A note on autoregressive-moving average identification

Abstract

are left prime, that h-1 g is analytic and nonsingular for I z I < 1 and that G is nonsingular. Nevertheless (1) is not uniquely specified. However (Hannan, 1969), if, for prescribed q, r, [A(q) *. B(r)] is of rankp then (1) is uniquely specified. This will be called the rank condition. Some structures (1) cannot be brought to a form where this condition is satisfied (Hannan, 1971), so that the condition is overidentifying. The set, Cq,r, of all structures satisfying the rank condition for given q and r is mapped, by using the elements of A(j) and B(j) and the on and above diagonal elements of C, onto an open set in Euclidean space, if it is required that h-1 g is nonsingular for I z I < 1, and hence constitutes an analytic manifold. For q and r fixed the set of structures (1) not in Cq,r is evidently of lower dimension than Cq,r Tuan (1978, end of ? 1) states a number of objections to the use Of Cq,r and we wish to discuss these, and the general problem of parameterizing (1). For this last purpose Tuan (1978) used a family of canonical forms, called by him the quasiautoregressive-moving average representation. With any structure (1) is associated a set of integers mj (j = 1, ...,p) which determine the form of this representation (Tuan, 1978, p. 101). Thus given any structure (1) there is associated a set of mj and a matrix of polynomials u(z), with unit determinant, such that ug and uh are in the canonical form. Of course the matrix function u is an extremely complicated function of g and h. Using K as a symbol for the m1 (j = 1, .. . p), we may also map the set CK of all structures with these mj into an open set in Euclidean space, if h-' g is nonsingular for I z I < 1. Now there is no overidentification. However, there is a major problem if there are prior constraints imposed on (1) for it seems almost impossible, in general, to translate these constraints into constraints on the canonical forms because u is such a complicated function of g and h. On the other hand, the set of all structures (1) for given q and r is very complicated to parameterize. However, for the reason mentioned at the end of the previous paragraph, almost all of that set is constituted by Cq,r, which is easily parameterized. Tuan (1978) objects to Cq,r because of overidentification and because constraints may cause the rank condition to fail, and for other reasons to be discussed below. It is most unlikely that constraints would make [A (q) .B(r)] identically of rank less than p,