PRODUCT Function

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PRODUCT Function

multiplies matrices of polynomials

PRODUCT( a, b<, dim>)

The inputs to the PRODUCT function are as follows:

a: is an m ×(ns) numeric matrix. The first m ×n submatrix contains the constant terms of the polynomials, the second m ×n submatrix contains the first order terms, and so on.
b: is an n ×(pt) matrix. The first n ×p submatrix contains the constant terms of the polynomials, the second n ×p submatrix contains the first order terms, and so on.
dim: is a 1 ×1 matrix, with value p>0. The value of this matrix is used to set p above. If omitted, the value of p is set to 1.

The PRODUCT function multiplies matrices of polynomials. The value returned is the m ×(p(s+t-1)) matrix of the polynomial products. The first m ×p submatrix contains the constant terms, the second m ×p submatrix contains the first order terms, and so on.

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

$\Phi_1(B) \Phi_2(B) Y_t = \Theta_1(B) \Theta_2(B) \epsilon_t$

where $\Phi_1(B)$ , $\Phi_2(B)$ , $\Theta_1(B)$ , and $\Theta_2(B)$ are matrix polynomial operators whose first matrix coefficients are identity matrices. Often $\Phi_2(B)$ and $\Theta_2(B)$ represent seasonal components that are isolated in the modeling process but multiplied with the other operators when forming predictors or estimating parameters. The RATIO function is often employed in a time series context as well.

For example, the statement

   r=product({1 2 3 4,
              5 6 7 8},
             {1 2 3,
              4 5 6}, 1);

produces the result

                R             2 rows      4 cols    (numeric)

                           9        31        41        33
                          29        79       105        69

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