The Monitoring Structural Change Tests viaCUSQ Tests for Long Memory Processes

Shin-Huei Wang † and Cheng Hsiao††

December 31, 2008

Abstract

This paper considers the monitoring CUSUM of squares (CUSQ)-type tests via an au-

toregressive (AR) spectral estimators when data generating process (DGP)is a long memory

processes.Under suitable regularity conditions, we show that the limiting distribution of

CUSQ -type test follows Brownian bridge and is free of long memory parameters for both

stationary and non-stationary ARFIMA (p, d, q). The test can detect one or more change-

points of an ARFIMA process, even though the exact order of the ARFIMA(p, d, q) process

is unknown. The spurious breaks considered in Kuan and Hsu (1998) and Granger and

Hyung (2004) will not arise in our tests. Neither does our test need to use the bootstrap

procedure to deliver a very promising size performance.Our monitoring test can also be used

to test for stationary short memory processes (I(0)process) against a change to stationary

or non-stationary long memory processes or to test for a change in the order of integration

of a time series either from I(d′) to I(1 + d′) or from I(1 + d′) to I(d′), d ∈ (−0.5, 0.5)

without known direction. Monte Carlo simulations confirm these theoretical findings. Fi-

nally, we applied our procedure to monitor the real time pattern of volatilities of three daily

nominal dollar exchange rates-Euro, Japanese Yen and British Pound covering the current

US subprime crisis.

Keywords: CUSUM of Squares Tests; Long Memory; Structural Change

† CORE, Universite Catholique de Louvain, and CEREFIM,University of Namur, Belgium.

† Email:shin-huei.wang@uclouvain.be; Tel: (32)-10-47-4329†† Department of Economics, University of Southern California, U.S.A.,

City University of Hong Kong and Wang Yanan Institute for Studiesin Economics, Xiamen University, China.

† Corresponding Author, E-Mail: shin-huei.wang@uclouvain.be

1

1. Introduction

Structural breaks tests often misleadingly infer the presence of spurious breaks. This

paper follows the pioneering work of Chu et al. (1996) to check whether the incoming data

are consistent with the estimated model based on the monitoring CUSUM of squares tests

(CUSQ) (Brown, Burbin and Evans,1975), when the underlying process is a long memory

process. Monitoring approaches have been widely used in statistical quality control for a

long time. In contrast to the classical one-shot tests to detect a structural break within the

data span (e.g. Ploberger et al., 1989; Hansen, 1992; Andrews, 1993), Chu et al. (1996)

provide an online procedure to monitor structural breaks that has asymptotic size close to

actual size.

Some existing structural break tests for long memory processes could have severe size dis-

tortion hence misleadingly pointing to structural breaks. For instance, Kuan and Hsu (1998)

show that the statistics of Hildago and Robinson (1996) could have large size distortions even

though their tests are designed specifically for autoregressive fractionally integrated moving

average process of order p,d,q, denoted as ARFIMA (p, d, q) or I(d) process. Extending the

results in Bai (1994) and Nunes et al. (1995), they show that many well-known structural-

change tests for the stationary long memory process may suggest a change has occurred,

even though there is no change at all. As well, with the current break estimation methods,

Granger and Hyung (2004) pointed out as the degree of integration of DGP increases, more

breaks are inferred in a finite sample.

We develop the new monitoring CUSUM of squares test via autoregressive spectral es-

timators, namely, MCUSQAR. Following the idea of Perron and Ng(1998), Poskitt (2007)

and Wang and Hsiao (2008), we first approximate a long memory process by an AR model,

then construct the CUSUM -type test statistics. We show that the limiting distribution of

MCUSQAR follows a Brownian bridge which is free of the long memory parameters. The

monitoring CUSQ tests based on the AR spectral estimators are easy to implement, even

though the exact order of the ARFIMA(p, d, q) process is unknown. Neither the issue of

spurious breaks for long memory processes will arise, nor our MCUSQAR tests need to rely

on the bootstrap procedure to deliver a very promising size performance. Further, our moni-

toring test can be used as a test for stationary short memory processes (I(0) process) against

a change in stationary or non-stationary long memory processes as well. Our theoretical

findings also are supported through Monte Carlo experiments clearly.

The outline of this paper is as follows. Section 2 presents the model and the test statistics

for stationary long memory processes. Section 3 shows the test statistics for non-stationary

long memory case. In section 4, we evaluate the size and power of our method by Monte

Carlo experiments. Section 5 illustrates the empirical usefulness of our monitoring tests.

Concluding remarks are in Section 6.

2

2. The Model and the Test

In this section, we suggest two monitoring CUSUM -based tests under long memory

processes via AR approximation. We first define the underlying process then give the details

of the proposed detecting test statistics.

2.1. Autoregressive Spectral Estimators under Long Memory Processes

In this subsection, we establish the autoregressive spectral estimators for the long mem-

ory process.

Suppose the DGP ηt satisfies the following Assumption 1.

Assumption 1. ηt is generated as:

φ(L)(1− L)dηt = θ(L)et, (1)

where (i) d ∈ (−0.5, 0.5); (ii) φ(L), and θ(L) are finite degree polynomials, and the zeroes

of φ(L), and θ(L) all lie outside the unit circle; (iii) φ(L) and θ(L) have no common zeroes;

(iv) et is an independently and identically distributed process, with E(et) = 0, E(e2t ) = σ2e ,

and E(e4t

)<∞ (v) σ2

t = limT→∞T−1E(η2

t ).

