heteroskedasticity prepared by vera tabakova, east carolina university

Chapter 8

Heteroskedasticity

Prepared by Vera Tabakova, East Carolina University

Chapter 8: Heteroskedasticity

8.1 The Nature of Heteroskedasticity

8.2 Using the Least Squares Estimator

8.3 The Generalized Least Squares Estimator

8.4 Detecting Heteroskedasticity

Slide 8-2Principles of Econometrics, 3rd Edition

8.1 The Nature of Heteroskedasticity

Slide 8-3Principles of Econometrics, 3rd Edition

(8.1)

(8.2)

(8.3)

1 2( )E y x

1 2( )i i i i ie y E y y x

1 2i i iy x e

8.1 The Nature of Heteroskedasticity

Figure 8.1 Heteroskedastic Errors

Slide 8-4Principles of Econometrics, 3rd Edition

8.1 The Nature of Heteroskedasticity

Slide 8-5Principles of Econometrics, 3rd Edition

(8.4)

2( ) 0 var( ) cov( , ) 0i i i jE e e e e

var( ) var( ) ( )i i iy e h x

ˆ 83.42 10.21i iy x

ˆ 83.42 10.21i i ie y x

8.1 The Nature of Heteroskedasticity

Figure 8.2 Least Squares Estimated Expenditure Function and Observed Data Points

Slide 8-6Principles of Econometrics, 3rd Edition

8.2 Using the Least Squares Estimator

The existence of heteroskedasticity implies:

The least squares estimator is still a linear and unbiased estimator, but

it is no longer best. There is another estimator with a smaller

variance.

The standard errors usually computed for the least squares estimator

are incorrect. Confidence intervals and hypothesis tests that use these

standard errors may be misleading.

Slide 8-7Principles of Econometrics, 3rd Edition

8.2 Using the Least Squares Estimator

Slide 8-8Principles of Econometrics, 3rd Edition

(8.5)

(8.6)

(8.7)

21 2 var( )i i i iy x e e

2

22

1

var( )( )

N

ii

bx x

21 2 var( )i i i i iy x e e

8.2 Using the Least Squares Estimator

Slide 8-9Principles of Econometrics, 3rd Edition

(8.8)

(8.9)

2 2

2 2 12 2

1 2

1

( )var( )

( )

N

i iNi

i iNi

ii

x xb w

x x

2 2

2 2 12 2

1 2

1

ˆ( )ˆvar( )

( )

N

i iNi

i iNi

ii

x x eb w e

x x

8.2 Using the Least Squares Estimator

Slide 8-10Principles of Econometrics, 3rd Edition

ˆ 83.42 10.21

(27.46) (1.81) (White se)

(43.41) (2.09) (incorrect se)

i iy x

2 2

2 2

White: se( ) 10.21 2.024 1.81 [6.55, 13.87]

Incorrect: se( ) 10.21 2.024 2.09 [5.97, 14.45]

c

c

b t b

b t b

8.3 The Generalized Least Squares Estimator

Slide 8-11Principles of Econometrics, 3rd Edition

(8.10)

1 2

2( ) 0 var( ) cov( , ) 0

i i i

i i i i j

y x e

E e e e e

8.3.1 Transforming the Model

Slide 8-12Principles of Econometrics, 3rd Edition

(8.11)

(8.12)

(8.13)

2 2var i i ie x

1 2

1i i i

i i i i

y x e

x x x x

1 2

1 i i i

i i i i i

i i i i

y x ey x x x e

x x x x

8.3.1 Transforming the Model

Slide 8-13Principles of Econometrics, 3rd Edition

(8.14)

(8.15)

1 1 2 2i i i iy x x e

2 21 1var( ) var var( )i

i i ii ii

ee e x

x xx

8.3.1 Transforming the Model

To obtain the best linear unbiased estimator for a model with

heteroskedasticity of the type specified in equation (8.11):

1. Calculate the transformed variables given in (8.13).

2. Use least squares to estimate the transformed model given in (8.14).

Slide 8-14Principles of Econometrics, 3rd Edition

8.3.1 Transforming the Model

The generalized least squares estimator is as a weighted least

squares estimator. Minimizing the sum of squared transformed errors

that is given by:

When is small, the data contain more information about the

regression function and the observations are weighted heavily.

When is large, the data contain less information and the

observations are weighted lightly.Slide 8-15Principles of Econometrics, 3rd Edition

22 1/2 2

1 1 1

( )N N N

ii i i

i i ii

ee x e

x

ix

ix

8.3.1 Transforming the Model

Slide 8-16Principles of Econometrics, 3rd Edition

(8.16)ˆ 78.68 10.45

(se) (23.79) (1.39)i iy x

2 2ˆ ˆse( ) 10.451 2.024 1.386 [7.65,13.26]ct

8.3.2 Estimating the Variance Function

Slide 8-17Principles of Econometrics, 3rd Edition

(8.17)

(8.18)

2 2var( )i i ie x

2 2ln( ) ln( ) ln( )i ix

2 2

1 2

exp ln( ) ln( )

exp( )

i i

i

x

z

8.3.2 Estimating the Variance Function

Slide 8-18Principles of Econometrics, 3rd Edition

(8.19)

