Sequential Estimation of an R-Vine Copula Model

This function sequentially estimates the pair-copula parameters of a d-dimensional R-vine copula model as specified by the corresponding RVineMatrix() object.

RVineSeqEst(
  data,
  RVM,
  method = "mle",
  se = FALSE,
  max.df = 30,
  max.BB = list(BB1 = c(5, 6), BB6 = c(6, 6), BB7 = c(5, 6), BB8 = c(6, 1)),
  progress = FALSE,
  weights = NA,
  cores = 1
)

Arguments

data: An N x d data matrix (with uniform margins).
RVM: An RVineMatrix() object including the structure, the pair-copula families and the pair-copula parameters (if they are known).
method: indicates the estimation method: either maximum likelihood estimation (method = "mle"; default) or inversion of Kendall's tau (method = "itau"). For method = "itau" only one parameter families and the Student t copula can be used (family = 1,2,3,4,5,6,13,14,16,23,24,26,33,34 or 36). For the t-copula, par2 is found by a crude profile likelihood optimization over the interval (2, 10].
se: Logical; whether standard errors are estimated (default: se = FALSE).
max.df: Numeric; upper bound for the estimation of the degrees of freedom parameter of the t-copula (default: max.df = 30; for more details see BiCopEst()).
max.BB: List; upper bounds for the estimation of the two parameters (in absolute values) of the BB1, BB6, BB7 and BB8 copulas
(default: max.BB = list(BB1=c(5,6),BB6=c(6,6),BB7=c(5,6),BB8=c(6,1))).
progress: Logical; whether the pairwise estimation progress is printed (default: progress = FALSE).
weights: Numerical; weights for each observation (optional).
cores: integer; if cores > 1, estimation will be parallelized within each tree (using parallel::parLapply()). However, the overhead caused by parallelization is likely to make the function run slower unless sample size is really large and method = "itau".

Value

An RVineMatrix() object with the sequentially estimated parameters stored in RVM$par and RVM$par2. The object is augmented by the following information about the fit:

se, se2: standard errors for the parameter estimates (if se = TRUE); note that these are only approximate since they do not account for the sequential nature of the estimation,
nobs: number of observations,
logLik, pair.logLik: log likelihood (overall and pairwise)
AIC, pair.AIC: Aikaike's Informaton Criterion (overall and pairwise),
BIC, pair.BIC: Bayesian's Informaton Criterion (overall and pairwise),
emptau: matrix of empirical values of Kendall's tau,
p.value.indeptest: matrix of p-values of the independence test.

Details

The pair-copula parameter estimation is performed tree-wise, i.e., for each R-vine tree the results from the previous tree(s) are used to calculate the new copula parameters using BiCopEst().

Note

For a comprehensive summary of the fitted model, use summary(object); to see all its contents, use str(object).

Author

Ulf Schepsmeier, Jeffrey Dissmann, Thomas Nagler

Examples


# define 5-dimensional R-vine tree structure matrix
Matrix <- c(5, 2, 3, 1, 4,
            0, 2, 3, 4, 1,
            0, 0, 3, 4, 1,
            0, 0, 0, 4, 1,
            0, 0, 0, 0, 1)
Matrix <- matrix(Matrix, 5, 5)

# define R-vine pair-copula family matrix
family <- c(0, 1, 3, 4, 4,
            0, 0, 3, 4, 1,
            0, 0, 0, 4, 1,
            0, 0, 0, 0, 3,
            0, 0, 0, 0, 0)
family <- matrix(family, 5, 5)

# define R-vine pair-copula parameter matrix
par <- c(0, 0.2, 0.9, 1.5, 3.9,
         0, 0, 1.1, 1.6, 0.9,
         0, 0, 0, 1.9, 0.5,
         0, 0, 0, 0, 4.8,
         0, 0, 0, 0, 0)
par <- matrix(par, 5, 5)

# define second R-vine pair-copula parameter matrix
par2 <- matrix(0, 5, 5)

# define RVineMatrix object
RVM <- RVineMatrix(Matrix = Matrix, family = family,
                   par = par, par2 = par2,
                   names = c("V1", "V2", "V3", "V4", "V5"))

# simulate a sample of size 300 from the R-vine copula model
set.seed(123)
simdata <- RVineSim(300, RVM)

# sequential estimation
summary(RVineSeqEst(simdata, RVM, method = "itau", se = TRUE))
#> tree       edge | family  cop          par  par2 |  tau   utd   ltd 
#> ------------------------------------------------------------------- 
#>    1        4,5 |      4    G  4.17 (0.28)     - | 0.76  0.82     -
#>             1,2 |      1    N  0.90 (0.01)     - | 0.72     -     -
#>             1,3 |      1    N  0.49 (0.05)     - | 0.33     -     -
#>             1,4 |      3    C  4.82 (0.52)     - | 0.71     -  0.87
#>    2      1,5;4 |      4    G  1.48 (0.08)     - | 0.32  0.40     -
#>           4,2;1 |      4    G  1.62 (0.08)     - | 0.38  0.47     -
#>           4,3;1 |      4    G  1.83 (0.11)     - | 0.45  0.54     -
#>    3    3,5;1,4 |      3    C  1.10 (0.17)     - | 0.35     -  0.53
#>         3,2;4,1 |      3    C  1.33 (0.19)     - | 0.40     -  0.59
#>    4  2,5;3,1,4 |      1    N  0.17 (0.06)     - | 0.11     -     -
#> ---
#> type: R-vine    logLik: 1197.32    AIC: -2374.63    BIC: -2337.59    
#> ---
#> 1 <-> V1,   2 <-> V2,   3 <-> V3,   4 <-> V4,   5 <-> V5
summary(RVineSeqEst(simdata, RVM, method = "mle", se = TRUE))
#> tree       edge | family  cop          par  par2 |  tau   utd   ltd 
#> ------------------------------------------------------------------- 
#>    1        4,5 |      4    G  4.01 (0.19)     - | 0.75  0.81     -
#>             1,2 |      1    N  0.90 (0.01)     - | 0.71     -     -
#>             1,3 |      1    N  0.46 (0.04)     - | 0.30     -     -
#>             1,4 |      3    C  5.02 (0.31)     - | 0.71     -  0.87
#>    2      1,5;4 |      4    G  1.48 (0.07)     - | 0.33  0.40     -
#>           4,2;1 |      4    G  1.59 (0.07)     - | 0.37  0.46     -
#>           4,3;1 |      4    G  1.87 (0.09)     - | 0.47  0.55     -
#>    3    3,5;1,4 |      3    C  0.97 (0.11)     - | 0.33     -  0.49
#>         3,2;4,1 |      3    C  1.24 (0.12)     - | 0.38     -  0.57
#>    4  2,5;3,1,4 |      1    N  0.18 (0.06)     - | 0.11     -     -
#> ---
#> type: R-vine    logLik: 1198.03    AIC: -2376.06    BIC: -2339.03    
#> ---
#> 1 <-> V1,   2 <-> V2,   3 <-> V3,   4 <-> V4,   5 <-> V5