Chapter 8

ECON 6002 Econometrics Memorial University of Newfoundland

Adapted from Vera Tabakova’s notes

8.1 The Nature of Heteroskedasticity

8.2 Using the Least Squares Estimator

8.3 The Generalized Least Squares Estimator

8.4 Detecting Heteroskedasticity

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

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

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.

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

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

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: regress food_exp income, robust

1 2

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

i i i

i i i i j

y x e

E e e e e

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

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

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.

22 1/2 2

1 1 1

( )N N N

ii i i

i i ii

ee x e

x

ix

ix

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]

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

21 2 2exp( )i i s iSz z

21 2ln( )i iz

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

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

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

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

ˆ 76.05 10.63

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

For our food expenditure example:

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]

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):

???

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

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

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

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

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:

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.

8.4.1 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.

ˆiy

8.4.2 The Goldfeld-Quandt Test

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


2

2

ˆ 31.8242.09

ˆ 15.243M

R

F

* --------------------------------------------* 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


21ˆ 3574.8

22ˆ 12,921.9

2221

ˆ 12,921.93.61

ˆ 3574.8F

For the food expenditure data

You should now be able to obtain this test statistic

And check whether it exceeds the critical value

8.4.3 Testing the Variance Function

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


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


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

21 2 2i i S iS ie z z v

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 .1846SSE

RSST

2 2 40 .1846 7.38N R

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

whitetst

Or

estat imtest, white

Slide 8-41Principles 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


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


(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


(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


(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


(8B.8)

2

2

/

1

SST SSE

SST N

SSEN

SST

N R

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Chapter 8