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TESTING STATIONARITY AND TREND STATIONARITY

AGAINST THE UNIT ROOT HYPOTHESIS

Herman J. BIERENS

Department of Economics, Southern Methodist University, Dallas

Shengyi GUO

Department of Economics, Southern Methodist University, Dallas

and Federal Reserve Bank, Dallas *

JEL Classification: C10, C12, C15, C22

ABSTRACT

In this paper we propose a family of relatively simple nonparametric tests for a

unit root in a univariate time series. Almost all the tests proposed in the literature

test the unit root hypothesis against the alternative that the time series involved is

stationary or trend stationary. In this paper we take the (trend) stationarity

hypothesis as the null and the unit root hypothesis as the alternative. The other

difference with most of the tests proposed in the literature is that in all four cases

the asymptotic null distribution is of a well-known type, namely standard Cauchy.

In the first instance we propose four Cauchy tests of the stationarity hypothesis

against the unit root hypothesis. Under H1 these four test statistics involved,

divided by the sample size n, converge weakly to a non-central Cauchy

distribution, to one, and to the product of two normal variates, respectively.

Hence, the absolute values of these test statistics converge in probability to

infinity (at order n). The tests involved are therefore consistent against the unit

root hypothesis. Moreover, the small sample performance of these test are

compared by Monte Carlo simulations. Furthermore, we propose two additional

Cauchy tests of the trend stationarity hypothesis against the alternative of a unit

root with drift.

*) The views expressed in this paper are not necessarily those of the Federal Reserve Bank of

Dallas or the Federal Reserve System.

2

(1.1) H0: y

t' µ % u

twith µ ' E [y

t]

(1.2) H1: )y

t' u

t,

1. INTRODUCTION

In this paper we propose a family of relatively simple nonparametric tests for a

unit root in a univariate time series. Almost all tests proposed in the literature test

the unit root hypothesis against the alterative that the time series involved is

stationary or trend stationary. See Fuller (1976), Dickey and Fuller (1979, 1981),

Evans and Savin (1981, 1984), Said and Dickey (1984), Phillips (1987), Phillips

and Perron (1988), Kahn and Ogaki (1990) and Bierens (1991), among others.

For further related references, see Phillips (1987) and Haldrup and Hylleberg

(1989, Table 1). Moreover, see Schwert (1989) and DeJong et al. (1991) for

power problems of unit root tests. Only Park's (1990) approach allows for a

reversal of the hypotheses involved.

In this paper we first take the stationarity hypothesis as the null and the

unit root hypothesis as the alternative, i.e., denoting the time series process by yt

we test the null hypothesis

against the alternative

where ut is a stationary process. The other difference with most of the tests in the

literature is that the asymptotic null distribution is of a well-known type, namely

standard Cauchy (c.q. the Student distribution with one degree of freedom).

Under H1 the four test statistics involved, divided by the sample size n, converge

weakly to a non-central Cauchy distribution, to one, and to the product of two

normal variates, respectively. Hence, the absolute values of these test statistics

converge in probability to infinity (at order n). The tests involved are therefore

consistent against the unit root hypothesis. Moreover, the small sample

performance of these tests is compared by limited Monte Carlo simulations, for

sample sizes 100 and 300, using 1000 replications.

In section 8 of this paper we generalize the third Cauchy test to two tests

of the trend stationarity hypothesis against the unit root with drift hypothesis, with

similar asymptotic properties.

3

(2.1) $n' [j

n

t'1

(t&t)(yt&y)] / [j

n

t'1

(t&t)2] ,

(2.2) t ' (1/n)jn

t'1

t '1

2(n%1) and y ' (1/n)j

n

t'1

yt

,

(b) suptE*u

t*6 < 4 for some 6 > 2 ;

(c) F2 ' limn64

E[(1/ n)jn

t'1

ut]2 exists and F2 > 0 ;

(d) (ut) is "&mixing with mixing coefficients "(s) that satisfy

j4

s'1

"(s)1&2/6 < 4.

(2.3) Wn(8) ' (1/ n)j

[8n]

j'1

uj/F if 8n $ 1, W

n(8) ' 0 if 8n < 1 ,

2. LINEAR TIME TREND REGRESSION

The intuition behind our tests is that under H1 the process yt has a stochastic trend

and therefore behaves (more or less) as if there is a deterministic linear trend.

This suggests to regress yt on time t, i.e., estimate the auxiliary "model" yt = µ + βt

+ vt on the basis of the observations t=1,...,n, and use the least squares estimate of

β,

with

as a basis for a test statistic. The further intuition is that the rate of convergence in

distribution of βn is different under Ho and H1, and that this difference can be

exploited to distinguish between Ho and H1.

Following Phillips (1987, Assumption 2.1) we assume that the process (ut)

is such that:

ASSUMPTION 1:

(a) E(ut) = 0 for all t;

Note that this assumption allows for a fair amount of heterogeneity.

Next, denote for λ ε [0,1],

4

(2.4) D(f,g) ' sup0#8#1

*f (8)&g(8)*.

(2.5) W(8) - N(0,8), W(8%*)&W(8) - N(0,*),

W(8) and W(8%*)&W(8) are mutually independent.

M(Wn) ' m

1

0

8m Wn(8)d8; m $ 0.

(1/n)jn

t'1

(t/n)m Wn(t/n) ' m

1

0

8mWn(8)d8 % O

p(1/n),

(2.6a) (1/n)jn

t'1

(t/n)mWn(t/n) Y m

1

0

8m W(8)d8,

(2.6b) (1/n)j[µ n]

t'1

(t/n)m Wn(t/n) Y m

µ

0

8m W(8)d8,

(2.6c) (1/n) jn

t'[µ n]%1

(t/n)m Wn(t/n) Y m

1

µ

8m W(8)d8.

where σ is defined in Assumption 1(c) and [x] means truncation to the nearest

integer # x. Then Wn(λ) is a stochastic element of the metric space D[0,1] of

functions on [0,1] with countably many discontinuities, endowed with the

Skohorod topology [cf. Billingsley (1968)]. The Skohorod norm is dominated by

the sup norm

Herrndorf (1984) shows that under Assumption 1, Wn converges weakly to a

standard Wiener process W (denoted by Wn Y W), which is a stochastic element

of the metric space C[0,1] with norm (2.4) of continuous functions on [0,1] such

that for 0 # λ# 1 and 0 # δ # 1-λ,

Moreover, for any continuous mapping Φ from D[0,1] into C[0,1] we have Φ(Wn)

Y Φ(W). A special case of such a mapping is the integral

Since,

we thus have

and similarly for 0 < µ < 1,

5

(2.7) (1/ n)jn

t'1

(t/n)m ut/F ' W

n(1)&mm

1

0

x m&1 Wn(x)dx

Y W(1)&mm1

0

x m&1W(x)dx.

