Testing against a Change from Short to Long Memory

Testing against a Change from

Short to Long Memory ^∗

Uwe Hassler

and Jan Scheithauer

Goethe-University Frankfurt

This version: December 19, 2007

Abstract

This paper studies some well-known tests for the null hypothesis of short memory against a change to nonstationarity. We show that they are also applicable for a change from I(0) to a fractional order of integration I(d) with d > 0 (long memory) in that the tests are consistent. The rates of divergence of the test statistics are derived

as T^2d. Experimentally, we explore the power properties of the tests

against fractional alternatives under various specifications and exam- ine for which settings the tests have satisfactory power. Further, we study the limiting behaviour under the assumption of constant order of integration d.

Keywords: Change in persistence; Unknown change point; Fractional integration

JEL classification: C12; C22

∗The authors are grateful for clarifying comments from the participants of the 18th EC² conference (Faro, December 2007) where an earlier version was presented.

†Corresponding author. Tel.: +49 69 798 25330; fax: +49 69 798 25662. E-mail address: [email protected]

‡Faculty of Business and Economics, D-60054 Frankfurt, Germany.

1 Introduction

In a recent paper Hassler and Nautz (2007) argue that the spread between the European overnight interest rate and the key policy rate of the ECB has changed from a short memory to a long memory series. They do so by fixing the potential break point a priori as the date where the ECB changed its operational framework. There are not many situations where applied workers are willing to assume a known break date. Hence, the question how to test against a change from short to long memory with unknown change point suggests itself.

Kim, Belaire-Franch and Amador (2002) and Busetti and Taylor [BT]

(2004) discussed tests for the null hypothesis of stationarity (or more precisely integration of order zero, I(0)) against alternatives of a change from I(0) to I(1). Their procedures are variants of the ratio tests by Kim (2000) who relates the sum of squared partial sum processes of subsamples, which is in spirit of the locally most powerful test proposed by Nyblom and M¨akel¨ainen (1983) and discussed by Kwiatkowski, Phillips, Schmidt and Shin (1992) [KPSS]. Inspired by the finding of Lee and Schmidt (1996) that the KPSS test has power against alternatives of fractional integration we investigate the behaviour of the ratio type tests for I(0) under fractional alternatives. BT (2004) discuss a variety of further change test. In particular, they derive and advocate the locally best invariant (LBI) test against a change from white noise to a random walk.

In this paper we add three aspects to this literature. First, we derive the rates of divergence of the mentioned test statistics in the presence of breaks from I(0) to I(d), thus establishing consistency. Second, the limiting distributions are derived under constant d > 0. Third, the size and power properties are investigated in an extensive Monte Carlo experiment.

The next section briefly presents the hypotheses considered and the as- sumptions. Section 3 contains the asymptotic results for Kim’s ratio tests, while in the fourth section the limiting results for the LBI tests by BT (2000) are derived. Section 5 is dedicated to the experimental evidence. Concluding

remarks are collected in the final section. Technical derivations are relegated to the Appendix.

2 Hypotheses and assumptions

The null hypothesis to be tested is that the univariate process y_tis integrated of order zero with mean µ (t = 1, 2, . . . , T ) :

H₀ : y_t= µ + z_t, z_t∼ I(0). (1) For simplicity we restrict the analysis to the simplest case of constant deter- ministics, although the extension to polynomials in time would be possible.

We focus on the alternative hypothesis that a change from I(0) to I(d) occurs at [λT ], where [·] denotes the integer part:

H₁^(d) : y_t=

½ µ₀ + z_0,t, t = 1, . . . , [λT ]

µ₁ + z_1,t, t = [λT ] + 1, . . . , T . (2) As a special case of (2) we will consider the case of constant persistence, where the process is fractionally integrated with d possibly different from zero. This is embedded in (2) for λ = 0. To become more precise on the stochastic properties allowed for, we assume that z_i,t, i = 0, 1, satisfy usual invariance principles.

