize
Difference Between OLS and GLS
221 2 )ˆˆ(ˆ
iii XYu bb (11.3.10)
• In GLS we minimize the expression (11.3.7):
2
2
0
1
2
ˆˆˆ
i
i
i
i
i
i
i
i XXYu
s b
s b
ss (11.3.7)
13
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 37
The Method of Generalized Least Squares (GLS)
Difference Between OLS and GLS
which can also be written as
2
2
0
1
2
ˆˆˆ
i
i
i
i
i
i
i
i XXYu
s b
s b
ss (11.3.7)
2201 2 )ˆˆ(ˆ
iiiiii XXYwuw bb (11.3.11)
where .2/1 ii
w s
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 38
The Method of Generalized Least Squares (GLS)
• Thus, in GLS we minimize a weighted sum of residual squares with acting as the weights.
Difference Between OLS and GLS
• But in OLS we minimize an unweighted or (what amounts to the same thing) equally weighted RSS.
2/1 ii
w s
• To see the difference between the two, consider the hypothetical scattergram in Fig. 11.7.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 39
FIGURE 11.7: Hypothetical scattergram
14
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 40
Consequences of Using OLS in the Presence of Heteroscedasticity
• Suppose we use and use the variance formula given in (11.2.2), which takes into account heteroscedasticity explicitly:
OLS Estimation Allowing For Heteroscedasticity
2 b̂
22
22
2 )( )ˆvar(
i
ii
x
x s b (11.2.2)
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 41
Consequences of Using OLS in the Presence of Heteroscedasticity
• Using this variance, and assuming are known, can we establish confidence intervals and test hypotheses with the usual t and F tests?
OLS Estimation Allowing For Heteroscedasticity
2
i s
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 42
Consequences of Using OLS in the Presence of Heteroscedasticity
• The answer is generally no because
OLS Estimation Allowing For Heteroscedasticity
• This means that the confidence intervals based on the latter will be unnecessarily larger.
)ˆvar()ˆvar( 22
bb
15
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 43
Consequences of Using OLS in the Presence of Heteroscedasticity
• Therefore, the t and F tests are likely to give us inaccurate results.
OLS Estimation Allowing For Heteroscedasticity
• What appears to be a statistically insignificant coefficient may in fact be significant if the correct confidence intervals were established on the basis of GLS.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 44
Consequences of Using OLS in the Presence of Heteroscedasticity
• The situation can become serious if we not only use OLS to estimate the slope but continue to use the homoscedastic variance formula despite heteroscedasticity being present or suspected.
OLS Estimation Disregarding Heteroscedasticity
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 45
Consequences of Using OLS in the Presence of Heteroscedasticity
• If we persist in using the usual testing procedures despite heteroscedasticity, whatever conclusions we draw or inferences we make may be very misleading.
OLS Estimation Disregarding Heteroscedasticity
• To throw more light on this topic, we refer to a Monte Carlo study conducted by Davidson and MacKinnon.
16
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 46
Consequences of Using OLS in the Presence of Heteroscedasticity
• Davidson and MacKinnon consider the following simple model, which in our notation is
OLS Estimation Disregarding Heteroscedasticity
iii uXY
21 bb (11.4.1)
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 47
Consequences of Using OLS in the Presence of Heteroscedasticity
• They assume that
OLS Estimation Disregarding Heteroscedasticity
1 1 b 1
2 b ) ,0(~
ii XNu
• As the last expression shows, they asssume that the error variance is heteroscedastic and is related to the value of the regressor X with power .
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 48
Consequences of Using OLS in the Presence of Heteroscedasticity
OLS Estimation Disregarding Heteroscedasticity
) ,0(~ ii
XNu
• If, for example, = 1, the error variance is proportional to the value of X.
• If = 2, the error variance is proportional to the square of X, and so on.
17
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 49
Consequences of Using OLS in the Presence of Heteroscedasticity
• Based on 20,000 replications and allowing for various values for , they obtain the standard errors of the two regression coefficients using
OLS Estimation Disregarding Heteroscedasticity
OLS [see Eq. (11.2.3)], OLS allowing for heteroscedasticity [see Eq. (11.2.2)], GLS [see Eq. (11.3.9)].
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 50
Note: OLShet means OLS allowing for heteroscedasticity. Slope variances:
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 51
Detection of Heteroscedasticity
• Nature of the problem. Very often the nature of the problem under consideration suggests whether heteroscedasticity is likely to be encountered.