Sn(x)'j

[xn]

t'1

vt

if x , [n &1,1], Sn(x) ' 0 if x , [0 ,n &1) ,

jn

t'1

F(t/n)vt' F(1)S

n(1) & m

1

0

f (x)Sn(x)dx .

rn' [(n%1)3 & 3(n%1)2 % 2(n%1)] / 12 .

(2.8) $n'

jn

t'1

(t & ½(n%1))ut

[(n%1)3 & 3(n%1)2 % 2(n%1)] /12'

0n

rn

2,

Furthermore, it follows from lemma 1 below that

LEMMA 1: For real numbers v1, ..., vn, let

and let F be a differentiable real function on [0,1], with derivative f. Then

PROOF: Easy and therefore left to the reader.

With these results at hand we can now prove lemmas 2 and 3 below:

LEMMA 2: Under Assumption 1 and H0 , rnβn Y N(0,σ2), where

PROOF: Observe that

say. It follows from (2.7) and (2.8) that

6

(2.9) 0n/(n n) ' (1/ n)j

n

t'1

[ (t/n) & ½]ut% O

p(1/n)

' ½FWn(1) & Fm

1

0

Wn(x)dx % O

p(1/n)

Y ½FW(1) & Fm1

0

W(x)dx .

½W(1) & m1

0

W(x)dx - N(0,1/12).

jn

t'1

(t&t)(yt&y) ' j

n

t'1

tjt

j'1

uj& ½(n%1)j

n

t'1j

t

j'1

uj

' Fn 2 n(1/n)jn

t'1

(t/n)Wn(t/n) & F½(n%1)n n(1/n)j

n

t'1

Wn(t/n).

(2.10) [r2

n / (n 2 n)]$nY Fm

1

0

(8& ½)W(8)d8,

(2.11) E[Fm1

0

(8&½)W(8)d8]2 ' F2/120 .

The lemma now easily follows from the fact that

LEMMA 3: Under Assumption 1 and H1, (rn/n)βn Y N(0,σ2/10).

PROOF: Observe that

Hence by (2.6a),

where rn is defined in Lemma 2.

The limiting distribution in (2.10) is normal with zero mean. It is not hard to

show that

Observing that 12r2n/n

3 6 1, it follows from (2.10) and (2.11) that

7

(2.12) (rn/n)$

nY F 12m

1

0

(8&½)W(8)d8 - N(0,F2/10) .

(3.1) >n' (1/n)j

n

t'1

yt& (1/[½n])j

[½n]

t'1

yt

,

Comparing the results of Lemmas 2 and 3 we see that under H0 the

asymptotic rate of convergence in distribution of βn is of order n%n, whereas under

H1 the asymptotic rate of convergence is of order %n. Thus, if σ2 were known, the

test rnβn/σ would be a consistent standard normal test of the stationarity hypothesis

against the unit root hypothesis, since ,rnβn/σ, 6 4 in probability under H1. However,

in practice we cannot use this test statistic because the variance σ2 is unknown.

3. CAUCHY TEST I

3.1 The Test

We may think of estimating σ2 in a similar way as in White and Domowitz (1984),

Newey and West (1987), and in particular Phillips (1987) and Phillips and Perron

(1988). Actually, this is the approach taken by Park (1990), who uses a Newey-West

(1987) type variance estimator. However, due to the behavior of the Newey-West

estimator under the unit root hypothesis this approach will lead to substantial loss of

power. Park's test statistic G, which is comparable with G* = [rnβn]2/σ2, with σ2

replaced by a Newey-West type estimator, is such that under the stationarity

hypothesis, G converges weakly to a chi-square distribution, whereas under the unit

root hypothesis, n-1+δ G 6 4 for some δ > 0, where δ depends on the truncation lag

of the Newey-West estimator. Cf. Park (1990, Lemma 3.3(b)). On the other hand,

the statistic G* is such that under the stationarity hypothesis, G* Y χ2(1), whereas

under the unit root hypothesis, n-2G* Y 0.1χ2(1). Cf. Lemma 3 for the latter result.

Thus, loosely speaking, the Newey-West estimator eats up a substantial amount of

asymptotic power. Therefore we propose a different approach. The idea is to

construct a statistic with equal rates of convergence under H0 and H1 that also

depends on σ in a similar way as above. Then we combine rnβn with this statistic

such that σ cancels out. There are many ways to construct such a statistic. In the first

instance, the following statistic will be used in the construction of the tests:

where [½n] denotes the integer part of ½n. Later on, in Section 8, we consider some

8

(3.2) S '1 ½ 3

½ 3 1.

(3.3)r

n$

n

>n

n' F

12[½Wn(1) & m

1

0

Wn(x)dx]

Wn(1) & 2((½n)/[½n])W

n(½)

% Op(1/n)

Y F12[½W(1) & m

1

0

W(x)dx]

W(1) & 2((½n)/[½n])W(½)

' Fx

1

x2

,

(3.4) ' '1/10 5/(32 3)

5/(32 3) 1/12.

(3.5) >n' (1/n)j

n

t'1j

t

j'1

uj& (1/[½n])j

[½n]

t'1j

t

j'1

uj

' (F n)(1/n)jn

t'1

Wn(t/n) & 2(1/n)j

[½n]

t'1

Wn(t/n) % o

p( n) ,

alternatives to (3.1).

The following two lemmas describe the limiting joint distribution of βn and

ξn under H0 and H1.

LEMMA 4: Under Assumption 1 and H0, (rnβn, ξn%n)T Y Nn[0,σ2Ω], where

PROOF: From (2.9) and (3.1) and it easily follows

say. Now (x1,x2)T is normally distributed with zero mean vector. We have already

seen in Lemma 2 that E[x21] = 1. It is easy to verify that E[x2

2] = 1 and E [x1x2] = ½%3.

LEMMA 5: Under Assumption 1 and H1 , (rnβn/n, ξn/%n)T Y N2(0,Γ), where

PROOF: Observe that

9

(3.6) >n/ n Y Fm

1

0

W(8)d8 & 2m½

0

W(8)d8.

(3.7) z1' 12m

1

0

(8&½)W(8)d8; z2' m

1

0

W(8)d8 & 2m½

0

W(8)d8 .

(3.8) ) '1/40 3/48

3/48 1/12.

(3.9) .1,n

' [1&exp(&( n(1& $"))4)](2n

% exp(&( n(1& $"))4)(2n

/n.

hence by (2.6b-c),

Together with (2.12) this result establishes that ((rn/n)βn , ξn/%n) | σ(z1,z2), where

We have already seen in (2.11) that E[z12] = 1/10. It is easy to verify that E[z2

2] =

1/12 and, E[z1z2] = 5/(32%3). Q.E.D.

