Baltagi - Estimating Error Component Models With General MA(q) Disturbances

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Estimating Error Component Models with General MA(q) DisturbancesAuthor(s): Badi H. Baltagi and Qi Li

Source: Econometric Theory, Vol. 10, No. 2 (Jun., 1994), pp. 396-408Published by: Cambridge University PressStable URL: http://www.jstor.org/stable/3532874Accessed: 23-01-2016 16:12 UTC

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Econometricheory,0, 1994, 96-408. rintedntheUnited tates fAmerica.

ESTIMATING ERROR COMPONENT

MODELS WITH GENERAL

MA(q) DISTURBANCES

BADI

H.

BALTAGI

Texas

A

&M University

Qi

Li

IndianaUniversity

This aper rovides simplestimationethodor n erroromponentegres-

sion

modelwith eneral A(q) remainderisturbances.he stimationethod

utilizes

he ransformationerived yBaltagi nd Li [3] for n error ompo-

nentmodelwith utoregressiveemainderisturbances,nd standardrthog-

onalizinglgorithm

or he

eneral A(q) model.

his stimation ethods

computationallyimple tilizingnlyeast-squaresegressions.his s mportant

for aneldataregressionshere rute orceGLS is n many asesnotfeasi-

ble.This stimationethod erforms

ell

elative

o

true LS

in

Monte-Carlo

experiments.

1.

INTRODUCTION

MacDonald and MacKinnon [14] argued that the AR(p) specifications

more

popular

han

he

MA(q) specification

n

empirical pplications,

ot

be-

cause t s more

plausible, ut

rather

ecause

t

s easier o compute.This pa-

per focuses

n

the

moving verageprocess,

which

s

a

viable

alternative

hat

naturally

rises

n

several conomic

xamples,

ee

Nicholls,Pagan,

and

Ter-

rell 17]. For the panel data error omponent egressionmodelwith emain-

der

disturbances ollowing

n

MA(q) process,

the

estimationmethod

s

further

omplicated y

the

presence

f

random

ndividual ffects.

ecently

Baltagi

nd

Li

[3]

obtained

simple ransformation

hat ransforms

he

uto-

regressiverror omponent

isturbancesnto

spherical

isturbances.

his

s

important

or

panel data studies

where

rute

orceGLS

is in

many

ases

not

feasible, specially

or

argepanels.Baltagi

nd

Li

[3]

showedhow this rans-

formation

an be

applied

when

he

remainder

isturbances

vit's)

ollow

n

AR(1),

an

AR(2),

or a

special AR(4) process

for

quarterly

ata.1

n

fact,

BaltagiandLi [3] argued hat,for heerror omponentmodel,their rans-

formation eneralizes

o

any

error

rocessgenerating

he

remainder istur-

We

thankMax King, Peter chmidt, eterPhillips, nd

two anonymous eferees

orproviding elpful om-

ments nd

suggestions hat mproved

he

paper.

396

( 1994Cambridge

University ress

0266-4666/94 5.00 +

.00






ERROR COMPONENTS WITH MA(q) DISTURBANCES 397

bances s long s there xists simple nownmatrix,ayC, that ransforms

the emainderisturbancesnto phericalisturbances.

n

order o operation-

alizethis ransformation,stimatesf the erial orrelationarametersre

needed. ortheAR(p) model, hese arameterstimatesreeasily btain-

able. However, or heMA(q) process, he stimation ethods more

n-

volved ven

n

conventionalime eriesmodels.Recently, acDonald nd

MacKinnon 14]proposed convenient ethod or stimatingheMA(1)

model,

ut

n

extension

f

this

methodo the

general A(q) process

s dif-

ficult. llah,Vinod, ndSingh 23]proposed new stimation ethod

or

linearmodelswithmoving verage isturbances

f

any order, ut this

method equires onlineareast-squaresegressions.dditionally,

or he r-

ror omponents odel, hismethods furtheromplicatedythepresence

of ndividualffects

hich

re

hard

oseparate

rom he

movingverage

f-

fects.

his

paper roposes simple pproach

o the stimationferror

om-

ponents egression

odel

with

emainder

isturbancesollowing general

MA(q) process.2

n

essence,

e show hat

nly he utocovarianceunction

estimatesf the

omposite

rror erm reneeded

n

thefirst

tep

o

obtain

GLS estimatesf regressionoefficients

n

the

econd tep.

The

computa-

tional teps re as

follows:

i) estimate

he

utocovarianceunctionf the

composite

rror erm

sing

LS

residuals;ii) retrievestimates

f

thegen-

eralMA(q) autocovarianceunctionswell sestimatesfthe arianceom-

ponents;

nd

iii) apply

he

ransformationroposed yBaltagi

nd

Li

[3]

that tilizes

ecursive

ormulas

or he

movingverage rocess

nd

perform

OLS. This

procedure

esults

n

asymptotically

fficientstimates

f

the

e-

gression

oefficientsnd

requires nly east-squaresegressions.

