# VineCopula class

The class of vine copula models (yet C-Vines and D-Vines only)

## Properties

### dimension

The dimension (=d) of the vine copula.

### type

The type of the vine copula, i.e., 'C-Vine', 'CVine' or 0 and 'D-Vine' ,'DVine' or 1.

### simplified

A logical or vector of logicals, which provides the information, whether the simplifying assumption is fulfilled or not. Either the whole vine is set to simplified, i.e., all conditional copulas are chosen to be unconditional bivariate copulas (pair- copulas) or each conditional copula being part of the vine copula model is set to 1 for a pair-copula, i.e., an unconditional bivariate copula or 0 for a conditional copula.

Possible values for simplified are 1 (for unconditional bivariate copulas; pair-copulas) and 0 for conditional copulas. If simplified is set to 1 then every pair-copula being part of the PCC is set to be a partial copula. In contrast, if the simplified property is chosen to be 0 then every conditional copula is set to be a conditional copula with functional parameter (note that this can also be an unconditional copula whenever the functional parameter is a constant). Alternatively, the simplified property can also be given as a 1 x ((d-1)*(d-2)/2) matrix, where each entry is either 0 or 1 and therefore every copula is set to be a pair-copula or a conditional copula with functional parameter, separately.

### structure

A vector containing the information about the structure of the vine copula model.

Possible values for the structure in the case of a C-Vine copula are all permutations of the (1,...,d) sequence. The first entry of the structure is the root node of tree 1 (i.e., the unique node of degree d-1), the second entry is the root node of degree d-2 in tree 2, etc. For a D-Vine possible values for the structure vector are also all permutations of the (1,...,d) sequence. The choice [1 2 3 4 5] corresponds to the D-Vine consisting of the pair-copulas C_12, C_23, C_34, C_45, C_13|2, C_24|3, C_35|4, C_14|23, C_25|34 and C_15|234.

### families

A 2 x (d*(d-1)/2) cell array or matrix, where in the first row the copula family for each pair-copula is stored and in the second row, one can optionally store the degree of rotation for the pair-copula.

Possible values for families is either
a 1 x (d*(d-1)/2) cell array (if there
are no rotated pair-copulas within the
PCC) or a 2 x (d*(d-1)/2) cell array,
where the first row consists of the
pair-copula families and the second
one of the degrees of rotation of the
different pair-copulas. If rotation is
empty for a pair-copula, then it is
the standard (unrotated) copula of
the specified family.
The copula families and rotation
degrees can also be given in the
numerical coding (e.g. 7 for the
Clayton copula and 90 for 'r90').
The order of the entries into the
columns is tree by tree and within
the tree it is ordered according to
the structure. For example if the
structure is [2,1,3,4] then the
entries in the family cell array
correspond to the pair-copulas in the
following ordering (for the C-Vine):
C_21, C_23, C_24, C_13|2, C_14|2 and
C_34_21. Here it is important to note
that the root in each tree is always
the first variable of each pair-
copula, which is important for the
specification of pair-copulas with
asymetric dependence. For a
five-dimensional D-Vine copula with
structure [1 2 3 4 5], the ordering
of the families has to correspond to
the pair-copulas C_12, C_23, C_34,
C_45, C_13|2, C_24|3, C_35|4,
C_14|23, C_25|34 and C_15|234.

### parameters

A 1 x (d*(d-1)/2) cell array containing all the parameters of the unconditional pair-copulas. Alternatively, the same information (the parameters) can also be given simply in vectorized format, where the parameters are contained in the vector in the same ordering as for example the pair-copula families.

Possible values for parameters are either a 1 x (d*(d-1)/2) cell array, where each entry is a vector of the corresponding pair-copula parameter(s). Alternatively, one can also specify the parameters by vectorizing the information contained in the cell array of parameters.

### condparameterfunctionals

A 1 x ((d-1)*(d-2)/2) cell array containing parameter functionals for the conditional copulas being part of the vine. The parameter functionals can also be provided in vectorized form.