Assumption 1 (I(0)) Let z0,t, t = 1, . . . , T, be an I(0) process with zero mean satisfying (as T → ∞)

T^−0.5

[sT ]

X

t=1

z0,t ⇒ B0(s), s ∈ [0, 1], (3)

T⁻¹ XT

t=1

z²_0,t →^p σ₀²

where B₀ is a Brownian motion, and “⇒” and “→” stand for weak conver-^p gence and convergence in probability, respectively.

Assumption 2 (I(d)) Let z1,t, t = 1, . . . , T, be an I(d) process with zero mean satisfying (as T → ∞)

T^−0.5−d

[sT ]

X

t=1

z_1,t ⇒ B_d(s), |d| < 0.5 , (4)

T⁻¹ XT

t=1

z_1,t² →^p σ₁²

where s ∈ [0, 1] and B_d is a fractional Brownian motion (of “type I” in the terminology of Marinucci and Robinson, 1999).

Technical assumptions yielding a functional central limit theorem (3) are well known from the literature. For a corresponding weak convergence result (4) see e.g. Chan and Terrin (1995) or Davidson and de Jong (2000), compare also Marinucci and Robinson (1999, eq. (2.16)). The fractional invariance principle in (4) could be extended to cover all possible values d > −0.5 fol- lowing Marinucci and Robinson (2000) who work with a “type II” fractional Brownian motion as limiting process. The limiting process Bd(s) from (4) may be factorized into a positive constant ω and a stochastic integral W_d(s), which is a functional of a standard Brownian motion and depending on d only (W_d(s) being defined e.g. in Marinucci and Robinson, 1999, eq. (2.1)):

Bd(s) = ωWd(s).

The final assumption refers to the break fraction λ in (2), which is typ- ically not known in practice. Hence, we work with set of potential break fractions τ that are assumed to lie in a certain range.

Assumption 3 Let τ ∈ T = [τ₁, τ₂] ⊂ (0, 1), and define T^∗ = [τ2T ] − [τ1T ] + 1 .

For applications we choose τ1 = 0.2 and τ2 = 0.8.

3 Kim’s ratio tests

The tests rely on splitting the sample and cumulating the respectively de- meaned observations:

S0,t(τ ) = Xt

j=1

(yj − y₀) , y₀ = 1 [τ T ]

[τ T ]

X

t=1

yt, t = 1, . . . [τ T ]

S_1,t(τ ) =

Xt j=[τ T ]+1

(y_j− y₁) , y₁ = 1 T − [τ T ]

XT t=[τ T ]+1

y_t, t = [τ T ] + 1, . . . T.

Here, S0,t(τ ) is computed for the first subsample, t = 1, . . . , [τ T ] , while S_1,t(τ ) is obtained from the second one, t = [τ T ] + 1, . . . , T. Following Kim (2000), Kim et al. (2002) and BT (2004) suggest to evaluate the ratio KT (τ ) for given τ :

K_T (τ ) =

[(1 − τ ) T ]⁻² P^T

t=[τ T ]+1

S_1,t² (τ )

[τ T ]⁻²

[τ T ]P

t=1

S_0,t² (τ )

. (5)

With the break fraction τ being unknown, one considers the maximum statis- tic

H₁(K_T) = max

τ ∈T K_T (τ ) ,

the mean score statistic and the mean-exponential statistic, Z

τ ∈T

K_T (τ ) dτ and log Z

τ ∈T

exp(K_T (τ ))dτ.

The null hypothesis will be rejected for too large values. For actual applica- tions the integrals of those functionals will be replaced by averages (see BT, 2004, footnote 2),

H₂(K_T) = 1 T^∗

[τX2T ] t=[τ1T ]

K_T µt

T

¶ ,

H3(KT) = log



 1 T^∗

[τX2T ] t=[τ1T ]

exp µ

KT

µt T

¶¶





with T^∗ from Assumption 3. To perform the tests we apply the corrections made by Hassler and Scheithauer (2007) to the critical values in Kim et al.

(2002). Notice that BT (2004) discuss a slightly different mean-exponential statistic,

He₃(K_T) = log



 1 T^∗

[τX2T ] t=[τ1T ]

exp µ1

2K_T µt

T

¶¶



.

