ONE MODIFICATION OF THE CUSUM TEST FOR DETECTION
EARLY STRUCTURAL CHANGES
Julia Bondarenko
Department of Economic and Social Sciences, Helmut-Schmidt University Hamburg
University of the Federal Armed Forces Hamburg, Holstenhofweg 85, 22043 Hamburg, Germany
Keywords:
CUSUM test for structural breaks, recursive residuals, power of test, mixture distribution, Monte Carlo simu-
lation.
Abstract:
Structural shifts in time series can occur as consequences of complex processes, arising in system. An ig-
norance of such structural changes can cause an associated regression model misspecification. In practice,
early detection and response to outbreaks, causing the changes in a process, is highly important. The famous
CUSUM test of Brown, Durbin & Evans, has a poor power in detecting the structural breaks in parameters
occurring early (and also late) in the sample. In this paper, we propose CUSUM-similar test which, due to the
transformation of recursive residuals forces the detection of temporal dependence structure in linear regres-
sion model and has a larger power for the early structural breaks. Here our interest centres on the detection of
single breaks occurring in parameters of the linear model. Distribution and other probabilistic characteristics
of the transformed residuals are provided, the boundaries for the new test are derived. The new test can be
considered then as a complement to the standard CUSUM test.
1 INTRODUCTION
Structural shifts in time series can occur as con-
sequences of complex processes, arising in system
(Basseville and Nikiforov, 1993). An ignorance of
such structural changes can cause an associated re-
gression model misspecification. The evidence of
the parameters instability in linear models can be
detected by the number of corresponding diagnos-
tic tests. Particularly, the famous and important of
them include fluctuation tests, with the CUSUM and
CUSUMQ tests of Brown, Durbin & Evans (BDE
tests) (R. L. Brown and Evans, 1975), standing first
on the list. These tests are easy for implement-
ing and based on the calculating of the cumulative
sums of recursive residuals (CUSUM) and the cu-
mulative sums of the squares of recursive residuals
(CUSUMQ) for regression. The CUSUM test is gen-
erally used to detect the systematic movements of pa-
rameters, whereas CUSUMQ test tends to capture the
sudden, haphazard movements.
A number of extensions of the standard BDE
CUSUM tests have been developed, like the CUSUM
and CUSUM of squares tests using OLS (ordinary
least squares) residuals instead of recursive resid-
uals (Proberger and Kraemer, 1992), the CUSUM
tests with lagged dependent variables in regression
(W. Proberger and Alt, 1989), the CUSUM tests with
non-stationary regressors (Inder and Hao, 1996), the
MOSUM and MOSUM of squares tests putting em-
phasis on moving averages rather than cumulative
sums (C.-S. J. Chu and Kuan, 1995a), the moving-
estimates test (C.-S. J. Chu and Kuan, 1995b), etc.
It is a well-known fact, that, CUSUM tests, both
with recursive and OLS residuals, have a poor power
in detecting the structural breaks in parameters occur-
ring early and late in the sample, as well as the ones
orthogonal to the mean regressor (see (W. Kraemer
and Alt, 1988); (Kraemer and Sonnberg, 1986), pp.
50-51; (Proberger and Kraemer, 1990); (Proberger
and Kraemer, 1992)). A modification of recursive
residuals CUSUM test, which is robust to the later
problem, was proposed in (Luger, 2001). The rea-
son for earlier problem is that CUSUM has no chance
to cumulate for the such kind of breaks. In particular,
some improving, due to alternative test boundaries de-
veloping, was suggested, in (Zeileis, 2000) for OLS
CUSUM test.
In practice, early detection and response to out-
breaks, causing the changes in a process, is highly
important. In this paper, we propose a CUSUM-
similar test which, due to the transformation of re-
cursive residuals forces the detection of temporal de-
pendence structure in linear regression model and has
18
Bondarenko J. (2008).
ONE MODIFICATION OF THE CUSUM TEST FOR DETECTION EARLY STRUCTURAL CHANGES.
