Testing against a Change from Short to Long Memory

Transcription

1 Testing against a Change from Short to Long Memory Uwe Hassler and Jan Scheithauer Goethe-University Frankfurt This version: January 2, 2008 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 examine 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: C2; C22 The authors are grateful for clarifying comments from the participants of the 8th EC 2 conference (Faro, December 2007) where an earlier version was presented. Corresponding author. Tel.: ; fax: address: hassler@wiwi.uni-frankfurt.de Faculty of Business and Economics, D Frankfurt, Germany.

2 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(). 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 Mäkeläinen (983) and discussed by Kwiatkowski, Phillips, Schmidt and Shin (992) [KPSS]. Inspired by the finding of Lee and Schmidt (996) 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 assumptions. 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 2

3 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 t is integrated of order zero with mean µ (t =, 2,..., T ) : H 0 : y t = µ + z t, z t I(0). () For simplicity we restrict the analysis to the simplest case of constant deterministics, 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: H (d) : y t = { µ0 + z 0,t, t =,..., [λt ] µ + z,t, t = [λt ] +,..., 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,, satisfy usual invariance principles. Assumption (I(0)) Let z 0,t, t =,..., T, be an I(0) process with zero mean satisfying (as T ) T 0.5 [st ] T z 0,t B 0 (s), s [0, ], (3) t= T t= z 2 0,t p σ 2 0 where B 0 is a Brownian motion, and and p stand for weak convergence and convergence in probability, respectively. 3

4 Assumption 2 (I(d)) Let z,t, t =,..., T, be an I(d) process with zero mean satisfying (as T ) T 0.5 d [st ] T z,t B d (s), d < 0.5, (4) t= T t= z 2,t p σ 2 where s [0, ] and B d is a fractional Brownian motion (of type I in the terminology of Marinucci and Robinson, 999). 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 (995) or Davidson and de Jong (2000), compare also Marinucci and Robinson (999, eq. (2.6)). The fractional invariance principle in (4) could be extended to cover all possible values d > 0.5 following Marinucci and Robinson (2000) who work with a type II fractional Brownian motion as limiting process. The limiting process B d (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, 999, eq. (2.)): B d (s) = ωw d (s). The final assumption refers to the break fraction λ in (2), which is typically 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 = [τ, τ 2 ] (0, ), and define T = [τ 2 T ] [τ T ] +. For applications we choose τ = 0.2 and τ 2 =

5 3 Kim s ratio tests The tests rely on splitting the sample and cumulating the respectively demeaned observations: t S 0,t (τ) = (y j y 0 ), y 0 = [τt ] y t, t =,... [τt ] [τt ] S,t (τ) = j= t j=[τt ]+ (y j y ), y = t= T [τt ] T t=[τt ]+ y t, t = [τt ] +,... T. Here, S 0,t (τ) is computed for the first subsample, t =,..., [τt ], while S,t (τ) is obtained from the second one, t = [τt ] +,..., T. Following Kim (2000), Kim et al. (2002) and BT (2004) suggest to evaluate the ratio K T (τ) for given τ: T [( τ) T ] 2 S,t 2 (τ) K T (τ) = [τt ] [τt ] 2 t=[τt ]+ t= S 2 0,t (τ). (5) With the break fraction τ being unknown, one considers the maximum statistic H (K T ) = max K T (τ), τ T the mean score statistic and the mean-exponential statistic, K T (τ) dτ and log exp(k T (τ))dτ. τ T The null hypothesis will be rejected for too large values. For actual applications the integrals of those functionals will be replaced by averages (see BT, 2004, footnote 2), H 2 (K T ) = T H 3 (K T ) = log [τ 2 T ] τ T t=[τ T ] [τ 2 T ] T t=[τ T ] 5 K T ( t T ), ( ( )) t exp K T T

6 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, H 3 (K T ) = log [τ 2 T ] T t=[τ T ] ( ( )) t exp 2 K T 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 2d K 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 =, 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 =, but for 0 < d < 0.5 our derivation becomes more complicated. Details of proof are relegated to the Appendix. Proposition Let Assumptions through 3 and the alternative (2) hold true with λ T and µ 0 = µ. Then for d > 0, H i (K T ) diverge with T 2d, i =, 2, 3. Proof: See Appendix. Remark We did not allow for breaks in the mean (µ 0 = µ ) in order to isolate the effect of changes in d and to simplify the proof. The power effect of level shifts (µ 0 µ ) 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 H 0 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. 6

