nu <- 3 # d.o.f. of the standardized distribution of Z_t
fixed.p <- list(mu = 0, # our mu (intercept)
                ar1 = 0.5, # our phi_1 (AR(1) parameter of mu_t)
                ma1 = 0.3, # our theta_1 (MA(1) parameter of mu_t)
                omega = 4, # our alpha_0 (intercept)
                alpha1 = 0.4, # our alpha_1 (GARCH(1) parameter of sigma_t^2)
                beta1 = 0.2, # our beta_1 (GARCH(1) parameter of sigma_t^2)
                shape = nu) # d.o.f. nu for standardized t_nu innovations
armaOrder <- c(1,1) # ARMA order
garchOrder <- c(1,1) # GARCH order
varModel <- list(model = "sGARCH", garchOrder = garchOrder)
spec <- ugarchspec(varModel, mean.model = list(armaOrder = armaOrder),
                   fixed.pars = fixed.p, distribution.model = "std") # t standardized residuals

plot(X,   type = "l", xlab = "t", ylab = expression(X[t]))

plot(sig, type = "h", xlab = "t", ylab = expression(sigma[t]))

plot(eps, type = "l", xlab = "t", ylab = expression(epsilon[t]))

## Fit an ARMA(1,1)-GARCH(1,1) model
spec <- ugarchspec(varModel, mean.model = list(armaOrder = armaOrder),
                   distribution.model = "std") # without fixed parameters here
fit <- ugarchfit(spec, data = X) # fit

## Extract the resulting series
mu. <- fitted(fit) # fitted hat{mu}_t (= hat{X}_t)
sig. <- sigma(fit) # fitted hat{sigma}_t

## Sanity checks (=> fitted() and sigma() grab out the right quantities)
stopifnot(all.equal(as.numeric(mu.),  fit@fit$fitted.values),
          all.equal(as.numeric(sig.), fit@fit$sigma))