Vuong Test Comparing Two R-Vine Copula Models
This function performs a Vuong test between two d-dimensional R-vine copula
models as specified by their RVineMatrix() objects.
RVineVuongTest(data, RVM1, RVM2)
data |
An N x d data matrix (with uniform margins). |
RVM1, RVM2 |
|
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 θ_1 and θ_2. We then compute the standardized sum, ν, of the log differences of their pointwise likelihoods m_i:=log[c_1(u_i|θ_1) / c_2(u_i|θ_2) ] for observations u_i in [0,1],i=1,...,N , i.e.,
statistic := ν = (1/n∑_{i=1}^N m_i) / ((∑_{i=1}^N (m_i - \bar{m} )^2)^0.5).
Vuong (1989) shows that ν 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 α if
ν > Φ^{-1}(1-α/2),
where Φ^{-1} denotes the inverse of the standard normal distribution function. If ν<-Φ^{-1}(1-α/2) we choose model 2. If, however, |ν| <= Φ^{-1}(1-α/2), 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.
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. |
Jeffrey Dissmann, Eike Brechmann
Vuong, Q. H. (1989). Ratio tests for model selection and non-nested hypotheses. Econometrica 57 (2), 307-333.
# 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
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