## RATIO Function

**divides matrix polynomials**
returns a matrix containing the terms of
considered as a matrix of rational
functions in *B* that have been expanded as power series

**RATIO(** *ar, ma, terms<, dim>***)**

The inputs to the RATIO function are as follows:
*ar*
- is an
*n* ×(*ns*) matrix representing a matrix polynomial
generating function, , in the variable *B*.
The first *n* ×*n* submatrix represents the constant
term and must be nonsingular, the second *n* ×*n*
submatrix represents the first order coefficients, and so on.

*ma*
- is an
*n* ×(*mt*) matrix representing a matrix polynomial
generating function, , in the variable *B*.
The first *n* ×*m* submatrix represents the
constant term, the second *n* ×*m* submatrix
represents the first order term, and so on.

*terms*
- is a scalar containing the number of terms to be
computed, denoted by
*r* in the discussion below.
This value must be positive.

*dim*
- is a scalar containing the value of
*m* above.
The default value is 1.

The RATIO function multiplies a matrix of polynomials
by the inverse of another matrix of polynomials.
It is useful for expressing univariate and multivariate ARMA
models in pure moving-average or pure autoregressive forms.

Note that the order of the first two arguments is
reversed from the corresponding PROC MATRIX function.

The value returned is an *n* ×(*mr*) matrix containing the
terms of considered as a matrix of
rational functions in *B* that have been expanded as power series.

**Note:**
The RATIO function can be used to consolidate the matrix operators
employed in a multivariate time-series model of the form

where and are matrix polynomial
operators whose first matrix coefficients are identity matrices.
The RATIO function can be used to compute a truncated
form of
for the equivalent infinite order model

The RATIO function can also be employed for simple
scalar polynomial division, giving a truncated
form of for two scalar
polynomials and .The cumulative sum of the elements of a
column vector **x** can be obtained using

ratio({ 1 -1} ,x,ncol(x));

Consider the following example for multivariate ARMA(1,1):
ar={1 0 -.5 2,
0 1 3 -.8};
ma={1 0 .9 .7,
0 1 2 -.4};
psi=ratio(ar,ma,4,2);

The matrix produced in
PSI
1 0 1.4 -1.3 2.7 -1.45 11.35
: -9.165
0 1 -1 0.4 -5 4.22 -12.1
: 7.726

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.