monitoring correlation change in a sequence of random ... · dominik wied, pedro galeano tu...

Monitoring correlation change in a

sequence of random variables

Dominik Wied, Pedro Galeano∗

TU Dortmund and Universidad Carlos III de Madrid

This Version: June 8, 2012

Abstract

We propose a monitoring procedure to test for the constancy of the correlation

coefficient of a sequence of random variables. The idea of the method is that a

historical sample is available and the goal is to monitor for changes in the correlation

as new data become available. We introduce a detector which is based on the

first hitting time of a CUSUM-type statistic over a suitably constructed threshold

function. We derive the asymptotic distribution of the detector and show that the

procedure detects a change with probability approaching unity as the length of the

historical period increases. The method is illustrated by Monte Carlo experiments

and the analysis of a real application with the log-returns of the Standard & Poor’s

500 (S&P 500) and IBM stock assets.

Keywords: Correlation changes; Gaussian process; Online detection; Threshold func-

tion.

∗TU Dortmund, Fakultat Statistik, D-44221 Dortmund, Germany. Email: [email protected], Phone: +49 231 755 3869 (D. Wied). Universidad Carlos III de Madrid, Departamentode Estadıstica, E-28903 Getafe, Madrid, Spain. Email: [email protected], Phone: +34 91 6248901 (P. Galeano). Financial support by Deutsche Forschungsgemeinschaft (SFB 823) and MCI grantMTM2008-03010 is gratefully acknowledged. We are grateful to helpful comments from two anonymousreferees which improved the paper.

1

1. Introduction

The correlation coefficient is the most widely used method to measure dependence be-

tween a sequence of two random variables. In the particular case of financial time series,

the analysis of the correlations between returns are very important in risk management.

Indeed, there is compelling empirical evidence that the correlation structure of financial

returns cannot be assumed to be constant over time, see e.g. Longin and Solnik (1995) and

Krishan et al. (2009). Consequently, in periods of financial crisis, investors are extremely

concerned about changes on correlations because in such periods, the correlation often

increases, a phenomenon which is referred to as “Diversification Meltdown” (Campbell

et al., 2008).

In order to construct an adequate model and to forecast future data, structural stability

is a key point. Testing for structural stability has recently become one of the principal

objectives of statistical analysis. There are two distinctly different approaches to tackle

this problem. On the one hand, the main goal of retrospective procedures is to look

for the presence of change points given an historical dataset of fixed size. On the other

hand, the main goal of sequential detection procedures is to detect as soon as possible

the presence of a change point once new data become available. This article is concerned

with the latter kind of procedures. We adopt the framework in Chu et al. (1996) in which

a historical sample is available and the goal is to monitor for a change point as new

data become available. In particular, we analyze the case of changes in the correlation

structure of a sequence of random variables. Other papers analyzing related problems

under this framework are Chu et al. (1996), Horvath et al. (2004), Aue et al. (2006), Aue

et al. (2009a), Aue et al. (2009b) and Aue et al. (2011), among others.

The paper is organized as follows. Section 2 proposes a monitoring procedure for de-

tecting a correlation change and presents its asymptotic properties under the null and

alternative hypothesis as well. Section 3 analyzes the finite sample properties of the

proposed procedure via Monte Carlo experiments. Section 4 illustrates the procedure by

analyzing log-returns of the S&P 500 and IBM stock assets. Finally, all proofs are given

2

in an appendix.

2. The monitoring procedure

Let (Xt, Yt), for t ∈ Z, be a sequence of bivariate random variables with finite 4-th

moments and correlation

ρt =Cov(Xt, Yt)√Var(Xt)Var(Yt)

.

We are interested in the hypothesis of correlation stability of the sequence. For that,

assume that we have observed a sequence of the bivariate random vector (Xt, Yt) of size

m. Since we are interested in sequentially monitoring whether or not the correlation

coefficient remains stable over time, we require that the correlation is constant over the

historical period of length m, i.e.:

Assumption 1. ρ1 = . . . = ρm, where m is a positive integer.

Although Assumption 1 may appear a strong assumption, in practice, if a sufficient

amount of historical data is available, it can be analyzed with the retrospective change

point method proposed by Galeano and Wied (2012). Given the results of this procedure,

one can make necessary adjustments to ensure correlation stability. Now, we want to test

the null hypothesis given by:

H0 : ρ1 = . . . = ρm = ρm+1 = . . . .

versus the alternative H1 that ρt changes at some t ≥ m+ 1, i.e.:

H1 : ∃k∗ ≥ 1 : ρ1 = . . . = ρm = . . . = ρm+k∗−1 6= ρm+k∗ = ρm+k∗+1 = . . . ,

where k∗ is referred to as the change point and is assumed unknown.

