11.5 Computation of Forecast
Use of difference equations, if the model equation is known then \(\widehat {x}(n,1)\) can be obtained from the model
equation by replacing,
- future values of \(Z\) by zero.
- future values of \(X\) by there conditional expectation.
- past values of \(X\) and \(Z\) by their observed values.
Example 11.5.1. Consider the SARIMA \((0,0,1)\times (1,1,0)_{12}\) used where \[X_t=X_{t-12}-\alpha (X_{t-1}-X_{t-13})+Z_t+\theta Z_{t-12}\]
\[\widehat {x}(n,1)=X_{n-11}-\alpha (X_n-X_{n-12})+\theta Z_{n-11}\]
and
\begin {align*} \widehat {x}(n,2) &=X_{n-10}-\alpha (X_{t+1}-X_{n-11})+\theta Z_{n-10}\\ &=X_{n-10}+\alpha (\widehat {x}(n,1)-X_{n-11})+\theta Z_{n-10} \end {align*}
forecasting further in the future can be calculated recursively when \(X_{n+1}\) is known.
\begin {align*} \widehat {x}(n+1,1) &=X_{n-10}+\alpha (X_{n+1}-X_{n-11})+\theta Z_{n-10}\\ &=\widehat {x}(n,2)+\alpha (X_{n+1}-\widehat {x}(n,1))+\theta Z_{n-10}\\ &=\widehat {x}(n,2)+\alpha Z_{n+1} \end {align*}
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