It follows now straightforwardly from Lemmas 4 and 5:

LEMMA 6: Let Assumption 1 hold and denote γ1n = 2[rnβn - ½(%3)(%n)ξn]; γ2n =

(%n)ξn. Under H0 , (γ1n,γ2n)T Y N2[0,σ2I2], whereas under H1 , (γ1n,γ2n)

T/n Y

N2[0,σ2∆], where

Now if we use γ1n/γ2n as a test statistic the asymptotic null distribution is

standard Cauchy, and therefore does not depend on σ2. On the other hand, the rate

of convergence under H0 and H1 is the same so that the test will be inconsistent.

However, the following lemma provides a solution to this consistency problem.

LEMMA 7: Let Assumption 1 hold, let α^ be the OLS estimator of α in the regression

model yt = F + αyt-1 + ut and let

Under H0, (γ1n,ζ1,n)T Y N2[0,σ2I2], whereas under H1, (γ1n/n,ζ1,n)

T Y N2[0,σ2∆], where

∆ is defined in (3.8).

PROOF: It follows from Theorem 1 of Phillips and Perron (1988) that under H1,

10

(3.10) n( $"&1) Y

½[W(1)2&F2

u /F2] & W(1)m1

0

W(8)d8

m1

0

W(8)2d8 & [m1

0

W(8)d8]2

(3.11) plimn64

$" < 1.

(3.12) plimn64

, n(1& $") , ' 4 if H0

is true,

' 0 if H1

is true.

(3.13) plimn64

[1&exp(&( n(1& $"))4)] ' 1;

plimn64

exp(&( n(1& $"))4) ' 0,

(3.14) [1&exp(&( n(1& $"))4)] ' Op(n &2); plim

n64exp(&( n(1& $"))4) ' 1.

(3.15) S1,n

' (1n

/.1,n

,

whereas it is easy to verify that under H0,

Hence,

It follows now from (3.10)-(3.12) that under H0,

whereas under H1,

Combining (3.13) - (3.14) with Lemmas 6, the lemma under review easily follows.

Q.E.D.

REMARK: The power 4 in expression (3.9) is somewhat arbitrary. It follows from

the proof of Lemma 7 that any power larger than 2 will do.

Denoting

the theorem below follows straightforwardly from Lemma 7:

11

Table 1: Monte Carlo Simulation of the Test in Theorem 1

J 2 n

5 and 10 % rejection frequencies

D = l

5% 10%

D = 0.95

5% 10%

D = 0.9

5% 10%

D = 0.8

5% 10%

D = 0

5% 10%

0.95

0.5

100

300

92.4 96.5

96.5 98.2

86.8 94.5

96.1 98.0

83.4 90.9

96.2 98.0

73.2 85.6

87.7 93.3

3.2 6.3

4.5 8.7

0.95

0

100

300

90.8 95.4

98.0 99.2

86.6 93.4

96.3 98.9

85.6 92.6

95.6 97.4

83.8 92.0

95.6 97.9

15.3 26.1

8.4 15.4

0.9

0.5

100

300

90.4 95.1

96.3 98.1

85.7 92.3

95.4 97.1

80.8 89.4

93.3 96.3

60.7 76.1

56.0 70.2

3.0 6.5

6.1 10.4

0.9

0

100

300

89.7 94.8

97.3 98.8

86.2 94.2

95.9 97.8

85.7 93.5

96.1 98.1

84.0 91.9

92.1 96.2

5.8 12.8

5.8 10.6

0.8

0.5

100

300

84.3 91.9

96.9 98.6

74.0 84.2

88.8 94.6

62.3 76.6

55.3 71.2

27.1 42.6

7.8 16.6

4.7 9.4

5.3 9.2

0.8

0

100

300

87.8 94.0

97.3 98.5

84.4 92.6

95.8 97.6

82.3 91.1

92.2 96.5

70.6 82.0

60.1 77.2

4.9 9.8

4.5 9.2

0

0.5

100

300

4.0 8.0

12.7 19.3

3.4 6.2

2.5 5.8

3.5 7.2

5.7 10.0

4.3 8.6

4.7 10.4

4.9 9.6

4.9 10.0

0

0

100

300

49.6 60.5

77.4 84.6

17.0 28.2

9.2 16.8

6.2 11.8

5.3 9.8

4.4 6.8

5.0 9.8

6.1 10.8

5.2 9.5

THEOREM 1: Let Assumption 1 hold. Under H0 , S1,n | Cauchy (0,1), whereas

under H1 , S1,n/n | 0.25%3 + (1.5%5)g , where g is distributed as Cauchy(0,1).

Note that under H1 the limiting distribution of S1,n/n is continuous, hence plimn64*S1,n*

= 4. Thus the two-sided Cauchy test involved is consistent against the unit root

hypothesis. Moreover, in comparing Park's (1990) test with our test we should

compare Park's test statistic G with S21,n, i.e., under the unit root hypothesis, n-1+δG 6

4 for some δ > 0, whereas n-2S21,n Y [0.25%3+ (1.5%5)g]2. This means that,

asymptotically, our test is more powerful than

Park's test, due to the fact that we have avoided the use of the Newey-West (1988)

estimator of σ2. The same applies to the other three Cauchy tests introduced below.

12

(3.16) yt' Dy

t&1%u

t; (1&JL)u

t' (1&2L)e

t; e

t- NID(0,1) ,

Table 2: Mean of "^ and w^ 1 for D=1, J = 0 (1000 replications)

2 = 0 2 = ½

n "^ w^ 1 "^ w^ 1

100

300

0.9478

0.9818

0.1904

0.0600

0.8011

0.9168

0.7945

0.6399

$w1' 1 & exp(&( n(1& $"))4)

3.2. Monte Carlo Results

In order to see how the test works in practice we have conducted the test on 1000

replications of data generating processes of the type:

where ρ, τ ε 0, .8, .9, .95 and θ ε 0,½. The sample sizes we use correspond to

25 years of quarterly and monthly data: n ε 100, 300. The results are given in

Table 1.

It is clear from Table 1 that the test is hardly able to distinguish a near-unit

root from a genuine one when the error process ut has a long memory namely in the

cases ρ, τ = 0.95, 0.9 and 0.8. On the other hand, these cases are close to I(2)

processes, so that we may expect a substantial size distortion. The cases ρ=0, τ =

0.95, 0.9 and 0.8 are more modest near unit root cases, and the actual size of the test

in these cases is quite close to the theoretical size, apart from the case ρ=0, τ=0.95,

θ=0, where the size is much too high in comparison with the case θ=0.5. A reason

for the latter may be that the ARMA process yt=0.95yt-1+et-0.5et-1 is less a near unit

root process than the AR process yt=0.95yt-1+et .

