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Bootstrap testing multiple changes in persistence for a heavy-tailed sequence
Computational Statistics and Data Analysis 56 (2012) 2303–2316
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Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda
Bootstrap testing multiple changes in persistence for a heavy-tailed sequence Zhanshou Chen a,b,∗ , Zi Jin c , Zheng Tian b,d , Peiyan Qi b a
Department of Mathematics, Qinghai Normal University, Xining, Qinghai 810008, PR China
b
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China
c
Department of Statistics, University of British Columbia, 333-6356 Agricultural Road, Vancouver, BC V67T 1Z2, Canada
d
National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing, 100101, PR China
article
info
Article history: Received 5 April 2010 Received in revised form 23 December 2011 Accepted 13 January 2012 Available online 25 January 2012 Keywords: Multiple changes in persistence Moving ratio test Bootstrap Heavy tailed
abstract This paper tests the null hypothesis of stationarity against the alternative of changes in persistence for sequences in the domain of attraction of a stable law. The proposed moving ratio test is valid for multiple changes in persistence while the previous residual based ratio tests are designed for processes displaying only a single change. We show that the new test is consistent whether the process changes from I (0) to I (1) or vice versa. And it is easy to identify the direction of detected change points. In particular, a bootstrap approximation method is proposed to determine the critical values for the null distribution of the test statistic containing unknown tail index. We also propose a two step approach to estimate the change points. Numerical evidence suggests that our test performs well in finite samples. In addition, we show that our test is still powerful for changes between short and long memory, and displays no tendency to spuriously over-reject I (0) null in favor of a persistence change if the process is actually I (1) throughout. Finally, we illustrate our test using the US inflation rate data and a set of high frequency stock closing price data. © 2012 Elsevier B.V. All rights reserved.
1. Introduction During the past two decades, there is a growing body of evidence showing that economic and financial time series display changes in persistence. This has been an issue of substantial empirical interest, especially concerning inflation rate series, short-term interest rates, government budget deficits and real output. Recently, a number of testing procedures have been suggested to distinguish such behavior. These include, inter alia, ratio tests (Kim, 2000; Kim et al., 2002; Leybourne and Kim, 2003; Harvey et al., 2006), LBI tests (Busetti and Taylor, 2004; Leybourne and Taylor, 2006), CUSUM of squaresbased tests (Leybourne et al., 2007a). As a general discussion about ratio tests and LBI tests, we refer the reader to Perron (2006). More recently, Carvaliere and Taylor (2008) have considered persistence change tests under the non-stationary volatility innovation case, Sibbertsen and Kruse (2009) have studied the long-range dependence innovation case, Hassler and Scheithauer (2009) have applied ratio tests and LBI tests to detect change points from short to long memory. Since the multiple change point test is also an important issue in change point analysis, Leybourne et al. (2007b) proposed a multiple change point detection procedure based on sequences of doubly-recursive implementations of the regression-based unit root statistic of Elliott et al. (1996). All of the works above are concentrated on the case where variances of the sequences are finite. Guillaume et al. (1997) and Rechev and Mittnik (2000) have argued that many types of data from economics and finance have the same character:
∗
Corresponding author at: Qinghai Normal University, Department of Mathematics, 810008 Xining, Qinghai, PR China. E-mail address: [email protected] (Z. Chen).
0167-9473/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2012.01.011
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a heavier tail than the normal variables. Motivated by this, Han and Tian (2007a,b) considered persistence change with innovations in the domain of attraction of κ -stable law. The tail index κ can reflect the heaviness of the data and the variance of κ -stable processes is infinite when κ < 2. However, these tests are not consistent against processes which display more than one change in persistence. Consequently, these tests cannot be used to consistently separate the data into its I (0) and I (1) regimes when multiple changes in persistence occur. Moreover, even for a single change model these procedures require different tests depending on the direction of change which the data display. In the light of these important drawbacks, this paper considers the testing and dating problems of multiple changes in persistence for sequences in the domain of attraction of κ -stable law. For more details about this heavy-tailed sequence we refer the reader to Horváth and Kokoszka (2003), Kokoszka and Wolf (2004) and Rosadi (2009), among many others. Our test procedure is based on a residual-based moving ratio statistic with a bandwidth parameter h ∈ (0, 1/2). The new statistic can be used to detect a single change point in persistence as well as multiple changes in persistence if there are large enough samples between two change points. We show that the new test will diverge to infinite if the sequence shows I (0) to I (1) change, and will converge to zero if the sequence shows I (1) to I (0) change. Therefore, this result can help us to identify the direction of the change points. Moreover, a two step approach is proposed to estimate the detected change points. In order to determine the critical values of the test statistic which containing unknown tail index κ , we propose a bootstrap approximation method. The bootstrap method can circumvent this nuisance parameter and obtain asymptotically correct critical values. Although our test procedure is based on I (0) null hypothesis, we also evaluate the size and power performances of the test under I (1) as well as fractional integration cases, and check its robustness if there are structural breaks in the trend function by simulation. The rest of the paper is organized as follows. Section 2 introduces the model and some necessary assumptions. Section 3 contains the testing and dating procedures and the requisite limiting results. In Section 4, we use Monte Carlo methods to test the finite sample performance of our test and illustrate it by the US inflation rate data and a set of high frequency stock data. We conclude the paper in Section 5. All technical proofs of the theoretical results are gathered in Section 6. 2. Model and assumptions Let y1 , y2 , . . . , yT be an observed time series that can be decomposed as yt = µt + εt , εt = ρ t εt − 1 + e t ,
(1) t = 1, 2, . . . , T ,
(2)
where µt = E (yt ) = δ dt is a deterministic component modeled as a linear combination of a vector of nonrandom regressors dt . Typical components of dt are a constant, a time trend or dummy variables. et is the stochastic part of the process with further discussion satisfy the following assumption. T
Assumption 2.1. The strictly stationary symmetrical sequences et are in the domain of attraction of a stable law with tail index κ ∈ (1, 2) and Eet = 0. Lemma 2.1. If Assumption 2.1 holds, then
−1
aT
[T τ ]
et , aT
−2
t =1
[T τ ]
e2t
d
−→ (U1 (τ ), U2 (τ )),
(3)
t =1
where aT = inf{x : P (|et | > x) ≤ T −1 }, d
and the random variable U1 (τ ) is κ -stable and U2 (τ ) is κ/2-stable Lévy process in [0, 1]. The notation → stands for convergence in distribution. Remark 2.1. This result is obtained by Kokoszka and Wolf (2004). The exact definition of the Lévy process (U1 (τ ), U2 (τ )) appearing in Lemma 2.1 is not needed in the following, but we recall that the quantities aT can be represented as aT = T 1/κ L(T ) for some slowly varying function L. Within the model (1), the process yt is I (0) if |ρt | < 1, the process yt is I (1) if |ρt | = 1. In this paper, we test whether the process yt is I (0) throughout the sample period or there contain changes between I (0) and I (1), that is, we consider the following null hypothesis: H0 : yt ∼ I (0),
t = 1, . . . , T ,
against the alternative hypothesis H1 : yt displays at least one regime switch between I (0) and I (1).
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3. Testing and dating algorithms 3.1. The test procedure Let uˆ 1,t be the OLS residuals from the regression of yt on dt , t = [T τ ] + 1, . . . , [T τ ] + [Th], and uˆ 0,t be the OLS residuals from the regression of yt on dt , t = [T τ ] − [Th] + 1, . . . , [T τ ]. When dt = 0, we denote uˆ t = yt . Our test procedure is based on the following moving ratio statistic
γˆ02 (τ ) RTh (τ ) =
γˆ12 (τ )
[T τ ]+[Th]
t =[T τ ]+1
i=[T τ ]+1
[ Tτ]
2
t
t =[T τ ]−[Th]+1
uˆ 1,i
2 ,
t i=[T τ ]−[Th]+1
(4)
uˆ 0,i
where h < τ ≤ 1 − h with some h ∈ (0, 1/2),
γˆ02 (τ ) =
1
[T τ ]
[Th] t =[T τ ]−[Th]+1
uˆ 20,t ,
1
γˆ12 (τ ) =
[T τ ]+[Th]
[Th] t =[T τ ]+1
uˆ 21,t .
Note that the above moving ratio statistic is different from the traditional ratio statistic of Kim (2000) who constructs the statistic based on full samples. Moreover, we introduce a modified component γˆ02 (τ )/γˆ12 (τ ) to attenuate the size distortion caused by higher persistence sequence as argued by Harvey et al. (2006). Although this modification will lose some power, we demonstrate that rejecting H0 for large values of RTh (τ ) provides a consistent test against I (0) to I (1) changes, and rejecting H0 for small values of RTh (τ ) provides a consistent test against I (1) to I (0) changes. As a consequence, we can easily identify the direction of changes. Throughout the paper, we assume that there are at least [Th] samples between two change points in persistence to ensure these change points can be distinguished. Theorem 3.1. Suppose that the Assumption 2.1 holds, then under the null hypothesis H0 , RTh (τ ) ⇒
(U2 (τ ) − U2 (τ − h)) (U2 (τ + h) −
τ +h
τ τ U2 (τ )) τ −h
2
V j ,1 ( r )
dr
≡ R∞ (τ ), 2 Vj,0 (r ) dr
(5)
where ‘‘ ⇒’’ stands for weak convergence and j = 1, 2, 3, V1,1 (r ) = U1 (r ) − U1 (τ );
V1,0 = U1 (r ) − U1 (τ − h);
V2,1 (r ) = U1 (r ) − U1 (τ ) − (r − τ )h−1 (U1 (τ + h) − U1 (τ )); V2,0 (r ) = U1 (r ) − U1 (τ − h) − (r − τ + h)h−1 (U1 (τ ) − U1 (τ − h)); V3,1 (r ) = (U1 (r ) − U1 (τ )) − K1−1 (r − τ ){4((τ + h)3 − τ 3 ) − 6h(2τ + h)
+ 3h(2 − 2τ − h)(τ + r )}(U1 (τ + h) − U1 (τ )) + 6K1−1 h(r − τ )(r − τ − h)
τ +h
τ
V3,0 (r ) = (U1 (r ) − U1 (τ )) − K2−1 (r − τ + h){4(τ 3 − (τ − h)3 ) − 6h(2τ − h)
+ 3h(2 − 2τ + h)(r + τ − h)}(U1 (τ ) − U1 (τ − h)) + 6K2 h(r − τ )(r − τ + h) −1
U1 (s)ds; τ
τ −h
U1 (s)ds;
and K1 = 4h2 (3τ 2 + 3τ h + h2 ) − 3h2 (2τ + h)2 , K2 = 4h2 (3τ 2 − 3τ h + h2 ) − 3h2 (2τ − h)2 . Theorem 3.1 indicates, under the null hypothesis H0 , we can find two constants c1 = c1 (α) and c2 = c2 (α) such that P {min RTh (τ ) < c1 } = P {max RTh (τ ) > c2 } = α τ ∈Ω
τ ∈Ω
for a given nominal level α and interval Ω = [h, 1 − h]. The following Theorem 3.2 shows the consistency of the test. Theorem 3.2. Suppose that the Assumption 2.1 holds, then under the alternative hypothesis H1 we have P { lim min RTh (τ ) < c1 } = 1,
if there is I (1) to I (0) change,
P { lim max RTh (τ ) > c2 } = 1,
if there is I (0) to I (1) change.
T →∞ τ ∈Ω
T →∞ τ ∈Ω
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3.2. Bootstrap approximation The drawback of statistic RTh (τ ) is that its asymptotic distribution depends on the tail index κ . Mandelbrot (1963) suggested a cursory way to estimate κ , but the accuracy is not enough. In order to avoid nuisance parameter κ , we now resort to the bootstrap methodology. The object of this section is to develop an approximation to the null distribution of statistic RTh (τ ), even if κ is unknown. The algorithm is as follows: 1. Compute the centered residuals
εj∗ = εˆ j −
T 1
T i=1
εˆ i ,
1 ≤ j ≤ T.
2. For a fixed N ≤ T , select with replacement a bootstrap sample {˜εi , i = 1, . . . , N } from {εi∗ , i = 1, . . . , T }. 3. Construct the bootstrap process y˜ i = δˆ T di + ε˜ i ,
i = 1, . . . , N ,
and calculate the statistic
γˆ0∗2 (τ ) R˜ Nh (τ ) =
γˆ1∗2 (τ )
[N τ ]+[Nh]
t =[N τ ]+1
i=[N τ ]+1
[ Nτ ]
2
t
t =[N τ ]−[Nh]+1
uˆ ∗1,i
t i=[N τ ]−[Nh]+1
2 ,
(6)
uˆ ∗0,i
where uˆ ∗0,i are the residuals from the regression of y˜ i on di , i = [N τ ] − [Nh] + 1, . . . , [N τ ], uˆ ∗1,i are the residuals from the regression of y˜ i on di , i = [N τ ] + 1, . . . , [N τ ] + [Nh].
