ARMA-GARCH model parameters and forecasting

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?
• How is it, did you get your problems clarified or do you need further elaboration? Oct 24, 2017 at 11:17

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 $\sqrt{\hat\sigma_t^2}$: $$\hat z_t:=\frac{\hat\varepsilon_t}{\sqrt{\hat\sigma_t^2}}.$$
• 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 $\hat\sigma_{t+h}^2$. Oct 24, 2017 at 8:02
• 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). Oct 24, 2017 at 8:27