There are extreme outliers of the real data found in many financial markets and telecommunication networks that play a key role to provide future forecasts and optimal services. Extremal events such as high flood levels of rivers or extremal values of environmental indicators are also often more informative than central parts. Statistical inference on the tail of a distribution has been one of the most important issues for extreme value analysis. The tail index of a time series, which is used as a measure of the thickness of the tail, gives more information than other volatility measures such as variance. For example, it determines the existence of higher-order moments and the limit distribution of sample maximum and minimum.
There is empirical evidence that the tail behavior of real data has changed at some point during the observation period, as mentioned by Hoga (2017), who discussed change point tests for the tail index of
We adopt the idea of Jureckova and Picek (2001), rather than Hill’s estimate approach, and propose a simpler test for the tail change in weakly-dependent processes. We consider a wide class of general weakly-dependent time series sequences, called
In the
The remainder of the paper is organized as follows. Section 2 presents the weak dependence and main results and Section 3 deals with GARCH(1, 1) processes. In Section 4, a Monte Carlo study and real data analysis are conducted. In Section 5 a concluding remark is stated while in Section 6 a proof is given along with the existing proposition.
We consider stationary and weakly dependent sequences of a time series satisfying weak dependence conditions in the following, proposed by Doukhan and Louhichi (1999) to unify all classes of times series under natural conditions on parameters. The following is summarized in the same way as in Doukhan
For
Define the Lipschitz coefficients on
We also set ℳ = ∪
In this work we are interested in change of tail behavior of a stationary weakly dependent process {
In order to construct the test statistics, we apply the empirical distribution functions of the extremes, based on splitting the set of observations into
Here
A class of CUSUM test statistics
We will reject the null hypothesis
A functional central limit theorem (FCLT) establishes the convergence in distribution:
We may use
Our main result for the asymptotic of
(A1) {
(A2) {
The concentration assumption in (A2) is not restrictive as mentioned by Doukhan
Its proof is drawn in Section 6. Now the following remark explains that the arguments of Jureckova and Picek (2001) can be applied in the case of weakly dependent sequences due to the existence of the phantom distribution function.
As mentioned above, in the case of iid random variables with cdf
As a specific model of the
In this section, we consider a GARCH(1, 1) process to present a simulation study for the proposed CUSUM tests. The GARCH process {
(A1)′ The GARCH parameters satisfy
(A2)′ The noise process {
Condition (A1)′ is for the strict stationarity of the GARCH process under the null hypothesis. The strict stationarity along with the bounded density in (A2)′ imply the concentration assumption in (A2). Two conditions (A1)′ and (A2)′ imply conditions (A1) and (A2) as well as ensure the existence of the phantom distribution function for the GARCH process. We discuss these details as follows. Assume that
This implies that the GARCH(1, 1) process is a Bernoulli shift with innovation {
Now we present simulations for the proposed CUSUM tests to see the size and power performances. We consider GARCH(1, 1) models (3.1) with several noises {
Table 1 gives the numerical illustration for values of the test statistics sup_{0≤}
Finite sample size and power properties of the proposed CUSUM tests sup_{0≤}
As a real data application, the proposed CUSUM tests are applied to log-return sets of the Korean stock price index (KOSPI). We adopt financial data from KOSPI-C (Construction) and KOSPI-F (Finance) between 08–02–2013 and 08–27–2017. Figure 2 provides the plots of the financial real data and their standardized log-returns.
For
We have proposed a class of CUSUM tests for tail behavior of the weakly-dependent processes. CUSUM test statistics are constructed using empirical distribution functions with sample extremes and its limiting distribution is derived to be a Brownian bridge. Powers and sizes are computed to see the performance of CUSUM tests via simulations. Also real data analysis is given. From the analysis of our CUSUM tests it might be possible to estimate the tail index of the weakly dependent time series models. The estimation needs refinements of the test statistics, to find the optimal
In order to prove Theorem 1 we need to state the following proposition that is related to the existence of the phantom distribution function of the weakly dependent process with conditions (A1)–(A2) under the null hypothesis. In case of
By the consistency of the estimator in (2.7) it suffices to show that for 0 ≤
According to Proposition 1, there exists a function
Let
Therefore
This work was supported by Research Fund of Gachon University (GCU-2018-0337).