Section reviewshe

Baltagi

ndLi

[3]

transformationor he rror om-

ponent

modelwith

utoregressive

emainder

isturbances,

nd

Section gen-

eralizes he transformation

o the error

omponent

model

with

general

MA(q)

remainder

isturbances.ection

performs

onte-Carlo

xperi-

ments ndconcludes hepaper.

2. THE MODEL

Consider he

followinganel

data

regression odel,

yit

x, to+uit

i

=

1,.

..,9N;

t

=

1,. ..,

T, (1)

whereB s

a

K

x

1

vector f

regressionoefficients,ncluding

he

ntercept.

The disturbancesollow one-wayrroromponent odelseeHsiao [9]),

Uit

=

Ai

+

Pit

(2)

with ndividual

ffects

Ai

i.i.d.

0,

,2)

and

remainder

isturbances

it

hat

follow

stationary-dependentrocess

where






398

BADI H. BALTAGI

AND Ql LI

'Ys

?

<

Isl

c

q

E

(pitPi,

t-s)

=(3

t/0

otherwise,

with

_-

=

Pys*Note that

a standard

MA(q)

process is

equivalent

to a sta-

tionary

-dependent

rocess.

In

fact, the standard

MA(q) process,

vit

=

Cit

+

XIei t-I

+

*.

+

XqEi,t-q,

where

it

-

i.i.d.(0,a,),

is a

special

ase

of

the

tationary

-dependent

rocess efined

n

3). On the

other

and,one

can

also

prove hat

ny

stationary

-dependent

rocess

an be represented

s a

standard

MA(q) process

see Brockwell

nd

Davis

[4]). In what

follows,

we

adopt a general

pproach,

assuming

he

vit's

ave

an arbitrary

erial

corre-

lation

form.

By using

the results

f Baltagi

and

Li [3], we derive

he spec-

traldecompositionf thevariance-covariance atrix fthecomposite rror

and then pply

t specifically

o thegeneral

MA(q) process

given

n (3).3

In

vector

form, quation

1)

can be written

s

y

=

X:

+ u

(4)

with

U

(IN

?&

T)L

+

P

(5)

where

N

is an

identity

matrix

f dimension

N,

eT

is a vector

of 1's of

di-

mension T,

,'

=

(Al,

. . . ,,IN),

and

v'

= (Pl1,

. .

,Y1T,

..

*., N1,

*

.

,PNT).

The variance-covariance

atrix

f u

is

?

2

IN

0

JT

+

g2

IN

0O

Q

=

IN

(&

A,

(6)

where

A

= or

JT

+

o20,

and

E(

vi

)

=

a20

is the variance-covariance

a-

trix

f the remainder

rror

erm

for ach individual 4

=

(Pv1I,

i2,

PiT),

for

=

1,...

,

N. Because

0

is a real

symmetricositive-definite

atrix,

here

exists

T x

T matrix

,

such thatCDC' =

IT.

Therefore,

we

propose

the

following

ransformation.

Definey*=

(IN

( C)Y; X* and u* are similarly efined. hetransformed

error

becomes

U*=

(IN

(D CeT)A

+

(IN

C)V

=

(IN

0

e')/

+

(IN

C)P,

(7)

where '

=

CeT

=

(011,.. T,

Y)

is

a

T x

1

vector,

epending

n the

specific

serial

correlation

rocess.

The

variance-covariance

matrixfor the

trans-

formed

isturbance

*

becomes

*=

IN

0

[CAC']

=

IN

0

*

(8)

withA*

=

CAC. Because C has thepropertyhat CDC'

= IT,

bywriting

JT

=

eTeT

and

noting

hat

CeT

=

e'2,

we

get

from he

definition

f A that

A

=

'2eTeT

+

o2IT

=

2,JT

+

(IT

(9)

whereJ

a=

ece.

This can

be rewritten

s

A*

=

02d2(eTea/d2)

+ or2IT

=

Or2d2JT

J2IT

(10)






ERROR COMPONENTS

WITH MA(q) DISTURBANCES

399

where 2

= e'

e'.

Thisreplaces

T eTeT by ts dempotent

ounterpart

JT= eTeT/d2.

By

following

ansbeek

nd Kapteyn24,25],one also re-

places IT by ET +

JT,

hereET = IT

-J.

From collectingike terms, ne

gets he pectral

ecompositionf A*:

A*

=

2JA

+

2

Ec

(11)

where

2,

u2d2+

u2.

BecauseJT nd

ET are dempotent

atriceshat re

orthogonal

o each

other,

we

have

A*P

=

(uf) JT+ (a2)PET

(12)

where is

an arbitrarycalar.

n particular,

= -1

gives *` (andhence

1*-1)

andp

= I-

gives A*1/2 (and hence E*-1/2). Therefore,

orS

=

(IN

(0

JT)

+

(IN

(?