Denote with ρlk the empirical correlation coefficient calculated from the observations k to

3

l with k < l, given by:

ρlk =

∑lt=k(Xt −Xk,l)(Yt − Y k,l)√∑l

t=k(Xt −Xk,l)2√∑l

t=k(Yt − Y k,l)2

where Xk,l = 1l−k+1

∑lt=kXt and Y k,l = 1

l−k+1

∑lt=k Yt. The sequential procedure is based

on the detector:

Vk = Dk√m

(ρm+km+1 − ρm1

), k ∈ N, (1)

where D is an estimator which is calculated from the first m observations and is given in

the appendix, see also Wied et al. (2012). We stop and declare H0 to be invalid at the

first time k such that the detector Vk exceeds the value of a scaled threshold function w,

therefore yielding the stopping rule:

τm = min

{k ≤ [mT ] : |Vk| > c · w

(k

m

)}, (2)

where T is a positive constant, c is a suitably chosen constant such that under H0,

limm→∞ P(τm < ∞) = α, with α ∈ (0, 1), and w is a positive and continuous function.

Here, we write τm < ∞ to indicate that the monitoring has been terminated during

the testing period, i.e., the detector Vk has crossed the boundary c · w(k/m) for some

k ≤ [mT ]. We write τm = ∞ if the detector has not crossed the boundary during the

testing period (compare Aue et al., 2011). Note that the stopping time τm need not be

the change point; in fact the change point might be before τm. Some comments on the

issue of estimating the change point once H0 has been declared invalid will be given at

the end of this section.

For deriving asymptotic results under H0, some additional assumptions are necessary.

The next three assumptions correspond to (A1), (A2) and (A3) in Wied et al. (2012).

Assumption 2. For

Ut :=

(X2t − E(X2

t ), Y 2t − E(Y 2

t ), Xt − E(Xt), Yt − E(Yt), XtYt − E(XtYt)

)′4

and Sj :=∑j

t=1 Ut, we have

limm→∞

E

(1

mSmS

′m

)=: D1 (finite and positive definite).

Assumption 3. The r-th absolute moments of the components of Ut are uniformly

bounded for some r > 2.

Assumption 4. The vector (Xt, Yt) is L2-NED (near-epoch dependent) with size − r−1r−2 ,

where r from Assumption 3, and constants (ct), t ∈ Z, on a sequence (Vt), t ∈ Z, which

is α-mixing of size φ∗ := − rr−2 , i.e.

||(Xt, Yt)− E ((Xt, Yt)|σ(Vt−l, . . . , Vt+l))||2 ≤ ctvl

with liml→∞ vl = 0, such that

ct ≤ 2||Ut||2

with Ut from Assumption 3 and the L2-norm || · ||2.

Furthermore, we impose a stationarity condition. This condition might be slightly relaxed

to allow for some fluctuations in the first and second moments (see (A4) and (A5) in Wied

et al., 2012), but for ease of exposition and because the procedure would remain exactly

the same we stick to this notation.

Assumption 5. (Xt, Yt), t ∈ Z, is weak-sense stationary.

Our main result is then:

Theorem 1. Under H0, Assumptions 1, 2, 3, 4 and 5 and for any T > 0,

limm→∞

P(τm <∞) = limm→∞

P

(sup

0≤b≤T

|V[m·b]+2|w(b)

> c

)= P

(sup

0≤b≤T

|G(b)|w(b)

> c

), (3)

where G(·) is a mean zero Gaussian process with covariance E(G(k)G(l)) = min(k, l)+kl.

5

Theorem 1 establishes the asymptotic behavior of the monitoring procedure based on

the stopping rule τm in Eq. (2). Following Aue et al. (2011), the limiting probability

in Eq. (3) can be written in an alternative way that allows for finite sample statistical

inference. First, it is easy to see that {G (b) : b ∈ [0, T ]} =d {W (b) + bξ : b ∈ [0, T ]},

where {W (b) : b ≥ 0} is a standard Brownian Motion independent of the standard Gaus-

sian random variable ξ. Then, it is also easy to see that {W (b) + bξ : b ∈ [0, T ]} =d

{(1 + b)W (b/ (1 + b)) : b ∈ [0, T ]} just by comparing their covariance structures. There-

fore,

sup0≤b≤T

|G(b)|w(b)

=d sup0≤b≤T

1 + b

w (b)

∣∣∣∣W (b

1 + b

)∣∣∣∣ . (4)

Eq. (4) leads to an obvious choice of the threshold function: take w (b) = 1 + b, because

in this case:

sup0≤b≤T

|G(b)|w(b)

=d sup0≤b≤T

∣∣∣∣W (b

1 + b

)∣∣∣∣ .With this expression, quantiles of interest can be easily simulated with Monte Carlo

methods. However, once there occurs a change point, it is very important to quickly

detect it. Therefore, we consider a kind of generalization of this threshold function,

previously considered in Horvath et al. (2004), which is given by:

w (b) = (1 + b) ·max

{(b

1 + b

)γ, ε

}(5)

where 0 ≤ γ < 12

and ε > 0 is a fixed constant which can be chosen arbitrarily small in

applications.