Informal methods
• For example, following the work of Prais and Houthakker on family budget studies, they found that the residual variance around the regression of consumption on income increased with income.
18
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 52
Detection of Heteroscedasticity
• Graphical method If there is no a priori or empirical information about the nature of heteroscedasticity, in practice one can do the regression analysis on the assumption that there is no heteroscedasticity.
Informal methods
• Then a postmortem examination of the squared residuals can show if they exhibit any systematic pattern.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 53
Detection of Heteroscedasticity
• Although are not the same thing as , they can be used as proxies especially if the sample size is reasonably large.
Informal methods
• An examination of the may reveal patterns. may be plotted against or against the explanatory variables.
2ˆ i
u 2 i
u
2ˆ i
u 2ˆ iu
i Ŷ
• Figures 11.8 and 11.9 show possible patterns.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 54
FIGURE 11.8: Hypothetical patterns of estimated squared residuals
19
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 55
FIGURE 11.9: Scattergram of estimated squared residuals against X
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 56
Detection of Heteroscedasticity
• Park Test Park formalizes the graphical method by suggesting that is some function of the explanatory variable Xi.
Formal methods
2
i s
• The functional form he suggested was
iv
ii eX bss 22
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 57
Detection of Heteroscedasticity
Formal methods
iv
ii eX bss 22
or
iii vX lnlnln 22 bss (11.5.1)
where vi is the stochastic disturbance term.
20
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 58
Detection of Heteroscedasticity
Formal methods
• Since is generally not known, Park suggests using the estimated as a proxy and running the following regression:
2
i s
2ˆ i
u
ii
iii
vX
vXu
ln
lnln ˆln 22
b
bs (11.5.2)
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 59
Detection of Heteroscedasticity
Formal methods
ii
iii
vX
vXu
ln
lnln ˆln 22
b
bs (11.5.2)
• If b turns out to be statistically significant, it would suggest that heteroscedasticity is present in the data. If not, we may accept the assumption of homoscedasticity.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 60
Detection of Heteroscedasticity
• The Park test has some problems.
Formal methods
• Goldfeld and Quandt have argued that the error term vi entering into (11.5.2) may not satisfy the OLS assumptions and may itself be heteroscedastic.
21
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 61
Detection of Heteroscedasticity
• Glejser Test After obtaining the residuals from the OLS regression, Glejser suggests regressing the absolute values of on the X variable that is thought to be closely associated with .
Formal methods
i û
i û
2
i s
• Glejser used the following functional forms:
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 62
Detection of Heteroscedasticity
iii
iii
i
i
i
i
i
i
iii
iii
vXu
vXu
v X
u
v X
u
vXu
vXu
2
21
21
21
21
21
21
|ˆ|
|ˆ|
1 |ˆ|
1 |ˆ|
|ˆ|
|ˆ|
bb
bb
bb
bb
bb
bb
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 63
Detection of Heteroscedasticity
• The error term vi has some problems in that its expected value is nonzero, it is serially correlated, and ironically it is heteroscedastic.
Formal methods
• Plus, these models are nonlinear in the parameters:
iii vXu
21 |ˆ| bb
iii vXu 2
21 |ˆ| bb
22
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 64
Detection of Heteroscedasticity
• Spearman Rank Correlation Test The Spearman’s rank corellation coefficient is defined as
Formal methods
)1(
61 2
2
nn
d r i
S (11.5.6)
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 65
Detection of Heteroscedasticity
Formal methods
)1(
61 2
2
nn
d r i
S (11.5.6)
• Where di = difference in the ranks assigned to the two different characteristics of the ith individual or phenomenon.
n = number of individuals or phenomena ranked.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 66
Detection of Heteroscedasticity
• Assume
Formal methods
iii uXY
10 bb
Step 1. Fit the regression to the data on Y and X and obtain the residuals
i û .
23
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 67
Detection of Heteroscedasticity
Formal methods
Step 2. Ignoring the sign of i
û , that is, taking their
absolute value |ˆ| i
u , rank both |ˆ| i
u and Xi (or iŶ ) according to an ascending or descending order and compute the Spearman’s rank correlation coefficient given previously.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 68
Detection of Heteroscedasticity
Formal methods
Step 3. Assuming that the population rank correlation coefficient s is zero and n > 8, the significance of the sample rs can be tested by the t test as follows:
21
2
S
S
r
nr t
(11.5.7)
with df = n –2.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 69
Detection of Heteroscedasticity
• If the computed t value exceeds the critical t value, we may accept the hypothesis of heteroscedasticity; otherwise we may reject it.
