Author: Marius Hofert
Date: 2020-12-11

# The Copula GARCH Model

#### Marius Hofert

#### 2020-12-11

    require(copula)
    require(rugarch)

In this vignette, we demonstrate the copula GARCH approach (in general). Note that a special case (with normal or student \(t\) residuals) is also available in the `rmgarch` package (thanks to Alexios Ghalanos for pointing this out).

## 1 Simulate data

First, we simulate the innovation distribution. Note that, for demonstration purposes, we choose a small sample size. Ideally, the sample size should be larger to capture GARCH effects.

    ## Simulate innovations
    n <- 200 # sample size
    d <- 2 # dimension
    nu <- 3 # degrees of freedom for t
    tau <- 0.5 # Kendall's tau
    th <- iTau(ellipCopula("t", df = nu), tau) # corresponding parameter
    cop <- ellipCopula("t", param = th, dim = d, df = nu) # define copula object
    set.seed(271) # reproducibility
    U <- rCopula(n, cop) # sample the copula
    nu. <- 3.5 # degrees of freedom for the t margins
    Z <- sqrt((nu.-2)/nu.) * qt(U, df = nu.) # margins must have mean 0 and variance 1 for ugarchpath()!

Now we simulate two ARMA(1,1)-GARCH(1,1) processes with these copula-dependent innovations. To this end, recall that an ARMA(\(p_1\),\(q_1\))-GARCH(\(p_2\),\(q_2\)) model is given by \[\begin{align} X_t &= \mu_t + \epsilon_t\ \text{for}\ \epsilon_t = \sigma_t Z_t,\\ \mu_t &= \mu + \sum_{k=1}^{p_1} \phi_k (X_{t-k}-\mu) + \sum_{k=1}^{q_1} \theta_k \epsilon_{t-k},\\ \sigma_t^2 &= \alpha_0 + \sum_{k=1}^{p_2} \alpha_k (X_{t-k}-\mu_{t-k})^2 + \sum_{k=1}^{q_2} \beta_k \sigma_{t-k}^2. \end{align}\]

    ## Fix parameters for the marginal models
    fixed.p <- list(mu  = 1,
                    ar1 = 0.5,
                    ma1 = 0.3,
                    omega  = 2, # alpha_0 (conditional variance intercept)
                    alpha1 = 0.4,
                    beta1  = 0.2)
    meanModel <- list(armaOrder = c(1,1))
    varModel <- list(model = "sGARCH", garchOrder = c(1,1)) # standard GARCH
    uspec <- ugarchspec(varModel, mean.model = meanModel,
                        fixed.pars = fixed.p) # conditional innovation density (or use, e.g., "std")

    ## Simulate ARMA-GARCH models using the dependent innovations
    ## Note: ugarchpath(): simulate from a spec; ugarchsim(): simulate from a fitted object
    X <- ugarchpath(uspec,
                    n.sim = n, # simulated path length
                    m.sim = d, # number of paths to simulate
                    custom.dist = list(name = "sample", distfit = Z)) # passing (n, d)-matrix of *standardized* innovations

    ## Extract the resulting series
    X. <- fitted(X) # X_t = mu_t + eps_t (simulated process)
    sig.X <- sigma(X) # sigma_t (conditional standard deviations)
    eps.X <- X@path$residSim # epsilon_t = sigma_t * Z_t (residuals)

    ## Basic sanity checks :
    stopifnot(all.equal(X.,    X@path$seriesSim, check.attributes = FALSE),
              all.equal(sig.X, X@path$sigmaSim,  check.attributes = FALSE),
              all.equal(eps.X, sig.X * Z,        check.attributes = FALSE))

    ## Plot (X_t) for each margin
    matplot(X., type = "l", xlab = "t", ylab = expression(X[t]~"for each margin"))

## 2 Fitting procedure based on the simulated data

We now show how to fit an ARMA(1,1)-GARCH(1,1) process to `X` (we remove the argument `fixed.pars` from the above specification for estimating these parameters):

    uspec <- ugarchspec(varModel, mean.model = meanModel, distribution.model = "std")
    fit <- apply(X., 2, function(x) ugarchfit(uspec, data = x))

Check the (standardized) `Z`, i.e., the pseudo-observations of the residuals `Z`:

    Z. <- sapply(fit, residuals, standardize = TRUE)
    U. <- pobs(Z.)
    par(pty = "s")
    plot(U., xlab = expression(hat(U)[1]), ylab = expression(hat(U)[2]))

Fit a \(t\) copula to the standardized residuals `Z`. For the marginals, we also assume \(t\) distributions but with different degrees of freedom; for simplicity, the estimation is omitted here.

    fitcop <- fitCopula(ellipCopula("t", dim = 2), data = U., method = "mpl")

    ## Warning in var.mpl(copula, u): the covariance matrix of the parameter estimates
    ## is computed as if 'df.fixed = TRUE' with df = 2.31168141320139

    nu. <- rep(nu., d) # marginal degrees of freedom; for simplicity using the known ones here
    est <- cbind(fitted = c(fitcop@estimate, nu.), true = c(th, nu, nu.)) # fitted vs true
    rownames(est) <- c("theta", "nu (copula)", paste0("nu (margin ",1:2,")"))
    est

    ##                  fitted      true
    ## theta         0.6724201 0.7071068
    ## nu (copula)   2.3116814 3.0000000
    ## nu (margin 1) 3.5000000 3.5000000
    ## nu (margin 2) 3.5000000 3.5000000

## 3 Simulate from the fitted time series model

Simulate from the fitted copula model.

    set.seed(271) # reproducibility
    U.. <- rCopula(n, fitcop@copula)
    Z.. <- sapply(1:d, function(j) sqrt((nu.[j]-2)/nu.[j]) * qt(U..[,j], df = nu.[j]))
    ## => Innovations have to be standardized for ugarchsim()
    sim <- lapply(1:d, function(j)
        ugarchsim(fit[[j]], n.sim = n, m.sim = 1,
                  custom.dist = list(name = "sample",
                                     distfit = Z..[,j, drop = FALSE])))

and plot the resulting series (\(X_t\)) for each margin

    X.. <- sapply(sim, function(x) fitted(x)) # simulated series X_t (= x@simulation$seriesSim)
    matplot(X.., type = "l", xlab = "t", ylab = expression(X[t]))