require(copula)
require(rugarch)
set.seed(271)
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$residSim)) # plot of Z's pseudo-observations => seem fine
We now show how to fit an ARMA(1,1)-GARCH(1,1) process to X
(remove ‘fixed.pars’ from specification to be able to fit).
uspec <- ugarchspec(varModel, mean.model = list(armaOrder=c(1,1)),
distribution.model = "norm")
fit <- apply(X@path$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
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))