12
Economics 310 Lecture 17 Seemingly Unrelated Regressions

Economics 310

Embed Size (px)

DESCRIPTION

Economics 310. Lecture 17 Seemingly Unrelated Regressions. Seemingly Unrelated Regression Models. Have several regressions using different data sets that span the same period of time. - PowerPoint PPT Presentation

Citation preview

Page 1: Economics 310

Economics 310

Lecture 17Seemingly Unrelated

Regressions

Page 2: Economics 310

Seemingly Unrelated Regression Models Have several regressions using

different data sets that span the same period of time.

Believe that their can be external forces that simultaneously affect the errors contemporaneously for each model.

Page 3: Economics 310

Seemingly Unrelated Regressions

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

222222

2222

111111

1111

,0

0,

0

0

.mod

.11,,1

:2

.11,,1

:1

e

eeand

X

XX

y

yy

WhereeXy

ore

e

X

X

y

y

eloneasunitstwotheWrite

TxiseandxkisTxkisXTxisy

whereeXyUnit

TxiseandxkisTxkisXTxisy

whereeXyUnit

Page 4: Economics 310

No Contemporaneous Covariance

.estimators OLS the..)(

)(

1

1

)(0

0)(1

1

10

01

10

01

0

0

0

01

0

01

0

0')'(ˆ

0

0'

0

0

)(

)()(

2

1

2'2

12

'2

1'1

11

'1

2'22

2

1'12

11

2'2

22

11

'1

21

2'22

2

1'12

1

1

2'22

2

1'12

1

2

1

22

21

'2

'1

1

2

1

22

21

'2

'1111

22

21

'2

'2

'21

'11

'

2

1

2

1

2

1

2

1

21

eib

b

yXXX

yXXX

yX

yX

XX

XX

yX

yX

XX

XX

y

y

I

I

X

X

X

X

I

I

X

XyWXXWX

WI

I

eeee

eeeeE

e

e

e

eEeeE

eE

eE

e

eEeE

T

T

T

T

T

T

Page 5: Economics 310

Contemporaneous Covariance

T

TTTT

T

T

tt

I

eeeeee

eeeeee

eeeeee

Ee

ttwheneande

12

12

12

12

21221121

22122211221

21122111211

'21

2t1122t1

00

00

00

eE

formmatrix In

'0)E(e)E(e

s,other wordIn periods. timeacrossnot

but period, timesame e within thcorrelated areunit each

of errors the,covariance neousContemporaWith

'

Page 6: Economics 310

Contemporaneous Covariance

2'22

1222

21

21

1'22

1222

21

21

2'12

1222

21

121

'12

1222

21

22

1

2'22

1222

21

21

1'22

1222

21

21

2'12

1222

21

121

'12

1222

21

22

2

1

212

22

21

21

212

22

21

21

212

22

21

12212

22

21

22

'2

'1

1

2

1

212

22

21

21

212

22

21

21

212

22

21

12212

22

21

22

'2

'1

2

1

1

2221

1221

'2

'1

1

2

1

1

2221

1221

'2

'1111

2112

2221

1221

'2

'2

'21

'11

'

2

1

2

1

0

0

0

0

0

0

0

0

0

0

0

0')'(ˆ

becomesestimator SquaresLeast dGeneralizeOur

'21

yXyX

yXyX

XXXX

XXXX

y

y

II

II

X

X

X

X

II

II

X

X

y

y

II

II

X

X

X

X

II

II

X

XyWXXWX

Note

WII

II

eeee

eeeeE

e

e

e

eEeeE

TT

TT

TT

TT

TT

TT

TT

TT

TT

TT

Page 7: Economics 310

Covariance Matrix of Estimator

11

111111111111

111111

)'(

)'(')'()'()'(')'(

)'('')'()'ˆ)(ˆ(

XWX

XWXXWWWXXWXXWXXWeeEWXXWX

XWXXWeeWXXWXEE

Page 8: Economics 310

Estimated Generalized Least-Squares estimator

yWXXWX

eiKT

ee

KT

ee

KT

ee

KT

e

KT

ee

KT

e

T

ttt

T

tt

T

tt

111

21121

12

221

22

22

111

21

21

ˆ')ˆ'(ˆ̂

..

'ˆˆ

ˆ

ˆ

ˆ

estimates.their

with scovariance and variances theReplace

Page 9: Economics 310

Fuel Consumption Have data for 3 countries on fuel

consumption per capita and per capita income.

The data covers the period from 1971 to 1992

Page 10: Economics 310

Restricted EstimateSame coefficients

yWZZWZb

isestimatorstricted

WeeE

X

XZwhereeZy

or

e

e

X

X

111

2

1

2

1

2

1

2

1

')'(ˆ

Re

)'(

,

y

y

get we,implicitlyaccount inton restrictio Taking

Page 11: Economics 310

Testing Restriction

KJand

jRRXWXRR

IIRH

e

e

X

X

y

y

elourHave

KK

)(~ˆ')ˆ'(ˆq

is statisticOur test

0:

0

0

mod

2111'

212

10

2

1

2

1

2

1

2

1

Page 12: Economics 310

Testing for Contemporaneous Covariance

.

deg21)-M(Mwith ~)rr(r*T

is statisticOur test

0:

test want to weexample,equation our threeFor

2223

213

212

2313120

jregressionwith

iregressionfromresidualsofncorrelatiorWhere

freedomofrees

H

ij

BREUSCH-PAGAN LM TEST FOR DIAGONAL COVARIANCE MATRIX