In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - SPSMC, pages 18-22
DOI: 10.5220/0001482900180022
Copyright
c
SciTePress
a larger power for the early structural breaks. Here
our interest centres on the detection of single breaks
occurring in parameters of the linear model. Distri-
bution and other probabilistic characteristics of the
transformed residuals are also provided, the bound-
aries for the new test are derived. The new test can
be considered then as a complement to the standard
CUSUM test. The paper is organized as follows. Sec-
tion 2 presents the BDE’s original formulation of the
CUSUM test, discusses its main features. Section 3
offers our modification of the CUSUM testing pro-
cedure. In the Section 4 we conduct comparison of
two tests via Monte Carlo simulation study. Section 5
summarizes and concludes.
2 STANDARD CUSUM TEST
Consider the linear regression model
y
t
= x
0
t
β
t
+ ε
t
, t = 1, ..., T , (1)
where y
t
is the tth observation of the dependent vari-
able, x
t
is the k ×1 vector of covariates at time t, β
t
are unknown parameters, and ε
t
are independent and
normally distributed with zero mean and variance σ
2
.
We are interested in testing for a discrete jump in at
least one of the β
t
at the unknown time point. The
null hypothesis can be formulated as
H
0
: β
t
= β, t = 1, ..., T , (2)
with alternative H
1
that at time T
at least one of the
β
t
changes its value:
H
1
: β
t
= β, t = 1, ..., T
< T ;β
t
6= β, t = T
+1, ...,T.
(3)
The CUSUM test suggested by Brown, Durbin
and Evans (BDE) is a standard and commonly used
diagnostic test for this kind of situations. BDE’s
CUSUM test uses the standardized residuals defined
as
w
t
=
y
t
x
0
t
b
β
t1
q
1 + x
t
X
0
t1
X
t1
1
x
0
t
for t = k + 1, k + 2, ..., T,
(4)
where
b
β
t1
is the OLS-estimator of β based on the
first t 1 observations,
b
β
t1
=
X
0
t1
X
t1
1
X
t1
Y
t1
, and X
t1
and Y
t1
are the (t 1) ×k and (t 1) ×1
matrices that obtain by stacking x
s
and y
s
, respec-
tively, for s = 1, 2, ...,t 1.
The advantage of working with w
t
s defined by (4): it
can be shown, that under H
0
(2), they are independent
normally distributed with zero mean and variance σ
2
.
BDE CUSUM test is based on the cumulated sums of
standardized residuals:
C
t
=
1
b
σ
t
s=k+1
w
s
, (5)
where
b
σ is OLS-estimate of σ,
c
σ
2
=
1
T k
t
s=k+1
w
2
s
.
We cannot obtain the explicit distribution of C
t
. But,
under null hypothesis of parameter stability, the ex-
pected value and variance of the statistic C
t
should be
equal to zero and the number of normalized residu-
als being summed. Hence, the continuous Brownian
motion process Z
t
can be considered as a good ap-
proximation of the discrete path of C
t
. Under H
0
the
sequence C
t
is thus a sequence of the approximately
normal variables, where E (C
t
) = 0, Var (C
t
) = t k,
Cov (C
t
,C
r
) = min(t,r) k. Confidence bounds for
the cumulated sums C
t
are then obtained by plot-
ting the two straight lines connecting the points k ±
a
T k and T ±3a
T k, where a is a parameter
depending on the α - significance level chosen, and
is calculated on the basis of results for the Gaussian
process. Namely, a = 1.143 for α = 0.01; a = 0.948
for α = 0.05; a = 0.850 for α = 0.1 (see (R. L. Brown
and Evans, 1975)).
Unfortunately, the CUSUM test suffers from low
power, which decreases dramatically if the change
point is close to the early beginning or to the end of
the sample. In the next section we consider one mod-
ification of the CUSUM test based on the sconced cu-
mulative sums.
3 MODIFICATION OF CUSUM
TEST
Let’s take into account the temporal structure of the
residuals w
t
, and construct a new sequence of random
variables u
t
, t = k + 1, ..., T :
u
t
= w
t
+ c w
t
I
{
w
t
w
t1
> 0
}
, w
k
= 0. (6)
where, as before, w
t
˜ N
0, σ
2
and independent, I
{
·
}
is indicator function, and c is some constant, c > 0,
which can be considered as a penalty magnitude for
upward or downward trends in CUSUM values. We
will obtain now a distribution of the variables u
t
.