Note that, if a correlation change occurs soon after the historical dataset, then, choosing

γ as large as possible, the stopping rule τm will stop nearly instantaneously. Note that

γ = 1/2 is excluded, since H0 would else be rejected with probability one regardless

whether it is true or not because of the law of the iterated logarithm for Brownian

Motions at zero, see Aue et al. (2009b). Using the threshold function in Eq. (5) and

6

calling u = b/ (1 + b), Eq. (4) leads to:

sup0≤b≤T

|G(b)|w(b)

=d sup0≤u≤ T

T+1

1

max{uγ, ε}|W (u)|. (6)

Finally, calling s = u (T + 1) /T and taking into account that {W (u) , u ∈ [0, T/(T +1)]}

has the same covariance structure as{√

T1+T

W (s) , s ∈ [0, 1]}

, Eq. (6) transforms into:

sup0≤b≤T

|G(b)|w(b)

=d

(T

1 + T

) 12−γ

sup0≤s≤1

1

max {sγ, ε((T + 1)/T )γ}|W (s)|. (7)

Therefore, under the conditions in Theorem 1:

limm→∞

P(τm <∞) = P

((T

1 + T

) 12−γ

sup0≤s≤1

1

max {sγ, ε((T + 1)/T )γ}|W (s)| > c

)

and Monte Carlo simulations can be used to obtain the constant c (α) such that:

P

((T

1 + T

) 12−γ

sup0≤s≤1

1

max {sγ, ε((T + 1)/T )γ}|W (s)| > c (α)

)= α,

for any α ∈ (0, 1). In this way, the probability of a false alarm is approximately α if m is

large enough.

For a local power analysis, we impose the assumption

Assumption 6. (Xt, Yt), t ∈ Z, is weak-sense stationary with the difference that Cov(Xt, Yt) =

1√mg(tm

)with a bounded function g that can be approximated by step functions such that

g(z) = 0, z ∈ [0, 1], and∫ T+1

1|g(z)|dz > 0.

Theorem 2 yields consistency of the monitoring procedure. Therefore, a correlation

change will be detected with high probability if the historical period is large enough.

Theorem 2. Under a sequence of local alternatives, Assumptions 1, 2, 3, 4, 5 and 6 and

7

for any T > 0,

limm→∞

P(τm <∞) = limm→∞

P

(sup

0≤b≤T

|V[m·b]+2|w(b)

> c

)= P

(sup

0≤b≤T

|G(b) + h(b)|w(b)

> c

),

where G(·) is as in Theorem 1 and h(b) = H(∫ b+1

1g(z)dz − b ·

∫ 1

0g(z)dz

)for a constant

H depending on the data generating process.

Once the presence of a correlation change is detected, an estimate of its location is

provided by using the statistic proposed in Wied et al. (2012). The estimate of the

change point is k = arg max1≤j≤τm−1

Dτm with

Dτm := Dj√τm

∣∣ρm+jm+1 − ρm+τm−1

m+1

∣∣ . (8)

Note that, except for the estimator D, we do not use the historical period to compute

the value of the statistic Dτm but only the observations from m+ 1 to m+ τm−1. Monte

Carlo experiments have shown that the inclusion of the historical period severely distorts

the estimates of the change point location. A theoretical analysis of this estimator is

beyond the scope of this paper.

3. Simulations

In this section, we report the results of the Monte Carlo experiments that we have per-

formed to assess the finite sample performance of the proposed monitoring procedure. In

all the experiments, we consider three different values of the parameter γ of the threshold

function w(b), b ∈ [0, T ] in Eq. (5), γ = 0, 0.25 and 0.45, while we have taken ε = 10−10.

Figure 1 shows the plot of the three threshold functions considered (which does not yet

give concrete information about the critical values used in the procedure). Note that the

larger the values of γ, the smaller the values of w(b), b ∈ [0, T ]. The threshold function

with γ = 0.45 is expected to allow for a quick detection of early change points. We

consider three different values of the size of the historical sample, m = 250, 500 and

8

1000. Note that these values are specially designated for financial returns in which we

can consider large historical samples. Finally, we consider four values of the parameter

T , T = 0.5, 1, 2 and 4. Note that these values cover a large number of sample sizes of the

generated bivariate series which is given by n = m + [Tm]. For instance, for m = 500,

the sample sizes of the series generated are 750, 1000, 1500 and 2500, respectively.

Figure 1 goes around here

First, we obtain critical values to apply the monitoring procedure for the different values

considered of γ and T . Table 1 shows the critical values at level α = 0.05 based on 10000

standard Brownian Motion processes approximated on a grid of 10000 equispaced points

in the interval [0, 1]. Note that the critical values increases with T and/or with γ as

expected.

Table 1 goes around here

Second, in order to obtain empirical sizes of the monitoring procedure, we generate 1000

bivariate series (Xt, Yt), for t = 1, . . . , n, and any choice of γ, T and m, as follows.

