Detecting Multiple Breaks in Financial Market Volatility Dynamics

from T = 500 to 3000. For the high-persistent process (DGP2) the test suffers
more serious distortions up to 20%. The power of the K&L test is evaluated
by a number of alternative hypotheses as defined in the previous section. The
results in Table 1 suggest that the tests have good power in detecting breaks
under the following alternative hypotheses: Break in the constant {Hβ~) or
dynamics f //1' j of volatility. The power of the test is demonstrated even
for small changes (e.g. a 0.1 increase) in β₀ for all DGPs. Similar results
apply to the alternative of a small change in the error term ∏∕fj∙ The
power of the tests increases with T in all DGPs. Note that the high nominal
power for the persistent GARCH process (DGP2) needs to be weighted by
the size distortions for this process. In Hβ the presence of outliers or short-
lived jumps, which are evident in financial markets, do not seem to have an
adverse effect on the test. The power of the K&L test is also evaluated for
early change-points for /∕_l'. /∕_l% Hβ and the results show that the K&L test
can detect breaks that occur as early as at π = 0.3 of the sample.

The size and power properties of the K&L test are compared with those
of I&T. The latter is derived for independent series but has been applied
to processes that exhibit dependence (Aggarawal et al., 1999, Granger and
Hyung, 1999). Therefore we examine the properties of the test for (r_t)², ∣r_t∣
as well as the errors of the GARCH process (w√j² where u_t = r_t∕y∕h_t yields
an independent series. Table 2 presents the nominal size and power simu-
lation results of the I&T test under the same null hypothesis of a Normal-
GARCH(I₅I) and all the alternative hypotheses discussed above. Let us
first compare the performance of the I&T for (r_t)² and ∣r_t∣. The I&T test
for (r_t)² suffers from size distortions (above 10%) for all DGPs and sample
sizes but appears to have good power in detecting even small changes in the
GARCH coefficients or the error process (shown by the alternative hypotheses
Hβ, Il'¹, //j j for large T. Nevertheless, its performance is adversely affected
by outliers which appear to be consistently detected as change-points. If in-
stead we adopt the ∖r_t∖ transformation we note some interesting differences.
The I&T test for ∖r_t∖ appears seriously under-sized and with relatively less
power, when compared with (r_t)² for any of the alternative hypotheses. How-
ever, it is interesting to note that for large T (e.g. 1000, 3000) and highly
persistent GARCH processes (e.g. DGP2) the I&T test has good power
properties and is not susceptible to outliers as opposed to (r_t)². Finally we
examine the I&T test for (w√j² which is by design an independent series. The
size of the I&T statistic for (wj² is near the nominal 5% level. The I&T
test for (w√)² has power in detecting even small changes in the variance of

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