Extreme joint dependencies with copulas A new approach for the structure of C-Vines

Transcription

1 U.U.D.M. Project Report 2015:24 Extreme joint dependencies with copulas A new approach for the structure of C-Vines Johannes Krouthén Examensarbete i matematik, 30 hp Handledare och examinator: Erik Ekström Juni 2015 Department of Mathematics Uppsala University

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3 Abstract This paper provides an introduction to copulas, why they are useful and methods on how to choose appropriate copulas for empirical data. Dierent Archimedean and Elliptical copulas are compared with approaches on how to best calibrate the copula parameters. Furthermore, the paper puts focus on a Canonical Vine-Garch method where a new algorithm for the tree structure is proposed. This new algorithm aims to catch in particular the dependencies in the lower tails of the distributions. The background for this new algorithm is that in nancial applications, especially in risk management and portfolio theory, one would often want to see extreme joint dependencies rather than overall correlations. The algorithm is tested on empirical data from four dierent European stock exchanges. It is concluded that the algorithm catches the dependencies quite well, but that more research in the area must be done to further improve the algorithm.

4 Contents Contents 1 Introduction Background to Copulas Vine Copulas Purpose with this thesis Basics about bivariate copulas and useful dependencies Denition of a Copula Bivariate Cdf Sklar's theorem Kendall's τ Tail dependence Copula Families Archimedean copulas Tail dependence for Archimedean copulas Elliptical copulas Gaussian copula Student- T copula ARMA/GARCH 12 5 Choosing a bivariate copula Selecting parameters Comparing and choosing between dierent copulas Vine copulas Pair copula constructions Vines Choosing C-vine structure Algorithm 1: C-Vine construction Sample from the C-vine structure Algorithm 2: C-Vine sampling Practical example on European stock exchanges Fitting the marginal distributions Building the model Tree Tree Tree Sampling Summary 36 9 Thoughts for future research 36 References Appendix: R-Codes 39

5 1 Introduction 1.1 Background to Copulas Copulas are a method used to describe the dependencies between two or more cumulative distribution functions based on their marginal distributions. When working with copulas, these marginal distributions are always transformed into uniform distributions. An advantage with copulas is that it allows us to describe the marginal distributions and their joint dependencies (the copulas) separately. This is not possible when working with general multivariate distributions. Take, for example, the multivariate normal distribution. In order to have a multivariate normal distribution, all the univariate marginal distributions have to be normally distributed. This is not the case when working with copulas since the margins and the dependence structure are described independently. We can, for example, have a Gaussian copula where the margins are gamma or exponentially distributed. The other thing that makes copulas so benecial is that they don't restrict the dependence structure to be linear. This gives way more options when explaining relationships between dierent variables instead of, for example, just using a correlation coecient. Its name is derived from the word "coupling" which means "a connection or a joint between things" and has, in the last decades, been increasingly used in nancial applications. For example, Embrechts et al. (2002) discussed the problems that can arise when people in risk management don't take into consideration the fact that the dependence structure between non-elliptical distributions can't be seen as linear. However, when the nance crisis hit the world of 2008, copulas received bad reputation as many people claimed it was one of the major reasons for the crisis. The paper with the most criticism was the paper by Li (2000) which aimed to show that copulas can be used to specify the joint distribution of survival times when marginal distributions are derived from market information, such as risky bond prices or asset swap spreads. The gaussian formula which the paper is based on was named by Wired Magazine "The formula that killed Wall Street" (Salmon 2009). Nevertheless, the research about copulas, especially within the area of quantitative nance, still seems to be a growing topic. 1.2 Vine Copulas One of the most dicult tasks within the copula area is to estimate copulas in higher dimensions. Both the Elliptical copulas and the Archimedean copulas can be used in higher dimensions, but because they use few parameters, they lack exibility. Another general problem with many of the multivariate copulas is that they don't have tail dependence, meaning that they understate the risk of extreme events. Gaussian, which is one of the most used copulas, is an example of a copula that lacks such a dependence. One way to make the model more exible is to use so called "Vine copulas". Vine copulas follow a nested tree structure where every node in the trees is a bivariate (two dimensional) copula. Conditional on the lower trees, the vine structure is then built up based on conditional, bivariate copulas. There are many dierent types of vine structures but in this paper we will focus on the C-vine copula. The C-vine copula uses a tree structure where every tree has a key node called the root. In the rst tree, this root is basically a single variable. The edges in tree one are then built up by all bivariate copulas that includes that variable. The next tree will then have one of those bivariate copulas as their root node and so on (Joe 1997; Schirmacher, D. & Schirmacher, E. 2008). The selection of the key node in every tree is a problem by itself. One common approach is to estimate all Kendalls τ's values and see which variable (or conditional copula in the higher trees) that has the biggest dependence with the other variables (conditional copulas) based on these values (Lu 2013). This approach seems to be reasonable if one only want to focus on creating copulas with the overall best t. However, if one mainly want to build a model that focuses on being as accurate as possible on describing dependencies during extreme events, other ways of choosing the tree 1

6 roots might be of interest. In this paper, we will create a new algorithm for a C-vine copula that aims to describe joint dependencies during extreme downfalls. 1.3 Purpose with this thesis The purpose with this paper is to give a general introduction to copulas and, in particular, the C-Vine copula approach. To build up the pairwise copulas, ve of the most commonly used copula families are tested where the marginal distributions are ltered through an Arma-Garch lter. However, instead of choosing the key roots in the trees based on Kendalls Tau, we propose a new algorithm where more focus is put on the lower tail dependencies. The background for this is that extreme joint downfalls is an interesting topic especially since the nancial crisis. A practical example will be done by applying the model to four dierent European stock exchanges. 2 Basics about bivariate copulas and useful dependencies 2.1 Denition of a Copula A distribution function on [0,1]*[0,1] where the marginal distributions are standard uniform is called a two-dimensional copula. More formal, a function C(u, v) is called a twodimensional copula function C(u, v) from I 2 to I if it has the following two properties; 1. For every u and v in I, C(u, 0) = C(0, v) = 0, C(u, 1) = u and C(1, v) = v. 2. For every u 1, u 2, v 1, v 2 in I such that u 1 u 2 and v 1 v 2, C(u 2, v 2 ) C(u 2, v 1 ) C(u 1, v 2 ) + C(u 1, v 1 ) 0. (Nelsen, 2006, pp 10-11) 2.2 Bivariate Cdf The joint cumulative distribution function of two random variables (Y,X) with corresponding marginal cumulative distribution functions F Y (y) and F X (x) is called a bivariate cdf and is dened as F (y, x) = P r[y y, X x]. (1) Once one has the bivariate cdf, it is easy to dene the marginal distribution functions as F X (x) = lim F (x, y) and F Y (y) = lim F (x, y) (2) y x and the conditional distribution functions as F X Y (x y) = F (x, y) y and F Y X (y x) = F (x, y). (3) x The denitions are completely analogous in the discrete case. survival function as We also dene the joint F (x, y) = P r(x > x, Y > y) = 1 F X (x) F Y (y) + F (x, y) (4) (Trivedi. Zimmer 2005, pp 7-8). 2