7 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 ): K T (τ) d [ ( τ) 2 Wd (s) W d (τ) s τ (W τ d() W d (τ)) ] 2 ds τ τ [ Wd (s) s W τ d(τ) ] 2 ds τ 2 0 K(τ; d), (6) where d stands for convergence in distribution. Consequently the limiting distributions of H i (K T ) depend on d only, asymptotically. The continuous mapping theorem [CMT] provides the following result. Proposition 2 Let Assumptions through 3 and (2) with λ = 0 hold true and K(τ; d) from (6). Then H (K T ) H 2 (K T ) H 3 (K T ) d d d sup K (τ; d), τ T K (τ; d) dτ, τ 2 τ τ T log exp(k (τ; d))dτ τ 2 τ, τ T 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 () increases. Consequently, the rejection of () could be indicative of a break in persistence as well as of a constant d > 0. 7

8 Remark 4 From Proposition 2 we learn that Kim s ratio tests for the null hypothesis () 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() 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 S (λ) = σ 2 (T [λt ]) 2 T St 2 (7) t=[λt ]+ with where S t = y = T T t (y j y) = (y j y) j=t T y t, t= σ 2 = T j= T (y t y) 2. t= In practice, λ is unknown so that BT (2004) suggest H (S ) = max τ τ τ 2 S (τ), H 2 (S ) = T [τ 2 T ] t=[τ T ] H 3 (S ) = log S ( t T [τ 2 T ] T t=[τ T ] ) ( ( )) t exp 2 S 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 σ 2 converges to a well-defined limit as long as d < 0.5. Further, S 2 t = O p ( T 2d+ ), such that S (τ) = O p ( T 2d ). Hence, H i (S ), i =, 2, and H 3 (S ) diverge 8

9 as given in the following proposition. Details of proof are provided in the Appendix. Proposition 3 Let Assumptions through 3 and the alternative (2) hold true and µ 0 = µ. Then for d > 0, H i (S ), i =, 2, and H 3 (S ) 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), σ 2 B(m) = σ T m T w B (j) (y t y) (y t j y), j= t=j+ where m but m/t 0. For this choice Lee and Schmidt (996, Theorem 3) prove under Assumption 2: σ B 2 (m) = O ( ) p m 2d. Hence, usage of σ B 2 (m) will reduce the rate of divergence given in Proposition 3 to H i (S ) = O p ( (T/m) 2d ). In practice, the so-called quadratic spectral kernel by Andrews (99) 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. 9

10 For both tests, we consider the maximum statistic (H ), the mean score statistic (H 2 ) and the mean exponential statistic (H 3 and H 3, 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) = { j/(m + ) j =,..., m 0 otherwise, where the bandwidth m is a truncation parameter. We choose m = m B (4) = [4(T/00) /3 ], where the choice of the bandwidth parallels Kwiatkowski et al. (992), only that we replace their power /4 by /3, which corresponds to the optimal rate derived by Andrews (99). In addition to the Bartlett kernel, we employ the quadratic spectral (QS) window, w QS (j) = 25m2 2π 2 j 2 such that: ( sin(6πj/5m) 6πj/5m ) cos(6πj/5m), j =,..., T, σ 2 QS(m) = σ T T T w QS (j) (y t y) (y t j y). j= t=j+ Here we choose m = m QS (4) = [4(T/00) /5 ] where the power /5 corresponds to the optimal rate derived by Andrews (99). 5. Empirical size In Table, we present empirical size results for white noise and an AR() process with a moderate coefficient ρ = In case of the LBI test, we provide results for the variance estimator σ 2 from (7), as well as for the 0

11 Table : Empirical size of LBI-type and ratio test: rejection frequencies. ρ = 0 LBI test with σ 2 LBI test with σ B 2 LBI test with σ QS 2 Kim s ratio test α T = H(S) H2(S) H3(S) H(S) H2(S) H3(S) H(S) H2(S) H3(S) H(KT ) H2(KT ) H3(KT ) ρ = 0.75 LBI test with σ 2 LBI test with σ B 2 LBI test with σ QS 2 Kim s ratio test α T = H(S) H2(S) H3(S) H(S) H2(S) H3(S) H(S) H2(S) H3(S) H(KT ) H2(KT ) H3(KT ) AR() process as in (8), 2000 replications, nominal size α.

12 long run variance estimators with Bartlett window ( σ B 2 ) and QS window ( σ QS 2 ). For white noise the experimental size is very close to the nominal one throughout. For the AR() 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 + ε t t =,..., [λt ], (8) with break fraction λ = 0.5, ρ {0, 0.75} and innovations ε t iid N(0, 0.0). For the fractionally integrated part of the process after the change point we assumed a smooth transition to long memory : y t = ρy [λt ] + t [λt ] j=0 d j ε t j, t = [λt ] +,..., T, (9) where d j = j +d d j j, d 0 =. Only innovations after the change point (ε [λt ]+,..., ε T ) are included to simulate the fractionally integrated part of the process so that long memory only slowly evolves after the break. For d =, (9) boils down to a random walk after the change point [λt ]: y t = y t + ε t, t = [λt ] + 2,..., T. 2 We consider ρ = 0 in (8). For the maximum statistics H ( ) we report in Tables 2 and 3 the rejection frequencies for a break fraction λ = 0.5. We observe considerable power for d = 0.2 and T = 250 already. The power is growing with d and T as expected for Propositions and 3. Moreover, We also obtained (typically more powerful) results in the less realistic case of immediate changes not reported here. 2 Note 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). 2

