g13adf

Purpose

Synopsis

[par,rv,isf,ifail] = g13adf(mr,r,xv<,ifail>)

Description

 
 Preliminary estimates of the p non-seasonal autoregressive 
 parameters (phi) ,(phi) ,...,(phi)  and the q non-seasonal moving
                 1      2          p                        
 average parameters (theta) ,(theta) ,...,(theta)  may be obtained
                           1        2            q        
 from the sample autocorrelations relating to lags 1 to p+q, i.e.,
                                        d       D              
 r ,...,r   , of the differenced (nabla) (nabla) x  where x  is 
  1      p+q                                    s t        t   
 assumed to follow a (possibly) seasonal ARIMA model.
 
 Taking r =1 and r  =r , the (phi) , for i=1,2,...,p are the 
         0        -k  k           i                         
 solutions to the equations
 
  r     (phi) +r     (phi) +...+r     (phi) =r   ,fori=1,2,...,p.
   q+i-1     1  q+i-2     2      q+i-p     p  q+i
 
 The (theta) , for j=1,2,...,q are obtained from the solutions to 
            j                                                    
 the equations
 
 
 c =(tau) (tau) +(tau) (tau)   +.....+(tau)   (tau) ,forj
  j      0     j      1     j+1            q+j     q
 =0,1,...,q
 
 
 (Cramer Wold-factorization) by setting
 
                                    (tau) 
                                         j
                         (theta) =- ------
                                j   (tau) 
                                         0
 
 where c  are the 'covariances' modified in a 2-stage process by 
        j                                                       
 the autoregressive parameters.
 
 Stage 1:
 
     d =r -(phi) r   -...-(phi) r   ,     for j=0,1,...,q;
      j  j      1 j-1          p j-p
 
     d =0,                                for j=q+1,q+2,...,p+q.
      j                   
 
 Stage 2:
 
     c =d -(phi) d   -(phi) d   -...-(phi) d   ,  for j=0,1,...,q.
      j  j      1 j+1      2 j+2          p j+p
 
 The P seasonal autoregressive parameters (Phi) ,(Phi) ,...,(Phi) 
                                               1      2          P
 and the Q seasonal moving average parameters 
 (Theta) ,(Theta) ,...,(Theta)  are estimated in the same way as 
        1        2            Q                                 
 the non-seasonal parameters, but each r  is replaced in the 
                                        j                   
 calculation by r   , where s is the seasonal period.
                 s*j                          
 
 An estimate of the residual variance is obtained by successively 
 reducing the sample variance, first for non-seasonal, and then 
 for seasonal, parameter estimates. If moving average parameters 
 are estimated, the variance is reduced by a multiplying factor of
      2                     
 (tau) , but otherwise by c .
      0                    0
 

Parameters

g13adf

Required Input Arguments:

mr (7)                                integer
r (:)                                 real
xv                                    real

Optional Input Arguments:                       <Default>

ifail                                 integer  -1

Output Arguments:

par (:)                               real
rv                                    real
isf (4)                               integer
ifail                                 integer