A more puzzling result is that the small sample power of the test in the case

τ = 0 differs dramatically between θ=0 and θ=0.5. This is possibly due to

`misbehavior' of α^ , by which the weight

remains too large in small samples. This can be observed from Table 2, where we

have calculated the mean of α^ and for the case ρ=1, τ = 0 on the basis of 1000w1

^

replications. The reason for this phenomenon is probably the size of σ2u /σ

2 in the

limiting distribution (3.10) of n(α^ -1), where σ2u = E[u2

t]. See also Perron (1991) and

Nabeya and Perron (1991). It is not hard to verify that the numerator of the

13

(3.17) F2

u / F2 '1&J

1%J1 %

22(1&J)

(1&2)2.

Table 3: F2u/F

2

J 2 F2u/F

2 J 2 F2u/F

2

0.95 0

0.95 ½

0.9 0

0.9 ½

0.0256

0.0308

0.0526

0.0737

0.8 0

0.8 ½

0 0

0 ½

0.1111

0.2000

1.0000

5.0000

(4.1) S2,n

' (1n

/.2,n

.

(4.2) .2,n

' [1&exp(&( n(1& $"))4)](2n

% [exp(&( n(1& $"))4)](1n

/n .

asymptotic distribution (3.10) has mathematical expectation -½σ2u /σ

2, and that for ut

= τut-1 + et - θet-1,

The values of σ2u /σ

2 for the cases under review are given in Table 3. We see that in

the case τ=0, θ=½ the negative bias -½σ2u /σ

2 = -2.5 is relatively large. This may

explain the low value of α^ under H1 in the case τ=0, θ=½.

4. CAUCHY TEST II

The argument in Section 3.1 also suggests the following alternative Cauchy test. Let

[Cf. (3.15)], where now

[Cf.(3.9)]. Then it is easy to prove along the lines of Sections 2 and 3 that the

following theorem holds.

THEOREM 2: Let Assumption 1 hold. Under H0 , S2,n Y Cauchy(0,1), whereas

under H1 , plimn64S2,n/n = 1.

14

Table 4: Monte Carlo Simulation of the Test in Theorem 2

J

2

n

5 and 10 % rejection frequencies

D = 1

5% 10%

D =0.95

5% 10%

D =0.9

5% 10%

D =0.8

5% 10%

D = 0

5%10%

0.95

0.5

100

300

97.6 99.1

99.9 99.9

97.6 98.9

99.8 99.8

96.2 97.5

99.2 99.8

83.6 89.2

91.2 94.9

3.9 6.3

3.1 6.6

0.95

0

100

300

98.3 99.2

99.4 99.9

97.4 98.7

99.7 99.8

97.1 98.2

99.8 99.9

96.3 97.9

99.0 99.4

16.1 26.1

9.6 9.0

0.9

0.5

100

300

96.4 98.3

99.5 99.6

95.8 97.9

99.4 99.8

91.0 95.0

94.6 97.0

70.6 79.8

61.0 73.0

3.3 6.2

4.9 9.9

0.9

0

100

300

98.2 99.0

99.8 99.8

98.0 98.7

99.6 99.8

95.3 97.1

99.6 99.7

93.8 96.5

94.7 97.6

4.9 10.6

3.7 7.9

0.8

0.5

100

300

92.7 95.9

99.8 99.8

86.5 91.3

91.1 95.9

71.1 80.5

57.0 71.2

26.6 40.6

8.1 15.0

4.1 8.0

5.1 9.7

0.8

0

100

300

96.7 98.4

99.8 99.9

95.7 97.4

99.2 99.6

92.7 95.5

95.2 97.8

79.1 87.0

64.0 77.6

4.7 8.8

5.6 11.7

0

0.5

100

300

5.6 9.0

13.7 21.3

2.3 4.7

4.8 9.1

4.6 7.9

5.4 10.8

3.7 8.1

4.6 9.1

5.1 10.7

6.4 11.5

0

0

100

300

55.5 66.0

80.3 86.7

18.4 30.4

10.1 18.2

6.5 13.2

4.5 8.5

5.4 10.5

5.2 9.5

5.9 11.5

6.1 11.9

(5.1) S1,n

' $w(1n

/(2n

' $wDn

,

The result under H1 implies that the test may be conducted one-sided. However, for

reasons of comparison we have in the numerical applications below conducted the

test two-sided.

We have conducted a similar Monte Carlo analysis for this test as for the one

in Theorem 1. The results are given in Table 4. Comparing Tables 1 and 4 we see

that the small sample properties of the two tests are about the same.

5. CAUCHY TEST III

A conceptual disadvantage of the tests in Theorems 1 and 2 is that the power of the

test is mainly determined by the behavior of the Dickey-Fuller-type test statistic α^ .

In particular, the test statistic (3.15) can be written as

say, where

15

(5.2) $w ' 1 / [1 % (n &1&1)exp(&( n(1& $"))4) ] .

(5.3) DnY g if H

0is true; D

nY .25 3 % (1.5 5)g if H

1is true ,

(5.4) plimn64

$w ' 1 if H0

is true; plimn64

$w/n ' 1 if H1

is true .

(5.5) nn' (

2n/ n,

(5.6) $F2

u ' (1/(n&1))jn

t'2

(yt&y

t&1)2

(5.7) .3,n

'n

nn

1 % n2

n / $F2

u

.

(5.8) .3,n

'n

nn

1 % n2

n / $F2

u

'n

n/ n

1/n % (n2

n /n) / $F2

u

YF

2

u

Fv2

(5.9) ((1n

/n,.3,n

) Y (Fv1, F

2

u / (Fv2)),

It follows from the proof of Theorem 1 that

where g is Cauchy (0,1) distributed, whereas

Thus, the actual power of the test is determined by w via α^ , and therefore the test

does not differ much from the Phillips-Perron (1988) test of H1 against H0. The main

difference is that we do not need to estimate σ2/σ2u, whereas in the latter case one

needs to modify n(α^ -1) in order to account for the unknown ratio σ2/σ2u.

We now propose a third Cauchy test that does not employ the estimator α^ . Let

and

The reason for using σ^ 2u in (5.7) is to make ζ3,n invariant for linear transformations of

yt . Since under H0, nn%n Y N(0,σ2) [Cf. Lemma 4] and σ^ 2u Y E(u2-u1)

2, it follows that

ζ3,n - nn%n Y 0. Moreover, it follows from the proof of Lemma 7 that under H1,

and

where (v1,v2)T is distributed as N2[0,∆], with ∆ defined by (3.8). These results suggest

16

(5.10) S3,n

Y (1n

/.3,n

.

