Hybrid gof test

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Description

gofHybrid combines all tests in this package to perform the hybrid test presented in Zhang et al. (2015). The test gives the possibility to combine several single tests which is helpful since in different test scenarios are different tests the most powerful.

Usage

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gofHybrid(copula, x, testset = c("gofPIOSRn", "gofKernel"), margins = "ranks", 
          dispstr = "ex", M = 1000, execute.times.comp = T, param = 0.5, 
          param.est = T, df = 4, df.est = T, m = 1, MJ = 100, delta.J = 0.5, 
          nodes.Integration = 12, m_b = 0.5, zeta.m = 0, b_Rn = 0.05, 
          processes = 1)

Arguments

copula

The copula to test for. Possible are "normal", "t", "clayton", "gumbel" and "frank".

x

A matrix containing the residuals of the data. Take into account that the dimension of the matrix must fit the required dimensions from the single tests.

testset

The tests to be used. Possible are "gofPIOSRn", "gofPIOSTn", "gofKernel", "gofRosenblattSnB", "gofRosenblattSnC", "gofRosenblattChisq", "gofRosenblattGamma", "gofSn", "gofKendallCvM", "gofKendallKS", "gofWhite", "gofRn".

margins

Specifies which estimation method shall be used in case that the input data are not in the range [0,1]. The default is "ranks", which is the standard approach to convert data in such a case. Alternatively can the following distributions be specified: "beta", "cauchy", Chi-squared ("chisq"), "f", "gamma", Log normal ("lnorm"), Normal ("norm"), "t", "weibull", Exponential ("exp").

dispstr

A character string specifying the type of the symmetric positive definite matrix characterizing the elliptical copula. Implemented structures are "ex" for exchangeable and "un" for unstructured, see package copula.

M

Number of bootstrapping samples in the single tests.

execute.times.comp

Logical. Defines if the time which the estimation most likely takes shall be computed. It'll be just given if M is at least 100.

param

The copulae parameters to use for each test, if it shall not be estimated.

param.est

Shall be either TRUE or FALSE. TRUE means that param will be estimated.

df

The degrees of freedom, if not meant to be estimated. Only necessary if tested for "t"-copula.

df.est

Indicates if df shall be estimated. Has to be either FALSE or TRUE, where TRUE means that it will be estimated.

m

Length of blocks. Only necessary if the test gofPIOSTn is part of testset.

MJ

Size of bootstrapping sample. Only necessary if the test gofKernel is part of testset.

delta.J

Scaling parameter for the matrix of smoothing parameters. Only necessary if the test gofKernel is part of testset.

nodes.Integration

Number of knots of the bivariate Gauss-Legendre quadrature. Only necessary if the test gofKernel is part of testset.

m_b

The power of the statistic. Only necessary if the test gofRn is part of testset.

zeta.m

The adjustment parameter. Only necessary if the test gofRn is part of testset.

b_Rn

The bandwidth for the estimation of the first-order partial derivatives based on the empirical copula. Only necessary if the test gofRn is part of testset.

processes

The number of parallel processes which are performed to speed up the bootstrapping. Shouldn't be higher than the number of logical processors. Please see the details.

Details

In most of scenarios for goodness-of-fit tests, including the one for copula models (e.g. Genest et al. (2009)) there exists no single dominate optimal test. Zhang et al. (2015) proposed a hybrid test which performed in their simulation study more desirably compared to the applied single tests.

The p-value is a combination of the single tests in the following way:

pn^(hybrid) = min(q x min(pn^(1), ..., pn^(q)), 1)

where q is the number of tests and pn^(i) the p-value of the test i. It is ensured that the hybrid test is consistent as long as at least one of the tests is consistent.

The computation of the individual p-values is performed as described in the details of this tests. Note that the derivation differs.

For small values of M, initializing the parallization via processes does not make sense. The registration of the parallel processes increases the computation time. Please consider to enable parallelization just for high values of M.

Value

A object of the class gofCOP with the components

method

a character which informs about the performed analysis

erg.tests

a matrix with the p-values and test statistics of the hybrid and the individual tests

References

Zhang, S., Okhrin, O., Zhou, Q., and Song, P.. Goodness-of-fit Test For Specification of Semiparametric Copula Dependence Models. Journal of Econometrics, 193, 2016, pp. 215-233 dx.doi.org/10.1016/j.jeconom.2016.02.017

Christian Genest, Bruno Remillard, David Beaudoin (2009). Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics, Volume 44, Issue 2, April 2009, Pages 199-213, ISSN 0167-6687. http://dx.doi.org/10.1016/j.insmatheco.2007.10.005

Examples

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data(IndexReturns)

gofHybrid("normal", IndexReturns[c(1:100),c(1:2)], 
testset = c("gofRosenblattSnB", "gofRosenblattSnC"), M = 10)