ET)

=

(IN

(0

IT)

-

6(IN

(092)

(13)

where

= 1-

(/ia). Thus

we can premultiply

he C-transformed

odel

by

UE*-1/2

to make the error pherical.y*

=

u*-l/2y*

and

X**

and u**

are

similarly

efined.

he

typical

lementsf

y**

re given y

T

s=1

Yit

=Y*-

a

(14)

Za

2

This

s a

generalized

ersionfthe

uller ndBattese

5]transformationor

theerroromponent

odelwith n

arbitrary

ariance-covarianceatrix,

E( i', )

=

a 2O,

on

the remainder isturbances. he

OLS regression

n the

(**)

transformedquation

s

equivalent

o theGLS

regression

n the rigi-

nal equation

4).

Equation12) also suggestsaturalstimatorsfthe arianceomponents.

Baltagi

nd

Li

[3] proposed

stimating

2

nd

o,2u

y

=

U*/(IN

0

JT)U*/N

and

2

=

U*J(IN

0

ET)u*/N(T

-

1).

(15)

These re

best

uadratic

nbiased stimators

f a and

2

if

the rue is-

turbances

*

are

known. hetrue esidualsre

generally

otknown.

n

this

case,

one can

replace

*

byu*,

the

OLS residuals n the

*)

transformed

equation.

3. THE GENERALMA(q) PROCESS

In

this ection

e

give

simple ecursiveransformation

or he rrorom-

ponent

model

with emainder

isturbancesollowingn MA(q) process.

Let

ys

=

E(viti,

t-s)

denote the

autocovariance

functionfor

Pit

for s

=

0,1,..

,q.

Let

S

=

ys/-Y

for = 0,1,...

,q, which mplies

hat

O

?

1. Note

that

E

(Pi,

Pis)

s the same

for ll i

=

1,... ,N,

therefore e do not

write he






400 BADI H.

BALTAGI AND

Ql

LI

subscript

in

7y.

The

Appendix ives

standard

rthogonalizinglgorithm

that ransformshe

general A(q)

disturbancesnto

pherical

isturbances

(see Li

[11]).'

With hisnotation, he *) transformationorourmodel,de-

fined

n

7),

is

given ecursively

y

Y7i

Yi,

Yit

=

[yit

-

*t*y*

-b

-

bt,jyi

]I.a

for t

=

2,...,q

Y*

=

[

Yi,

-bt,

t-qYt,

-q]

I

for t

=

q +1,...,T.

(16)

Both

,

and

bt,5

re

n

turn eterminedecursively,s follows:

a,1

=

at= 1-b

-

-b722-r2

1

for

t=2,...,q

atf-

1

-

b

tt

2

for

t=q+

1,...,T. (17)

The recursiveelation or

bt,,

s

given y

(i)

for

<

q,

bt,

=

rtI

bt,,_5s=

[r.

-

bt- s

-t-s-I

b,s-I

bt,s,I

bt,I

/

for s

=

1,...,t-2; (18)

(ii)

for

q

+

1,..

,T

bt,t-q

=

rt,q/atq

b,

t-s=

[r-

bs,s-Ibt1s,1-

-

bs,t-qbt,t-q]

/

for s= 1,...,t-2. (19)

By replacing * by

at

and

replacing

it

y

1

n

(16),

we

can

get

e

=

CT =

(a

,1

C

2,

Ce

T))

as

follows:

1 =

1,

oet

[1

-

b,tbt-

t-.

-bt,IceI]/

/d-

t

=

2,

,q

ot

=

[1

-

bt,t-aoet-I

-

bt,t-qat-q]/v

t= q

+

1,...,

T.

(20)

They.*andX,* are the ame s in 14). The OLS regressionn the **)

transformedquation s equivalento theGLS regression

n theoriginal

equation 4).

The

above

ransformationepends n the utocovarianceunctionf

Pit,

that

s,

-ys(s

0,

1,

.

,q). Therefore, or he

transformationo be feasible,

we must

et

stimates

f

ys.

This

ooks ike difficultroblem ecause t s






ERROR COMPONENTS

WITH

MA(q)

DISTURBANCES 401

not lear

how

one can

separate

he ndividualffect

,i

rom

he

emainder

error erm

it.

We

make

he

followingimplification

hat nables s to

get

consistentstimatesf the

y7's

as well s

u,2

yusing nly east-squarese-

siduals.

Denote by Q5

=

E(uitui,

-s)

the autovariancefunction f

ui,.

Then

we

have

Ur2

+

T,S

=

0,

1, ... .,

q

QS

=

or2

ss

q +

1.

(21)

Qs

can

be consistently

stimated

y5

N T

Qs

=

E(uituiut-s)

E N O0's?

T-1

(22)

i=1I

t=s?

IJV(T

-s)

where he

uit's

enote he

OLS

residuals.

rom

21)

and

22),

we

get onsis-

tent

stimates

f

ys,

2

=a

yo

nd

u2

as follows:

es

=

Qs

-

Q

s

q;

for some r>

q

Uyy2

Q,

for some r

> q. (23)

We eethat

he

stimation

fthe ariance

omponent

2

andthemovingv-

erage arameters

s

s simple, equiringnlyOLS residuals,ndonedoes

notneed o recoverhe

Xt's

f the

MA(q) process.

fter

etting,

Ts

and

"2=

or

=yo one computes

-=

o

for

=1,...

