CDVine (version 1.4)

CDVineVuongTest: Vuong test comparing two vine copula models

Description

This function performs a Vuong test between two d-dimensional C- or D-vine copula models, respectively.

Usage

CDVineVuongTest(data, Model1.order=1:dim(data)[2], Model2.order=1:dim(data)[2], Model1.family, Model2.family, Model1.par, Model2.par, Model1.par2=rep(0,dim(data)[2]*(dim(data)[2]-1)/2), Model2.par2=rep(0,dim(data)[2]*(dim(data)[2]-1)/2), Model1.type, Model2.type)

Arguments

data

An N x d data matrix (with uniform margins).

Model1.order, Model2.order

Two numeric vectors giving the order of the variables in the first D-vine trees or of the C-vine root nodes in models 1 and 2 (default: Model1.order and Model2.order = 1:dim(data)[2], i.e., standard order).

Model1.family, Model2.family

Two d*(d-1)/2 numeric vectors of the pair-copula families of models 1 and 2, respectively, with values 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)

Model1.par, Model2.par

Two d*(d-1)/2 numeric vectors of the (first) copula parameters of models 1 and 2, respectively.

Model1.par2, Model2.par2

Two d*(d-1)/2 numeric vectors of the second copula parameters of models 1 and 2, respectively; necessary for t, BB1, BB6, BB7 and BB8 copulas. If no such families are included in Model1.family/Model2.family, these arguments do not need to be specified (default: Model1.par2 and Model2.par2 = rep(0,dim(data)[2]*(dim(data)[2]-1)/2)).

Model1.type, Model2.type

Type of the respective vine model: 1 or "CVine" = C-vine 2 or "DVine" = D-vine

Value

statistic, statistic.Akaike, statistic.Schwarz

Test statistics without correction, with Akaike correction and with Schwarz correction.

p.value, p.value.Akaike, p.value.Schwarz

P-values of tests without correction, with Akaike correction and with Schwarz correction.

Details

The likelihood-ratio based test proposed by Vuong (1989) can be used for comparing non-nested models. For this let $c_1$ and $c_2$ be two competing vine copulas in terms of their densities and with estimated parameter sets $\theta_1$ and $\theta_2$. We then compute the standardized sum, $\nu$, of the log differences of their pointwise likelihoods $m_i:=log[c_1(u_i|\theta_1) / c_2(u_i|\theta_2) ]$ for observations $u_i in [0,1],i=1,...,N$ , i.e., $$ \texttt{statistic} := \nu = \frac{\frac{1}{n}\sum_{i=1}^N m_i}{\sqrt{\sum_{i=1}^N\left(m_i - \bar{m} \right)^2}}. $$ Vuong (1989) shows that $\nu$ is asymptotically standard normal. According to the null-hypothesis $$ H_0: E[m_i] = 0\ \forall i=1,...,N, $$ we hence prefer vine model 1 to vine model 2 at level $\alpha$ if $$ \nu>\Phi^{-1}\left(1-\frac{\alpha}{2}\right), $$ where $\Phi^{-1}$ denotes the inverse of the standard normal distribution function. If $\nu<-\Phi^{-1}(1-\alpha/2)$ we choose model 2. If, however, $|\nu| <= \phi^{-1}(1-\alpha="" 2)$,="" no="" decision="" among="" the="" models="" is="" possible.<="" p="">

Like AIC and BIC, the Vuong test statistic may be corrected for the number of parameters used in the models. There are two possible corrections; the Akaike and the Schwarz corrections, which correspond to the penalty terms in the AIC and the BIC, respectively.

References

Vuong, Q. H. (1989). Ratio tests for model selection and non-nested hypotheses. Econometrica 57 (2), 307-333.

See Also

CDVineClarkeTest, CDVineAIC, CDVineBIC

Examples

## Not run: 
# # load data set
# data(worldindices)
# d = dim(worldindices)[2]
# 
# # select the C-vine families and parameters
# cvine = CDVineCopSelect(worldindices,c(1:6),type="CVine")
# 
# # select the D-vine families and parameters
# dvine = CDVineCopSelect(worldindices,c(1:6),type="DVine")
# 
# # compare the two models based on the data
# vuong = CDVineVuongTest(worldindices,1:d,1:d,cvine$family,dvine$family,
#                         cvine$par,dvine$par,cvine$par2,dvine$par2,
#                         Model1.type=1,Model2.type=2)
# vuong$statistic
# vuong$statistic.Schwarz
# vuong$p.value
# vuong$p.value.Schwarz
# ## End(Not run)