VineCopula (version 1.0)

RVineClarkeTest: Clarke test comparing two R-vine copula models

Description

This function performs a Clarke test between two d-dimensional R-vine copula models as specified by their RVineMatrix objects.

Usage

RVineClarkeTest(data, RVM1, RVM2)

Arguments

data

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

RVM1, RVM2

RVineMatrix objects of models 1 and 2.

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 $\hat{\boldsymbol{\theta}}_1$ and $\hat{\boldsymbol{\theta}}_2$. The null hypothesis of statistical indistinguishability of the two models is $$H_0: P(m_i > 0) = 0.5\ \forall i=1,..,N,$$ where $m_i:=\log\left[\frac{c_1(\boldsymbol{u}_i|\hat{\boldsymbol{\theta}}_1)}{c_2(\boldsymbol{u}_i|\hat{\boldsymbol{\theta}}_2)}\right]$ for observations $\boldsymbol{u}_i,\ i=1,...,N$. 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 $\mathbf{1}$ 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 $Np = \frac{N}{2}$. 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

RVineVuongTest, RVineAIC, RVineBIC

Examples

# define first 5-dimensional RVineMatrix object
Matrix1 = 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)
Matrix1 = matrix(Matrix1,5,5)

family1 = 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)
family1 = matrix(family1,5,5)

par1 = 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)
par1 = matrix(par1,5,5)

RVM1 = RVineMatrix(Matrix=Matrix1,family=family1,par=par1,
                   par2=matrix(0,5,5),names=c("V1","V2","V3","V4","V5"))
                  
# define second 5-dimensional RVineMatrix object
Matrix2 = c(5,4,3,2,1,0,4,3,2,1,0,0,3,2,1,0,0,0,2,1,0,0,0,0,1)
Matrix2 = matrix(Matrix2,5,5)

family2 = c(0,3,1,3,2,0,0,1,5,3,0,0,0,2,3,0,0,0,0,1,0,0,0,0,0)
family2 = matrix(family2,5,5)

par2 = c(0,0.8,0.3,1.5,0.8,0,0,-0.4,1.6,1.2,0,0,0,-0.4,1.5,
         0,0,0,0,0.6,0,0,0,0,0)
par2 = matrix(par2,5,5)

nu2 = c(0,0,0,0,5,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0)
nu2 = matrix(nu2,5,5)

RVM2 = RVineMatrix(Matrix=Matrix2,family=family2,par=par2,par2=nu2,
                   names=c("V1","V2","V3","V4","V5"))                  

# simulate a sample of size 300 from the first R-vine copula model
simdata = RVineSim(300,RVM1)

# compare the two models based on this sample
clarke = RVineClarkeTest(simdata,RVM1,RVM2)
clarke$statistic
clarke$statistic.Schwarz
clarke$p.value
clarke$p.value.Schwarz