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assuming mean and variance equal to unity and innovation term taken from
Normal (0, 1) distribution whose variance appropriately scaled for each selected
lag-1 autocorrelation considered. Fig. 10.2(a) suggests that the MK test correctly
converges to the nominal significance level for independent cases (i.e. when
lag-1 autocorrelation is nearly zero) and it considerably deviates from the
nominal significance level as the degree of serial association departs from
zero resulting in biased tests. For example, when the autocorrelation is of the
order of 0.6, the rejection rate is about 30% compared to the nominal rate of
5%. In addition, the rejection rate appears to be independent of the sample
size but it shows a slight increasing tendency as the sample size increases [see
the inset in Fig. 10.2(a)]. The MMK1 and MMK2 tests, applied with the
assumption of an autoregressive process of order-1 [i.e., AR(1)], are able to
address the influence of serial dependence on trend significance for a larger
part of the range of autocorrelations [Figs 10.2(b, c
)
]. On overall basis, the
MMK1 test performs relatively better than the MMK2 test. Thus, the results
of simulation experiments suggest that when the assumption of an AR(1)
process holds true, the MMK1 test would provide better estimates of trend
significance than the MK and MMK2 tests.
Two other approaches namely the pre-whitening (PW) (von Storch, 1995)
and trend-free pre-whitening (TFPW) (Yue et al., 2002b) were also suggested
to address the influence of serial dependence of type AR(1) on the significance
of trends. Yue et al. (2002b) and Fleming and Clark (2002) found that if both
trend and autocorrelation are present in a time series then the PW approach
renders a positively (negatively) autocorrelated time series appearing less
(more) trendy. To address this issue, Wang and Swail (2001) introduced a
modified iterative PW approach. Though the procedure of the TFPW approach
appears to be plausible, it has serious difficulties in preserving the nominal
rejection rates of the null hypothesis at a given significance level. This could
be due to the influence of autocorrelation on trend and vice versa. The PW and
TFPW approaches are not considered further in this chapter. In addition, a
resampling based approach, block bootstrap (BBS) (Kundzewicz and Robson,
Fig. 10.3.
Proportion of time series with significant trends identified with the (a)
MK, (b) MMK1 and (c) MMK2 tests for eight sample sizes ranging from 30 to
100, with an interval of 10. For the MMK tests, only the first autocorrelation,
whether found significant or non-significant, is considered. Figure adopted from
Khaliq et al. (2009b).
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