7 2.3 Sklar's theorem The pseudo inverse or the inverse quantile function of a distribution function F X (x) is dened as F 1 X (u) = inf{x : F X(x) u, 0 < u < 1}. (5) This comes from the fact that if U is a random uniform variable, U[0,1] then P r(f 1 X (u) x) = P r(u F X (x)). Therefore, F 1 X (u) F X(x). Furthermore for 0 u 1, P r(f X (x) u) = P r(x F 1 1 X (u)) = F (FX (u)) = u. So if we want to generate a random number X from its distribution function F X (x), we can use the fact that X = F 1 X (u) F X (Devroye 1986, p 28). By using this, we are now ready to state Sklar's theorem, which gives us the unique copula from the distribution function. It states that every joint distribution function F (x 1,..., x n ) can be written as C(F 1 (x 1 ),..., F n (x n )) where C is a copula (Nelsen, 2006, p 21). So if F (x, y) is the bivariate distribution function of the two random variables X and Y, then the unique copula C associated with that cdf can be written as where U 1 and U 2 are U(0,1). F (x, y) = P r[x x, Y y] = P r(f 1 X (U 1) x, F 1 Y (U 2) y) = P r[u 1 F X (x), U 2 F X (x)] = C(F X (x), F Y (y)) Also, the joint survival function is dened as C = P r[u 1 > u 1, U 2 > u 2 ] = 1 u 1 u 2 + C(u 1, u 2 ) (7) for two uniform random variables with a joint distribution function as the copula C (Nelsen, 2006, pp 32-33). 2.4 Kendall's τ Kendall's τ measures the dierence between the probability of concordance and discordance between the observed pairs (x 1.y 1 ),...,(x n.y n ). They are concordant if x 1 > x 2 whenever y 1 > y 2, and x 1 < x 2 whenever y 1 < y 2. If the opposite is true, they are called discordant. So, τ(x, Y ) = P r((x 1 X 2 )(Y 1 Y 2 ) > 0) P r((x 1 X 2 )(Y 1 Y 2 ) < 0). However, kendall's τ can also be expressed by the copulas which is shown below. First, τ = P r((x 1 X 2 )(Y 1 Y 2 ) > 0) P r((x 1 X 2 )(Y 1 Y 2 ) < 0) = P r((x 1 X 2 )(Y 1 Y 2 ) > 0) (1 P r((x 1 X 2 )(Y 1 Y 2 ) > 0)) = 2P r((x 1 X 2 )(Y 1 Y 2 ) > 0)) 1. (6) Furthermore, P r((x 1 X 2 )(Y 1 Y 2 ) > 0) = P r(x 1 > X 2, Y 1 > Y 2 ) + P r(x 1 < X 2, Y 1 < Y 2 ), and by integrating over the distribution of one of the vectors (X 1, Y 1 ) or (X 2, Y 2 ) we can calculate these probabilities. Then, let C 1 be the copula of (X 1, Y 1 ) and C 2 be the copula of (X 2, Y 2 ). The rst term on the right hand side in the above expression P r(x 1 > X 2, Y 1 > Y 2 ) = P r(x 2 > X 1, Y 2 > Y 1 ) 3

8 and by integrating over (X 1, Y 1 ) we get that P r(x 2 > X 1, Y 2 > Y 1 ) = P r(x 2 x, Y 2 y)dc 1 (F X (x), F Y (y)) = C 2(F X (x), F Y (y))dc 1 (F X (x), F Y (y)) By using the probability transform, u 1 = F X (x), u 2 = F Y (y) we have P r(x 1 > X 2, Y 1 > Y 2 ) = C 2(u 1, u 2 )dc 1 (u 1, u 2 ). In a similar way, the second term can be dened as, P r(x 1 < X 2, Y 1 < Y 2 ) = P r(x 2 > x, Y 2 > y)dc 1 (F X (x), F Y (y)) = (1 F X(x) F Y (y) + C 2 (F X (x), F Y (y))dc 1 (F X (x), F Y (y)) = (1 u 1 u 2 + C 2 (u 1, u 2 )dc 1 (u 1, u 2 )) = (1 u 1 u 2 )dc 1 (u 1, u 2 ) C 2(u 1, u 2 )dc 1 (u 1, u 2 ). However, the rst term, so, (1 u 1 u 2 )dc 1 (u 1, u 2 ) = 1 1/2 1/2 = 0 since E(U 1 ) = E(U 2 ) = 1/2 so, so, = 1 0 P r(x 1 < X 2, Y 1 < Y 2 ) = C 2(u 1, u 2 )dc 1 (u 1, u 2 ) P r((x 1 X 2 )(Y 1 Y 2 ) > 0)) 1 0 C 2(u 1, u 2 )dc 1 (u 1, u 2 ) C 2(u 1, u 2 )dc 1 (u 1, u 2 ) = C 2(u 1, u 2 )dc 1 (u 1, u 2 ) 0 2P r((x 1 X 2 )(Y 1 Y 2 ) > 0) 1 = C 2(u 1, u 2 )dc 1 (u 1, u 2 ) 1. However, since the integral in the later term can be seen as the expected value of C(U 1, U 2 ), τ can nally be dened as τ = 4E(C(U 1, U 2 )) 1. (8) (Nelsen, 2006, p 159) An unbiased estimator of τ, also called Kendalls sample τ can be calculated as: τ = (number of concordant pairs)-(number of discordant pairs) ( n 2) (9) where n is the amount of pair observations. We also dene Kendall's distribution function } K C (t) = µ C {(u 1, u 2 ) I 2 C(u 1, u 2 ) t, (10) 4

9 where µ C is the measured induced on I 2 by C (Nelsen, et al 2003, pp ). 2.5 Tail dependence When extreme events occur one is often interested in how this aects other events. One of the most useful things with copulas is that many of them can capture those joint extremes. This is often measured by the tail dependence. The tail dependence is the probability of joint events in the upper and lower tails of a bivariate distribution. The upper tail dependence can be written as P r(y > F 1 1 Y (t) X > FX (t)) for some t close to 1 and the lower tail dependence as P r(y < F 1 1 Y (t) X < FX (t)) for some t close to 0. Joe (1997) p 33, found that this can be showed in terms of the copula functions. For the lower tail dependence we have λ L = lim t 0 + P r[y F 1 1 (t) X F (t)] = lim t 0 + And for the upper tail dependence we have, = lim t 1 Y X P r[x F 1 1 X (t),y FY (t)] P r[x F 1 X (t)] = lim t 0 + C(t,t) t. (11) λ U = lim t 1 P r[y > F 1 1 (t) X > F (t)] Y X P r[x F = lim 1 1 X (t),y FY (t)] t 1 P r[x F 1 X (t)] 1 (P r[x F X (t)]+p r[y FY (t)] P r[x FX (t),y FY (t)]) 1 P r[x F 1 X (t)] = lim t 1 1 2t+C(t,t) 1 t. (12) If λ L = 0 it means that tail events in the lower tail are asymptotically independent of each other. If λ L > 0 we will see a bigger dependence between the variables as the uniformed variables get close to 0. Graphically, this means a more compact cluster close to (0,0) in the dependence structure of the the two uniformed variables. Same is true for λ U but at (1,1). 3 Copula Families Most copulas studied in the literature belong to a specic copula family, with one or more parameters that control the strength and properties of the dependence. In this paper we will investigate six specic families. Four Archimedean copulas and two Elliptical copulas. 3.1 Archimedean copulas The bivariate copula C(u 1.u 2 ) is called an Archimedean copula if it can be written as C(u 1.u 2 ; θ) = φ 1 (φ(u 1, θ) + φ(u 2, θ)) (13) where θ is the parameter of the one parameter Archimedean copula and φ(t, θ), the generator, satises the following conditions; φ is a strictly decreasing convex function with φ(1, θ) = 0, φ is completely monotone i.e ( 1) k (φ(x) s ) 0, k = s = 0, 1, 2... where s is the order of derivitave. Further, the pseudo inverse of φ, φ [ 1] is dened as φ [ 1] (t, θ) = { φ 1 (t, θ) if 0 t φ(0, θ). 0 if φ(0, θ) t. (14) 5

