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MGMG 522 : Session #8MGMG 522 : Session #8HeteroskedasticityHeteroskedasticity

(Ch. 10)(Ch. 10)

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HeteroskedasticityHeteroskedasticity Heteroskedasticity is a violation of the classical Heteroskedasticity is a violation of the classical

assumption #5 (observations of the error term are assumption #5 (observations of the error term are drawn from a distribution that has a constant drawn from a distribution that has a constant variance). (See Figure 10.1 on p. 348)variance). (See Figure 10.1 on p. 348)

Heteroskedasticity can be categorized into 2 kinds:Heteroskedasticity can be categorized into 2 kinds:– Pure heteroskedasticity (Heteroskedasticity that Pure heteroskedasticity (Heteroskedasticity that

exists in a exists in a correctly specifiedcorrectly specified regression model). regression model).– Impure heteroskedasticity (Heteroskedasticity Impure heteroskedasticity (Heteroskedasticity

that is caused by specification errors: omitted that is caused by specification errors: omitted variables or incorrect functional form).variables or incorrect functional form).

Heteroskedasticity mostly happens in a cross-Heteroskedasticity mostly happens in a cross-sectional data set. This, however, doesn’t mean that sectional data set. This, however, doesn’t mean that time-series data cannot have heteroskedasticity.time-series data cannot have heteroskedasticity.

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Pure HeteroskedasticityPure Heteroskedasticity

Classical assumption #5 implies that the Classical assumption #5 implies that the observations of the error term are drawn from observations of the error term are drawn from a distribution with a constant variance, or a distribution with a constant variance, or VAR(VAR(εεii) = ) = 22..

In other words, the error term must be In other words, the error term must be homoskedastic.homoskedastic.

For a heteroskedastic error term, VAR(For a heteroskedastic error term, VAR(εεii) = ) = ii

22.. Heteroskedasticity can take on many forms Heteroskedasticity can take on many forms

but we will limit our discussion to one form of but we will limit our discussion to one form of heteroskedasticity only.heteroskedasticity only.

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Heteroskedasticity of the Form: Heteroskedasticity of the Form: VAR(VAR(εεii) = ) = 22ZZii

22

Z is called a proportionality factor because the Z is called a proportionality factor because the variance of the error term changes proportionally to variance of the error term changes proportionally to the Z factor (or variable). (See Figure 10.2 on p. 350 the Z factor (or variable). (See Figure 10.2 on p. 350 and Figure 10.3 on p. 351)and Figure 10.3 on p. 351)

The variance of the error term increases as Z The variance of the error term increases as Z increases.increases.

Z may or may not be one of the explanatory Z may or may not be one of the explanatory variables in the regression model.variables in the regression model.

Heteroskedasticity can occur in at least 2 situations:Heteroskedasticity can occur in at least 2 situations:1.1. When there is significant change in the dependent When there is significant change in the dependent

variable of a variable of a time-seriestime-series model. model.2.2. When there are different amounts of measurement When there are different amounts of measurement

errors in the sample of different periods or different errors in the sample of different periods or different sub-samples.sub-samples.

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Impure HeteroskedasticityImpure Heteroskedasticity

Caused by specification errors, Caused by specification errors, especially the omitted variables.especially the omitted variables.

Incorrect functional form is less Incorrect functional form is less likely to cause heteroskedasticity.likely to cause heteroskedasticity.

Specification errors should be Specification errors should be corrected first by way of corrected first by way of investigating the independent investigating the independent variables and/or functional form.variables and/or functional form.

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How can omitted variables cause How can omitted variables cause heteroskedasticity?heteroskedasticity?

Correct model: Y = Correct model: Y = 00++11XX11++22XX22++εε If XIf X22 is omitted: Y = is omitted: Y = 00++11XX11++εε**

Where, Where, εε** = = 22XX22++εε If If εε is small compared to is small compared to 22XX22, and X, and X22

is heteroskedastic, the is heteroskedastic, the εε** will also be will also be heteroskedastic.heteroskedastic.

Both the bias and impure Both the bias and impure heteroskedasticity will disappear heteroskedasticity will disappear once the model gets corrected.once the model gets corrected.

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Consequences of Consequences of HeteroskedasticityHeteroskedasticity

1.1. Pure heteroskedasticity does not cause Pure heteroskedasticity does not cause bias in the coefficient estimates.bias in the coefficient estimates.

