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So the formula for the first forecast with a ARMA(2,2)-GARCH(1,2) and a differenced time series looks like this:

Y(t+1)=Y(t)+Alpha(1)*(Y(t)-Y(t-1))+Alpha(2)*(Y(t-1)-Y(t-2)) - Beta(1)*e(t) - Beta(2)*e(t-1) + e(t+1) 

with

e(t+1) = Sigma(t+1)*Z(t+1)  ,   Z(t+1)=N(0,1) 

and

Sigma^2 (t+1) = Omega + a(1)*u^2(t) + b(1)*Sigma^2(t) + b(2)*Sigma^2(t-1)

My questions are:

  1. What is u(t) in this equation?
  2. Am i right that Sigma^2(t) and Sigma^2(t-1) are the variances of the timeseries up to time t and up to time t-1?
  3. To get the residual in the ARMA model I have to take the root of Sigma^2(t+1) and multiply it with a random variable?
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  • $\begingroup$ How is it, did you get your problems clarified or do you need further elaboration? $\endgroup$ – Richard Hardy Oct 24 '17 at 11:17
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  1. $u_t$ is $\varepsilon_t$, i.e. $u_t$ should be replaced with $\varepsilon_t$.
  2. Not up to but at (unless they mean the same for you), i.e. $\sigma_t^2$ is the conditional variance of $\varepsilon_t$.
  3. To obtain the residual $\hat\varepsilon_t$, take the actual observation $y_t$ and subtract the fitted value from the ARMA model $\hat y_t$: $$\hat\varepsilon_t:=y_t-\hat y_t.$$ To obtain the standardized residual $\hat z_t$, divide $\hat\varepsilon_t$ by its estimated standard deviation $\sqrt{\hat\sigma_t^2}$: $$\hat z_t:=\frac{\hat\varepsilon_t}{\sqrt{\hat\sigma_t^2}}.$$

(Residuals and fitted values have hats to denote they are estimated quantities, in contrast to errors/shocks/innovations and theoretical quantities.)

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  • $\begingroup$ Point 1 is now clear to me: Point 2: But sigma^2(t) is computed through the time series up to time t? thats what i meant. Point 3: How do i get the residuals for a n-step ahead forecast where i don't have the actual values of the future time series? $\endgroup$ – user2968163 Oct 24 '17 at 7:52
  • $\begingroup$ Point 2: data from $\tau=1$ to $\tau=t-1$ is used for estimating the model coefficients and producing fitted values such as $\hat\sigma_t^2$, but the meaning of $\hat\sigma_t^2$ is just as stated in my point 2. Point 3: Considering forecasts, you do not talk about residuals but rather forecast errors. When forecasting the actual future, there is no way to get forecast errors until the future actually happens (time passes). The forecasted "residual" or the forecasted forecast error $\hat\varepsilon_{t+h}$ is zero with a forecasted variance $\hat\sigma_{t+h}^2$. $\endgroup$ – Richard Hardy Oct 24 '17 at 8:02
  • $\begingroup$ Point 3 is now almost clear. So e(t+h) is a random variable with zero mean and variance sigma^2(t+h). But which distribution does it follow? I assume normal distribution like in the ARMA type models? To point 3: I want to estimate the coefficients once at time t and keep them fixed throughout the forecast. So for computing sigma°2(t+h) I need the time series from 1 to t plus the forecasted time series from t+1 to t+h-1? $\endgroup$ – user2968163 Oct 24 '17 at 8:16
  • $\begingroup$ In the above comment, you got almost everything right. You can choose which distribution $z_t$ follows (Normal, $t$, generalized error distribution or some other) when specifying the model likelihood, and then $\varepsilon_t$ follows that one but scaled by $\sigma_t$. When forecasting more than one step ahead, you do not refit model parameters, they stay the same. But you use the forecast for $h-1$ to iteratively obtain the forecast for $h$ (I think you got this right). $\endgroup$ – Richard Hardy Oct 24 '17 at 8:27
  • $\begingroup$ I read in your profile that you work with R. My forecasting will be done in R also. Do you know by any chance if there is a package for this type of forecast or do I have to do it by my own. I found ugarchforecast in the package rugarch but it seems, that it gives constant forecasts with no random error terms. $\endgroup$ – user2968163 Oct 24 '17 at 8:32

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