Formal methods
24
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 70
Detection of Heteroscedasticity
• The data in Table 11.2 pertain to the average annual return (E, %) and the standard deviation (si, %) of 10 mutual funds.
Example: Illustration Of The Rank Correlation Test
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 71
Detection of Heteroscedasticity
Example: Illustration Of The Rank Correlation Test
• The capital market line (CML) of portfolio theory postulates a linear relationship between expected return (Ei) and risk (as measured by si) of a portfolio as follows:
ii E sbb
21
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 72
* Obtained from the regression i
Ê = 5.8194 + 0.4590si. † Absolute value of the residuals.
Note: The ranking is in ascending order of values.
TABLE 11.2: RANK CORRELATION TEST
25
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 73
Detection of Heteroscedasticity
Example: Illustration Of The Rank Correlation Test
• Applying formula (11.5.6), we obtain
3333.0 )1100(10
110 6 1
S r (11.5.8)
)1(
61 2
2
nn
d r i
S (11.5.6)
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 74
Detection of Heteroscedasticity
Example: Illustration Of The Rank Correlation Test
• Applying the t test given in (11.5.7), we obtain
(11.5.9)9998.0 1110.01
)8)(3333.0(
t
21
2
S
S
r
nr t
(11.5.7)
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 75
Detection of Heteroscedasticity
• Goldfeld–Quandt Test This method is applicable if one assumes that the heteroscedastic variance, , is positively related to one of the explanatory variables in the regression model.
Formal methods
2
i s
26
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 76
Detection of Heteroscedasticity
• For simplicity, consider the usual two–variable model:
Formal methods
iii uXY
21 bb
• Suppose is positively related to Xi as 2
i s
222
ii Xss (11.5.10)
where s2 is a constant.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 77
Detection of Heteroscedasticity
Formal methods
222
ii Xss (11.5.10)
• This postulates that is proportional to the square of the X variable.
• Such an assumption has been found quite useful by Prais and Houthakker in their study of family budgets.
2
i s
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 78
Detection of Heteroscedasticity
Formal methods
Step 1. Order or rank the observations according to the values of
i X , beginning with the lowest X value.
Step 2. Omit c central observations, where c is specified a priori, and divide the remaining (n–c) observations into two groups each of (n–c) / 2 observations.
27
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 79
Detection of Heteroscedasticity
Step 3. Fit separate OLS regressions to the first (n–c)/2 observations and the last (n–c)/2 observations, and obtain the respective residual sums of squares RSS1 and RSS2.
RSS1 is the RSS from the regression corresponding to the smaller Xi values (the small variance group) and RSS2 is that from the larger Xi values (the large variance group).
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 80
Detection of Heteroscedasticity
Step 3. (cont.)
These R SS each h ave
k cn
2
or
2 2 kcn
df
where k is the num ber of param eters to be estim ated , includin g the intercept.
For the tw o–variable case k is of course 2.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 81
Detection of Heteroscedasticity
Step 4. Compute the ratio
df/RSS
df/RSS
1
2 (11.5.11)
If the errors are assumed to be normally distributed (which we usually do), and if the assumption of homoscedasticity is valid, then it can be shown that the of (11.5.11) follows the F distribution with numerator and denominator df each of (n – c – 2k)/2.
28
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 82
Detection of Heteroscedasticity
• The c central observations are omitted to sharpen or accentuate the difference between the small variance group and the large variance group.
Formal methods
• But the ability of the Goldfeld–Quandt test to do this successfully depends on how c is chosen.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 83
Detection of Heteroscedasticity
• For the two–variable model the Monte Carlo experiments done by Goldfeld and Quandt suggest that
c is about 8 if the sample size is about 30 c is about 16 if the sample size is about 60.
Formal methods
• But Judge et al. note that c = 4 if n = 30 and c = 10 if n = 60 have been found satisfactory in practice.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 84
Detection of Heteroscedasticity
Example: The Goldfeld-Quandt Test
• We present in Table 11.3 data on consumption expenditure in relation to income for a cross section of 30 families.
• Reordering the data and dropping the middle 4 observations, the OLS regressions based on the first 13 and the last 13 observations are obtained.
