CDVine (version 1.4)

CDVineClarkeTest: Clarke test comparing two vine copula models

Description

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

Usage

CDVineClarkeTest(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 test proposed by Clarke (2007) allows to compare 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 $\theta_2$. The null hypothesis of statistical indistinguishability of the two models is $$ H_0: P(m_i > 0) = 0.5\ \forall i=1,..,N, $$ where $m_i:=log[ c_1(u_i|\theta_1) / c_2(u_i|\theta_2) ]$ for observations $u_i, i=1,...,N$.

Since under statistical equivalence of the two models the log likelihood ratios of the single observations are uniformly distributed around zero and in expectation $50\%$ of the log likelihood ratios greater than zero, the tets statistic $$ \texttt{statistic} := B = \sum_{i=1}^N \mathbf{1}_{(0,\infty)}(m_i), $$ where $1$ is the indicator function, is distributed Binomial with parameters $N$ and $p=0.5$, and critical values can easily be obtained. Model 1 is interpreted as statistically equivalent to model 2 if $B$ is not significantly different from the expected value $np=N/2$.

Like AIC and BIC, the Clarke 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

Clarke, K. A. (2007). A Simple Distribution-Free Test for Nonnested Model Selection. Political Analysis, 15, 347-363.

See Also

CDVineVuongTest, 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
# clarke = CDVineClarkeTest(worldindices,1:d,1:d,cvine$family,dvine$family,
#                           cvine$par,dvine$par,cvine$par2,dvine$par2,
#                           Model1.type=1,Model2.type=2)
# clarke$statistic
# clarke$statistic.Schwarz
# clarke$p.value
# clarke$p.value.Schwarz
# ## End(Not run)