## 1. Analog Data

First, let’s model the distribution of innovation. We chose a small sample size. Ideally, the sample size should be larger, making it easier to detect the GARCH effect.

``Kendall's Tau Copula(" T ", Param = Th, Dim = D, df = Nu) SQRT ((nu.-2)/nu.) * qt(U, df = nu) # For ugarchpath(), the edge must have mean 0 and variance 1!``

Now we simulate two ARMA(1,1)-GARCH(1,1) processes using the distribution of innovation that these copulas rely on.

``P < -list (mu = 1, spec(varModel, meanModel, fix.pars)) # condition innovation density (or use, e.g., "STD") ## Simulate the ARMA-GARCH model with innovation ## Note: ugarchpath(): Simulate from spec; < -fitted (X) # X\_t = mu\_t + eps_t (simulated process) # StopIfNot (all.equal(X., X@path\$seriesSim, check. Attributes = FALSE), ## plot(X., type = "l", xlab = "t")``

## 2. Fitting program based on simulated data

We now show how to fit the ARMA(1,1)-GARCH(1,1) process for X (we remove the parameter fix.pars to estimate these parameters).

``````spec(varModel, mean.model = meanModel)
ugarchfit(uspec, data = x))``````

Check the (standardized) Z, the pseudo-observed value of the residuals Z.

``plot(U.)``

For the edge distribution, we also assume a T-distribution, but with different degrees of freedom.

``fit("t", dim = 2), data = U., method = "mpl")``
``Fitted = C (Estimate, nu.), True = C (Th, nu, nu.)``

## 3. Simulate from the fitting time series model

The simulation was carried out from the fitted Copula model.

``Set. The seed (271) # repeatability sapply (1: d, the function SQRT (j) ((nu \ [j \] - 2)/nu \ [j \]) * qt (U \ [j \], Df = nu \ [j \])) # # = > innovation must be standardized garch () sim (fit \ [\ [j \] \], n.s im = n, Margaret spellings im = 1,``

Each resulting sequence (XtXt) is plotted.

``Apply (Sim,fitted(X)) # Plot (X.. , type = "l")``

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