g13ajf

Purpose

Synopsis

[rms,st,nst,fva,fsd,isf,ifail] = g13ajf(mr,par,c,x,nfv<,kfc,ifail>)

Description

 
 The time series x ,x ,...,x  supplied to the routine is assumed 
                  1  2      n                                   
 to follow a seasonal autoregressive integrated moving average 
 (ARIMA) model with known parameters.
 
 The model is defined by the following relations:
 
              d       D      
 (a)   (nabla) (nabla) x -c=w 
                      s t    t
       
                    d       D                            
       where (nabla) (nabla) x  is the result of applying non-
                            s t                          
       seasonal differencing of order d and seasonal differencing 
       of seasonality s and order D to the series x , and c is a 
                                                   t            
       constant.
 
 (b)   w =(Phi) w   +(Phi) w     +...+(Phi) w     +e -(Theta) e   
        t      1 t-s      2 t-2*s          P t-P*s  t        1 t-s
 
          -(Theta) e     -...-(Theta) e     .
                  2 t-2*s            Q t-Q*s
       
       This equation describes the seasonal structure with 
       seasonal period s; in the absence of seasonality it reduces
       to
                                 w =e .
                                  t  t
 
 (c)   e =(phi) e   +(phi) e   +...+(phi) e   +a -(theta) a   
        t      1 t-1      2 t-2          p t-p  t        1 t-1
       
          -(theta) a   -...-(theta) a   .
                  2 t-2            q t-q
 
       This equation describes the non-seasonal structure.
 
 Given the series, the constant c, and the model parameters (Phi),
 (Theta), (phi), (theta), the routine computes:
 
 (a)   The state set required for forecasting. This contains the 
       minimum amount of information required for forecasting and 
       comprises:
       (i)   the differenced series w , for (N-s*P)<=t<=N,
                                     t                   
       
       (ii)  the (d+D*s) values required to reconstitute the 
             original series x  from the differenced series w ,
                              t                              t
       
       (iii) the intermediate series e , for N-max(p,Q*s)<t<=N,
                                      t                       
       
       (iv)  the residual series a , for (N-q)<t<=N where N=n-
                                  t                      
             (d+D*s).
 
 (b)   A set of L forecasts of x , for t=n+1,n+2,...,n+L (L may be
                                t                               
       zero).
       
       The forecasts are generated from the state set, and are 
       identical to those that would be produced from the same 
       state set by G13AHF.
       
       Use of G13AJF should be confined to situations in which the
       state set for forecasting is unknown. Forecasting from the 
       series requires recalculation of the state set and this is 
       relatively expensive.
 

Parameters

g13ajf

Required Input Arguments:

mr (7)                                integer
par (:)                               real
c                                     real
x (:)                                 real
nfv                                   integer

Optional Input Arguments:                       <Default>

kfc                                   integer  1
ifail                                 integer  -1

Output Arguments:

rms                                   real
st (:)                                real
nst                                   integer
fva (:)                               real
fsd (:)                               real
isf (4)                               integer
ifail                                 integer