Assumption 1 guarantees that the conditions in Theorem 3 of Hosking (1996) hold, and

allows us to represent an ARFIMA process ηt as:

ηt =∞∑

j=0

ψjet−j , where ψj = O(jd−1

)as j →∞, (2)

or

ηt =∞∑

j=1

βjηt−j + et, where βj = O(j−d−1

)as j →∞. (3)

Hosking (1996) also show that a stationary and invertible ARFIMA process with d 6= 0 has

an autocovariance function that satisfies

γj ∼σ2fη(0)Γ(1− 2d)

Γ(d)Γ(1− d)j2d−1, as j →∞,

where Γ(.) is the Gamma function, and

fη(0) =(1 + θ1 + . . .+ θq)

2

(1− φ1 − . . .− φp)2 .

Therefore, the presence of the short memory parameters (φ’s and θ’s) does not have much

impact on the asymptotic behavior of the autocovariance functions of an ARFIMA(p, d, q)

process.

3

When ηt is a stationary short memory process, based on the analysis of Berk (1974),

Perron and Ng (1998) proposed the autoregressive spectral estimator of σ2η = limT→∞E(η2

t )

(long run variance), defined by

s2AR = s2ek/(1− b(1))2, (4)

where b(1) =∑k

j=1 bj and s2ek = T−1∑T

t=k+1 e2tk with bj and etk obtained from the following

autoregression:

ηt =

k∑j=1

bj ηt−j + etk. (5)

where

η = yt − µt.

When the underlying process is a long memory process, by the Theorem 5.1 of Poskitt

(2007), we can also construct s2AR by the autoregression (5). Thus, before presenting the

main results of our monitoring tests, we first establish the following Theorem 1.

THEOREM 1. Given that all the conditions in Theorem 5.1 of Poskitt (2007) hold, then

the exact order of magnitude of s2AR is Op(k2d) when d ∈ (−0.5, 0.5), where S2

AR is defined

as in (4) when the underlying process is I(d)

Theorem 1 shows that s2ARp−→ ∞ as d > 0 and s2AR

p−→ 0 as d < 0. This pattern is

identical to those of the population long run variance of an I(d) process, see Lee and Schmidt

(1996). Moreover, the convergence (or divergence) rate of s2AR is different from those of the

alternative long run variance estimator Ωl of Newey and West (1987) and Andrews (1991),

where l denotes the bandwidth parameter of the kernel function and is assumed to increase

with the sample size. In fact, under very mild restrictions on the kernel functions, Tsay (1999)

proved that the exact order of magnitude of Ωl is Op(l2d) when the data generating process

is an I(d) process. Moreover, because the asymptotic properties of s2AR depend on the lag

length k, while those of Ωl depend on the bandwidth parameter l, the power performance of

many existing unit root tests and stationarity tests under fractional alternatives definitely

depend on which type of long run variance estimator is chosen. Therefore, Theorem 1 helps

us to establish the test statistics to monitor shifts from I(0) to I(d) and vice versa or changes

from I(d1) to I(d2) and vice versa when d1, d2 ∈ (−0.5, 0.5). Those will be discussed in the

following sections.

2.2 The Monitoring Procedure

Most tests in the literature regarding change point problems are retrospective tests, i.e.,

given a set of observations, those tests decide if a change has occurred within the time span of

the data (Andrews, 1993 ; Inclan and Tiao, 1994). However, as noted in Chu et, al (1996),

repeating a retrospective test of structural change each time as we get new observations

4

leads to the rejection of the hypothesis of constancy with a large probability, even if no

break occurs actually. Accordingly, such a procedure will have poor size property and often

result in misleadingly suggesting spurious breaks.

One possible way to tackle this preceding problem is to use sequential updating proce-

dure to develop tests of structural stability for the real time monitoring of economic system.

As described in Chu et, al (1996), given a initial fixed time date m, the monitoring scheme

needs a stopping device, determined by a detecting statistics (detector),Tn and a boundary

function, g(n/m) to monitor whether the breaks occur after the data being increased to

n(n ≥ m). More precisely, under the hypothesis of no break, Tn may cross a boundary

function g(n/m), for some n ≥ m, with certain probability, say 0.05 0.10. On the other

hand, if a break indeed occurs, we expect Tn to cross g(n/m) with a large probability. Op-

erationally, the null hypothesis is rejected when |Tn| ≥ g(n,m) for some n > m. Conversely,

the monitoring process keeps us running until a break is observed.

2.3. MCUSQAR Tests

We consider the following model

yt = µ+ ηt (6)

where µ = E(yt), σ2t = E(ηt)

2, ηt is an I(d) process satisfying Assumption 1.

We assume

ASSUMPTION 2. E(yt) = µt = µ0, ηt = I(d) = I(d0) and σ2t = σ2

0 for t = 1, 2, . . . ,m.

Assumption 2 implies that the mean of yt, the long memory parameter d or the variance

of yt are stable over historical period from time 1 to time m. We consider two types of

break. The first is a shift in the mean. Therefore, the null hypothesis tested for DGP (6)

throughout this paper can be represented as:

H0 : µt = µ0 for t = m+ 1,m+ 2, . . .

versus the alternative

H1 : µt = µ∗ 6= µ0 for some t ≥ m+ 1 .

Under the assumption 2, Kuan and Hsu (1998) shows that many structural break tests will

lead to spurious break when the DGP is a stationary long memory process.