We now characterize the three test statistics under the alternative in (2).

Our result corresponds to Kim (2000, Theorem 3.4), although we do not have to employ sequential asymptotics and become precise on the rate of divergence. To that end we first establish that T^−2dK_T(τ ) has a well-defined limit distribution if τ ≤ λ, and second, we characterize the limit of K_T(τ ) if the true break occurs earlier than the assumed break point. Consequently, H_i(K_T), i = 1, 2, 3, behave as stated in the following proposition as long as λ ∈ T . Our results parallel BT (2004, Theorem 2.2) who treat the case d = 1, but for 0 < d < 0.5 our derivation becomes more complicated. Details of proof are relegated to the Appendix.

Proposition 1 Let Assumptions 1 through 3 and the alternative (2) hold true with λ ∈ T and µ₀ = µ₁. Then for d > 0, H_i(K_T) diverge with T^2d, i = 1, 2, 3.

Proof: See Appendix.

Remark 1 We did not allow for breaks in the mean (µ₀ = µ₁) in order to isolate the effect of changes in d and to simplify the proof. The power effect of level shifts (µ₀ 6= µ₁) under d = 0 has been discussed by Belaire-Franch (2005).

Remark 2 The interpretation of those findings is that a change from I(0) to long memory (d > 0) results in diverging test statistics rejecting H₀ with probability one asymptotically. Not surprisingly, the power will grow with T and d. For small d the rate T^2d is very slow. The effect of the break fraction λ is not so clear. For experimental evidence, see the next section.

Next we extend the limiting distribution of the test statistics provided by Kim et al. (2002) and BT (2004) by allowing for the hypothesis of constant d, i.e. λ = 0 in (2). Under our assumptions the following result is easy to establish (similar to proof of Proposition 1):

K_T (τ ) →^d

1 (1−τ )²

R1 τ

£W_d(s) − W_d(τ ) − _1−τ^s−τ (W_d(1) − W_d(τ ))¤₂ ds

1 τ²

Rτ 0

£W_d(s) −_τ^sW_d(τ )¤₂ ds

≡ K(τ ; d) , (6)

where “→” stands for convergence in distribution. Consequently the limiting^d distributions of H_i(K_T) depend on d only, asymptotically. The continuous mapping theorem [CMT] provides the following result.

Proposition 2 Let Assumptions 1 through 3 and (2) with λ = 0 hold true and K(τ ; d) from (6). Then

H1(KT) → sup^d

τ ∈T

K (τ ; d) , H₂(K_T) →^d 1

τ2− τ1

Z

τ ∈T

K (τ ; d) dτ,

H3(KT) → log^d



 1 τ₂− τ₁

Z

τ ∈T

exp(K (τ ; d))dτ



,

as T → ∞.

Proof: Omitted.

Remark 3 For d = 0 this corresponds to the results by Kim et al. (2002) and BT (2004). The effect of d on the quantiles of the distributions was studied through Monte Carlo experiments (not reported here). It turned out that they crucially hinge on d. Experimentally, we observed that with growing d the rejection rate of the null (1) increases. Consequently, the rejection of (1) could be indicative of a break in persistence as well as of a constant d > 0.

Remark 4 From Proposition 2 we learn that Kim’s ratio tests for the null hypothesis (1) are not consistent against I(d) without break. A similar point has been stressed by Harvey, Leybourne and Taylor (2006) when studying the tests under I(1) processes without break.

4 LBI type tests by BT (2004)

For known break fraction λ, BT (2004) derive the LBI test statistic against a change from white noise to a random walk as

S1(λ) = bσ⁻² (T − [λT ])²

XT t=[λT ]+1

S_t² (7)

with

S_t= XT

j=t

(y_j− y) = − Xt−1

j=1

(y_j − y) where

y = 1 T

XT t=1

y_t, σb² = 1 T

XT t=1

(y_t− y)². In practice, λ is unknown so that BT (2004) suggest

H₁(S₁) = max

τ1≤τ ≤τ2

S₁(τ ) ,

H₂(S₁) = 1 T^∗

[τX2T ] t=[τ1T ]

S₁ µ t

T

¶

He3(S1) = log



 1 T^∗

[τX2T ] t=[τ1T ]

exp µ1

2S1

µ t T

¶¶



.