13 Table 2: Rejection frequencies of LBI-type test σ 2 H (S ) (max. statistic) d = T = 250 T = 500 T = 000 T = σ B 2 H (S ) (max. statistic) d = T = 250 T = 500 T = 000 T = σ QS 2 H (S ) (max. statistic) d = T = 250 T = 500 T = 000 T = white noise, break fraction λ = 0.5, 2000 replications. Nominal size α = 0.05, critical values from Busetti and Taylor (2004). 3

14 the LBI-type tests are typically more powerful than Kim s test. However, the long-run variance estimation ( σ B 2 or σ2 QS ) reduces power, see Remark 6 above. Table 3: Rejection frequencies of Kim s ratio test H (K T ) (max. statistic) d = T = 250 T = 500 T = 000 T = white noise, break fraction λ = 0.5, 2000 replications. Nominal size: α = Critical values from Kim et al. (2002). 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 particular if the break fraction λ is smaller than 0.5, and T and d are small at the same time. 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. 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() but from I(0) to I(d), d <. 4

15 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 test statistics (maximum statistic, mean score statistic and mean exponential statistic). Further, we derive the limiting distributions of the tests for constant fractional order of integration d > 0. It turns out that only the LBI-type tests are consistent in this case. 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, rejection 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. Appendix Proof of Proposition 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 2d K 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 =, 2, 3. ) τ λ: For the denominator we consider S 0,t (τ) = t j= (z 0,j z 0 ), z 0 = [τt ] [τt ] z 0,t. t= 5

16 For t = [st ] [τt ] it holds with (3) and hence by the CMT: The numerator involves T 0.5 S 0,[sT ] (τ) B 0 (s) s τ B 0(τ) V 0 (s; τ), y = µ + [τt ] [τt ] 2 t= T [τt ] S0,t(τ) 2 d τ τ 2 [λt ] t=[τt ]+ 0 z 0,t + Further, it requires the distinction of two cases. (i) τ s λ: Here we obtain S,[s,T ] (τ) = with (where d > 0 and by (4)) (ii) s > λ: Here we obtain S,[s,T ] (τ) = with (where d > 0) [st ] j=[τt ]+ V 2 0 (s; τ)ds. (0) T t=[λt ]+ (µ + z 0,j y ) z,t T 0.5 d S,[sT ] (τ) s τ τ (B d() B d (λ)). [λt ] j=[τt ]+ (µ + z 0,j y ) + [st ] j=[λt ]+. (µ + z,j y ) T 0.5 d S,[sT ] (τ) B d (s) B d (λ) s τ τ (B d() B d (λ)). Defining V d (s; τ) for (i) and (ii) appropriately, the CMT yields for d > 0: [( τ)t ] 2 T 2d T t=[τt ]+ S,t(τ) 2 d ( τ) 2 6 τ V 2 d (s; τ)ds. ()

17 Collecting those results we observe T 2d K T (τ) d N(d; τ) D(d; τ), (2) where N(d; τ) and D(d; τ) are defined in () and (0), respectively. Consequently, K T (τ) diverges at rate T 2d for τ λ. 2) τ > λ: In this case the numerator satisfies S,t (τ) = and for s τ it holds t j=[τt ]+ (z,j z ), z = T [τt ] T t=[τt ]+ z,t, T 0.5 d S,[sT ] (τ) B d (s) B d (τ) s τ τ (B d() B d (τ)) V d (s; τ). For the denominator, two cases have to be distinguished. (i) s λ: Here we obtain where such that for d > 0: y 0 = [τt ] (ii) s > λ : Here we obtain [st ] S 0,[sT ] (τ) = (µ + z 0,j y 0 ) j= [λt ] (µ + z 0,t ) + t= [τt ] t=[λt ]+ (µ + z,t ), T 0.5 d S 0,[sT ] (τ) s τ (B d(τ) B d (λ)). [λt ] S 0,[sT ] (τ) = (µ + z 0,j y 0 ) + j= 7 [st ] j=[λt ]+ (µ + z,j y 0 )

