# VineCopulaFit

Estimating vine copulas

# Purpose

The function computes ML-estimates for the parameters of a C-Vine or D-Vine copula. Therefore, first starting values for the joint estimation are obtained by iteratively estimating the pair-copulas in the first trees and using those estimates to obtain the arguments for the copulas in the second tree. Then the pair- copulas in the second tree are estimated and so on. These sequentially estimated parameters from the sequential procedure are then used to obtain the ML-estimates, by minimizing the overall negative log-likelihood of the whole C-Vine (or D-Vine) numerically. Possible Vine copula types:

```
0 C-Vine
1 D-Vine
```

# Usage

```
Simplified standard C-Vine or D-Vine copula
ParamHat = VineCopulaFit('C-Vine',families,d,u)
[ParamHat, MaxLogLikes] = VineCopulaFit('C-Vine',families,d,u)
[ParamHat, MaxLogLikes, theta0] = VineCopulaFit('C-Vine',families,d,u)
Simplified standard C-Vine or D-Vine copula with rotated copulas
ParamHat = VineCopulaFit('C-Vine',families,d,u,rotation)
[ParamHat, MaxLogLikes] = VineCopulaFit('C-Vine',families,d,u,rotation)
[ParamHat, MaxLogLikes, theta0] = VineCopulaFit('C-Vine',families,d,u,rotation)
Simplified standard C-Vine or D-Vine copula (specified estimation method
(i.e., joint or sequential estimation)
ParamHat = VineCopulaFit('C-Vine',families,d,u,rotation,EstMethod)
[ParamHat, MaxLogLikes] = VineCopulaFit('C-Vine',families,d,u,rotation,EstMethod)
[ParamHat, MaxLogLikes, theta0] = VineCopulaFit('C-Vine',families,d,u,rotation,EstMethod)
Truncated simplified standard C-Vine or D-Vine copula
ParamHat = VineCopulaFit('C-Vine',families,d,u,rotation,EstMethod,CutOffTree)
[ParamHat, MaxLogLikes] = VineCopulaFit('C-Vine',families,d,u,rotation,EstMethod,CutOffTree)
[ParamHat, MaxLogLikes, theta0] = VineCopulaFit('C-Vine',families,d,u,rotation,EstMethod,CutOffTree)
```

# Inputs

```
type = The vine copula type.
families = A vector of the pair-copula families, which
are part of the PCC. The vector has to have
the length (d-1)*d/2. The first d-1 entries are
the copula families in the first tree and the
next d-2 entries are the copula families in the
second tree and so on. That means, for d=4 the
array should look similar to this {'Frank',
'Frank', 'Frank', 'AMH', 'AMH', 'Clayton'}, which
is the special case where all copulas in the
first tree are Frank copulas, all copulas in the
second tree are AMH copulas and all copulas in
the third tree are Clayton copulas.
The order of the families is:
* Exemplarily for the four-dimensional C-Vine):
C12, C13, C14, C23|1, C24|1, C34|12
* Exemplarily for the four-dimensional D-Vine):
C12, C23, C34, C13|2, C24|3, C14|23
Note: If families is a simple string/character,
e.g., 'Clayton', then all pair-copulas are
specified to be from this copula family.
d = The dimension of the C- or D-Vine.
u = A (n x d) dimensional vector of values lying in
[0,1] (the observations).
rotation = A vector of the same dimension as families in
which one can specify rotation levels.
EstMethod = The estimation method must be either 'joint' or
'sequential'. If it is not explicitly given, a
joint estimation is performed (default).
CutOffTree = The CutOffTree (or also called truncation level)
can be used to set all pair-copulas from the
(CutOffTree + 1)-th tree on to independence
copulas (i.e., ignore them in the joint
estimation).
```

# Outputs

```
ParamHat = The ML-estimates of the parameters for the
(d-1)*d/2 pair-copulas. These estimates are given
in the same order as the families vector, but
in a row-vector. If a pair-copula is an
independence copula, then there is no estimate
given. Furthermore, if a pair-copula has two or
more parameters, the estimates are given in same
order as they have to be provided if the pair-
copula is considered only. For example, for a t-
copula, the first estimate is for the parameter
rho and the second one for the degrees of freedom
parameter nu.
MaxLogLikes = The first entry is the value of copula-log-
likelihood evaluated at the sequentially estimated
ML-estimates. The second value is the value of
the copula-log-likelihood evaluated for the
joint ML-estimates.
theta0 = The vector of ML-estimates, which are obtained by
using the sequential estimation approach. Note
that these estimates are also used as starting
point for the global maximum likelihood
estimation.
```