(8.20)

21 2 2exp( )i i s iSz z

21 2ln( )i iz

1 2( )i i i i iy E y e x e

8.3.2 Estimating the Variance Function

Slide 8-19Principles of Econometrics, 3rd Edition

(8.21)2 21 2ˆln( ) ln( )i i i i ie v z v

2ˆln( ) .9378 2.329i iz

21 1ˆ ˆˆ exp( )i iz

1 2

1i i i

i i i i

y x e

8.3.2 Estimating the Variance Function

Slide 8-20Principles of Econometrics, 3rd Edition

(8.22)

(8.24)

(8.23)

22 2

1 1var var( ) 1i

i ii i i

ee

1 2

1ˆ ˆ ˆ

i ii i i

i i i

y xy x x

1 1 2 2i i i iy x x e

8.3.2 Estimating the Variance Function

Slide 8-21Principles of Econometrics, 3rd Edition

(8.25)

(8.26)

1 2 2i i k iK iy x x e

21 2 2var( ) exp( )i i i s iSe z z

8.3.2 Estimating the Variance Function

The steps for obtaining a feasible generalized least squares estimator

for are:

1. Estimate (8.25) by least squares and compute the squares of

the least squares residuals .

2. Estimate by applying least squares to the equation

Slide 8-22Principles of Econometrics, 3rd Edition

1 2, , , K

2ie

1 2, , , S

21 2 2ˆln i i S iS ie z z v

8.3.2 Estimating the Variance Function

3. Compute variance estimates .

4. Compute the transformed observations defined by (8.23),

including if .

5. Apply least squares to (8.24), or to an extended version of

(8.24) if .

Slide 8-23Principles of Econometrics, 3rd Edition

21 2 2ˆ ˆ ˆˆ exp( )i i S iSz z

3, ,i iKx x 2K

2K

(8.27)ˆ 76.05 10.63

(se) (9.71) (.97)iy x

8.3.3 A Heteroskedastic Partition

Slide 8-24Principles of Econometrics, 3rd Edition

(8.28)

(8.29b)

(8.29a)

9.914 1.234 .133 1.524

(se) (1.08) (.070) (.015) (.431)

WAGE EDUC EXPER METRO

1 2 3 1,2, ,Mi M Mi Mi Mi MWAGE EDUC EXPER e i N

1 2 3 1,2, ,Ri R Ri Ri Ri RWAGE EDUC EXPER e i N

1 9.914 1.524 8.39Mb

8.3.3 A Heteroskedastic Partition

Slide 8-25Principles of Econometrics, 3rd Edition

(8.30)2 2var( ) var( )Mi M Ri Re e

2 2ˆ ˆ31.824 15.243M R

1 2 3

1 2 3

9.052 1.282 .1346

6.166 .956 .1260

M M M

R R R

b b b

b b b

8.3.3 A Heteroskedastic Partition

Slide 8-26Principles of Econometrics, 3rd Edition

(8.31b)

(8.31a)1 2 3

1Mi Mi Mi MiM

M M M M M

WAGE EDUC EXPER e

1,2, , Mi N

1 2 3

1Ri Ri Ri RiR

R R R R R

WAGE EDUC EXPER e

1,2, , Ri N

8.3.3 A Heteroskedastic Partition

Feasible generalized least squares:

1. Obtain estimated and by applying least squares separately to

the metropolitan and rural observations.

2.

3. Apply least squares to the transformed model

Slide 8-27Principles of Econometrics, 3rd Edition

(8.32)

ˆ M ˆ R

ˆ when 1ˆ

ˆ when 0

M i

i

R i

METRO

METRO

1 2 3

1ˆ ˆ ˆ ˆ ˆ ˆ

i i i i iR

i i i i i i

WAGE EDUC EXPER METRO e

8.3.3 A Heteroskedastic Partition

Slide 8-28Principles of Econometrics, 3rd Edition

(8.33) 9.398 1.196 .132 1.539

(se) (1.02) (.069) (.015) (.346)

WAGE EDUC EXPER METRO

8.3.3 A Heteroskedastic Partition

Slide 8-29Principles of Econometrics, 3rd Edition

Remark: To implement the generalized least squares estimators

described in this Section for three alternative heteroskedastic

specifications, an assumption about the form of the

heteroskedasticity is required. Using least squares with White

standard errors avoids the need to make an assumption about the

form of heteroskedasticity, but does not realize the potential

efficiency gains from generalized least squares.

8.4 Detecting Heteroskedasticity

8.4.1 Residual Plots

Estimate the model using least squares and plot the least squares

residuals.

With more than one explanatory variable, plot the least squares

residuals against each explanatory variable, or against , to see if

those residuals vary in a systematic way relative to the specified

variable.

