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## 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


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