The second type of break we consider is a change in variance or long memory parameter

d. Then the null hypothesis tested for DGP (6) is

H∗0 : ηt = I(d) = I(d0) and σ2

t = σ20 for t = m+ 1,m+ 2, . . . ,

5

versus the alternative

H∗1 : σ2

t 6= σ20 or ηt = I(d) 6= I(d0) for some t ≥ m+ 1

Testing for H∗0 of the ” one-shot ” type was suggested by Beran and Terrin (1996).

Nevertheless, Beran and Terran’s test for structural changes cannot be applied to monitor

out of sample stability when new data arrive.

We now illustrate the implementation of the on-line CUSUM of squares tests based on

an AR approximation. We propose two detectors, Tn and T ∗n , as CUSUM of squares test

originally proposed by Brown, Durbin and Evans (1975). Inclan and Tiao (1994) suggest

using a centered version of this statistics for a retrospective test on a change of the variance

of an i.i.d sequence of centered random variable. Thus, we shall follow them to use a centered

version for our monitoring test. Let Cl =∑l

t=1 e2t be the cumulative sum of squares of a

series of uncorrelated random variable et with mean 0 and variance σ2e,t, t = 1, · · · , T and

denote the centered (for n ≤ m) cumulative sum of squares as

Dn =Cn

Cm− n− 2

m− 2.

The process Sn = 2−1/2mDn is equal to 2−1/2∑n

t=1

(e2t

σ2e,t

− 1

)− 21/2(n−m)/(m− 2),

where σ2e,t is the estimator of σ2 based on historical dataset , e1, e2, · · · , em, that is σ2

e,t =

m−1∑m

t=k+1 e2t .

Following Lemma 2.1, Theorem 5.1, Theorem 5.2 and Lemma 5.7 of Poskitt (2007),

we note that we can approximate ηt by an AR(k) model, ηt =∑k

j=1 bj ηt−j + etk. Let

etk = et +∑∞

j=k+1 bjηt−j −∑k

i=1(bj − bj)ηt−j and σ2tk = (T − k)−1

∑Tt=k+1 e

2tk = (T −

k)−1∑T

t=k+1 e2t +op((

kλmin(Γh))(

logTT )1−2d′

) = (T−k)−1∑T

t=k+1 e2t +op(1), where d′ = max

0, d. Further define

Sn = 2−1/2mDn,

it follows that

m−1/2Sn = 2−1/2m−1/2n∑

i=k+1

(e2iσ2

tk

− m

m− 2

)= m−1/2Sn + op(1). (7)

From the Proposition 6.1.2 and item (iii) of the Definition 6.1.4 of Brockwell and Davis

(1991), we note that

m−1/2Snp−→ m−1/2Sn.

As shown by Theorem 5.1 and Theorem 5.2 of Poskitt (2007), the residuals of an AR(k)

approximation of long memory processes are i.i.d. Hence, according to Theorem 1 of Inclan

and Tiao (1994), we know that under H0,

Tn = m−1/2S[mt] =⇒ W 0(t).

6

where ” =⇒ ” denotes weak convergence and where W 0 = W (t)− tW (1), with W being a

standard Wiener Process.

By the same reasoning as Chu et al. (1996), the crossing probability of W 0(t), t ≥ 1 can

be investigated in terms of the Wiener process. We choose g(t) = [t(a2 + ln(t))]1/2 to be

our boundary function, then as shown in (13) in Chu et al. (1996),

P|W 0(t)| = (t− 1)−1|W (t)| ≥ t(a2 + ln(t))]1/2

= 2[1−Φ(a) + aΦ(a)] for some t ≥ 1.(8)

When a2 = 7.78, the crossing probabilities is 5%. It is this approximation which we will

use in our simulation and empirical work below. Consequently, the asymptotic properties of

the detecting test Tn can be established and summarized in Theorem 2.

THEOREM 2. Let ηt and yt be the processes given by (1) and (6). Under the null

hypothesis of Assumption 2, as k, T → ∞, the following functional central limit (FCLT)

holds for MCUSQAR test Tn:

Tn ⇒ W 0(t) = W (t)− tW (1)

where W and W 0 are the (one-dimensional) Brownian motion and Brownian bridge such as

limm→∞P|Tn| ≥ g(t), for some t ≥ 1 = 2[1−Φ(a) + aΦ(a)]. (9)

The second detector we propose is T ∗n , which is based on the long run variance estimators

s2AR proposed by Perron and Ng (1998). We rewrite Cl as

C∗l =

∑lt=1 e

2t

(1−∑k

i=1 bi)2.

The process S∗n = 2−1/2mD∗n is equal to∑n

i=1

(e2i /(1−

∑k

i=1bi)

2

σ2/(1−∑k

i=1bi)2

− 1

)− 2(n − m)/(m −

2), where σ2et is the estimator of σ2 based on historical dataset , e1, e2, · · · , em, that is

σ2e,t = m−1

∑mt=k+1 e

2t . Denote S∗n = 2−1/2mD∗n. Following the arguments of Theorem 5.1,

Theorem 5.2 and Lemma 5.7 of Poskitt (2007), we can show that

m1/2S∗n = 2−1/2m−1/2n∑

i=k+1

e2i

(1−∑k

i=1bi)2

σ2tk

(1−∑k

i=1bi)2

− m

m− 2

= m−1/2S∗n + op(1)

Again, following the results of Inclan and Tiao (1994), under H0,

T ∗n = m−1/2S∗[mt] =⇒ W 0(t).

