Detecting Multiple Breaks in Financial Market Volatility Dynamics

Hence all the results in the paper are based on the VARHAC estimator for
σ appearing in (1.4).

The advantage of the K&L test is its validity under a wide class of
processes, including long memory, GARCH-type and nonlinear time series
models. In a study closely related to ours Granger and Hyung (1999) use
a different test, proposed by Inclan and Tiao (1994) for linear models with
breaks such as those proposed by Chen and Tiao (1990) and Engle and Smith
(1999). Aggarwal et al. (1999) also apply this test to GARCH models. The
Inclan and Tiao test (henceforth I&T test) applies in principle to indepen-
dent series and is designed to find a break in the (unconditional) variance
with unknown location. The test statistic is defined as:

IT = ∖Jτ/2 max ∖Dj_ς]

[(∑√=ι Xj/ ∑j=ι ɪə ~ к∕T] ■ ɪt ^is interesting to note that the

asymptotic distribution of the statistic in (1.5) is the same as in (1.4), that is
the supremum of Brownian bridge and hence the same Kolmogorov-Smirnov
type asymptotic distribution. In the Monte Carlo simulations we will exam-
ine how the Inclan and Tiao test performs in non-independent settings, trans-
formations that yield independent processes and compare it to the Kokoszka
and Leipus test.

The Kokoszka and Leipus test is also adapted for the multiple breaks
hypothesis. The number of breaks is determined following a sequential sam-
ple segmentation approach similar to that of Inclan and Tiao (1994) and
Bai (1997). The simulations present some encouraging results regarding the
performance of the test for further theoretical investigation in the multiple
breaks case. The test is applied to a few sample segments given the appro-
priate significance level adjustment.

1.2 Least squares type tests

The change-point literature has recently dealt with the unknown multiple
change points question in weakly dependent processes in a least-squares con-
text. For instance, Bai (1994), Bai and Perron (1998) and Liu et al. (1997)
use the Hajek-Rdnyi inequality to establish the asymptotic distribution of the
test procedure. Recent work by Lavielle and Moulines (2000) has greatly in-
creased the scope of testing for multiple breaks. They prove the Hajek-Rdnyi

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