# Bitcoin Analysis

#### Alexander J. McNeil and Martin Bladt

#### 2024-02-16

## The Bitcoin Log-Return Data

We first load the data and calculate log-returns.

    data(bitcoin)
    X <- (diff(log(bitcoin))[-1]) * 100 # log-returns (as percentages)
    length(X)
    #> [1] 1043
    plot(X, type = "h")

## Copula Processes

### ARMA Copulas

We start with the fulcrum profile-likelihood plot which indicates that a value \(\delta = 0.45\) is about right.

    U <- strank(X)

    copmod_Gauss <- armacopula(pars = list(ar = 0.95, ma = -0.85))
    pts <- seq(from=0.01, to=0.99, length=51) + 0.005
    profilefulcrum(U, copmod_Gauss, locations = pts)
    abline(v = 0.45)

Our baseline model is a model with ARMA(1,1) copula and v-transform with fulcrum at \(\delta=0.45\).

    mod_Gauss <- vtscopula(copmod_Gauss, Vlinear(0.45))
    fit_Gauss <- fit(mod_Gauss, U, tsoptions = tsoptions)
    fit_Gauss
    #> object class: vtscopula
    #> ____________
    #> Base copula:
    #> object class: armacopula
    #> name: ARMA(1,1)
    #> parameters:
    #>        ar1        ma1
    #>  0.9631285 -0.8424401
    #> ____________
    #> V-transform:
    #> name: Vlinear
    #> parameters:
    #> delta
    #>  0.45
    #> _____________________
    #> Summary of estimates:
    #>         ar.ar1      ma.ma1
    #> par 0.96312845 -0.84244010
    #> se  0.01146705  0.02698389
    #> convergence status: 0, log-likelihood: 91.87754

### D-Vine Copula (second kind)

In this next section we try a d-vine copula model of the second kind with a finite value for `maxlag` and a linear v-transform. We continue to use the Frank copula. The resulting model is superior to the ARMA copula model and has the same number of parameters. Note that an infinite value for `maxlag` makes next to no difference in the fit.

    copmod_Frank <- dvinecopula2(family = "frank",
                                 pars = list(ar = 0.95, ma = -0.85),
                                 maxlag = 30)
    mod_Frank <- vtscopula(copmod_Frank, Vlinear(0.45))
    fit_Frank <- fit(mod_Frank, U, tsoptions = tsoptions)

    AIC(fit_Gauss, fit_Frank)
    #>           df       AIC
    #> fit_Gauss  2 -179.7551
    #> fit_Frank  2 -186.4793

We provide plots for the model with Frank copula to show aspects of the fit.

    plot(fit_Frank, plottype = "residual")
    plot(fit_Frank, plottype = "kendall")

## Marginal Models

We fit 6 marginal distributions (3 symmetric and 3 skewed) of which the double Weibull gives the lowest AIC value.

    marg_st <- fit(margin("st"), X)
    marg_sst <- fit(margin("sst"), X)
    marg_lp <- fit(margin("laplace",
                          pars = c(mu = 0.2, scale = 2.7)), X)
    marg_slp <- fit(margin("slaplace",
                    pars = c(mu = 0.2, scale = 2.7, gamma = 0.9)), X)
    marg_dw <- fit(margin("doubleweibull",
                          pars = c(mu = 0.2, shape = 0.8, scale = 2.7)), X)
    marg_sdw <- fit(margin("sdoubleweibull",
                    pars = c(mu = 0.2, shape = 0.8, scale = 2.7, gamma = 0.9)), X)
    AIC(marg_st, marg_sst, marg_slp, marg_lp, marg_dw, marg_sdw)
    #>          df      AIC
    #> marg_st   3 5816.288
    #> marg_sst  4 5818.249
    #> marg_slp  3 5801.355
    #> marg_lp   2 5799.447
    #> marg_dw   3 5749.362
    #> marg_sdw  4 5751.137

## Full Models

### D-Vine Copula

We fit a full model combining the double Weibull margin with the VT-d-vine model of the second kind.

    fullmod <- tscm(fit_Frank, margin = marg_dw)
    fullmod <- fit(fullmod, as.numeric(X),
                   method = "full", tsoptions = tsoptions)
    fullmod
    #> object class: tscmfit
    #> _______
    #> MARGIN:
    #> name: doubleweibull
    #> parameters:
    #>        mu     shape     scale
    #> 0.1916831 0.8440733 2.6951432
    #> _______
    #> COPULA:
    #> object class: vtscopula
    #> ____________
    #> Base copula:
    #> object class: dvinecopula2
    #> name: type2-d-vine
    #> copula family: frank
    #> KPACF: kpacf_arma with max lag 30
    #> parameters:
    #> [1]  0.9597057 -0.8344573
    #> ____________
    #> V-transform:
    #> name: Vlinear
    #> parameters:
    #> delta
    #>  0.45
    #> _________________________
    #> summary of all estimates:
    #>             ar          ma    margin.mu margin.shape margin.scale
    #> par 0.95970573 -0.83445730 0.1916831037   0.84407334    2.6951432
    #> se  0.01419982  0.02912741 0.0006240765   0.03191155    0.2711454
    #> convergence status: 10 , log-likelihood: -2776.988
    AIC(marg_dw, fullmod)
    #>         df      AIC
    #> marg_dw  3 5749.362
    #> fullmod  5 5563.977

We show all the possible plots for the full model.

    plot(fullmod, plottype = "residual")
    plot(fullmod, plottype = "kendall")
    plot(fullmod, plottype = "margin")
    plot(fullmod, plottype = "vtransform")
    plot(fullmod, plottype = "volprofile")
    plot(fullmod, plottype = "volproxy")

    plot(fullmod, plottype = "glag")