Table 5: Monte Carlo Simulation of the Test in Theorem 3

J 2 n

5 and 10 % rejection frequencies

D = 1

5% 10%

D=0.95

5% 10%

D =0.9

5% 10%

D =0.8

5% 10%

D = 0

5% 10%

0.95

0.5

100

300

79.1 89.2

92.1 96.2

50.7 70.8

58.8 77.3

32.1 56.7

28.1 57.1

10.7 30.2

7.0 26.4

3.7 6.2

5.2 9.5

0.95

0

100

300

82.0 89.3

93.5 96.4

60.2 76.7

65.4 81.4

44.1 67.1

47.5 71.2

25.6 49.0

19.6 46.9

3.4 7.5

2.9 6.8

0.9

0.5

100

300

66.4 78.7

83.8 91.2

28.9 55.1

27.6 55.5

12.8 36.3

7.0 30.4

4.0 14.1

4.5 11.0

3.2 8.6

3.9 8.0

0.9

0

100

300

70.6 85.8

87.7 94.5

43.8 68.8

46.6 70.2

25.7 53.4

18.1 44.3

8.0 30.4

6.4 22.2

3.1 7.1

3.2 8.9

0.8

0.5

100

300

45.8 63.6

65.0 79.4

9.6 28.7

7.6 30.0

4.0 14.2

4.3 10.4

3.1 8.3

4.6 9.6

3.8 8.4

4.2 9.3

0.8

0

100

300

58.5 76.0

76.9 86.0

26.0 51.5

18.1 45.9

10.1 32.3

5.9 21.2

2.9 13.1

5.3 11.4

3.6 7.6

4.6 9.7

0

0.5

100

300

2.5 5.5

5.7 17.1

2.1 4.9

3.5 8.5

3.5 7.8

4.9 9.1

4.6 10.2

3.9 8.2

6.0 12.8

6.4 12.5

0

0

100

300

11.0 27.2

31.5 50.6

3.1 6.9

4.3 8.6

3.9 7.2

4.2 9.4

5.0 9.1

4.5 9.7

5.8 11.2

5.6 10.8

the following alternative (two-sided) Cauchy test:

THEOREM 3: Let Assumption 1 hold. Under H0 , S3,n Y Cauchy(0,1), whereas

under H1 , S3,n/n Y (σ2/σ2u)v1v2 , where (v1, v2)

T is distributed as N2(0,∆) with ∆ defined

by (3.8).

The results of a similar Monte Carlo simulation as in Tables 1 and 4 are

presented in Table 5. We see that in general the size distortion in the near unit root

cases is less (although still substantial) than for the tests in Theorems 1 and 2, but

that also the power against the unit root case ρ=1 is lower, in particular in the case

17

(6.1) $F2 ' (1/(n&1))jn

t'2

(yt& $"y

t&1)2

% 2(1/(n&1))jmn

j'1

wj,n j

n

t'j%2

(yt& $"y

t&1)(y

t&j& $"y

t&1&j).

(6.2) S4,n

' (1n

/.4,n

,

(6.3) .4,n

'n

nn

1 % n2

n / $F2,

τ=0, θ=0.

6. CAUCHY TEST IV

The result in Theorem 3 for the unit root hypothesis indicates that the power of the

test in Theorem 3 depends on the ratio σ2/σ2u. We can make the power independent

of this ratio by replacing σ^ 2u in (5.7) and (5.8) by a Newey-West (1987) type estimator

of σ2. Thus, replace (5.6) by

where wj,n = 1 - j/(mn+1), and mn 6 4 at rate o(n¼). In the numerical applications

below we have chosen mn = 1+[5n1/5]. Denoting

where now

it follows from Theorem 3 and the consistency of σ^ 2 that:

THEOREM 4: Let Assumption 1 hold. Under H0 , S4,n Y Cauchy(0,1), whereas

under H1 , S4,n/n Y v1v2 , where (v1, v2)T is distributed as N2(0,∆) with ∆ defined by

(3.8).

18

Table 6: Monte Carlo Simulation of the Test in Theorem 4

J 2

n

5 and 10 % rejection frequencies

D = 1

5% 10%

D =0.95

5% 10%

D =0.9

5% 10%

D = 0.8

5% 10%

D = 0

5% 10%

0.95

0.5

100

300

45.6 61.5

57.8 75.5

11.8 31.0

8.7 28.6

5.3 20.9

4.3 17.1

3.7 11.5

3.9 11.1

2.7 5.9

4.0 7.2

0.95

0

100

300

43.2 60.0

58.4 75.7

13.3 32.0

8.4 32.5

6.8 20.0

4.4 18.4

4.0 12.4

4.0 12.3

2.9 7.8

3.7 7.4

0.9

0.5

100

300

32.2 50.4

52.2 70.2

6.0 19.2

4.7 15.8

3.9 13.3

5.6 12.4

2.6 9.2

3.9 8.5

3.3 7.5

4.7 9.7

0.9

0

100

300

32.1 50.4

51.1 67.7

6.1 17.7

5.1 19.7

4.0 13.2

3.5 9.8

3.4 9.1

4.9 9.6

2.8 5.8

5.2 10.1

0.8

0.5

100

300

18.3 40.6

39.5 57.9

3.3 11.5

5.7 12.3

3.3 8.2

5.7 11.2

4.0 8.3

6.2 11.7

4.9 9.6

5.4 9.5

0.8

0

100

300

23.2 41.3

41.3 59.7

3.8 14.1

4.8 13.1

2.9 8.4

5.3 11.1

3.8 9.0

5.6 10.7

3.3 6.8

4.7 9.3

0

0.5

100

300

3.6 11.8

21.0 38.2

2.9 6.3

3.6 8.0

3.0 6.6

4.8 10.7

5.5 9.4

3.6 8.1

4.1 8.3

5.3 11.1

0

0

100

300

13.4 28.9

35.0 53.4

4.3 9.0

4.5 9.7

3.0 6.9

4.6 8.7

3.2 6.5

5.2 9.8

6.3 12.2

5.2 9.8

In Table 6 we present the Monte Carlo results involved. Comparing Tables

5 and 6 we see that indeed the power in the case τ = 0, θ = ½, has been improved

(although less than expected on the basis of Table 3), but at the expense of lower

power in the cases with τ > 0. Thus, in general the test in Theorem 4 does not

perform better than the one in Theorem 3.

Summarizing, the tests in Theorems 1 and 2 perform the best when the

process ut has a short memory, and the test in Theorem 3 performs the best in the

near unit root cases.

7. THE NEAR-STATIONARITY CASE

Since the null hypothesis of our tests is stationarity, it makes sense to compare their

performance for the following near-stationarity cases:

19

(7.1) yt' y

t&1% e

t& 2e

t&1, e

t- NID(0,1) , 2 ' 0.8, 0.9, 0.95, 0.99.

