simpA.param: 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 = NULL,
  numericalInt = list(kind = "legendre", nGrid = 10)
)

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.

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. Note that truncVal must be in the interval $[0, 0.5)$, i.e. $0$ is allowed but not $0.5$.

The default is truncVal = NULL, which actually means that truncVal = h if h < 0.5 and truncVal = 0 else.

numericalInt

parameters to be given to statmod::gauss.quad, including the number of quadrature points and the type of interpolation.

References

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

Examples

Run this code

# 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.

# \donttest{
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)
# }