., q. Next, one computes

t

and

bt,s

from

17), 18),

and

19)

and

performs

he

C

transformation

e-

scribed

n

16). Finally,

ne obtains

2

=

l2

from20) andestimate

2

by

c2

=

2

+

2. By using

=

1

-

(4r-y/a,>),

thefinal ransformation

(**)

canbe

performed

s

described

n

14).

The

OLS

estimatef

B

rom he

(**)

transformedquation

s

thus

quivalent

o theGLS estimatefthe

rig-

inal

equation 4).

The advantages

f our

approach

are

by

now evident: 2

-

yo

is

trivially

obtainedfrom

OLS residuals.

This s

because

we

did

not

choose

a,2

as

in

Baltagi nd

Li

[3].

Next,

weestimated

's

atherhanX's.The 's

require

only

inear

east

quares,

whereasheX's

require

onlineareast

quares.

i-

nally,

ur

proposed

stimation

rocedureequiresimple

ecursive

ransfor-

mations hat re

very asy

o

program.

his hould

rove

seful

or

anel

data users.

4. MONTE-CARLORESULTS

This ection

eports

onte-Carloesults

n the

erformance

f

our

proposed

feasible

LS estimatorelative

o

true

GLS.

Weusethe

ollowingimple

e-

gressionquation

Yit

=

a

+

fXit

+

Uit






402

BADI H.

BALTAGI AND Ql Li

where

=

5 and

/3

0.5, nd similarata-generating

rocesss that nNer-

love 16],namely

it

=

0.1 +

0.5Xi,t_1 wit

where

it

s uniformlyistrib-

uted n the nterval-0.5,0.5]. The nitial aluesXi0were hosen s (5 +

10wio). it ,ui vit

with

i

-

IIN(0,o,2)

nd

vit

=

Eit

+

XjEi,t,j

+

.. .

+

XqEi,t-q

with

it

IIN(0,o2).

We focus

n the

MA(1)

and

MA(2)

models

(q

=

1 and 2) in ourexperiments.e fix

he otalvariance cross xperi-

ments o be

0r2

+

o2

=

20.

Note

hat

2

=

(1

+

X2

+

...

+

X2)r2. Next,we

vary

=

or2/(u2

+ ur2)

to take

the

values

0.2,0.5,0.8),respectively.

or

sample ize,

we

choose

N, T)

=

(100,6), 100,12),

250,6),

nd

250,12) e-

causeN>>

T in

a

typicalonsumer

r abor

panel.

or

each

xperiment,

e

calculate

hemean

quare

rror

MSE)

of an estimator

by

MSE(3) =

Z1)2

As

00)2/M,

where

0

= 0.5 s the rue alue f

A3,

is the stimator

of

in

the

th

replication,

nd

M

=

1000s thenumberfreplications.6or

each

replication,

he

following

stimators

f

B

re considered:

i)

the

OLS

estimator;ii)

our

proposed

easible LS estimatorescribed

n

Section

;

(iii)the rueGLS estimator

TGLS;

(iv)

the

onventionalne-way

rror om-

ponents LS estimatorf assuming o

MA(q) process

n the emainder

disturbances;

nd

(v)

the

within

stimatorf/3,

hich

reats

he

<i's

s

fixed ffects.We then ompute he

relative

fficiency

f

/3

y MSE(3)/

MSE(OTGLS).

These elative SE's aredenotedyROLS, RGLS, RCGLS,

andRW correspondingoOLS, ourproposed easibleGLS, the onventional

feasible

GLS,

and the within

stimators,

espectively.

or the

MA(1) case,

we use

r

=

2 in

(23) to get

J2,

Q2, and

for he

MA(2)

case we use

r

=

3

in

(23) to get

2

=

Q

.7 The results re reported

n

Tables

1

and

2

below.

Table

1

gives the Monte-Carlo results

for the MA(1) case:

Pit

=

(it

+

Ei,t-l,

whereX

=

0, 0.5,

and 1.

First,

et us look at the case

of

(N, T)

=

(100,6). Our proposedfeasibleGLS estimator erforms uite

well

even

for

the case whereX

=

0).

Its MSE is at most 5

1 larger

han

that

of the true

GLS for

ll cases

of X and

p

considered. he

OLS

estimator

erforms ery

badlydue to the fact hat t gnores oththe ndividual ffects

,i

ndthese-

rial

correlation

n

the remainder rror

erm

vit.

As

expected,

he

perfor-

manceof OLS deterioratess (i) X ncreases the arger

he erial

orrelation)

for fixed

alue

of

p,

or as

(ii) p increases the arger

he ndividual

ffect)

for fixed alue of X. For theconventional

rror

omponents,

easible

GLS,

which

gnores

he

MA(1) term,

as

the ame

MSE

(up

to

the

econd

decimal

in

relative

fficiency)

s that

f

trueGLS

(when

X

=

0)

for

ll

valuesof

p

con-

sidered.