Simulated test statistics sup_{0≤}
Normal | Cauchy | Pareto | N + |
N +C | N+ P | |||||
---|---|---|---|---|---|---|---|---|---|---|
1.00 | 0.0998 | 0.4596 | 0.0981 | 1.4273 | 0.0 | 1.4669 | 2.9695 | 1.7541 | 1.4378 | 1.3734 |
1.25 | 0.0999 | 0.1745 | 0.1945 | 1.2267 | 0.0 | 1.8400 | 2.3199 | 1.6881 | 1.7785 | 1.0353 |
1.50 | 0.0996 | 0.4235 | 0.0998 | 0.8843 | 0.0 | 1.5763 | 2.4109 | 2.0522 | 1.6138 | 1.2572 |
1.75 | 0.0995 | 0.6771 | 0.1866 | 1.0368 | 0.0 | 1.9319 | 2.6258 | 1.4158 | 1.3595 | 1.4746 |
2.00 | 0.0998 | 0.4112 | 0.2407 | 1.0507 | 0.0 | 1.9406 | 3.1334 | 1.8647 | 1.8917 | 1.5973 |
2.25 | 0.0991 | 0.1749 | 0.3150 | 0.7745 | 0.0 | 2.1645 | 3.7539 | 1.8528 | 1.7461 | 1.1337 |
2.50 | 0.0997 | 0.3782 | 0.2420 | 0.7692 | 0.0 | 1.9278 | 2.9041 | 1.6773 | 2.0532 | 1.8058 |
2.75 | 0.0994 | 0.3857 | 0.1525 | 0.3469 | 0.0 | 1.9457 | 3.3271 | 1.7049 | 1.6039 | 0.3581 |
3.00 | 0.0993 | 0.0574 | 0.3995 | 0.8770 | 0.0 | 2.0149 | 3.3641 | 1.9841 | 1.7784 | 1.1438 |
Size of 5% level tests in GARCH(1, 1) model with
Noise distribution | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1.00 | 1.25 | 1.50 | 1.75 | 2.00 | 2.25 | 2.50 | 2.75 | 3.00 | ||||
800 | 4 | 0.25 | 0.040 | 0.043 | 0.052 | 0.059 | 0.046 | 0.051 | 0.055 | 0.050 | 0.052 | |
0.50 | 0.053 | 0.046 | 0.046 | 0.056 | 0.047 | 0.047 | 0.047 | 0.054 | 0.050 | |||
1200 | 6 | 0.25 | 0.012 | 0.009 | 0.016 | 0.010 | 0.008 | 0.008 | 0.021 | 0.011 | 0.011 | |
0.50 | 0.007 | 0.012 | 0.016 | 0.010 | 0.011 | 0.013 | 0.010 | 0.011 | 0.014 | |||
800 | 4 | 0.25 | 0.049 | 0.066 | 0.061 | 0.066 | 0.065 | 0.064 | 0.055 | 0.074 | 0.056 | |
0.50 | 0.052 | 0.045 | 0.064 | 0.066 | 0.052 | 0.063 | 0.071 | 0.060 | 0.062 | |||
1200 | 6 | 0.25 | 0.080 | 0.079 | 0.082 | 0.076 | 0.077 | 0.081 | 0.077 | 0.090 | 0.086 | |
0.50 | 0.084 | 0.080 | 0.085 | 0.083 | 0.086 | 0.083 | 0.078 | 0.074 | 0.078 | |||
Cauchy | 800 | 4 | 0.25 | 0.064 | 0.066 | 0.076 | 0.059 | 0.073 | 0.057 | 0.057 | 0.061 | 0.064 |
0.50 | 0.076 | 0.062 | 0.065 | 0.065 | 0.055 | 0.067 | 0.072 | 0.073 | 0.076 | |||
1200 | 6 | 0.25 | 0.083 | 0.094 | 0.076 | 0.068 | 0.092 | 0.070 | 0.081 | 0.092 | 0.082 | |
0.50 | 0.082 | 0.082 | 0.073 | 0.084 | 0.081 | 0.071 | 0.085 | 0.097 | 0.080 | |||
Pareto | 800 | 4 | 0.25 | 0.081 | 0.050 | 0.091 | 0.098 | 0.081 | 0.057 | 0.054 | 0.064 | 0.057 |
0.50 | 0.065 | 0.040 | 0.046 | 0.080 | 0.071 | 0.054 | 0.068 | 0.072 | 0.057 | |||
1200 | 6 | 0.25 | 0.067 | 0.076 | 0.079 | 0.056 | 0.075 | 0.064 | 0.092 | 0.047 | 0.112 | |
0.50 | 0.113 | 0.092 | 0.117 | 0.068 | 0.077 | 0.085 | 0.085 | 0.100 | 0.093 |
