Las fases, hitos y restricciones del proyecto

We now turn to the task of modeling the joint distribution of stock returns, bid-ask spreads, and default intensities of multiple firms. To capture both linear dependences and potential non-linearities in the dependence structure, we rely on copulas in our modeling approach. More precisely, we employ dynamic R-vine copulas which pro- vide us with a powerful tool to model high-dimensional distributions and to capture complex and time-varying dependences in an extremely flexible way. Subsequently, we discuss R-vine copulas and present our dynamization approach. We start with a brief review on copulas and pair-copula constructions.

2.2.2.1 Copulas and pair-copula constructions

Generally speaking, a d-dimensional copula function is a multivariate distribution function on the unit cube _r0,1_sd with standard uniform margins. More precisely, a copula specifies the link between a multivariate distribution and its one-dimensional marginal distributions (see Nelsen, 2006). Formally, with _pX1, ...,Xdq denoting a d-

dimensional random vector with joint density f _{“ p}f1, ..., fdqand distribution function

F_{“ p}F1, ...,Fdq, the copula C of the distribution F is given by

C_pu1, ...,udq “ FpF₁´1pu1q, ...,F_d´1pudqq, (2.5)

where F´_i 1 is the generalized inverse of Fi and ui P r0,1s, i “1, ...,d. The theoretical

framework of copulas goes back to Sklar (1959) who shows that, under certain condi- tions, every copula is a joint distribution function and vice versa (see Nelsen, 2006, for a detailed discussion). Using (2.5), the joint density, f , can be expressed as

f_px1, ...,xdq “ cpF1px1q, ...,Fdpxdqq d

ź

i“1

fipxiq, (2.6)

where c denotes the density of C. Hence, we can separate the dependence structure from the marginal structure and thus model the joint distribution by first modeling the

2.2. ECONOMETRIC METHODOLOGY 22

marginal distributions and then specifying a model for the dependence structure.11

In case of bivariate data (i.e., d _“ 2), there is a wide range of Archimedean and elliptical copulas available that allow for flexible dependence modeling.12 _{In case of}

multivariate data sets (that is, d _ą 2), however, this becomes much more difficult so that existing studies in the econometrics and statistics literature emphasize the need for flexible copula models in high dimensions (see Chollete et al., 2009, Aas et al., 2009, Dißmann et al., 2013).13 While some papers attempt to construct multivariate ex- tensions of (bivariate) Archimedean copulas (Embrechts et al., 2003, Savu and Trede, 2010, Hofert, 2011), another strand in the literature aims to construct flexible mul- tivariate dependence models by splitting up the copula density, c, into a cascade of bivariate (unconditional and conditional) copulas.14 _{The resulting expression is called}

a pair-copula construction (PCC hereafter) and can be derived as follows.

Let fj|k “ fj|kpxj|xkq, Fj|k “ Fj|kpxj|xkq, ci j|k “ ci j|kpFi|k,Fj|kq, and beηapd´1q-

dimensional vector satisfyingηℓ P t1, ...,du z tiuandηℓ1 ‰ ηℓ2 forℓ1 ‰ ℓ2. Then, we

can decompose the multivariate density, f , in the following way

f _“ fd d´1

ź

i“1

fd´i|d´i`1,...,d. (2.7)

Further, as stated in Aas et al. (2009), the conditional density, fj|η, can be factorized as

fj|η “ cjηm|η´mfj|η´m, (2.8)

whereηm is an arbitrarily chosen component of ηand η´mresults from removing ηm

from η, m _{P t}1, ...,d_´1_u. Combining the two factorizations in (2.7) and (2.8) then

11_{Note that the expression in (2.6) provides the theoretical basis for our modeling strategy since we first}

model the marginal densities using GARCH processes and then model the dependence structure with R-vine copulas.

12_{See Nelsen (2006) for a detailed overview.}

13_{Note that, in high dimensions, the choice of copulas is virtually reduced to elliptical copulas such as}

the normal and the t copula which are only useful if the assumption of elliptical dependence is valid.

14_{For further details, see the seminal works by Joe (1997), Bedford and Cooke (2001, 2002), Whelan}

2.2. ECONOMETRIC METHODOLOGY 23

yields the following expression for a PCC

f _“ d ź k“1 fk d´1 ź h“1 d´h ź i“1 ciηm|η´m, (2.9)

where h_“dim_pη_qand m_“m_ph,i_{q P t}1, ...,h_uis arbitrarily chosen.15

Based on the pioneering works by Joe (1996, 1997) and Bedford and Cooke (2001, 2002), Aas et al. (2009) introduced the concept of pair-copulas to the finance literature and spurred a surge in empirical applications of PCCs (see, e.g., Heinen et al., 2009, Aas and Berg, 2009, Chollete et al., 2009, Min and Czado, 2010, 2011). For our mod- eling framework, the use of PCCs is especially appropriate in many respects. First, splitting up the multivariate density according to (2.9) results in a computationally fea- sible density for likelihood estimation and, therefore, enables us to handle the high dimensionality of our modeling approach. Moreover, PCCs provide us with an ex- tremely flexible tool to capture the presumably intricate dependences between stock returns, bid-ask spreads, and default intensities. Using PCCs, we are able to choose each pair-copula from a different parametric copula family and, further, PCCs permit the modeling of not only the pairs of the original variables but also pairs of conditional distributions of recomputed variables (see Weiß and Supper, 2013).16 Since we follow Patton (2006) and estimate dynamic processes for the parameters of the pair-copulas, the dynamic PCCs are also capable of accounting for potentially time-varying patterns in the dependence structure.

