# An Algorithm for the Fisher Information Matrix of a VARMAX Process
## 4 A Sufficient Condition of Identifiability  
The conditions for the identifiability of a VARMAX model have been studied by [25], among others. They are repeated later. A clear indication of identifiability is the invertibility of the Fisher information matrix. In this section, we derive a sufficient condition of invertibility for the case where \(m=n\) and we enforce that the condition is not necessary, as could be deduced from the wrong arguments of [17]. For an ARMAX model, i.e., when \(n=m=1\), [14] proved that a necessary and sufficient condition for invertibility of the Fisher information matrix is that the three polynomials have no common root. Indeed, as will be seen, the result of [14] cannot be extended to vector processes. We have already been reminded of the concepts of reciprocal polynomials and eigenvalues of matrix polynomials. There remains to define what are a unimodular polynomial matrix and the common left divisor of matrix polynomials, see, e.g., [39]. A unimodular polynomial matrix \(U(z)\) is a polynomial square matrix such that its determinant is a non-zero constant, instead of being a polynomial in \(z\). Consequently, \(U^{-1}(z)\) is also a polynomial matrix. In general, the inverse of a square polynomial matrix is a matrix with rational elements, not polynomial elements. Three matrix polynomials \(A(z)\), \(B(z)\), and \(C(z)\) with the same number of rows \(n\) have a common left divisor if there exist \(n\times n\) polynomial matrices \(F(z)\), \(A^{\prime}(z)\), \(B^{\prime}(z)\), and \(C^{\prime}(z)\), such that \(A(z)=F(z)A^{\prime}(z)\), \(B(z)=F(z)B^{\prime}(z)\), and \(C(z)=F(z)C^{\prime}(z)\). In matrix form, we can write \((A(z)\ B(z)\ C(z))=F(z)(A^{\prime}(z)\ B^{\prime}(z)\ C^{\prime}(z))\). Then \((A(z)\ B(z)\ C(z))\) is called a right multiple of \(F(z)\). A left divisor \(F(z)\) is called the greatest common left divisor of \((A(z)\ B(z)\ C(z))\) if it is a right multiple of all left divisors. Multiplying a greatest common left divisor to the right by any unimodular polynomial matrix yields another greatest common left divisor. As indicated by (Section 2.2, [25]), a greatest common left divisor can be constructed by elementary column operations: interchange any two columns, multiply any column by a real number different from 0, and add a polynomial multiple of any column to any column. Also, it corresponds to the right multiplication of \((A(z)\ B(z)\ C(z))\) by an appropriate unimodular matrix. The concept can also be defined for rectangular matrices with the same number of rows although we will not consider that generalization. Lemma 1: _Assume that \(A(z)\), \(B(z)\), and \(C(z)\) have the prescribed degrees, respectively, \(p\), \(q\), and \(r-1\), with \(\det(A(z))\
eq 0\) and \(\det(B(z))\
eq 0\), for \(z\) in the unit circle of \(\mathbb{C}\), and \(\Sigma\) is non-singular. Assume also that \(x_{t}\) has an absolutely continuous spectrum with spectral-density non-zero on a set of positive measure on \((-\pi,\pi]\). Then a necessary and sufficient condition of identifiability of a stationary VARMAX process satisfying (1) is (i) \(A(z)\), \(B(z)\), and \(C(z)\) have \(I_{n}\) as greatest common left divisor, (ii) the matrix \((A_{p}\ B_{q}\ C_{r-1})\) has rank \(n\)._Proof. : Since \(\Sigma\) is non-singular and given the assumptions on \(A(z)\) and \(B(z)\), the conditions (8a), (8b), and (8c) of [40] are satisfied for \(A^{-1}(z)B(z)\), and the lemma is a special case of (Theorem 2, [40]) in the case (iii)\({}_{2}\). Since identifiability is equivalent to the inversibility of the Fisher information matrix, we are interested in finding a sufficient condition for identifiability. Equivalently, we are looking for a simple necessary condition for the lack of identifiability or the singularity of the Fisher information matrix, when \(m=n\). We can state the following theorem.