Question

Derive multistep-ahead forecasts for a GARCH(1,2) model at the forecast origin h.

Answer 1

Answer :

Suppose that the forecast origin is h.

For 1-step ahead forecast :
σ^2h (1) = α0 + α1ε^2h + β1σ^2h

For 2-step ahead forecast :
σ^2h (2) = α0 + (α1+β1)σ^2h(1)

For l-step ahead forecast :
σ^2h(l) = α0(1-(α1+β1)^l-1)/(1-α1-β1+(α1+β1)^l-1σ^2h(l)

So, Therefore :-
σ^2h(l) ---->α0/(1-α1-β1)

The multistep ahead volatility forecasts of GARCH(1,1) model converge to the unconditional.

Variance of ε as the forecast horizon increases to infinity provided that Var(εt) exists.

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