error term (demonstrated by //fj and is not seriously affected by outliers
f//(9J for large samples, T = 3000. This evidence complements the results in
Inclan and Tiao for i.i.d. series in that it shows that this test can be applied
to the residuals of a GARCH for which it would have power to detect breaks
only in the error term for very large samples. This statement is supported by
the simulation results for the and for (w√)2 which show that it lacks
power in detecting breaks in the conditional variance. The reason ut lacks
power is due to the standardization of the returns process rt∕y∕ht that offsets
the corresponding changes in rt and y∕ht and yields an i.i.d. error process,
ut ∙
In Table 3 we report the Lavielle and Moulines test simulation results
for a single break. This table reports the frequency distribution of the num-
ber of change points - to make it comparable with the simulation results for
multiple breaks that follow. The test appears to have good size properties
for either information criterion (BIC or LWZ) and for both DGPs and re-
turn transformations. For the persistent GARCH (DGP2) the LWZ criterion
performs better than the BIC in terms of size. The test appears to have
good power overall for a single break except when there is a small change
in the coefficients of either the constant or dynamics of the GARCH model.
This results is evident in the low persistent GARCH and in particular when
the LWZ is used. However, the L&M test with the LWZ criterion seems to
have relatively more power in detecting changes in the GARCH error than
the BIC. Both criteria appear robust to the outliers. Comparing the L&M
and K&L simulation results we observe that the latter performs relatively
better when the size of change is small (e.g. 0.1 increase in the parame-
ters of DGPl) but suffers from relatively higher size distortions in persistent
GARCH processes.
The remaining simulation analysis addresses the multiple breaks hypothe-
ses given in model (2.2). Table 4 reports the K&L results using a sequential
sample segmentation approach. The frequency distribution of the number of
breaks under the alternative hypotheses is reported. The results show that
the K&L has good power only for large and non-monotone (rather than small
and gradual) changes in the GARCH parameters for any of the DGPs but
for the absolute rather than the squared returns transformations. Similarly
it shares good power for detecting changes in the variance of the error term
in the GARCH process. As the sample size (T) increases the performance of
the test improves even for small change points. The Lavielle and Moulines
multiple breaks results are reported in Table 5. The frequency distribution
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