Financial Econometrics and Empirical Market Microstructure

Examples of Using the Test

It is worth noting that, in general, the result may significantly affect the use of data in levels or differences, since levels and differences often correspond to different levels of rank correlation. In both the examples below, the data is used in absolute increases of what is displayed on the relevant charts. Also, during computation, abnormally high values in the beginning or end of the series are not noticed, allowing us to use the same weight for observation as in Brodsky et al. (2009) or Penikas (2012).

Following Penikas (2012), the test statistic has been applied to determine the structural shift to quarterly observations of U. S. GDP in the first quarter of 1947 to the second quarter of 2012. There are a total of 262 observations and 261 observations for differences. Tests were made for time series in differences (absolute

increases). In Fig. 4, we present the graphs and Kolmogorov-Smirnov and Cramer - von Mises statistics.

We can see that the two statistics vary in results: Kolmogorov-Smirnov statistic takes the maximum value of 0.2546 in 150th observation, which corresponds to the second quarter of 1982. Kendall’s rank correlations between current and lagged observation before and after 150th observation are respectively 0.290 and 0.251; Cramer-von Mises’ statistic takes the maximum value of 0.01794 at 173th observation 173, which corresponds to the first quarter of 1990; with corresponding rank correlations 0.278 and 0.243. Two statistics give different results, showing the same behavior. Changes in rank correlation at the supposed structural break points are 0.039 and 0.035 respectively. Therefore, to detect structural break, critical values for length N = 250 and zero change in rank correlation should be applied. After comparing obtained values with the table’s critical numbers, we could ascertain existence of a structural break with or without change in copula at significance levels

of 10 and 5 %, but at significance level of 1 %, structural breaks are not revealed (however, estimations of structural break’s moments are different for two statistics).

The test procedure was also applied to 823 observations of the S&P 500 stock index, based on market capitalization of the 500 largest public companies traded in USA stock markets.

Data represents weekly closing values of index from the 17th of August, 1995 to 30th of May, 2011 (source: http://finance. yahoo. com/). Results are presented in Fig. 5.

Both statistics attain the maximum at the 244th observation (0.07623 and 0.010328 respectively for Kolmogorov-Smirnov and Cramer-von Mises), which corresponds to the 24th of April, 2000. Rank correlations of current and lagged values before and after the 244th observation are —0.1216 and —0.0317; difference is 0.0899, which corresponds to zero change in rank correlation. Critical values for N = 1,000 were used, and results of the test procedure do not reveal existence of a structural break for this observations of time series for all statistics and all significance levels, whether or not the copula changes. That corresponds with the
findings of Patton (2012), whose method also didn’t detect a structural break in the same data, thus assuming that the structural break’s moment is unknown (we used the test for structural break in rank correlation, based on a parametric bootstrap).