CDVine (version 1.4)

BiCopEst: Parameter estimation for bivariate copula data using inversion of Kendall's tau or maximum likelihood estimation

Description

This function estimates the parameter(s) for a bivariate copula using either inversion of empirical Kendall's tau for single parameter copula families or maximum likelihood estimation for one and two parameter copula families supported in this package.

Usage

BiCopEst(u1, u2, family, 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)))

Arguments

u1,u2

Data vectors of equal length with values in [0,1].

family

An integer defining the bivariate copula family: 0 = independence copula 1 = Gaussian copula 2 = Student t copula (t-copula) 3 = Clayton copula 4 = Gumbel copula 5 = Frank copula 6 = Joe copula 7 = BB1 copula 8 = BB6 copula 9 = BB7 copula 10 = BB8 copula 13 = rotated Clayton copula (180 degrees; ``survival Clayton'') 14 = rotated Gumbel copula (180 degrees; ``survival Gumbel'') 16 = rotated Joe copula (180 degrees; ``survival Joe'') 17 = rotated BB1 copula (180 degrees; ``survival BB1'') 18 = rotated BB6 copula (180 degrees; ``survival BB6'') 19 = rotated BB7 copula (180 degrees; ``survival BB7'') 20 = rotated BB8 copula (180 degrees; ``survival BB8'') 23 = rotated Clayton copula (90 degrees) 24 = rotated Gumbel copula (90 degrees) 26 = rotated Joe copula (90 degrees) 27 = rotated BB1 copula (90 degrees) 28 = rotated BB6 copula (90 degrees) 29 = rotated BB7 copula (90 degrees) 30 = rotated BB8 copula (90 degrees) 33 = rotated Clayton copula (270 degrees) 34 = rotated Gumbel copula (270 degrees) 36 = rotated Joe copula (270 degrees) 37 = rotated BB1 copula (270 degrees) 38 = rotated BB6 copula (270 degrees) 39 = rotated BB7 copula (270 degrees) 40 = rotated BB8 copula (270 degrees)

method

Character indicating the estimation method: either maximum likelihood estimation (method = "mle"; default) or inversion of Kendall's tau (method = "itau"). For method = "itau" only one parameter bivariate copula families can be used (family = 1,3,4,5,6,13,14,16,23,24,26,33,34 or 36).

se

Logical; whether standard error(s) of parameter estimates is/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).

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))).

Value

par, par2

Estimated copula parameter(s).

se,se2

Standard error(s) of the parameter estimate(s) (if se = TRUE).

Details

If method = "itau", the function computes the empirical Kendall's tau of the given copula data and exploits the one-to-one relationship of copula parameter and Kendall's tau which is available for many one parameter bivariate copula families (see BiCopPar2Tau and BiCopTau2Par). The inversion of Kendall's tau is however not available for all bivariate copula families (see above). If a two parameter copula family is chosen and method = "itau", a warning message is returned and the MLE is calculated.

For method = "mle" copula parameters are estimated by maximum likelihood using starting values obtained by method = "itau". If no starting values are available by inversion of Kendall's tau, starting values have to be provided given expert knowledge and the boundaries max.df and max.BB respectively.

A warning message is returned if the estimate of the degrees of freedom parameter of the t-copula is larger than max.df. For high degrees of freedom the t-copula is almost indistinguishable from the Gaussian and it is advised to use the Gaussian copula in this case. As a rule of thumb max.df = 30 typically is a good choice. Moreover, standard errors of the degrees of freedom parameter estimate cannot be estimated in this case.

References

Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.

See Also

BiCopPar2Tau, BiCopTau2Par, CDVineSeqEst, BiCopSelect

Examples

## Example 1: bivariate Gaussian copula
dat = BiCopSim(500,1,0.7)
u1 = dat[,1]
v1 = dat[,2]

# empirical Kendall's tau
tau1 = cor(u1,v1,method="kendall")

# inversion of empirical Kendall's tau 
BiCopTau2Par(1,tau1)
BiCopEst(u1,v1,family=1,method="itau")$par

# maximum likelihood estimate for comparison
BiCopEst(u1,v1,family=1,method="mle")$par


## Example 2: bivariate Clayton and survival Gumbel copulas
# simulate from a Clayton copula
dat = BiCopSim(500,3,2.5)
u2 = dat[,1]
v2 = dat[,2]

# empirical Kendall's tau
tau2 = cor(u2,v2,method="kendall")

# inversion of empirical Kendall's tau for the Clayton copula
BiCopTau2Par(3,tau2)
BiCopEst(u2,v2,family=3,method="itau",se=TRUE) 

# inversion of empirical Kendall's tau for the survival Gumbel copula
BiCopTau2Par(14,tau2)
BiCopEst(u2,v2,family=14,method="itau",se=TRUE)

# maximum likelihood estimates for comparison
BiCopEst(u2,v2,family=3,method="mle",se=TRUE)
BiCopEst(u2,v2,family=14,method="mle",se=TRUE)

## Example 3: fit of a t-copula to standardized residuals of
## S&P 500 and DAX returns
data(worldindices)
BiCopEst(worldindices[,1],worldindices[,4],family=2,method="mle",se=TRUE)