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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
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