10 There are several classes of Archimedean copulas. They all dier in their dependence structure and, in this study, four of the most used ones will be described namely; Clayton's, Gumbel's, Frank's and Joe's Copulas. All bivariate Archimedean copulas have a direct relationship to Kendall's tau with an explicit formula which makes them very easy to work with. It's been proven that when the generator φ of an Archimedean copula is continuously dierentiable, then K C (t) described in (10) can be denoted as K C (t) = t φ(t) φ (t). From (8) we have that τ = 4E(C(U 1, U 2 )) 1. Since U 1 and U 2 are uniform variables with joint distribution function C(U, V ), then K C (t) is the distribution function of C(U 1, U 2 ) and Furthermore, τ = 4E(C(U 1, U 2 )) 1 = tdk C (t) 1. so, 1 0 tdk C(t) = tk C (t) K C(t)dt = K C (1) 1 0 K C(t)dt = (1 φ(1) φ (1) ) 1 0 K C(t)dt = K C(t)dt τ = 4(1 1 0 K C(t)dt) 1 = K C(t)dt = φ(t) (t 0 = 3 4( t = φ(t) φ (t) φ(t) φ (t) ) φ (t) )dt And therefore, Kendall's τ c of an Archimedean, where φ(t) is the generator of an Archimedean copula can be dened as 1 φ(t) τ CA = φ dt (15) (t) (Nelsen, 2006, pp ; Cherubini el alt (2004), pp ) In table 1, one can see the denitions of the four Archimedean copulas. Family Denition of C θ (u 1, u 2 ) Generator φ(t, θ) θ limits Clayton copula max([u θ 1 + u θ 2 1] 1/θ 1, 0) θ (t θ 1) θ [ 1, ), θ 0 Gumbel copula exp( [( ln(u ( 1 )) θ + ( ln(u 2 )) ) θ ] 1/θ ) ( ln(t)) θ θ [1, ) Frank copula 1 θ ln 1 + (e θu 1 1)(e θu 2 1) -ln exp( θt) 1 e θ 1 exp( θ) 1 θ (, ), θ 0) Joe copula 1 ((1 u 1 ) θ + (1 u 2 ) θ (1 u 1 )(1 u 2 ) θ ) 1/θ -ln(1 (1 t) θ )) θ [1, ) Table 1: Four dierent Archimedean copulas. 6

11 3.1.1 Tail dependence for Archimedean copulas It is fairly straightforward to calculate the tail dependencies of the Archimedean copulas. Clayton, Gumbel and Joe are all asymmetric copulas. Clayton has a greater dependence in the lower tail than the upper tail and the Gumbel and Joe copulas are vice versa. The Frank copula is the only one of the four that is a symmetric copula. Below, one can see how to calculate the upper and lower tail dependencies for the Gumbel copula. Since we have that λ U = lim t 1 an expression for C(t, t). 1 2t+C(t,t) 1 t C(t,t) and that λ L = lim t 0 + t we rst need to get C Gumbel (u 1, u 2 ) = exp( [( ln(u 1 )) θ + ( ln(u 2 )) θ ] 1 θ so, C Gumbel (t, t) = exp( [( ln(t)) θ + ( ln(t)) θ ] 1 θ = exp( [ 2ln(t) θ ] 1 θ ) = exp( [ 2 1 θ ln(t)]) = t 2 1 θ. So, C(t,t) λ L = lim t 0 + t = dc(0) dt = [ d dt t2 1 θ [ ] ]t=0 = 2 1 θ t 2 θ 1 1 = 0. t=0 When calculating λ U we rst realize that λ U = lim So, 1 2t+C(t,t) t 1 1 t [ ] λ U = dc(0) dt 2t 1 + (1 t) 2 θ 1 [ ] = 2 + dc(0) dt (1 t) 2 θ 1 t=0 [ ] = 2 + ( 1)(1 t) 2 θ θ t=0 = θ > 0 for θ > 1. t=0 2t 1+C(1 t,1 t) = lim t 0 + t. The upper and lower tail dependencies for all of the investigated Archimedean copulas are shown in table 2. In gure 1, one can see the samples of 2000 random variables generated by the four dierent Archimedean copulas with three dierent θ's each. It's clear how the dierent copulas produce dierent patterns of correlated pairs, and also what kind of impact the parameter θ has on the strength of the dependence. (Nelsen, 2006, pp ). Family λ U λ L Clayton copula 0 2 1/θ (θ > 0) Gumbel copula 2 2 1/θ 0 Frank copula 0 0 Joe copula 2 2 1/θ 0 Table 2: Upper and Lower tail depedncies for the Archimedean copulas Clayton, Gumbel, Frank and Joe. 7

12 Figure 1: Scatterplots of random variables produced by the four described Archimedean copulas. For each of the copulas, three dierent θ's has been tested. It is clear that when θ is getting further away from 0, the correlation gets stronger. One can also see the cluster around (0,0) for the Clayton copula and around (1,1) for the Gumbel and Joe coplas. This is due to the lower respective upper tail dependencies that the two copulas possess. One can also see that the Frank copula has a negative correlation for θ's<0 and a positive correlation for θ's >0. 8

13 3.2 Elliptical copulas. A Elliptical copula is basically the copula of a specic Elliptical distribution. All Elliptical distributions are radially symmetric i.e. λ U (X, Y ) = λ L (X, Y ). Since Elliptical copulas don't have closed form expressions, a dierent approach is usually used to calculate their tail dependencies than the approach used for Archimedean copulas. One way to deal with this is by using the regularly varying index, a method described by Schimdt. A measurable function f : R (+) R (+) is called regularly varying (at ) with index αɛr if for any t > 0, = tα (i.e if they can be seen as power functions). lim x f(tx) f(x) It is called O-regularly varying if for any t 1, 0 < lim x inf f(tx) f(x) lim x sup f(tx) f(x) <. Schmidt shows that if the generator g of the distributions probability density function (pdf) is regularly varying, then a closed form expression for the tail dependence can be calculated as λ = λ U = λ L = h(p) u α u 2 1 du u α du (16) u 2 1 where Pearsons correlation coecient ρ = σ12 σ11σ 22 and h(ρ) := ( 1 + (1 ρ)2 1 ρ ). Furthermore, 2 he proves that if the distribution has tail dependence, the density generator g must be O-regularly varying (Schmidt, 2002, p 322) Gaussian copula The bivariate Gaussian copula is dened as C Guassian (u 1, u 2 ) = φ R12 (φ 1 (u 1 ), φ 1 (u 2 )). Here φ R12 is the bivariate cumulative distribution function with correlation matrix R 12 of the standard normal distribution function and φ 1 is the inverse cumulative distribution function of the standard normal distribution function. The copula can be written as C Gaussian (u, v) = φ 1 (u) φ 1 (v) 1 2π(1 R 2 12 )1/2 exp ( s2 2R 12 st + t 2 2(1 R 2 12 ) ) dsdt (17) (Meyer, 2013, p 2403). The pdf generator of the bivariate gaussian distribution is dened as, Since lim g(tu) g(u) = lim g(u) = (2π) 1 exp( 1 2 u). (2π) 1 exp( 1 2 tu) (2π) 1 exp( 1 1 exp( 2 u(t 1)) = 0 and therefore not O- = lim u u 2 u) u regularly varying, the Gaussian copula doesn't have any tail dependence. In gure 2, one can see samples of 2000 random variables generated by the Gaussian copula for four dierent ρ's. 9

