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)
e(t+1) = Sigma(t+1)*Z(t+1) , Z(t+1)=N(0,1)
Sigma^2 (t+1) = Omega + a(1)*u^2(t) + b(1)*Sigma^2(t) + b(2)*Sigma^2(t-1)
My questions are:
- What is
u(t)in this equation?
- Am i right that
Sigma^2(t-1)are the variances of the timeseries up to time t and up to time t-1?
- 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?