The asymptotic properties of the detecting test T ∗n is summarized in Theorem 3.

7

THEOREM 3. Let ηt and yt be the processes given by (1) and (6). Under the null

hypothesis of Assumption 2, as k, T → ∞, the following functional central limit (FCLT)

holds for MCUSQAR test T ∗n :

T ∗n ⇒ W 0(t) = W (t)− tW (1)

where W and W 0 are the (one-dimensional) Brownian motion and Brownian bridge such as

limm→∞P|T ∗n | ≥ g(t), for some t ≥ 1 = 2[1−Φ(a) + aΦ(a)]. (10)

3. Non-stationary Long Memory Processes

In this section, we analyze the limiting distribution of the MCUSQAR test when the

time series yt are the non-stationary long memory processes. We specify the data generating

process as:

yt = yt−1 + ηt. (11)

.

When ηt is the stationary short memory process and yt is unit root, Perron and Ng

(1996, 1998) suggest the following long run variance estimator

s2AR∗ = s2ek∗/(1− b(1)∗)2 (12)

that is from the following autoregression estimated by OLS:

4yt = b∗0yt−1 +k∑

j=1

b∗j4yt−j + etk

where s2ek∗ = T−1∑T

t=k+1 e2tk, b(1)

∗ =∑k

j=1 b∗j , b

∗0 −→ 0. Thus, following the reasoning

Perron and Ng (1998), when ηt satisfies assumption 1 and yt is the non-stationary long mem-

ory, we note that S2AR∗ in (12) and s2AR in (4) are asymptotically equivalent. Accordingly,

the asymptotic properties of Tn, T∗n is established in Theorem 4.

THEOREM 4. Let ηt and yt be the processes given by (1) and (11). Under the null

hypothesis of Assumption 2 , as k, T → ∞, the following functional central limit (FCLT)

holds for MCUSQAR tests Tn, T∗n :

Tn, T∗n ⇒ W 0(t) = W (t)− tW (1)

where W and W 0 are the (one-dimensional) Brownian motion and Brownian bridge such as

limm→∞P|Tn, T∗n | ≥ g(t), for some t ≥ 1 = 2[1−Φ(a) + aΦ(a)]. (13)

Theorem 4 shows that the limiting distribution of MCUSQAR test T ∗n also follows the

Brownian bridge when the series are non-stationary long memory processes. From Theorem

2, Theorem 3 and Theorem 4, we can see that the MCUSQAR test can be also used as a

test for a change in the order of integration of a time series either from I(d) to I(1+d) or

from I(1+d) to I(d), d ∈ (−0.5, 0.5) without known direction.

8

4. Monte Carlo Evidence

This section present results from simulation experiments to examine the finite sample

behavior of the proposed monitoring procedure. Following Cheung (1993), we consider

differencing parameters d = (0.1, 0.2, 0.3, 0.4, 0.45, 0.49, 0.499) to two ARFIMA (p, d, q)

processes of the form:

DGP (a). (1− L)dηt = et,

DGP (b). (1− φ1L)(1− L)dηt = (1 + θ1)et (φ1 = −0.7, θ1 = 0.5).

We also follow the experiment design of Chu et, al (1996). The considered nominal size

of the test is at 5%, which corresponds to choosing a2 = 7.78 in (8) and (9). We compute

empirical crossing probabilities under H0 for historical sample size m = 50, 100, 200 and

300. Three different monitoring horizons, q = 2m, 3m and 4m are considered. The lag

length of AR(k) approximation of long memory processes for each m are chosen by AIC as

suggested in Poskitt (2007) and Wang and Hsiao (2008). All the simulations are based on

2500 replications.

The frequency of rejection of the null hypothesis for some n lying between m and q for

both DGPs and tests are summarized in Table 1 and 3. The results for each test reported in

Table 1 and 3 gives no hint of improper size distortion except several cases when m = 50.

As a matter of fact, with the increase of m and q, these two monitoring procedures are a

little conservative no matter what the d value is.

To investigate the finite sample power performance of our monitoring test procedure,

we create an artificial out-of-sample structural break in the mean at time t = m × 1.1,

where the mean shifts from 2 to 2.8. Our MCUSQAR indeed signals the structural change

eventually. Table 2 and 4 show that the power performance for the case of a break in mean

only. It appears that with the increase of d, the empirical power is getting closer to 1.

For example, as m ≥ 100 and q ≥ 2m, the empirical power is close to 1 when d ≥ 0.45.

Additionally, we are concerned with the case of a break in the parameter d (or in the

variance of yt), at which d shifts from 0.3 to 0.45 or 1.3 to 1.45 for both DGP. Furthermore,

the cases of shifting the structure of GDP from model (a) and model (b) for two monitoring

procedures are also included in our experiments in Table 5. We denote the GDP before

the break and after the break as GDP (I) and GDP (II) respectively. Those results are

reported in Table 6 and 7. The results for Table 6 and Table 7 suggest that when DGPs are

non-stationary, the power performances for Tn and T ∗n are better than those for stationary

DGPs. However, size performance is better when DGPs are stationary. Furthermore, taking

the experiment 1 and 2 as examples, the size and power performance of our monitoring

tests are promising. It implies that our monitoring test also can be a test for for stationary

short memory processes (I(0)process) against a change to stationary long memory processes.

Overall, for most cases, the finite sample performance of T ∗n outperform Tn, especially when

m ≥ 100. As a consequence, these finite results suggest that it is important to keep the

values of m and q in mind, when conducting our monitoring tests.

