7/26/2019 Elliot Rottenberg Stock 1996 Efficient tests for an autoregressive unit root.pdf

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Y"t,lo4

.

Econometica,

Vol.64,

No.

(July,

1996),

813-836

EFFICIENT

TESTS FOR AN

AUTOREGRESSIVE

UNIT

ROOT

By Gnasav

ELLrorr,

Tnouas

J.

RorHpNspRG, AND

Javes

H. Srocxl

The

asymptotic

power

envelope

is derived

for

point-optimal

tests

of a unit root

in

the

autoregressive

representation of

a

Gaussian

time

series under various

trend

specifications.

We

propose a

family

of

tests whose asymptotic

power functions are tangent to the

power

envelope at

one

point and are never far below the

envelope.

When the

series

has

no

detdrministic

component, some

previously

proposed

tests

are

shown

to be asymptotically

equivalent

to members

of this

family. When the

series has

an unknown mean or

linear

trend,

commonly used

tests

are found

to

be dominated

by

members of the

family of

point-optimal

invariant

tests.

We propose a modified

version

of

the Dickey-Fuller

,

test

which has

substantially

impioved

power

when an unknown mean

or

trend

is

present.

A

Monte

Carlo experiment

indicates

that the modified

test works

well in

small samples.

KEywoRDs:

Power

envelope,

point

optimal

tests,

nonstationarity, Ornstein-Uhlenbeck

processes.

1. lNrRooucrroru

ForrowrNc

rHE

sEMINAL

woRK

of

Fuller

(1976)

and Dickey and

Fuller

(1979),

econometricians

have developed

numerous alternative

procedures for testing

the

hypothesis

that

a

univariate

time series is

integrated

of order

one against

the

hypothesis

that

it

is

integrated

of order

zero.

The

procedures

typically

are based

on

second-order

sample

moments,

but employ

various testing

principles and

a

variety

of'methods

to

eliminate

nuisance

parameters.

Banerjee et

al.

(L993)

and

Stock

(1994)

survey

many

of the

most

popular

of thesg

tests. Although

numerical

calculations

(e.g.,

Nabeya

and Tanaka

(1990))

suggest

that the

power functions

for

the

tests

can

differ

substantially,

no

general optimality theory has been

developed.

In

particular,

there

are few

general

results

(even

.asymptotic)

con-

cerning the relative

merits

of

the competing

testing

principles and

of

the

various

methods

for eliminating

trend

parameters.

Emptoying

a model common

in

the

previous

literature,

we

assume

that the

data

y1,,,,,/r

woro

generated

as

(1)

yt

--

dt

+

ul

(t:t,...,7),

.

ttt:

dllt-1+

Dt

where

{d,}

is a deterministic

component

and

(u,}

is

an

unobserved stationary

zero-mean error

'process

whose spectral

density

function

is

positive

at zero

frequency.

dur

interest

is

in the

null hypothesis

a: 1

(which

implies

the

y,

ate

integrated

of

order one)

versus

lalrrr

r-t1"zvtlRAlz

,

E(s2)

:

2T-

t

ltr(A)

-

.-2

tr(

Ee)l

:

-2

o-2

7-

tDI:

t(k)Q -

k)

dk-

|

-

o-2y(0)

-

t,

Var(S2

)

:

T-2o-

2

4

fil

At R

El

/

2At

R

>

|

/

2

+ RA

EA' Rl

-srr)A]:2L

y(k)lar(&)

-a7(0)l

-

0,

t-1

T-l

T-2trlA,

(V

-

rrl)

A)

:

2

|

p(k)ta7(k)

_

a1(0)l

-

0.

k:1

Cf. Anderson

(1971,

1,0.2.3).

Using

(A?),

we have

(43)

T-2

I

RAf

-

T-

2

fi

At

R2A

:

T-

2

tr[

@2A,

U-

1A

+

o)-

2At

>A

-

2 A,

A]

+

0.

Leuva

A3:

If the

data are generated

by

(7)

under

Conditions

A

and B,

then

(A4)

limT-2d'_1'-td_t:tirr,T-t

Ad,

E-td_1:0,

(As)

plimT-2

d'

-

1

Z-

I

u

-,

:

plim

I-

I

d,_

r

S-

r

4a

:

plim

T-

|

Ad,

E-

1

u

_,

:

g,

where

d_,

:

(0,

db

. .

., dr_

)

and

Atl

:

(db

d2

-

dr,

. . .

,

dr

-

dr_

rl

.

Pnoor:

Under

Cond.ition_B,

T-2d,_t>-rd_1-t)T-2dt_i_t-

tA

ZA, E-

1

d

-

t

x-.2x'x)+0,so?-llRxl2-0,whereR

is

deflned

in

the

proof

of Lemma A2. It follows that

53:T-3/2X'(>-'

-

r-'I)u-,

Lg,

So:7-rtz*'rr-t

-.-211u

O,

since

tr[E(S3S )]:

(d-2T-ltrlXtR>-t/2A>At>-t/2RX'l