Slide 8-30Principles of Econometrics, 3rd Edition

ˆiy

8.4 Detecting Heteroskedasticity

8.4.2 The Goldfeld-Quandt Test

Slide 8-31Principles of Econometrics, 3rd Edition

(8.34)

(8.35)

2 2

( , )2 2

ˆ

ˆ M M R R

M MN K N K

R R

F F

2 2 2 20 0: against :M R M RH H

2

2

ˆ 31.8242.09

ˆ 15.243M

R

F

8.4 Detecting Heteroskedasticity

8.4.2 The Goldfeld-Quandt Test

Slide 8-32Principles of Econometrics, 3rd Edition

21ˆ 3574.8

22ˆ 12,921.9

2221

ˆ 12,921.93.61

ˆ 3574.8F

8.4 Detecting Heteroskedasticity

8.4.3 Testing the Variance Function

Slide 8-33Principles of Econometrics, 3rd Edition

(8.36)

(8.37)

1 2 2( )i i i i K iK iy E y e x x e

2 21 2 2var( ) ( ) ( )i i i i S iSy E e h z z

1 2 2 1 2 2( ) exp( )i S iS i S iSh z z z z

21 2( ) exp ln( ) ln( )i ih z x

8.4 Detecting Heteroskedasticity

8.4.3 Testing the Variance Function

Slide 8-34Principles of Econometrics, 3rd Edition

(8.38)

(8.39)

1 2 2 1 2 2( )i S iS i S iSh z z z z

1 2 2 1( ) ( )i S iSh z z h

0 2 3

1 0

: 0

: not all the in are zero

S

s

H

H H

8.4 Detecting Heteroskedasticity

8.4.3 Testing the Variance Function

Slide 8-35Principles of Econometrics, 3rd Edition

(8.40)

(8.41)

2 21 2 2var( ) ( )i i i i S iSy E e z z

2 21 2 2( )i i i i S iS ie E e v z z v

(8.42)2

1 2 2i i S iS ie z z v

(8.43)2 2 2

( 1)SN R

8.4 Detecting Heteroskedasticity

8.4.3a The White Test

Slide 8-36Principles of Econometrics, 3rd Edition

1 2 2 3 3( )i i iE y x x

2 22 2 3 3 4 2 5 3z x z x z x z x

8.4 Detecting Heteroskedasticity

8.4.3b Testing the Food Expenditure Example

Slide 8-37Principles of Econometrics, 3rd Edition

4,610,749,441 3,759,556,169SST SSE

2 1 .1846SSE

RSST

2 2 40 .1846 7.38N R

2 2 40 .18888 7.555 -value .023N R p

Keywords

Slide 8-38Principles of Econometrics, 3rd Edition

Breusch-Pagan test generalized least squares Goldfeld-Quandt test heteroskedastic partition heteroskedasticity heteroskedasticity-consistent

standard errors homoskedasticity Lagrange multiplier test mean function residual plot transformed model variance function weighted least squares White test

Chapter 8 Appendices

Slide 8-39Principles of Econometrics, 3rd Edition

Appendix 8A Properties of the Least Squares

Estimator

Appendix 8B Variance Function Tests for

Heteroskedasticity

Appendix 8A Properties of the Least Squares Estimator

Slide 8-40Principles of Econometrics, 3rd Edition

(8A.1)

1 2i i iy x e

2( ) 0 var( ) cov( , ) 0 ( )i i i i jE e e e e i j

2 2 i ib w e

2i

i

i

x xw

x x

Appendix 8A Properties of the Least Squares Estimator

Slide 8-41Principles of Econometrics, 3rd Edition

2 2

2 2

i i

i i

E b E E w e

w E e

Appendix 8A Properties of the Least Squares Estimator

Slide 8-42Principles of Econometrics, 3rd Edition

(8A.2)

2

2

2 2

2 2

22

var var

var cov ,

( )

( )

i i

i i i j i ji j

i i

i i

i

b w e

w e w w e e

w

x x

x x

Appendix 8A Properties of the Least Squares Estimator

Slide 8-43Principles of Econometrics, 3rd Edition

(8A.3)

2

2 2var( )i

bx x

Appendix 8B Variance Function Tests for Heteroskedasticity

Slide 8-44Principles of Econometrics, 3rd Edition

(8B.2)

(8B.1)21 2 2i i S iS ie z z v

( ) / ( 1)

/ ( )

SST SSE SF

SSE N S

2

2 2 2

1 1

ˆ ˆ ˆ and N N

i ii i

SST e e SSE v

Appendix 8B Variance Function Tests for Heteroskedasticity

Slide 8-45Principles of Econometrics, 3rd Edition

(8B.4)

(8B.3)2 2( 1)( 1)

/ ( ) S

SST SSES F

SSE N S

2var( ) var( )i i

SSEe v

N S

(8B.5)2

2var( )i

SST SSE

e

Appendix 8B Variance Function Tests for Heteroskedasticity

Slide 8-46Principles of Econometrics, 3rd Edition

(8B.6)

(8B.7)

24ˆ2 e

SST SSE

22 2 4

2 4

1var 2 var( ) 2 var( ) 2i

i i ee e

ee e

2 2 2 2

1

1ˆ ˆvar( ) ( )

N

i ii

SSTe e e

N N

Appendix 8B Variance Function Tests for Heteroskedasticity

Slide 8-47Principles of Econometrics, 3rd Edition

(8B.8)

2

2

/

1

SST SSE

SST N

SSEN

SST

N R