General forecasting equation for ARIMA(p,d,q)(P,D,Q)s

Question

what is general forecasting equation for ARIMA(p,d,q)(P,D,Q)s?

I wrote this equation, can someone confirm if it is a correct one?

If not, can someone correct it?

$$\overline{y}_{t+m}= \varphi_{1}y_{t} + \varphi_{2}y_{t-1}+...+ \varphi_{p}y_{t-p+m}+\phi_{1}y_{t} + \phi_{2}y_{t-s}+...+ \phi_{P}y_{t-P+m}-\theta_{1}e_{t}-\theta_{2}e_{t-1}-...-\theta_{q}e_{t-q+m}-...-\Theta_{1}e_{t}-\Theta_{2}e_{t-s}-...-\Theta_{Q}e_{t-Q+m}$$

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but I need it in a form $\overline{y}_{t+m}= \ldots$

accordinng to your links, is it then proper:

$$\overline{y}_{t+m}=\frac{ (\varphi_{1}y_{t} + \varphi_{2}y_{t-1}+...+ \varphi_{p}y_{t-p+m})(\phi_{1}y_{t} + \phi_{2}y_{t-s}+...+ \phi_{P}y_{t-P+m})}{(\theta_{1}e_{t}-\theta_{2}e_{t-1}-...-\theta_{q}e_{t-q+m})(\Theta_{1}e_{t}-\Theta_{2}e_{t-s}-...-\Theta_{Q}e_{t-Q+m})}$$

Answer 1

There's more than one way of formulating ARIMA. Matlab implements this form: $\phi(L)\Phi(L^S)(1-L)^D(1-L^S)^{D_s}y_t=c+\theta(L)\Theta(L^S)\varepsilon_t$ See the lag operator specifications here.

This is a multiplicative seasonality. You don't seem to be using this one. Maybe you're using some form of additive seasonality, otherwise you'd have $\theta\Theta$ terms in your equation, see the example here