simpA.param: Semiparametric testing of the simplifying assumption

In CondCopulas: Estimation and Inference for Conditional Copula Models

Semiparametric testing of the simplifying assumption

Description

This function tests the “simplifying assumption” that a conditional copula

C_{1,2|3}(u_1, u_2 | X_3 = x_3)

does not depend on the value of the conditioning variable x_3 in a semiparametric setting, where the conditional copula is of the form

C_{1,2|3}(u_1, u_2 | X_3 = x_3) = C_{\theta(x_3)}(u_1,u_2),

for all 0 <= u_1, u_2 <= 1 and all x_3. Here, (C_\theta) is a known family of copula and \theta(x_3) is an unknown conditional dependence parameter. In this setting, the simplifying assumption can be rewritten as “\theta(x_3) does not depend on x_3, i.e. is a constant function of x_3”.

Usage

simpA.param(
  X1,
  X2,
  X3,
  family,
  testStat = "T2c",
  typeBoot = "boot.NP",
  h,
  nBootstrap = 100,
  kernel.name = "Epanechnikov",
  truncVal = h,
  numericalInt = list(kind = "legendre", nGrid = 10)
)

Arguments

X1	vector of n observations of the first conditioned variable
X2	vector of n observations of the second conditioned variable
X3	vector of n observations of the conditioning variable
family	the chosen family of copulas (see the documentation of the class VineCopula::BiCop() for the available families).
testStat	name of the test statistic to be used. The only choice implemented yet is 'T2c'.
typeBoot	the type of bootstrap to be used. (see Derumigny and Fermanian, 2017, p.165). Possible values are "boot.NP": usual (Efron's) non-parametric bootstrap "boot.pseudoInd": pseudo-independent bootstrap "boot.pseudoInd.sameX3": pseudo-independent bootstrap without resampling on X_3 "boot.pseudoNP": pseudo-non-parametric bootstrap "boot.cond": conditional bootstrap "boot.paramInd": parametric independent bootstrap "boot.paramCond": parametric conditional bootstrap
h	the bandwidth used for kernel smoothing
nBootstrap	number of bootstrap replications
kernel.name	the name of the kernel
truncVal	the value of truncation for the integral, i.e. the integrals are computed from truncVal to 1-truncVal instead of from 0 to 1.
numericalInt	parameters to be given to statmod::gauss.quad, including the number of quadrature points and the type of interpolation.

Value

a list containing

• true_stat: the value of the test statistic computed on the whole sample

• vect_statB: a vector of length nBootstrap containing the bootstrapped test statistics.

• p_val: the p-value of the test.

References

Derumigny, A., & Fermanian, J. D. (2017). About tests of the “simplifying” assumption for conditional copulas. Dependence Modeling, 5(1), 154-197. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1515/demo-2017-0011")}

See Also

Other tests of the simplifying assumption:

• simpA.NP in a nonparametric setting

• simpA.kendallReg: test based on the constancy of conditional Kendall's tau

• the counterparts of these tests in the discrete conditioning setting: bCond.simpA.CKT (test based on conditional Kendall's tau) bCond.simpA.param (test assuming a parametric form for the conditional copula)

Examples

# We simulate from a conditional copula
set.seed(1)
N = 500
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1], mean = Z)
X2 = qnorm(simCopula[,2], mean = - Z)

result <- simpA.param(
   X1 = X1, X2 = X2, X3 = Z, family = 1,
   h = 0.03, kernel.name = "Epanechnikov", nBootstrap = 5)
print(result$p_val)
# In practice, it is recommended to use at least nBootstrap = 100
# with nBootstrap = 200 being a good choice.


set.seed(1)
N = 500
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = 0.8
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1], mean = Z)
X2 = qnorm(simCopula[,2], mean = - Z)

result <- simpA.param(
   X1 = X1, X2 = X2, X3 = Z, family = 1,
   h = 0.08, kernel.name = "Epanechnikov", nBootstrap = 5)
print(result$p_val)

CondCopulas documentation built on Sept. 11, 2024, 9:10 p.m.