testing equality between sets of coefficients in two linear ... - j-stage

The Economic Studies Quarterly Vol. 40, No. 1, March 1989

TESTING

EQUALITY

COEFFICIENTS

BETWEEN

IN TWO LINEAR

SETS OF

REGRESSIONS

WHEN ERROR TERMS ARE AUTOCORRELATED* By

1. When

structural (1960)

Sickles

variances

be

significantly

nominal

change

has

(1977)

the

is examined

widely

been

among

AND

the

terms

different

TOSHIHISA

this

is heteroscedasticity,

alternative

tests

ticulary,since justified

by

TOYODA

by

many

large

Although

See,

that In

size

this

paper,

we

when

Consider

(1)

also

Prakken

Statistic a linear

a test

Equality

regression

t=1, t=n1+1,

are grateful

can by

by

we

Monte

Carlo

with

under first-order

can

be

test

time

series time

by

different

in

statistic

be

have

in final

when

been

Chow

Cairo the

data,

this

view

test

when

experiments

nominal

size.

in two

examine

comparison

(1985).

section

From

of the Monte

two Par

studied

Toyoda

cross

data.

and

so-called

tested.

of coefficients

process,

can

is admittedly

and

from

equality

the

the

which

and/or

properties

test

proposed

Ohtani

series

showed

experiments

test

when

Chow

misleading

can

Wald

Wald

test

the

by and

Namely, of

subsequently

and

and

is shown

Coefficients

model

the

sampling

can

proposed Schmidt

researchers,

have

autoregressive

example

of

2,...,

which

test and

size

test

(1985)

significantly

a first-order

applied

economic

the

be

by

of

economic

many

Chow (1974)

heteroscedasticity

Silver

process

statistic

statistic

yt=xtƒÀ2+ur;

autors

in

actual

(1979)

of the

and

studied

test

empirical

yt=xtƒÀ1+ut;

The

Ali

(1982)

obey

test an

for

properties

the

Chow

is a kind

sample

in

the

under

(1982), appears

of

Watt

(1979)

appears

Chow

terms

Also,

and

autoregressive

the

derive

of the

test.

Test

of

error

properties

nalChow

terms

result

the

Toyoda

to heteroscedasticity.

unequal,

coefficients

Watt

often

and

actual

sample

small

robust

by

is pre-assigned the

(1977)

Honda

a first-order

the

regressions

2.

obey

which that

by

the e.g.,

in error Pollock

terms

are

of

proposed

heteroscedasticity

point,Corsi,

is not

size

equality

samples,

autocorrelation

error

test

authors.

test

means

equation,

as shown

regressions

Jayatissa

which

the

the

fact

economic

However,

Chow

in two

from

Since

in some

employed.

others,

of error

size.

there

*

OHTANI

Introduction

Chow

by

KAZUHIRO

some with

the

linear small origi

section.

Autocorrelated

Errors

autocorrelated

errors:

n1 n2+2,...,

to an anonymous

n1+n2,

referee -35-

for his comments

on an earlier

version

of this

paper .

The Economic where

yt

tionsof

is the

t-th

independent

obeying

the

(2)

is an

t=1,

remain

approach

autocorrelated

constant to the

errors.

assumption The

model

(3)

y=XƒÀ+u

2,...,

with

(1)

can

so be

the

P=[(1-ƒÏ2)1/2 -p 0

multiplying

(4)

where variable

model

xt

vectors

is an

1•~k

vector

of coefficients

and

and

two

actual

and ƒÃt is the constant

sample equality

data

used

error

term

variance ƒÐ2.

t-th

observa

ut is an error

regimes

may

be

of coefficients in the

last

which

term

is normally

Although

section

the

restrictive,

in two

and

assumption

it may

linear

at least

expressed

in

a matrix

form

as

be

allowed

regressions seems

to

under show

0...0 0...0 xn1+1...xn1+n2], u=[u1...un1 un1+1...un1+n2], ƒÀ=[ƒÀ1 ƒÀ2].Defining the matrix P as

0

0 ...

1

0 ...0

0

0

1 ...

(3)

by

0

0

P

from

0

the

left,

we

obtain

the

Note

that

the

trandformed

transformed

model:

y*=X*ƒÀ+u*,

y*=Py, is a

X*=PX usual

and u*=Pu.

form:

the transformed

that

follows:

y*=[ (1-ƒÏ2)1/2y1 y2-ƒÏy1...yn-ƒÏyn-1] (n=n1+n2),

while

of the

unreasonable.

-ƒÏ

the

k•~1

(•bƒÏ•b0

(or

examine the

case,

the

the

4.

An

As

an

equation

selection

of the

modified

testing

sample

Chow

tests,

Empirical empirical Japan

The

procedure

followed

the

the

of selecting

terms

tests

main-test

for

a two-stage

scope

in

the

and

paper.

At

the

sign

consists

the

we

tests

depending

be

of the

can

estimate

pre-test

for

it is important

to

selection

present,

will

autocor

of the

Although to give

Chow section

or negatively

on

which

H0:ƒÀ1=ƒÀ2.

modified next

positively

depend

test

test

of this the

are

may

a two-stage

among

analysis

error

Chow

of such

it is beyond

empirical

the

yields

by

properties

procedure of ƒÏ.

for

whether

H00: ƒÏ