Heteroskedasticity

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Chapter 5

• Heteroskedasticity

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What is in this Chapter?

• How do we detect this problem
• What are the consequences of this problem?
• What are the solutions?

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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

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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.

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5.1 Introduction

• The homoskedasticityvariance of the error terms
  is constant
• The heteroskedasticityvariance 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)

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5.1 Introduction
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5.1 Introduction
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5.1 Introduction
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5.1 Introduction
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5.1 Introduction
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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.

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5.2 Detection of Heteroskedasticity
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5.2 Detection of Heteroskedasticity
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5.2 Detection of Heteroskedasticity
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5.2 Detection of Heteroskedasticity
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5.2 Detection of Heteroskedasticity
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5.2 Detection of Heteroskedasticity
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5.2 Detection of Heteroskedasticity

• Some Other Tests
• Likelihood Ratio Test
• Goldfeld and Quandt Test
• Breusch-Pagan Test

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5.2 Detection of Heteroskedasticity

• Likelihood Ratio Test

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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.

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5.2 Detection of Heteroskedasticity

• Breusch-Pagan Test

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5.2 Detection of Heteroskedasticity
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5.2 Detection of Heteroskedasticity

• Illustrative Example

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5.2 Detection of Heteroskedasticity
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5.2 Detection of Heteroskedasticity
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5.2 Detection of Heteroskedasticity
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5.2 Detection of Heteroskedasticity
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5.2 Detection of Heteroskedasticity
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5.3 Consequences of Heteroskedasticity
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5.3 Consequences of Heteroskedasticity
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5.3 Consequences of Heteroskedasticity
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5.3 Consequences of Heteroskedasticity
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5.3 Consequences of Heteroskedasticity
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5.3 Consequences of Heteroskedasticity
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5.3 Consequences of Heteroskedasticity
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5.3 Consequences of Heteroskedasticity
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5.3 Consequences of Heteroskedasticity
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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 si.
• General solutions.
• We first discuss category 1. Here we have two
  methods of estimation weighted least squares
  (WLS) and maximum likelihood (ML).

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5.4 Solutions to the Heteroskedasticity Problem

• WLS

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5.4 Solutions to the Heteroskedasticity Problem
Thus the constant term in this equation is the
slope coefficient in the original equation.
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5.4 Solutions to the Heteroskedasticity Problem
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5.4 Solutions to the Heteroskedasticity Problem
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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

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5.4 Solutions to the Heteroskedasticity Problem
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5.4 Solutions to the Heteroskedasticity Problem
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5.4 Solutions to the Heteroskedasticity Problem

• Illustrative Example

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5.4 Solutions to the Heteroskedasticity Problem
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5.4 Solutions to the Heteroskedasticity Problem
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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."

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5.5 Heteroskedasticity and the Use of Deflators
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5.5 Heteroskedasticity and the Use of Deflators
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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 inferencescalculation of the
  residuals, prediction of future values,
  calculation of elasticities at the means, etc.,
  from the original equationnot the equation in
  the deflated variables.

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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

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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

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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

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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.

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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."

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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.

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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.

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5.5 Heteroskedasticity and the Use of Deflators

• Illustrative Example

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5.5 Heteroskedasticity and the Use of Deflators
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5.5 Heteroskedasticity and the Use of Deflators
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5.5 Heteroskedasticity and the Use of Deflators
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5.5 Heteroskedasticity and the Use of Deflators
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5.5 Heteroskedasticity and the Use of Deflators
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5.6 Testing the Linear Versus Log-Linear
Functional Form
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5.6 Testing the Linear Versus Log-Linear
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.

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5.6 Testing the Linear Versus Log-Linear
Functional Form

• The Box-Cox Test
• One solution to this problem is to consider a
  more general model of which both the linear and
  log-linear forms are special cases. Box and Cox
  consider the transformation

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5.6 Testing the Linear Versus Log-Linear
Functional Form
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5.6 Testing the Linear Versus Log-Linear
Functional Form
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5.6 Testing the Linear Versus Log-Linear
Functional Form
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5.6 Testing the Linear Versus Log-Linear
Functional Form
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5.6 Testing the Linear Versus Log-Linear
Functional Form
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5.6 Testing the Linear Versus Log-Linear
Functional Form
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5.6 Testing the Linear Versus Log-Linear
Functional Form
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5.6 Testing the Linear Versus Log-Linear
Functional Form
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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.

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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

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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.

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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.