However,

s

X

departs

rom and

increases,

ts

performance

ete-

riorates.This is because this estimatorgnores

he

serialcorrelation

n

Pit.

For a fixed alue of X (> 0), theperformancef this onventional easible

GLS estimator

mproves

s

p

increases

or

most ases.

This s because

when

p

is

large,

the

ndividual ffect

i

will

dominate he remainder erm ffect

Pft,

iven

hatwe fixed hetotalvariance

cross

experiments.

ut evenwhen

p

=

0.8,

the

efficiency

oss due to

ignoring

heserial orrelation

n

vit

s

quite

largecompared o our proposedfeasibleGLS

estimator. inally, heperfor-

mance of the within

stimator

s

better han

thatof OLS, and a littleworse






ERROR

COMPONENTS

WITH MA(q) DISTURBANCES

403

TABLE 1.

The MA(1)

case

p X ROLS RGLS RCGLS RW

N=

100,

T=6

.2 0 1.19

1.01

1.00

1.06

.2

.5 1.38

1.05

1.27

1.39

.2 1

1.85

1.02

1.64

1.77

.5

0 1.68

1.00

1.00

1.05

.5 .5

1.86

1.01

1.18

1.23

.5

1

2.83

1.01

1.65

1.69

.8

0

4.25

1.04

1.00

1.00

.8 .5 5.78 1.03 1.25 1.27

.8

1

5.78

1.02 1.56

1.58

N= 100,

T= 12

.2

0 1.08

1.01 1.00

1.01

.2

.5 1.19

1.00 1.11

1.12

.2

1

1.66

1.04

1.57 1.59

.5

0 1.44

1.00

1.00

1.01

.5

.5 1.45

1.00

1.14 1.14

.5 1 1.76 1.00 1.42 1.42

.8

0 2.97

1.00

1.00

1.00

.8

.5

2.40

1.01 1.13

1.14

.8

1

2.92

1.03

1.48

1.48

N=

250,

T=

6

.2 0 1.15

1.03

1.00

1.08

.2 .5

1.26 1.02

1.15

1.26

.2

1

1.94

1.01 1.67

1.76

.5

0 1.86 1.00 1.00

1.04

.5 .5 1.99 1.00 1.18 1.22

.5 1

2.47

1.01

1.58

1.64

.8 0 4.82 1.00

1.00

1.00

.8 .5

4.76 1.01 1.21

1.25

.8 1 5.66 1.01

1.64 1.66

N

-

250,

T

=

12

.2 0

1.10

1.00

1.00

1.01

.2 .5

1.18

1.01

1.11

1.12

.2 1 1.62 1.01 1.53 1.54

.5

0

1.49

1.00

1.00

1.00

.5

.5

1.42

1.00

1.15

1.15

.5

1 1.86

1.02

1.47

1.48

.8

0 2.88

1.00 1.00

1.00

.8

.5

2.39

1.00

1.11

1.11

.8

1

3.03

1.02 1.45

1.45






404 BADI H. BALTAGI

AND

Ql LI

TABLE

2.

