gofKernel {gofCopula}

2 dimensional gof test of Scaillet (2007)

Description

gofKernel tests a 2 dimensional dataset with the Scaillet test for a copula. The possible copulae are "normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos", "huslerReiss", "tawn", "tev", "fgm" and "plackett". 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

gofKernel(
  copula = c("normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos",
    "huslerReiss", "tawn", "tev", "fgm", "plackett"),
  x,
  param = 0.5,
  param.est = TRUE,
  df = 4,
  df.est = TRUE,
  margins = "ranks",
  flip = 0,
  M = 1000,
  MJ = 100,
  dispstr = "ex",
  delta.J = 0.5,
  nodes.Integration = 12,
  lower = NULL,
  upper = NULL,
  seed.active = NULL,
  processes = 1
)

Arguments

copula

The copula to test for. Possible are "normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos", "huslerReiss", "tawn", "tev", "fgm" and "plackett".

x

A matrix containing the data with rows being observations and columns being variables.

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 for the margins shall be used. The default is "ranks", which is the standard approach to convert data in such a case. Alternatively the following distributions can be specified: "beta", "cauchy", Chi-squared ("chisq"), "f", "gamma", Log normal ("lnorm"), Normal ("norm"), "t", "weibull", Exponential ("exp"). Input can be either one method, e.g. "ranks", which will be used for estimation of all data sequences. Also an individual method for each margin can be specified, e.g. c("ranks", "norm", "t") for 3 data sequences. If one does not want to estimate the margins, set it to NULL.

flip

The control parameter to flip the copula by 90, 180, 270 degrees clockwise. Only applicable for bivariate copula. Default is 0 and possible inputs are 0, 90, 180, 270 and NULL.

M

Number of bootstrapping loops.

MJ

Size of bootstrapping sample.

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.

delta.J

Scaling parameter for the matrix of smoothing parameters.

nodes.Integration

Number of knots of the bivariate Gauss-Legendre quadrature.

lower

Lower bound for the maximum likelihood estimation of the copula parameter. The constraint is also active in the bootstrapping procedure. The constraint is not active when a switch to inversion of Kendall's tau is necessary. Default NULL.

upper

Upper bound for the maximum likelihood estimation of the copula parameter. The constraint is also active in the bootstrapping procedure. The constraint is not active when a switch to inversion of Kendall's tau is necessary. Default NULL.

seed.active

Has to be either an integer or a vector of M+1 integers. If an integer, then the seeds for the bootstrapping procedure will be simulated. If M+1 seeds are provided, then these seeds are used in the bootstrapping procedure. Defaults to NULL, then R generates the seeds from the computer runtime. Controlling the seeds is useful for reproducibility of a simulation study to compare the power of the tests or for reproducibility of an empirical study.

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(\mathbf{u}, \theta), the corresponding kernel estimator is given by

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

where U_{ij} for i = 1, \dots,n; j = 1, \dots,d are the pseudo observations, \mathbf{u} \in [0,1]^d and K_H(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} \hat{\Sigma}^{1/2} with \hat{\Sigma} the sample covariance matrix. The test statistic is then given by

T = \int_{[0,1]^d} \{c_n(\mathbf{u}) - K_H * c(\mathbf{u}, \theta_n)\} \omega(\mathbf{u}) d \mathbf{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_{b=1}^M \mathbf{I}(|T_b| \geq |T|) / M,

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

Value

An object of the class gofCOP with the components

method

a character which informs about the performed analysis

copula

the copula tested for

margins

the method used to estimate the margin distribution.

param.margins

the parameters of the estimated margin distributions. Only applicable if the margins were not specified as "ranks" or NULL.

theta

dependence parameters of the copulae

df

the degrees of freedem of the copula. Only applicable for t-copula.

res.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 doi: 10.1016/j.jeconom.2016.02.017

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

Examples

data(IndexReturns2D)

gofKernel("normal", IndexReturns2D, M = 5, MJ = 5)