14 Figure 2: Scatter plot of random univariate variables produced by the Gaussian copula for four dierent ρ's. Obviously, negative correlation creates a negatived correlated copula, and the closer you get to 1 and -1 the stronger correlated copula we get Student- T copula The bivariate student-t copula is dened as C T (u 1.u 2 ) = t ρ,m (t 1 m (u 1, u 2 )) = t 1 v (u1) t 1 v (u2) 1 2π ( s 2 + t 2 2ρst) m ρ 2 m(1 p 2 dsdt ) (18) where t 1 v denotes the quantile function of a standard univariate t v distribution, m the degrees of freedom and ρ the correlation coecient described earlier (Jäckel, 2002, p 117). The pdf generator of the student-t distribution is dened as g(u) = (1 + t m ) 2+m 2 (Schmidt, 2002, g(tu) g(u) = lim u (1 + tu m ) (1+ m 2 ) (1 + tu m )(1+ m 2 ) = t (1+ m 2 ) the student- p 324). And since lim u distribution has a regularly varying pdf generator. We can therefore us (16) to calculate the close form of the tail dependence with α = (1 + m 2 ). However, Embrechts et al. (2002, p 19) shows that for a T-distribution, the following expression can be used for calculating the the tail dependence 10

15 λ = 2t ρ,m+1 m p 1 + p. (19) In gure 3, six dierent student t copulas are shown and in table 3, one can see dierent λ for dierent combinations of p and m. Figure 3: Scatter plots of samples of 2000 random uniform variables generated by the Student-T copula with six dierent combinations of ρ's and m's (degrees of freedom). As with the Gaussian copula, changing the sign of ρ, changes the direction of copula and ρ's closer to -1 and 1 gives us a copula that is stronger correlated. However, the Student-T copula has more points in the tails than the Gaussian copula and produce pseudo observations that appear in a somewhat star liked shape. 11

16 ρ\m Table 3: Tail dependence for the student T-copula for various combination of ρ and m. It is clear that as the degree of freedom increases (i.e getting closer to the Gaussian copula) the tail dependence gets smaller. One can also see that higher correlation means higher tail dependence. 4 ARMA/GARCH One of the main purposes with copulas is that the models can link the marginal distributions together to create the joint distributions. Therefore, before one can start to t a copula to the data, we rst need to estimate the marginal distributions. When copulas rst were created, the margins were usually from i.i.d observations taken straight from the raw data. However, a common approach more recently is to t each of the univariate distributions to individual time series and use the error terms as the margins. By doing this, we can assume that the observations of the margins are independent over time. This method has showed to be very useful especially when working with nancial data where time dependencies are very common. See for example (Jondeau & Rockinger, 2006). In this project, we rst t an ARMA(p, q) to specify the conditional mean of the process and then t an GARCH(1,1) for the conditional variance. An ARMA(p, q) model is describing the conditional mean of a process X t as X t = α 0 + p α j X t j + j=1 where, p is the number of autoregressive terms. α j is the autoregressive parameters. q is the number of moving average terms. β j is the moving average parameters. ɛ t N(0, σ 2 ) is the white noise. q βɛ t k (20) GARCH stands for generalized autoregressive conditional heteroskedasticity and is used when a non-constant variance is assumed for the model. The GARCH therefore species the conditional variance for the process i.e the variance, σ 2 for the ɛ t in the ARMA(p, q) model and the GARCH(1, 1) is dene as, k=1 σ 2 t = ωδ 0 + δɛ 2 t 1 + γσ 2 t 1 (21) where, ω is a constant δ is the autoregressive parameter. 12

17 γ is the moving average parameter. A deeper explanation of the ARMA(p, q)-garch(1, 1) model will not be explained in this paper. For further reading please see for example (Francq, Zakoian, 2006) where information about model estimation is covered as well. In this paper, the R package GarchFit with a Quasi-Maximum Likelihood Estimation has been used to t appropriate models to the data. Once the residuals of the time series have been ltered out, we use these residuals to create the marginal distributions. This can be done either with a parametric approach, or with a non-parametric approach. In the parametric approach, we t parametric distributions for the residuals. This could be done with a lot of dierent types of distributions. Some of the most common once are the normal distribution, student-t distribution and skewed normal distribution. The parameters for these distributions are usually estimated by maximum likelihood (ML) ˆθ m = ArgMax θm T t=1 lnf(ɛ t ; θ m ) (22) where ˆθ m is the estimated parameter or parameters for the marginal distribution function f(ɛ t ) and ɛ t is the residual at time t from the times series. Comparing the dierent distributions can be done using for example Bayesian information criterion (BIC). For the non-parametric approach, instead of tting the residuals to a parametric distribution, the goal is to t the residuals with the sample empirical distribution function (Patton 2012, pp. 6-7). ˆF (ɛ) = 1 T + 1 T 1{ˆɛ i ɛ} (23) i=1 5 Choosing a bivariate copula Selecting a appropriate bivariate copula to the data normally consist of two steps. In the rst step parameters are estimated for each of the tested copulas based on the marginal distributions. In the second step, which is usually the more dicult one, we choose which of the copulas to use. 5.1 Selecting parameters Selecting the parameters for the dierent copulas are usually made by maximum likelihood estimation. Two dierent, but very similar maximum likelihood estimations are used depending on if we have the margins estimated by a nonparametric approach or a parametric approach. When the margins are estimated by the parametric approach we estimate the copula parameter(s) ˆθ C based on the following MLE: ˆθ C = ArgMax θc T t=1 ln C(F 1 (x 1t ; ˆθ M1 ), F 2 (x 2t ; ˆθ M2 ); θ C ) (24) where, F 1 and F 2 represents the cumulative distribution functions of the marginal distributions with parameters ˆθ 1 and ˆθ 2 estimated as in (22). 13

