39
Kendall’r rank correlation caused it to be used frequently in the real data analysis [2–4]. It is
especially important in the analysis of variables with non-Gaussian behaviour often demonstrated in the real time series. The other measure of relationship that need to be mentioned
is the Spearman correlation. The Spearman correlation between two variables is equal to the
Pearson correlation between the rank values of those two variables; while Pearson’s correlation assesses linear relationships, Spearman’s correlation assesses monotonic relationships
(whether linear or not) [2].
In the more advanced statistical analysis one can also consider other measures of dependence adequate especially for processes with infinite variance. We only mention here the codifference [5–7], covariation [5,8–10] and fractional lower order covariance [11–12]. All of them
are used in the practical applications and can be alternatives to the classical measures of
dependence.
Beyond the aforementioned measures of dependence, in the literature there are considered
weighted correlation coefficients [13–14], to obtain reliable full-rank dependence measures
which take under consideration the fact that present observations weight more than past
measurements. Therefore the weighted correlation coefficients can characterise the dynamics of the dependence structure in the better way than the classical measures [13]. It should
be mentioned, the idea of weighted correlations is not new [14–17]. The study of the optimal
weights, aimed at avoiding unwelcome side effects has been mostly overlooked in the literature.
In this paper we analyse three selected correlation coefficients presented in [13], namely classical Pearson correlation coefficient, Kendall’s rank correlation and exponentially weighted
Pearson correlation coefficient. We apply them in order to find the relationship between main
market risk factors of KGHM capital group. We indicate that three mentioned correlation
coefficients can give different information about the structure of dependence dynamics of
the selected risk factors. We analyse the relationship as a function of time in order to indicate
it changes when the unexpected events on the market appear. The main result of the paper
is to demonstrate that the main market risk factors of KGHM capital group are strongly
correlated and different measures of relationship change over time. However, the weighted
correlation coefficient is more sensitive on the market changes with respect to the measures
that take under consideration all observations with the same weights. Thus in order to enable
using the examined relation for forecasting purposes it is crucial to build models taking under
account this kind of correlation measures.
The rest of the paper is organized as follows: in section 2 we present the main market
risk factors of the KGHM capital group and indicate their specific behaviour. Section 3 is
devoted to the measuring of the dependence dynamics of the selected marker risk factors. We
introduce three analysed measures of dependence and in section 4 indicate their behaviour
for the analysed time series taking into account parameters of selected measures such as the
weight factor in the weighted Pearson correlation coefficient or the way of their determining
(data frequency). All analysed factors influence the behaviour of the structure of dependence
dynamics. Last section concludes the paper.
2 MAIN MARKET RISK FACTORS OF KGHM CAPITAL GROUP
In this section we present briefly the main market risk factors of KGHM capital group that
are analysed in the following sections. In our analysis we take under consideration the following factors: copper (Cu) price, gold (Au) and silver (Ag) prices, nickel (Ni) price, EURUSD
and USDPLN exchange rates and LIBOR. Due to lack of space we present only selected
time series of the risk factors that will be analysed in following sections, namely Cu, Ag, Au
and USDPLN. In Fig. 1 we demonstrate their behaviour along time. In this paper we examine
the daily real time series from 1st of January 2000 to 31st of March 2017.
As one can see, the data exhibit very specific behaviour. In Fig. 2, we present the logarithmic rates of return for selected risk factors. They change dynamically over time and in
many cases the variance of the data is very high, especially for metals. Moreover, one can see
non-Gaussian behaviour of the analysed time series. Volatility clustering appears in similar
Kendall’r rank correlation caused it to be used frequently in the real data analysis [2–4]. It is
especially important in the analysis of variables with non-Gaussian behaviour often demonstrated in the real time series. The other measure of relationship that need to be mentioned
is the Spearman correlation. The Spearman correlation between two variables is equal to the
Pearson correlation between the rank values of those two variables; while Pearson’s correlation assesses linear relationships, Spearman’s correlation assesses monotonic relationships
(whether linear or not) [2].
In the more advanced statistical analysis one can also consider other measures of dependence adequate especially for processes with infinite variance. We only mention here the codifference [5–7], covariation [5,8–10] and fractional lower order covariance [11–12]. All of them
are used in the practical applications and can be alternatives to the classical measures of
dependence.
Beyond the aforementioned measures of dependence, in the literature there are considered
weighted correlation coefficients [13–14], to obtain reliable full-rank dependence measures
which take under consideration the fact that present observations weight more than past
measurements. Therefore the weighted correlation coefficients can characterise the dynamics of the dependence structure in the better way than the classical measures [13]. It should
be mentioned, the idea of weighted correlations is not new [14–17]. The study of the optimal
weights, aimed at avoiding unwelcome side effects has been mostly overlooked in the literature.
In this paper we analyse three selected correlation coefficients presented in [13], namely classical Pearson correlation coefficient, Kendall’s rank correlation and exponentially weighted
Pearson correlation coefficient. We apply them in order to find the relationship between main
market risk factors of KGHM capital group. We indicate that three mentioned correlation
coefficients can give different information about the structure of dependence dynamics of
the selected risk factors. We analyse the relationship as a function of time in order to indicate
it changes when the unexpected events on the market appear. The main result of the paper
is to demonstrate that the main market risk factors of KGHM capital group are strongly
correlated and different measures of relationship change over time. However, the weighted
correlation coefficient is more sensitive on the market changes with respect to the measures
that take under consideration all observations with the same weights. Thus in order to enable
using the examined relation for forecasting purposes it is crucial to build models taking under
account this kind of correlation measures.
The rest of the paper is organized as follows: in section 2 we present the main market
risk factors of the KGHM capital group and indicate their specific behaviour. Section 3 is
devoted to the measuring of the dependence dynamics of the selected marker risk factors. We
introduce three analysed measures of dependence and in section 4 indicate their behaviour
for the analysed time series taking into account parameters of selected measures such as the
weight factor in the weighted Pearson correlation coefficient or the way of their determining
(data frequency). All analysed factors influence the behaviour of the structure of dependence
dynamics. Last section concludes the paper.
2 MAIN MARKET RISK FACTORS OF KGHM CAPITAL GROUP
In this section we present briefly the main market risk factors of KGHM capital group that
are analysed in the following sections. In our analysis we take under consideration the following factors: copper (Cu) price, gold (Au) and silver (Ag) prices, nickel (Ni) price, EURUSD
and USDPLN exchange rates and LIBOR. Due to lack of space we present only selected
time series of the risk factors that will be analysed in following sections, namely Cu, Ag, Au
and USDPLN. In Fig. 1 we demonstrate their behaviour along time. In this paper we examine
the daily real time series from 1st of January 2000 to 31st of March 2017.
As one can see, the data exhibit very specific behaviour. In Fig. 2, we present the logarithmic rates of return for selected risk factors. They change dynamically over time and in
many cases the variance of the data is very high, especially for metals. Moreover, one can see
non-Gaussian behaviour of the analysed time series. Volatility clustering appears in similar