The Results of Evaluation on Generated Data

A suggested method of detecting and estimation of structural breaks in a time series was used on a generated time series with a specified pattern of dependence.

For analysis we can use six types of copulas, described in Table 1. Dependence from lagged value was generated at different levels, corresponding to Kendall’s rank correlations —0.8, —0.6, —0.4, —0.2, 0, 0.2, 0.4, 0.6, 0.8. Since not all copulas can describe all of the above levels of dependence, we only used 34 copulas.

Copula

Value of rank correlation

Clayton

-0.8, -0.6, -0.4, -0.2, 0.2, 0.4, 0.6, 0.8

Frank

-0.8, -0.6, -0.4, -0.2, 0.2, 0.4, 0.6, 0.8

Gumbel

0, 0.2, 0.4, 0.6, 0.8

Product

0

FGM

-0.2, 0, 0.2

Plackett

-0.8, -0.6, -0.4, -0.2, 0, 0.2, 0.4, 0.6, 0.8

Table 1 Generated copulas

For a fixed length of time series N = 1,000 and for every three parameters of moment of structural break в = 0.3, в = 0.5, в = 0.7 where в = m, m – observation, in which structural break occurs, we generated a time series for all possible pairs of copulas from the 34 described above. In total we turned 3*34*34 = 3,468 various time series. In 3*34 = 102 of them, the copula before structural break are the same as after, i. e. there is no structural break. In 3*74= 222, rank correlation changed without changing of copula. In 3*210 = 630, rank correla­tion didn’t changed but copula changed. In 3*838 = 2,514 changing occurred both in rank correlation and copula. For every time series we calculated the suggested statistics of Kolmogorov-Smirnov and Cramer-von Mises and Andrews-Zivot test (all three modifications). For nonlinear statistics, values were found on a uniform grid of a unit square of dimension of 50 at 50 nodes.

The Andrews-Zivot test, in general, did not lead to accurate results. The null hypothesis of this unit root test (which means rejection structural break existence) was not rejected only in two of 3,468 cases. In other time series tests, statistics have pointed to existence of a structural break in the beginning or end of the sample, and this result didn’t depend on whether or at what point there was a structural break. Regardless, it is the specification of a structural shift test (in trend, intercept, or both).

In Fig. 1, histograms of Andrews-Zivot statistic maximum’s distribution are introduced (for every moment of break (in 300-th, 500-th and 700-th observation of 1,000) and every specification of the test).

Подпись: Anrews-Zivot test (trend), break point at 300 0 200 400 600 800 1000 observation Anrews-Zivot test (intercept), break point at 300 Подпись: Anrews-Zivot test (trend), break point at 500 0 200 400 600 800 1000 observation Anrews-Zivot test (intercept), break point at 500 Подпись: Anrews-Zivot test (trend), break point at 700 0 200 400 600 800 1000 observation Anrews-Zivot test (intercept), break point at 700

For nonlinear copula-based tests, the same trend of large quantity of false signals about the structural break in both ends of sample are observed only with small enough changes in rank correlation, and more for Kolmogorov-Smirnov statistic. Overall, the Cramer-von Mises statistic gives more acceptable results. With the Kolmogorov-Smirnov statistic, the anomaly of high values in edges occurs more often. Using a grid with larger number of nodes decreases this effect, but doesn’t

eliminate it, so the values of statistics have been adjusted for otherwise than suggested by Brodsky et al. (2009). The correction factor equals the squared original factor, which means more weight for values from the middle of sample.

This a reasonable compromise: decreasing the probability of receiving false signals about structural breaks at the edges, thereby decreasing the chance to determine if an actual shift occurred in the same place. Due to the small number of observations, the statistic will show significantly less accurate approximations of real dependence. Moreover, in time series without structural breaks, the large values in the beginning or end of the row also have been observed, so the statistic’s maximum in that range couldn’t be interpreted as an indicator of structural break for small changes in rank correlation.

As an example, maximum’s histograms of Kolmogorov-Smirnov and Cramer – von Mises statistic for structural breaks in the 700th observation and different rank correlation changes are introduced in Figs. 2 and 3. The statistics do not count large values in the first and last 5 % of observations.

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Critical values in absolute scale for different pairs of copulas differ slightly, which lets us calculate critical values as corresponding quantiles of the statistic’s values sample in all different points of time and all possible combinations of copulas before and after the structural break. Values are calculated for different levels in rank correlation changes, and applied in time series generation: there are only 9 values from 0 to 1.6 increments by 0.2. Additionally, some calculations have been performed by the same scheme for time series of length N = 250 and N = 500 observations. We traced the same character of revealed results, and concluded that with equal change in rank correlation, the statistic value is bigger as a rule if there is a change in copula. Critical values were calculated separately for observations with and without change in copula for significance levels of 90, 95 and 99 %. Obtained critical values were larger than found by Brodsky et al. (2009) approximately two to three times. This explains the considerable difference in the problem statement,

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Подпись: Injk-

because now the components of the multi-dimensional vectors of observations are not independent.

These values reveal one interesting fact: for almost all levels of rank correlation, change corresponds to a bigger critical value, but for zero this value is large enough.

Thus, the model procedure of structural break detecting is as follows: first, determine the maximal value of the test statistic and corresponding observation. Then estimate the difference between Kendall’s tau of current and lagged values before and after the suspected moment of structural. According to the critical value of the nearest rank correlation change from the critical values tables and for given significance level, we can make a decision about existence of structural break. If the critical value is exceeded, the moment of the statistic’s maximum is taken as a estimation of structural break point.

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