INFLUENCE OF THE KURTOSIS VALUE ON THE RESULT OF GOODNESS OF FIT TEST FOR THE DAILY R ATE OF RETURN ON SELECTED WIG20 AND WIG30 COMPANIES

In the paper, the modeling of empirical distributions of return rates on WIG20 and WIG30 companies was conducted. The validity of the hypothesis was tested, which stated that the fitting of theoretical distributions to empirical distributions of return rates – where the fitting was tested by the chi-squared test – depends on the value of kurtosis. In order to prove the validity of the hypothesis, studies on four sets of diversified data were conducted.


Introduction
This work has been inspired by the observations made while working on paper (Bednarz-Okrzyńska, 2016), where modeling of empirical distributions of return rates on WIG20 companies was conducted by means of theoretical distributions.
On the basis of the studies conducted in the aforementioned work, the dependence of the value of excess kurtosis on the time interval applied in measuring return rates was observed. Namely, for the daily data, the excess kurtosis equaled 4.112, and for the monthly data -1.232. At the same time, the influence of the time interval used in return rate measurements on the results of modeling empirical distributions of return rates using the Gaussian and Laplace distributions was also observed. Namely, the normal distribution provided a better fit for the monthly data, and the Laplace distribution proved more useful for the daily data.
The results obtained in paper (Bednarz-Okrzyńska, 2016) point toward the relationship between the value of excess kurtosis and the result of goodness-of-fit test applied to the above mentioned theoretical distributions. Therefore, this paper concentrates on the issue of influence of the value of excess kurtosis on the result of a goodness-of-fit test.
Below, the basic relations, used later throughout the paper, are presented. First order moment m (mean value) is derived from formula where: x i -observations; i = 1, 2, 3, ..., N.
The variance is given by: Fourth central moment is given by: Standardized fourth central moment KU (kurtosis) is derived from formula (Krishnamoorthy, 2006;Sobczyk, 2004): Another measure derived from the kurtosis is excess ex (Sobczyk, 2004): Yet another relation describing concentration K can be found in (Tarczyński, 2002): where: KU is given by (4).
For the normal distribution, kurtosis takes the value K=0, however for the Laplace distribution K=3. This means that the value of kurtosis should have an impact on the results of the goodness-of-fit test when modeling empirical distributions of return rates using the Gaussian and Laplace distributions.
The primary goal of this paper is to determine the validity of the hypothesis which states that the fitting of theoretical distributions to empirical distributions of return rates -measured with the chi-squared goodness-of-fit test -depends on the value of kurtosis.
In order to validate the hypothesis, appropriate studies will be conducted on the sets of data of diversified nature.
Therefore, four sets of data were investigated including return rates on selected WIG companies. Set A included daily, weekly and monthly data on: WIG, WIG20, MWIG40, SWIG80 indexes and WIG20 companies from the period of one year, 2013. It comprised in total 72 observations.
Set B included return rates on the same indexes and companies as Set A, however it comprised only the daily data from the period 2010-2013 including 89 observations. Set C included daily return rates on WIG and WIG20 indexes as well as WIG20 companies covering the period 4.10.2001-30.08.2017. Within this time interval, four boom subintervals (H4, H5, H6, H7) and three slump subintervals (B4, B5, B6) were identified. The analysis comprised WIG20 companies which were listed on the WIG20 index during each slump and boom period under study. The following boom and slump periods were

Results of calculations
The data set A comprised the values of daily, weekly and monthly return rates on WIG20 companies in 2013. For the sake of clarity, Figure 1  . Generally ratio NEGG is given by: where: LG -number of cases when K > BG in total, LN -number of cases when K > BG and Hpn = 0.    Figure 3 made for Set A corresponds to the Laplace distribution -labeling the same as in Figure 1. This time it was assumed that for the concentration smaller than the value of threshold BL, a negative result of the chi-squared test would be obtained. Ratio NEGL is given by: LG (8) where: LG -number of cases when K < BL in total, LN -number of cases when K < BL and HpL = 0

LN NEGL
LG Subsequent calculations were made for Set B -values of daily return rates on WIG20 companies in the period 2010-2013.

LN NEGG
LG and for BG2 = 5.3 ratio NEGG = 1 (LN = LG = 6). For threshold value BG2, the largest value of NEGG can be observed, and for threshold value B1, the largest number of observations -15 out of 16 -can be observed.   In Figure 6 two threshold values can be found BL1 = 0.7 and BL2 = 1.1 for which NEGL = 0.5. However taking into account the number of observations, value BL2 should be rated more highly, for which = 9 0.5 18 can be observed, while for BL1, only = 4 0.5 8 can be observed. Figure 7 was made for Set C of data. In Figure 7 threshold value BG = 3.9 can be found for which ratio NEGG assumes the highest value, NEGG = 0.933.

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Influence of the kurtosis value on the result of goodness of fit test for the daily rate of return on selected WIG20 and WIG30 companies  Figure 6. Values of ratio NEGL as a function of threshold value B2 (Set B) Source: author's own study.  Figure 7. Values of ratio NEGG as a function of threshold value B1 (Set C) Source: author's own study.
In Figure 8 threshold value BL = 2.6 can be found for which ratio NEGL assumes the highest value, NEGL = 0.552.
The last set of data (D) comprised the values of daily return rates on WIG30 bank sector stocks. Figure 9 presents the values of ratio NEGG as a function of threshold value B1. From the results presented in Figure 9,   Vol. 27/2, 3/2018 Influence of the kurtosis value on the result of goodness of fit test for the daily rate of return on selected WIG20 and WIG30 companies The results in Table 1 prove that the order of the data sets: A, B, C, D was not coincidental -the order corresponds to increasing maximum values of ratios NEGG and NEGL. The worst results for Set A stem from the heterogeneity of the data: daily, weekly and monthly values of return rates. As it was already mentioned in Introduction, the value of kurtosis for the monthly data was 3.34 times larger than for the daily data. Set A was included in the paper on purpose, however, so as to warn against combining return rates with different measure time intervals.

Conclusions
Sets B and C yield similar maximum values of ratios NEGG and NEGL, which results from the fact that they comprise homogenous (daily) data.
The best results were obtained for Set D, which comprised the data of doubled homogeneity, since they were daily data and the companies represented the same sector (banking).
As far as the hypothesis put forward in Introduction is concerned, it should be noticed that the hypothesis was validated for the normal distribution, since the maximum values of ratio NEGG > 0.9 -obtained for Sets B, C, D -are satisfying. In the case of the Laplace distribution, no unequivocal answer was found. Sets B and C did not provide decisive results. Only in the case of Set D, the value of ratio NEGL = 0.875 can be accepted as supportive evidence for the hypothesis.