29
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 85
TABLE 11.3: HYPOTHETICAL DATA ON CONSUMPTION EXPENDITURE Y ($) AND INCOME X ($) TO ILLUSTRATE THE GOLDFELD– QUANDT TEST
Data ranked by X values
Y X Y X 55 80 55 80 65 100 70 85 70 85 75 90 80 110 65 100 79 120 74 105 84 115 80 110 98 130 84 115 95 140 79 120 90 125 90 125 75 90 98 130 74 105 95 140
110 160 108 145 113 150 113 150 125 165 110 160 108 145 125 165 Middle 4 115 180 115 180 observations 140 225 130 185 120 200 135 190 145 240 120 200 130 185 140 205 152 220 144 210 144 210 152 220 175 245 140 225 180 260 137 230 135 190 145 240 140 205 175 245 178 265 189 250 191 270 180 260 137 230 178 265 189 250 191 270
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 86
Detection of Heteroscedasticity
• Regression based on the first 13 observations:
• Regression based on the last 13 observations:
11df 17.377RSS 8887.0 )0744.0( )7049.8(
6968.0 4094.3 ˆ
1
2
r
XY ii
11df 8.536,1RSS 7681.0 )1319.0( )6421.30(
7941.0 0272.28 ˆ
2
2
r
XY ii
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 87
Detection of Heteroscedasticity
Example: The Goldfeld-Quandt Test
• From these results we obtain
• The 5% critical F value for 11 numerator and 11 denominator df is 2.82. Since the estimated F ( ) of 4.07 > 2.82, we conclude that there is heteroscedasticity in the error variance.
07.4 11/17.377 11/8.536,1
df/RSS df/RSS
1
2
30
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 88
Detection of Heteroscedasticity
• Breusch–Pagan–Godfrey Test The success of the Goldfeld–Quandt test depends not only on the value of c (the number of central observations to be omitted) but also on identifying the correct X variable with which to order the observations.
Formal methods
• This limitation of this test can be avoided if we consider the Breusch–Pagan–Godfrey (BPG) test.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 89
Detection of Heteroscedasticity
• To illustrate this test, consider the k–variable linear regression model
Formal methods
ikikii uXXY bbb
221 (11.5.12)
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 90
Detection of Heteroscedasticity
• Assume that the error variance is described as
Formal methods
(11.5.13)
2
i s
)( 221
2
mimii ZZf s
• That is, is some function of the nonstochastic variables Z’s; some or all of the X’s can serve as Z’s.
2
i s
31
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 91
Detection of Heteroscedasticity
• Specifically, assume that
Formal methods
(11.5.14)
• If 2 = 3 = … = m = 0, then (a constant).
mimii ZZ s
221
2
1
2 s i
• Therefore, to test whether is homoscedastic, we can test the hypothesis that 2 = 3 = … = m = 0.
2
i s
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 92
Detection of Heteroscedasticity
Formal methods
Step 1. Estimate (11.5.12) by OLS and obtain the residuals
1 û ,
2 û , …,
n û .
Step 2. Obtain nui /ˆ~ 22s . This is the maximum
likelihood (ML) estimator of s2.
[Note: The OLS estimator is )/(ˆ 2 knui ].
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 93
Detection of Heteroscedasticity
Step 3. Construct variables pi defined as
22 ~/ˆ s ii
up
which is simply each residual squared divided by
2~s .
Formal methods
32
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 94
Detection of Heteroscedasticity
Step 4. Regress the pi thus constructed on the Z’s as
imimii
vZZp 221
(11.5.15)
where vi is the residual term of this regression.
Formal methods
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 95
Detection of Heteroscedasticity
Step 5. Obtain the ESS from (11.5.15) and define
)ESS( 2 1
Θ (11.5.16)
Formal methods
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 96
Detection of Heteroscedasticity
• Assuming the ui are normally distributed, one can show that if there is homoscedasticity and if the sample size increases indefinitely, then
Formal methods
that is, Q follows the chi–square distribution with (m–1) df.
2
1asy ~ Θ
m (11.5.17)
33
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 97
Detection of Heteroscedasticity
• Therefore, if in an application the computed Q (=2) exceeds the critical 2 value at the chosen level of significance, one can reject the hypothesis of homoscedasticity; otherwise one does not reject it.
Formal methods
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 98
Detection of Heteroscedasticity
• White’s General Heteroscedasticity Test The general test of heteroscedasticity proposed by White does not rely on the normality assumption and is easy to implement.