γˆ0∗2 (τ ) =
[N τ ]
1
[Nh] t =[N τ ]−[Nh]+1
uˆ ∗0,2t ,
γˆ1∗2 (τ ) =
1
[Nh]
[N τ ]+[Nh] t =[N τ ]+1
uˆ ∗1,2t .
4. Repeat the Step 2 and Step 3 B times, approximating the asymptotic critical value of statistic RTh (τ ) by the empirical quantile of R˜ Nh (τ ). In order to prove the convergence of R˜ Nh (τ ), we need the following assumption: Assumption 3.1. As N → ∞, T → ∞ and N /T → 0. Theorem 3.3. If Assumption 3.1 and the null hypothesis H0 hold, then for every real x, p
Pϵ (R˜ Nh (τ ) ≤ x) −→ P (R∞ (τ ) ≤ x), p
where ϵ = σ (εi , i ≥ 1), and → stands for convergence in probability. Theorem 3.4. If Assumption 3.1 holds, then under the alternative hypothesis H1 , we have R˜ Nh (τ ) = Op (1). Theorem 3.3 implies that the bootstrap test has asymptotically correct size. And Theorem 3.4 shows the consistency of bootstrap test. 3.3. Estimation of change points In this section we concentrate on the estimation of change points in persistence. We estimate I (0) to I (1) change point by
τ˜ = arg max Λh (τ ), τ
and estimate I (1) to I (0) change point by
λ˜ = arg min Λh (τ ), τ
Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316
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where
[Th]−1
[T τ ]+[Th] t =[T τ ]+1
Λh (τ ) =
[ Tτ] t =[T τ ]−[Th]+1
|ˆu1,t | .
|ˆu0,t |
However, a lot of unreported simulations indicate that the above estimator cannot capture the true change points very well. ˜ are crude estimation of true break dates respectively. τ˜ is replaced In order to obtain a more precise estimator, let τ˜ and λ by
ˆ (τ ), τˆ = arg max Λ
(7)
τ ∈D(τ˜ )
where [T τ ]+[Th] t =[T τ ]+1
ˆ τ˜ (τ ) = Λ
|ˆu1,t |/[T (τ˜ + h − τ )]2
[ Tτ] t =[T τ˜ ]−[Th]+1
, |ˆu0,t |/[T (τ − τ˜ + h)]
and D(τ˜ ) = τ˜ − h + 2hδ, τ˜ + h − 2hδ
is a closed interval with some constant δ > 0. Throughout this paper we set δ = 0.2. Using the above two step estimation method, we can obtain a more precise estimator of true change points. For change ˜ we can replace it similarly by point λ
ˆ λ˜ (τ ). λˆ = arg min Λ
(8)
˜ τ ∈D(λ)
In summary, we test and estimate change points in persistence by the following steps. Step 1.
(1)
Let A1 = [h, 1 − h], if max{RTh (τ )} > c2 , set τ˜1 = arg max Λh (τ ), τ ∈A1
ˆ τ˜1 (τ ), else let τˆ1 = 0. τˆ1 = arg max Λ (2)
τ ∈A1
τ ∈D(τ˜1 )
˜ 1 = arg min Λh (τ ), Let B1 = A1 − (τˆ1 − h, τˆ1 + h), if min{RTh (τ )} < c1 , set λ τ ∈B1
τ ∈B1
ˆ λ˜ (τ ), else let λˆ 1 = 0. λˆ 1 = arg min Λ 1 τ ∈D(λ˜ 1 )
(3)
ˆ 1 = 0, jump to Step 4. If τˆ1 = 0, or λ
(1)
ˆ 1 − h, λˆ 1 + h), if max{RTh (τ )} > c2 , set τ˜2 = arg max Λh (τ ), Let A2 = B1 − (λ
Step 2. τ ∈A2
ˆ τ˜2 (τ ), else let τˆ2 = 0. τˆ2 = arg max Λ (2)
τ ∈A2
τ ∈D(τ˜2 )
˜ 2 = arg min Λh (τ ), Let B2 = A2 − (τˆ2 − h, τˆ2 + h), if min{RTh (τ )} < c1 , set λ τ ∈B2
τ ∈B2
ˆ λ˜ (τ ), else let λˆ 2 = 0. λˆ 2 = arg min Λ 2 τ ∈D(λ˜ 2 )
(3)
ˆ 2 = 0, jump to Step 4. If τˆ2 = 0, or λ
Step 3. Using the same approach as Step 2 continues judging in the residual subset and stopping until one cannot detect any change point more.
ˆ i > 0, i = 1, 2, . . .} and concludes: for every element in H, we say there is a Step 4. Let H = {τˆi > 0, i = 1, 2, . . .}, G = {λ persistence change from I (0) to I (1); for every element in G, we say there is a persistence change from I (1) to I (0). Obviously, the above test procedure will stop after a finite number of steps. Recall that there are [Th] samples at least between two change points, and |#H − #G| ≤ 1, (where #H denotes the number of elements in H), therefore we could capture every such change point.
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Table 1 Empirical sizes at 5% nominal level.
κ
ρ\h
T = 200
T = 500
Max
1.14
1.43
1.87
2
0 0.5 0.9 1 0 0.5 0.9 1 0 0.5 0.9 1 0 0.5 0.9 1
Min
Max
Min
0.2
0.3
0.2
0.3
0.2
0.3
0.2
0.3
6.1 5.5 7.7 8.9 5.2 5.6 7.0 6.4 5.8 5.1 3.5 4.0 6.6 4.8 3.3 3.6
5.1 4.9 6.9 9.2 5.0 4.1 5.4 5.1 5.0 4.8 4.1 3.3 5.0 4.5 3.6 3.1
5.8 5.5 6.9 8.6 4.9 4.2 5.5 6.0 5.2 4.6 4.6 4.5 6.4 5.1 3.7 4.1
5.2 4.5 4.1 9.1 5.2 4.6 3.9 5.3 5.0 4.2 2.9 3.6 4.4 4.0 3.1 3.2
5.7 5.1 5.9 14.2 5.4 5.1 5.9 7.4 5.9 4.8 3.8 3.4 5.6 4.8 2.6 2.6
5.4 4.9 5.6 9.6 6.0 5.8 5.3 5.1 6.0 5.5 4.8 3.6 6.2 5.0 2.9 2.3
4.8 4.5 6.5 13.4 5.6 4.9 4.7 8.3 5.7 4.9 3.4 3.3 5.9 4.1 2.8 2.5
4.7 4.2 4.7 9.3 4.7 3.9 3.8 4.9 4.4 3.8 2.6 3.3 6.4 5.4 3.1 2.4
4. Simulations and empirical application 4.1. Simulations In this section we use Monte Carlo simulation methods to investigate the finite sample performance of our test. Our simulation study is based on samples of size T = 200, 500 for h = 0.2, 0.3. Since the optimal bootstrap frequency N is difficult to select, it is not given in this paper and will be researched in future work. However, a lot of previous simulations indicate that N = 4T / log(T ) is a desirable choice to control the empirical size well. Throughout this section we fix the bootstrap frequency B = 500. All results are obtained by 2500 replications at nominal level α = 5%. To save space we just report the results for dt = 1, the other cases have similar results. The first model we use for the simulations is an ARMA(1,1) process yt = r0 + εt , εt = ρεt −1 + et + β et −1 ,
t = 1, . . . , T ,
(9)
where r0 , ρ and β are parameters. The parameter values are chosen to be r0 = 0.1 and β = 0.5. The innovation process {et } satisfies Assumption 2.1 with tail index κ varying among {1.14, 1.43, 1.87, 2}. The innovation sequence used in this section is generated by the program of STABLE. The program STABLE is available from J. P. Nolan’s website: academic2.american.edu/˜jpnolan. To investigate the size performance of the test and evaluate its robustness we vary ρ among 0, 0.5, 0.9 and 1. Table 1 reports the empirical sizes of our test. The size distortion is slight in this example. Particularly, even for ρ = 0.9 case, the empirical sizes are still close to the nominal level. This supports the motivation to reduce the size distortion by introducing a modified component. Moreover, we observe that smaller tail index does not induce more serious size distortion than larger tail index. This might owe to the bootstrap method. However, we cannot find strong evidence to say that the empirical size will control better as sample size increases. Since the size distortion is still not significant if ρ = 1, we guess that our test displays no tendency to spuriously over-reject I (0) null in favor of a change in persistence when a series is actually I (1) throughout. To investigate the power property of the test and the change point estimator we consider the same model as above, allowing changes in parameter ρ as the following four different change cases. Case 1: ρ changes at τ = 0.5 from 0.3 to 1. Case 1 displays a single change in persistence from I (0) to I (1). Case 2: ρ changes at τ = 0.5 from 1 to 0.3. Case 2 displays a single change in persistence from I (1) to I (0). Case 3: ρ changes at τ1 = 0.3 from 1 to 0.3, and changes from 0.3 to 1 at τ2 = 0.65. Case 3 is an I (1)− I (0)− I (1) persistence change profile. Case 4: ρ changes at τ1 = 0.3 from 0.5 to 1, and changes from 1 to 0.5 at τ2 = 0.65. Case 4 is an I (0)− I (1)− I (0) persistence change profile. Table 2 shows the simulated empirical powers for each persistence change profile. From Table 2, three conclusions can be obtained on power properties. First, the empirical power increases as sample size or bandwidth h increases. This is an expected result for our moving ratio test which is a consistent test. Second, the empirical power decreases as the tail index κ decreases. It is mainly because the time series yt has more ‘‘outliers’’ when the tail index decreases. This conclusion is similar to the finding of Han and Tian (2007b). Finally, the break fraction near 0.5 gives the highest rejection frequency, and the empirical power decreases as the change size |ρ − 1| decreases. This is a common conclusion like what other change point test procedures can get.
Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316
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Table 2 Empirical powers in each persistence change case. T
h\κ
Max
0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3
Case 1 Case 2 200 Case 3 Case 4 Case 1 Case 2 500 Case 3 Case 4
Min
1.14
1.43
1.87
2
1.14
1.43
1.87
2
47.48 75.52 4.16 4.20 51.28 57.96 31.64 31.20 85.52 94.40 6.84 7.76 84.48 90.20 68.23 64.76
49.28 76.12 2.80 2.78 51.98 58.60 32.04 30.88 86.84 94.72 4.60 3.56 85.16 90.60 69.64 64.20
52.36 76.56 2.42 1.26 52.76 59.02 32.68 28.92 87.44 95.48 2.72 1.20 86.08 91.36 70.68 65.40
53.30 76.84 2.26 0.48 53.04 59.64 33.72 28.56 88.04 96.16 2.36 0.88 86.40 91.80 71.72 65.32
3.84 2.64 49.76 73.96 46.36 49.86 33.20 36.36 6.08 5.08 88.40 94.84 87.88 88.16 75.20 77.28
2.92 1.56 52.36 74.60 47.84 45.88 34.16 37.04 3.62 2.36 89.28 95.24 88.16 89.36 76.16 78.52
2.12 0.96 57.28 75.68 50.76 45.48 34.68 39.32 2.52 0.92 90.16 96.28 88.20 90.24 76.68 79.16
2.68 0.90 58.64 76.80 51.72 46.52 35.88 40.60 2.76 0.78 91.68 96.40 89.52 90.40 77.88 79.44
Table 3 Empirical mean and standard deviation of change point estimators in Case 1 and 2. T
h
\κ
0.2
τˆ
0.3
τˆ
200
0.2 500 0.3
Case 1
se(τˆ ) se(τˆ )
τˆ
se(τˆ )
τˆ
se(τˆ )
Case 2
1.14
1.43
1.87
2
1.14
1.43
1.87
2
0.5291 0.1392 0.5119 0.1233 0.5224 0.1154 0.4996 0.1025
0.5407 0.1290 0.5211 0.1120 0.5272 0.1003 0.5094 0.0872
0.5468 0.1073 0.5304 0.0970 0.5261 0.0806 0.5122 0.0718
0.5494 0.1005 0.5283 0.0893 0.5234 0.0704 0.5107 0.0651
0.55576 0.0981 0.5843 0.0873 0.5574 0.0653 0.5768 0.0736
0.5436 0.0923 0.5661 0.0767 0.5431 0.0535 0.5612 0.0633
0.5299 0.0778 0.5474 0.0755 0.5343 0.0518 0.5483 0.0585
0.5248 0.0739 0.5385 0.0664 0.5282 0.0494 0.5455 0.0541
Table 4 Empirical mean and standard deviation of change point estimators in Case 3 and 4. T
h
0.2 200 0.3
0.2 500 0.3
\κ
Case 3
τˆ1 se(τˆ1 ) τˆ2 se(τˆ2 ) τˆ1 se(τˆ1 ) τˆ2 se(τˆ2 ) τˆ1 se(τˆ1 ) τˆ2 se(τˆ2 ) τˆ1 se(τˆ1 ) τˆ2 se(τˆ2 )
Case 4
1.14
1.43
1.87
2
1.14
1.43
1.87
2
0.4154 0.1478 0.5889 0.1810 0.3724 0.1026 0.6444 0.1047 0.3803 0.1178 0.6171 0.1421 0.3524 0.0794 0.6430 0.0837
0.3862 0.1342 0.6091 0.1696 0.3482 0.0842 0.6610 0.0953 0.3596 0.0970 0.6383 0.1279 0.3343 0.0624 0.6578 0.0710
0.3553 0.1083 0.6312 0.1476 0.3242 0.0675 0.6718 0.0791 0.3408 0.0741 0.6456 0.1026 0.3214 0.0482 0.6593 0.0601
0.3457 0.0940 0.6344 0.1438 0.3161 0.0641 0.6730 0.0732 0.3323 0.0559 0.6526 0.0854 0.3178 0.0481 0.6628 0.0581
0.3862 0.1534 0.6647 0.1518 0.3725 0.1511 0.6664 0.1436 0.3426 0.1153 0.6888 0.0948 0.3223 0.1035 0.6928 0.0884
0.3837 0.1437 0.6693 0.1207 0.3616 0.1299 0.6746 0.1159 0.3372 0.1001 0.6890 0.0695 0.3195 0.0889 0.6852 0.0701
0.3701 0.1160 0.6739 0.0884 0.3535 0.1083 0.6702 0.0885 0.3322 0.0806 0.6807 0.0557 0.3206 0.0712 0.6732 0.0576
0.3640 0.1091 0.6676 0.0762 0.3498 0.0989 0.6662 0.0832 0.3309 0.0778 0.6757 0.0476 0.3212 0.0659 0.6703 0.0570
To find the accuracy of our two step estimation method, we evaluate the empirical mean and the standard deviation of change point estimators. The results are shown in Table 3 (for Case 1 and Case 2) and Table 4 (for Case 3 and Case 4). It is not surprising that the estimations appear to converge to the true change points as sample size T or bandwidth h increases. Furthermore, the larger tail index, the smaller is the standard deviation. It is also not surprising for smaller tail index implies more ‘‘outliers ’’. An interesting finding is that the deviation increases as the tail index increases for I (0) to I (1) change, but it decreases as the tail index increases for I (1) to I (0) change. Next, we consider the properties of the test when the data-generating process (DGP) exhibits long-range dependences. The data are generated by the ARFIMA(0,d,0) process y t = r 0 + εt ,
(1 − B)d εt = et ,
t = 1, . . . , T ,
(10)
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Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316
Table 5 Empirical sizes with fractional integration innovations.
