2 and 3 dimensional gof test of Scaillet

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Description

gofKernel tests a 2 or 3 dimensional dataset with the Scaillet test for a copula. The possible copulae are "normal", "t", "gumbel", "clayton" and "frank". The parameter estimation is performed with pseudo maximum likelihood method. In case the estimation fails, inversion of Kendall's tau is used. The approximate p-values are computed with a parametric bootstrap, which computation can be accelerated by enabling in-build parallel computation.

Usage

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gofKernel(copula, x, M = 1000, param = 0.5, param.est = T, df = 4, df.est = T, 
          margins = "ranks", dispstr = "ex", MJ = 100, delta.J = 0.5, 
          nodes.Integration = 12, execute.times.comp = T, processes = 1)

Arguments

copula

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

x

A 2 or 3 dimensional matrix containing the residuals of the data.

M

Number of bootstrapping loops.

param

The parameter to be used.

param.est

Shall be either TRUE or FALSE. TRUE means that param will be estimated with a maximum likelihood estimation.

df

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.

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.

MJ

Size of bootstrapping sample.

delta.J

Scaling parameter for the matrix of smoothing parameters.

nodes.Integration

Number of knots of the bivariate Gauss-Legendre quadrature.

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.

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

The Scaillet test is a kernel-based goodness-of-fit test with a fixed smoothing parameter. For the copula density c(u, theta), the corresponding kernel estimator is given by

c_n(\mathbf{u}) = \frac{1}{n} ∑_{i=1}^n K_H[\mathbf{u} - (U_{i1}, …, U_{id})^{\top}],

where U[ij] for i = 1, ...,n; j = 1, ...,d are the pseudo observations, u in [0,1]^d and KH(y) = K(H^(-1)y)/det(H) for which a bivariate quadratic kernel is used, as in Scaillet (2007). The matrix of smoothing parameters is H = 2.6073n^{-1/6} {Sigma_hat}^{1/2} with Sigma_hat the sample covariance matrix. The test statistic is then given by

int_([0,1]^d) {c_n(u) - K_H * c(u, theta_n)} omega(u) d u,

where * denotes the convolution operator and omega is a weight function, see Zhang et al. (2015). The bivariate Gauss-Legendre quadrature method is used to compute the integral in the test statistic numerically, see Scaillet (2007).

The approximate p-value is computed by the formula

sum(|T[b]| >= |T|, b=1, .., M) / M,

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

statistic

value of the test statistic

p.value

the approximate p-value

References

Zhang, S., Okhrin, O., Zhou, Q., and Song, P.. Goodness-of-fit Test For Specification of Semiparametric Copula Dependence Models. under revision in Journal of Econometrics from 15.01.2014 http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2013-041.pdf

Scaillet, O. (2007). Kernel based goodness-of-fit tests for copulas with fixed smoothing parameters. Journal of Multivariate Analysis, 98:533-543

Examples

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data(IndexReturns)

gofKernel("normal", IndexReturns[c(1:100),c(1:2)], M = 5, MJ = 5)