Chapter 8

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


(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


Figure 8.1 Heteroskedastic Errors




(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


Figure 8.2 Least Squares Estimated Expenditure Function and Observed Data Points


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.




(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



(8.8)

(8.9)

2 2

2 2 12 2

1 2

1

( )var( )

( )

N

i iNi

i i Nii

i

x xb w

x x

2 2

2 2 12 2

1 2

1

ˆ( )ˆvar( )

( )

N

i iNi

i i Nii

i

x x eb w e

x x



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


(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


(8.11)

(8.12)

(8.13)

2 2var i i ie x

1 21i i i

i i i i

y x ex x x x

1 21 i i i

i i i i ii i i i

y x ey x x x ex x x x



(8.14)

(8.15)

1 1 2 2i i i iy x x e

2 21 1var( ) var var( )ii i i

i ii

ee e xx xx


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).



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 Ni

i i ii i ii

ee x ex

ix

ix



(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


(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



(8.19)

(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



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

2ˆln( ) .9378 2.329i iz

21 1ˆ ˆˆ exp( )i iz

1 21i i i

i i i i

y x e



(8.22)

(8.24)

(8.23)

22 2

1 1var var( ) 1ii i

i i i

e e

1 21

ˆ ˆ î i

i i ii i i

y xy x x

1 1 2 2i i i iy x x e



(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


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


1 2, , , K

2ie

1 2, , , S

21 2 2ˆln i i S iS ie z z v


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 .


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


(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



(8.30)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



(8.31b)

(8.31a)1 2 3

1Mi Mi Mi MiM

M M M M M

WAGE EDUC EXPER e

1,2, , Mi N

1 2 31Ri Ri Ri Ri

RR R R R R

WAGE EDUC EXPER e

1,2, , Ri N


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


(8.32)

ˆ M ˆ R

ˆ when 1ˆ

ˆ when 0

M i

i

R i

METRO

METRO

1 2 31

ˆ ˆ ˆ ˆ ˆ î i i i i

Ri i i i i i

WAGE EDUC EXPER METRO e



(8.33) 9.398 1.196 .132 1.539(se) (1.02) (.069) (.015) (.346)WAGE EDUC EXPER METRO



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.


îy


8.4.2 The Goldfeld-Quandt Test


(8.34)

(8.35)

2 2

( , )2 2

ˆˆ 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 31.824 2.09ˆ 15.243

M

R

F


8.4.2 The Goldfeld-Quandt Test


21ˆ 3574.8

22ˆ 12,921.9

2221

ˆ 12,921.9 3.61ˆ 3574.8

F


8.4.3 Testing the Variance Function


(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




(8.38)

(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




(8.40)

(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


8.4.3a The White Test


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


8.4.3b Testing the Food Expenditure Example


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

2 1 .1846SSERSST

2 2 40 .1846 7.38N R

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

Keywords


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


Appendix 8A Properties of the Least Squares

Estimator Appendix 8B Variance Function Tests for

Heteroskedasticity

Appendix 8A Properties of the Least Squares Estimator


(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

ii

x xwx x



2 2

2 2

i i

i i

E b E E w e

w E e



(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



(8A.3)

2

2 2var( )i

bx x

Appendix 8B Variance Function Tests for Heteroskedasticity


(8B.2)

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

( ) / ( 1)/ ( )

SST SSE SFSSE N S

2

2 2 2

1 1ˆ ˆ ˆ and

N N

i ii i

SST e e SSE v



(8B.4)

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

/ ( ) SSST SSES F

SSE N S

2var( ) var( )i iSSEe v

N S

(8B.5)2

2var( )i

SST SSE

e



(8B.6)

(8B.7)

24ˆ2 e

SST SSE

22 2 4

2 4

1var 2 var( ) 2 var( ) 2ii i e

e e

e e e

2 2 2 2

1

1 ˆ ˆvar( ) ( )N

i ii

SSTe e eN N



(8B.8)

2

2

/

1

SST SSESST N

SSENSST

N R

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Chapter 8