κ
T
h\ω
Max
0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3
1.14 1.43 200 1.87 2 1.14 1.43 500 1.87 2
Min
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
13.9 13.8 13.9 11.3 10.3 9.2 9.3 7.9 19.4 18.2 16.8 16.4 12.8 8.7 9.2 9.8
14.1 14.0 13.6 10.6 9.7 7.8 8.4 8.0 19.2 18.8 15.6 14.3 12.4 7.9 10.8 9.9
14.5 14.3 13.4 11.1 9.3 7.7 7.9 8.1 19.2 19.3 15.8 14.9 11.2 8.2 8.8 8.9
15.5 1.35 13.6 11.5 9.3 7.4 8.3 7.1 18.4 19.2 15.2 15.6 10.8 8.4 8.6 7.4
13.8 10.7 14.0 9.4 9.4 8.3 11.2 6.8 14.0 10.3 12.3 8.4 9.6 8.9 10.0 9.2
13.6 10.5 13.4 8.7 9.6 8.4 11.0 7.1 13.6 10.7 11.2 7.8 9.2 8.6 9.6 9.2
13.0 10.0 13.3 8.1 9.5 8.7 10.2 6.1 13.6 11.1 10.8 8.1 9.2 8.2 9.5 8.6
12.8 10.2 12.7 8.5 9.1 7.7 10.0 6.2 12.4 11.2 11.2 7.5 8.6 9.1 8.4 7.7
Table 6 Empirical powers with fractional integration innovations. h\κ
T
I 200 II I 500 II
0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3
Max
Min
1.14
1.43
1.87
2
1.14
1.43
1.87
2
67.32 51.60 19.72 19.40 85.68 71.04 30.86 31.20
64.12 49.44 18.20 19.28 84.64 66.44 24.44 27.36
58.96 46.32 13.76 15.40 79.82 62.60 21.48 24.02
59.76 43.88 15.24 15.04 79.38 58.88 19.82 22.96
73.52 62.28 19.12 12.64 88.06 77.64 31.86 20.94
72.84 60.88 16.44 12.36 87.62 78.78 24.84 17.48
69.68 59.80 13.20 10.08 86.14 74.96 21.26 13.96
68.56 60.60 13.96 10.72 86.08 76.20 17.88 15.08
where r0 = 0.1, and the innovation process {et } is treated the same as GDP (9). Since the range of permissible long memory parameters for a given tail index κ is restricted to the interval 0 < d < 1 − 1κ (see e.g. Kokoszka and Taqqu, 1996). We set the long memory parameter d = ω(1 − 1/κ), and vary ω among {0.2, 0.4, 0.6, 0.8}. Similar to Kokoszka and Taqqu (1996) we approximate the innovation εt by
εt =
1000
cj et −j ,
t = 1, . . . , T ,
j =1
in which c0 = 1, cj+1 = j+1 cj . Table 5 presents the simulated empirical sizes. It can be seen that the size distortions are not negligible under the fractional integration case especially for small κ . The long memory parameter d also has some influence to the empirical size. This indicates that our test procedure does not robust enough in such an innovation case. After evaluating the size performance of the test, we consider the power performance. We use the same model as DGP (10), allowing changes in long memory parameter d as the following two different change cases. j+d
I: d changes at τ1 = 0.3 from 0 to 0.5(1 − 1/κ), and changes from 0.5(1 − 1/κ) to 0 at τ2 = 0.65. Case I is an I (0)− I (d)− I (0) persistence change profile. II: d changes at τ = 0.3 from 1 to 0.5(1 − 1/κ), and changes from 0.5(1 − 1/κ) to 1 at τ2 = 0.65. Case II is an I (1)− I (d)− I (1) persistence change profile. As we can seen from Table 6 that our test performs well if the process changes from short to long memory, or vice versa (see the case I). It is interesting to note that our test has better power property when using a bandwidth of h = 0.2 than using h = 0.3. In addition, the smaller the tail index, the higher the empirical power. This finding is quite different from the conclusion obtained in the short memory case. However, this outcome is somewhat artificial, the reason for this phenomenon lying in the size distortion. Since the power, as the short memory case, is growing with sample size T , we guess that our test is still consistent for this change profile. Unfortunately, although the test seems still sensitive if the process changes between I (1) and fractional integration, the empirical power is poor. Finally, we use a small simulation study to check whether our test is robust when there is a mean change point in DGP. Table 7 reports the results when r0 in DGP (10) changes at τ = 0.5 from 0 to 1. The bandwidth was set to be h = 0.2. It can be seen that the empirical size is growing with decreasing long memory parameter or increasing tail index. In particular,
Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316
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Table 7 Empirical sizes when r0 in DGP (21) changes at τ = 0.5 from 0 to 1. T
200
500
κ \d 1.14 1.43 1.87 2 1.14 1.43 1.87 2
Max
Min
0
0.5(1 − 1/κ)
1
0
0.5(1 − 1/κ)
1
7.32 10.34 15.48 32.86 6.0 12.86 36.28 68.08
13.48 12.52 10.26 8.60 21.42 18.96 13.64 8.46
12.26 10.0 7.14 6.02 13.70 7.72 3.72 5.12
8.86 9.40 14.72 32.06 11.44 11.72 29.04 68.86
13.68 10.98 11.60 12.06 15.14 13.36 8.74 7.58
10.74 10.68 7.0 6.08 13.72 10.08 5.42 5.48
14 12 10 8 6 4 2 0 2
0
50
100
150
200
250
300
Fig. 1. US inflation rate data from May 1952 to April 1977.
the test suffers from serious size distortions if d = 0. This indicates that our test is not robust if there are structural breaks in the trend function. In conclusion, our test performs well in the short memory case. Moreover, although our test is based on I (0) null hypothesis, the empirical size is still controlled well when a series is actually I (1) throughout. However, our test is not robust enough and powerful when the series is long memory or contains structural breaks in the trend function. In these cases, a more robust and powerful test is necessary. A small bandwidth could capture short range changes while it has lower power for the small sample size. An ideal way is that a larger bandwidth is chosen for a smaller sample size, and a smaller one for a larger sample size. Although the test with bandwidth h = 0.3 performs a little better than the test with bandwidth h = 0.2, it has poor performance if the location of the change point does not lie in the interval (0.3, 0.7). Further, a large bandwidth leads to the estimation procedure having poor accuracy. Consequently, we suggest to use bandwidth h = 0.2 if the sample size is not too large or too small. In most cases, h = 0.2 is a reasonable choice to control the empirical size well and to obtain a satisfactory empirical power. 4.2. Empirical application In this section, we illustrate our test using two sets of financial time series data. The first set contains the US inflation rate monthly data which was observed from May 1952 until April 1977 with samples of 300 observations. The second set contains the stock closing price data of BOE (BOE Technology Group Co., Ltd.) in China stock market observed every five minutes from Oct. 26, 2010 to Nov. 9, 2010 with samples of 528 observations. We apply our test analyze these two data sets by setting N = 4T / log(T ), B = 500, h = 0.2, and nominal level α = 5%. Fig. 1 reports the observations in the first data set. Chen et al. (2010) has considered this data set using a sequential monitoring procedure, and concluded that the data contains an I (1) to I (0) change point before October 1961, and an I (0) to I (1) change before April 1966. We apply our test procedure to check this conclusion. The simulated upper tail and lower tail critical values by the bootstrap method are 21.469 and 0.0492 respectively. The maximum and minimum values of statistic RTh (τ ) in interval τ ∈ [0.2, 0.8] are 22.643 and 0.0481 respectively. This indicates that the data undergoes both I (0) to I (1) and I (1) to I (0) changes. The estimated I (0) to I (1) change period is May 1965 (see the right vertical line in Fig. 1), and the estimated I (1) to I (0) change period is July 1959 (see the left vertical line in Fig. 1). Then, by continued searching the maximum and minimum value of statistic RTh (τ ) in the residual interval τ ∈ [0.72, 0.8], we found that the maximum value does not exceed the upper tail critical value and the minimum value does not exceed the lower tail critical value. This implies that there does not exist any other change point in persistence. This conclusion coincided with Chen et al. (2010). Now, we consider the second data set. The observations of this data set are shown in Fig. 2. We can see that there are many ‘‘outliers’’. Shi et al. (2008) has argued that it is more suitable to fit such high frequency data by a stable distribution with tail index 1 < κ < 2 than a normal distribution. We employed the method of Mandelbrot (1963) to estimate the tail index κ and obtained κ = 1.86. The simulated upper tail and lower tail critical values by the bootstrap method are 23.848 and 0.0414 respectively. We found that the statistic RTh (τ ) has the maximum value 30.6805 at observation 415, and the minimum value 0.0363 is obtained at observation 301. Therefore, we can conclude that the data undergoes a change from
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Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316 3.85 3.8 3.75 3.7 3.65 3.6 3.55 3.5 3.45 3.4 3.35
0
100
200
300
400
500
600
Fig. 2. BOE stock closing price data observed every five minutes from Oct. 26, 2010 to Nov. 9, 2010.
I (1) to I (0) at observation 301, and a change from I (0) to I (1) at observation 415. Since the maximum and minimum values of statistic RTh (τ ) in the residual interval [0.2, 0.38] do not exceed the upper tail critical value and the lower tail critical value respectively, we say that there just only exist two change points in persistence. In this example, we also considered h = 0.3 case. However, we just only captured an I (1) to I (0) change point at observation 301. The reason that it failed to find the I (0) to I (1) change point is that the sample sizes 114 between two change points are smaller than the subsample size [Th] = 158. 5. Conclusions In this paper, we have proposed a moving ratio procedure to test and estimate multiple changes in persistence for a heavy-tailed sequence. We derived the asymptotic distribution of the moving ratio statistic which is a complicated function of the Lévy process under the I (0) null hypothesis. However, the asymptotic distribution depends on the unknown tail index κ . To overcome the problem, an approach based on residual bootstrap is proposed. Unlike the asymptotic test, the bootstrap procedure does not require any knowledge of the often very complex asymptotic distribution. We have also proposed a two step procedure to estimate the change points in persistence. Neither the direction of change nor the number of change points needs to be assumed known. Numerical results showed that the moving ratio test based on residual bootstrap constitutes a functional tool for detecting and locating multiple changes in persistence in short memory heavy-tailed sequence. In addition, the proposed test displays no tendency to spuriously over-reject I (0) null in favor of a change in persistence when a series is actually I (1) throughout. Finally, we illustrated our test using the US inflation rate data and a set of high frequency stock closing price data. Since the test has poor performance when the DGP exhibits long memory or structural breaks in the trend function, a more powerful and robust test can be more interesting for further study. 6. Mathematical proofs Proof of Theorem 3.1. Let t = [Tr ], then Lemma 2.1 gives that, if dt = 0, 1 a− T
[Tr ] i=[T τ ]+1
1 a− T
[Tr ]
1 uˆ 1,i = a− T
εi ⇒ ψ∞ (U1 (r ) − U1 (τ )) ≡ ψ∞ V1,1 (r )
(11)
i=[T τ ]+1
[Tr ]
[Tr ]
1 uˆ 0,i = a− T
i=[T τ ]−[Th]+1
εi ⇒ ψ∞ (U1 (r ) − U1 (τ − h)) ≡ ψ∞ V1,0 (r )
(12)
i=[T τ ]−[Th]+1
2 2 a− ˆ02 = a− T [Th]γ T
[T τ ]
εi2 ⇒ Ψ22 (U2 (τ ) − U2 (τ − h))
(13)
i=[T τ ]−[Th]+1 2 2 a− ˆ12 = a− T [Th]γ T
[T τ ]+[Th]
εi2 ⇒ Ψ22 (U2 (τ + h) − U2 (τ )).