Now we can prove a result corresponding to BT (2004, Theorem 2.4). We consider the alternative of a break in persistence as in (2) or constant long memory (λ = 0) at the same time. It is straightforward to show that bσ² converges to a well-defined limit as long as d < 0.5. Further, S_t² = O_p¡

T^2d+1¢ , such that S₁(τ ) = O_p¡

T^2d¢

. Hence, H_i(S₁) , i = 1, 2, and eH₃(S₁) diverge

as given in the following proposition. Details of proof are provided in the Appendix.

Proposition 3 Let Assumptions 1 through 3 and the alternative (2) hold true and µ0 = µ1. Then for d > 0, Hi(S1), i = 1, 2, and eH3(S1) diverge with T^2d. The result continues to hold if λ = 0 in (2).

Proof: See Appendix.

Remark 5 Note that under constant persistence, λ = 0 in (2), the LBI type tests by BT (2004) are consistent, which contrasts the behaviour of Kim’s ratio tests characterized in Proposition 2.

Remark 6 To allow for short-run dependence, the variance estimator in (7) has to be replaced by a spectral estimator of the long-run variance. BT (2004, eq. (6.5)) propose Bartlett weights w_B(j),

b

σ_B²(m) = bσ²+ 2 T

Xm j=1

w_B(j) XT t=j+1

(y_t− y) (y_t−j− y) ,

where m → ∞ but m/T → 0. For this choice Lee and Schmidt (1996, Theorem 3) prove under Assumption 2: bσ_B² (m) = O_p¡

m^2d¢

. Hence, usage of bσ_B² (m) will reduce the rate of divergence given in Proposition 3 to

H_i(S₁) = O_p¡

(T /m)^2d¢ .

In practice, the so-called quadratic spectral kernel by Andrews (1991) may be superior to simple Bartlett weights. Experimental evidence is provided in the next section.

5 Monte Carlo Evidence

This section provides simulation results on size and power of both Kim’s ratio tests and BT’s LBI tests. In order to confront both tests with the same type of simulated process, we employ the simulation setup of Kim (2000, Section 4)throughout, setting the constant to zero without loss of generality.

For both tests, we consider the maximum statistic (H1), the mean score statistic (H₂) and the mean exponential statistic (H₃ and eH₃, respectively).

In case of the LBI test, we apply the Bartlett window as the first of our two alternative weighting schemes for the auto-covariances:

w_B(j) =

½ 1 − j/(m + 1) j = 1, . . . , m

0 otherwise ,

where the bandwidth m is a truncation parameter. We choose m = m_B(4) = [4(T /100)^1/3],

where the choice of the bandwidth parallels Kwiatkowski et al. (1992), only that we replace their power 1/4 by 1/3, which corresponds to the optimal rate derived by Andrews (1991).

In addition to the Bartlett kernel, we employ the quadratic spectral (QS) window,

wQS(j) = 25m² 12π²j²

µsin(6πj/5m)

6πj/5m − cos(6πj/5m)

¶

, j = 1, . . . , T − 1, such that:

b

σ_QS² (m) = bσ²+ 2 T

XT −1 j=1

w_QS(j) XT t=j+1

(y_t− y) (y_t−j − y) .

Here we choose

m = mQS(4) = [4(T /100)^1/5]

where the power 1/5 corresponds to the optimal rate derived by Andrews (1991).