18 When we use the observed margins as in the nonparametric approach we use the following MLE for estimating the copula parameter(s) ˆθ C : ˆθ C = ArgMax θc T t=1 ln C((û 1t, û 2t ); θ C ). (25) Here, û 1t and û 2t, t = 1, 2.., T are the pseudo inverses of the empirical distribution functions from (23). (Cherubini el alt, 2004, pp ) 5.2 Comparing and choosing between dierent copulas After choosing parameters for each of the investigated couplas, the last step of creating a bivariate copula is to decide which of them that best t our data. Genest et al. (2009) made a rigorous study comparing dierent proposed approaches. With a monte carlo experiment, they compare seven dierent goodness of t tests based on the null hypothesis H 0 : C C 0 where C 0 is a parametric copula, and C the underlying copula. They generated samples from seven dierent underlying copulas. Three Archimedean copulas (Frank, Gumbel and Clayton copula), the two Elliptical copulas described in 3.2 (The normal and the studentt copula) and the placket family of copulas. The test was performed several times with dierent degrees of dependencies. They show that a Cramer-von Mises test was the overall best test. This test can be made in several dierent ways. The most common approach is to T compare the tted copula to the empirical copula C E (u 1, u 2 ) = 1 T 1(Û1 t u 1, Û2 t u 2 ), where Û1 t = ˆF 1 (ɛ t ) and Û2 t = ˆF 2 (ɛ t ). ˆF1 (ɛ) and ˆF 2 (ɛ) are computed as in (22). The rank-based version of the Cramer-von Mises S n test statistic can then be calculated as S t = ( T (C E C θ )) 2 dc E (u 1, u 2 ) = [0,1] 2 t=1 T (C E C θ ) 2 (26) where C θt is the observations from our tted copula. A similar also rank-based test is the the Kolmogorov-Smirnov test. Its test statistic can be calculated as t=1 T t = sup T (C E C θ ) (27) Another Cramer-von Mises and Kolmogorov-Smirnov method is one that uses the so called Kendall's process. The Kendall's process is dened as with its empirical version K θ (t) = P r(c(u 1, U 2 ) t) (28) K E (v) = 1 T T 1(C E (Û1t, Û2t) v), v (0, 1) (29) t=1 The Cramer-von Mises and the Kolmogorov-Smirnov test statistic can then be calculated as S n (K) = T (KE (v) K θ (t)) 2 dk E (v) (30) [0,1] 2 14

19 and T (K) t = sup T (K E (v) K θ (t)) (31) respectively. The later GOF method was proven in the monte carlo experiment to work well for all copulas but especially when testing Archimedean copulas. When having an Archimedean copula with generator φ, then the Kendall's function can easily be calculated as K θ (t) = K φ (v) = t φ(v) φ (v). A very nice feature with the Kendall's function is that we easly can create a graphical representation of the empirical observations and compare it with the suggested copulas. By setting Z = C(Û1, Û2) T we can compute Z i = 1 T 1 1(U 1j < j=1,j i U 1i, U 2j < U 2i ). We can then graphically compare the cumulative distribution function of the Z i s with the distribution function of K θ. 6 Vine copulas 6.1 Pair copula constructions From Sklar's theoreme described in 2.2, we saw that the joint distribution function F (x 1,..., x n ) can be written as C(F 1 (x 1 ),..., F n (x n )). By using the chain rule, we also see that the joint density function f with marginal densities F 1,..., F n can be written as f(x 1,..., x n ) = C 1,...,n (F 1 (x 1 ),..., F n (x n )) f 1 (x 1 ),..., f n (x n )) (32) by taking the partial derivatives on all arguments in C(F 1 (x 1 ),..., F n (x n ). In the bivariate case this gives us f(x, y) = C x,y (F x (x), F y (y)) f X (x) f Y (y)). (33) We can also dene the conditional density as f y x (x y) = f(x y) f X (x) = C 12(F x (x), F y (y)) f y (y). (34) The interesting thing is that this method can be used in more than two dimensions by combining several bivariate copulas and creating so called pair-copula constructions. For example, in three dimensions with the random variables X 1, X 2, X 3, the conditional density f(x 1 x 2, x 3 ) can be written as f(x 1 x 2, x 3 ) = C 12 3 (F (x 1 x 3 ), F (x 2 x 3 )) f(x 1 x 3 ). = C 12 3 (F (x 1 x 3 ), F (x 2 x 3 )) C 13 (F (x 1 ), F (x 3 )), where C 12 3 (F (x 1 x 3 ), F (x 2 x 3 )) = C(F (x 1 x 3 ), F (x 2 x 3 )) f 1 (x 1 ). More generally, f(x v) = C xvj v j (F (x v j ), F (v j v j )) f(x v j ). (35) Here v is a d-dimensional vector and v j denotes the v vector not including the j th component. C x,vj v j is therefore a bivariate conditional copula. As one can see, this model is built up by marginal conditional distributions F (x v) and it is further proven that F (x v) = C x,v j v j (F (x v j),f (v j v j) F (v j v j) for every j. If v only has one component, and x and v are both uniform, the function is usually dened as the h-function h(x, v, C xv (θ)) = F (x v) = C xv(f x (x), F v (v)) F v (v) (36) 15

20 where C xv is the copula with parameter(s) θ describing the dependence between x and v. The h-function is therefore the partial derivative of the copulas distribution function with respect to v. When we later on will simulate from dierent decompositions, we will nd that the inverse of the h-function, h 1 (x, v, C xv (θ)) also is necessary (Aas et al. 2009, p 183). 6.2 Vines It is clear that these pair copula constructions are not unique. For example, the conditional density f(x 1 x 2, x 3 ) doesn't need to be denoted as C 12 3 (F (x 1 x 3 ), F (x 2 x 3 )) f(x 1 x 3 ). Another decomposition would be f(x 1 x 2, x 3 ) = C 13 2 (F (x 1 x 3 ), F (x 2 x 3 )) f(x 1 x 2 ) = C 13 2 (F (x 1 x 3 ), F (x 2 x 3 )) C 12 (F (x 1 ), F (x 2 )) f 1 (x 1 ). As the dimensions goes up, the amount of possible dierent constructions of pair-copulas increase very fast. Therefore so called vines have been introduced to structure these bivariate copulas. Basically vines follow a nested tree structure with edges and nodes. The edges in tree T i are the nodes in tree T i+1. All possible combination of pair copula constructions for a certain dimension can be described in a so called regular vine described by (Bedford, T. & Cooke, R, 2002). However, the big amount of dierent compositions related to regular vines still makes them hard to work with and dierent sub classes from the regular vines are often used instead. Two of the most common ones are C-vines and D-vines and in this study we will focus on the C-Vines. C-vines (Canonical vines) are built up so that each tree T i has one specic node called it's root with degree i 1 (Aas et al. 2009, p 185). Even though we limit the amount of dierent decompositions by only considering C-vines, we still have several dierent ones to choose from. In gure 4 one can see all possible decomposition in each tree for a C-vine in four dimensions. 1, , , , Tree 1: Four choices for a root. Variable 2 chosen. Bivariate conditional and unconditional coupulas. Nodes Edges Tree 2: Three choices for a root when having variable 2 as the root in tree 1. Variable 3 chosen ,4 23 Same copula 4, Tree 3: Two choices for a root when having variables 2 and 3 as roots in tree 1 and 2 respectively. The choice of root in the last tree does not matter for the structure of the tree but it matters for simulation purposes. Figure 4: Four dimension C-Vine decomposition. 16

