VineCopula (version 1.3)

RVineVuongTest: Vuong test comparing two R-vine copula models

Description

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

Usage

RVineVuongTest(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 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 $\hat{\boldsymbol{\theta}}_1$ and $\hat{\boldsymbol{\theta}}_2$. We then compute the standardized sum, $\nu$, of the log differences of their pointwise likelihoods $m_i:=\log\left[\frac{c_1(\boldsymbol{u}_i|\hat{\boldsymbol{\theta}}_1)}{c_2(\boldsymbol{u}_i|\hat{\boldsymbol{\theta}}_2)}\right]$ for observations $\boldsymbol{u}_i\in[0,1],\ i=1,...,N$ , 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 $\nu<-\Phi^{-1}\left(1-\frac{\alpha}{2}\right)$ we choose model 2. If, however, $|\nu|\leq\Phi^{-1}\left(1-\frac{\alpha}{2}\right)$, no decision among the models is possible. 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

RVineClarkeTest, RVineAIC, RVineBIC

Examples

# load data set
data(daxreturns)

# select the R-vine structure, families and parameters
RVM = RVineStructureSelect(daxreturns[,1:5],c(1:6))

# select the C-vine structure, families and parameters
CVM = RVineStructureSelect(daxreturns[,1:5],c(1:6),type="CVine")

# compare the two models based on the data
vuong = RVineVuongTest(daxreturns[,1:5],RVM,CVM)
vuong$statistic
vuong$statistic.Schwarz
vuong$p.value
vuong$p.value.Schwarz