Modeling dependence structure among European markets and among Asian-Pacific markets: a regime switching regular vine copula approach

Abstract

This paper investigates the structure of dependence among twelve European markets and among twelve Asian-Pacific markets. The dynamic of the dependence structure is described by a two-state regime switching model. The dependence structure during a bull phase is modelled by the Gaussian copula, while dependence during a bear phase is modelled by the regular vine copula. We analyze the regular vine structure in the second regime precisely. We perform a simplification procedure using a likelihood-ratio test and discuss the substitution of general regular vines by canonical vines or drawable vines. The analysis confirms the two-state nature of financial markets in addition to asymmetric and heavy-tailed dependences. Additionally, the European market has proven to be more strongly connected than the Asian-Pacific market, and European dependences are deeper in terms of conditional dependences. The results can be used by international investors by taking into account differences of both analyzed regions. Additionally, the analysis may help with the crisis prediction. The shift time to the market phase describing crisis times occurs significantly before the crisis itself.

Appendix

One may choose different weights in line 1. We have chosen Prim’s algorithm in lines 2 and 8, although it is possible to perform a different algorithm e.g. Kruskal’s algorithm. Note that the constructed graph is not necessarily a tree in line 6. It is possible that in a single step during the stepwise procedure conditional variables ${\varvec{x}}_{i\left( {e_1 } \right) |D\left( {e_1 } \right) }$and ${\varvec{x}}_{i\left( {e_2 } \right) |D\left( {e_2 } \right) }$ occur, for which $i\left( {e_1 } \right) =i\left( {e_2 } \right)$ and $D\left( {e_1 } \right) \ne D\left( {e_2 } \right)$. So the operations in lines 2 and 7 are not the same. It is not possible to simply perform operation in line 2 and in line 7 for transformed observations. Additionally, the operations in lines 3 and 9 are not the same. In line 9, the use of Algorithm 1 is not straightforward, as with Kendall’s tau calculations, the pseudo-observations obtained by formula (5) might be different in different pairs. In fact, in the case of a D-vine structure, each variable at each level is transformed differently in each pair (there are at most two pairs in which a single variable is involved), while in the case of a C-vine, at each level all conditioning sets are the same. Thus, essentially procedures in lines 2 and 7 are the same as are procedures in lines 3 and 9 in the case of C-vines.

Algorithm 4 can be modified in such a way that in line 1, initial copulae may be set with parameters in place of probabilities. In this case lines 3, 4 and lines 5, 6 are interchanged. One may choose smoothed probabilities in line 1 of Algorithm 4 diversely.