In this section we present the so-called copula approach. It is performed in two steps. In the first step one analyzes the marginal (univariate) distributions. In the second step one analyzes the dependence between components of the random vector. Therefore the analysis of dependence is ``independent'' from the analysis of marginal distributions. This idea is different from the one present in the classical approach, where multivariate analysis is performed ``jointly'' for marginal distributions and dependence structure by considering the complete covariance matrix, like e.g. in the MGARCH approach. So one can think that instead of analyzing the whole covariance matrix, where the off diagonal elements contain information about scatter and dependence) one analyzes only the main diagonal (scatter measures) and then the structure of dependence ``not contaminated'' by scatter parameters.
The fundamental concept of copulas becomes clear by Sklar theorem (Sklar; 1959). The multivariate joint distribution function is represented as a copula function linking the univariate marginal distribution functions:
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(2.8) |
It is worth mentioning some properties of copulas for modeling dependence. The most important ones are the following:
There are very possible copulas. A popular family contains the so-called Archimedean copulas, defined on the base of strictly decreasing and convex function, called generator. In the bivariate case it is given as:
An often used copula function is the so-called normal (Gaussian) copula, which links the distribution function of multivariate normal distribution with the distribution functions of the univariate normal distributions. This means that:
Consider different pairs of stock market indices and exchange rates, studied in Section 2.1.3. For each pair we fitted a bivariate copula, namely the Clayton, Frank, Ali-Mikhail-Haq, and the Farlie-Gumbel-Morgenstern.
We present here the results obtained for Frank copula. Table
2.3 presents selected results for pairs of exchange
rates and Table 2.4 for pairs of stock indices.
Bivariate data | ![]() |
||
. | |||
USD/PLN and EUR/PLN | 2 | . | 730 |
USD/PLN and EUR/USD | -2 | . | 563 |
EUR/PLN and EUR/USD | 3 | . | 409 |
Bivariate data | ![]() |
||
. | |||
WIG and WIG20 | 11 | . | 462 |
WIG and DJIA | 0 | . | 943 |
WIG and FTSE-100 | 2 | . | 021 |
WIG and DAX | 2 | . | 086 |
There is positive extreme dependence between all stock indices. As could have been expected, there is strong dependence between indices of the WSE and much lower between WSE and the other exchanges, with weaker dependence between WSE and NYSE than between WSE and large European exchanges. The copula approach can be applied in the so-called tail dependence coefficients. The detailed description of tail dependence is given in Chapter 3.
The copula approach also gives the possibility to analyze extreme values in the general multivariate case. This is possible by linking this approach to univariate extreme value analysis. In order to make this possible, we concentrate on the multivariate distribution of extrema, where the extremum is taken for each component of a random vector.
The main result in the multivariate extreme value analysis is given for the limiting distribution of normalized maxima:
Therefore, to obtain the MEV distribution one has to apply the EVC to univariate GEV distributions (Fréchet, Weibull, or Gumbel). Since there are many possible extreme value copulas, we get many possible multivariate extreme value distributions.
The EVC is a copula satisfying the following relation:
The main application of multivariate extreme value approach is the estimation of the maximum loss of each component of the portfolio. We get then the limiting distribution of the vector of maximal losses. The limiting distributions for the components are univariate GEV distributions and the relation between the maxima is reflected through extreme value copula.
As in Section 2.2.2 we consider different pairs of stock market indices and exchange rates. In the first step we analyze separate components in each pair to get estimates of generalized extreme value distributions. In the second step, we use empirical distribution functions obtained in the first step and estimate three copulas belonging to EVC family: Gumbel, Gumbel II, and Galambos. We present here the results obtained for Galambos copula (Table 2.5) and Gumbel copula (Table 2.6)
It turns out that in the case of exchange rates we obtained the
best fit for the Galambos copula, see Table 2.5. In
the case of stock indices the best fit was obtained for different
copulas. For the comparison we present the results obtained for
the Gumbel copula, see Table 2.6.
Bivariate data | ![]() |
||
. | |||
USD/PLN and EUR/PLN | 34 | . | 767 |
USD/PLN and EUR/USD | 2 | . | 478 |
EUR/PLN and EUR/USD | 2 | . | 973 |
Bivariate data | ![]() |
||
. | |||
WIG and WIG20 | 21 | . | 345 |
WIG and DJIA | 14 | . | 862 |
WIG and FTSE-100 | 2 | . | 275 |
WIG and DAX | 5 | . | 562 |
One of the basic models applied in the classical (mean-oriented) approach in the analysis of multivariate time series was the multivariate GARCH model (MGARCH) aimed at modeling of conditional covariance matrix. One of the disadvantages of this approach was the joint modeling of volatilities and correlations, as well as relying on the correlation coefficient as a measure of dependence.
