vuongtest: Model comparison using a Vuong test
In CopulaRegression: Bivariate Copula Based Regression Models

Description Usage Arguments Details Value Author(s) References See Also Examples

This function compares two bivariate copula-based regression models

1	vuongtest(model1,model2,selection="AIC")

`model1`	`copreg` object returned from `copreg`
`model2`	`copreg` object returned from `copreg`
`selection`	model selection criterion. Options are AIC and BIC. Default ist AIC.

Let us denote by { \ell}^{(1)},\,{\bm \ell}^{(2)} the vectors of pointwise loglikelihoods for a model with copula family 1 and 2 respectively. Here, we assume that both models have the same degrees of freedom, i.e. the same number of parameters. We now compute the differences of the pointwise loglikelihood as

m_i:=\ell^{(1)}_i - \ell^{(2)}_i,\ i=1,…,n\,.

Denote by

\overline{m}=\frac{1}{n}∑_{i=1}^n m_i

the mean of the differences. The test statistic

T_V:= \frac{√{n}\cdot\overline{m}}{ √{∑_{i=1}^n ≤ft(m_i - \overline{m}\right)^2}},

is asymptotically normally distributed with zero mean and unit variance. Hence, we prefer copula family 1 to copula family 2 at level α if

T_V> Φ^{-1}≤ft(1-\frac{α}{2}\right)\,,

where Φ denotes the standard normal distribution function. If

T_V< Φ^{-1}≤ft(\frac{α}{2}\right)\,,

we prefer copula family 2. Otherwise, no decision among the two copula families is possible. If the models contain different numbers of estimated parameters, the test statistic is corrected using either the AIC or BIC criterion.

value of the test statistic

Nicole Kraemer

N. Kraemer, E. Brechmann, D. Silvestrini, C. Czado (2013): Total loss estimation using copula-based regression models. Insurance: Mathematics and Economics 53 (3), 829 - 839.

Vuong, Q. H. (1989). Ratio tests for model selection and non-nested hypotheses. Econometrica 57 (2), 307-333.

copreg

n<-200 # number of examples
 R<-S<-cbind(rep(1,n),rnorm(n)) # design matrices with intercept 
 alpha<-beta<-c(1,-1) # regression coefficients
 exposure<-rep(1,n) # constant exposure
 delta<-0.5 # dispersion parameter
 tau<-0.3 # Kendall's tau
 family=3 # Clayton copula
 # simulate data
 my.data<-simulate_regression_data(n,alpha,beta,R,S,delta,tau,family,TRUE,exposure)
 x<-my.data[,1]
 y<-my.data[,2]
 
 # joint model without standard errors
 my.model.clayton<-copreg(x,y,R,S,family=3,exposure,FALSE,TRUE)
 my.model.gauss<-copreg(x,y,R,S,family=1,exposure,FALSE,TRUE)
 #
 vuongtest(my.model.clayton,my.model.gauss)