Formal methods
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 99
Detection of Heteroscedasticity
Formal methods
• Consider the following three–variable regression model (the generalization to the k–variable model is straightforward):
iiii uXXY
33221 bbb (11.5.21)
• The White test proceeds as follows:
34
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 100
Detection of Heteroscedasticity
Formal methods
Step 1. Given the data, we estimate (11.5.21) and obtain the residuals,
i û .
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 101
Detection of Heteroscedasticity
Formal methods
Step 2. We then run the following (auxiliary) regression:
iiiiiiii vXXXXXXu
326
2
35
2
2433221
2ˆ
(11.5.22)
Higher powers can also be introduced. Obtain the R2 from this (auxiliary) regression.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 102
Detection of Heteroscedasticity
Formal methods
Step 3. Under the null hypothesis that there is no heteroscedasticity, the sample size (n) times the R2
obtained from the auxiliary regression asymptotically follows the chi–square distrubution with df equal to the number of regressors (excluding the constant term) in the auxiliary regression.
35
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 103
Detection of Heteroscedasticity
Formal methods
Step 3. (cont.)
That is,
2
dfasy
2 ~ Rn (11.5.23)
where df is as defined previously. In our example, there are 5 df since there are 5 regressors in the auxiliary regression.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 104
Detection of Heteroscedasticity
Formal methods
Step 4. If the chi–square value obtained in (11.5.23) exceeds the critical chi–square value, we conclude that there is heteroscedasticity.
If it does not exceed it, there is no heteroscedasticity.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 105
Detection of Heteroscedasticity
Formal methods
Step 4. (cont.)
That is, in the auxiliary regression (11.5.22)
iiiiiiii vXXXXXXu
326
2
35
2
2433221
2ˆ
0 65432
36
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 106
Detection of Heteroscedasticity
Formal methods
• Warning: Care must be exercised in applying the White test due to the degrees of freedom.
• What does this mean?
• The White test can be a test of pure heteroscedasticity, specification error, or both.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 107
Detection of Heteroscedasticity
• There are several other tests of heteroscedasticity, each based on certain assumptions.
Other Tests of Heteroscedasticity
• One such test, which is simple, is the Koenker– Bassett (KB) test.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 108
Detection of Heteroscedasticity
• For the KB test, assume the original model is
Other Tests of Heteroscedasticity
ikikiii uXXXY bbbb
33221 (11.5.26)
• Obtain the estimated residuals and the fitted Y values from this model.
37
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 109
Detection of Heteroscedasticity
• The next step is to estimate
Other Tests of Heteroscedasticity
(11.5.27)
• The null hypothesis is that 2 = 0. If this is not rejected, then we conclude that there is no heteroscedasticity.
iii vYu 2
21
2 )ˆ(ˆ
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 110
Remedial Measures
• There are two approaches to remediation:
(1) When is known;2 i
s
(2) When is not known.2 i
s
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 111
Remedial Measures
When is Known: The Method Of Weighted Least Squares
2
i s
• As we have seen, if is known, the most straightforward method of correcting heteroscedasticity is by means of weighted least squares.
• The estimators thus obtained are BLUE.
38
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 112
Remedial Measures
When is Not Known2 i
s
• White’s Heteroscedasticity–Consistent Variances and Standard Errors. White has shown that this estimate can be performed so that asymptotically valid (i.e., large–sample) statistical inferences can be made about the true parameter values.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 113
Remedial Measures
When is Not Known2 i
s
• Several computer packages present White’s heteroscedasticity–corrected variances and standard errors along with the usual OLS variances and standard errors.
• White’s approach retains the OLS regression but changes only the standard errors and t values.
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 114
Remedial Measures
• As an example, we quote the model due to Greene.
• In this model, Y = per capita expenditure on public schools by state in 1979 and income = per capita income by state in 1979.
• The data consisted of 50 states plus Washington, D.C.
39
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 115
Remedial Measures
• The results are:
)91.1( )48.1( )81.1(
)0.830( )0.243,1( )9.460( se White
)06.3( )21.2( )54.2(
)1.519( )0.829( )3.327( se OLS
)Income(04.587,1)Income(2.834,191.832 ˆ 2
t
t
Y i
(11.6.4)
CHAPTER 11: Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
ANDREW PAIZIS-QC BASIC ECONOMETRICS 5th Ed. 116
Remedial Measures
When is Not Known2 i
s
• Plausible Assumptions about the Heteroscedasticity Pattern. The White procedure is a large–sample procedure.
• One drawback of the White procedure is that the White estimators may not be so efficient as those obtained by methods that transform data to reflect specific types of heteroscedasticity.
