11.3 Box-Jenkins Seasonal (SARIMA) Model

In practice main time series contain a seasonal periodic component which repeats every ’s’ observations. e.g with monthly observations \(s=12\). \(X_t\) depends on terms such that \(X_{t-12}\) and perhaps \(X_{t-24}\). As well as \(X_{t-1},X_{t-2},\cdots \cdots \)

Box-Jenkins have generalized the ARIMA model to deal with seasonality and find the general multiplicative ARIMA model (SARIMA)as \[\phi _p(B) \Phi _P(B^s) W_t=\theta _q(B) \Theta _Q(B^s)Z_t\] where \(\hspace {0.3cm} W_t=\nabla ^d\, \nabla ^D_s X_t\), B denotes the backward shift operator, \(\,\phi _p \,,\,\Phi _P ,\,\theta _q \,\text {and} \hspace {0.2cm}\Theta _Q\) are polynomials of order p,P,q,Q respectively. \(Z_t\) denotes a purely random process.
If \(P=1\) the term \(\Phi _p(B^s)\) will be \((1- \text {constant}\, XB^s)\) which means that \(W_t\) will depend on \(W_{t-s}\) since \(B^s\,W_t=W_{t-s}\). The variables \(\{W_t\}\) are formed from the original series \(\{X_t\}\) not only by simple difference to remove trend but also by seasonal difference to remove seasonality.

If \(d=D=1\) and \(s=12\) \begin {align*} W_t &=\bigtriangledown \bigtriangledown _{12} X_t\\ &=\bigtriangledown _{12} X_t-\bigtriangledown _{12} X_{t-1}\\ &=(X_t-X_{t-12})-(X_{t-1}-X_{t-13})\\ \end {align*}

The model equation \(\,\phi _p(B) \Phi _P(B^s) W_t=\theta _q(B) \Theta _Q(B^s) Z_t\,\) and \(\,W_t=\nabla ^d \,\nabla ^D_s X_t\) is said to be a SARIMA model of order \(\,(p,d,q)\times (P,D,Q)_s\).

So we note that \(s=12\) then the model
\(\phi _p(B) \Phi _P(B^s) W_t=\theta _q(B) \Theta _Q(B^s) Z_t\) and \(W_t=\nabla ^d\, \nabla ^D X_t\) can be written as

\((1-\alpha B)W_t=(1+\theta B^{12})\) where \(W_t=\nabla _{12} X_t\)

we find \(X_t=X_{t-12}+\alpha (X_{t-1}-X_{t-13})+Z_t+\theta Z_{t-12}\) so that \(X_t\) depends on \(X_{t-1},X_{t-12},X_{t-13}\) as the innovation time.

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