In this section we present another approach, where volatilities and dependences in multivariate time series, both conditional, are modeled separately. This is possible due to the application of copula approach directly to univariate time series, being the components of multivariate time series. Our presentation is based on the idea presented by Jondeau and Rockinger (2002), which combines the univariate time series modeling by GARCH type models for volatility with copula analysis. The proposed model is given as:
We also give the description of the method, which was used in the empirical example. We describe this procedure for the case of bivariate time series. The proposed procedure consists of two steps. In the first step, the models for univariate time series are built for both time series. Here the combined procedure of ARIMA models for conditional mean and GARCH models for conditional variance was used. In the second step, the values of the distribution function for residuals obtained after the application of univariate models were subject to copula analysis.
In this example we study three pairs of time series, namely WIG
and WIG20, WIG and DJIA, USD/PLN and EUR/PLN. First of all, to get
the best fit: an AR(10)-GARCH (1,1) model was built for each
component of bivariate time series. Then the described procedure
of fitting copula and obtaining conditional dependence parameter
was applied. In order to do this, the interval [0, 1] of the
values of univariate distribution function was divided into 4
subintervals: [0, 0.15), [0.15, 0.5); [0.5, 0.85); [0.85, 1]. Such
a selection of subintervals allows us to concentrate on tails of
the distributions. Therefore we obtained 16 disjoint areas. For
each area the conditional dependence parameter was estimated using
different copula function.
[ 0, 0.15) | [0.15, 0.5) | . | [0.5, 0.85) | . | [0.85, 1] | . | ||||||
. | . | . | . | |||||||||
&lsqb#lbrack;0, 0.15) | 15 | . | 951 | 4 | . | 426 | 5 | . | 010 | 1 | . | 213 |
&lsqb#lbrack;0.15, 0.5) | 6 | . | 000 | 18 | . | 307 | 8 | . | 704 | 1 | . | 524 |
&lsqb#lbrack;0.5, 0.85) | -0 | . | 286 | 8 | . | 409 | 19 | . | 507 | 5 | . | 133 |
&lsqb#lbrack;0.85, 1&rsqb#rbrack; | 0 | . | 000 | 2 | . | 578 | 1 | . | 942 | 19 | . | 202 |
[ 0, 0.15) | [0.15, 0.5) | . | [0.5, 0.85) | . | [0.85, 1] | . | ||||||
. | . | . | . | |||||||||
&lsqb#lbrack;0, 0.15) | 2 | . | 182 | 1 | . | 169 | 0 | . | 809 | 2 | . | 675 |
&lsqb#lbrack;0.15, 0.5) | 1 | . | 868 | 0 | . | 532 | 0 | . | 954 | 2 | . | 845 |
&lsqb#lbrack;0.5, 0.85) | 1 | . | 454 | 1 | . | 246 | 0 | . | 806 | 0 | . | 666 |
&lsqb#lbrack;0.85, 1&rsqb#rbrack; | -0 | . | 207 | 0 | . | 493 | 1 | . | 301 | 1 | . | 202 |
[ 0, 0.15) | [0.15, 0.5) | . | [0.5, 0.85) | . | [0.85, 1] | . | ||||||
. | . | . | . | |||||||||
&lsqb#lbrack;0, 0.15) | 3 | . | 012 | 2 | . | 114 | 2 | . | 421 | 0 | . | 127 |
&lsqb#lbrack;0.15, 0.5) | 3 | . | 887 | 2 | . | 817 | 2 | . | 824 | 5 | . | 399 |
&lsqb#lbrack;0.5, 0.85) | 2 | . | 432 | 3 | . | 432 | 2 | . | 526 | 3 | . | 424 |
&lsqb#lbrack;0.85, 1&rsqb#rbrack; | 7 | . | 175 | 3 | . | 750 | 4 | . | 534 | 4 | . | 616 |
The values on ``the main diagonal'' of the presented tables correspond to the same subintervals of univariate distribution functions. Therefore, the values for the lowest interval (upper left corner of the table) and highest interval (lower right corner of the table) correspond to the notion of lower tail dependence and upper tail dependence. Also, the higher are values concentrated along ``the main diagonal'', the stronger conditional dependence is observed.
From the results presented in Tables 2.7-2.9, we can see, that there is a strong conditional dependence between returns on WIG and WIG20; the values of conditional dependence parameter ``monotonically decrease with the departure from the main diagonal.'' This property is not observed in the other two tables, where no significant regular patterns can be identified.
We presented here only some selected non-classical methods of the analysis of financial time series. They proved some usefulness for real data. It seems that the plausible future direction of the research would be the integration of econometric methods, aimed at studying the dynamic properties, with statistical methods, aimed at studying the distributional properties.