Theorem. Let w
t
are independent identically dis-
tributed random variables, having a common symmet-
ric about zero distribution, where F
w
(x) is a probabil-
ity function, and c > 0. Then
ONE MODIFICATION OF THE CUSUM TEST FOR DETECTION EARLY STRUCTURAL CHANGES
19
F
u
t
(x) =
1
2
F
w
t
(x) +F
w
t
x
1 +c

. (7)
Proof. We can perform F
u
t
(x) as a sum of the fol-
lowing probabilities:
F
u
t
(x) = P [u
t
< x] = P [w
t
+ c w
t
I
{
w
t
w
t1
> 0
}
< x] = (8)
P[w
t
+ c w
t
< x ; w
t
w
t1
> 0] +
P[w
t
< x ; w
t
w
t1
0].
For simplicity let’s consider two cases:
1) x 0. Then (8) can be written as:
P[w
t
+ c w
t
< x ; w
t1
< 0] +
P[w
t
x ; w
t1
0] = P
w
t
<
x
1+c
; w
t1
< 0
+
P[w
t
x ; w
t1
0] = F
w
t
x
1+c
F
w
t
(0) +
F
w
t
(x)(1 F
w
t
(0)) = F
w
t
x
1+c
F
w
t
(0) + F
w
t
(x)
F
w
t
(x)F
w
t
(0) =
1
2
F
w
t
x
1+c
+ F
w
t
(x)
.
2) x > 0. (8) has a form:
P[w
t
+ c w
t
< 0 ;w
t
w
t1
> 0] +
P[0 w
t
+ c w
t
< x ; w
t
w
t1
> 0]+
P[w
t
< x ; w
t
w
t1
0] = P [w
t
< 0 ; w
t1
< 0] +
P
0 w
t
<
x
1+c
; w
t1
> 0
+
P[w
t
< x ; w
t
w
t1
0] = F
w
t
(0)F
w
t
(0) +
F
w
t
x
1+c
F
w
t
(0)
F
w
t
(0)+
P[w
t
< 0 ;w
t1
0] + P [0 w
t
< x ; w
t1
< 0] =
F
w
(0)F
w
(0)+
F
w
t
x
1+c
F
w
t
(0)
F
w
t
(0) + F
w
t
(0)(1 F
w
t
(0)) +
(F
w
t
(x) F
w
t
(0))F
w
t
(0) =
1
4
+
1
2
F
w
t
x
1+c
1
2
+
1
4
+
1
2
F
w
t
(x)
1
2
=
1
2
F
w
t
x
1+c
+ F
w
t
(x)
.
Thus, the CDF (PDF) of the random variables
u
t
is a mixture of two normal distributions. It fol-
lows from the Theorem that u
t
s have zero expectation,
E (u
t
) = 0, and variance Var (u
t
) =
σ
2
2
1 +(1 + c)
2
.
The joint CDF function obtained can be obtained by
analogy with (7).
Covariance, C ov (u
t
, u
t+1
)=
E [(u
t
E (u
t
))(u
t+1
E (u
t+1
))]=E (u
t
u
t+1
), equals
to zero. In addition, variables u
t
s have zero skewness
s(u
t
) = 0, and kurtosis κ(u
t
) =
6
[
1+(1+c)
4
]
(
1+(1+c)
2
)
2
, revealing
fatter tails.
The new statistic, which we will call ”penalized”
CUSUM (PCUSUM), can be written as following:
C
p
t
=
1
c
σ
p
t
s=k+1
u
s
=
1
c
σ
p
t
s=k+1
w
s
+ c
t
s=k+1
w
s
I
{
w
s
w
s1
> 0
}
!
,
(9)
where
c
σ
p
2
=
1
T k
t
s=k+1
u
2
s
. It is follows from the prop-
erties of u
t
, that E
C
p
t
= 0 but
c
σ
p
b
σ, thus, we ex-
pect that the PCUSUM test has a chance to cumulate
well and to detect structural shifts only in very begin-
ning of the sample.
Like the original CUSUM test (5), the modified
test (9) is only an asymptotic test and has no any
certain distribution. Obviously, under c 0, the
sequence C
p
k+1
, ...,C
p
t
may be approximated by the
Brownian motion process mentioned in Section 1, and
the bounds for cumulated sums C
p
t
are calculated then
as the ones for C
t
. But for larger values c this can not
be applied. Hence, we have here a subproblem of sim-
ulation of the presented test boundaries.
