RVineVuongTest: Vuong Test Comparing Two R-Vine Copula Models

In tnagler/VineCopula: Statistical Inference of Vine Copulas

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.

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.

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.

Author(s)

Jeffrey Dissmann, Eike Brechmann

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

# vine structure selection time-consuming (~ 20 sec)

# 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

tnagler/VineCopula documentation built on March 6, 2024, 5 a.m.