Heteroskedasticity Prepared by Vera Tabakova, East Carolina University.
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Transcript of 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