5.1 Empirical size

In Table 1, we present empirical size results for white noise and an AR(1) process with a moderate coefficient ρ = 0.75. In case of the LBI test, we provide results for the variance estimator bσ² from (7), as well as for the

Table1:EmpiricalsizeofLBI-typeandratiotest:rejectionfrequencies. ρ=0 LBItestwithbσ2 LBItestwithbσ^{2 B}LBItestwithbσ^{2 QS}Kim’sratiotest αT=H1(S1)H2(S1) e H(S)H(S)H(S)³11121

e H(S)H(S)H(S)³11121

e HH(S)H)(K)H(K(K)³T3T2T11 0.0110.0110.0120.0110.0110.0110.010.0100.0100.0110.0100.0100.009500 0.0540.0540.0540.0510.0530.0530.050.0540.0530.0540.0500.0500.054500 0.0990.1020.1040.1060.1070.1070.1080.1060.1040.1050.1060.1075000.10 0.0090.0120.0080.0120.0110.0130.0100.01010000.010.0120.0140.0120.011 0.0510.0590.0510.0510.0550.0460.05110000.0530.050.0450.0540.0520.059 0.1130.1150.1000.1120.0950.1150.1140.1130.1130.1160.1160.11610000.10 75ρ0.= 2^{2 B2 QS}bσwithwithratioKim’sLBIbσtesttesttestbσwithtestLBILBI (S(SH)H=Tα)1121

e H(S)H(S)H(S)³11121

e H(S)H(S)H(S)³11121

e HH(S)H)(K)H(K(K)³T3T2T11 0.0630.0180.0230.0180.0610.0580.0610.0610.4970.4870.5295000.010.058 0.1560.1800.0810.0710.0810.1600.1550.6900.1610.1810.6820.7275000.05 0.2580.2640.1330.1370.1390.2730.2610.2600.8070.8070.8215000.100.277 0.0530.0150.0200.0140.0460.0440.0430.0440.0490.5220.5050.56410000.01 0.1470.0590.0710.0560.1510.1600.1510.1460.7220.7150.75710000.050.155 0.2300.1090.1210.1020.2360.2480.2350.2270.8300.8310.85110000.100.242 2000αsizenominalreplications,cess(8),inasproAR(1).

long run variance estimators with Bartlett window (bσ_B²) and QS window (bσ_QS² ). For white noise the experimental size is very close to the nominal one throughout. For the AR(1) case a different picture arises. While the size distortion is still acceptable for Kim’s test, the LBI test is seriously oversized even for the two long run variance estimators. All results are mainly irrespective of the particular test statistic H_i(·) utilized.

5.2 Power

In order to investigate the power properties of Kim’s ratio test, we generate y_t before the change point ([λT ]) under the alternative hypothesis as

y_t= ρy_t−1+ ε_t t = 1, . . . , [λT ], (8) with break fraction λ = 0.5, ρ ∈ {0, 0.75} and innovations ε_t ∼ iid N(0, 0.01).

For the fractionally integrated part of the process after the change point, y_t, t = [λT ] + 1, . . . , T , we consider two different types of processes. We refer to the first type of process as “immediate change” to long memory:

yt= Xt−1 j=0

djεt−j, t = [λT ] + 1, . . . , T, (9) where d_j = ^j−1+d_j d_j−1, d₀ = 1. After the change point, the process jumps immediately to long memory, as all innovations ε₁, . . . , ε_t (including those before the break point [λT ]) enter observation y_t, t = [λT ], . . . , T . Thus, even the first observation after the change point is computed as the finite sample analogue to fractionally integrated noise, comprising the current and all available past innovations of the simulated process.

The second type of process given in (10) is labelled “smooth transition”

to long memory:

y_t= ρy_{[λT ]}+

t−1−[λT ]X

j=0

d_jε_t−j, t = [λT ] + 1, . . . , T. (10)

