
(a)  ${\nabla}^{d}{\nabla}_{s}^{D}{x}_{t}c={w}_{t}$ where ${\nabla}^{d}{\nabla}_{s}^{D}{x}_{t}$ is the result of applying nonseasonal differencing of order $d$ and seasonal differencing of seasonality $s$ and order $D$ to the series ${x}_{t}$, and $c$ is a constant. 
(b)  ${w}_{t}={\Phi}_{1}{w}_{ts}+{\Phi}_{2}{w}_{t2\times s}+\cdots +{\Phi}_{P}{w}_{tP\times s}+{e}_{t}{\Theta}_{1}{e}_{ts}{\Theta}_{2}{e}_{t2\times s}\cdots {\Theta}_{Q}{e}_{tQ\times s}\text{.}$ This equation describes the seasonal structure with seasonal period $s$; in the absence of seasonality it reduces to ${w}_{t}={e}_{t}$. 
(c)  ${e}_{t}={\varphi}_{1}{e}_{t1}+{\varphi}_{2}{e}_{t2}+\cdots +{\varphi}_{p}{e}_{tp}+{a}_{t}{\theta}_{1}{a}_{t1}{\theta}_{2}{a}_{t2}\cdots {\theta}_{q}{a}_{tq}\text{.}$ This equation describes the nonseasonal structure. 
(a)  The state set required for forecasting. This contains the minimum amount of information required for forecasting and comprises:


(b)  A set of $L$ forecasts of ${x}_{t}$ and their estimated standard errors, ${s}_{t}$, for $\mathit{t}=n+1,\dots ,n+L$ ($L$ may be zero).
The forecasts and estimated standard errors are generated from the state set, and are identical to those that would be produced from the same state set by G13AHF. 
$1$  On entry the set of parameter values of this type does not satisfy the stationarity or invertibility test conditions. 
$\phantom{}0$  No parameter of this type is in the model. 
$\phantom{}1$  Valid parameter values of this type have been supplied. 
On entry,  ${\mathbf{NPAR}}\ne p+q+P+Q$, 
or  the orders vector MR is invalid (check the constraints in Section 5), 
or  ${\mathbf{KFC}}\ne 0$ or $1$. 
On entry,  ${\mathbf{IFV}}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NFV}}\right)$. 