2.2. Pure heteroskedasticity increases Pure heteroskedasticity increases variances of the coefficient estimates.variances of the coefficient estimates.

3.3. Pure heteroskedasticity causes OLS to Pure heteroskedasticity causes OLS to underestimate the standard errors of the underestimate the standard errors of the coefficients. (Hence, pure coefficients. (Hence, pure heteroskedasticity overestimates the t-heteroskedasticity overestimates the t-values.)values.)

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Detection of HeteroskedasticityDetection of Heteroskedasticity

Because heteroskedasticityBecause heteroskedasticity can take can take on many forms, therefore, there is on many forms, therefore, there is no specific test to test for no specific test to test for heteroskedasticity.heteroskedasticity.

However, we will discuss two tests However, we will discuss two tests as a tool for detecting as a tool for detecting heteroskedasticity of a certain form.heteroskedasticity of a certain form.

The two tests are (1) Park Test, and The two tests are (1) Park Test, and (2) White Test.(2) White Test.

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Detection of Heteroskedasticity: Detection of Heteroskedasticity: CaveatsCaveats

Before you go on and conduct one or Before you go on and conduct one or both of these tests, ask yourself both of these tests, ask yourself these 3 questions:these 3 questions:

1.1. Is your model already correctly specified? If the Is your model already correctly specified? If the answer is “No”, correct your model first.answer is “No”, correct your model first.

2.2. Does the topic you are studying have a known Does the topic you are studying have a known heteroskedasticity problem as pointed out by heteroskedasticity problem as pointed out by other researchers?other researchers?

3.3. When you plot a graph of the residuals against When you plot a graph of the residuals against the Z factor, does it indicate a potential the Z factor, does it indicate a potential heteroskedasticity problem?heteroskedasticity problem?

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The Park TestThe Park Test

The Park test tests for heteroskedasticity The Park test tests for heteroskedasticity of the form, VAR(of the form, VAR(εεii) = ) = 22ZZii

22.. One difficulty with the Park test is the One difficulty with the Park test is the

specification of the Z factor.specification of the Z factor. The Z factor is usually one of the The Z factor is usually one of the

explanatory variables, but not always. The explanatory variables, but not always. The Z factor could very well be other variable Z factor could very well be other variable not included in the regression model.not included in the regression model.

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3 Steps of the Park Test3 Steps of the Park Test

1.1. Run the OLS and get the residuals:Run the OLS and get the residuals:

eeii = Y = Yii–b–b00–b–b11XX1i1i–b–b22XX2i2i

2.2. Regress the ln(e)Regress the ln(e)22 on the ln(Z): on the ln(Z):

ln(eln(eii22) = b) = b00+b+b11ln(Zln(Zii))

3.3. Check to see if bCheck to see if b11 is significant or not. is significant or not.– If bIf b11 is significant (meaning b is significant (meaning b11 ≠ 0), this ≠ 0), this

implies implies heteroheteroskedasticity.skedasticity.– If bIf b11 is not significant (meaning b is not significant (meaning b1 1 == 0), this 0), this

implies that implies that heteroheteroskedasticity is unlikely. skedasticity is unlikely. (However, it may still exist in other form or (However, it may still exist in other form or exist with other Z factor.)exist with other Z factor.)

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The White TestThe White Test

In stead of having to specify the Z In stead of having to specify the Z factor and heteroskedasticity of the factor and heteroskedasticity of the form VAR(form VAR(εεii) = ) = 22ZZii

22, the White test , the White test takes a more general approach.takes a more general approach.

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3 Steps in the White Test3 Steps in the White Test

1.1. Run the OLS and get the residuals:Run the OLS and get the residuals:

eeii = Y = Yii–b–b00–b–b11XX1i1i–b–b22XX2i2i–b–b33XX3i3i

2.2. Regress eRegress e22 on each X, square of each X, cross product of on each X, square of each X, cross product of the Xs. For a 3-independent variable model, run OLS ofthe Xs. For a 3-independent variable model, run OLS of

3.3. Test the overall significance of the model above with Test the overall significance of the model above with 22 test. The test statistic is nRtest. The test statistic is nR22..