The

MA(2) case

P

X1

X2

ROLS RGLS

RCGLS

RW

N=

100,

T=

6

.2 0 0

1.19 1.01 1.00 1.06

.2 .5

.5 1.64 1.03

1.46 1.56

.2

1

1

2.46

1.01 2.14

2.27

.5 0 0

1.68

1.01

1.00 1.05

.5 .5

.5 2.22

1.02

1.40 1.44

.5 1 1

3.56 1.02 1.97 2.01

.8 0

0 4.25 1.03

1.00

1.01

.8 .5 .5 5.74 1.03 1.43 1.45

.8

1 1

7.00 1.05 1.94

1.96

N=

100,

T=

12

.2

0

0 1.08 1.01

1.00

1.01

.2

.5 .5

1.33 1.02 1.26 1.27

.2

1

1

2.08 1.03 2.01 2.02

.5 0

0

1.44 1.01 1.00 1.01

.5

.5 .5

1.67 1.02 1.38 1.39

.5 1 1 2.32 1.02 1.96 1.96

.8

0 0

2.75

1.00

1.00 1.00

.8 .5 .5 2.59

1.05 1.26 1.26

.8 1 1

3.63 1.06

2.01

2.01

N=250,

T=6

.2 0

0

1.15 1.02

1.00

1.08

.2

.5

.5 1.59

1.03 1.41

1.51

.2 1

1

2.40

1.01

1.99 2.09

.5 0

0

1.86

1.00

1.00 1.01

.5 .5 .5 2.20 1.02 1.41 1.47

.5 1 1

3.50

1.02 2.02

2.06

.8

0

0

4.99 1.02

1.00

1.01

.8

.5 .5

5.49

1.02

1.45 1.45

.8

1

1

6.89

1.03

2.14

2.18

N=250,

T =

12

.2 0 0

1.10 1.01

1.00

1.01

.2

.5

.5

1.40

1.02

1.33 1.34

.2 1 1 2.01 1.02 1.94 1.97

.5 0

0

1.46 1.00

1.00 1.00

.5 .5

.5 1.63

1.02

1.33

1.33

.5 1 1

2.55 1.01

2.06

2.06

.8

0

0

3.06 1.01

1.00 1.00

.8 .5

.5

2.46

1.00

1.35

1.35

.8

1

1

3.56 1.03

1.94

1.95






ERROR OMPONENTS

WITH

MA(q) DISTURBANCES

405

than hat f theconventional

easible LS estimator. owever,

n most

cases, hewithinstimatorhows big oss

n

efficiencyue

to gnoringhe

serial orrelationn theremaindererm.

We

keep

N fixed

nd double

hevalueof T, that s, (N,T)

=

(100,12).

While

n

general

ll the stimators

mprove

nrelative SEperformance,

he

OLS estimatormproves

he

most.However,

he

rankings

f these

stima-

tors ccording

o

their

elative SE

and

the

eneral attern

bservedbove

remain nchanged. ext,wekeep

T

fixed nd ncreasehe

value fN from

100

o 250. n

general,

heresultsre quite imilar

o

the ase

of

N

=

100.

Note hat he elative SE performancefOLS, conventional

easible

LS,

and thewithinstimatorsid not how ystematicmprovement

s we n-

creased for fixed alue fT. However, urproposed easible LS esti-

mator erforms

ell

ompared

o trueGLS

and ts

relative

fficiency

s

at

most .03

for

ll

cases onsidered.

n

general,

he

ankings

f

these

stima-

tors ccording

o their elative

SE

and

the

eneral attern

bserved

bove

remain

nchanged.

Table

2

gives

he

Monte-Carlo

esults

orthe

MA(2)

case

vit

=

cit

+

XiEi,t-l

+

X2ijt-2

We

varied

I

and

X2

over

,

0.5,

and

1.

As

clear

rom

his

table,

he

onsequences

f gnoringhe erial orrelation

n

theremainder

term remuchmore ronounced

or heMA(2) case. Therelativefficiency

for = 0.8,X1

=2=

1,T= 6,andN= 100, s 7.00forOLS, 1.96for he

within

stimator,

.94 for

he onventionalrror omponent

easible LS

estimatorhat gnoresheMA(2)

process,

nd 1.05for

ur

proposed easi-

ble GLS estimator.

n

general,

ur

proposed stimatorerforms

ell or

ll

experiments,ielding

relative

SE

nolarger

han

Wo

bove hat f true

GLS. The

weak

erformance

f

OLS, within,

ndconventionaleasible LS

warns he esearcherbout he

high rice f gnoringerial orrelation

n

the

remaindererm

ven

forpanels

with

arge

N.

NO TES

1. Lillard ndWillis 12]werehe irsto considern error omponent

odelwith erially

correlated

emainderisturbances

f

heAR(l) type. lternative

ormulationsf he rrorom-

ponentmodel

with

eriallyorrelated

rrors ere onsideredyRevankar21],

Lee

[10],

nd

MaCurdy13]for

he

ingle quation

model, ndbyMagnusndWoodland15]for hemulti-

variate egression odel.

2. Thispaper ssumes hat

he rder ftheMA processs known. owever,fonefollows

themaximumikelihoodoute,ne anuse Hannan ndRissanen

8]type frecursionhich

buildsnthe

chwarz

22]BIC

criterion

t

each tage odeterminehe rder f he

ARMA

pro-

cess.More ecently,hillipsndPloberger19]derivedBayesmodel" estswhichn case of

posterior

dds ead to a

scaling

hat

s

equivalent

o theuse

of a conditionalype

f

Jeffreys

prior ndresultna generalizationftheBIC criterion.hisnew

IC criterionanbe used s

inPhillipsndPloberger20]

for

electingoth he agorder nd

deterministicrend egreen

the lassof ARMA p, q)

models

with

eterministicrend.

3. Wethank eter chmidtor uggesting

his

eneralpproach

o

our

ransformation.

4. Alternativeransformations

or he

MA(l) process

re

given

y

Pesaran

18]

ndBales-

tra

2]. Also,Ansley1]

and

Galbraithnd

Zinde-Walsh

6,7]

for

he

ARMA

p, q) process.






406 BADI H. BALTAGI

AND Ql

LI

5.

Throughout

he

paper, consistency

s forN

-+

oo,

withT finite r T

-*

oo.

6. We also calculate he MSE for

e,

he ntercept. he results re quite similar o thosefor

(,

and are not reported ere to save space.

7. For theMA(q) case, one can also use a linear ombination f

QT's

or

=

q

+.1,.

