2.2 Multivariate Time Series


2.2.1 Copula Approach

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:

$\displaystyle H(x_1 ,...,x_n )=C \{ F_1 (x_1 ),...,F_n (x_n )\}$ (2.8)

where $ H$ the multivariate distribution function; $ F_i$ the distribution function of the $ i$-th marginal distribution; $ C$ is a copula. The copula describes the dependence between components of a random vector.

It is worth mentioning some properties of copulas for modeling dependence. The most important ones are the following:

The lower and upper limits for the copula function have important consequences for modeling the dependence. It can be explained in the simplest, bivariate case. Suppose there are two variables $ X$ and $ Y$ and there exists a function (not necessarily a linear one), which links these two variables. One speaks about the so-called total positive dependence between $ X$ and $ Y$, when $ Y=T(X)$ and $ T$ is the increasing function. Similarly, one speaks about the so-called total negative dependence between $ X$ and $ Y$, when $ Y=T(X)$ and $ T$ is the decreasing function. Then: The introduction of the copula leads to a natural ordering of the multivariate distributions with respect to the strength and the direction of the dependence. This ordering is given as:

$\displaystyle C_1 (u_1 ,...,u_n )\le C_2 (u_1 ,...,u_n )
$

and then we have:

$\displaystyle C^-(u_1 ,...,u_n )\le C^\neg (u_1 ,...,u_n )\le C^+(u_1 ,...,u_n).
$

The presented properties are valid for any type of the dependence, not just linear dependence. More facts of copulas is given in Franke, Härdle and Hafner (2004); Rank and Siegl (2002) and Kiesel and Kleinow (2002).

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:

$\displaystyle C(u_1 ,u_2 )=\psi ^{-1}\{\psi (u_1 )+\psi (u_2 )\},$ (2.9)

where $ \psi :[0;1]\to [0;\infty )$, and $ \psi (1)=0$. The most popular and well-studied Archimedean copulas are:
  1. The Clayton copula:

    $\displaystyle \psi (t)=\left\{ {{\begin{array}{*{20}c} {(t^{-\theta }-1)/\theta...
... =0,\,\,\,\,\,\,\,\,} \hfill \\ \end{array} }} \right. \theta \in [-1,\infty ).$ (2.10)

  2. The Frank copula:

    $\displaystyle \psi (t)=\left\{ {{\begin{array}{*{20}c} {-\log \left( {\frac{\ex...
...,\,\,\,\,\,\,\,\,\,\theta =0.\,\,\,\,\,\,\,\,} \hfill \\ \end{array} }} \right.$ (2.11)

  3. The Ali-Mikhail-Haq copula:

    $\displaystyle \psi (t)=\log \left( {\frac{1-\theta (1-t)}{t}} \right), \theta \in [-1;1].$ (2.12)

Among other copulas, which do not belong to Archimedean family, it is worth to mention the Farlie-Gumbel-Morgenstern copula, given in the bivariate case as:

$\displaystyle C_\theta (u,v)=uv+\theta uv(1-u)(1-v), \theta \in [-1;1].$ (2.13)

In all these copulas there is one parameter, which can be interpreted as dependence parameter. Here the dependence has a more general meaning, presented above and described by a monotonic function.

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:

$\displaystyle C(u_1 ,...,u_n )=\Phi _R^n \{\Phi ^{-1}(u_1 ),...,\Phi ^{-1}(u_n )\}$ (2.14)

The other commonly used example is the Gumbel copula, which for the bivariate case is given as:

$\displaystyle C(u_1 ,u_2 )=\exp [-\{(-\log u_1 )^\delta +(-\log u_2 )^\delta \}^{1/\delta }]$ (2.15)

Figure 2.4 presents an example of the shape of the copula function. In this case it is a Frank copula (see (2.11)), with parameters $ \theta$ taken from results presented in Section 2.2.2. The estimation of the copula parameters can be performed by using maximum likelihood given the distribution function of marginals. As the simplest approach to the distribution function of marginals one can take just the empirical distribution function.

Figure 2.4: Plot of $ C(u, v)$ for the Frank copula for $ \theta = -2,563$ in left panel, and $ \theta = 11.462$ in right panel.

\includegraphics[width=0.86\defpicwidth]{STFeva01.ps} \includegraphics[width=0.86\defpicwidth]{STFeva02.ps}


2.2.2 Examples

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.

Table 2.3: The estimates of the Frank copula for exchange rates.
Bivariate data $ \theta$  
    . 
USD/PLN and EUR/PLN 2 .730
USD/PLN and EUR/USD -2 .563
EUR/PLN and EUR/USD 3 .409

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Table 2.4: The estimates of the Frank copula for stock indices.
Bivariate data $ \theta$  
    . 
WIG and WIG20 11 .462
WIG and DJIA 0 .943
WIG and FTSE-100 2 .021
WIG and DAX 2 .086

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The important conclusion to be drawn from Table 2.3 is that one pair, namely USD/PLN and EUR/USD, shows negative dependence, whereas the other two show positive dependence. This is particularly important for the entities that are exposed to exchange rate risk and they want to decrease it by appropriate management of assets and liabilities.

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.


