# Copula Processes with V-Transforms

#### Alexander J. McNeil and Martin Bladt

#### 2024-02-16

All types of copula process can be combined with a v-transform to model volatile time series.

## 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)

## 2. VT-D-Vine Copula Processes (type 2)

VT-D-Vine processes are created by adding a v-transform to an `dvinecopula2` object using the command `vtscopula`. The generic commands `sim`, `fit` and `plot` also work for these processes.

### Construction

We add a 2-parameter V-transform.

    copmod <- dvinecopula2(family = "joe",
                           kpacf = "kpacf_arma",
                           pars = list(ar = 0.9, ma = -0.85),
                           maxlag = 20)
    vcopmod <- vtscopula(copmod,
      Vtransform = V2p(delta = 0.6, kappa = 0.8)
    )
    vcopmod
    #> object class: vtscopula
    #> ____________
    #> Base copula:
    #> object class: dvinecopula2
    #> name: type2-d-vine
    #> copula family: joe
    #> KPACF: kpacf_arma with max lag 20
    #> parameters:
    #> [1]  0.90 -0.85
    #> ____________
    #> V-transform:
    #> name: V2p
    #> parameters:
    #> delta kappa
    #>   0.6   0.8

### Simulation

    set.seed(13)
    data2 <- sim(vcopmod, n = 2000)
    hist(data2)
    ts.plot(data2)

### Estimation

    copspec_Joe <- dvinecopula2(family = "joe",
                                pars = list(ar = 0, ma = 0),
                                maxlag = 30)
    vcopspec <- vtscopula(copspec_Joe, V2p(delta = 0.6))
    vcopfit <- fit(vcopspec, data2,
                   tsoptions = list(hessian = TRUE),
                   control = list(maxit = 1000))
    vcopfit
    #> object class: vtscopula
    #> ____________
    #> Base copula:
    #> object class: dvinecopula2
    #> name: type2-d-vine
    #> copula family: joe
    #> KPACF: kpacf_arma with max lag 30
    #> parameters:
    #> [1]  0.8973846 -0.8434568
    #> ____________
    #> V-transform:
    #> name: V2p
    #> parameters:
    #>     delta     kappa
    #> 0.6000000 0.7743815
    #> _____________________
    #> Summary of estimates:
    #>             ar         ma   vt.kappa
    #> par 0.89738462 -0.8434568 0.77438147
    #> se  0.01948553  0.0253182 0.06339235
    #> convergence status: 0, log-likelihood: 48.57659
    coef(vcopfit)
    #>                            delta      kappa
    #>  0.8973846 -0.8434568  0.6000000  0.7743815
    coef(vcopmod)
    #>             delta kappa
    #>  0.90 -0.85  0.60  0.80

### Plotting

We can plot the estimated v-transform and well as the goodness-of-fit plots for the `dvinecopula2` object based on Kendall rank correlations.

    plot(vcopfit, plottype = "vtransform")
    plot(vcopfit, plottype = "kendall")
    plot(vcopfit, plottype = "residual")