21 6.3 Choosing C-vine structure When tting a C-vine, one usually start of by looking at the rst tree to see which variable i, i = (1, 2,..., d) to use as a root. An approach used by Dibmann, J et al. (2013, pp ), is to estimate all pairwise Kendall's τ s and see which variable i that maximizes: Ŝ i := d j=1 τ i,j, where τ i,j are the estimated τ s from all the pairwise dependencies. This variable is then considered the root of tree 1. For trees T i, i>1 the root is again based on Kendall's τ 's. However, the roots are now based on the conditional copulas. To estimate these conditional copulas, the h-function described dened in (32) is used. Czado (2009), pp proves that F (x j x D ) = h(f (x j x D v ), F (x v x D v ), C jv D v (θ)) (37) where D is a vector of variables, v D and θ jv D v parameter(s). And since, is the conditional bivariate copula F (x j x D v ) = h(f (x j x D v w ), F (x w x D v w ), C jw D v w (θ)) and, F (x v x D v ), θ jv D v )) = h(f (x v x D v w ), F (x w x D v w ), C vw D v w (θ)) where w D, we will be able to calculate the h-functions for the copulas in tree i + 1 as long as we know the copulas in tree i. In the Kendall's τ approach, the conditional copula which has the total highest value of τ 's with the other conditional copulas in tree i is then considered the root in that tree. The conditional copulas and their parameters are selected with methods shown in previous sections. This steps follows until the whole C-vine has been constructed. Since the model above choose the roots in the trees based on absolute Kendall's τ, it is choosing these trees based on a general dependence with no particular regards to the tails. In this paper we mainly wanna study copula models that can catch lower tail dependencies and therefore, a new model is investigated. Instead of selecting the edge that maximize the sum of absolute empirical Kendall's tau's, we wanna see which edges that maximize the sum of the lower tail dependencies for the copulas at each tree. A dierence from the approach above is that in order to calculate the tail dependencies, one must rst t every single copula in that tree. Since we want to nd copulas that can catch lower tail dependencies, we also try to choose these types of copulas. Therefore, we not only chose the copula with the best t according to the Cramer-von Mises test in (30). Instead, we choose the copula with the highest λ U. However, to be sure that the copulas that are being used are tting the data well, we still only considering copulas that are signicant based on the Cramer-von Mises test. The proposed algorithm for constructing the C-vine structure is described below Algorithm 1: C-Vine construction Input: Data: (X 1,...X d ), where variable X i is a vector with n U(0,1) observations and d the dimension. Output: Complete C-vine composition with suggested copulas and their respective parameter(s). C.1: For tree 1: C.1.1: For all possible vector pairs: C Select parameters to all dierent copulas according to section 5.1. C Calculate a p-value for all copulas according to section 5.2. C Keep only the copulas where the p-value is higher than some number β. C Calculate the theoretical lower tail dependence λ L according to table 2 17

22 (Archimedean copulas) and (Student- T copula). C Choose the copula with the biggest λ L. These copulas now correspond to all the possible edges in Tree 1. C.1.2 Among these edges, nd the variable i 1, whose corresponding pair copulas C i,2,..., C i,n maximizes Ŝi 1 := d j=1 λ L i1,j. The variable i 1 is therefore the root in tree 1. The edges in tree 1 is the corresponding copulas C i 1,2,..., C i 1,n. C.2: For tree α = 2,.., (d 1): C.2.1: By using the h-function and the copulas in trees 1,.., α 1, compute conditional observations for tree α C.2.2: For all possible conditional copula pairs in tree α: C Select parameters to all dierent conditional copulas according to section 5.1. C Calculate a p-value for all conditional copulas according to section 5.2. C Keep only the conditional copulas where the p-value is higher than β C Calculate the theoretical lower tail dependence λ L according to table 2 (Archimedean copulas) and (Student- T copula). C Choose the copula with the biggest λ L. These conditional copulas now correspond to possible edges in Tree α. C.2.3 Among these edges, nd the variable i α, whose corresponding conditional pair copulas C iα,d(1) i,..., C 1,...,i α 1 i α,d(m) i maximizes 1,...,i Ŝi α 1 α := d α j=1 λ L i α,d(j) i 1,...,i α 1 where D(1,.., m) is the vector of all variables excluding variables i 1,..., i α 1 and variable i α. The variable i α is therefore the root in tree α. The edges in tree α are the corresponding conditional copulas C i α,d(1) i,...,iα 1,..., C 1 i,...,iα 1. α,d(d) i 1 C.3: The whole C-vine tree is now constructed. 6.4 Sample from the C-vine structure Sampling from a C-vine can be done by using the the h-function and it's inverse, h 1 (x, v, θ). A general algorithm to generate random variables from a C-vine can be found in (Aas et al. 2009, p 187). However, below is a simplied algorithm of how to generate N observations from each variable in a four dimensional vine Algorithm 2: C-Vine sampling S.1 Generate w 1, w 2, w 3, w 4 where w 1,.., w 4 are N independent U[0, 1]. S.2 X 1 = V 1,1 = w 1 S.3 V 2,1 = w 2 S.4 V 2,1 = h 1 (V 2,1, V 1,1, C 1,2 (θ)) S.5 X 2 = V 2,1 S.6 V 2,2 = h(v 2,1, V 1,1, C 1,2 (θ)) S.7 V 3,1 = w 3 S.8 V 3,1 = h 1 (V 3,1, V 2,2, C 2,3 1 (θ)) S.9 V 3,1 = h 1 (V 3,1, V 1,1, C 1,3 (θ)) S.10 X 3 = V 3,1 S.11 V 3,2 = h(v 3,1, V 1,1, C 1,3 (θ)) S.12 V 3,3 = h(v 3,2, V 2,2, C 2,3 1 (θ)) S.13 V 4,1 = w 4 S.14 V 4,1 = h 1 (V 4,1, V 3,3, C 3,4 1,2 (θ)) S.15 V 4,1 = h 1 (V 4,1, V 2,2, C 2,4 1 (θ)) S.16 V 4,1 = h 1 (V 4,1, V 1,1, C 1,4 (θ)) S.17 X 4 = V 4,1 18

23 7 Practical example on European stock exchanges As a practical experiment, four of the biggest European exchange indexes are investigated in order to see if their dependence structure can be desrcibed by the copula methods described earlier. The four Indexes investigated are the following. EURONEXT 100: The blue chip for pan-eurpoean exchange Euronext. Euronext is the largest exchange market in Europe and is based in France, Netherlands, Belgium, Luxemburg and Portugal. Its blue chip index contains the largest stocks traded on the market. DB1: This is the stock market index for the Frankfurt stock exchange, the largest stock exchange in Germany. OMX30: OMX30 contains the 30 most traded stocks on the Stockholm stock exchange. LSEL: LSEL is the market index for the Londond Exchange Group, the fth largest stock exchange in the world. The data is coming from Yahoo Finance and are 2850 historical daily closing prices for each index between the fall of 2002 to the summer of Since the markets have dierent holidays with no trading, only days where all four markets were operating are investigated. In gure 5 one can see the time series for the four markets. It is clear that the indexes follow similar patterns. Figure 5: The time series of the four market indexes over the eleven years studied. 19

24 7.1 Fitting the marginal distributions For each of the time series, we t an Arma(p,q) model. The Arma models are selected based on the information criterion BIC. On these Arma models, we t a Garch(1,1) model to the error terms. In table 4 one can see what model that has been selected for each of the time series along with their corresponding BIC values. Index ARMA(p,q)+ GARCH(1,1) BIC EURONEXT 100 ARMA(3,1)+ GARCH(1,1) -619 DB1 ARMA(2,3)+ GARCH(1,1) -510 OMX30 ARMA(1,1)+ GARCH(1,1) -598 LSEL ARMA(1,1)+ GARCH(1,1) -479 Table 4: Fited Arma(p,q)+Garch(1,1) model for the four time series with their corresponding BIC values. It is now time to t the residuals into marginal distributions. We rst try to t the residuals to parametric distributions. The residuals from each index are tted to three distributions; the normal, the skew normal and the t-distribution. In gures 6-9 one can see the QQ plots where the residuals are compared to each of the three distributions. Figure 6: QQ plots of the EURONEXT 100 and three tted distributions. Figure 7: QQ plots of the DB1 and three tted distributions. Figure 8: QQ plots of the OMX30 and three tted distributions. 20