sample size = 800, 1200;
Power of 5% level tests in GARCH(1,1) model with
Noise distribution | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1.00 | 1.25 | 1.50 | 1.75 | 2.00 | 2.25 | 2.50 | 2.75 | 3.00 | ||||
800 | 4 | 0.25 | 1.000 | 1.000 | 0.997 | 0.989 | 0.984 | 0.979 | 0.968 | 0.932 | 0.916 | |
0.50 | 0.999 | 0.994 | 0.984 | 0.974 | 0.957 | 0.929 | 0.916 | 0.897 | 0.882 | |||
1200 | 6 | 0.25 | 1.000 | 0.995 | 0.997 | 0.997 | 0.974 | 0.962 | 0.938 | 0.935 | 0.912 | |
0.50 | 0.998 | 0.993 | 0.986 | 0.963 | 0.952 | 0.933 | 0.890 | 0.866 | 0.839 | |||
800 | 4 | 0.25 | 1.000 | 0.996 | 0.997 | 0.994 | 0.981 | 0.990 | 0.952 | 0.955 | 0.925 | |
0.50 | 0.999 | 0.993 | 0.984 | 0.975 | 0.957 | 0.929 | 0.917 | 0.897 | 0.882 | |||
1200 | 6 | 0.25 | 1.000 | 1.000 | 0.998 | 0.993 | 0.986 | 0.973 | 0.958 | 0.937 | 0.914 | |
0.50 | 1.000 | 0.987 | 0.970 | 0.985 | 0.943 | 0.925 | 0.893 | 0.867 | 0.833 | |||
Cauchy + Pareto | 800 | 4 | 0.25 | 0.852 | 0.759 | 0.664 | 0.600 | 0.508 | 0.455 | 0.432 | 0.402 | 0.351 |
0.50 | 0.724 | 0.633 | 0.546 | 0.450 | 0.429 | 0.371 | 0.324 | 0.275 | 0.289 | |||
1200 | 6 | 0.25 | 0.893 | 0.785 | 0.704 | 0.592 | 0.555 | 0.487 | 0.489 | 0.399 | 0.372 | |
0.50 | 0.774 | 0.669 | 0.590 | 0.496 | 0.425 | 0.401 | 0.331 | 0.296 | 0.260 | |||
Normal + Pareto | 800 | 4 | 0.25 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
0.50 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |||
1200 | 6 | 0.25 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |
0.50 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
sample size = 800, 1200;
Computed test statistics sup_{0≤}
KOSPI-C | KOSPI-F | KOSPI-C | KOSPI-F | |||||
---|---|---|---|---|---|---|---|---|
2.00 | 2.2357 | 3.1586 | 2.2340 | 3.1590 | 2.2309 | 3.1622 | 2.2356 | 3.1616 |
2.25 | 2.2356 | 3.1600 | 2.2356 | 3.1599 | 2.2321 | 3.1572 | 2.2325 | 3.1570 |
2.50 | 2.2318 | 3.1586 | 2.2343 | 3.1588 | 2.2318 | 3.1576 | 2.2355 | 3.1617 |
2.75 | 2.2347 | 3.1617 | 2.2347 | 3.1605 | 0.2934 | 3.1604 | 2.2350 | 3.1621 |
3.00 | 2.2341 | 3.1571 | 2.2355 | 3.1613 | 0.3034 | 0.4094 | 2.2347 | 3.1615 |
3.25 | 2.2339 | 3.1601 | 2.2353 | 3.1622 | 0.3017 | 0.4112 | 0.2617 | 3.1568 |
3.50 | 0.3057 | 0.3838 | 2.2301 | 3.1618 | 0.2296 | 0.4096 | 0.1799 | 0.3464 |
3.75 | 0.3014 | 0.3970 | 2.2327 | 3.1603 | 0.1862 | 0.3057 | 0.2195 | 0.2378 |
4.00 | 0.2986 | 0.4042 | 2.2344 | 3.1597 | 0.1961 | 0.2620 | 0.1909 | 0.2320 |
4.25 | 0.2956 | 0.4044 | 0.2627 | 0.3609 | 0.2034 | 0.2673 | 0.1827 | 0.2809 |
4.50 | 0.2758 | 0.3505 | 0.1774 | 0.2523 | 0.1821 | 0.2755 | 0.1824 | 0.2528 |
sample size = 1000;