require(copula)
require(rugarch)
set.seed(271)

1) Simulate data

First, we simulate the innovation distribution.

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
U <- rCopula(n, cop) # sample the copula
Z <- qnorm(U) # adjust margins
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
## Set parameters
fixed.p <- list(mu  = 1,
                ar1 = 0.5,
                ma1 = 0.3,
                omega = 2, # alpha_0 (conditional variance intercept)
                alpha1= 0.4,
                beta1 = 0.2)
varModel <- list(model = "sGARCH", garchOrder=c(1,1)) # standard GARCH
uspec <- ugarchspec(varModel, mean.model = list(armaOrder=c(1,1)),
                    fixed.pars = fixed.p,
                    distribution.model = "norm") # conditional innovation density
## 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 sample (n x d)-matrix
str(X, max.level=2) # => @path$sigmaSim, $seriesSim, $residSim
## Formal class 'uGARCHpath' [package "rugarch"] with 3 slots
##   ..@ path :List of 3
##   ..@ model:List of 10
##   ..@ seed : int 310025978
matplot(X@path$sigmaSim,  type="l") # plot of sigma's (conditional standard deviations)

matplot(X@path$seriesSim, type="l") # plot of X's

matplot(X@path$residSim,  type="l") # plot of Z's

par(pty="s")
plot(pobs(X@path$residSim)) # plot of Z's pseudo-observations => seem fine

2) Fitting procedure based on the simulated data

We now show how to fit an ARMA(1,1)-GARCH(1,1) process to X (remove ‘fixed.pars’ from specification to be able to fit).

uspec <- ugarchspec(varModel, mean.model = list(armaOrder=c(1,1)),
                    distribution.model = "norm")
fit <- apply(X@path$seriesSim, 2, function(x) ugarchfit(uspec, x))
str(fit, max.level=3)
## List of 2
##  $ :Formal class 'uGARCHfit' [package "rugarch"] with 2 slots
##   .. ..@ fit  :List of 25
##   .. ..@ model:List of 11
##  $ :Formal class 'uGARCHfit' [package "rugarch"] with 2 slots
##   .. ..@ fit  :List of 25
##   .. ..@ model:List of 11
str(fit[[1]], max.level=2) # for first time series
## Formal class 'uGARCHfit' [package "rugarch"] with 2 slots
##   ..@ fit  :List of 25
##   ..@ model:List of 11
stopifnot(identical(fit[[1]]@fit$residuals, residuals(fit[[1]]@fit))) # => the same

Check the residuals.

Z. <- sapply(fit, function(fit.) residuals(fit.@fit))
U. <- pobs(Z.)
par(pty="s")
plot(U., # plot of Z's pseudo-observations => seem fine
     xlab=expression(italic(hat(U)[1])),
     ylab=expression(italic(hat(U)[2])))

Fit a t copula to the residuals Z.

fitcop <- fitCopula(ellipCopula("t", dim=2), data=U., method="mpl")
rbind(est = fitcop@estimate, true = c(th, nu)) # hat{rho}, hat{nu}; close to th, nu
##           [,1]     [,2]
## est  0.6766895 2.696942
## true 0.7071068 3.000000

3) Simulate from the fitted time series model

Simulate from the fitted copula model.

U.. <- rCopula(n, fitcop@copula)
Z.. <- qnorm(U..)
X..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]))@simulation)
str(X..sim, max.level=3)
## List of 2
##  $ :List of 3
##   ..$ sigmaSim : num [1:200, 1] 1.51 1.49 1.88 1.65 2.37 ...
##   ..$ seriesSim: num [1:200, 1] 1.74 3.35 1.88 3.99 -2.05 ...
##   ..$ residSim : num [1:200, 1] 0.244 1.657 -0.996 2.63 -5.533 ...
##  $ :List of 3
##   ..$ sigmaSim : num [1:200, 1] 2.11 2.11 2.11 2.11 2.11 ...
##   ..$ seriesSim: num [1:200, 1] 1.204 1.792 0.835 2.944 -2.498 ...
##   ..$ residSim : num [1:200, 1] 0.25 0.643 -0.708 2.277 -5.099 ...
X..Z <- sapply(X..sim, `[[`, "residSim")
X.. <- sapply(X..sim, `[[`, "seriesSim")
plot(X..Z, main="residSim"); abline(h=0,v=0, lty=2, col=adjustcolor(1, .5))

plot(X.., main="seriesSim"); abline(h=0,v=0, lty=2, col=adjustcolor(1, .5))

matplot(pobs(X..), type="l")

par(pty="s")
plot(pobs(X..), main="pobs(series..)"); rect(0,0,1,1, border=adjustcolor(1, 1/2))