ARMALIK Call
computes the log likelihood and
residuals for an ARMA model
 CALL ARMALIK( lnl, resid, std, x, phi,
theta);
The inputs to the ARMALIK subroutine are as follows:
 x
 is an n ×1 or 1 ×n matrix containing
values of the time series (assuming mean zero).
 phi
 is a 1 ×(p+1) matrix containing
the autoregressive parameter values.
The first element is assumed to have the value 1.
 theta
 is a 1 ×(q+1) matrix containing the movingaverage
parameter values. The first element is assumed to have the
value 1.
The ARMALIK subroutine returns the following values:
 lnl
 specifies a 3 ×1 matrix containing the log
likelihood concentrated with respect to the innovation
variance; the estimate of the innovation variance (the
unconditional sum of squares divided by n); and the
log of the determinant of the variance matrix, which
is standardized to unit variance for the innovations.
 resid
 specifies an n ×1 matrix
containing the standardized residuals.
These values are uncorrelated with a constant variance
if the specified ARMA model is the correct one.
 std
 specifies an n ×1 matrix containing the scale factors used
to standardize the residuals.
The actual residuals from the onestepahead predictions using
the past values can be computed as std#resid.
The ARMALIK subroutine computes the concentrated
log likelihood function for an ARMA model.
The unconditional sum of squares is readily available,
as are the onestepahead prediction residuals.
Factors that can be used to generate confidence limits associated
with prediction from a finite past sample are also returned.
The notational conventions for the ARMALIK subroutine
are the same as those used by the ARMACOV subroutine.
See the description of the ARMACOV call for the model employed.
In addition, the condition
for z<1 should
be satisfied to guard against floatingpoint overflow.
If the column vector x contains n values
of a time series and the variance matrix is denoted
, where is the variance
of the innovations, then, up to additive constants, the log
likelihood, concentrated with respect to , is
The matrix V is a function
of the specified ARMA model parameters.
If L is the lower Cholesky root of V (that is,
V = LL'), then the standardized
residuals are computed as resid= L^{1}x.
The elements of std are the diagonal elements of L.
The variance estimate is x'V^{1}x/n, and
the log determinant is .See Ansley (1979) for further details.
To compute the loglikelihood for the model

y_{t}  y_{t}1 + 0.25y_{t2} = e_{t} + 0.5e_{t1}
use the following IML code:
proc iml;
phi={ 1 1 0.25} ;
theta={ 1 0.5} ;
x={ 1 2 3 4 5} ;
call armalik(lnl,resid,std,x,phi,theta);
print lnl resid std;
The printed output is
LNL RESID STD
0.822608 0.4057513 2.4645637
0.8721154 0.9198158 1.2330147
2.3293833 0.8417343 1.0419028
1.0854175 1.0098042
1.2096421 1.0024125
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.