(14)
i=[T τ ]+1
In which ψ∞ = −1
aT
∞
[Tr ] i=[T τ ]+1
j =0
ϕj , Ψ2 denotes l2 norm of sequence {ϕj }, and ϕj = ρ j . If dt = 1, −1
uˆ 1,i = aT
[Tr ] i=[T τ ]+1
εi − [Th]
−1
[Tr ]
[T τ ]+[Th]
εj
i=[T τ ]+1 j=[T τ ]+1 h−1 U 1
⇒ ψ∞ (U1 (r ) − U1 (τ ) − (r − τ ) ≡ ψ∞ V2,1 (r )
( (τ + h) − U1 (τ ))) (15)
Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316
[Tr ]
1 a− T
[Tr ]
1 uˆ 0,i = a− T
⇒ ψ∞ (U1 (r ) − U1 (τ − h) − (r − τ + ) ≡ ψ∞ V2,0 (r ) 2 [T τ ] [T τ ] 1 −2 −2 2 εi − = aT εj a [Th] T j=[T τ ]−[Th]+1 i=[T τ ]−[Th]+1
2 a− ˆ02 T [Th]γ
εj
i=[T τ ]−[Th]+1 j=[T τ ]−[Th]+1 h h−1 U 1 U1
i=[T τ ]−[Th]+1
i=[T τ ]−[Th]+1
[T τ ]
[Tr ]
εi − [Th]−1
( (τ ) −
2313
(τ − h))) (16)
⇒ Ψ22 (U2 (τ ) − U2 (τ − h)) 2 2 a− ˆ12 = a− T [Th]γ T
[T τ ]+[Th]
εi2 −
i=[T τ ]+1
1
[Th]
(17)
2 a− T
[T τ ]+[Th]
2 εj
j=[T τ ]+1
⇒ Ψ22 (U2 (τ + h) − U2 (τ )).
(18)
If dt = (1, t )T , let δ = (α, β)T , then by the definition of LS we have
αˆ − α βˆ − β
−1
=
t
t
t2
εt
t εt
,
[T τ ]
where = t =[T τ ]−[Th]+1 , if we estimate δ using the samples y[T τ ]−[Th]+1 , . . . , y[T τ ] , and using the samples y[T τ ]+1 , . . . , y[T τ ]+[Th] . Hence, by a tedious calculation we have
1 a− T
[Tr ]
[Tr ]
1 uˆ 1,i = a− T
i=[T τ ]+1
εi − (δˆ0 − δ0 ) − (δˆ1 − δ1 )i
=
[T τ ]+[Th]
t =[T τ ]+1 ,
if we estimate δ
i=[T τ ]+1
⇒ ψ∞ U1 (r ) − U1 (τ ) − K1−1 (r − τ ){4((τ + h)3 − τ 3 ) − 6h(2τ + h) + 3h(2 − 2τ − h)(τ + r )}(U1 (τ + h) − U1 (τ )) + 6K1 h(r − τ )(r − τ − h) −1
≡ ψ∞ V3,1 (r ),
τ +h
τ
U1 (s)ds (19)
with K1 = 4h2 (3τ 2 + 3τ h + h2 ) − 3h2 (2τ + h)2 . Similar arguments gives that 1 a− T
[Tr ]
uˆ 0,i ⇒ ψ∞ U1 (r ) − U1 (τ ) − K2−1 (r − τ + h){4(τ 3 − (τ − h)3 )
i=[T τ ]−[Th]+1
− 6h(2τ − h) + 3h(2 − 2τ + h)(r + τ − h)}(U1 (τ ) − U1 (τ − h)) τ −1 + 6K2 h(r − τ )(r − τ + h) U1 (s)ds τ −h
≡ ψ∞ V3,0 (r ),
(20)
with K2 = 4h2 (3τ 2 − 3τ h + h2 ) − 3h2 (2τ − h)2 . According to the above proof we can see that αˆ − α = Op (aT T −1 ), and βˆ − β = Op (aT T −2 ). Therefore 2 2 a− ˆ02 = a− T [Th]γ T
[T τ ]
εi2 + Op (T −1 ) ⇒ Ψ22 (U2 (τ ) − U2 (τ − h)).
(21)
i=[T τ ]−[Th]+1 2 2 a− ˆ12 = a− T [Th]γ T
[T τ ]+[Th] i=[T τ ]+1
εi2 + Op (T −1 ) ⇒ Ψ22 (U2 (τ + h) − U2 (τ )).
(22)
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Combining (11)–(22), Theorem 3.1 follows immediately from the continuous mapping theorem and continuity of the functionals. For the remainder of this section we omit proofs for the dt = (1, t )T case; these are straightforward but tedious and follow the same logical development as those presented for the dt = 1 case. Proof of Theorem 3.2. Since we have [Th] samples at least between two persistence change points, then for any such change point τ0 , it will not have the same direction persistence change point as τ0 in interval Ω = (τ0 − 2h, τ0 + 2h) ∩ [0, 1]. According to the test procedure, we just only need to prove P { lim max ΞTh (τ ) > c2 } = 1, T →∞ τ ∈Ω
if the direction of change point τ0 is I (0) to I (1), and P { lim min ΞTh (τ ) < c1 } = 1, T →∞ τ ∈Ω
if the direction of change point τ0 is I (1) to I (0). Suppose that the I (1) sequence yt changes to I (0) at τ0 , then Lemma 2.1 gives that, if τ − h < τ0 ≤ τ , t
uˆ 1,i = Op (aT );
γˆ02 (τ ) = Op (a2T );
i=[T τ ]+1 t
uˆ 0,i = Op (TaT );
γˆ12 (τ ) = Op (T −1 a2T ).
i=[T τ ]−[Th]+1
These results combining with the continuous mapping theorem indicate that the numerator of statistic RTh (τ ) is
γˆ (τ ) 2 0
[T τ ]+[Th]
t =[T τ ]+1
t
2 = Op (Ta4T ),
uˆ 1,i
i=[T τ ]+1
the denominator [T τ ]
γˆ (τ ) 2 1
t =[T τ ]−[Th]+1
2
t
= Op (T 2 a4T ).
uˆ 0,i
i=[T τ ]−[Th]+1
Hence RTh (τ ) = Op (T −1 ).
(23)
Similarly, when yt changes at τ0 from I (0) to I (1), then if τ ≤ τ0 < τ + h, t
uˆ 1,i = Op (TaT );
γˆ02 (τ ) = Op (T −1 a2T );
i=[T τ0 ]+1 t
uˆ 0,i = Op (aT );
γˆ12 (τ ) = Op (a2T ).
i=[T τ ]−[Th]+1
These results combining with the continuous mapping theorem indicate that the numerator of statistic RTh (τ ) is
γˆ (τ ) 2 0
[T τ ]+[Th] t =[T τ ]+1
t
2 uˆ 1,i
= Op (T 2 a4T ),
i=[T τ ]+1
the denominator
γˆ (τ ) 2 1
[T τ ] t =[T τ ]−[Th]+1
t
2 uˆ 0,i
= Op (Ta4T ).
i=[T τ ]−[Th]+1
Hence RTh (τ ) = Op (T ). From (23) and (24), we establish Theorem 3.2.
(24)
Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316
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Proof of Theorem 3.3. Observing that when we select one of ε1∗ , . . . , εT∗ , we select the corresponding unobservable noise variable which will be denoted as ε i . This means that [Nt ] 1
aN i = 1
[Nt ] 1
ε˜ i =
aN i=1
εi −
T [Nt ]
TaN i=1
εi .
By (3) and Assumption 3.1 we have T [Nt ]
TaN i=1
εi =
1−1/κ−δ T T [Nt ] T 1/κ L(N ) 1 N 1 ε ≤ 2 εi = op (1). i 1 /κ T N L(N ) aT i=1 T aT i=1
The above inequality follows from L(T )/L(N ) ≤ 2(T /N )δ (cf. Theorem 1.5.6 of Bigham et al., 1987), δ > 0 be chosen so small that 1 − 1/κ − δ > 0. Horváth and Kokoszka (2003) showed that for any bounded continuous functional g on D[0, 1]
Pϵ
−1
g
aM
[Mt ]
εi
p
≤ x −→ P (g (U1 (t )) ≤ x).
(25)
i=1
Hence [Nt ] 1
aN i = 1
ε˜ i ⇒ U1 (t ).
(26)
On the other hand [Nt ] 1
a2N i=1
[Nt ] 1
ε˜ = 2 i
a2N i=1 [Nt ]
1
=
a2N i=1
ε − 2 i
2
Ta2N
ε 2i + Op (1)
[Nt ]
2 N T a2T 1 εi + 2 2 εi εi
a2T T 2 a2N
T aN
i =1
i=1
−
2aT
TaN
aT i = 1
.
It is easy to see that aT
lim
T →∞
2
≤ lim
TaN
T →∞
N
1−1/κ−δ N T
= 0.
Using the same proof line of assertion (25) we have
Pϵ
g
−2
aN
[Nt ]
ε
2 i
p
≤ x −→ P (g (U2 (t )) ≤ x).
(27)
i=1
Consequently, [Nt ] 1
a2N i=1
ε˜ i2 ⇒ U2 (t ).
(28)
According to (26) and (28), we can complete the proof of Theorem 3.3 by the same proof line of Theorem 3.1, we omit to report here. Proof of Theorem 3.4. Recall that the residual εˆ i = Op (aT ) if the corresponding observation yi ∼ I (1). Consequently, whether yi is I (0) or I (1), the corresponding centered residual εi∗ = Op (aT ). From the construction of the bootstrap step we can see that the bootstrap process y˜ i ∼ I (1), i = 1, . . . , N. Therefore, for any τ ∈ (0, 1), the residuals uˆ ∗1,i = Op (aN ) = uˆ ∗0,i . Continuous mapping theorem gives that both the denominator and the numerator of statistic R˜ Nh (τ ) is Op (N 3 a4N ). This completes the proof of Theorem 3.4. Acknowledgments We are grateful to the associate editor and two anonymous referees for their detailed comments and valuable suggestions. In addition, we would like to thank Prof. John Nolan who provided the software for generating the stable innovations in Section 4. This work was supported by the National Natural Science Foundation of China (No. 60972150, No. 10926197 and No. 61065009).