Initially, we generate two series(Xt, Yt

), for t = 1, . . . , n, independently, following the

GARCH(1, 1) models given by:

Xt =√h1,tε1,t

h1,t = 0.01 + 0.05X2t−1 + 0.8h1,t−1

and,

Yt =√h2,tε2,t

h2,t = 0.01 + 0.1Y 2t−1 + 0.75h2,t−1

9

respectively, where ε1,t and ε2,t are standard Gaussian distributed. Then, we transform

the bivariate series(Xt, Yt

)into (Xt, Yt) by multiplying each value of the pair

(Xt, Yt

)with Σ1/2 where Σ is a square symmetric matrix with ones in the main diagonal and with

ρ = 0.5 outside the main diagonal. Then, the correlation between Xt and Yt is ρ = 0.5.

Afterwards, for each simulated dataset, we apply the monitoring procedure from time

m + 1 until time n, with level α = 0.05. Table 2 reports the simulated empirical sizes

for the monitoring procedure based on the detector Vk. In most cases, the simulated

empirical sizes slightly exceed the nominal sizes; for γ = 0.45 the empirical sizes are

at least twice the nominal size. However, empirical and nominal sizes get closer as m

increases which is reasonable based on the results in Section 2. Also, larger empirical sizes

are found as γ gets larger and m is small. Therefore, if a correlation change is expected

to occur not shortly after the historical period and we want to minimize the type I error,

the choice of the threshold function with γ = 0 appears to be appropriate. However, if

a correlation change is expected to occur shortly after the historical period and we want

to detect it as soon as possible even if a false alarm can happen, it is better to use the

threshold function with γ = 0.45.


Third, in order to estimate the power of the monitoring procedure, the Monte Carlo

setup is similar to the one described previously, but the series are generated with a single

change point in the correlation at two different positions k = [0.05mT ] and k = [0.5mT ],

in which ρ = 0.5 increases to ρ = 0.75. Therefore, the first m observations have the same

correlation coefficient, that changes after k observations of the monitoring time. The first

change point is at the initial 5% of the monitoring time, so that it is specially designated to

estimate the power of the procedure in situations in which the change point occurs shortly

after the historical period. The second change point is at the middle of the monitoring

time, so that it is specially designated to estimate the power of the procedure in situations

10

in which the change point does not occur shortly after the historical period. Tables 3,

4, 5, 6, 7 and 8 show the results for the three possible values of the γ parameter, γ = 0,

0.25 and 0.45, and the two possible change points, k = [0.05mT ] and k = [0.5mT ]. These

tables show the empirical power of the procedure and a summary of both, the empirical

stopping time distribution and the estimated change points, including the quartiles, the

mean, the standard deviation and the coefficient of variation. The tables show that the

power increases with m and it can be large except in cases in which m and T are small.

Besides, the power for early changepoints is larger than the power for changes at the

middle of the monitoring period. Regarding the empirical stopping time distribution, if

a change occurs shortly after the beginning of the monitoring period, then the threshold

function with γ = 0.45 have the shortest detection delay time. However, for a change

point at the middle of the monitoring period with m = 250 and m = 500, the first

quartiles of the empirical stopping times with γ = 0.45 are very small indicating that

is more likely to falsely detect a correlation change even before it occurred. On the

other hand, regarding the change point estimates, we can observe that the estimates of

the change point at the beginning of the monitoring period are upward biased, while

the estimates of the change point at the middle of the monitoring period are downward

biased. However, in both cases, the bias reduces substantially if m and/or T increases. In

any case, the precision of the change point detection estimate is quite acceptable specially

when the power is large.

In summary, if the bivariate series is going to be monitored for a long time and the

type I error is to be avoided, or if a change in the correlation is expected to occur not

shortly after the beginning of monitoring period, the threshold function with γ = 0 may

be a good choice. However, if the focus is to detect a change point in the correlation as

soon as possible, even if a false alarm is accepted, and if the change point is expected to

occur shortly after the beginning of monitoring period, then it is better to use γ = 0.45.

Alternately, the threshold function with γ = 0.25 appears to be a good compromise

between the previous frameworks.

11







4. Real data example

In this section, we apply the proposed monitoring procedure discussed in Section 2 to

a real data example. Galeano and Wied (2012) analyzed the log-return series of two

U.S. assets: the Standard & Poors 500 Index and the IBM stock using a posteriori

change point tests. In particular, Galeano and Wied (2012) considered the sample period

starting from January 2, 1997 to December 31, 2010 consisting of n = 3524 observations,

that are plotted in Figure 2. The binary segmentation procedure proposed in that paper

detected a first change point at August 19, 1999 (observation number 664), that can be

associated with the collapse of the dot-com bubble started at the end of the 1990s and

the beginning of the 2000s, and a second change point at November 12, 2007 (observation

number 2734), that can be associated with the beginning of the Global Financial Crisis

around the end of 2007, which is considered by many economists the worst financial crisis

since the Great Depression of the 1930s.