Table 7: The near-stationarity case

5 and 10 % rejection frequencies; D=1, J=0

n

2 = 0.8

5% 10%

2 = 0.9

5% 10%

2 = 0.95

5% 10%

2 = 0.99

5% 10%

Test

100

300

2.0 4.5

1.8 3.6

2.2 4.0

2.0 4.3

3.5 6.9

1.9 4.4

4.9 10.3

3.1 6.5

Th.1

100

300

1.9 3.4

2.0 4.7

2.0 4.3

2.0 2.8

3.1 4.9

2.4 4.5

6.1 12.0

4.1 7.4

Th.2

100

300

1.7 3.4

1.4 3.3

2.4 5.0

0.8 2.9

3.2 6.6

1.5 3.8

4.8 9.3

4.5 9.1

Th.3

100

300

1.1 2.7

1.4 3.0

2.1 4.3

2.2 3.4

2.9 6.3

1.9 4.3

4.1 8.9

4.2 8.8

Th.4

(8.1) H+

0 : yt' µ % $t % u

t,

Note that for θ = 1 this process is stationary, as then the unit root cancels out. Thus

for θ close to one we may consider (7.1) as a near-stationary process.

The Monte Carlo results for our four tests on the basis of 1000

replications are presented in Table 7. We see from Table 7 that our tests have only

trivial power against these near-stationarity cases.

8. TREND STATIONARITY VERSUS THE UNIT ROOT HYPOTHESIS

The stationarity hypothesis is often too restrictive for macro economic time series.

Time series like real GNP usually seem to have a deterministic linear trend rather

than being stationary. Such a linear deterministic trend may occur because the series

is trend stationary, or because it is unit root process with drift. In this section we

shall therefore extend the test in Theorem 3 to testing the null hypothesis of trend

stationarity,

20

(8.2) H+

1 : yt& y

t&1' µ % u

t,

(8.4) s2

1 (<) ' (2<%1)(<%1)(<%2)

<(<&1)

2

.

against the unit root hypothesis with (or without) drift,

where in both cases the process (ut) satisfies Assumption 1.

Again we base our test on the OLS estimates of the parameters in an auxiliary

trend regession, but because our null is now trend stationarity we can no longer use

the OLS estimate of the parameter of time t. Therefore we propose to run the

following auxiliary regression:

(8.3) yt = F + βt + δtν + et ,

where ν>0, ν … 1. A natural choice of ν would be ν = 2, but any positive ν unequal

to 1 will work too.

Now let δn(ν) be the OLS estimator of δ in the auxiliary regression (8.3). In

the Appendix we shall prove:

LEMMA 8: Let Assumption 1 hold. Under H+0 , n

ν+½δn(ν) Y N(0,σ2s21(ν)), where

Under H+1 , n

ν-½δn(ν) Y N(0,σ2s22(ν)), where s2

2(ν) can be calculated as indicated in the

Appendix.

In order to construct a Cauchy test, we need a second statistic having the same

rate of convergence under H0 and H1, with a limiting normal distribution with zero

mean and variance proportional to σ2 under H0. Note that the statistic (3.1) is no

longer suitable, due to the fact that the null hypothesis is now trend stationarity. A

possible solution to this problem is to replace the yt in (3.1) by the OLS residuals ^gt

of the auxiliary regression

(8.5) yt = F + βt + gt .

21

(8.6) >n(J) '

1

[nJ]j[nJ]

t'1

$gt.

gt' j

t

j'1

uj.

Q1(<,J) '

s2

1 (<) s13

(<,J)

s13

(<,J) s2

3 (J),

s2

3 (J) ' J&1 & 3(J&1)2 & 1, s13

(<,J) ' s2

1 (<)2(<&1)&3J<

(<%1)(<%2)%

J<

<%1.

Thus, let for τ ε (0,1),

Note that under H+0, gt = ut , whereas under H1 ,

The joint asymptotic distribution of δn(ν) and ξn(τ) is given in Lemma 9:

LEMMA 9: Let Assumption 1 hold. Under H+0, (nν+½δ(ν), n½ξn(τ))

T Y

N2(0,σ2Q1(ν,τ)), where

with

Under H+1, (nν-½δn(ν),n

-½ζn(τ))T Y N2(0,σ2Q2(ν,τ)), where Q2 can be calculated as

indicated in the Appendix.

Now let the matrix L(ν,τ) be such that

(8.7) L(ν,τ)L(ν,τ)T = Q1(ν,τ),

and define

(8.8) (γ1n(ν,τ), γ2n(ν,τ))T = L(ν,τ)-1(nν+½δn(ν), n

½ζn(τ))T.

Note that s13(ν,τ) = 0 for ν = 2, τ = ½, so that then

22

(8.9) L(2,½) '

s2

1 (2) 0

0 s2

3 (½)

'6 ½ 0

0 ½.

(8.10) $F2

)$g'

1

n&1j

n

t'2

($gt&$g

t&1)2 ,

P0,n

(t) ' 1; P1,n

(t) 't

n&

n%1

2n,

P2,n

(t) ' (t

n&a

2,n)P

1,n(t) & b

2,nP

0,n(t) ,

P3,n

(t) ' (t

n&a

3,n)P

2,n(t) & b

3,nP

1,n(t) ,

Similarly to Lemma 6, it follows from Lemma 9:

LEMMA 10: Let Assumption 1 hold. Under H+0 , (γ1n(ν,τ),γ2n(ν,τ))

T Y N2(0,σ2I2),

whereas under H+1 , (γ1n(ν,τ),γ2n(ν,τ))

T/n Y σ(v1(ν,τ),v2(ν,τ))T, where (v1(ν,τ), v2(ν,τ))

T

is bivariate normally distributed.

With this result at hand we can now easily modify the tests in Theorems 1-4

to tests of the trend stationarity hypothesis against the unit root hypothesis. Here we

shall only focus on the modification of the test in Theorem 3. Thus define the test

statistic S5,n(ν,τ) similarly to S3,n [cf. (5.10)], where ^σ2u in (5.7) is replaced by

with the ^gt's the OLS residuals of the auxiliary regression (8.5). Then

THEOREM 5: Let Assumption 1 hold. Under H+0, S5,n(ν,τ) Y Cauchy(0,1), whereas

under H+1, S5,n(ν,τ)/n Y (σ2/σ2

u)v1(ν,τ)v2(ν,τ).

An alternative to the above approach is to base the test on the OLS estimators

of two superfluous regressions tν and tη, say, and orthogonalise these OLS estimators

in order to obtain the statistics γ1n and γ2n. If we choose for ν and η natural numbers,

say 2 and 3, respectively, the orthogonalization and estimation can be done jointly

by using orthogonal polynomials. Thus let for t=1,..,n,

23

ai,n

'

jn

t'1

(t/n)P2

i&1,n(t)

jn

t'1

P2

i&1,n(t)

, bi,n

'

jn

t'1

(t/n)[Pi&2,n

(t)Pi&1,n

(t)]

jn

t'1

P2

i&2,n(t)

(i ' 2,3) .

limn64

P0,n

([nx]) ' q0(x) ' 1; lim

n64P

1,n([nx]) ' q

1(x) ' x&

1

2;

limn64

P2,n

([nx]) ' q2(x) ' (x&

1

2)2&

1

12;

limn64

P3,n

([nx]) ' q3(x) ' (x&

1

2)3&

3

20(x&

1

2) .