Possible values for the condparameterfunctionals are either a 1 x ((d-1)*(d-2)/2) cell array, where each entry is a vector of the corresponding pair-copula parameter functional(s). Alternatively one can also specify the parameter functional(s) by vectorizing the information contained in the cell array of parameter functional(s).

### MaxLLs

The values of the maximized log- likelihoods of the PCC evaluated at the sequentially and jointly estimated parameter vectors.

The first entry always corresponds to the maximized value of the vine copula log-likelihood evaluated at the sequentially estimated parameters, while the second entry corresponds to the jointly estimated parameters.

### SeqEstParameters

The sequentially estimated parameters for the vine copulas.

The estimates from estimating the vine copula by applying the sequential estimation approach.

## Methods

### VineCopula (VineCopulaObject)

The "function" VineCopula can be used to construct members of the VineCopula class. At least the properties dimension, type and simplified have to be specified.

### Fit (VineCopulaObject)

The method Fit can be used to estimate a vine copula model, specified as an object of the VineCopula class, using a dataset by maximizing the log-likelihood of the vine copula.

### Sim (VineCopulaObject)

The method Sim can be used to simulate from a vine copula, specified as an object of the VineCopula class.

### StructureSelect (VineCopulaObject)

The method StructureSelect can be used to find an "adequate" structure and pair-copula families (and parameters) for a given data set.

### GetPseudoObsFromVine

The method GetPseudoObsFromVine can be used to obtain pseudo- observations from the conditional copulas being part of the specified PCC.

### SeqTestOnSimplified (VineCopulaObject)

The method SeqTestOnSimplified can be used to sequentially test on the simplifying assumption by applying a vectorial independence test.

## References

[1] Aas, K., C. Czado, A. Frigessi and H. Bakken (2009), "Pair-copula constructions of multiple dependence", Insurance: Mathematics and Economics 44(2), pp. 182-198.

[2] Acar, E. F., C. Genest and J. Neslehová (2012), "Beyond simplified pair-copula constructions", Journal of Multivariate Analysis 110, pp. 74-90.

[3] Bedford, T. and R. M. Cooke (2001), "Probability density decomposition for conditionally dependent random variables modeled by vines", Annals of Mathematics and Artificial Intelligence 32 (1), pp. 245-268.

[4] Bedford, T. and R. M. Cooke (2002), "Vines -- A new graphical model for dependent random variables", The Annals of Statistics 30(4), pp. 1031-1068.

[5] 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.

[6] Hobaek-Haff, I., K. Aas and A. Frigessi (2010), "On the simplified pair-copula construction -- Simply useful or too simplistic?", Journal of Multivariate Analysis 101(5), pp. 1296- 1310.

[7] Joe, H. (1996), "Families of m-Variate Distributions With Given Margins and m(m-1)/2 Bivariate Dependence Parameters", Distributions with Fixed Marginals and Related Topics, ed. by L. Rüschendorf, B. Schweizer, and M. D. Taylor, Hayward, CA: Institute of Mathematical Statistics.

[8] Joe, H. (1997), Multivariate models and dependence concepts, 1. ed., reprint., Monographs on statistics and applied probability; 73, Boca Raton, Fla. [u.a.]: Chapman & Hall/CRC.

[9] Kurowicka, D. and H. Joe (Eds.) (2011), "Dependece Modeling -- Vine Copula Handbook", Singapore: World Scientific Publishing Co. Pte. Ltd.

[10] Kurz, M. (2012), "On the simplified pair-copula construction -- Graphical and computational illustrations", Unpublished Term Paper, Master Seminar on Financial Econometrics, Department of Statistics, Ludwig-Maximilians-University Munich.

[11] Kurz, M. (2013), "Tests on the partial copula", Unpublished Master's Thesis, Department of Statistics, Ludwig-Maximilians-University Munich.

[12] 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.

[13] 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.

[14] Spanhel, F. and M. Kurz (2014), "Simplified vine copula approximations -- Properties and consequences", submitted for publication.

[15] Stöber, J., H. Joe and C. Czado (2013), "Simplified pair copula constructions -- Limitations and extensions", Journal of Multivariate Analysis 119, pp. 101-118.