of the Kokoszka and Leipus and Lavielle and Moulines tests for GARCH type
processes that can be considered as representative models of financial asset
returns. Kokoszka and Leipus (2000) report simulation results that focus on
the sampling distribution of the change-point estimator к for an ARCH(I)
process. They find that its sampling distribution depends on the location
of the change-point, the size of the variance change and its source. The ex-
tensive results presented in this section complement some early simulation
evidence of these tests in establishing their power for univariate and bivari-
ate GARCH processes and a number of alternative change-point hypotheses
often encountered in asset returns. The robust character of the test is also ex-
amined in the presence of outliers given the stylized fact of jumps or extreme
observations observed in volatility and absolute returns which may lead to
spurious nonlinearities or IGARCH effects (e.g. Lamoureux and Lastrapes
(1990), van Dijk et al. (1999)).
The apparent similarity of the CUSUM-type statistics in K&L and I&T
calls for an interesting comparison which brings about the connection be-
tween these two tests and their power in detecting change-points in GARCH
processes as well as jumps in financial markets. For comparison purposes
both tests in (1.4) and (1.5) are evaluated for absolute and squared returns
whereas the I&T test is also applied to the residuals of a GARCH process
given that this test is originally designed for independent processes. First we
discuss the simulation design followed by an analysis of the results.
2.1 Simulation design
The simulated returns processes are generated from a univariate Normal-
GARCH process given by:
1∖t = h∙i,t (2 1)
hi,t = ωi + aiujt-1 + βihi,t-γ, t = l,...,T and г = 0,1.
where ri,t is the returns process generated by the product of ui,t which is
i.i.d.(0,1) and the volatility hi,t that has a GARCH(I5I) specification. This
process without change points is denoted by i = 0 whereas a break in any
of the parameters of the process is symbolized by г = 1. The models used
in the simulation study are representative of financial markets data with the
following set of parameters that capture a range of degrees of volatility per-
sistence (measured by a0 + β0). The vector parameters (ω0, α0, β0) in (2.1)