Chapter 5
Heteroskedasticity
What is in this Chapter?
• How do we detect this problem
• What are the consequences of this
problem?
• What are the solutions?
What is in this Chapter?
• First, We discuss tests based on OLS
residuals, likelihood ratio test, G-Q test
and the B-P test. The last one is an LM
test.
• Regarding consequences, we show that
the OLS estimators are unbiased but
inefficient and the standard errors are also
biased, thus invalidating tests of
significance
What is in this Chapter?
• Regarding solutions, we discuss solutions
depending on particular assumptions
about the error variance and general
solutions.
• We also discuss transformation of
variables to logs and the problems
associated with deflators, both of which
are commonly used as solutions to the
heteroskedasticity problem.
5.1 Introduction
• The homoskedasticity=variance of the
error terms is constant
• The heteroskedasticity=variance of the
error terms is non-constant
• Illustrative Example
– Table 5.1 presents consumption expenditures
(y) and income (x) for 20 families. Suppose
that we estimate the equation by ordinary
least squares. We get (figures in parentheses
are standard errors)
5.1 Introduction
5.1 Introduction
5.1 Introduction
5.1 Introduction
5.1 Introduction
5.1 Introduction
• The residuals from this equation are
presented in Table 5.3
• In this situation there is no perceptible
increase in the magnitudes of the
residuals as the value of x increases
• Thus there does not appear to be a
heteroskedasticity problem.
5.2 Detection of Heteroskedasticity
5.2 Detection of Heteroskedasticity
5.2 Detection of Heteroskedasticity
5.2 Detection of Heteroskedasticity
5.2 Detection of Heteroskedasticity
5.2 Detection of Heteroskedasticity
5.2 Detection of Heteroskedasticity
• Some Other Tests
– Likelihood Ratio Test
– Goldfeld and Quandt Test
– Breusch-Pagan Test
5.2 Detection of Heteroskedasticity
• Likelihood Ratio Test
5.2 Detection of Heteroskedasticity
• Goldfeld and Quandt Test
– If we do not have large samples, we can use the
Goldfeld and Quandt test.
– In this test we split the observations into two
groups — one corresponding to large values of x and
the other corresponding to small values of x —
– Fit separate regressions for each and then apply an
F-test to test the equality of error variances.
– Goldfeld and Quandt suggest omitting some
observations in the middle to increase our ability to
discriminate between the two error variances.
5.2 Detection of Heteroskedasticity
• Breusch-Pagan Test
5.2 Detection of Heteroskedasticity
5.2 Detection of Heteroskedasticity
• Illustrative Example
5.2 Detection of Heteroskedasticity
5.2 Detection of Heteroskedasticity
5.2 Detection of Heteroskedasticity
5.2 Detection of Heteroskedasticity
5.2 Detection of Heteroskedasticity
5.3 Consequences of
Heteroskedasticity
5.3 Consequences of
Heteroskedasticity
5.3 Consequences of
Heteroskedasticity
5.3 Consequences of
Heteroskedasticity
5.3 Consequences of
Heteroskedasticity
5.3 Consequences of
Heteroskedasticity
5.3 Consequences of
Heteroskedasticity
5.3 Consequences of
Heteroskedasticity
5.3 Consequences of
Heteroskedasticity
5.4 Solutions to the
Heteroskedasticity Problem
• There are two types of solutions that have been
suggested in the literature for the problem of
heteroskedasticity:
– Solutions dependent on particular assumptions about
σi.
– General solutions.
• We first discuss category 1. Here we have two
methods of estimation: weighted least squares
(WLS) and maximum likelihood (ML).
5.4 Solutions to the
Heteroskedasticity Problem
• WLS
5.4 Solutions to the
Heteroskedasticity Problem
Thus the constant term in this equation is the
slope coefficient in the original equation.
5.4 Solutions to the
Heteroskedasticity Problem
5.4 Solutions to the
Heteroskedasticity Problem
5.4 Solutions to the
Heteroskedasticity Problem
• If we make some specific assumptions
about the errors, say that they are normal
• We can use the maximum likelihood
method, which is more efficient than the
WLS if errors are normal
5.4 Solutions to the
Heteroskedasticity Problem
5.4 Solutions to the
Heteroskedasticity Problem
5.4 Solutions to the
Heteroskedasticity Problem
• Illustrative Example
5.4 Solutions to the
Heteroskedasticity Problem
5.4 Solutions to the
Heteroskedasticity Problem
5.5 Heteroskedasticity and the Use
of Deflators
• There are two remedies often suggested
and used for solving the heteroskedasticity
problem:
– Transforming the data to logs
– Deflating the variables by some measure of
"size."
5.5 Heteroskedasticity and the Use
of Deflators
5.5 Heteroskedasticity and the Use
of Deflators
5.5 Heteroskedasticity and the Use
of Deflators
• One important thing to note is that the purpose
in all these procedures of deflation is to get more
efficient estimates of the parameters
• But once those estimates have been obtained,
one should make all inferences—calculation of
the residuals, prediction of future values,
calculation of elasticities at the means, etc.,
from the original equation—not the equation
in the deflated variables.
5.5 Heteroskedasticity and the Use
of Deflators
• Another point to note is that since the purpose of
deflation is to get more efficient estimates, it is
tempting to argue about the merits of the
different procedures by looking at the standard
errors of the coefficients.
• However, this is not correct, because in the
presence of heteroskedasticity the standard
errors themselves are biased, as we showed
earlier
5.5 Heteroskedasticity and the Use
of Deflators
• For instance, in the five equations presented
above, the second and third are comparable and
so are the fourth and fifth.