9

5. Application to U.S. Subprime Crisis

In this section, we illustrate the application of the monitoring procedure T ∗n to a foreign

exchange rate study. Since Larouche, the political action community, announced that the

financial crash occurred on June 25, 2007, many U.S. and European banks bankrupted

in the aftermath. In the mean time, for the sake of stimulating the economy, the U.S.

Federal Reserve Bank (FED), European Central bank (ECB) and Bank of England (BOE)

cut interest rates. We would therefore like to examine the real time pattern of Euro/USD,

GBP/USD and USD/YEN under the current U.S. subprime crisis.

The data series under this study are daily spot exchange rates from January 3 , 2005

to January 12, 2008 with 1015 observations total. Define the volatilities of the EUR (YEt),

GBP(YGt) and the JPY(YY t) as squared returns of those curriencies,

YEt =(ln(PEt,t/PEt,t−1)

)2

YY t =(ln(PY t,t/PY t,t−1)

)2

YGt =(ln(PGt,t/PGt,t−1)

)2

where PEt , PY t and PGt are the daily prices of three important nominal U.S dollar exchange

rates-the Euro , the Japanese Yen and British Pound respectively.

As a preliminary data analysis, our first task is to check the statistics properties of YEt,

YY t, and YGt. We use the Maximum Likelihood Estimation (MLE) method by OX Metrics.

The results of estimating ARFIMA models for YEt, YY t and YGt are reported in Table 8.

The volatility of the Euro, Japanese Yen and British Pound are estimated as I(0.3386),

I(0.2469) and I(0.4023) with highly significant AR and MA parameter estimates (φ, θ).

Table 9 and Table 10 present the timetable for U.S. and European bank crash and rate

cut dates of FED, ECB and BOE, respectively. The results for Table 11 show that structural

breaks are detected by T ∗n monitoring test for three volatility series. Because more structural

breaks were detected after July 2007, that means Euro and Pound became more volatile after

Larouche announced that the financial crash occurred on June 25, 2007. The volatility of

euro and pound also increased around Dec 6, 2007 when BOE started to announce the rate

cut. Particularly, more structural breaks are detected by our monitoring test for pound than

those for euro. This appears to be consistent with the observation that BOE made more and

earlier decisions on rate cutting policy than ECB did. Volatility of euro, pound and yen also

change much more dramatically after August 2008 when several European and U.S banks

bankrupted and ECB also started to cut interest rate. More interestingly, the results in Table

11 appear to match events in Table 9 and Table 10. For instance, we monitored the breaks at

Auguest, 8, 2007, July,7, 2007, and July, 27, 2007 for Euro, Yen and Pound,respectively, after

Countrywide crashed in July which was the first major bankrupted bank in U.S.. Likewise,

BOE announced the rate cut on Dec 6 , 2007. We also detect breaks at Dec 2, 2007 and

Dec 5, 2007 for Euro and Pound, respectively. The evolving events appear to corroborate

the usefulness of our monitoring tests.

10

6. Concluding Remarks

In this paper, we extend the on-line scheme of Chu et, al (1996) to the stationary

long memory processes via autoregresssion approximation. We proposed two monitoring

detectors, Tn and T ∗n , to check whether incoming data are consistent with a previously

estimated model. Through our simulation study, we note that Tn and T ∗n perform well.

They are able to detect the structural change. The tests also have nominal size close to

actual size without the need to use the bootstrap procedure.

We also applied our methodology to monitor the real time pattern of volatilities of three

daily nominal dollar exchange rates-Euro, Japanese Yen and British Pound before and after

the recent US subprime crisis. The empirical findings demonstrate our monitoring tests

indeed can capture the real time pattern of exchange rate volatilities.

11

Table 1. Empirical Size of Tn, When ηt are (a) and (b)

GDP m q 0.1 0.2 0.3 0.4 0.45 0.49 0.499

(a) 50 2m 4.9 5.3 5.9 6.2 6.4 6.6 6.7

3m 5.3 5.8 6.0 6.2 6.3 6.7 7.0

4m 5.7 6.1 6.2 6.4 6.6 7.0 7.2

100 2m 4.5 4.7 5.0 4.8 4.0 5.5 5.4

3m 4.6 4.7 5.2 5.2 5.5 5.9 6.1

4m 4.9 4.9 5.3 5.4 5.7 6.1 6.0

200 2m 4.0 4.2 4.2 4.5 4.5 4.6 4.9

3m 4.1 4.3 4.2 4.5 4.7 4.8 4.8

4m 4.2 4.4 4.5 4.7 4.6 4.9 5.2

300 2m 3.0 3.2 3.4 3.6 4.1 4.4 4.5

3m 3.2 3.3 3.6 3.8 4.2 4.8 4.7

4m 3.5 3.7 3.9 4.1 4.6 4.7 4.9

(b) 50 2m 5.4 5.7 6.0 6.2 6.4 7.0 7.1

3m 5.6 6.0 6.3 6.5 6.9 7.3 7.5

4m 6.0 6.2 6.5 6.6 7.0 7.4 7.7

100 2m 4.6 4.7 5.1 4.9 5.1 5.7 5.7

3m 4.8 4.8 5.2 5.2 5.5 5.9 6.0

4m 5.0 5.1 5.3 5.4 5.7 6.1 6.3

200 2m 4.2 4.4 4.6 4.5 4.5 4.6 4.9

3m 4.2 4.5 4.6 4.6 4.7 4.8 4.8

4m 4.5 4.6 4.7 4.9 4.6 4.9 5.2

300 2m 3.7 3.2 3.4 3.6 4.1 4.4 4.5

3m 3.6 3.3 3.7 3.9 4.2 4.8 4.7

4m 3.9 3.7 4.2 4.1 4.6 4.7 4.9

12

Table 2. Power of Tn, When ηt are (a) and (b)

