This function performs a goodness-of-fit test for R-vine copula models. There are 15 different goodness-of-fit tests implemented, described in Schepsmeier (2013).
RVineGofTest(
data,
RVM,
method = "White",
statistic = "CvM",
B = 200,
alpha = 2
)An N x d data matrix (with uniform margins).
RVineMatrix() objects of the R-vine model under the
null hypothesis.
Only the following copula families are allowed in
RVM$family due to restrictions in RVineGrad() and
RVineHessian() 0 = independence copula 1 = Gaussian copula 2 = Student t copula (t-copula)3 = Clayton copula 4 = Gumbel copula 5 = Frank copula 6 = Joe copula 13 = rotated Clayton copula (180 degrees; survival Clayton'') \cr `14` = rotated Gumbel copula (180 degrees; survival Gumbel'') 16 = rotated Joe copula (180 degrees; ``survival Joe'') 23 = rotated Clayton copula (90 degrees)
`24` = rotated Gumbel copula (90 degrees)
`26` = rotated Joe copula (90 degrees)
`33` = rotated Clayton copula (270 degrees)
`34` = rotated Gumbel copula (270 degrees)
`36` = rotated Joe copula (270 degrees)
A string indicating the goodness-of-fit method:"White" = goodness-of-fit test based on White's information matrix
equality (default) "IR" = goodness-of-fit test based on the
information ratio "Breymann" = goodness-of-fit test based on the
probability integral transform (PIT) and the aggregation to univariate data
by Breymann et al. (2003). "Berg" = goodness-of-fit test based on
the probability integral transform (PIT) and the aggregation to univariate
data by Berg and Bakken (2007). "Berg2" = second goodness-of-fit
test based on the probability integral transform (PIT) and the aggregation
to univariate data by Berg and Bakken (2007). "ECP" =
goodness-of-fit test based on the empirical copula process (ECP) "ECP2" = goodness-of-fit test based on the combination of probability
integral transform (PIT) and empirical copula process (ECP) (Genest et al.
2009)
A string indicating the goodness-of-fit test statistic
type:"CvM" = Cramer-von Mises test statistic (univariate for
"Breymann", "Berg" and "Berg2", multivariate for
"ECP" and "ECP2") "KS" = Kolmogorov-Smirnov test
statistic (univariate for "Breymann", "Berg" and
"Berg2", multivariate for "ECP" and "ECP2") "AD" = Anderson-Darling test statistic (only univariate for
"Breymann", "Berg" and "Berg2")
an integer for the number of bootstrap steps (default B = 200)
For B = 0 the asymptotic p-value is returned if available,
otherwise only the test statistic is returned.
WARNING: If B is chosen too large, computations will take very long.
an integer of the set 2,4,6,... for the "Berg2"
goodness-of-fit test (default alpha = 2)
For method = "White":
test statistic
p-value, either asymptotic for B = 0 or bootstrapped
for B > 0
For method = "IR":
test statistic (raw version as stated above)
So far no p-value is returned nigher a asymptotic nor a bootstrapped one. How to calculated a bootstrapped p-value is explained in Schepsmeier (2013). Be aware, that the test statistics than have to be adjusted with the empirical variance.
For method = "Breymann", method = "Berg"
and method = "Berg2":
test statistic according to
the choice of statistic
p-value, either asymptotic
for B = 0 or bootstrapped for B > 0. A asymptotic p-value is
only available for the Anderson-Darling test statistic if the R-package
ADGofTest is loaded.
Furthermore, a asymptotic p-value can be
calculated for the Kolmogorov-Smirnov test statistic. For the Cramer-von
Mises no asymptotic p-value is available so far.
For method = "ECP" and method = "ECP2":
test statistic according to the
choice of statistic
bootstrapped p-value
Warning: The code for all the p-values are not yet approved since some of them are moved from R-code to C-code. If you need p-values the best way is to write your own algorithm as suggested in Schepsmeier (2013) to get bootstrapped p-values.
method = "White":
This goodness-of fit test uses the information
matrix equality of White (1982) and was original investigated by Huang and
Prokhorov (2011) for copulas.
Schepsmeier (2012) enhanced their approach
to the vine copula case.
The main contribution is that under correct
model specification the Fisher Information can be equivalently calculated as
minus the expected Hessian matrix or as the expected outer product of the
score function.
The null hypothesis is
$$ H_0: \boldsymbol{H}(\theta) + \boldsymbol{C}(\theta) = 0 $$
against the alternative
$$ H_1: \boldsymbol{H}(\theta) +
\boldsymbol{C}(\theta) \neq 0 , $$
where
\(\boldsymbol{H}(\theta)\) is the expected Hessian matrix and
\(\boldsymbol{C}(\theta)\) is the expected outer product of the
score function.
For the calculation of the test statistic we use the
consistent maximum likelihood estimator \(\hat{\theta}\) and the sample
counter parts of \(\boldsymbol{H}(\theta)\) and
\(\boldsymbol{C}(\theta)\).