Table 2: Rejection frequencies of LBI-type test b

σ² H₁(S₁) (max. statistic)

d = T = 250 T = 500 T = 1000 T = 2000

0.2 0.790 0.878 0.922 0.963

0.3 0.867 0.924 0.957 0.983

0.4 0.892 0.960 0.986 0.996

0.5 0.952 0.980 0.994 1.000

0.6 0.976 0.996 0.999 1.000

0.7 0.991 1.000 1.000 1.000

0.8 0.997 0.999 1.000 1.000

0.9 1.000 1.000 1.000 1.000

b

σ²_B H₁(S₁) (max. statistic)

d = T = 250 T = 500 T = 1000 T = 2000

0.2 0.708 0.818 0.877 0.926

0.3 0.733 0.822 0.899 0.928

0.4 0.743 0.847 0.912 0.952

0.5 0.793 0.894 0.941 0.971

0.6 0.846 0.925 0.962 0.985

0.7 0.880 0.949 0.978 0.996

0.8 0.928 0.971 0.986 0.998

0.9 0.950 0.983 0.991 0.999

b

σ²_QS H₁(S₁) (max. statistic)

d = T = 250 T = 500 T = 1000 T = 2000

0.2 0.714 0.820 0.879 0.928

0.3 0.752 0.834 0.892 0.936

0.4 0.761 0.858 0.913 0.960

0.5 0.823 0.898 0.940 0.973

0.6 0.851 0.928 0.973 0.983

0.7 0.895 0.957 0.986 0.992

0.8 0.934 0.974 0.989 1.000

0.9 0.956 0.984 0.997 1.000

white noise, break fraction λ = 0.5, “smooth tran- sition to long memory”, 2000 replications. Nominal size α = 0.05, critical values from Busetti and Taylor

Only innovations after the change point (ε_{[λT ]+1}, . . . , εT) are included to sim- ulate the fractionally integrated part of the process so that long memory only slowly evolves after the break. For d = 1, both (9) and (10) boil down to a random walk after the change point [λT ]: y_t= y_t−1+ε_t, t = [λT ]+2, . . . , T .¹ Clearly, in cases of “immediate change” more power is to be expected. The

“smooth transition”, however, seems to be more realistic, and here we focus on this scenario.

We consider ρ = 0 in (8), and a “smooth transition” from white noise to fractional noise of order d. For the maximum statistics H₁(·) we report in Tables 2 and 3 the rejection frequencies for a break fraction λ = 0.5. First, we observe considerable power for d = 0.2 and T = 250 already. The power is growing with d and T as expected for Propositions 1 and 3. Second, the LBI-type tests are typically more powerful than Kim’s test. However, the long-run variance estimation (bσ²_Bor bσ_QS² ) reduces power, see Remark 6 above.

Moreover, we wish to comment on some results not documented here in detail (they are available upon request). First, rejection frequencies may be lower, in case the break fraction λ is different from 0.5. Second, for ρ = 0.75 in (8) and a change to I(d), the power of Kim’s ratio tests is much smaller compared to the results from Table 3 under white noise. Third, of course, rejection frequencies for the LBI test are remarkably higher than for Kim’s ratio test in case that ρ = 0.75 – however, they come at the expense of a too serious size distortion, see Table 1.

6 Conclusions

In the present paper, we explore whether tests of change in persistence are applicable if the change is not from I(0) to I(1) but from I(0) to I(d), d < 1.

We show that both the ratio test by Kim (2000) in the form of Kim et al. (2002) and the locally best (LBI) invariant test by Busetti and Taylor (2004) are consistent, as they diverge with rate T^2d for all three variants of

1Note that this is virtually the same setup as in Kim (2000) and Kim et al. (2002), but different from the setup in Busetti and Taylor (2004).

test statistics (maximum statistic, mean score statistic and mean exponen- tial statistic). Further, we derive the limiting distributions of the tests for constant fractional order of integration d > 0.

In addition, we study size and power of both LBI and ratio tests for various setups. Simulations with autoregressive processes show that Kim’s test is moderately oversized. The size distortion of the LBI test may be a lot more severe than for Kim’s test. Hence, the ratio test seems to be more robust than the LBI test.

Three findings concerning power do not come as a surprise. First, rejec- tion frequencies rise in both sample size T and order of fractional integration d after the change point. Second, the LBI-type tests are more powerful than Kim’s tests under changes from white noise to fractional integration. Third, rejection frequencies also depend on the short-run dynamics of the I(0)-part before the change point, and on the type of transition from short to long memory that is considered.