– If nRIf nR22 > critical > critical 2 2 value, this implies value, this implies heteroheteroskedasticity.skedasticity.– If nRIf nR22 < critical < critical 2 2 value, this implies that value, this implies that heteroheteroskedasticity is skedasticity is

unlikely.unlikely. n = number of observations, Rn = number of observations, R22= the value of the = the value of the

unadjusted Runadjusted R22, the DF for the critical , the DF for the critical 22 value is the value is the number of the independent variables in the model above.number of the independent variables in the model above.

i329318217236

225

2143322110

2i

uXXbXXbXXbXb

XbXbXbXbXbb e

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Correcting HeteroskedasticityCorrecting Heteroskedasticity

If heteroskedasticity still is present in If heteroskedasticity still is present in your correctly specified regression model your correctly specified regression model (pure heteroskedasticity), consider one (pure heteroskedasticity), consider one of these three remedies.of these three remedies.

1.1. Use Weighted Least SquaresUse Weighted Least Squares

2.2. Use Heteroskedasticity-Corrected Use Heteroskedasticity-Corrected Standard ErrorsStandard Errors

3.3. Redefine the variables (See an example Redefine the variables (See an example on p. 366-369)on p. 366-369)

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Weighted Least SquaresWeighted Least Squares

Take an original model with a Take an original model with a heteroskedastic error term,heteroskedastic error term,

Suppose the variance is of the form,Suppose the variance is of the form,

Eq.1 can be shown to be equal to,Eq.1 can be shown to be equal to,

Divide Eq.2 with ZDivide Eq.2 with Zii, obtaining, obtaining

1) (Eq.εXβXββY i2i21i10i

2i

22ii ZσσεVAR

2) (Eq.uZXβXββY ii2i21i10i

3) (Eq.uZ

Xβ

Z

Xβ

Z

β

Z

Yi

i

2i2

i

1i1

i

0

i

i

See 10-6 See 10-6 on p. 598 on p. 598 for proof.for proof.

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Weighted Least Squares (Con..)Weighted Least Squares (Con..) If Z happens to be one of the independent If Z happens to be one of the independent

variables, use OLS with Eq.3 specification to variables, use OLS with Eq.3 specification to obtain the coefficient estimates.obtain the coefficient estimates.

If Z is not one of the independent variables, we If Z is not one of the independent variables, we must add an intercept term because Eq.3 has must add an intercept term because Eq.3 has no intercept term. We will run OLS using Eq.4 no intercept term. We will run OLS using Eq.4 specification.specification.

The interpretation of the slope coefficients The interpretation of the slope coefficients becomes a bit tricky.becomes a bit tricky.

There are two problems with WLS: (1) what is There are two problems with WLS: (1) what is Z?, and (2) how Z relates to VAR(Z?, and (2) how Z relates to VAR(εε)?)?

4) (Eq.uZ

Xβ

Z

Xβ

Z

βα

Z

Yi

i

2i2

i

1i1

i

00

i

i

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White’s Heteroskedasticity-Corrected Standard White’s Heteroskedasticity-Corrected Standard Errors (a.k.a. Robust Standard Errors)Errors (a.k.a. Robust Standard Errors)

The idea of this remedy goes like this.The idea of this remedy goes like this. Since, there is no bias in the coefficient estimates.Since, there is no bias in the coefficient estimates. But, the standard errors of the coefficient estimates But, the standard errors of the coefficient estimates

are larger with heteroskedasticity than without it.are larger with heteroskedasticity than without it. Therefore, why not fix the standard errors and leave Therefore, why not fix the standard errors and leave

the coefficient estimates alone?the coefficient estimates alone? This method is referred to as HCCM This method is referred to as HCCM

(heteroskedasticity-consistent covariance matrix). (heteroskedasticity-consistent covariance matrix). In EViews, you will choose “LS” and click on In EViews, you will choose “LS” and click on

“Options”, then select “Heteroskedasticity-“Options”, then select “Heteroskedasticity-Consistent Coefficient Covariance” and select Consistent Coefficient Covariance” and select “White”.“White”.

The standard errors of the coefficient estimates can The standard errors of the coefficient estimates can be bigger or smaller than those from the OLS.be bigger or smaller than those from the OLS.

This method works best in a large sample data.This method works best in a large sample data.

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Redefine the VariablesRedefine the Variables

1.1. For example, take log of some For example, take log of some variables.variables.

2.2. Or, normalize some variables.Or, normalize some variables.