P(

i,n(t) 'P

i,n(t)

jn

j'1

P2

i,n(j)

.

limn64

nP(

i,n([nx]) 'q

i(x)

m1

0

q2

i (z)dz

' q(

i (x), i ' 0,1,2,3 .

(8.11) yt' j

3

i'0

biP

(

i,n(t) % et,

with

These polynomials are exactly orthogonal. It can be shown that limn64a2,n = limn64a3,n

= ½, limn64b2,n = 1/12, limn64b3,n = 1/15. For x ε [0,1] we then have:

Next, define the corresponding system of orthonormal polynomials by:

Note that

Now run the orthonormal regression

and let ^bi be the OLS estimator of bi. Stacking the ^bi and bi in vectors ^b and b,

respectively, we have:

24

1

n( $b&b) Y F m

1

0

W(r)dr , m1

0

q(

1 (r)W(r)dr , m1

0

q(

2 (r)W(r)dr , m1

0

q(

3 (r)W(r)dr

T

.

(8.12) (1n

' $b2, (

2n' $b

3,

(8.13) $F2

) $e '1

n&1j

n

t'2

( $et& $e

t&1)2 ,

S6,n

/n Y (F2 /F2

u )m1

0

q(

2 (x)W(x)dxm1

0

q(

3 (x)W(x)dx .

LEMMA 11: Let Assumption 1 hold. Under H+0 ,

^b - b Y N4(0,σ2I4), whereas under

H+1 ,

Since under the null hypothesis H+0, bi = 0 for i = 2, 3, we may now choose

and again we can construct a Cauchy test S6,n similarly to (5.10), where now σ2u in

(5.7) is replaced by

with the ^et's the OLS residuals of (8.11).

THEOREM 6: Let Assumption 1 hold. Under H+0 , S6,n Y Cauchy(0,1), whereas

under H+1 ,

We have conducted a similar Monte Carlo analysis as for the test in Theorem

3. The small sample properties of the tests in Theorems 5 and 6 appear to be very

similar to those of the test in Theorem 3. The results are not reported here, but are

available from the second author on request.

9. SUMMARY AND DISCUSSION

In this paper we have proposed four Cauchy tests of the stationarity hypothesis H0:

yt = µ + ut, against the unit root hypothesis H1: yt = yt-1 + ut, where ut is a zero mean

α-mixing process. These tests have better asymptotic power properties than Park's

(1991) stationarity test. All four tests are based on the OLS estimator βn of the

25

(nn

n)/(1%8n2

n) Y N(0,F2) under H0,

Y 1/(8z) under H1

parameter β in the auxiliary trend model yt = µ + βt + ut . In particular, we exploit the

fact that the speed of convergence in distribution of βn under H0 and H1 differs by a

factor equal to the sample size n, i.e., rnβn | N(0,σ2) under H0, rnβn/n | N(0,σ2/10)

under H1, where rn is given in Lemma 1 and σ2 is the long run variance of ut. The

main novelty of the approach in this paper is the way we have avoided the use of the

Newey-West (1987) estimator of σ2 in constructing the tests, namely by constructing

a second statistic ζn which has the same rate of convergence in distribution under H0

and H1 and is, under H0, asymptotically N(0,σ2) distributed. The statistics rnβn and

ζn can then be combined into a test statistic Sn which under H0 takes, asymptotically,

the form of a ratio of two independent N(0,σ2) distributed variates. Hence, σ then

cancels out like in the case of t- and F- statistics, and the asymptotic distribution of

Sn is thus standard Cauchy. Moreover, if H1 is true then Sn/n converges weakly to a

(possibly degenerated) distribution.

The tests in Theorems 1 and 2 use the OLS estimator αn of the coefficient α

in the auxiliary model yt = µ + αyt-1 + ut to equalize the speed of convergence of the

statistic ζn under H0 and H1. Although the finite sample performance of these tests

in comparison with the Monte Carlo results in Phillips-Perron (1988) and Schwert

(1989) is not bad at all, the fact that they use the OLS estimate αn is a conceptual

disadvantage because the performance of the tests involved is almost completely

determined by the behavior of αn under H0 and H1.

The tests in Theorems 3 and 4 equalize the speed of convergence of the

statistic ζn under H0 and H1 in a completely different way. The trick is actually very

simple: if nn%n Y N(0,σ2) under H0 and nn/%n Y z under H1 then for any λ > 0,

Cf. (5.7) and (5.8). Thus, in this way we can construct a statistic ζn that has the same

speed of convergence under H0 and H1 and is asymptotically N(0,σ2) distributed

under H0.

The main difference between the tests in Theorems 3 and 4 is that under H1,

Sn/n Y (σ2/σ2u)v1v2 for the test in Theorem 3 and Sn/n Y v1v2 for the test in Theorem

4, where v1 and v2 are normal variates and σ2u is the variance of ut . Since the test in

Theorem 3 is the simplest one of the two and σ2/σ2u will likely be larger than one for

economic time series, the test in Theorem 3 is our favorite. Moreover, the Monte

26

Carlo results involved show that the tests in Theorems 3 and 4 are less sensitive to

size distortion caused by a near unit root than the tests in Theorems 1 and 2, and that

in general the power of the test in Theorem 3 is better than the power of the test in

Theorem 4.

In Section 8 we generalize the test in Theorem 3 to testing the trend

stationarity hypothesis against the unit root hypothesis with drift, in two directions.

The first extension employs instead of βn the OLS estimator δn of the parameter δ in

the auxiliary regression yt = F + βt + δtν + et, where ν … 1, together with a statistic

similar to ζn. The second extension is based on the OLS estimates of the regression

of yt on second and third order orthonormal polynomials in t.

Finally, one may wonder what the practical significance is of testing (trend)

stationarity against the unit root hypothesis, rather than testing the unit root

hypothesis against (trend) stationarity. First, there are situations where the

stationarity hypothesis is a more natural null hypothesis that the unit root hypothesis,

for example in testing for cointegration. The same applies to trend stationarity. But

even if the unit root hypothesis is the natural null, it makes sense to conduct (trend)

stationarity tests as well. As is shown by Phillips and Perron (1988) and Schwert

(1989), the power of unit root tests is often quite low and the size distortion can be

substantial. Therefore, conducting (trend) stationarity tests after having conducted

unit root tests provides a double check of the decision to reject or accept the unit root

hypothesis. Of course, the same applies the other way around. If the natural null

hypothesis is (trend) stationarity, one should also conduct unit root tests as a double

check, because the Monte Carlo results in this paper show that also our Cauchy tests

may suffer from size distortion and low power.