Appendix

Proof of Proposition 1

The proof procedes in three steps. First, we analyse K_T(τ ) from (5) under τ ≤ λ, and establish that K_T (τ ) diverges at rate T^2d by characterising the limiting distribution of T^−2dK_T (τ ). Second, we show that K_T (τ ) alone converges to a nondegenerate random variable for τ > λ. Third, we draw conclusions about H_i(K_T), i = 1, 2, 3.

1) τ ≤ λ: For the denominator we consider

S_0,t(τ ) = Xt

j=1

(z_0,j − z₀), z₀ = 1 [τ T ]

[τ T ]

X

t=1

z_0,t.

For t = [sT ] ≤ [τ T ] it holds with (3) T^−0.5S_{0,[sT ]}(τ ) ⇒ B₀(s) − s

τB₀(τ ) ≡ V₀(s; τ ),

and hence by the CMT:

[τ T ]⁻²

[τ T ]

X

t=1

S_0,t² (τ )→^d 1 τ²

Zτ

0

V₀²(s; τ )ds. (11)

The numerator involves y₁ = µ + 1

T − [τ T ]





[λT ]

X

t=[τ T ]+1

z_0,t+ XT t=[λT ]+1

z_1,t



 .

Further, it requires the distinction of two cases.

(i) τ ≤ s ≤ λ: Here we obtain

S_{1,[s,T ]}(τ ) =

[sT ]

X

j=[τ T ]+1

(µ + z_0,j− y₁)

with (where d > 0 and by (4))

T^−0.5−dS1,[sT ](τ ) ⇒ −s − τ

1 − τ (Bd(1) − Bd(λ)) . (ii) s > λ: Here we obtain

S_{1,[s,T ]}(τ ) =

[λT ]

X

j=[τ T ]+1

(µ + z_0,j − y₁) +

[sT ]

X

j=[λT ]+1

(µ + z_1,j− y₁)

with (where d > 0)

T^−0.5−dS_{1,[sT ]}(τ ) ⇒ B_d(s) − B_d(λ) − s − τ

1 − τ (B_d(1) − B_d(λ)) . Defining V_d(s; τ ) for (i) and (ii) appropriately, the CMT yields for d > 0:

[(1 − τ )T ]⁻²T^−2d XT t=[τ T ]+1

S_1,t² (τ )→^d 1 (1 − τ )²

Z1

τ

V_d²(s; τ )ds. (12)

Collecting those results we observe

T^−2dKT(τ )→^d N(d; τ )

D(d; τ ), (13)

where N(d; τ ) and D(d; τ ) are defined in (12) and (11), respectively. Conse- quently, K_T(τ ) diverges at rate T^2d for τ ≤ λ.

2) τ > λ: In this case the numerator satisfies

S_1,t(τ ) = Xt j=[τ T ]+1

(z_1,j− z₁), z₁ = 1 T − [τ T ]

XT t=[τ T ]+1

z_1,t,

and for s ≥ τ it holds

T^−0.5−dS_{1,[sT ]}(τ ) ⇒ B_d(s) − B_d(τ ) − s − τ

1 − τ (B_d(1) − B_d(τ ))

≡ V_d(s; τ ).

For the denominator, two cases have to be distinguished.

(i) s ≤ λ: Here we obtain

S_{0,[sT ]}(τ ) =

[sT ]

X

j=1

(µ + z_0,j− y₀)

where

y₀ = 1 [τ T ]





[λT ]

X

t=1

(µ + z0,t) +

[τ T ]

X

t=[λT ]+1

(µ + z1,t)



 ,

such that for d > 0:

T^−0.5−dS_{0,[sT ]}(τ ) ⇒ −s

τ (B_d(τ ) − B_d(λ)) . (ii) s > λ : Here we obtain

S_{0,[sT ]}(τ ) =

[λT ]

X

j=1

(µ + z_0,j− y₀) +

[sT ]

X

j=[λT ]+1

(µ + z_1,j− y₀)

with

T^−0.5−dS_{0,[sT ]}(τ ) ⇒ B_d(s) − B_d(λ) − s

τ (B_d(τ ) − B_d(λ)) .

