Maximum Likelihood Estimation of an R-Vine Copula Model

This function calculates the maximum likelihood estimate (MLE) of the R-vine copula model parameters using sequential estimates as initial values (if not provided).

RVineMLE(
  data,
  RVM,
  start = RVM$par,
  start2 = RVM$par2,
  maxit = 200,
  max.df = 30,
  max.BB = list(BB1 = c(5, 6), BB6 = c(6, 6), BB7 = c(5, 6), BB8 = c(6, 1)),
  grad = FALSE,
  hessian = FALSE,
  se = FALSE,
  ...
)

Arguments

data: An N x d data matrix (with uniform margins).
RVM: An RVineMatrix() object including the structure and the pair-copula families and parameters (if known).
start: Lower triangular d x d matrix with zero diagonal entries with starting values for the pair-copula parameters (optional; otherwise they are calculated via
RVineSeqEst(); default: start = RVM$par).
start2: Lower triangular d x d matrix with zero diagonal entries with starting values for the second parameters of pair-copula families with two parameters (optional; otherwise they are calculated via RVineSeqEst(); default: start2 = RVM$par2).
maxit: The maximum number of iteration steps (optional; default: maxit = 200).
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))).
grad: If RVM$family only contains one parameter copula families or the t-copula the analytical gradient can be used for maximization of the log-likelihood (see RVineGrad(); default: grad = FALSE).
hessian: Logical; whether the Hessian matrix of parameter estimates is estimated (default: hessian = FALSE). Note that this is not the Hessian Matrix calculated via RVineHessian() but via finite differences.
se: Logical; whether standard errors of parameter estimates are estimated on the basis of the Hessian matrix (see above; default: se = FALSE).
...: Further arguments for optim (e.g. factr controls the convergence of the "L-BFGS-B" method, or trace, a non-negative integer, determines if tracing information on the progress of the optimization is produced.)
For more details see the documentation of optim().

Value

RVM: RVineMatrix() object with the calculated parameters stored in RVM$par and RVM$par2. Additional information about the fit is added (e.g., log-likelihood, AIC, BIC).
value: Optimized log-likelihood value corresponding to the estimated pair-copula parameters.
convergence: An integer code indicating either successful convergence (convergence = 0) or an error:
1 = the iteration limit maxit has been reached
51 = a warning from the "L-BFGS-B" method; see component message for further details
52 = an error from the "L-BFGS-B" method; see component message for further details
message: A character string giving any additional information returned by optim(), or NULL.
counts: A two-element integer vector giving the number of calls to fn and gr respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to fn to compute a finite-difference approximation to the gradient.
hessian: If hessian = TRUE, the Hessian matrix is returned. Its calculation is on the basis of finite differences (output of optim).

Note

RVineMLE uses the L-BFGS-B method for optimization.
If the analytical gradient is used for maximization, computations may be up to 10 times faster than using finite differences.

References

Dissmann, J. F., E. C. Brechmann, C. Czado, and D. Kurowicka (2013). Selecting and estimating regular vine copulae and application to financial returns. Computational Statistics & Data Analysis, 59 (1), 52-69.

Stoeber, J. and U. Schepsmeier (2013). Estimating standard errors in regular vine copula models. Computational Statistics, 1-29 https://link.springer.com/article/10.1007/s00180-013-0423-8#.

Author

Ulf Schepsmeier, Jeffrey Dissmann

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)

# compute the MLE
mle <- RVineMLE(simdata, RVM, grad = TRUE, trace = 0)

# compare parameters
round(mle$RVM$par - RVM$par, 2)
#>       [,1] [,2]  [,3]  [,4] [,5]
#> [1,]  0.00 0.00  0.00  0.00    0
#> [2,] -0.03 0.00  0.00  0.00    0
#> [3,]  0.04 0.08  0.00  0.00    0
#> [4,]  0.02 0.04 -0.01  0.00    0
#> [5,]  0.17 0.00 -0.01 -0.02    0