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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 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)

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 si. - 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 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.

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 Log-Linear

Functional Form

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.

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

5.6 Testing the Linear Versus Log-Linear

Functional Form

5.6 Testing the Linear Versus Log-Linear

Functional Form

5.6 Testing the Linear Versus Log-Linear

Functional Form

5.6 Testing the Linear Versus Log-Linear

Functional Form

5.6 Testing the Linear Versus Log-Linear

Functional Form

5.6 Testing the Linear Versus Log-Linear

Functional Form

5.6 Testing the Linear Versus Log-Linear

Functional Form

5.6 Testing the Linear Versus Log-Linear

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.