Figure 2 goes around here

Here, we apply the proposed monitoring procedure as follows. The analysis in Galeano

and Wied (2012) indicated that the correlations between both log-returns remained con-

stant for the period starting from January 2, 1997 to August 19, 1999. Then, we use the

12

log-returns from January, 2, 1997 until May, 28, 1999, as the historical period, i.e., we

take m = 607. If no correlation changes are found after n−m = 2917 observations (then,

T = 4.8056) the procedure would be terminated. Otherwise, a change point is detected

and a new historical period is defined with m = 607. Then, the monitoring procedure is

applied again in a similar fashion. The results of our analysis are summarized in Table 9

for the three threshold functions with γ = 0, 0.25 and 0.45, for which the corresponding

critical values at 5% level are 2.0510 for γ = 0, 2.2630 for γ = 0.25, and 2.7435 for

γ = 0.45, respectively. The proposed procedure with the three values of the threshold

functions detects four change points sequentially. Regarding the first hitting times, the

procedure with γ = 0.45 has the shortest detection delay time whereas the procedure

with γ = 0 the longest. This is in accordance with the Monte Carlo experiments in

Section 3. Regarding the estimated change points, the procedure with the three values

gives very similar estimates. Indeed, the first and the last detected change points coincide

with the ones given in Galeano and Wied (2012). Finally, Table 10 shows the empirical

correlations between the Standard & Poors 500 and IBM log-returns in the periods given

by the monitoring procedure. As it can be seen, there are substantial differences between

correlations at different periods.



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13

Aue, A., L. Horvath, M. Huskova, and P. Kokoszka (2006): “Change-point

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14

A. Appendix

A.1. The scalar D from the test statistic in Eq. (1)

The scalar D from our test statistic in Eq. (1) based on observations from t = 1, . . . , r

can be written as

D = (F1D3,1 + F2D3,2 + F3D3,3)− 1

2

where F1

F2

F3

=

D3,1E11 + D3,2E21 + D3,3E31

D3,1E12 + D3,2E22 + D3,3E32

D3,1E13 + D3,2E23 + D3,3E33

,

E11 = D1,11 − 4µxD1,13 + 4µ2xD1,33,

E12 = E21 = D1,12 − 2µxD1,23 − 2µyD1,14 + 4µxµyD1,34,

E22 = D1,22 − 4µyD1,24 + 4µ2yD1,44,

E13 = E31 = −µyD1,13 + 2µxµyD1,33 − µxD1,14 + 2µ2xD1,34 + D1,15 − 2µxD1,35,

E23 = E32 = −µyD1,23 + 2µxµyD1,44 − µxD1,24 + 2µ2yD1,34 + D1,25 − 2µyD1,45,

E33 = µ2yD1,33 + 2µxµyD1,34 − 2µyD1,35 + µ2

xD1,44 + D1,55 − 2µxD1,45,

D1 = (D1,ij)5i,j=1 =

r∑t=1

r∑u=1

k

(t− uδr

)VtVu

′,

Vt =1√rU∗∗∗t , δr = [log r],

15

U∗∗∗t =

(X2t − (X2)r Y 2

t − (Y 2)r Xt − Xr Yt − Yr XtYt − (XY )r

)′,

k(x) =

1− |x|, |x| ≤ 1

0, otherwise

,

µx = Xr, µy = Yr, D3,1 = −1

2

σxyσyσ−3x , D3,2 = −1

2

σxyσx

σ−3y , D3,3 =1

σxσy

and

σ2x = (X2)r − (Xr)

2, σ2y = (Y 2)r − (Yr)

2, σxy = (XY )r − XrYr.

This is the same expression as in Appendix A.1 in Wied et al. (2012).

A.2. Proofs

Proof of Theorem 1

Let D[d1, d2] be the space of cadlag-functions on the interval [d1, d2] equipped with the

supremum norm.

The proof is mainly based on the fact that for fixed c ≥ 0, and m → ∞ the process

{Pm(d), c ≤ d ≤ T}, defined by

Pm(d) = D[m · d]− [m · c]√

m(ρ

[m·d][m·c] − ρ1),

converges in distribution to the process {W (d)−W (c), c ≤ d ≤ T} on D[c, T ] with W (·)

being a standard Brownian Motion. This result is a generalization of Lemma A.3 in Wied

et al. (2012):

First, under Assumptions 2-5, we obtain with Lemma A.1 in Wied et al. (2012) that the

16

process {Qm(d), 0 ≤ d ≤ T}, defined by

Qm(d) :=1√m

[m·d]∑t=1

Ut

with

Ut =

(X2t − E(X2

t ) Y 2t − E(Y 2

t ) Xt − E(Xt) Yt − E(Yt) XtYt − E(XtYt)

)′,

converges to D1W5(·), where W5 is a 5-dimensional Brownian Motion.