DGP m q 0.1 0.2 0.3 0.4 0.45 0.49 0.499

(a) 50 2m 19.5 21.0 20.4 24.7 26.4 41.1 55.2

3m 22.6 24.9 22.9 28.9 33.8 50.2 58.2

4m 26.4 37.1 29.2 37.5 41.2 54.8 60.1

100 2m 34.3 44.7 53.0 62.2 68.5 69.9 70.1

3m 50.2 57.5 67.7 74.9 77.2 83.4 92.6

4m 66.7 70.9 72.3 81.2 85.7 90.1 93.0

200 2m 54.1 64.2 70.1 77.9 85.2 89.1 92.2

3m 62.3 70.1 77.2 84.5 90.6 94.8 100

4m 72.6 78.2 81.3 91.7 98.2 100 100

300 2m 63.7 70.2 79.1 88.0 94.1 92.4 96.5

3m 77.5 77.3 87.2 93.2 97.9 98.8 99.5

4m 80.0 82.9 90.9 98.8 100 100 100

(b) 50 2m 21.2 22.0 23.4 28.7 30.4 43.1 58.2

3m 27.6 28.9 28.9 30.9 33.8 52.2 62.4

4m 29.1 30.1 33.2 33.5 41.2 57.9 63.1

100 2m 37.1 48.2 54.1 63.7 67.2 70.9 74.5

3m 56.3 59.3 70.0 77.9 80.1 86.0 93.1

4m 67.2 72.9 74.3 82.2 88.8 95.1 94.7

200 2m 56.0 65.2 72.1 77.9 66.2 93.2 96.0

3m 66.1 71.1 80.2 89.6 91.6 95.1 100

4m 74.0 80.2 84.3 94.7 99.9 100 100

300 2m 65.1 72.1 81.5 89.6 95.2 97.4 99.5

3m 78.2 79.6 88.1 95.1 99.9 100 100

4m 83.2 86.1 92.1 98.8 100 100 100

13

Table 3. Empirical Size of T ∗n , When ηt are (a) and (b)

GDP m q 0.1 0.2 0.3 0.4 0.45 0.49 0.499

(a) 50 2m 5.1 5.6 6.0 6.2 6.6 6.9 6.9

3m 5.5 5.9 6.2 6.4 6.9 6.9 7.4

4m 5.8 6.3 6.5 6.9 7.0 7.2 7.5

100 2m 4.5 4.7 5.0 4.8 4.0 5.5 5.4

3m 4.6 4.7 5.2 5.2 5.5 5.9 6.1

4m 4.9 4.9 5.3 5.4 5.7 6.1 6.0

200 2m 4.0 4.2 4.2 4.5 4.5 4.6 4.9

3m 4.1 4.3 4.2 4.5 4.7 4.8 4.8

4m 4.2 4.4 4.5 4.7 4.6 4.9 5.2

300 2m 3.0 3.2 3.4 3.6 4.1 4.4 4.5

3m 3.2 3.3 3.6 3.8 4.2 4.8 4.7

4m 3.5 3.7 3.9 4.1 4.6 4.7 4.9

(b) 50 2m 5.6 5.8 6.2 6.4 6.6 7.2 7.6

3m 5.7 6.3 6.5 6.6 6.9 7.5 8.0

4m 6.2 6.4 6.8 6.9 7.2 7.7 8.2

100 2m 4.8 4.9 5.5 5.0 5.1 5.9 6.0

3m 5.0 5.1 5.7 5.3 5.5 5.9 6.4

4m 5.5 5.4 5.9 5.5 5.7 6.2 6.7

200 2m 4.2 4.8 4.6 4.5 4.5 4.8 5.0

3m 4.6 4.5 4.6 4.6 4.7 4.8 5.6

4m 4.6 4.6 4.7 4.9 4.8 5.1 5.9

300 2m 3.4 3.4 3.7 3.8 4.5 4.6 4.8

3m 3.5 3.1 3.9 4.2 4.7 4.7 4.6

4m 3.7 3.5 4.5 4.4 4.9 5.2 5.1

14

Table 4. Power of T ∗n , When ηt are (a) and (b)