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References Bigham, N.H., Goldie, C.M., Teugels, J.L., 1987. Regula Variation. Cambridge University Press, Cambridge. Busetti, F., Taylor, A.M.R., 2004. Tests of stationarity against a change in persistence. Journal of Econometrics 123, 33–66. Carvaliere, G., Taylor, A.M.R., 2008. Testing for a change in persistence in the presence of a non-stationary volatility. Journal of Econometrics 147, 84–98. Chen, Z., Tian, Z., Wei, Y., 2010. Monitoring change in persistence in linear time series. Statistics & Probability Letters 80, 1520–1527. Elliott, G., Rothenberg, T.J., Stock, J.H., 1996. Efficient tests for an autoregressive unit root. Econometrica 64, 813–836. Guillaume, D.M., et al., 1997. From the bird’s eye to the microscope: a survey of new stylized facts of the intra-daily foreign exchange markets. Finance and Stochastics 1, 95–129. Han, S., Tian, Z., 2007a. Subsampling change-point detection in persistence with heavy-tailed innovations. Journal of Applied Mathematics and Computing 23, 57–71. Han, S., Tian, Z., 2007b. Bootstrap testing for changes in persistence with heavy-tailed innovations. Communications in Statistics—Theory and Methods 36, 2289–2299. Harvey, D., Leybourne, S., Taylor, A.M.R., 2006. Modified tests for a change in persistence. Journal of Econometrics 134, 441–469. Hassler, U., Scheithauer, J., 2009. Detecting changes from short to long memory. Statistical Papers. doi:10.1007/s00362-009-0292-y. Horváth, L., Kokoszka, P., 2003. A bootstrap approximation to a unit root test statistics for heavy-tailed observations. Statistics & Probability Letters 62, 163–173. Kim, J.Y., 2000. Detection of change in persistence of a linear time series. Journal of Econometrics 95, 97–116. Kim, J.Y., Belaire-Franch, J., Amador, R.B., 2002. Corrigendum to ‘‘detection of change in persistence of a linear time series’’. Journal of Econometrics 109, 389–392. Kokoszka, P., Taqqu, M.S., 1996. Parameter estimation for infinite variance fractional ARIMA. The Annals of Statistics 24, 1880–1913. Kokoszka, P., Wolf, M., 2004. Subsampling the mean of heavy-tailed dependent observations. Journal of Time Series Analysis 25, 217–234. Leybourne, S.J., Kim, T.H., 2003. Tests for a change in persistence against the null of difference stationarity. The Econometrics Journal 6, 290–310. Leybourne, S.J., Kim, T.H., Taylor, A.M.B., 2007b. Detecting multiple changes in persistence. Studies in Nonlinear Dynamics & Econometrics 11, 1370–1384. Leybourne, S.J., Taylor, A.M.B., 2006. Persistence change tests and shifting stable autoregressions. Economics Letters 91, 44–49. Leybourne, S.J., Taylor, R., Kim, T.H., 2007a. CUSUM of squares-based tests for a change in persistence. Journal of Time Series Analysis 28, 408–433. Mandelbrot, B.B., 1963. The variation of certain speculative prices. Journal of Business 36 (3), 394–491. Perron, P., 2006. Dealing with structural breaks. In: Mills, T.C., Patterson, K. (Eds.), New Palgrave Handbook of Econometrics, vol. 1. Macmillan, London, pp. 278–352. Rechev, S.T., Mittnik, S., 2000. Stable Paretian Models in Finance. Wiley, New York. Rosadi, D., 2009. Testing for independence in heavy-tailed time series using the codifference function. Computational Statistics and Data Analysis 53 (12), 4516–4529. Shi, X., Miao, B., Ge, C., 2008. Consistency of change point estimators for symmetrical stable distribution with parameters shift. Science in China, Series A 51 (5), 1–10. Sibbertsen, P., Kruse, R., 2009. Testing for a break in persistence under long-range dependencies. Journal of Time Series Analysis 30, 263–285.
Contents lists available at SciVerse ScienceDirect
Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda
Bootstrap testing multiple changes in persistence for a heavy-tailed sequence Zhanshou Chen a,b,∗ , Zi Jin c , Zheng Tian b,d , Peiyan Qi b a
Department of Mathematics, Qinghai Normal University, Xining, Qinghai 810008, PR China
b
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China
c
Department of Statistics, University of British Columbia, 333-6356 Agricultural Road, Vancouver, BC V67T 1Z2, Canada
d
National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing, 100101, PR China
article
info
Article history: Received 5 April 2010 Received in revised form 23 December 2011 Accepted 13 January 2012 Available online 25 January 2012 Keywords: Multiple changes in persistence Moving ratio test Bootstrap Heavy tailed
abstract This paper tests the null hypothesis of stationarity against the alternative of changes in persistence for sequences in the domain of attraction of a stable law. The proposed moving ratio test is valid for multiple changes in persistence while the previous residual based ratio tests are designed for processes displaying only a single change. We show that the new test is consistent whether the process changes from I (0) to I (1) or vice versa. And it is easy to identify the direction of detected change points. In particular, a bootstrap approximation method is proposed to determine the critical values for the null distribution of the test statistic containing unknown tail index. We also propose a two step approach to estimate the change points. Numerical evidence suggests that our test performs well in finite samples. In addition, we show that our test is still powerful for changes between short and long memory, and displays no tendency to spuriously over-reject I (0) null in favor of a persistence change if the process is actually I (1) throughout. Finally, we illustrate our test using the US inflation rate data and a set of high frequency stock closing price data. © 2012 Elsevier B.V. All rights reserved.
1. Introduction During the past two decades, there is a growing body of evidence showing that economic and financial time series display changes in persistence. This has been an issue of substantial empirical interest, especially concerning inflation rate series, short-term interest rates, government budget deficits and real output. Recently, a number of testing procedures have been suggested to distinguish such behavior. These include, inter alia, ratio tests (Kim, 2000; Kim et al., 2002; Leybourne and Kim, 2003; Harvey et al., 2006), LBI tests (Busetti and Taylor, 2004; Leybourne and Taylor, 2006), CUSUM of squaresbased tests (Leybourne et al., 2007a). As a general discussion about ratio tests and LBI tests, we refer the reader to Perron (2006). More recently, Carvaliere and Taylor (2008) have considered persistence change tests under the non-stationary volatility innovation case, Sibbertsen and Kruse (2009) have studied the long-range dependence innovation case, Hassler and Scheithauer (2009) have applied ratio tests and LBI tests to detect change points from short to long memory. Since the multiple change point test is also an important issue in change point analysis, Leybourne et al. (2007b) proposed a multiple change point detection procedure based on sequences of doubly-recursive implementations of the regression-based unit root statistic of Elliott et al. (1996). All of the works above are concentrated on the case where variances of the sequences are finite. Guillaume et al. (1997) and Rechev and Mittnik (2000) have argued that many types of data from economics and finance have the same character:
∗
Corresponding author at: Qinghai Normal University, Department of Mathematics, 810008 Xining, Qinghai, PR China. E-mail address: [email protected] (Z. Chen).
0167-9473/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2012.01.011
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a heavier tail than the normal variables. Motivated by this, Han and Tian (2007a,b) considered persistence change with innovations in the domain of attraction of κ -stable law. The tail index κ can reflect the heaviness of the data and the variance of κ -stable processes is infinite when κ < 2. However, these tests are not consistent against processes which display more than one change in persistence. Consequently, these tests cannot be used to consistently separate the data into its I (0) and I (1) regimes when multiple changes in persistence occur. Moreover, even for a single change model these procedures require different tests depending on the direction of change which the data display. In the light of these important drawbacks, this paper considers the testing and dating problems of multiple changes in persistence for sequences in the domain of attraction of κ -stable law. For more details about this heavy-tailed sequence we refer the reader to Horváth and Kokoszka (2003), Kokoszka and Wolf (2004) and Rosadi (2009), among many others. Our test procedure is based on a residual-based moving ratio statistic with a bandwidth parameter h ∈ (0, 1/2). The new statistic can be used to detect a single change point in persistence as well as multiple changes in persistence if there are large enough samples between two change points. We show that the new test will diverge to infinite if the sequence shows I (0) to I (1) change, and will converge to zero if the sequence shows I (1) to I (0) change. Therefore, this result can help us to identify the direction of the change points. Moreover, a two step approach is proposed to estimate the detected change points. In order to determine the critical values of the test statistic which containing unknown tail index κ , we propose a bootstrap approximation method. The bootstrap method can circumvent this nuisance parameter and obtain asymptotically correct critical values. Although our test procedure is based on I (0) null hypothesis, we also evaluate the size and power performances of the test under I (1) as well as fractional integration cases, and check its robustness if there are structural breaks in the trend function by simulation. The rest of the paper is organized as follows. Section 2 introduces the model and some necessary assumptions. Section 3 contains the testing and dating procedures and the requisite limiting results. In Section 4, we use Monte Carlo methods to test the finite sample performance of our test and illustrate it by the US inflation rate data and a set of high frequency stock data. We conclude the paper in Section 5. All technical proofs of the theoretical results are gathered in Section 6. 2. Model and assumptions Let y1 , y2 , . . . , yT be an observed time series that can be decomposed as yt = µt + εt , εt = ρ t εt − 1 + e t ,
(1) t = 1, 2, . . . , T ,
(2)
where µt = E (yt ) = δ dt is a deterministic component modeled as a linear combination of a vector of nonrandom regressors dt . Typical components of dt are a constant, a time trend or dummy variables. et is the stochastic part of the process with further discussion satisfy the following assumption. T
Assumption 2.1. The strictly stationary symmetrical sequences et are in the domain of attraction of a stable law with tail index κ ∈ (1, 2) and Eet = 0. Lemma 2.1. If Assumption 2.1 holds, then
−1
aT
[T τ ]
et , aT
−2
t =1
[T τ ]
e2t
d
−→ (U1 (τ ), U2 (τ )),
(3)
t =1
where aT = inf{x : P (|et | > x) ≤ T −1 }, d
and the random variable U1 (τ ) is κ -stable and U2 (τ ) is κ/2-stable Lévy process in [0, 1]. The notation → stands for convergence in distribution. Remark 2.1. This result is obtained by Kokoszka and Wolf (2004). The exact definition of the Lévy process (U1 (τ ), U2 (τ )) appearing in Lemma 2.1 is not needed in the following, but we recall that the quantities aT can be represented as aT = T 1/κ L(T ) for some slowly varying function L. Within the model (1), the process yt is I (0) if |ρt | < 1, the process yt is I (1) if |ρt | = 1. In this paper, we test whether the process yt is I (0) throughout the sample period or there contain changes between I (0) and I (1), that is, we consider the following null hypothesis: H0 : yt ∼ I (0),
t = 1, . . . , T ,
against the alternative hypothesis H1 : yt displays at least one regime switch between I (0) and I (1).
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3. Testing and dating algorithms 3.1. The test procedure Let uˆ 1,t be the OLS residuals from the regression of yt on dt , t = [T τ ] + 1, . . . , [T τ ] + [Th], and uˆ 0,t be the OLS residuals from the regression of yt on dt , t = [T τ ] − [Th] + 1, . . . , [T τ ]. When dt = 0, we denote uˆ t = yt . Our test procedure is based on the following moving ratio statistic
γˆ02 (τ ) RTh (τ ) =
γˆ12 (τ )
[T τ ]+[Th]
t =[T τ ]+1
i=[T τ ]+1
[ Tτ]
2
t
t =[T τ ]−[Th]+1
uˆ 1,i
2 ,
t i=[T τ ]−[Th]+1
(4)
uˆ 0,i
where h < τ ≤ 1 − h with some h ∈ (0, 1/2),
γˆ02 (τ ) =
1
[T τ ]
[Th] t =[T τ ]−[Th]+1
uˆ 20,t ,
1
γˆ12 (τ ) =
[T τ ]+[Th]
[Th] t =[T τ ]+1
uˆ 21,t .
Note that the above moving ratio statistic is different from the traditional ratio statistic of Kim (2000) who constructs the statistic based on full samples. Moreover, we introduce a modified component γˆ02 (τ )/γˆ12 (τ ) to attenuate the size distortion caused by higher persistence sequence as argued by Harvey et al. (2006). Although this modification will lose some power, we demonstrate that rejecting H0 for large values of RTh (τ ) provides a consistent test against I (0) to I (1) changes, and rejecting H0 for small values of RTh (τ ) provides a consistent test against I (1) to I (0) changes. As a consequence, we can easily identify the direction of changes. Throughout the paper, we assume that there are at least [Th] samples between two change points in persistence to ensure these change points can be distinguished. Theorem 3.1. Suppose that the Assumption 2.1 holds, then under the null hypothesis H0 , RTh (τ ) ⇒
(U2 (τ ) − U2 (τ − h)) (U2 (τ + h) −
τ +h
τ τ U2 (τ )) τ −h
2
V j ,1 ( r )
dr
≡ R∞ (τ ), 2 Vj,0 (r ) dr
(5)
where ‘‘ ⇒’’ stands for weak convergence and j = 1, 2, 3, V1,1 (r ) = U1 (r ) − U1 (τ );
V1,0 = U1 (r ) − U1 (τ − h);
V2,1 (r ) = U1 (r ) − U1 (τ ) − (r − τ )h−1 (U1 (τ + h) − U1 (τ )); V2,0 (r ) = U1 (r ) − U1 (τ − h) − (r − τ + h)h−1 (U1 (τ ) − U1 (τ − h)); V3,1 (r ) = (U1 (r ) − U1 (τ )) − K1−1 (r − τ ){4((τ + h)3 − τ 3 ) − 6h(2τ + h)
+ 3h(2 − 2τ − h)(τ + r )}(U1 (τ + h) − U1 (τ )) + 6K1−1 h(r − τ )(r − τ − h)
τ +h
τ
V3,0 (r ) = (U1 (r ) − U1 (τ )) − K2−1 (r − τ + h){4(τ 3 − (τ − h)3 ) − 6h(2τ − h)
+ 3h(2 − 2τ + h)(r + τ − h)}(U1 (τ ) − U1 (τ − h)) + 6K2 h(r − τ )(r − τ + h) −1
U1 (s)ds; τ
τ −h
U1 (s)ds;
and K1 = 4h2 (3τ 2 + 3τ h + h2 ) − 3h2 (2τ + h)2 , K2 = 4h2 (3τ 2 − 3τ h + h2 ) − 3h2 (2τ − h)2 . Theorem 3.1 indicates, under the null hypothesis H0 , we can find two constants c1 = c1 (α) and c2 = c2 (α) such that P {min RTh (τ ) < c1 } = P {max RTh (τ ) > c2 } = α τ ∈Ω
τ ∈Ω
for a given nominal level α and interval Ω = [h, 1 − h]. The following Theorem 3.2 shows the consistency of the test. Theorem 3.2. Suppose that the Assumption 2.1 holds, then under the alternative hypothesis H1 we have P { lim min RTh (τ ) < c1 } = 1,
if there is I (1) to I (0) change,
P { lim max RTh (τ ) > c2 } = 1,
if there is I (0) to I (1) change.