We will partially implement the techniques used
in (Tanizaki, 1995) for confidence intervals calcula-
tion. The simulation algorithm may be carried out
as follows. Let us generate L replicates. In each
replicate i, i = 1, 2, ..., L, we simulate T K random
variables u
i,t
, t = k + 1,k + 2,...,T , pairwise depen-
dent and uncorrelated, drawn from the mixture of
two bivariate normal distributions (7). Then the cu-
mulated sums, s
p
i,t
=
1
c
σ
p
t
s=k+1
u
i,s
, are calculated. At
the significance level α we have for statistic (9) that
P[L
p
k+1
< C
p
k+1
< U
p
k+1
,...,L
p
T
< C
p
T
< U
p
T
] = 1 α,
where L
p
t
and U
p
t
are correspondingly the lower and
upper bounds for the value C
p
t
, and
P
L
p
k+1
< C
p
k+1
< U
p
k+1
, ..., L
p
T
< C
p
T
< U
p
T
6=
P
L
p
t
< C
p
t
< U
p
t

T k
for any t = k + 1, ..., T , since C
p
t
are not indepen-
dent. Let assume that P
L
p
t
< C
p
t
< U
p
t
= 1 α
p
,
and denote α = f (α
p
), where f is some unknown
function (in the case of independence one has f (x) =
1 x
T k
), which we will obtain by simulation. Ap-
plying the Newton-Raphson algorithm, we calculate
our α
p
following the scheme:
α
( j)
p
= α
( j1)
p
+ d
( j)
α f
α
( j1)
p

,
where j is iteration number, d
( j)
= δd
( j1)
with
δ = 0.5 and d
(0)
= 1, α
(0)
p
= α. The conver-
gence criterion is
α
( j)
p
α
( j1)
p
< 0.0001. Re-
peat, that the function f
α
( j)
p
is derived at jth it-
eration as following: f
α
( j)
p
equals to the num-
ber of sequences
n
s
p
i,k+1
, ..., s
p
i,T
o
L
i=1
within inter-
vals
L
p
i,k+1
,U
p
i,k+1
( j)
,...,
L
p
i,T
,U
p
i,T
( j)
divided by
L, where intervals
L
p
i,t
,U
p
i,t
( j)
are obtained for
the value α
( j)
p
. Note, that unlike the standard
CUSUM statistic, which has symmetric lower and
upper boundaries, we don’t claim here L
p
t
= U
p
t
for PCUSUM statistic. To be fair to the standard
ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics
20
CUSUM, we use a linear regression to obtain the lin-
ear boundaries from the curved ones:
e
L
p
t
= a
1L
+a
2L
t
and
e
U
p
t
= a
1U
+ a
2U
t, where a
1L
, a
1U
are intercepts,
a
2L
, a
2U
are slopes,
e
L
p
t
,
e
U
p
t
are the best fitting lines.
4 MONTE CARLO STUDY
In this section we present a Monte Carlo study of the
new CUSUM test in comparison with the classical
one. In order to see the performance of C
p
t
, we con-
sider the model (1), where the matrix of independent
variables x
t
has the following design:
X
MC
=
n
1, (1)
t
0
o
T
t=1
, (10)
which was also used in simulation study in (In-
der and Hao, 1996) and (Kraemer and Sonnberg,
1986). Values ε
t
, as before, are independent nor-
mal, generated with parameters 0 and 1. We sim-
ulate our responses y
t
for three sample sizes, T =
20 (small sample), T = 50 (medium sample) and
T = 500 (large sample), with β = [10, 2]
0
under the
null hypothesis of parameters constancy. The signif-
icance level α = 0.05, the computed empirical levels
α
p
for samples under c = {0.2; 0.5; 1.5} are cor-
respondingly α
p
=
{
0.0075; 0.0049; 0.0054
}
(T =
20), α
p
=
{
0.0055; 0.0036; 0.0029
}
(T = 50), α
p
=
{
0.0011; 0.0013; 0.0016
}
(T = 500).
Empirical (actual) test sizes, based on the simu-
lated data, at the nominal size of 5%, and c = {0.2;
0.5; 1.5} in PCUSUM test, are presented in the Table
1. Generally estimated sizes, calculated as rejection
rates under null hypothesis, with Monte-Carlo repli-
cations number N = 5000, are either below the nom-
inal size for CUSUM test, resulting in more ”liberal”
test, or almost equal to the nominal size. However,
the empirical sizes for PCUSUM test under c = 1.5
are larger than 0.05, declaring the more ”sensitive”
test.