DGP m q 0.1 0.2 0.3 0.4 0.45 0.49 0.499

(a) 50 2m 21.3 23.6 27.7 29.6 32.1 42.6 52.1

3m 24.1 29.1 26.1 36.1 41.1 54.4 60.2

4m 30.2 34.2 38.3 40.1 49.3 57.1 62.1

100 2m 34.3 44.7 53.0 62.2 68.5 69.9 70.1

3m 50.2 57.5 67.7 74.9 77.2 83.4 92.6

4m 66.7 70.9 72.3 81.2 85.7 90.1 93.0

200 2m 55.2 66.2 71.1 79.9 56.1 90.1 91.2

3m 66.1 72.1 78.5 85.6 92.6 95.8 100

4m 77.4 78.2 83.4 92.7 99.1 100 100

300 2m 60.1 71.3 81.1 89.1 92.2 91.4 99.1

3m 68.4 79.1 88.2 94.2 96.5 99.2 100

4m 74.6 82.1 90.1 96.2 100 100 100

(b) 50 2m 22.9 24.7 27.4 30.7 33.4 47.1 59.2

3m 30.6 31.8 35.9 38.9 40.8 50.2 61.2

4m 32.2 35.7 39.2 44.5 46.2 59.8 67.1

100 2m 34.3 44.7 53.0 62.2 68.5 69.9 72.1

3m 50.2 57.5 67.7 74.9 77.2 83.4 96.6

4m 66.7 70.9 72.3 81.2 85.7 90.1 97.0

200 2m 59.1 64.2 70.1 77.9 55.2 89.1 98.2

3m 67.3 70.1 77.2 84.5 90.6 94.8 100

4m 76.6 78.2 81.3 91.7 98.2 100 100

300 2m 63.7 70.2 80.1 88.0 94.1 97.4 99.5

3m 77.5 77.3 87.2 93.2 99.9 100 100

4m 80.0 83.7 91.9 98.8 100 100 100

15

Table 5. Breakpoint Specifications by Experiments (EX)

EX DGP (I) DGP (II) d1 d2 µ1 µ2

1 SD (a) 0.0 0.1 0.0 0.02 SD (b) 0.0 0.1 0.0 0.03 (a) (a) 0.3 0.45 0.0 0.04 (a) (a) 0.3 0.45 0.3 0.95 (b) (b) 0.3 0.45 0.3 0.96 (b) (b) 0.3 0.45 0.0 0.07 (a) (b) 0.3 0.45 0.0 0.08 (a) (a) 1.3 1.45 0.0 0.09 (b) (b) 1.3 1.45 0.0 0.010 (a) (b) 1.3 1.45 0.0 0.0

Table 6. The Size and Power of Tn testfor Experiments Designed in Table 5.

m q EX1 2 3 4 5 6 7 8 9 10

Size 50 2m 5.4 5.6 5.8 5.9 6.3 6.0 6.6 6.2 6.7 6.93m 5.6 5.7 6.0 5.9 6.3 6.2 7.0 6.3 6.7 7.24m 5.9 6.1 6.1 6.2 6.7 6.4 7.2 6.6 7.0 7.5

100 2m 5.1 5.3 5.5 5.6 6.0 5.6 6.3 6.0 6.4 6.93m 5.2 5.5 5.9 5.7 6.2 5.8 6.7 6.3 6.9 7.04m 5.2 5.7 5.9 5.9 6.4 6.0 7.0 6.5 6.9 7.2

200 2m 4.7 5.3 5.3 5.2 5.6 5.3 5.6 5.5 5.9 6.13m 5.2 5.4 5.5 5.5 5.9 5.6 5.8 5.6 6.1 6.24m 5.2 5.7 5.6 5.7 5.8 5.8 5.9 6.0 5.9 6.3

300 2m 3.9 4.2 4.0 4.2 4.7 4.5 5.1 5.7 5.3 5.83m 4.0 4.6 4.5 4.6 5.0 4.8 5.4 5.3 5.7 6.04m 4.0 4.5 4.6 4.7 5.0 4.8 5.6 4.9 5.8 6.1

Power 50 2m 18.2 20.0 26.4 22.1 36.4 44.1 51.2 18.5 51.1 58.83m 19.6 20.9 30.0 29.9 38.8 47.2 53.2 29.4 58.2 60.14m 22.1 27.1 36.9 39.4 42.2 52.7 63.1 32.5 60.1 67.2

100 2m 21.2 28.4 29.2 30.2 40.1 50.1 56.7 31.9 60.1 64.63m 29.3 30.9 34.7 34.1 42.2 59.4 60.6 32.2 66.5 69.44m 31.7 36.2 38.3 38.2 44.3 67.9 72.2 39.5 69.9 76.1

200 2m 24.2 30.1 40.1 47.9 52.7 59.1 62.2 38.2 66.7 70.23m 30.2 38.1 44.2 54.5 56.6 64.8 69.7 50.7 69.5 76.54m 32.6 40.2 52.3 61.7 61.2 70.2 79.1 59.8 77.5 78.4

300 2m 33.6 45.2 55.1 60.2 72.7 77.4 89.1 69.1 80.1 89.03m 37.5 57.1 69.2 71.2 80.0 85.4 90.2 72.6 84.2 96.54m 40.2 67.9 75.9 78.3 84.2 89.0 100.0 82.1 89.0 100

16

Table 7. The Size and Power of T ∗n testfor Experiments Designed in Table 5.

m q Ex1 2 3 4 5 6 7 8 9 10

Size 50 2m 5.8 6.1 6.4 6.6 6.7 7.0 7.2 6.6 6.7 7.13m 6.1 6.2 6.3 6.8 6.9 7.2 7.1 6.7 6.9 7.34m 6.3 6.6 6.8 7.0 7.3 7.4 7.6 7.0 7.0 7.4

100 2m 5.6 5.8 6.0 5.6 6.0 5.6 5.8 6.3 6.2 6.63m 5.4 5.9 6.2 5.7 6.2 5.8 6.0 6.4 6.5 6.74m 5.5 6.1 6.3 5.9 6.4 6.0 6.3 6.6 6.6 6.9

200 2m 5.0 5.3 5.3 5.2 5.6 5.4 5.7 5.6 6.0 6.43m 5.2 5.4 5.5 5.5 5.9 5.6 5.8 5.9 6.1 6.54m 5.2 5.7 5.6 5.7 6.0 5.8 6.0 6.0 6.3 6.7

