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Chapter 8

ECON 4550 Econometrics Memorial University of

Newfoundland

- Heteroskedasticity

Adapted from Vera Tabakovas notes

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

8.1 The Nature of Heteroskedasticity

(8.1)

(8.2)

(8.3)

8.1 The Nature of Heteroskedasticity

- Figure 8.1 Heteroskedastic Errors

8.1 The Nature of Heteroskedasticity

(8.4)

Food expenditure example

8.1 The Nature of Heteroskedasticity

- 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.

8.2 Using the Least Squares Estimator

(8.5)

(8.6)

(8.7)

8.2 Using the Least Squares Estimator

(8.8)

(8.9)

8.2 Using the Least Squares Estimator

We can use a robust estimator regress food_exp

income, robust

8.2 Using the Least Squares Estimator

In SHAZAM the HETCOV option on the OLS command

reports the White's heteroskedasticity-consistent

standard errors OLS FOOD INCOME / HETCOV

Confidence interval estimate using the White

standard errors CONFID INCOME / TCRIT2.024 But

the White standard errors reported by SHAZAM have

numeric differences compared to our textbook

results. There is a correction we could apply,

but we will not worry about it for now

Slide 8-11

Principles of Econometrics, 3rd Edition

8.3 The Generalized Least Squares Estimator

(8.10)

8.3.1 Transforming the Model

(8.11)

(8.12)

(8.13)

8.3.1 Transforming the Model

(8.14)

(8.15)

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) - Calculate the transformed variables given in

(8.13). - Use least squares to estimate the transformed

model given in (8.14).

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.

8.3.1 Transforming the Model

Food example again, where was the problem coming

from? regress food_exp income aweight

1/income

(8.16)

8.3.1 Transforming the Model

Food example again, where was the problem coming

from? Specify a weight variable (SHAZAM works

with the inverse) GENR W1/INCOME OLS FOOD

INCOME / WEIGHTW 95 confidence interval, p.

205 CONFID INCOME / TCRIT2.024

(8.16)

Slide 8-18

Principles of Econometrics, 3rd Edition

Note that

- The residual statistics reported in a WLS

regression (SIGMA2, STANDARD ERROR OF THE

ESTIMATE-SIGMA, and SUM OF SQUARED ERRORS-SSE)

are all based on the transformed (weighted)

residuals - You should remember when making model

comparisons, that the high-variance observations

are systematically underweighted by this

procedure - This may be a good thing, if you want to avoid

having these observations dominate the model

selection comparisons. - But if you want model selection to be based on

how well the alternative models fit the original

(untransformed) data, you must base the model

selection tests on the untransformed residuals.

8.3.2 Estimating the Variance Function

(8.17)

(8.18)

8.3.2 Estimating the Variance Function

(8.19)

(8.20)

8.3.2 Estimating the Variance Function

(8.21)

8.3.2 Estimating the Variance Function

(8.22)

(8.23)

(8.24)

8.3.2 Estimating the Variance Function

(8.25)

(8.26)

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

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 .

(8.27)

8.3.2 Estimating the Variance Function

For our food expenditure example gen z

log(income) regress food_exp income predict ehat,

residual gen lnehat2 log(ehatehat) regress

lnehat2 z -------------------------------------

------- Feasible GLS -------------------------

------------------- predict sig2, xb gen wt

exp(sig2) regress food_exp income aweight 1/wt

Slide 8-27

Principles of Econometrics, 3rd Edition

8.3.2 Estimating the Variance Function

The HET command can be used for Maximum

Likelihood Estimation of the model given in

Equations (8.25) and (8.26), p. 207. This

method is an alternative estimation method to the

GLS method discussed in the text (so the results

will also be different) HET FOOD INCOME

(INCOME) / MODELMULT

Slide 8-28

Principles of Econometrics, 3rd Edition

8.3.3 A Heteroskedastic Partition

Using our wage data (cps2.dta)

(8.28)

(8.29a)

(8.29b)

???

8.3.3 A Heteroskedastic Partition

(8.30)

8.3.3 A Heteroskedastic Partition

(8.31a)

(8.31b)

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

(8.32)

8.3.3 A Heteroskedastic Partition

(8.33)

8.3.3 A Heteroskedastic Partition

--------------------------------------------

Rural subsample regression ---------------------

----------------------- regress wage educ exper

if metro 0 scalar rmse_r e(rmse) scalar

df_r e(df_r) ---------------------------------

----------- Urban subsample regression

-------------------------------------------- regre

ss 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 - metro gen wt(rmse_r2rural)

(rmse_m2metro) regress wage educ exper metro

aweight 1/wt

STATA Commands

Slide 8-34

Principles of Econometrics, 3rd Edition

8.3.3 A Heteroskedastic Partition

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.

