Language Reference: RATIO Function :: SAS/IML(R) 9.22 User's Guide

Language Reference

RATIO Function

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

The RATIO function divides matrix polynomials.

The arguments to the RATIO function are as follows:

ar: is an $\text{[math]}$ matrix that represents a matrix polynomial generating function, $\text{[math]}$ , in the variable $\text{[math]}$ . The first $\text{[math]}$ submatrix represents the constant term and must be nonsingular, the second $\text{[math]}$ submatrix represents the first-order coefficients, and so on.
ma: is an $\text{[math]}$ matrix that represents a matrix polynomial generating function, $\text{[math]}$ , in the variable $\text{[math]}$ . The first $\text{[math]}$ submatrix represents the constant term, the second $\text{[math]}$ submatrix represents the first-order term, and so on.
terms: is a scalar that contains the number of terms to be computed, denoted by $\text{[math]}$ in the following discussion. This value must be positive.
dim: is a scalar that contains the value of $\text{[math]}$ , a dimension of the matrix ma. 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.

The value returned is an $\text{[math]}$ matrix that contains the terms of $\text{[math]}$ considered as a matrix of rational functions in $\text{[math]}$ that have been expanded as power series.

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

$\text{[math]}$

where $\text{[math]}$ and $\text{[math]}$ are matrix polynomial operators whose first matrix coefficients are identity matrices. The RATIO function can be used to compute a truncated form of $\text{[math]}$ for the equivalent infinite-order model

$\text{[math]}$

The RATIO function can also be employed for simple scalar polynomial division, giving a truncated form of $\text{[math]}$ for two scalar polynomials $\text{[math]}$ and $\text{[math]}$ .

The cumulative sum of the elements of a column vector $\text{[math]}$ can be obtained by using the following statement:

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

Consider the following example for multivariate ARMA(1,1):

   ar = {1 0 -0.5  2,
         0 1  3   -0.8};
   ma = {1 0  0.9  0.7,
         0 1  2   -0.4};
   psi = ratio(ar, ma, 4, 2);

The matrix produced is as follows:

           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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