2.2.3 Multivariate Extreme Value Approach

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:

$\displaystyle \mathop {\lim }\limits_{n\to \infty } P\left( {\frac{X_{n:n}^1 -b...
...1 }\le x^1,...,\frac{X_{n:n}^m -b_n^m }{a_n^m }\le x^m} \right)=G(x^1,...,x^m),$ (2.16)

It was shown by Galambos (1978) that this limiting distribution can be presented in the following form:

$\displaystyle G(x^1,...,x^m)=CG\{G_1 (x^1),...,G_m (x^m)\}.$ (2.17)

where $ CG$ is the so-called Extreme Value Copula (EVC). This is the representation of the multivariate distribution of maxima, called here Multivariate Extreme Value distribution (MEV), in the way it is presented in the Sklar theorem. It is composed of two parts and each part has a special meaning: univariate distributions belong to the family of GEV distributions, therefore they are the Fréchet, Weibull or Gumbel distributions.

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:

$\displaystyle C(u_1 ^t,...,u_n ^t)=C^t(u_1 ,...,u_n )$    for  $\displaystyle t>0.$ (2.18)

It can be shown that the bivariate extreme value copula can be represented in the following form:

$\displaystyle C(u_1 ,u_2 )=\exp \{\log (u_1 u_2 )A(\log (u_1 ))/\log (u_1 u_2 )\}.$ (2.19)

Here $ A$ is a convex function satisfying the following relations:
    $\displaystyle A(0)=A(1)=1,$  
      (2.20)
    $\displaystyle \max (w,1-w)\le A(w)\le 1.$  

The most common extreme value copulas are:
  1. Gumbel copula, where:
        $\displaystyle C(u_1 ,u_2 )=\exp [-\{(\log u_1) ^\theta +(\log u_2) ^\theta
\}^{1/\theta }],$  
        with $\displaystyle A(w)=\{w^\theta
+(1-w)^\theta \}^{1/\theta },$ (2.21)
        and $\displaystyle \theta \in
[1,\infty ).$  

  2. Gumbel II copula, where:
        $\displaystyle C(u_1 ,u_2 )=u_1 u_2 \exp \{\theta (\log u_1 \log u_2 )/(\log
u_1 +\log u_2)\},$  
        with $\displaystyle A(w)=\theta
w^2-\theta w+1,$ (2.22)
        and $\displaystyle \theta \in [0,1].$  

  3. Galambos copula, where:
        $\displaystyle C(u_1 ,u_2 )=u_1 u_2 \exp [\{(\log u_1 )^{-\theta }+(\log u_2
)^{-\theta }\}^{-1/\theta }],$  
        with $\displaystyle A(w)=1-\{w^{-\theta }+(1-w)^{-\theta }\}^{-1/\theta },$ (2.23)
        and $\displaystyle \theta \in [0,\infty ).$  

All three presented copulas are one parameter functions. This parameter can be interpreted as dependence parameter. The important property is that for these copulas, as well as for other possible extreme value copulas, there is positive dependence between the two components of the random vector.

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.


2.2.4 Examples

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.

Table 2.5: The estimates of the Galambos copula for exchange rates.
Bivariate data $ \theta$  
    . 
USD/PLN and EUR/PLN 34 .767
USD/PLN and EUR/USD 2 .478
EUR/PLN and EUR/USD 2 .973

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The dependence parameter of the Galambos copula takes only non-negative values. The higher the value of this parameter, the stronger the dependence between maximal losses of respective variables. We see that there is strong extreme dependence between the exchange rates of USD/PLN and EUR/PLN and rather weak dependence between EUR/PLN and EUR/USD as well as for USD/PLN and EUR/USD.

Table 2.6: The estimates of the Gumbel copula for stock indices.
Bivariate data $ \theta$  
    . 
WIG and WIG20 21 .345
WIG and DJIA 14 .862
WIG and FTSE-100 2 .275
WIG and DAX 5 .562

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The dependence parameter for Gumbel copula takes values higher or equal to 1. The higher the value of this parameter, the stronger the dependence between maximal losses of respective variables. The results given in this table indicate strong dependence (as could have been expected) between stock indices of the Warsaw Stock Exchange. It also shows stronger extreme dependence between WSE and NYSE than between WSE and two large European exchanges.


2.2.5 Copula Analysis for Multivariate Time Series

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:

$\displaystyle \log(\theta_t)=\sum\limits_{j=1}^{16}{d_jI\{(u_{t-1}, v_{t-1}) \in {\cal A}_j\}},$ (2.24)

where $ {\cal A}_j$ is the $ j$th element of the unit-square grid. To each parameter $ d_j$, an area $ {\cal A}_j$ is associated. For instance, $ {\cal A}_1 = \lbrack 0, p_1 \rbrack \times \lbrack 0,
q_1 \rbrack$ and $ {\cal A}_2 = \lbrack p_1, p_2 \rbrack \times
\lbrack 0, q_1 \rbrack$, where $ p_1=q_1=0.15$, $ p_2=q_2=0.5$, and $ p_3=q_3=0.85$. The choice of 16 subintervals is, according to Jondeau and Rockinger (2002), somewhat arbitrary. Therefore the dependence parameter is conditioned on the lagged values of univariate distribution functions, where the 16 possible sets of pairs of values are taken into account. The larger value of parameter $ d_j$, the stronger dependence on the past values.

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.


2.2.6 Examples

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.

Table 2.7: Conditional dependence parameter for time series WIG, WIG20.
  [ 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

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Table 2.8: Conditional dependence parameter for time series WIG, DJIA.
  [ 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

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Table 2.9: Conditional dependence parameter for time series USD/PLN, EUR/PLN.
  [ 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

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For the purpose of comparison, we present the results obtained in the case of the Frank copula. These results are given in the Tables 2.7-2.9.

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.