25 Figure 9: QQ plots of the LSEL and three tted distributions It is clear that all of them appear to have somewhat t-distributed errors. However, the residuals still seems to be dierently distributed in the tails comparing to the tted t-distributions. Therefore, we use a non parametric approach and t the residuals to its sample empirical distribution function to get the pseudo observations. In gure 10 one can see the scatter plots of the dierent pairs of empirical pseudo observations. 21

26 Figure 10: Scatter plots of the empirical pseudo observations. 22

27 7.2 Building the model 7.3 Tree 1 As a rst step of building the C-vine model, we will take a closer look at the bivariate dependencies (see C C in 6.3.1) and try to t appropriated copulas to each of the pairs. By maximum likelihood, we estimate parameters to all of the copulas based on the marginal distributions that were estimated from the sample empirical distributions in the previous section. It is clear that a lot of the copulas appear to catch the distributions quite well. To limit the amount of potential copulas, we apply the Cramer-von Mises test to to see which copulas that we can reject. By parametric bootstrap, we conclude the p-values for each of the bivariate copulas. We reject the hypothesis H 0 : C C 0 for p-values lower than 0.05 where C 0 is a parametric copula, and C is the scatter plots of the empirical pseudo observations. Since we want to choose copulas that have a strong lower tail dependence, we calculate the lower tail dependencies for all of the suggested copulas. For each of the pairs we then choose the copula that has the strongest lower tail dependence as long as is not rejected based on the Cramer-von Mises test. The result is shown in tables 5-10 with the chosen copula in red. By using the approach described in 5.2, we can also graphically compare the cumulative distribution functions created by the copulas and the empirical ones. These comparisons are shown in gures To nally choose our root in tree one, we have to nd out which variable that has the highest total tail dependence with the other variables, based on the selected copulas (see C.1.2). The result can be found in table 11. Copula ˆθ1 ˆθ2 P-value (5 % level ) lower λ L Gumbel (Accepted) 0 Clayton (Accepted) 0.45 Frank (Accepted) 0 Joe (Rejected) 0 Normal (Accepted) 0 Student T (v, degrees of freedom) 0.33 (Accepted) 0.12 Table 5: EURONEXT 100,DB1: Estimated parameters for the copulas, p-values from the Cramer-Von Mises test and lower tail dependence for the specic copula with the estimated parameter. Based on the test, the Clayton copula is chosen as the copula between EU- RONEXT 100 and DB1. Figure 11: EURONEXT 100,DB1: Cumulative distribution function for the empirical data and for the estimated copulas. 23

28 Copula ˆθ 1 ˆθ 2 P-value (5 % level ) lower λ L Gumbel (Accepted) 0 Clayton (Rejected) 0.74 Frank (Accepted) 0 Joe (Rejected) 0 Normal (Accepted) 0 Student T (v, degrees of freedom) 0.64 (Accepted) Table 6: EURONEXT 100,OMX30: Estimated parameters for the copulas, p-values from the Cramer-Von Mises test and lower tail dependence for the specic copula with the estimated parameter. Based on the test, the Student-T copula is chosen as the copula between EURONEXT 100 and OMX30. Figure 12: EURONEXT 100,OMX30: Cumulative distribution function for the empirical data and for the estimated copulas. Copula ˆθ 1 ˆθ 2 P-value (5 % level ) lower λ L Gumbel (Accepted) 0 Clayton (Accepted) 0.40 Frank (Accepted) 0 Joe (Rejected) 0 Normal (Accepted) 0 Student T (v, degrees of freedom) 0.08 (Accepted) Table 7: EURONEXT 100,LSEL: Estimated parameters for the copulas, p-values from the Cramer-Von Mises test and lower tail dependence for the specic copula with the estimated parameter. Based on the test, the Clayton copula is chosen as the copula between EU- RONEXT 100 and LSEL. Figure 13: EURONEXT 100,LSEL: Cumulative distribution function for the empirical data and for the estimated copulas. 24

29 Copula ˆθ1 ˆθ2 P-value (5 % level ) lower λ L Gumbel (Accepted) 0 Clayton (Accepted) 0.39 Frank (Accepted) 0 Joe (Rejected) 0 Normal (Accepted) 0 Student T (v, degrees of freedom) 0.88 (Accepted) Table 8: DB1,OMX30: Estimated parameters for the copulas, p-values from the Cramer-Von Mises test and lower tail dependence for the specic copula with the estimated parameter. Based on the test, the Clayton copula is chosen as the copula between DB1 and OMX30. Figure 14: DB1,OMX30: Cumulative distribution function for the empirical data and for the estimated copulas. Copula ˆθ1 ˆθ2 P-value (5 % level ) lower λ L Gumbel (Accepted) 0 Clayton (Accepted) 0.28 Frank (Accepted) 0 Joe (Accepted) 0 Normal (Accepted) 0 Student T (v, degrees of freedom) 0.81 (Accepted) Table 9: DB1,LSEL: Estimated parameters for the copulas, p-values from the Cramer-Von Mises test and lower tail dependence for the specic copula with the estimated parameter. Based on the test, the Clayton copula is chosen as the copula between DB1 and LSEL. Figure 15: DB1,LSEL: Cumulative distribution function for the empirical data and for the estimated copulas. 25

30 Copula ˆθ 1 ˆθ 2 P-value (5 % level ) lower λ L Gumbel (Accepted) 0 Clayton (Accepted) 0.31 Frank (Accepted) 0 Joe (Accepted) 0 Normal (Accepted) 0 Student T (v, degrees of freedom) 0.24 (Accepted) 0.04 Table 10: OMX30,LSEL: Estimated parameters for the copulas, p-values from the Cramer- Von Mises test and lower tail dependence for the specic copula with the estimated parameter. Based on the test, the Clayton copula is chosen as the copula between OMX30 and LSEL. Figure 16: OMX30,LSEL: Cumulative distribution function for the empirical data and for the estimated copulas. 26