Defining eVd(s; τ ) in accordance with (i) and (ii) the CMT hence provides

K_T(τ ) =

[(1 − τ )T ]⁻²T^−2d P^T

t=[τ T ]+1

S_1,t² (τ )

[τ T ]⁻²T^{−2d[τ T ]}P

t=1

S_0,t² (τ )

→d 1 (1−τ )²

R1 τ

V_d²(s; τ )ds

1 τ²

Rτ 0

Ve_d²(s; τ )ds

. (14)

3) Finally, we consider the behaviour of the statistics H_i(K_T), i = 1, 2, 3, for T = [τ₁, τ₂]. With K_T(τ ) diverging on [τ₁, λ], the supremum statistic diverges at the same rate because of (13), (14) and the CMT:

T^−2dH₁(K_T) = max

τ1≤τ ≤τ2

T^−2dK_T(τ ) → sup^d

τ1≤τ ≤λ

N(d; τ ) D(d; τ ). Analogously we obtain for H₂(K_T):

T^−2dH₂(K_T) = 1 T^∗

[τX2T ] t=[τ1T ]

KT

¡_t

T

¢ T^2d = T

T^∗

[τX2T ] t=[τ1T ]

KT

¡_t

T

¢ T^2d

1 T

→d 1

τ₂− τ₁ Z _λ

τ1

N(d; τ ) D(d; τ )dτ . Similarly, it is holds for H₃(K_T):

T^−2dH₃(K_T) = T^−2dlog



 T T^∗

[τX2T ] t=[τ1T ]

exp µ

K_T µt

T

¶¶ 1 T





≈ T^−2dlog



 1 τ2− τ1

τ2

Z

τ1

exp

µK_T (τ ) T^2d T^2d

¶ dτ





≈ T^−2dlog

½ 1

τ₂− τ₁ Z _λ

τ1

exp

µN(d; τ ) D(d; τ )T^2d

¶ dτ

¾ .

Due to the mean-value theorem there exists τ^∗ with τ^∗ ∈ [τ1, λ] such that we may further conclude

T^−2dH₃(K_T) ≈ T^−2dlog

½λ − τ₁ τ₂− τ₁ exp

µN(d; τ^∗) D(d; τ^∗)T^2d

¶¾

= T^−2dlog

½λ − τ₁ τ₂− τ₁

¾

+ N(d; τ^∗) D(d; τ^∗), which establishes the convergence of T^−2dH₃(K_T).

Proof of Proposition 3

By Assumptions 1 and 2 we observe b

σ² → λσ^{p 2}₀+ (1 − λ) σ²₁. Further, consider

S_{[rT ]+1} = −

[rT ]

X

j=1

y_j+ [rT ] y

= −

[rT ]

X

j=1

z_i,j +[rT ] T





[λT ]

X

t=1

z_0,t+ XT t=[λT ]+1

z_1,t





where

T^−0.5−d[rT ] T

XT t=[λT ]+1

z_1,t ⇒ r (B_d(1) − B_d(λ)) .

Hence, S_t² = O_p¡

T^2d+1¢ and

XT t=[τ T ]+1

S_t² = O_p¡

T^2d+2¢ ,

which proves that S₁(τ ) diverges with T^2d. The proof is completed by discussing H_i(S₁), i = 1, 2, and eH₃(S₁) as for Proposition 1.

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Table 3: Rejection frequencies of Kim’s ratio test H₁(K_T) (max. statistic)

d = T = 250 T = 500 T = 1000 T = 2000

0.2 0.625 0.740 0.816 0.908

0.3 0.702 0.827 0.906 0.949

0.4 0.807 0.904 0.952 0.988

0.5 0.896 0.966 0.989 0.997

0.6 0.948 0.988 0.998 1.000

0.7 0.981 0.997 1.000 1.000

0.8 0.995 1.000 1.000 1.000

0.9 0.999 1.000 1.000 1.000

white noise, break fraction λ = 0.5, “smooth transi- tion” to long memory, 2000 replications. Nominal size:

α = 0.05. Critical values from Kim et al. (2002).