Polynomial Functions

RATS provides a suite of functions for manipulating polynomials. These are represented as a VECTOR of the form

\(\left\| {{\kern 1pt} {\kern 1pt} v\left( 1 \right), \ldots ,v\left( {n + 1} \right){\kern 1pt} {\kern 1pt} } \right\|\) ,

which denotes the polynomial:

\(v\left( 1 \right) + v\left( 2 \right)x + \ldots + v\left( {n + 1} \right){x^n}\)

%eqnlagpoly(eqn,s)	Extract lag polynomial from an equation
%polyadd(V1,V2)	Add two polynomials
%polycxroots(V)	Complex roots of a polynomial
%polydiv(V1,V2,d)	Divide two polynomials
%polymult(V1,V2)	Multiply two polynomials
%polyroots(V)	(Real) roots of a polynomial
%polysub(V1,V2)	Subtract two polynomials
%polyvalue(V1,x)	Evaluate a polynomial at a given value for x

The most important of these is %EQNLAGPOLY(eqn,series), which pulls a lag polynomial out of an equation or regression (the most recent regression if eqn=0). If series is the dependent variable, it extracts the standard normalization for the autoregression polynomial, that is, 1–lag polynomial from the right side.

For example, this implements an ARDL (autoregressive-distributed lag) model and does calculations of the long-run effect of x (LOGY) on y (LOGC) and the first eight terms of the expansion of the y on x.

linreg(define=ardl) logc

# logc{1 to 3} logy{0 to 3} seas{0 to -3}

compute arpoly = %eqnlagpoly(0,logc)

compute dlpoly = %eqnlagpoly(0,logy)

* Compute the long run effect

compute ardllt=%polyvalue(dlpoly,1)/%polyvalue(arpoly,1)

* Compute 8 lags of the expansion (degree 7 polynomial)

compute ardlest=%polydiv(dlpoly,arpoly,7)