1. VT-ARMA Copula Processes
VT-ARMA processes are created by adding a v-transform to an
armacopula process using the command vtscopula. The generic
commands sim, fit and plot also
work for these processes.
VT-ARMA(1,1) Example
This example uses a ARMA(1,1) copula and an off-centre linear v-transform. We set up the model and generate some data.
vtarma11 <- vtscopula(armacopula(list(ar = 0.95, ma = -0.85)),
Vtransform = Vlinear(delta = 0.6))
vtarma11
#> object class: vtscopula
#> ____________
#> Base copula:
#> object class: armacopula
#> name: ARMA(1,1)
#> parameters:
#> ar1 ma1
#> 0.95 -0.85
#> ____________
#> V-transform:
#> name: Vlinear
#> parameters:
#> delta
#> 0.6
set.seed(19)
data <- sim(vtarma11, 2000)
ts.plot(data)
We now fit the model with a fixed value for the fulcrum parameter \(\delta\) and plot the results. (More on fulcrum choice next.)
vtarma11spec <- vtscopula(armacopula(list(ar = 0, ma = 0)), Vtransform = Vlinear(delta = 0.6))
vtarma11fit <- fit(vtarma11spec, data)
vtarma11fit
#> object class: vtscopula
#> ____________
#> Base copula:
#> object class: armacopula
#> name: ARMA(1,1)
#> parameters:
#> ar1 ma1
#> 0.9539846 -0.8525973
#> ____________
#> V-transform:
#> name: Vlinear
#> parameters:
#> delta
#> 0.6
#> _____________________
#> Summary of estimates:
#> ar.ar1 ma.ma1
#> 0.9539846 -0.8525973
#> convergence status: 0, log-likelihood: 104.0813
plot(vtarma11fit)
plot(vtarma11fit, plottype = "vtransform")
plot(vtarma11fit, plottype = "kendall" )
Optimization over the fulcrum parameter \(\delta\) does not take place. To identify a reasonable value for \(\delta\) we can carry a profile likelihood analysis using different fixed values for the fulcrum parameter.
profilefulcrum(data, tscopula = vtarma11spec, locations = seq(from = 0, to = 1, length = 11))
abline(v = 0.6)