• In both cases if we look at the standard errors of
the coefficient of X, the coefficient in the
undeflated equation has a smaller standard error
than the corresponding coefficient in the deflated
equation
• However, if the standard errors are biased, we
have to be careful in making too much of these
differences
5.5 Heteroskedasticity and the Use
of Deflators
• In the preceding example we have considered miles M
as a deflator and also as an explanatory variable
• In this context we should mention some discussion in the
literature on "spurious correlation" between ratios.
• The argument simply is that even if we have two
variables X and Y that are uncorrelated, if we deflate
both the variables by another variable Z, there could be
a strong correlation between X/Z and Y/Z because of the
common denominator Z
• It is wrong to infer from this correlation that there exists a
close relationship between X and Y
5.5 Heteroskedasticity and the Use
of Deflators
• Of course, if our interest is in fact the
relationship between X/Z and Y/Z, there is no
reason why this correlation need be called
"spurious."
• As Kuh and Meyer point out, "The question of
spurious correlation quite obviously does not
arise when the hypothesis to be tested has
initially been formulated in terms of ratios, for
instance, in problems involving relative prices.
5.5 Heteroskedasticity and the Use
of Deflators
• Similarly, when a series such as money value of
output is divided by a price index to obtain a
'constant dollar' estimate of output, no question
of spurious correlation need arise.
• Thus, spurious correlation can only exist when a
hypothesis pertains to undeflated variables and
the data have been divided through by another
series for reasons extraneous to but not in
conflict with the hypothesis framed an exact, i.e.,
nonstochastic relation."
5.5 Heteroskedasticity and the Use
of Deflators
• In summary, often in econometric work deflated
or ratio variables are used to solve the
heteroskedasticity problem
• Deflation can sometimes be justified on pure
economic grounds, as in the case of the use of
"real" quantities and relative prices
• In this case all the inferences from the estimated
equation will be based on the equation in the
deflated variables.
5.5 Heteroskedasticity and the Use
of Deflators
• However, if deflation is used to solve the
heteroskedasticity problem, any inferences we
make have to be based on the original equation,
not the equation in the deflated variables
• In any case, deflation may increase or decrease
the resulting correlations, but this is beside the
point. Since the correlations are not comparable
anyway, one should not draw any inferences
from them.
5.5 Heteroskedasticity and the Use
of Deflators
• Illustrative Example
5.5 Heteroskedasticity and the Use
of Deflators
5.5 Heteroskedasticity and the Use
of Deflators
5.5 Heteroskedasticity and the Use
of Deflators
5.5 Heteroskedasticity and the Use
of Deflators
5.5 Heteroskedasticity and the Use
of Deflators
5.6 Testing the Linear Versus LogLinear Functional Form
5.6 Testing the Linear Versus LogLinear Functional Form
• When comparing the linear with the log-linear
forms, we cannot compare the R2 because R2 is
the ratio of explained variance to the total
variance and the variances of y and log y are
different
• Comparing R2's in this case is like comparing
two individuals A and B, where A eats 65% of a
carrot cake and B eats 70% of a strawberry cake
• The comparison does not make sense because
there are two different cakes.
5.6 Testing the Linear Versus LogLinear Functional Form
• The Box-Cox Test
– One solution to this problem is to consider a more
general model of which both the linear and loglinear forms are special cases. Box and Cox
consider the transformation
5.6 Testing the Linear Versus LogLinear Functional Form
5.6 Testing the Linear Versus LogLinear Functional Form
5.6 Testing the Linear Versus LogLinear Functional Form
5.6 Testing the Linear Versus LogLinear Functional Form
5.6 Testing the Linear Versus LogLinear Functional Form
5.6 Testing the Linear Versus LogLinear Functional Form
5.6 Testing the Linear Versus LogLinear Functional Form
5.6 Testing the Linear Versus LogLinear Functional Form
Summary
• 1. If the error variance is not constant for all the
observations, this is known as the heteroskedasticity
problem. The problem is informally illustrated with an
example in Section 5.1.
• 2. First, we would like to know whether the problem
exists. For this purpose some tests have been
suggested. We have discussed the following tests:
–
–
–
–
–
–
(a) Ramsey's test.
(b) Glejser's tests.
(c) Breusch and Pagan's test.
(d) White's test.
(e) Goldfeld and Quandt's test.
(f) Likelihood ratio test.
Summary
• 3. The consequences of the heteroskedasticity
problem are:
– (a) The least squares estimators are unbiased but
inefficient.
– (b) The estimated variances are themselves biased.
– If the heteroskedasticity problem is detected, we can
try to solve it by the use of weighted least squares.
– Otherwise, we can at least try to correct the error
variances
Summary
• 4. There are three solutions commonly
suggested for the heteroskedasticity problem
–
–
–
–
(a) Use of weighted least squares.
(b) Deflating the data by some measure of "size.“
(c) Transforming the data to the logarithmic form.
In weighted least squares, the particular weighting
scheme used will depend on the nature of
heteroskedasticity.
Summary
• 5. The use of deflators is similar to the weighted least
squared method, although it is done in a more ad hoc
fashion. Some problems with the use of deflators are
discussed in Section 5.5.
• 6. The question of estimation in linear versus logarithmic
form has received considerable attention during recent
years. Several statistical tests have been suggested for
testing the linear versus logarithmic form. In Section 5.6
we discuss three of these tests: the Box-Cox test, the
BM test, and the PE test. All are easy to implement with
standard regression packages. We have not illustrated
the use of these tests.