., T- 1

to estimate

2

. This may givea better stimator f

2

in finite amples,but

should

not make

a differencesymptotically. e performedome Monte-Carlo xperimentsn two alternative

estimators

f

2, one based on

U2'=

t=2

(T

-

)Q/=2

(T

-

T) and another ne based

on

2

2

=

Q2.

Our results how thatforN

=

100, T

=

6, the MSE of

&21

is smaller han that

of

&,22

but the relativeMSE of the corresponding

3's withrespect o trueGLS are virtually

identical.

REFERENCES

1. Ansley,C.F. An algorithm or he exact ikelihood f a mixed utoregressive-movingv-

erage process. Biometrika

6

(1979): 59-65.

2.

Balestra,

P. A

note on the exacttransformationssociated

with

he first-order oving v-

erage process.

Journal

f Econometrics

4

(1980):

381-394.

3.

Baltagi,B.H. & Q.

Li. A

transformationhat

will

circumventhe problem

f

autocorrela-

tion

n

an error

omponentmodel.

Journal

f

Econometrics

8 (1991):

385-393.

4.

Brockwell, .J.

&

R.A. Davis. Time series: Theory nd Methods. New

York:

Springer-

Verlag, 199 .

5. Fuller,W.A. & G.E. Battese.Estimation f linearmodelswith ross-errortructure. our-

nal

of Econometrics (1974): 67-78.

6.

Galbraith,

J.W.

& V. Zinde-Walsh.The GLS transformation atrix nd a semi-recursive

estimator

or

he

inear egression

odelwithARMA

errors. conometric

heory (1992a):

95-111.

7.

Galbraith,

J.W.& V.

Zinde-Walsh. ransforming

he

error-component

odel

for

estima-

tion with

general

ARMA

disturbances. npublished aper,

McGill

University, ontreal,

Quebec, Canada,

1992b.

8.

Hannan, E.J. & J. Rissanen.Recursive stimation f

ARMA

order.Biometrika

9

(1982):

81-94.

9. Hsiao, C. Analysisof Panel Data. Cambridge:CambridgeUniversity ress, 1986.

10.

Lee, Lung-Fei.

Estimation

f autocorrelated rror omponentsmodel

with

panel

data.

Working aper, Department

f

Economics, University

f

Minnesota,

1979.

11. Li, Q. Estimating linear egression odelwith eneralMA(q) disturbances.Working a-

per, University f Guelph, Ontario, Canada, 1992.

12.

Lillard,

L.A.

& R.J. Willis.Dynamic spects f earningmobility.

conometrica

6

(1978):

985-1012.

13.

MaCurdy,

T.A.

The

use

of time eriesprocesses o model the error tructure

f

earnings

in a

longitudinal

ata

analysis.

Journal

f

Econometrics

8

(1982):

83-114.

14. MacDonald, G.M. & J.G. MacKinnon.Convenientmethods or stimationf inear egres-

sion models with

MA(1)

errors.Canadian Journal

f

Economics

18

(1985):

106-116.

15. Magnus, J.R. &

A.D.

Woodland.

On

the maximum

ikelihood

stimation

f

multivariate

regressionmodels ontaining erially

orrelated rror

omponents.

nternational conomic

Review 29

(1988):

707-725.

16. Nerlove,M. Furthervidence n the stimation fdynamic conomic elations rom time-

seriesof crosssections.

Econometrica 9

(1971):

359-382.

17.

Nicholls,D.F.,

A.R.

Pagan

& R.D.

Terrell.

The

estimation nd use of models

with

mov-

ing verage

isturbanceerms:

survey.

nternational

conomic

Review16

1975):

113-134.

18.

Pesaran,

M.H. Exact

maximum

ikelihood stimation f a

regressionquation

with first-

order

moving-average

rror.

Review

of

Economic Studies

40

(1973):

529-536.

19.

Phillips,

.C.B. & W.

Ploberger.

ime

series

modeling

with

Bayesian

frame f reference:






ERROR COMPONENTS

WITH

MA(q)

DISTURBANCES

407

concepts,llustrations

nd

asymptotics.

owles oundation iscussionaper

No. 1038,

1992a.

20. Phillips,.C.B.

& W.Ploberger.osteriordds

estingor unit oot

with ata-based odel

selection. owles oundationiscussionaperNo. 1017, 992b.

21. Revankar, .S.

Error

omponent

odelswith

eriallyorrelatedime ffects. ournal

f

the ndian StatisticalAssociation

17 (1979): 137-160.

22.

Schwarz,

. Estimatinghe imension

f model. nnals fStatistics

(1978): 61-464.

23.

Ullah,A.,

H.D. Vinod R.S.

Singh. stimationf inearmodels

ithmovingverageis-

turbances.Journal f Quantitative

conomics

2

(1986):

137-152.

24. Wansbeek,

. & A. Kapteyn.

simple ay o obtain he pectral

ecompositionf vari-

ance

components

odels orbalanced

ata. Communications

n Statistics

ll

(1982):

2105-2112.

25. Wansbeek,

. & A. Kapteyn. note n spectral

ecompositionnd

maximumikelihood

estimationfANOVAmodels ith alanced ata. tatisticsndProbabilityetters 1983):

213-215.