ACKNOWLEDGMENT

The very helpful comments of Lourens Broersma, Esfandiar Maasoumi and two

referees are gratefully acknowledged. A previous version of this paper, entitled

"Testing Stationarity Against the Unit Root Hypothesis", was presented by the first

author at the 6th World Congress of the Econometric Society, Barcelona, 1990. A

substantial part of this research was done while the first author was affiliated with the

Free University, Amsterdam.

27

Dn' n

1 0

0 n, x

1t' (1,t), x

2t'(t <), X

T

i ' [xT

i1 ,....,xT

in ], i'1,2 ,

u ' (u1,...,u

n)T, g ' (g

1,...,g

n)T, $g ' ($g

1,...,$g

n)T, e ' (e

1,...,e

n)T,

u ( ' (u1,u

1%u

2,....,j

n

t'1

ut)T ,

D&1

n (XT

1 X1)D

&1

n 61 1/2

1/2 1/3' M, say,

(n <%½Dn)&1X

T

1 X26

(<%1)&1

(<%2)&1' N, say,

n &(<%1/2)XT

2 u Y Fm1

0

r <dW(r) ' FS1, say,

n &(<%3/2)(XT

2 u () Y Fm1

0

r <W(r)dr ' FS(

1 , say,

D&1

n XT

1 u Y F m1

0

dW(r), m1

0

rdW(r)

T

' FS2, say,

(nDn)&1X

T

1 u ( Y F m1

0

W(r)dr, m1

0

rW(r)dr

T

' FS(

2 , say,

1T

[nJ]X1(n ½D

n)&1 6

JJ2/2

' P(J), say.

TECHNICAL APPENDIX

Denote

and let 1s be the n dimensional vector with 1's as the first s entries and zeros as the

rest. Using simple algebra and Lemma 1, it can be easily shown that

28

(a) E m1

0

f(r)dW(r)mJ

0

g(r)dW(r) ' mJ

0

f(r)g(r)dr ,

(b) E m1

0

f(r)W(r)drmJ

0

g(r)W(r)dr ' mJ

0

f(r) mr

0

g(s)sds % rmJ

r

g(s)ds dr

% m1

J

f(r) mJ

0

sg(s)ds dr .

E[dW(r)dW(s)] '0 if r … s

dr if r ' s; E [W(r)W(s)] ' min(r,s) .

(A1) E [S1S

2] ' N , E [S

2

1 ] ' (2<%1)&1 , E [S2S

T

2 ] ' M .

n <%1/2*n(<) '

d1,n

(<)

d2,n

(<), say ,

d1,n

(<) ' n &(<%1/2) XT

2 e & (n <%1/2Dn)&1(X

T

2 X1)[D

&1

n (XT

1 X1)D

&1

n ]&1(D&1

n X1e) ,

Before presenting the proofs, we first give the following lemma which is

useful for calculating the covariance matrix of functions of a Brownian motion.

LEMMA A: For any deterministic continuous functions f and g on [0,1], and any

τ ε [0,1],

Proof: Since the Brownian motion W has independent increments, it follows that

Using this result, Lemma A easily follows. Q.E.D.

As an application of Lemma A, we can calculate the following covariances:

Proof of Lemma 8:

Observe that

with

29

d2,n

(<) ' n &(2<%1)jn

t'1

t 2<

& (XT

2 X1n &(<%1/2)D

&1

n )[D&1

n (XT

1 X1)D

&1

n ]&1(n &(<%1/2)D&1

n XT

1 X2) .

limn64

d2,n

(<) ' (2<%1)&1 & N TM &1N .

d1,n

(<) Y F(S1&N TM &1S

2)

n (<%1/2)*n(<) Y F[(2<%1)&1 & N TM &1N]&1[S

1&N TM &1S

2] - N(0,F2s

2

1 ) ,

s2

1 ' [(2<%1)&1 & N TM &1N]&1 ' (2<%1)(<%1)(<%2)

<(<&1)

2

.

d1,n

(<) /n Y F(S(

1 & N TM &1S(

2 ) ,

n <&½*n(<) Y F [(2<%1)&1&N TM &1N]&1[S

(

1 &N TM &1S(

2 ]

' F(2<%1)(<%1)(<%2)

<(<&1)

2

m1

0

r <W(r)dr%2(<&1)

(<%1)(<%2)m1

0

W(r)dr

&6<

(<%1)(<%2)m1

0

rW(r)dr - N(0,F2s2

2 ) ,

Moreover it easily follows that

Under H0, e = u, so that

hence

where, using (A1),

Under H1 , e = u*, so that

hence

where s22 can be calculated using Lemma A. Q.E.D.

30

1T

s$g ' 1

T

sg & (1T

s X1D

&1

n )[D&1

n (XT

1 X1)D

&1

n ]&1(D&1

n XT

1 g)

n ½>n(J) '

n

[nJ]n &½1

T

[nJ]$g ,

n &½1T

[nJ]g ' n &½j[nJ]

t'1

utY FW(J) ,

n &½1T

[nJ]$g Y F(W(J) & P(J)TM &1S

2),

n ½>n(J) Y FJ&1[W(J) & P(J)TM &1S

2]

' FJ&1[mJ

0

dW(r)&(4J&3J2)m1

0

dW(r)%(6J&6J2)m1

0

rdW(r)]

- N(0,F2s2

3 ) ,

s2

3 ' J&2[J&P(J)TM &1P(J)] ' J&1&3(J&1)2&1.

Q1'

s2

1 s13

s13

s2

3

Proof of Lemma 9:

Observe that

and

Under H0, g = u, so that

hence

Consequently,

where

It then trivially follows that the joint asymptotic distribution is bivariate normal, with

covariance matrix,

31

s13

' J&1s2

1 [J<%1(<%1)&1&N TM &1P(J)] ' s2

1

2(<&1)&3J<

(<%1)(<%2)%

J<

<%1.

n &3/21T

s, ' n &3/2js

t'1j

t

j'1

ujY Fm

J

0

W(r)dr ,

n &½>n(J) Y FJ&1[m

J

0

W(r)dr & P(J)TM &1S(

2 ]

' FJ&1[mJ

0

W(r)dr&(4J&3J2)m1

0

W(r)dr%(6J&6J2)m1

0

rW(r)dr] ,

where

Under H1, g = u*, so that then

which is normally distributed. The covariance matrix Q2 can now be calculated

using Lemma A(b). Q.E.D.

The proofs of Lemmas 10 and 11 are easy and therefore left to the reader.

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