Let c2 ∈ [0, T ] and c3 ∈ [0, T ] be fixed. By applying the continuous mapping theorem

with the continuous functional

(x(t))0≤t≤T →

((x(t)− x(c1))1{t≥c1})0≤t≤T

(x(c3)− x(c2))

we have for d ≥ c1

Rm(d) :=1√m

∑[m·d]t=[m·c1]+1 Ut∑[m·c3]t=[m·c2]+1 Ut

=

Qm(d)−Qm(c1)

Qm(c3)−Qm(c2)

⇒d

D1W5(d)−D1W5(c1)

D1W5(c3)−D1W5(c2)

in D([c1, T ],R5)× R5.

Now, on {Rm(d), c1 ≤ d ≤ T} we can (separately for both components) apply all calcula-

tions from the proofs of Lemma A.2 and A.3 (using the delta method argument) in Wied

et al. (2012).

Then, we obtain, for 0 ≤ b ≤ T , with c3 = 0, c1 = c2 = 1, that

Sm(b) :=

D [m·b]+2√m

(ρm+[m·b]+2m+1 − ρ1

)D√m (ρm1 − ρ1)

⇒d

W (b+ 1)−W (1)

W (1)

,

where W (·) is a one-dimensional Brownian Motion.

17

Consequently,

V[m·b]+2 = D[m · b] + 2√

m

(ρm+[m·b]+2m+1 − ρm1

)= D

[m · b] + 2√m

(ρm+[m·b]+2m+1 − ρ1

)− D [m · b] + 2√

m(ρm1 − ρ1)

converges to the process

{(W (b+ 1)−W (1))− bW (1), 0 ≤ b ≤ T} = {W (b+ 1)− (b+ 1)W (1), 0 ≤ b ≤ T}.

Applying the continuous mapping theorem and calculating the covariance structure of

the limit process proves the result. �

Proof of Theorem 2

The proof uses the same arguments as Theorem 1 and is mainly based on the fact that

for fixed c ≥ 0, and m → ∞ the process {Pm(d), c ≤ d ≤ T} converges in distribution

to the process{W (d)−W (c) +H

∫ dcg(z)dz, c ≤ d ≤ T

}on D[c, T ] with W (·) being a

standard Brownian Motion. The constant H is, up to a constant, the limit of D under

the null hypothesis, compare the proof of Theorem 2 in Wied et al. (2012). This result

is a generalization of arguments used in Theorem 2 in Wied et al. (2012), executed in

basically the same way as presented in Theorem 1. �

18

Figure 1: Threshold functions for different values of γ and ε = 10−10

0.0 0.2 0.4 0.6 0.8 1.0

0.00.5

1.01.5

2.0

Threshold functions

t

w(t)

gamma=0gamma=0.25gamma=0.45

Table 1: Critical values.T γ = 0 γ = 0.25 γ = 0.450.5 1.2870 1.8001 2.62821 1.5578 1.9924 2.68442 1.8158 2.1684 2.72154 1.9980 2.2467 2.7660

Table 2: Empirical sizes.T m = 250 m = 500 m = 10000.5 0.059 0.058 0.050

γ = 0 1 0.077 0.069 0.0612 0.066 0.054 0.0574 0.063 0.071 0.060

0.5 0.075 0.079 0.047γ = 0.25 1 0.075 0.064 0.057

2 0.087 0.063 0.0524 0.073 0.077 0.064

0.5 0.169 0.125 0.109γ = 0.45 1 0.174 0.136 0.116

2 0.164 0.138 0.1094 0.161 0.128 0.106

19

Figure 2: Log-retuns of S&P 500 and IBM indexes

2000 2005 2010

−10

−50

510

Log−returns of SP500

Year

Leve

ls

2000 2005 2010

−15

−55

Log−returns of IBM

Year

Leve

ls

20

Tab

le3:

Em

pir

ical

pow

er,

stop

pin

gti

me

and

chan

gep

oint

esti

mat

efo

rγ

=0

andk

=[0.0

5mT

].

m=

250

Em

pir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(6)

0.88

465

8096

81.4

920

.27

0.24

622

40.2

526

.58

22.5

60.

841

(12)

0.97

294

115

142.

2512

1.83

38.1

10.

3112

3059

39.0

333

.22

0.85

2(2

5)0.

993

131

160

202

173.

7361

.61

0.35

2539

6752

.36

40.7

70.

774

(50)

0.99

817

822

428

4.75

242.

4491

.41

0.37

4861

9074

.87

47.2

50.

63m

=50

0E

mpir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(12)

0.99

795

112

137

118.

5031

.78

0.26

1230

5837

.99

31.6

70.

831

(25)

113

616

018

816

6.34

43.5

30.

2624

3969

50.6

137

.24

0.73

2(5

0)1

191

224

264

231.

5255

.53

0.23

4758

8369

.03

36.8

10.

534

(100

)1

268

319

376.

2532

8.61

81.7

80.