2.2.2.2 Regular vines

As can be seen from the expression in (2.9), there exist many different PCCs for a given multivariate distribution, F.17 To select a particular PCC and to determine the way in which the marginals are to be coupled, Bedford and Cooke (2001, 2002) introduce so- called (regular) vines. Vines are convenient tools with a graphical representation that

15_{We use the convention iη}

m|H “iηm. Thus, h“1 yields unconditional pair-copulas ciηm, i“1, ...,d´

1.

16_{That is, we are capable of specifying the conditional dependence structure for the joint distribution.} 17_{This results from the fact thatη}

2.2. ECONOMETRIC METHODOLOGY 24

facilitate the description of the conditional specifications made for the joint distribu- tion, F. More precisely, an R-vine is a graphical tree model that is based on a nested set of trees satisfying certain conditions.

To formally describe the concept of R-vines, we label the components of X from 1 to d and recall that a tree, T _{“ t}N,E_u, is an acyclical graph, where N _Ă N _and

E _Ă `N₂˘denote the set of nodes and edges, respectively. Bedford and Cooke (2002) define a regular vine on d elements, V_{, as a nested set of trees,} V _{“ tT1, ...,Td´1u,} that satisfies the following conditions

(c1) T1is a tree with nodes N1 “ t1, ...,duand a set of edges denoted E1.

(c2) For i_“2, ...,d, Ti is a tree with nodes Ni “Ei´1and|Ni| “i`1.

(c3) For i_“2, ...,d_´1 and_ta,b_{u P} Ei, it must hold that|aXb| “1.

To derive the PCC induced by V, each edge in Ti is associated with a bivariate (un-

)conditional copula, i _“ 1, ...,d _´1. The edges of the R-vine trees are computed according to (c1)-(c3) and on the basis of set operations on so-called conditioning and conditioned sets, which are given as follows.18 _{With U}

ei denoting the set of all indices

contained in ei “ ta,bu P Ei, the conditioning set, Dei, is given by Dei “UaXUb, and

the conditioned set, Cei, is defined to be Cei “Ua∆Ub, with∆denoting the symmetric

difference operator.19

As shown in Bedford and Cooke (2001, 2002), there is a unique PCC associated withV, which can be expressed as

f _“ d ź k“1 fk d´1 ź h“1 ź ePEh cCe|De. (2.10)

Hence, R-vine copulas as used in our modeling approach are particular PCCs, i.e. PCCs with a particular decomposition (2.9), which are determined according to the combinatorial rules presented above.20

18_{We follow the presentation in Dißmann et al. (2013).} 19_{Note that|C}

ei| “2 and CeiXDei “ H.

20_{A detailed description on the construction of R-vines and R-vine copulas as well as examples and il-}

2.2. ECONOMETRIC METHODOLOGY 25

2.2.2.3 Fitting an R-vine copula

Fitting an R-vine copula can be organized into three steps: (1) Selection of R-vine structure, (2) Selection of bivariate copula families, and (3) Estimation of copula pa- rameters. These steps are accomplished following the sequential method as proposed in Dißmann et al. (2013) and Hobæk Haff (2013), which exploits the tree-by-tree struc- ture of vines and under which selection and estimation are performed treewise, con- ditioning on the precedingly selected trees and estimated copula parameters.21 _More

precisely, for a given tree, Ti P V, we first calculate the empirical Kendall’s tau,

ˆ

τj,k, for all possible variable pairs, tj,ku, j,k “ 1, ...,d, and determine the edges of Ti by selecting the spanning tree that maximizes the sum of absolute empirical taus.22

Then, each of the resulting edges is associated with a bivariate (un-)conditional copula, which is selected according to the Akaike information criterion (AIC).23 We calculate the AIC for each copula family considered and choose the copula with the minimum AIC.24 _{Using the fitted copulas in tree T}

i, we now compute the transformed variables

by means of the corresponding h-functions and repeat the above procedure until we reach tree Td´1(see Dißmann et al., 2013, for details), resulting in a total of dpd´1q{2

dynamic (un-)conditional pair-copulas.

Since we need standard uniform data to consistently estimate copulas, fitting the R- vine copula in our econometric approach should be based on white-noise time series. Assuming that the GARCH processes discussed above correctly specify the marginal densities, we apply the R-vine copula to the corresponding GARCH residuals, εi,t.

The pseudo-observations used for estimation, ui, are then computed as the ranks of the

residuals, i.e. ui “Fipεiq.

21_{Note that this method does not necessarily lead to a global optimum. Most of the dependence is,}

however, captured in the first tree so that the model fit is considerably influenced by the fit of the copulas in the first tree.

22_{Actually, we use Prim’s (1957) algorithm and calculate the minimum spanning tree with weights´τˆ} j,k. 23_{As found in Manner (2007), the AIC provides a reliable criterion, especially when compared to alter-}

native criteria such as copula goodness-of-fit tests.

24_{We include dynamic extensions of the normal, t, (rotated) Clayton, (rotated) Gumbel, and (rotated)}