8.4 Detecting Heteroskedasticity

- 8.4.2 The Goldfeld-Quandt Test

(8.34)

(8.35)

8.4 Detecting Heteroskedasticity

- 8.4.2 The Goldfeld-Quandt Test

--------------------------------------------

Goldfeld Quandt test ---------------------------

----------------- scalar GQ rmse_m2/rmse_r2 s

calar crit invFtail(df_m,df_r,.05) scalar

pvalue Ftail(df_m,df_r,GQ) scalar list GQ

pvalue crit

Principles of Econometrics, 3rd Edition

Slide 8-38

8.4 Detecting Heteroskedasticity

- 8.4.2 The Goldfeld-Quandt Test

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 Detecting Heteroskedasticity

- 8.4.2 The Goldfeld-Quandt Test

Sort the data by income SORT INCOME FOOD / DESC

OLS FOOD INCOME On the DIAGNOS command the

CHOWONE option reports the Goldfeld-Quandt test

for heteroskedasticity (bottom of page 212) with

a p-value for a one-sided test. The HET option

reports the tests for heteroskedasticity reported

on page 215. DIAGNOS / CHOWONE20 HET

Principles of Econometrics, 3rd Edition

Slide 8-40

8.4 Detecting Heteroskedasticity

- 8.4.2 The Goldfeld-Quandt Test

SORT INCOME FOOD / DESC OLS FOOD INCOME On

the DIAGNOS command the CHOWONE option reports

the Goldfeld-Quandt test for heteroskedasticity

(bottom of page 212) with a p-value for a

one-sided test. The HET option reports the tests

for heteroskedasticity reported on page 215.

DIAGNOS / CHOWONE20 HET Of course, this

option also computes the Chow test statistic for

structural change (that is, tests the null of

parameter stability in the two subsamples).

Principles of Econometrics, 3rd Edition

Slide 8-41

8.4 Detecting Heteroskedasticity

- 8.4.2 The Goldfeld-Quandt Test

SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS N1

N2 SSE1 SSE2 CHOW PVALUE G-

Q DF1 DF2 PVALUE 20 20 0.23259E06 6

4346. 0.45855 0.636 3.615 18 18 0

.005 CHOW TEST - F DISTRIBUTION WITH

DF1 2 AND DF2 36

Principles of Econometrics, 3rd Edition

Slide 8-42

8.4 Detecting Heteroskedasticity

- 8.4.2 The Goldfeld-Quandt Test

SHAZAM considers that the alternative hypothesis

is smaller error variance in the second subset

relative to the first subset. Some authors

present the alternative as larger variance in the

second subset. Goldfeld and Quandt recommend

ordering the observations by the values of one of

the explanatory variables. This can be done

with the SORT command in SHAZAM. The DESC option

on the SORT command should be used if it is

assumed that the variance is positively related

to the value of the sort variable.

Principles of Econometrics, 3rd Edition

Slide 8-43

8.4 Detecting Heteroskedasticity

- 8.4.3 Testing the Variance Function

(8.36)

(8.37)

8.4 Detecting Heteroskedasticity

- 8.4.3 Testing the Variance Function

(8.38)

(8.39)

8.4 Detecting Heteroskedasticity

- 8.4.3 Testing the Variance Function

(8.40)

(8.41)

(8.42)

(8.43)

8.4 Detecting Heteroskedasticity

- 8.4.3a The White Test

8.4 Detecting Heteroskedasticity

- 8.4.3b Testing the Food Expenditure Example

whitetst Or estat imtest, white

Further testing in SHAZAM

- DIAGNOS / HET
- Will yield a battery of heteroskedasticity tests

using different specifications

SHAZAM for food example

- DIAGNOS\HET
- REQUIRED MEMORY IS PAR 7 CURRENT PAR 22

480 - DEPENDENT VARIABLE FOOD 40 OBSERVATI

ONS - REGRESSION COEFFICIENTS
- 10.2096426868 83.4160065402
- HETEROSKEDASTICITY TESTS
- CHI-SQUARE D.F.

P-VALUE - TEST STATISTIC
- E2 ON YHAT 7.384 1

0.00658 - E2 ON YHAT2 7.549 1

0.00600 - E2 ON LOG(YHAT2) 6.516 1

0.01069 - E2 ON LAG(E2) ARCH TEST 0.089 1

0.76544 - LOG(E2) ON X (HARVEY) TEST 10.654 1

0.00110 - ABS(E) ON X (GLEJSER) TEST 11.466 1

0.00071 - E2 ON X TEST
- KOENKER(R2) 7.384 1

0.00658 - B-P-G (SSR) 7.344 1

0.00673 - E2 ON X X2 (WHITE) TEST
- KOENKER(R2) 7.555 2

0.02288

Same in SLR

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)

Appendix 8A Properties of the Least Squares

Estimator

Appendix 8A Properties of the Least Squares

Estimator

(8A.2)

Appendix 8A Properties of the Least Squares

Estimator

(8A.3)

Appendix 8B Variance Function Tests for

Heteroskedasticity

(8B.1)

(8B.2)

Appendix 8B Variance Function Tests for

Heteroskedasticity

(8B.3)

(8B.4)

(8B.5)

Appendix 8B Variance Function Tests for

Heteroskedasticity

(8B.6)

(8B.7)

Appendix 8B Variance Function Tests for

Heteroskedasticity

(8B.8)