The correction of the
Covariance-Matrix in the test statistic for the uncertainty in the margins
is skipped. The implemented test assumes that there is no uncertainty in the
margins. The correction can be found in Huang and Prokhorov (2011) for
bivariate copulas and in Schepsmeier (2013) for vine copulas. It involves
multi-dimensional integrals.
method = "IR":
As the White test the information matrix ratio
test is based on the expected Hessian matrix
\(\boldsymbol{H}(\theta)\)
and the expected outer product of
the score function \(\boldsymbol{C}(\theta)\).
$$ H_0:-\boldsymbol{H}(\theta)^{-1}\boldsymbol{C}(\theta) = I_{p} $$
against the alternative
$$ H_1:
-\boldsymbol{H}(\theta)^{-1}\boldsymbol{C}(\theta) \neq I_{p} . $$
The test statistic can then be calculated as
$$ IR_n:=tr(\Phi(\theta))/p $$ with
\(\Phi(\theta)=-\boldsymbol{H}(\theta)^{-1}\boldsymbol{C}(\theta)\),
\(p\) is the number of parameters, i.e. the length of \(\theta\), and
\(tr(A)\) is the trace of the matrix \(A\)
For details see Schepsmeier (2013)
method = "Breymann", method = "Berg" and method = "Berg2":
These tests are based on the multivariate probability integral
transform (PIT) applied in RVinePIT(). The multivariate data
\(y_{i}\) returned form the PIT are aggregated to univariate data by
different aggregation functions \(\Gamma(\cdot)\) in the sum $$
s_t=\sum_{i=1}^d \Gamma(y_{it}), t=1,...,n $$.
In Breymann et al. (2003) the weight function is suggested as
\(\Gamma(\cdot)=\Phi^{-1}(\cdot)^2\), while in
Berg and Bakken (2007) the weight function is either
\(\Gamma(\cdot)=|\cdot-0.5|\) (method="Berg") or
\(\Gamma(\cdot)=(\cdot-0.5)^{\alpha},\alpha=2,4,6,...\)
(method="Berg2").
Furthermore, the "Berg" and
"Berg2" test are based on the order statistics of the PIT returns.
See Berg and Bakken (2007) or Schepsmeier (2013) for details.
method = "ECP" and method = "ECP2":
Both tests are test
for \(H_0: C \in C_0\) against \(H_1: C \notin C_0\) where C denotes the
(vine) copula distribution function and \(C_0\) is a class of parametric
(vine) copulas with \(\Theta\subseteq R^p\) being the parameter space of
dimension p. They are based on the empirical copula process (ECP)
$$
\hat{C}_n(u)-C_{\hat{\theta}_n}(u), $$
with
\(u=(u_1,\ldots,u_d)\in[0,1]^d\) and
\(\hat{C}_n(u) =
\frac{1}{n+1}\sum_{t=1}^n \boldsymbol{1}_{\{U_{t1}\leq u_1,\ldots,U_{td}\leq
u_d \}} \).
The ECP is utilized in a multivariate
Cramer-von Mises (CvM) or multivariate Kolmogorov-Smirnov (KS) based test
statistic. An extension of the ECP-test is the combination of the
multivariate PIT approach with the ECP. The general idea is that the
transformed data of a multivariate PIT should be "close" to the independence
copula Genest et al. (2009). Thus a distance of CvM or KS type between them
is considered. This approach is called ECP2. Again we refer to Schepsmeier
(2013) for details.
Berg, D. and H. Bakken (2007) A copula goodness-of-fit approach based on the conditional probability integral transformation. https://www.danielberg.no/publications/Btest.pdf
Breymann, W., A. Dias and P. Embrechts (2003) Dependence structures for multivariate high-frequency data in finance. Quantitative Finance 3, 1-14
Genest, C., B. Remillard, and D. Beaudoin (2009) Goodness-of-fit tests for copulas: a review and power study. Insur. Math. Econ. 44, 199-213.
Huang, w. and A. Prokhorov (2011). A goodness-of-fit test for copulas. to appear in Econometric Reviews
Schepsmeier, U. (2013) A goodness-of-fit test for regular vine copula models. Preprint https://arxiv.org/abs/1306.0818
Schepsmeier, U. (2015) Efficient information based goodness-of-fit tests for vine copula models with fixed margins. Journal of Multivariate Analysis 138, 34-52.
White, H. (1982) Maximum likelihood estimation of misspecified models, Econometrica, 50, 1-26.
## time-consuming example
# load data set
data(daxreturns)
# select the R-vine structure, families and parameters
RVM <- RVineStructureSelect(daxreturns[,1:5], c(1:6))
# White test with asymptotic p-value
RVineGofTest(daxreturns[,1:5], RVM, B = 0)
#> $White
#> [1] 120.5856
#>
#> $p.value
#> [1] 0.1418691
#>
# ECP2 test with Cramer-von-Mises test statistic and a bootstrap
# with 200 replications for the calculation of the p-value
RVineGofTest(daxreturns[,1:5], RVM, method = "ECP2",
statistic = "CvM", B = 200)
#> $CvM
#> [1] 0.1153089
#>
#> $p.value
#> [1] 0.69
#>