T →∞ τ ∈Ω
T →∞ τ ∈Ω
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3.2. Bootstrap approximation The drawback of statistic RTh (τ ) is that its asymptotic distribution depends on the tail index κ . Mandelbrot (1963) suggested a cursory way to estimate κ , but the accuracy is not enough. In order to avoid nuisance parameter κ , we now resort to the bootstrap methodology. The object of this section is to develop an approximation to the null distribution of statistic RTh (τ ), even if κ is unknown. The algorithm is as follows: 1. Compute the centered residuals
εj∗ = εˆ j −
T 1
T i=1
εˆ i ,
1 ≤ j ≤ T.
2. For a fixed N ≤ T , select with replacement a bootstrap sample {˜εi , i = 1, . . . , N } from {εi∗ , i = 1, . . . , T }. 3. Construct the bootstrap process y˜ i = δˆ T di + ε˜ i ,
i = 1, . . . , N ,
and calculate the statistic
γˆ0∗2 (τ ) R˜ Nh (τ ) =
γˆ1∗2 (τ )
[N τ ]+[Nh]
t =[N τ ]+1
i=[N τ ]+1
[ Nτ ]
2
t
t =[N τ ]−[Nh]+1
uˆ ∗1,i
t i=[N τ ]−[Nh]+1
2 ,
(6)
uˆ ∗0,i
where uˆ ∗0,i are the residuals from the regression of y˜ i on di , i = [N τ ] − [Nh] + 1, . . . , [N τ ], uˆ ∗1,i are the residuals from the regression of y˜ i on di , i = [N τ ] + 1, . . . , [N τ ] + [Nh].
γˆ0∗2 (τ ) =
[N τ ]
1
[Nh] t =[N τ ]−[Nh]+1
uˆ ∗0,2t ,
γˆ1∗2 (τ ) =
1
[Nh]
[N τ ]+[Nh] t =[N τ ]+1
uˆ ∗1,2t .
4. Repeat the Step 2 and Step 3 B times, approximating the asymptotic critical value of statistic RTh (τ ) by the empirical quantile of R˜ Nh (τ ). In order to prove the convergence of R˜ Nh (τ ), we need the following assumption: Assumption 3.1. As N → ∞, T → ∞ and N /T → 0. Theorem 3.3. If Assumption 3.1 and the null hypothesis H0 hold, then for every real x, p
Pϵ (R˜ Nh (τ ) ≤ x) −→ P (R∞ (τ ) ≤ x), p
where ϵ = σ (εi , i ≥ 1), and → stands for convergence in probability. Theorem 3.4. If Assumption 3.1 holds, then under the alternative hypothesis H1 , we have R˜ Nh (τ ) = Op (1). Theorem 3.3 implies that the bootstrap test has asymptotically correct size. And Theorem 3.4 shows the consistency of bootstrap test. 3.3. Estimation of change points In this section we concentrate on the estimation of change points in persistence. We estimate I (0) to I (1) change point by
τ˜ = arg max Λh (τ ), τ
and estimate I (1) to I (0) change point by
λ˜ = arg min Λh (τ ), τ
Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316
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where
[Th]−1
[T τ ]+[Th] t =[T τ ]+1
Λh (τ ) =
[ Tτ] t =[T τ ]−[Th]+1
|ˆu1,t | .
|ˆu0,t |
However, a lot of unreported simulations indicate that the above estimator cannot capture the true change points very well. ˜ are crude estimation of true break dates respectively. τ˜ is replaced In order to obtain a more precise estimator, let τ˜ and λ by
ˆ (τ ), τˆ = arg max Λ
(7)
τ ∈D(τ˜ )
where [T τ ]+[Th] t =[T τ ]+1
ˆ τ˜ (τ ) = Λ
|ˆu1,t |/[T (τ˜ + h − τ )]2
[ Tτ] t =[T τ˜ ]−[Th]+1
, |ˆu0,t |/[T (τ − τ˜ + h)]
and D(τ˜ ) = τ˜ − h + 2hδ, τ˜ + h − 2hδ
is a closed interval with some constant δ > 0. Throughout this paper we set δ = 0.2. Using the above two step estimation method, we can obtain a more precise estimator of true change points. For change ˜ we can replace it similarly by point λ
ˆ λ˜ (τ ). λˆ = arg min Λ
(8)
˜ τ ∈D(λ)
In summary, we test and estimate change points in persistence by the following steps. Step 1.
(1)
Let A1 = [h, 1 − h], if max{RTh (τ )} > c2 , set τ˜1 = arg max Λh (τ ), τ ∈A1
ˆ τ˜1 (τ ), else let τˆ1 = 0. τˆ1 = arg max Λ (2)
τ ∈A1
τ ∈D(τ˜1 )
˜ 1 = arg min Λh (τ ), Let B1 = A1 − (τˆ1 − h, τˆ1 + h), if min{RTh (τ )} < c1 , set λ τ ∈B1
τ ∈B1
ˆ λ˜ (τ ), else let λˆ 1 = 0. λˆ 1 = arg min Λ 1 τ ∈D(λ˜ 1 )
(3)
ˆ 1 = 0, jump to Step 4. If τˆ1 = 0, or λ
(1)
ˆ 1 − h, λˆ 1 + h), if max{RTh (τ )} > c2 , set τ˜2 = arg max Λh (τ ), Let A2 = B1 − (λ
Step 2. τ ∈A2
ˆ τ˜2 (τ ), else let τˆ2 = 0. τˆ2 = arg max Λ (2)
τ ∈A2
τ ∈D(τ˜2 )
˜ 2 = arg min Λh (τ ), Let B2 = A2 − (τˆ2 − h, τˆ2 + h), if min{RTh (τ )} < c1 , set λ τ ∈B2
τ ∈B2
ˆ λ˜ (τ ), else let λˆ 2 = 0. λˆ 2 = arg min Λ 2 τ ∈D(λ˜ 2 )
(3)
ˆ 2 = 0, jump to Step 4. If τˆ2 = 0, or λ
Step 3. Using the same approach as Step 2 continues judging in the residual subset and stopping until one cannot detect any change point more.
ˆ i > 0, i = 1, 2, . . .} and concludes: for every element in H, we say there is a Step 4. Let H = {τˆi > 0, i = 1, 2, . . .}, G = {λ persistence change from I (0) to I (1); for every element in G, we say there is a persistence change from I (1) to I (0). Obviously, the above test procedure will stop after a finite number of steps. Recall that there are [Th] samples at least between two change points, and |#H − #G| ≤ 1, (where #H denotes the number of elements in H), therefore we could capture every such change point.
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Table 1 Empirical sizes at 5% nominal level.
κ
ρ\h
T = 200
T = 500
Max
1.14
1.43
1.87
2
0 0.5 0.9 1 0 0.5 0.9 1 0 0.5 0.9 1 0 0.5 0.9 1
Min
Max
Min
0.2
0.3
0.2
0.3
0.2
0.3
0.2
0.3
6.1 5.5 7.7 8.9 5.2 5.6 7.0 6.4 5.8 5.1 3.5 4.0 6.6 4.8 3.3 3.6
5.1 4.9 6.9 9.2 5.0 4.1 5.4 5.1 5.0 4.8 4.1 3.3 5.0 4.5 3.6 3.1
5.8 5.5 6.9 8.6 4.9 4.2 5.5 6.0 5.2 4.6 4.6 4.5 6.4 5.1 3.7 4.1
5.2 4.5 4.1 9.1 5.2 4.6 3.9 5.3 5.0 4.2 2.9 3.6 4.4 4.0 3.1 3.2
5.7 5.1 5.9 14.2 5.4 5.1 5.9 7.4 5.9 4.8 3.8 3.4 5.6 4.8 2.6 2.6
5.4 4.9 5.6 9.6 6.0 5.8 5.3 5.1 6.0 5.5 4.8 3.6 6.2 5.0 2.9 2.3
4.8 4.5 6.5 13.4 5.6 4.9 4.7 8.3 5.7 4.9 3.4 3.3 5.9 4.1 2.8 2.5
4.7 4.2 4.7 9.3 4.7 3.9 3.8 4.9 4.4 3.8 2.6 3.3 6.4 5.4 3.1 2.4
4. Simulations and empirical application 4.1. Simulations In this section we use Monte Carlo simulation methods to investigate the finite sample performance of our test. Our simulation study is based on samples of size T = 200, 500 for h = 0.2, 0.3. Since the optimal bootstrap frequency N is difficult to select, it is not given in this paper and will be researched in future work. However, a lot of previous simulations indicate that N = 4T / log(T ) is a desirable choice to control the empirical size well. Throughout this section we fix the bootstrap frequency B = 500. All results are obtained by 2500 replications at nominal level α = 5%. To save space we just report the results for dt = 1, the other cases have similar results. The first model we use for the simulations is an ARMA(1,1) process yt = r0 + εt , εt = ρεt −1 + et + β et −1 ,
t = 1, . . . , T ,
(9)
where r0 , ρ and β are parameters. The parameter values are chosen to be r0 = 0.1 and β = 0.5. The innovation process {et } satisfies Assumption 2.1 with tail index κ varying among {1.14, 1.43, 1.87, 2}. The innovation sequence used in this section is generated by the program of STABLE. The program STABLE is available from J. P. Nolan’s website: academic2.american.edu/˜jpnolan. To investigate the size performance of the test and evaluate its robustness we vary ρ among 0, 0.5, 0.9 and 1. Table 1 reports the empirical sizes of our test. The size distortion is slight in this example. Particularly, even for ρ = 0.9 case, the empirical sizes are still close to the nominal level. This supports the motivation to reduce the size distortion by introducing a modified component. Moreover, we observe that smaller tail index does not induce more serious size distortion than larger tail index. This might owe to the bootstrap method. However, we cannot find strong evidence to say that the empirical size will control better as sample size increases. Since the size distortion is still not significant if ρ = 1, we guess that our test displays no tendency to spuriously over-reject I (0) null in favor of a change in persistence when a series is actually I (1) throughout. To investigate the power property of the test and the change point estimator we consider the same model as above, allowing changes in parameter ρ as the following four different change cases. Case 1: ρ changes at τ = 0.5 from 0.3 to 1. Case 1 displays a single change in persistence from I (0) to I (1). Case 2: ρ changes at τ = 0.5 from 1 to 0.3. Case 2 displays a single change in persistence from I (1) to I (0). Case 3: ρ changes at τ1 = 0.3 from 1 to 0.3, and changes from 0.3 to 1 at τ2 = 0.65. Case 3 is an I (1)− I (0)− I (1) persistence change profile. Case 4: ρ changes at τ1 = 0.3 from 0.5 to 1, and changes from 1 to 0.5 at τ2 = 0.65. Case 4 is an I (0)− I (1)− I (0) persistence change profile. Table 2 shows the simulated empirical powers for each persistence change profile. From Table 2, three conclusions can be obtained on power properties. First, the empirical power increases as sample size or bandwidth h increases. This is an expected result for our moving ratio test which is a consistent test. Second, the empirical power decreases as the tail index κ decreases. It is mainly because the time series yt has more ‘‘outliers’’ when the tail index decreases. This conclusion is similar to the finding of Han and Tian (2007b). Finally, the break fraction near 0.5 gives the highest rejection frequency, and the empirical power decreases as the change size |ρ − 1| decreases. This is a common conclusion like what other change point test procedures can get.
Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316
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Table 2 Empirical powers in each persistence change case. T
h\κ
Max
0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3
Case 1 Case 2 200 Case 3 Case 4 Case 1 Case 2 500 Case 3 Case 4
Min
1.14
1.43
1.87
2
1.14
1.43
1.87
2
47.48 75.52 4.16 4.20 51.28 57.96 31.64 31.20 85.52 94.40 6.84 7.76 84.48 90.20 68.23 64.76
49.28 76.12 2.80 2.78 51.98 58.60 32.04 30.88 86.84 94.72 4.60 3.56 85.16 90.60 69.64 64.20
52.36 76.56 2.42 1.26 52.76 59.02 32.68 28.92 87.44 95.48 2.72 1.20 86.08 91.36 70.68 65.40
53.30 76.84 2.26 0.48 53.04 59.64 33.72 28.56 88.04 96.16 2.36 0.88 86.40 91.80 71.72 65.32
3.84 2.64 49.76 73.96 46.36 49.86 33.20 36.36 6.08 5.08 88.40 94.84 87.88 88.16 75.20 77.28
2.92 1.56 52.36 74.60 47.84 45.88 34.16 37.04 3.62 2.36 89.28 95.24 88.16 89.36 76.16 78.52
2.12 0.96 57.28 75.68 50.76 45.48 34.68 39.32 2.52 0.92 90.16 96.28 88.20 90.24 76.68 79.16
2.68 0.90 58.64 76.80 51.72 46.52 35.88 40.60 2.76 0.78 91.68 96.40 89.52 90.40 77.88 79.44
Table 3 Empirical mean and standard deviation of change point estimators in Case 1 and 2. T
h
\κ
0.2
τˆ
0.3
τˆ
200
0.2 500 0.3
Case 1
se(τˆ ) se(τˆ )
τˆ
se(τˆ )
τˆ
se(τˆ )
Case 2
1.14
1.43
1.87
2
1.14
1.43
1.87
2
0.5291 0.1392 0.5119 0.1233 0.5224 0.1154 0.4996 0.1025
0.5407 0.1290 0.5211 0.1120 0.5272 0.1003 0.5094 0.0872
0.5468 0.1073 0.5304 0.0970 0.5261 0.0806 0.5122 0.0718
0.5494 0.1005 0.5283 0.0893 0.5234 0.0704 0.5107 0.0651
0.55576 0.0981 0.5843 0.0873 0.5574 0.0653 0.5768 0.0736
0.5436 0.0923 0.5661 0.0767 0.5431 0.0535 0.5612 0.0633
0.5299 0.0778 0.5474 0.0755 0.5343 0.0518 0.5483 0.0585
0.5248 0.0739 0.5385 0.0664 0.5282 0.0494 0.5455 0.0541
Table 4 Empirical mean and standard deviation of change point estimators in Case 3 and 4. T
h
0.2 200 0.3
0.2 500 0.3
\κ
Case 3
τˆ1 se(τˆ1 ) τˆ2 se(τˆ2 ) τˆ1 se(τˆ1 ) τˆ2 se(τˆ2 ) τˆ1 se(τˆ1 ) τˆ2 se(τˆ2 ) τˆ1 se(τˆ1 ) τˆ2 se(τˆ2 )
Case 4
1.14
1.43
1.87
2
1.14
1.43
1.87
2
0.4154 0.1478 0.5889 0.1810 0.3724 0.1026 0.6444 0.1047 0.3803 0.1178 0.6171 0.1421 0.3524 0.0794 0.6430 0.0837
0.3862 0.1342 0.6091 0.1696 0.3482 0.0842 0.6610 0.0953 0.3596 0.0970 0.6383 0.1279 0.3343 0.0624 0.6578 0.0710
0.3553 0.1083 0.6312 0.1476 0.3242 0.0675 0.6718 0.0791 0.3408 0.0741 0.6456 0.1026 0.3214 0.0482 0.6593 0.0601
0.3457 0.0940 0.6344 0.1438 0.3161 0.0641 0.6730 0.0732 0.3323 0.0559 0.6526 0.0854 0.3178 0.0481 0.6628 0.0581
0.3862 0.1534 0.6647 0.1518 0.3725 0.1511 0.6664 0.1436 0.3426 0.1153 0.6888 0.0948 0.3223 0.1035 0.6928 0.0884
0.3837 0.1437 0.6693 0.1207 0.3616 0.1299 0.6746 0.1159 0.3372 0.1001 0.6890 0.0695 0.3195 0.0889 0.6852 0.0701
0.3701 0.1160 0.6739 0.0884 0.3535 0.1083 0.6702 0.0885 0.3322 0.0806 0.6807 0.0557 0.3206 0.0712 0.6732 0.0576
0.3640 0.1091 0.6676 0.0762 0.3498 0.0989 0.6662 0.0832 0.3309 0.0778 0.6757 0.0476 0.3212 0.0659 0.6703 0.0570
To find the accuracy of our two step estimation method, we evaluate the empirical mean and the standard deviation of change point estimators. The results are shown in Table 3 (for Case 1 and Case 2) and Table 4 (for Case 3 and Case 4). It is not surprising that the estimations appear to converge to the true change points as sample size T or bandwidth h increases. Furthermore, the larger tail index, the smaller is the standard deviation. It is also not surprising for smaller tail index implies more ‘‘outliers ’’. An interesting finding is that the deviation increases as the tail index increases for I (0) to I (1) change, but it decreases as the tail index increases for I (1) to I (0) change. Next, we consider the properties of the test when the data-generating process (DGP) exhibits long-range dependences. The data are generated by the ARFIMA(0,d,0) process y t = r 0 + εt ,
(1 − B)d εt = et ,
t = 1, . . . , T ,
(10)
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Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316
Table 5 Empirical sizes with fractional integration innovations.
κ
T
h\ω
Max
0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3
1.14 1.43 200 1.87 2 1.14 1.43 500 1.87 2
Min
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
13.9 13.8 13.9 11.3 10.3 9.2 9.3 7.9 19.4 18.2 16.8 16.4 12.8 8.7 9.2 9.8
14.1 14.0 13.6 10.6 9.7 7.8 8.4 8.0 19.2 18.8 15.6 14.3 12.4 7.9 10.8 9.9
14.5 14.3 13.4 11.1 9.3 7.7 7.9 8.1 19.2 19.3 15.8 14.9 11.2 8.2 8.8 8.9
15.5 1.35 13.6 11.5 9.3 7.4 8.3 7.1 18.4 19.2 15.2 15.6 10.8 8.4 8.6 7.4
13.8 10.7 14.0 9.4 9.4 8.3 11.2 6.8 14.0 10.3 12.3 8.4 9.6 8.9 10.0 9.2
13.6 10.5 13.4 8.7 9.6 8.4 11.0 7.1 13.6 10.7 11.2 7.8 9.2 8.6 9.6 9.2
13.0 10.0 13.3 8.1 9.5 8.7 10.2 6.1 13.6 11.1 10.8 8.1 9.2 8.2 9.5 8.6
12.8 10.2 12.7 8.5 9.1 7.7 10.0 6.2 12.4 11.2 11.2 7.5 8.6 9.1 8.4 7.7
Table 6 Empirical powers with fractional integration innovations. h\κ
T
I 200 II I 500 II
0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3
Max
Min
1.14
1.43
1.87
2
1.14
1.43
1.87
2
67.32 51.60 19.72 19.40 85.68 71.04 30.86 31.20
64.12 49.44 18.20 19.28 84.64 66.44 24.44 27.36
58.96 46.32 13.76 15.40 79.82 62.60 21.48 24.02
59.76 43.88 15.24 15.04 79.38 58.88 19.82 22.96
73.52 62.28 19.12 12.64 88.06 77.64 31.86 20.94
72.84 60.88 16.44 12.36 87.62 78.78 24.84 17.48
69.68 59.80 13.20 10.08 86.14 74.96 21.26 13.96
68.56 60.60 13.96 10.72 86.08 76.20 17.88 15.08
where r0 = 0.1, and the innovation process {et } is treated the same as GDP (9). Since the range of permissible long memory parameters for a given tail index κ is restricted to the interval 0 < d < 1 − 1κ (see e.g. Kokoszka and Taqqu, 1996). We set the long memory parameter d = ω(1 − 1/κ), and vary ω among {0.2, 0.4, 0.6, 0.8}. Similar to Kokoszka and Taqqu (1996) we approximate the innovation εt by
εt =
1000
cj et −j ,
t = 1, . . . , T ,
j =1
in which c0 = 1, cj+1 = j+1 cj . Table 5 presents the simulated empirical sizes. It can be seen that the size distortions are not negligible under the fractional integration case especially for small κ . The long memory parameter d also has some influence to the empirical size. This indicates that our test procedure does not robust enough in such an innovation case. After evaluating the size performance of the test, we consider the power performance. We use the same model as DGP (10), allowing changes in long memory parameter d as the following two different change cases. j+d
I: d changes at τ1 = 0.3 from 0 to 0.5(1 − 1/κ), and changes from 0.5(1 − 1/κ) to 0 at τ2 = 0.65. Case I is an I (0)− I (d)− I (0) persistence change profile. II: d changes at τ = 0.3 from 1 to 0.5(1 − 1/κ), and changes from 0.5(1 − 1/κ) to 1 at τ2 = 0.65. Case II is an I (1)− I (d)− I (1) persistence change profile. As we can seen from Table 6 that our test performs well if the process changes from short to long memory, or vice versa (see the case I). It is interesting to note that our test has better power property when using a bandwidth of h = 0.2 than using h = 0.3. In addition, the smaller the tail index, the higher the empirical power. This finding is quite different from the conclusion obtained in the short memory case. However, this outcome is somewhat artificial, the reason for this phenomenon lying in the size distortion. Since the power, as the short memory case, is growing with sample size T , we guess that our test is still consistent for this change profile. Unfortunately, although the test seems still sensitive if the process changes between I (1) and fractional integration, the empirical power is poor. Finally, we use a small simulation study to check whether our test is robust when there is a mean change point in DGP. Table 7 reports the results when r0 in DGP (10) changes at τ = 0.5 from 0 to 1. The bandwidth was set to be h = 0.2. It can be seen that the empirical size is growing with decreasing long memory parameter or increasing tail index. In particular,
Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316
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Table 7 Empirical sizes when r0 in DGP (21) changes at τ = 0.5 from 0 to 1. T
200
500
κ \d 1.14 1.43 1.87 2 1.14 1.43 1.87 2
Max
Min
0
0.5(1 − 1/κ)
1
0
0.5(1 − 1/κ)
1
7.32 10.34 15.48 32.86 6.0 12.86 36.28 68.08
13.48 12.52 10.26 8.60 21.42 18.96 13.64 8.46
12.26 10.0 7.14 6.02 13.70 7.72 3.72 5.12
8.86 9.40 14.72 32.06 11.44 11.72 29.04 68.86
13.68 10.98 11.60 12.06 15.14 13.36 8.74 7.58
10.74 10.68 7.0 6.08 13.72 10.08 5.42 5.48
14 12 10 8 6 4 2 0 2
0
50
100
150
200
250
300
Fig. 1. US inflation rate data from May 1952 to April 1977.
the test suffers from serious size distortions if d = 0. This indicates that our test is not robust if there are structural breaks in the trend function. In conclusion, our test performs well in the short memory case. Moreover, although our test is based on I (0) null hypothesis, the empirical size is still controlled well when a series is actually I (1) throughout. However, our test is not robust enough and powerful when the series is long memory or contains structural breaks in the trend function. In these cases, a more robust and powerful test is necessary. A small bandwidth could capture short range changes while it has lower power for the small sample size. An ideal way is that a larger bandwidth is chosen for a smaller sample size, and a smaller one for a larger sample size. Although the test with bandwidth h = 0.3 performs a little better than the test with bandwidth h = 0.2, it has poor performance if the location of the change point does not lie in the interval (0.3, 0.7). Further, a large bandwidth leads to the estimation procedure having poor accuracy. Consequently, we suggest to use bandwidth h = 0.2 if the sample size is not too large or too small. In most cases, h = 0.2 is a reasonable choice to control the empirical size well and to obtain a satisfactory empirical power. 4.2. Empirical application In this section, we illustrate our test using two sets of financial time series data. The first set contains the US inflation rate monthly data which was observed from May 1952 until April 1977 with samples of 300 observations. The second set contains the stock closing price data of BOE (BOE Technology Group Co., Ltd.) in China stock market observed every five minutes from Oct. 26, 2010 to Nov. 9, 2010 with samples of 528 observations. We apply our test analyze these two data sets by setting N = 4T / log(T ), B = 500, h = 0.2, and nominal level α = 5%. Fig. 1 reports the observations in the first data set. Chen et al. (2010) has considered this data set using a sequential monitoring procedure, and concluded that the data contains an I (1) to I (0) change point before October 1961, and an I (0) to I (1) change before April 1966. We apply our test procedure to check this conclusion. The simulated upper tail and lower tail critical values by the bootstrap method are 21.469 and 0.0492 respectively. The maximum and minimum values of statistic RTh (τ ) in interval τ ∈ [0.2, 0.8] are 22.643 and 0.0481 respectively. This indicates that the data undergoes both I (0) to I (1) and I (1) to I (0) changes. The estimated I (0) to I (1) change period is May 1965 (see the right vertical line in Fig. 1), and the estimated I (1) to I (0) change period is July 1959 (see the left vertical line in Fig. 1). Then, by continued searching the maximum and minimum value of statistic RTh (τ ) in the residual interval τ ∈ [0.72, 0.8], we found that the maximum value does not exceed the upper tail critical value and the minimum value does not exceed the lower tail critical value. This implies that there does not exist any other change point in persistence. This conclusion coincided with Chen et al. (2010). Now, we consider the second data set. The observations of this data set are shown in Fig. 2. We can see that there are many ‘‘outliers’’. Shi et al. (2008) has argued that it is more suitable to fit such high frequency data by a stable distribution with tail index 1 < κ < 2 than a normal distribution. We employed the method of Mandelbrot (1963) to estimate the tail index κ and obtained κ = 1.86. The simulated upper tail and lower tail critical values by the bootstrap method are 23.848 and 0.0414 respectively. We found that the statistic RTh (τ ) has the maximum value 30.6805 at observation 415, and the minimum value 0.0363 is obtained at observation 301. Therefore, we can conclude that the data undergoes a change from
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Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316 3.85 3.8 3.75 3.7 3.65 3.6 3.55 3.5 3.45 3.4 3.35
0
100
200
300
400
500
600
Fig. 2. BOE stock closing price data observed every five minutes from Oct. 26, 2010 to Nov. 9, 2010.