18 with T 0.5 d S 0,[sT ] (τ) B d (s) B d (λ) s τ (B d(τ) B d (λ)). Defining Ṽd(s; τ) in accordance with (i) and (ii) the CMT hence provides K T (τ) = d [( τ)t ] 2 T 2d T ( τ) 2 τ τ 2 [τt ] [τt ] 2 T 2d τ 0 t=[τt ]+ t= Vd 2 (s; τ)ds S 2 0,t(τ) S 2,t(τ) Ṽd 2(s;. (3) 3) Finally, we consider the behaviour of the statistics H i (K T ), i =, 2, 3, for T = [τ, τ 2 ]. With K T (τ) diverging on [τ, λ], the supremum statistic diverges at the same rate because of (2), (3) and the CMT: T 2d H (K T ) = Analogously we obtain for H 2 (K T ): T 2d H 2 (K T ) = T d max T 2d K T (τ) d N(d; τ) sup τ τ τ 2 τ τ λ D(d; τ). [τ 2 T ] K T ( t T T 2d t=[τ T ] λ τ 2 τ Similarly, it is holds for H 3 (K T ): T 2d H 3 (K T ) = T 2d T log T T 2d log τ 2 τ { T 2d log τ 2 τ ) = T T N(d; τ) τ D(d; τ) dτ. [τ 2 T ] t=[τ T ] τ 2 8 τ λ τ [τ 2 T ] t=[τ T ] ( K t ) T T T 2d T ( ( )) t exp K T T T ( ) KT (τ) exp T 2d T 2d dτ ( ) } N(d; τ) exp D(d; τ) T 2d dτ.

19 Due to the mean-value theorem there exists τ with τ [τ, λ] such that we may further conclude { ( )} λ T 2d H 3 (K T ) T 2d τ N(d; τ ) log exp τ 2 τ D(d; τ ) T 2d { } λ = T 2d τ log + N(d; τ ) τ 2 τ D(d; τ ), which establishes the convergence of T 2d H 3 (K T ). Proof of Proposition 3 By Assumptions and 2 we observe Further, consider where σ 2 p λσ ( λ) σ 2. [rt ] S [rt ]+ = y j + [rt ] y j= [rt ] = 0.5 d [rt ] T T j= Hence, S 2 t = O p ( T 2d+ ) and z i,j + [rt ] T T t=[λt ]+ T t=[τt ]+ [λt ] z 0,t + t= T t=[λt ]+ z,t z,t r (B d () B d (λ)). S 2 t = O p ( T 2d+2 ), which proves that S (τ) diverges with T 2d. The proof is completed by discussing H i (S ), i =, 2, and H 3 (S ) as for Proposition. 9

20 References [] Andrews, D.W.K. (99), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation, Econometrica 59, [2] Belaire-Franch, J. (2005), A Proof of the Power of Kim s Test against Stationarity Processes with Structural Breaks, Econometric Theory 2, [3] Busetti, F., and A.M.R. Taylor (2004), Tests for Stationarity against a Change in Persistence, Journal of Econometrics 23, [4] Chan, N.H., and Terrin, N. (995), Inference for Unstable Long Memory Processes with Applications to Fractional Unit Root Autoregressions, The Annals of Statistics 23, [5] Davidson, J., and de Jong, R.M. (2000), The Functional Central Limit Theorem and Weak Convergence to Stochastic Integrals II: Fractionally Integrated Processes, Econometric Theory 6, [6] Harvey, D.I., S.J. Leybourne and A.M.R. Taylor (2006), Modified Tests for a Change in Persistence, Journal of Econometrics 34, [7] Hassler, U., and D. Nautz (2007), On the Persistence of the Eonia Spread, submitted. [8] Hassler, U., and J. Scheithauer (2007), Correct Usage of Percentiles of Kim s Tests against a Change in Persistence, manuscript Goethe- University Frankfurt. [9] Kim, J.-Y. (2000), Detection of Change in Persistence of a Linear Time Series, Journal of Econometrics 95, [0] Kim, J.-Y., J. Belaire-Franch and R.B. Amador (2002), Corrigendum to Detection of change in Persistence of a Linear Time Series, Journal of Econometrics 09,

21 [] Kwiatkowski, D., P.C.B. Phillips, P. Schmidt and Y. Shin (992), Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root, Journal of Econometrics 54, [2] Lee, D.L., and P. Schmidt (996), On the Power of the KPSS Test of Stationarity against Fractionally-integrated Alternatives, Journal of Econometrics 73, [3] Marinucci, D., and P.M. Robinson (999), Alternative Forms of Fractional Brownian Motion, Journal of Statistical Planning and Inference 80, -22. [4] Marinucci, D., and P.M. Robinson (2000), Weak Convergence of Multivariate Fractional Processes, Stochastic Processes and their Applications 86, [5] Nyblom, J., and T. Mäkeläinen (983), Comparison of Tests for the Presence of Random Walk Coefficients in a Simple Linear Model, Journal of the American Statistical Association 78,