300 2m 4.4 4.5 4.7 4.9 5.0 5.2 5.3 5.1 5.4 5.83m 4.3 4.8 5.0 5.2 5.3 5.3 5.6 5.3 5.7 6.14m 4.5 4.8 5.0 5.1 5.6 5.2 5.5 5.6 6.0 6.3

Power 50 2m 25.9 23.1 26.4 29.3 39.1 51.2 60.2 30.2 60.3 67.43m 28.7 29.9 30.1 39.9 42.7 57.4 69.2 34.5 62.3 70.24m 33.1 34.1 35.9 39.4 47.8 62.7 80.1 36.7 65.7 76.1

100 2m 33.1 28.4 34.5 40.0 45.2 56.1 70.6 40.2 62.2 72.33m 39.6 34.7 45.1 45.2 50.2 69.4 74.6 50.2 67.3 74.14m 41.9 38.1 45.6 45.1 53.3 67.9 93.2 53.2 70.3 76.2

200 2m 35.1 39.1 45.6 53.9 57.7 69.2 82.2 51.2 71.2 88.43m 39.5 44.0 50.2 65.2 61.7 78.1 89.7 55.7 79.4 93.14m 43.7 47.0 57.1 66.1 67.6 81.2 97.0 60.2 88.1 100

300 2m 43.1 58.2 60.1 70.1 79.1 87.1 91.1 66.7 83.4 90.23m 51.2 70.1 74.4 77.1 88.1 95.4 99.2 80.1 99.2 1004m 50.7 71.2 79.3 88.1 96.1 99.1 100 89.2 100 100

17

Table 8. Estimation of ARFIMA (p, d, q) Models

for the volatility of Euro and Japanese Yen

Estimates Yen Euro GBP

d 0.3386 0.2469 0.4023

φ n/a −0.1902 −0.2761

θ −0.2602 n/a −0.4027

td 6.7116 7.7203 8.2101

tφ n/a −5.0215 −5.9943

tθ −4.4503 n/a −6.0827

Table 9. Timetable for the News of U.S. and European Banks Crash

Time Bank Nationality

7/11/2007 Countrywide U.S

3/17/2008 BearStearns U.S

9/15/2008 MerrillLynch U.S

9/15/2008 LehmanBrothers U.S

9/25/2008 WashingtonMutual U.S

9/29/2008 Fortis Belgium

9/29/2008 B&B British

9/30/2008 Glinter Iceland

10/7/2008 Landsbanki Iceland

10/20/2008 FR−BNP France

18

Table 11. Timetable for the Interest Rate Cut News of Central Banks

Bank D1 D2 D3 D4 D5

U.S. FED 2007/9/18 2007/10/31 2007/12/11 2008/1/30 2008/3/18

−0.5% −0.25% −0.25% −1.25% −0.25%

BOE 2007/12/6 2008/2/7 2008/4/10 2008/10/2 2008/11/06

−0.25% −0.25% −0.25% −0.5% −50%

ECB 2008/10/2 2008/11/6 2008/12/4 2009/1/14/2009

−0.5% −0.5% −0.75% −0.5%

Series D6 D7 D8 D9 D10

U.S.FED 2008/10/29 2008/12/16

−1.00% −0.75%

BOE 2008/12/4 2008/1/08

−1.00% −0.50%

19

Table 10. Multiple Structural Changes Dates Detection

Series T1 T2 T3 T4 T5

EUR/USD 2005/3/18 2005/6/27 2006/7/7 2007/8/8 2007/12/2

USD/YEN 2005/5/13 2005/7/2 2005/10/14 2006/4/24 2007/2/26

GBP/USD 2005/6/5 2005/9/15 2006/7/7 2007/7/27 2007/10/22

Series T6 T7 T8 T9 T10

EUR/USD 2008/7/14 2008/10/23 2009/1/2

USD/YEN 2007/7/9 2008/3/13 2008/10/7

GBP/USD 2007/12/5 2008/4/7 2008/8/13 2008/10/22 2008/12/18

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APPENDIX

A.1. Proof of Theorem 1:

By Theorem 5.1, Theorem 5.2 and Lemma 5.7 of Poskitt (2007), we note that

1

T

T∑t=k+1

e2t,k =1

T

T∑t=k+1

e2t + op(1)p−→ σ2. (A.1)

and ∥∥∥β − β∥∥∥ = Op

((

k

λmin(Γh))1/2(

logT

T)1/2−d′

).

21

Where d = max0, d. Thus, by Cauchy-Schwarz inequality, we note that

k∑j=1

(βj − βj

)≤

k∑

j=1

(βj − βj

)2

1/2 k∑

j=1

12

1/2

= Op

((

k

λmin(Γh))1/2k1/2(

logT

T)1/2−d′

).

This implies that

k∑j=1

βj =k∑

j=1

βj +Op

((

k

λmin(Γh))1/2k1/2(

logT

T)1/2−d′

)

=

k∑j=1

βj + op (1) .

Therefore, as d < 0, we can show that

1−k∑

j=1

bj = 1−k∑

j=1

bj = O(k−d) and s2AR = Op(k2d).

As d > 0, we note that (1− L)dC = 0, for some constant C. This implies that

1−k∑

j=1

bj =∞∑

j=k+1

bj = O(k−d).

Therefore, as d > 0, we also prove s2AR = Op(k2d). Combining the preceding results, Theo-

rem 1 is established.

22