Given an ARIMA process (assuming $d=0$ for simplicity) of the form $\begin{align*} W_{t}:&= \underbrace{\alpha_{1}W_{t-1}+ \cdots + \alpha_{p}W_{t-p}}_{\text{AR}(p)}\\[0.2em] &\hphantom{blaj}+\underbrace{Z_{t}+\beta_{1}Z_{t-1}+ \cdots + \beta_{q}Z_{t-q}}_{\text{MA}(q)} \end{align*}$With different formulations of the model, it allows three ways of forecasting. For all three methods, past values of $W_{t}$ and $Z_{t}$ are replaced with observed values, future values replaced with their conditional expectations ($0$ for $Z_{t}$). ### Forecasting in ARIMA Form The most straightforward method is to iteratively compute new forecasts $\hat{W}_{N+h}$ using previous forecasts: $\hat{W}_{N+h}=\alpha_{1}\hat{W}_{N+h-1}+ \cdots + \alpha_{p}\hat{W}_{N+h-p}$where $\hat{W}_{i}$ is the forecasted value if $i >N$, and the observed value when $i \le N$. ### Forecasting in AR Form By rewriting the ARIMA process as an $\mathrm{AR}(\infty)$ process with weights $\pi_{1},\dots$, we can also iteratively compute the forecasts $\hat{W}_{N+h}=\pi_{1}\hat{W}_{N+h-1} + \pi_{2}\hat{W}_{N+h-2} +\cdots $ ### Variance Forecasting in MA Form Rewriting as an $\mathrm{MA}(\infty)$ process with weights $\phi_{1},\dots$ gives ${W}_{N+h}=Z_{N+h}+\phi_{1}Z_{N+h-1}+\cdots$ It does not give a useful point estimate, but it forecasts $\mathrm{Var}(W_{t})$ to be $\mathrm{Var}[\widehat{W_{N+h}}\,|\,(X_{t}),(Z_{t})]=\sigma^{2}_{Z}\sum_{i=0}^{h-1}\phi^{2}_{i}$which is the forecast variance given the model and past values -- it does not account for the variance in estimating model parameters.