Chapter 8

Chapter 8

Heteroskedasticity

Adapted from Vera Tabakova’s notes

ECON 4551 Econometrics IIMemorial University of Newfoundland

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

Food expenditure example:


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

We can use a robust estimator: GRETL offers several options…check the defaults

8.2 Using the robust estimator

The existence of heteroskedasticity implies: Why not use robust estimation all the time? Well, that is a good idea for large samples but for small samples,

homoskedasticity plus normality guarantees that the t ratios are

distributed as t But robust estimates do not guarantee that, so our inference could be

misleading! If you have a small sample, check whether there is homoskedasticity

or not!Slide 8-11Principles of Econometrics, 3rd Edition

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 8-16Principles 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

Food example again, where was the problem coming from?

regress food_exp income [aweight = 1/income]

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



For our food expenditure example (GRETL:

#Estimating the skedasticity function and GLS ols y const xgenr lnsighat = log($uhat*$uhat)genr z = log(x)

#Obtain prediction of variance:ols lnsighat const zgenr predsighat = exp($yhat)

#generate weights;genr w = 1/predsighat

wls w y const x



For our food expenditure example (STATA):

gen z = log(income)regress food_exp incomepredict ehat, residualgen lnehat2 = log(ehat*ehat)regress lnehat2 z

* --------------------------------------------* Feasible GLS* --------------------------------------------predict sig2, xbgen wt = exp(sig2)regress food_exp income [aweight = 1/wt]

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

Using our wage data (cps2.dta):

???



(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

_cons -9.398362 1.019673 -9.22 0.000 -11.39931 -7.397408 metro 1.538803 .3462856 4.44 0.000 .8592702 2.218336 exper .1322088 .0145485 9.09 0.000 .1036595 .160758 educ 1.195721 .068508 17.45 0.000 1.061284 1.330157 wage Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 36081.2155 999 36.1173328 Root MSE = 5.1371 Adj R-squared = 0.2693 Residual 26284.1488 996 26.3897076 R-squared = 0.2715 Model 9797.0667 3 3265.6889 Prob > F = 0.0000 F( 3, 996) = 123.75 Source SS df MS Number of obs = 1000

(sum of wgt is 3.7986e+01). regress wage educ exper metro [aweight = 1/wt]



* --------------------------------------------* Rural subsample regression* --------------------------------------------regress wage educ exper if metro == 0 scalar rmse_r = e(rmse)scalar df_r = e(df_r)* --------------------------------------------* Urban subsample regression* --------------------------------------------regress wage educ exper if metro == 1 scalar rmse_m = e(rmse)scalar df_m = e(df_r)* --------------------------------------------* Groupwise heteroskedastic regression using FGLS* --------------------------------------------gen rural = 1 - metrogen wt=(rmse_r^2*rural) + (rmse_m^2*metro)regress wage educ exper metro [aweight = 1/wt]

STATA Commands:



#Wage Exampleopen "c:\Program Files\gretl\data\poe\cps2.gdt"ols wage const educ exper metro

# Use only metro observationssmpl metro --dummyols wage const educ experscalar stdm = $sigma

#Restore the full samplesmpl full

GRETL Commands:



#Create a dummy variable for ruralgenr rural = 1-metro

#Restrict sample to rural observationssmpl rural --dummyols wage const educ experscalar stdr = $sigma

#Restore the full samplesmpl full

GRETL Commands:

#Generate standard deviations for each metro and rural obs

genr wm = metro*stdm genr wr = rural*stdr

#Make the weights (reciprocal) #Remember, Gretl's wls needs these to be variances so

you'll need to square them genr w = 1/(wm + wr)^2

#Weighted least squares wls w wage const educ exper metro

Principles 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.


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




2

2

ˆ 31.824 2.09ˆ 15.243

M

R

F

STATA:* --------------------------------------------* Goldfeld Quandt test* --------------------------------------------

scalar GQ = rmse_m^2/rmse_r^2scalar crit = invFtail(df_m,df_r,.05)scalar pvalue = Ftail(df_m,df_r,GQ)scalar list GQ pvalue crit

GRETL:

#Goldfeld Quandt statisticscalar fstatistic = stdm^2/stdr^2




21ˆ 3574.8

22ˆ 12,921.9

2221

ˆ 12,921.9 3.61ˆ 3574.8

F

More generally, the test can be based Simply on a continuous variable

Split the sample in halves (usually omittingsome from the middle) after orderingthem according to the suspected variable(income in our food example)




21ˆ 3574.8

22ˆ 12,921.9

2221

ˆ 12,921.9 3.61ˆ 3574.8

F

For the food expenditure data

You should now be able to obtain this test statistic

And check whether it exceeds the critical value

Remember that you can probably use the one-tail version of this testWhy?


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

For the mean:

For the variance, in general:

For example::




(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

S is the number of variables used

This is a large sample test It is a Lagrange Multiplier (LM) test,

which are based on an auxiliary regression

In this case named after Breusch and Pagan

Here (and in the textbook) we saw a test statistic based on a linear function of the squared residual, but the good thing about this test is that this form can be used to test for any form of heteroskedasticity

Principles of Econometrics, 3rd Edition


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

Since we may not know which variables explain heteroskedasticity…


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

whitetst

Or

estat imtest, whiteBreusch-Pagan test

White test

STATA:

GRETL: ols y const xmodtest --breusch-paganmodtest –white

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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