Table 1: Empirical Sizes of the Tests, α = 0.05.
Test T = 20 T = 50 T = 500
CUSUM 0.0154 0.0276 0.0458
PCUSUM, c = 0.2 0.0338 0.0453 0.0476
PCUSUM, c = 0.5 0.0512 0.0510 0.0444
PCUSUM, c = 1.5 0.0686 0.0586 0.0646
Now, we introduce at time T
= [λT ], where λ
can take any values between 0 and 1, some struc-
tural shift. Let’s consider a single structural shift
in parameters β is given by ∆β =
b
0
T
[cosφ, sinφ]
0
,
where φ is the angle between ∆β and mean regressor
r =
1
T
T
t=1
x
1t
,
1
T
T
t=1
x
2t
0
= [1, 0]
0
, b
0
is a constant
determining the intensity of the shift ,
k
∆β
k
=
|
b
0
|
T
.
We will take a number of different values of b
0
and φ,
namely, b
0
=
{
12; 8; 6; 3; 3; 6; 8; 12
}
(pos-
itive and negative values, as we don’t claim the
boundaries of PCUSUM to be symmetric) and φ =
{
0
}
. In other words, we are testing two hypotheses:
H
0
: parameters are constant for 1 t T against
H
1
: parameters have two different constant values,
for 1 t < T
and T
t T .
It is possible to show that, as we have expected,
the linear boundaries for PCUSUM test become nar-
rower than CUSUM test boundaries only at the begin-
ning of the sample. Hence, it makes sense to choose
λ corresponding to the structural changes at the be-
ginning of the sample, λ = 0.3, for example. Empiri-
cal power was calculated as a probability that the test
statistic under the alternative hypothesis exceeded the
significance threshold calculated from the distribution
under the null hypothesis (a frequency of the null hy-
pothesis rejection under the alternative hypothesis).
The obtained power plots, under N = 1000, λ = 0.3,
α = 5%, c {0.2; 0.5; 1.5} for T = 20, T = 50 and
T = 500 are presented on the Figures 1, 2 and 3, cor-
respondingly.
Figure 1: Power plots under T = 20, λ = 0.3.
Figure 2: Power plots under T = 50, λ = 0.3.
The obtained simulation results for the small sam-
ple with T = 20 reveal that the PCUSUM outper-
forms the CUSUM. For small values of shift inten-
sity b
0
and for φ = 90 this superiority is quite in-
sufficient, for all samples. For the medium sample
T = 50, PCUSUM with c = 1.5 has higher power ev-
ONE MODIFICATION OF THE CUSUM TEST FOR DETECTION EARLY STRUCTURAL CHANGES
21
Figure 3: Power plots under T = 500, λ = 0.3.
erywhere, but PCUSUM with c = 0.5 and c = 0.2 -
only for negative values of b
0
. In big sample with
T = 500 an advantage is exhibited only by PCUSUM
with c = 1.5, for positive b
0
s. By virtue of the bound-
aries non-symmetry, the results are different for posi-
tive and negative values of b
0
also among PCUSUM
tests: for example, for T = 20 PCUSUM with larger
parameter c outperforms PCUSUM with smaller one
for positive b
0
s, but for negative b
0
s PCUSUM with
smaller value of c remains more effective.
5 CONCLUSIONS
In this paper we proposed a modified, based on the
penalized residuals, version of the standard BDE
CUSUM test for single structural breaks in parame-
ters of linear regression. The new test, PCUSUM,
is recommended as a complement to the standard
CUSUM test for better detecting the structural shifts
occurring early in the samples. Simulation results
have shown, that the modified CUSUM test has the
better chance to cumulate parameter breaks, occurred
at the beginning of the sample.
The subjects of eventual future research are:
closer examination of properties of the PCUSUM
test boundaries, their comparison with derived
curved boundaries of the standard CUSUM test;
adoption of the modified residuals (6) by CUSUM
of squares test (CUSUMQ) for testing structural
changes in variance (serial correlation and het-
eroscedasticity);
appropriate transformation of the proposed new
test for both early and late structural breaks (in-
clusion of the more lagged residuals in (6), etc).
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