2495

105

126

113.

9137

.11

0.32

m=

1000

Em

pir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(25)

114

2.75

165

188.

2516

8.74

35.1

30.

2023

3869

49.4

236

.17

0.73

1(5

0)1

198

228

259.

2523

2.47

46.7

40.

2048

6190

.25

72.7

739

.66

0.54

2(1

00)

128

531

836

432

6.92

62.4

60.

1996

104

124

111.

4533

.47

0.30

4(2

00)

142

2.75

475

537.

2548

3.86

85.8

10.

1719

120

021

420

3.85

37.2

20.

18

21

Tab

le4:

Em

pir

ical

pow

er,

stop

pin

gti

me

and

chan

gep

oint

esti

mat

efo

rγ

=0

andk

=[0.5mT

].

m=

250

Em

pir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(62)

0.28

692

107

117

100.

6623

.28

0.23

2952

6346

.24

21.5

20.

461

(125

)0.

397

182

212

232

202.

8739

.51

0.19

9611

812

710

8.09

33.6

70.

312

(250

)0.

514

350.

2540

845

5.75

389.

8090

.70

0.23

208.

2524

1.5

251

217.

2063

.16

0.29

4(5

00)

0.58

669

5.5

809.

590

1.5

772.

1918

4.68

0.23

452.

2549

150

244

8.89

111.

890.

24m

=50

0E

mpir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(125

)0.

505

188

208

230

205.

5331

.18

0.15

9511

912

710

8.33

31.8

00.

291

(250

)0.

740

357

404

448

396.

8170

.90

0.17

211.

7524

325

022

3.17

51.4

60.

232

(500

)0.

856

696

774.

585

076

9.60

125.

250.

1645

749

050

146

6.11

71.9

60.

154

(100

0)0.

915

1323

1497

1679

.514

74.7

230

0.25

0.20

949

989

1000

930.

1316

6.93

0.17

m=

1000

Em

pir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(250

)0.

833

363

403

438

398.

3256

.30

0.14

215

242

251

227.

9041

.92

0.18

1(5

00)

0.95

867

674

482

374

5.32

116.

430.

1544

6.25

489

501

457.

0183

.14

0.18

2(1

000)

0.99

612

71.5

1395

1542

.513

98.9

322

6.79

0.16

925

985.

510

0092

6.62

155.

200.

164

(200

0)1

2438

.526

8829

16.5

2665

.75

435.

850.

1619

0519

7619

9818

70.8

830

0.12

0.16

22

Tab

le5:

Em

pir

ical

pow

er,

stop

pin

gti

me

and

chan

gep

oint

esti

mat

efo

rγ

=0.

25an

dk

=[0.0

5mT

].

m=

250

Em

pir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(6)

0.80

759

.576

9576

.89

24.4

60.

316

1837

24.5

422

.85

0.93

1(1

2)0.

964

80.7

510

513

9.25

113.

1843

.78

0.38

1127

5336

.99

33.0

70.

892

(25)

0.98

811

114

618

916

0.14

72.0

50.

4424

3767

51.0

544

.28

0.86

4(5

0)0.

998

150.

2519

625

421

5.77

96.0

80.

4446

5578

67.8

044

.56

0.65

m=

500

Em

pir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(12)

0.99

380

101

128

107.

0536

.75

0.34

1124

5234

.05

30.5

00.

891

(25)

111

113

517

214

4.28

49.0

30.

3324

3661

45.8

832

.29

0.70

2(5

0)1

159

190

233

202.

5070

.92

0.35

4755

7665

.50

38.0

40.

584

(100

)1

227

275

332.

2528

5.95

83.7

20.

2992

102

117

106.

9536

.24

0.33

m=

1000

Em

pir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(25)

111

313

516

314

0.62

38.5

30.

2723

3458

42.9

930

.75

0.71

1(5

0)1

156.

7518

721

919

2.01

49.1

80.

2546

.75

5677

.25

64.1

430

.84

0.48

2(1

00)

123

527

531

928

1.93

64.8

00.

2292

101.

511

710

6.19

31.4

40.

294

(200

)1

361

412

472

421.

4686

.12

0.20

187.

7519

921

219

7.38

33.1

10.

16

23

Tab

le6:

Em

pir

ical

pow

er,

stop

pin

gti

me

and

chan

gep

oint

esti

mat

efo

rγ

=0.

25an

dk

=[0.5mT

].

m=

250

Em

pir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(62)

0.24

184

102

115

91.1

733

.62

0.36

1850

6142

.75

24.1

60.

561

(125

)0.

350

167

205

225

185.

1460

.89

0.32

77.2

511

412

699

.30

42.0

30.

422

(250

)0.

472

350

409

455

384.

2710

3.04

0.26

210.

7524

1.5

252

217.

5763

.89

0.29

4(5

00)

0.55

668

2.75

804

880.

2573

9.39

230.

860.

3143

2.75

488

501

430.

