Selecting the structure and pair-copula families for a vine copula
The function can be used to find an adequate structure and to select pair-copula families for a given data set. By default (StructuringRule = 0) the nodes of the C-vine are chosen in a way that in each tree, the root (i.e. the node, which is connected by a copula to all other nodes) is the variable which has maximal dependence with all other variables. The maximal dependence is found by choosing the variable which has the maximal column sum in the matrix of absolute empirical Kendall’s τ (cf. Schepsmeier, Stöber, and Brechmann (2013) for an R-function (RVineStructureSelect) of exactly the same procedure and Czado, Schepsmeier, and Min (2012, p. 240) for the theoretical background of the approach).
Alternative methods for choosing a structure for C-Vine copulas (cf. Nikoloulopoulos, Joe and Li (2012)): The root node of the first tree is chosen in the same way as in the default method, i.e., the variable with the strongest dependence with all other variables. Then one can choose between three rules suggested in Nikoloulopoulos, Joe and Li (2012, p.3665): StructuringRule = 1: List the other variables by their dependence to the root node in decreasing order. StructuringRule = 2: List the other variables by their dependence to the root node in increasing order. StructuringRule = 3: List the other variables sequentially by choosing the variable which is least dependent with the previously selected one.
Furthermore, the copula families are chosen according to the AIC criterion and for each pair- copula an independence test is performed (cf. Schepsmeier, Stöber, and Brechmann (2013) and Brechmann and Schepsmeier (2013) for R-functions (RVineStructureSelect / RVineCopSelect / CDVineCopSelect) and Genest and Favre (2007, p. 351) for the independence test).
For D-Vine copulas there is no structuring rule implemented yet. Therefore, the function uses the specified structure and only selects the pair-copula families using the AIC criterion.
Select a simplified vine copula model VineCopulaHat = StructureSelect(VineCopulaObject,u) Select a simplified vine copula model, where the pair-copulas are from a specified set of families (possible choices 'all' (default), 'R', 'R-package', 'VineCopulaPackage' (they all are equivalent and correspond to the set of pair-copulas of the R-package VineCopula (Schepsmeier, Stöber, and Brechmann (2013), Version 1.2)) or a cell-array consisting of possible pair-copula families (i.e., a user selected list of possible pair-copula families). VineCopulaHat = StructureSelect(VineCopulaObject,u,'all') VineCopulaHat = StructureSelect(VineCopulaObject,u,'R') VineCopulaHat = StructureSelect(VineCopulaObject,u,'R-package') VineCopulaHat = StructureSelect(VineCopulaObject,u,'VineCopulaPackage') VineCopulaHat = StructureSelect(VineCopulaObject,u,familyset) Select a simplified vine copula model using an alternative structuring rule: VineCopulaHat = StructureSelect(VineCopulaObject,u,familyset,1) VineCopulaHat = StructureSelect(VineCopulaObject,u,familyset,2) VineCopulaHat = StructureSelect(VineCopulaObject,u,familyset,3)
VineCopulaObject= An object from the class VineCopula. u = A (n x d) dimensional vector of values lying in [0,1] (the observations). familyset = The set of possible pair-copula families. By setting it to the strings 'all', 'R', 'R-package' or 'VineCopulaPackage' one can choose one of the pre-defined sets. Alternatively one can choose an array containing a subset of the possible families: 0 Indep 1 AMH 2 AsymFGM 3 BB1 4 BB6 5 BB7 6 BB8 7 Clayton 8 FGM 9 Frank 10 Gaussian 11 Gumbel 12 IteratedFGM 13 Joe 14 PartialFrank 15 Plackett 16 Tawn1 17 Tawn2 18 Tawn 19 t
VineCopulaHat = An object from the class VineCopula. The selected vine copula structure can be found in VineCopulaHat.structure and the selected pair-copulas in VineCopulaHat.families. Furthermore, the sequential estimates, which are obtained during the selection procedure of the structure and copula families, are stored in VineCopulaHat.parameters.
 Brechmann, E. C. and U. Schepsmeier (2013), "Modeling Dependence with C- and D-Vine Copulas: The R-Package CDVine", Journal of Statistical Software 52(3), R package version 1.1-13, pp. 1-27, url: http://CRAN.R-project.org/package=CDVine.
 Czado, C., U. Schepsmeier, and A. Min (2012), "Maximum likelihood estimation of mixed C-vines with application to exchange rates", Statistical Modelling 12(3), pp. 229-255.
 Genest, C. and A. Favre (2007), "Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask", Journal of Hydrologic Engineering 12(4), pp. 347- 368.
 Nikoloulopoulos, A. K., H. Joe, H. Li (2012), "Vine copulas with asymmetric tail dependence and applications to financial return data", Computational Statistics & Data Analysis 56(11), pp. 3659-3673.
 Schepsmeier, U., J. Stöber, and E. C. Brechmann (2013), VineCopula: Statistical inference of vine copulas, R package version 1.2, url: http://CRAN.R-project.org/package=VineCopula.