APPENDIX

This

Appendixderives

standard

rthogonalizing

lgorithm

or

he

generalMA(q)

processdefined n 3). We seek a T x T matrix suchthat

v'

=

Cpiarespherical is-

turbances

with

'

-

(0,0yOIT).

To

keep

the notationmore

compact,

we

drop

the

i

subscript nd

write

t

for

Pit

and

v for

vi.

We define

r,

=

-ys/YO

with

ro

=1),

for

s

=

O,

. .

,

q.

Then

lE')

=

E

=

yo

with

=

Toeplitz

l,1,

l

..

,rp,O,.

.

,0).

De-

fine

wt

Pt/V/y

t

=

1,

. .

.,

T),

thenwe have w

-

(O,Q).

We

wantto find C ma-

trix uch

that

CDC'

=

IT.

The variance-covariance

matrix f w

(i.e., Q)

can be

summarized s follows:

rs

S

IsI

=

0,1 *...

,q

E

cot t-s)

=

0 otherwise (A.1)

with

O

5

I

and

r-s

=

rs.

Let

w*=CW,

we

want

t st=s

0

otherwise.

(A.2)

That s,we wantthetransformedrrors

,*

to be

orthogonal

o

each other

nd have

unitnorm.

We

write

w*

as a

linear ombination

f

wt

and

*

's for

r

< t. Note that

the

wt's

re not

correlated

f

they

re more han

q periods

part.Also,

note that

he

wt

dependonlyon currentndpastvaluesofcoss c t). Therefore,

*

houldhave

the

following

ecursive tructure:

'4

= coi/Vd

=

w[ct-bt,

t-14-I

-

I

-bt,c*1a]/c

(t=2,...,q)

CA)t =

[&)t

-

btI

.

-bt,t-qW

q]/Vat

(t

=

q

+

1,...

., T)

(A.3)






408

BADI H. BALTAGI ND Ql Li

where t s the

normalizationonstanto

ensure

(w*2)

1 for ll t. One can eas-

ily how hat

t,,

E(wtwc*)

or < t. Weoutlinehe roof ere.

Multiply

4

by

W*

forr

<

tandthen ake hemathematicalxpectation.y using (4C0*o) 0 for

t

*

s

and E(*o2)

= 1

for all s, we have

E(wco*)

=

[E(ct*)

-

t,-

-a

=

0.

Hence,

bt,7

E(wtco*)

for

maxtt

-

q,1]

c r<

t.

By therequirement

hat (c42)

=

1

for ll t, and using hefact

hat

(o2)-

=o

1

for

ll

t,

we

get

a1

=

1

tI-1-[E(wt*

1)2

[E(cotw*

]2

=

I b

2

t 1 b2,

1

~-

-t=

1

-

t-

(t=

2,..

.,q)

=

t

I

-

-[E(wt

-

1)]2

[E(ctw*

)]2 =

1-b-bt2q

(t

=

q

+

1,

.

.,

T).

(A.4)

To determinehe ransformation

ompletely,

e

also need o

know

t,,

E(Cotw4)

for < t. These re determined

ecursivelyy

the

following

ormulas.

Case (i).

For t

c q,

we

have

btj

=

E(wco)

=-E(wtco)/-a =rt

bt,t-, = E(wtw* s) = E[o?tlwt_s -bt-s,t-,-lUw* l-

b

-ts, I

*

I/at-s I

=

ts-

I

-

~

)%f;~

= [rs - bt-s,t-s- bt, -s- I bt-s, bt,,]

Ifd-a

(s

=

1, ..,

t-2). (A.5)

Case

(ii).

For

t

q

+

1,

we

have

bt,t-q

E(WtWt_q)

=

E[wt[Wt-q-

bt-q,t-q-C0t*-q-1

-bt-q,e

fl/

q]

where

Q

=

max(l,t

-

2ql.

Therefore,

bt,t-q

E((tC0t-q)/at-q

=

rq/latq

(A.6)

because

wt

and

wt-r

are

not

orrelated

or

>

q.

For

=

1,

t

2,

we have

bt,t-s

=

E(Wtw*

s) =

E[wt{cwts8

-

*s-

bt-s,t-q?t-q

-bt-s,t-q-

-

IW-

-bt-s,m(0'nJ/

ats]

where

m

=

max{l,t

-

s

-

qJ. Therefore,

bt,

-s

=

[

s

-

bt-s,

-s-

bt,

-s

-

.

-

bt-s,

-qbt,tq]

-q

*.

(A.7)

Therefore,

he

complete

ransformation

f

co*

Cw with

var(w*)

=

IT

is de-

scribed

ecursivelyy quationsA.3)

to

A.7).

Hence,

*

=

V/7_

=

v%CCv

=

can be

obtained

rom

A.3)

with'*and

vt

replacing

and

t,

respectively.

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Baltagi - Estimating Error Component Models With General MA(q) Disturbances

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