31 EUR DB1 OMX30 LSEL TOTAL EUR DB OMX LSEL Table 11: Total λ L for each of the variables in tree 1. EURONEXT 100 is giving us the highest total values of λ L and is therefore chosen as the root in tree Tree 2 To be able to create the second tree, we use the h-function to calculate conditional observations. Since we chose the EUR as the root in the rst tree, we estimate the other variables based on the EUR (C.2.1). I.e we are using the h-function to create observation for DB1 conditional on EUR, OMX30 conditional on EUR and LSEL conditional on EUR. We then estimate copula parameters for all these pairs (C.2.2.2) and calculate the p-values for all of these copulas (C.2.2.3). To nally choose our root in tree 2 we again chose the variable that has the highest absolute total lower tail dependence to the other variables. These results are shown in tables However, as one can see, non of the pair-copulas involving the OMX30 give a signicant lower tail dependence. The Student-T and the Clayton copula which are the only ones who normally have lower tail dependence have parameters that are to weak. Therefore, we choose the pair-copulas involving OMX30 arbitrary. We base it on the p-value and therefore use the Student T copula for the DB1,OMX30 EUR copula and the Frank copula for the OMX30,LSEL EUR copula. When choosing which of the variables to have as root in tree 2, this will also be arbitrary. Since non of the choosen copulas with OMX30 have lower tail dependence, DB1 EUR and LSEL EUR have the same total lower tail dependence (only the tail dependence against each other), see table 15. Here we choose LSEL EUR as our root in tree 2. Copula ˆθ 1 ˆθ 2 P-value (5 % level ) lower λ L Gumbel (Accepted) 0 Clayton (Accepted) Frank (Accepted) 0 Joe (Accepted) 0 Normal (Accepted) 0 Student T (v, degrees of freedom) 0.70(Accepted) 0.00 Table 12: DB1,OMX30 EUR. Since non of the copulas gives us a signicant λ L the Student- T copula has been chosen because of the highest P-value. Copula ˆθ1 ˆθ2 P-value (5 % level ) lower λ L Gumbel (Accepted) 0 Clayton (Accepted) Frank (Accepted) 0 Joe (Accepted) 0 Normal (Accepted) 0 Student T (v, degrees of freedom) 0.37(Accepted) Table 13: DB1,LSEL EUR. Based on the test, the Clayton copula is chosen as the copula. 27

32 Copula ˆθ1 ˆθ2 P-value (5 % level ) lower λ L Gumbel (Accepted) 0 Clayton (Accepted) Frank (Accepted) 0 Joe (Accepted) 0 Normal (Accepted) 0 Student T (v, degrees of freedom) 0.34(Accepted) 0.00 Table 14: OMX30,LSEL EUR. Since non of the copulas gives us a signicant λ L the Frank copula has been chosen because of the highest P-value. DB1 EUR OMX30 EUR LSEL EUR TOTAL DB1 EUR OMX30 EUR LSEL EUR Table 15: Total λ L for each of the variables in tree 2. Since DB1 EUR and OMX30 EUR have the same total λ L, the choice between them is arbitrary. LSEL EUR has been chosen as the root in tree Tree 3 For tree 3 we follow the same procedure as for tree 2. So, we now estimate OMX30 and DB1 where OMX30 and DB1 are conditioned on LSEL (which already is conditioned on EUR). Again, we are using the h-function. We then follow the same method as in the previous trees. However, since we here only will have two set of observations, we will only need to choose one copula. The result is found in table 20. Since we only have one pair-copula to construct, we can skip step 2.3 in the algorithm. Just as in the second tree, most of the estimated copulas are acceptable with the Cramer-von Mises test (all except the Gumbel copula). Again as in tree 2, non of the copulas gives us a signicant lower tail dependence and we therefore choose the copula arbitrary based on the highest p-value, i.e the Frank copula. Since DB1 had a higher lower tail dependence in tree 2 we choose that variable as our third variable and hence the OMX30 as our fourth and last variable. The choice of order of the two last variables does not matter for the C-vine tree, however it makes a dierence when we sample from it. The C-vine structure is now completed and the full structure can be found in gure 18. Copula ˆθ1 ˆθ2 P-value (5 % level ) lower λ L Gumbel (Rejected) 0 Clayton (Accepted) Frank (Accepted) 0 Joe (Accepted) 0 Normal (Accepted) 0 Student T (v, degrees of freedom) 0.64 (Accepted) 0.00 Figure 17: OMX30,DB1 LSEL,EUR. Since non of the copulas are given a signicant λ L, we base the copula based on the p-value. Therefore the Frank copula is chosen. 28

33 EUR,LSEL Clayton (0.74) EUR Tree 1: EURONEXT 100 chosen as the root in tree 1. DB1 LSEL OMX 30 EUR,LSEL Tree 2: LSEL chosen as the root in tree 2. EUR,DB1 EUR, OMX30 Tree 3: DB1 chosen as root in tree 3. LSEL,DB1 EUR DB1,OMX30 EUR,LSEL Frank (0.35) LSEL,OMX30 EUR Figure 18: Full C-Vine structure based on algorithm Sampling Using the sampling algorithm 2 described in 6.4.1, we can now produce random variables from the C-vine presented in gure 18. After all four variables have been sampled from this algorithm, we check all the pairwise dependencies of those variables. In gures one can see the scatter plots for each of the pairwise observations. Beside each plot are the empirical plots (from gure 10). The X-axis can be seen as the daily movements for the rst variable (in comparison to all other days on a 0-1 scale) where 1 being the biggest positive movement and 0 being the biggest negative movement. The Y axis represent the same thing but for the other variable. So a point in (1,1) would mean that the biggest daily movement for the rst variable occurred at the same time as the biggest movement for the second variable and so on. By checking the average distances between all pairwise points (in both the X and Y axis) we can nd the average error. Since we are particularly interested in the lower quadrant, we check the lowest 5% for both of the two variables and look at the average distances in that area. The results for all of the pairwise variables can be seen in the tables below the gures. To get an understanding of what the numbers mean, the average distance if we compared two uniformed random variables U[0,1] would be Also included in the tables are the distances from the empirical observations when using all the bivariate (unconditional) copulas as our samples. The copulas chosen are the six copulas from tree 1. However, keep in mind that these samples are not produced by the sample algorithm but instead directly using it's respectively copula. Since each of these copulas are tted completely independent of each other (in comparison to the C-vine where the dependence structure are tree-based), these bivariate samples should be more similar to the empirical observations than when applying the sample algorithm. It is clear from the gures and the tables that the C-vine catch the empirical pairwise observations from tree 1 in the lower quadrant (i.e the ones having EURONEXT 100 as one of the two variables) quite well. However, even in tree 2-3 signicantly similarities can be seen. It is also clear that the the C-vine catches the bivariate dependencies almost as good as when we sample the observations straight from the bivariate copulas. 29

34 EUR,LSEL Average distance between the randomly generated variables and the empirical observation Samples from C-Vine using algorithm 'S' in Samples straight from the estimated bivariate copula in tree 1 (Not conditional copulas). All points Lowest 5% Figure 19: Above: Scatter plots of the pseudo observations for EUR and LSEL. On the left side are samples from the C-vine and on the right hand side the empirical observations. Table below: Average distances between the empirical points and the ones from the dierent simulations. 30

35 EUR,DB1 Average distance between the randomly generated variables and the empirical observation Samples from C-Vine using algorithm 'S' in Samples straight from the estimated bivariate copula in tree 1 (Not conditional copulas). All points Lowest 5% Figure 20: Above: Scatter plots of the pseudo observations for EUR and DB1. On the left side the generated samples from the C-vine and on the right hand side the empirical observations. Table below: Average distances between the empirical points and the ones from the dierent simulations. 31

36 EUR,OMX30 Average distance between the randomly generated variables and the empirical observation Samples from C-Vine using algorithm 'S' in Samples straight from the estimated bivariate copula in tree 1 (Not conditional copulas). All points Lowest 5% Figure 21: Above: Scatter plots of the pseudo observations for EUR and OMX30. On the left side the generated samples from the C-vine and on the right hand side the empirical observations. Table below: Average distances between the empirical points and the ones from the dierent simulations. 32