I (1) to I (0) at observation 301, and a change from I (0) to I (1) at observation 415. Since the maximum and minimum values of statistic RTh (τ ) in the residual interval [0.2, 0.38] do not exceed the upper tail critical value and the lower tail critical value respectively, we say that there just only exist two change points in persistence. In this example, we also considered h = 0.3 case. However, we just only captured an I (1) to I (0) change point at observation 301. The reason that it failed to find the I (0) to I (1) change point is that the sample sizes 114 between two change points are smaller than the subsample size [Th] = 158. 5. Conclusions In this paper, we have proposed a moving ratio procedure to test and estimate multiple changes in persistence for a heavy-tailed sequence. We derived the asymptotic distribution of the moving ratio statistic which is a complicated function of the Lévy process under the I (0) null hypothesis. However, the asymptotic distribution depends on the unknown tail index κ . To overcome the problem, an approach based on residual bootstrap is proposed. Unlike the asymptotic test, the bootstrap procedure does not require any knowledge of the often very complex asymptotic distribution. We have also proposed a two step procedure to estimate the change points in persistence. Neither the direction of change nor the number of change points needs to be assumed known. Numerical results showed that the moving ratio test based on residual bootstrap constitutes a functional tool for detecting and locating multiple changes in persistence in short memory heavy-tailed sequence. In addition, the proposed test displays no tendency to spuriously over-reject I (0) null in favor of a change in persistence when a series is actually I (1) throughout. Finally, we illustrated our test using the US inflation rate data and a set of high frequency stock closing price data. Since the test has poor performance when the DGP exhibits long memory or structural breaks in the trend function, a more powerful and robust test can be more interesting for further study. 6. Mathematical proofs Proof of Theorem 3.1. Let t = [Tr ], then Lemma 2.1 gives that, if dt = 0, 1 a− T
[Tr ] i=[T τ ]+1
1 a− T
[Tr ]
1 uˆ 1,i = a− T
εi ⇒ ψ∞ (U1 (r ) − U1 (τ )) ≡ ψ∞ V1,1 (r )
(11)
i=[T τ ]+1
[Tr ]
[Tr ]
1 uˆ 0,i = a− T
i=[T τ ]−[Th]+1
εi ⇒ ψ∞ (U1 (r ) − U1 (τ − h)) ≡ ψ∞ V1,0 (r )
(12)
i=[T τ ]−[Th]+1
2 2 a− ˆ02 = a− T [Th]γ T
[T τ ]
εi2 ⇒ Ψ22 (U2 (τ ) − U2 (τ − h))
(13)
i=[T τ ]−[Th]+1 2 2 a− ˆ12 = a− T [Th]γ T
[T τ ]+[Th]
εi2 ⇒ Ψ22 (U2 (τ + h) − U2 (τ )).
(14)
i=[T τ ]+1
In which ψ∞ = −1
aT
∞
[Tr ] i=[T τ ]+1
j =0
ϕj , Ψ2 denotes l2 norm of sequence {ϕj }, and ϕj = ρ j . If dt = 1, −1
uˆ 1,i = aT
[Tr ] i=[T τ ]+1
εi − [Th]
−1
[Tr ]
[T τ ]+[Th]
εj
i=[T τ ]+1 j=[T τ ]+1 h−1 U 1
⇒ ψ∞ (U1 (r ) − U1 (τ ) − (r − τ ) ≡ ψ∞ V2,1 (r )
( (τ + h) − U1 (τ ))) (15)
Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316
[Tr ]
1 a− T
[Tr ]
1 uˆ 0,i = a− T
⇒ ψ∞ (U1 (r ) − U1 (τ − h) − (r − τ + ) ≡ ψ∞ V2,0 (r ) 2 [T τ ] [T τ ] 1 −2 −2 2 εi − = aT εj a [Th] T j=[T τ ]−[Th]+1 i=[T τ ]−[Th]+1
2 a− ˆ02 T [Th]γ
εj
i=[T τ ]−[Th]+1 j=[T τ ]−[Th]+1 h h−1 U 1 U1
i=[T τ ]−[Th]+1
i=[T τ ]−[Th]+1
[T τ ]
[Tr ]
εi − [Th]−1
( (τ ) −
2313
(τ − h))) (16)
⇒ Ψ22 (U2 (τ ) − U2 (τ − h)) 2 2 a− ˆ12 = a− T [Th]γ T
[T τ ]+[Th]
εi2 −
i=[T τ ]+1
1
[Th]
(17)
2 a− T
[T τ ]+[Th]
2 εj
j=[T τ ]+1
⇒ Ψ22 (U2 (τ + h) − U2 (τ )).
(18)
If dt = (1, t )T , let δ = (α, β)T , then by the definition of LS we have
αˆ − α βˆ − β
−1
=
t
t
t2
εt
t εt
,
[T τ ]
where = t =[T τ ]−[Th]+1 , if we estimate δ using the samples y[T τ ]−[Th]+1 , . . . , y[T τ ] , and using the samples y[T τ ]+1 , . . . , y[T τ ]+[Th] . Hence, by a tedious calculation we have
1 a− T
[Tr ]
[Tr ]
1 uˆ 1,i = a− T
i=[T τ ]+1
εi − (δˆ0 − δ0 ) − (δˆ1 − δ1 )i
=
[T τ ]+[Th]
t =[T τ ]+1 ,
if we estimate δ
i=[T τ ]+1
⇒ ψ∞ U1 (r ) − U1 (τ ) − K1−1 (r − τ ){4((τ + h)3 − τ 3 ) − 6h(2τ + h) + 3h(2 − 2τ − h)(τ + r )}(U1 (τ + h) − U1 (τ )) + 6K1 h(r − τ )(r − τ − h) −1
≡ ψ∞ V3,1 (r ),
τ +h
τ
U1 (s)ds (19)
with K1 = 4h2 (3τ 2 + 3τ h + h2 ) − 3h2 (2τ + h)2 . Similar arguments gives that 1 a− T
[Tr ]
uˆ 0,i ⇒ ψ∞ U1 (r ) − U1 (τ ) − K2−1 (r − τ + h){4(τ 3 − (τ − h)3 )
i=[T τ ]−[Th]+1
− 6h(2τ − h) + 3h(2 − 2τ + h)(r + τ − h)}(U1 (τ ) − U1 (τ − h)) τ −1 + 6K2 h(r − τ )(r − τ + h) U1 (s)ds τ −h
≡ ψ∞ V3,0 (r ),
(20)
with K2 = 4h2 (3τ 2 − 3τ h + h2 ) − 3h2 (2τ − h)2 . According to the above proof we can see that αˆ − α = Op (aT T −1 ), and βˆ − β = Op (aT T −2 ). Therefore 2 2 a− ˆ02 = a− T [Th]γ T
[T τ ]
εi2 + Op (T −1 ) ⇒ Ψ22 (U2 (τ ) − U2 (τ − h)).
(21)
i=[T τ ]−[Th]+1 2 2 a− ˆ12 = a− T [Th]γ T
[T τ ]+[Th] i=[T τ ]+1
εi2 + Op (T −1 ) ⇒ Ψ22 (U2 (τ + h) − U2 (τ )).
(22)
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Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316
Combining (11)–(22), Theorem 3.1 follows immediately from the continuous mapping theorem and continuity of the functionals. For the remainder of this section we omit proofs for the dt = (1, t )T case; these are straightforward but tedious and follow the same logical development as those presented for the dt = 1 case. Proof of Theorem 3.2. Since we have [Th] samples at least between two persistence change points, then for any such change point τ0 , it will not have the same direction persistence change point as τ0 in interval Ω = (τ0 − 2h, τ0 + 2h) ∩ [0, 1]. According to the test procedure, we just only need to prove P { lim max ΞTh (τ ) > c2 } = 1, T →∞ τ ∈Ω
if the direction of change point τ0 is I (0) to I (1), and P { lim min ΞTh (τ ) < c1 } = 1, T →∞ τ ∈Ω
if the direction of change point τ0 is I (1) to I (0). Suppose that the I (1) sequence yt changes to I (0) at τ0 , then Lemma 2.1 gives that, if τ − h < τ0 ≤ τ , t
uˆ 1,i = Op (aT );
γˆ02 (τ ) = Op (a2T );
i=[T τ ]+1 t
uˆ 0,i = Op (TaT );
γˆ12 (τ ) = Op (T −1 a2T ).
i=[T τ ]−[Th]+1
These results combining with the continuous mapping theorem indicate that the numerator of statistic RTh (τ ) is
γˆ (τ ) 2 0
[T τ ]+[Th]
t =[T τ ]+1
t
2 = Op (Ta4T ),
uˆ 1,i
i=[T τ ]+1
the denominator [T τ ]
γˆ (τ ) 2 1
t =[T τ ]−[Th]+1
2
t
= Op (T 2 a4T ).
uˆ 0,i
i=[T τ ]−[Th]+1
Hence RTh (τ ) = Op (T −1 ).
(23)
Similarly, when yt changes at τ0 from I (0) to I (1), then if τ ≤ τ0 < τ + h, t
uˆ 1,i = Op (TaT );
γˆ02 (τ ) = Op (T −1 a2T );
i=[T τ0 ]+1 t
uˆ 0,i = Op (aT );
γˆ12 (τ ) = Op (a2T ).
i=[T τ ]−[Th]+1
These results combining with the continuous mapping theorem indicate that the numerator of statistic RTh (τ ) is
γˆ (τ ) 2 0
[T τ ]+[Th] t =[T τ ]+1
t
2 uˆ 1,i
= Op (T 2 a4T ),
i=[T τ ]+1
the denominator
γˆ (τ ) 2 1
[T τ ] t =[T τ ]−[Th]+1
t
2 uˆ 0,i
= Op (Ta4T ).
i=[T τ ]−[Th]+1
Hence RTh (τ ) = Op (T ). From (23) and (24), we establish Theorem 3.2.
(24)
Z. Chen et al. / Computational Statistics and Data Analysis 56 (2012) 2303–2316
2315
Proof of Theorem 3.3. Observing that when we select one of ε1∗ , . . . , εT∗ , we select the corresponding unobservable noise variable which will be denoted as ε i . This means that [Nt ] 1
aN i = 1
[Nt ] 1
ε˜ i =
aN i=1
εi −
T [Nt ]
TaN i=1
εi .
By (3) and Assumption 3.1 we have T [Nt ]
TaN i=1
εi =
1−1/κ−δ T T [Nt ] T 1/κ L(N ) 1 N 1 ε ≤ 2 εi = op (1). i 1 /κ T N L(N ) aT i=1 T aT i=1
The above inequality follows from L(T )/L(N ) ≤ 2(T /N )δ (cf. Theorem 1.5.6 of Bigham et al., 1987), δ > 0 be chosen so small that 1 − 1/κ − δ > 0. Horváth and Kokoszka (2003) showed that for any bounded continuous functional g on D[0, 1]
Pϵ
−1
g
aM
[Mt ]
εi
p
≤ x −→ P (g (U1 (t )) ≤ x).
(25)
i=1
Hence [Nt ] 1
aN i = 1
ε˜ i ⇒ U1 (t ).
(26)
On the other hand [Nt ] 1
a2N i=1
[Nt ] 1
ε˜ = 2 i
a2N i=1 [Nt ]
1
=
a2N i=1
ε − 2 i
2
Ta2N
ε 2i + Op (1)
[Nt ]
2 N T a2T 1 εi + 2 2 εi εi
a2T T 2 a2N
T aN
i =1
i=1
−
2aT
TaN
aT i = 1
.
It is easy to see that aT
lim
T →∞
2
≤ lim
TaN
T →∞
N
1−1/κ−δ N T
= 0.
Using the same proof line of assertion (25) we have
Pϵ
g
−2
aN
[Nt ]
ε
2 i
p
≤ x −→ P (g (U2 (t )) ≤ x).
(27)
i=1
Consequently, [Nt ] 1
a2N i=1
ε˜ i2 ⇒ U2 (t ).
(28)
According to (26) and (28), we can complete the proof of Theorem 3.3 by the same proof line of Theorem 3.1, we omit to report here. Proof of Theorem 3.4. Recall that the residual εˆ i = Op (aT ) if the corresponding observation yi ∼ I (1). Consequently, whether yi is I (0) or I (1), the corresponding centered residual εi∗ = Op (aT ). From the construction of the bootstrap step we can see that the bootstrap process y˜ i ∼ I (1), i = 1, . . . , N. Therefore, for any τ ∈ (0, 1), the residuals uˆ ∗1,i = Op (aN ) = uˆ ∗0,i . Continuous mapping theorem gives that both the denominator and the numerator of statistic R˜ Nh (τ ) is Op (N 3 a4N ). This completes the proof of Theorem 3.4. Acknowledgments We are grateful to the associate editor and two anonymous referees for their detailed comments and valuable suggestions. In addition, we would like to thank Prof. John Nolan who provided the software for generating the stable innovations in Section 4. This work was supported by the National Natural Science Foundation of China (No. 60972150, No. 10926197 and No. 61065009).
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