5013

6.65

0.31

m=

500

Em

pir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(125

)0.

497

180

209

230

198.

3146

.30

0.23

9211

912

810

6.29

36.8

40.

341

(250

)0.

643

363

411

446

393.

3982

.88

0.21

213

243

252

223.

3353

.54

0.23

2(5

00)

0.80

369

7.5

789

876

771.

4015

7.22

0.20

453.

549

050

145

7.12

92.3

40.

204

(100

0)0.

897

1320

1503

1693

1459

.73

362.

230.

2493

698

810

0091

1.61

212.

270.

23m

=10

00E

mpir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(250

)0.

804

360

406

447

393.

4877

.74

0.19

212

241.

525

122

1.35

59.9

70.

251

(500

)0.

951

670.

575

083

3.5

744.

2913

3.66

0.17

453

489

500

456.

2189

.41

0.19

2(1

000)

0.99

312

9714

2715

7614

22.8

525

6.28

0.18

943

986

999

935.

7215

2.72

0.16

4(2

000)

124

84.5

2710

2962

.526

98.1

745

8.50

0.16

1914

1979

1999

1881

.67

298.

380.

15

24

Tab

le7:

Em

pir

ical

pow

er,

stop

pin

gti

me

and

chan

gep

oint

esti

mat

efo

rγ

=0.

45an

dk

=[0.0

5mT

].

m=

250

Em

pir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(6)

0.72

943

6890

64.2

835

.12

0.54

417

38.2

524

.64

23.7

80.

961

(12)

0.92

267

101.

514

310

5.05

61.5

00.

5810

2755

38.1

136

.54

0.95

2(2

5)0.

979

9113

719

714

6.50

94.3

30.

6421

3565

50.6

151

.01

1.00

4(5

0)0.

993

130

194

284

218.

1515

3.63

0.70

4356

8673

.00

62.2

80.

85m

=50

0E

mpir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(12)

0.98

063

9212

795

.96

54.1

30.

5610

2451

34.8

733

.70

0.96

1(2

5)0.

998

9212

416

612

9.43

66.5

00.

5121

3357

43.2

938

.22

0.88

2(5

0)1

127

169

219

171.

0286

.40

0.50

4153

7259

.20

39.1

60.

664

(100

)1

203

260

325

259.

1312

0.38

0.46

8610

011

498

.22

46.3

50.

47m

=10

00E

mpir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(25)

187

116

150

117.

6955

.93

0.47

19.7

530

5340

.03

33.7

30.

841

(50)

112

916

720

516

2.49

69.4

40.

4241

.75

5370

56.1

431

.80

0.56

2(1

00)

120

024

2.5

291

238.

4395

.57

0.40

8699

113

97.0

240

.77

0.42

4(2

00)

133

039

445

437

9.52

138.

110.

3617

319

720

817

9.24

61.4

10.

34

25

Tab

le8:

Em

pir

ical

pow

er,

stop

pin

gti

me

and

chan

gep

oint

esti

mat

efo

rγ

=0.

45an

dk

=[0.5mT

].

m=

250

Em

pir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(62)

0.23

13

896

.540

.12

47.9

91.

192

451

22.1

126

.54

1.20

1(1

25)

0.31

05

157

216

119.

2510

0.07

0.83

381

.512

268

.57

55.3

70.

802

(250

)0.

416

637

1.5

440

264.

3820

2.17

0.76

621

724

815

6.63

112.

020.

714

(500

)0.

427

21.5

748

884

557.

6939

1.11

0.70

45.5

474

498.

534

3.19

213.

240.

62m

=50

0E

mpir

ical

firs

thit

ting

tim

eC

han

gep

oint

esti

mat

eT

(k)

Pow

er1s

tQ

Med

ian

3rd

QM

ean

Std

CV

1st

QM

edia

n3r

dQ

Mea

nStd

CV

0.5

(125

)0.

403

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26

Table 9: Results of the monitoring procedure for three values of γ (EFHT stands for empiricalfirst hitting times).

γ = 0 γ = 0.25 γ = 0.45EFHT Est. changepoints EFHT Est. changepoints EFHT Est. changepoints

984 665 (1999/08/20) 808 682 (1999/09/15) 772 682 (1999/09/15)1580 1399 (2002/07/25) 1554 1399 (2002/07/25) 1529 1399 (2002/07/25)2222 2196 (2005/09/22) 2209 2053 (2005/03/01) 2208 2053 (2005/03/01)3014 2936 (2008/09/02) 2945 2733 (2007/11/09) 2890 2733 (2007/11/09)

Table 10: Empirical correlations at different periods.Period γ = 0 γ = 0.25 γ = 0.45

1 0.6274 0.6237 0.62372 0.5245 0.5264 0.52643 0.7249 0.7410 0.74104 0.6033 0.5364 0.53645 0.8021 0.7800 0.7800

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monitoring correlation change in a sequence of random ... · dominik wied, pedro galeano tu...

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