# An Algorithm for the Fisher Information Matrix of a VARMAX Process
## 4 A Sufficient Condition of Identifiability
Theorem 1: _Under the assumptions of Lemma 1 and if \(m=n\), a sufficient condition of invertibility of the Fisher information matrix, \(\mathcal{F}(\theta)\), associated with the VARMAX model with the matrix polynomials \(A(z),C(z)\), and \(B(z)\) of degree \(p,r-1,q\), respectively, is that the reciprocal matrix polynomials \(\tilde{A}(z)\), \(\tilde{B}(z)\), and \(\tilde{C}(z)\) have no common eigenvalue._  
Proof. : If the lack of identifiability occurs because (i) in Lemma 1 is not satisfied, that means that there exists a non-unimodular polynomial matrix \(F(z)\), i.e., with \(\det(F(z))\) different from a constant and matrices \(A^{\prime}(z)\), \(B^{\prime}(z)\), and \(C^{\prime}(z)\), such that \((A(z)\ B(z)\ C(z))=F(z)(A^{\prime}(z)\ B^{\prime}(z)\ C^{\prime}(z))\). Since the three polynomial matrices are square, we can consider their determinant stored in a row vector for which  
\[(\det(A(z))\ \det(B(z))\ \det(C(z)))=\det(F(z))(\det(A^{\prime}(z))\ \det(B^{ \prime}(z))\ \det(C^{\prime}(z))),\]  
where \(\det(F(z))\) is a polynomial different from a constant. Hence, the equations \(\det(A(z))=0\), \(\det(B(z))=0\), and \(\det(C(z))=0\) have at least one common root. Consequently, the matrix polynomials \(A(z)\), \(B(z)\), and \(C(z)\) have at least one common eigenvalue. The same is also true for the reciprocal matrix polynomials \(\tilde{A}(z)\), \(\tilde{B}(z)\), and \(\tilde{C}(z)\). Now, if the lack of identifiability occurs because (ii) in Lemma 1 is not satisfied, that means that the matrix \((A_{p}\ B_{q}\ C_{r-1})\) has a rank strictly smaller than \(n\). Consequently, row \(n\), say, of that matrix is a linear combination of the other \(n-1\) rows and the matrices \(A_{p}\), \(B_{q}\), and \(C_{r-1}\) do not have full rank. Hence the determinants \(\det(A(z))\), \(\det(B(z))\), and \(\det(C(z))\) are polynomials of degree strictly smaller than, respectively, \(np\), \(nq\), and \(n(r-1)\). Each of the reciprocal matrix polynomials \(\tilde{A}(z)\), \(\tilde{B}(z)\), and \(\tilde{C}(z)\) have at least an eigenvalue equal to \(0\), hence at least one common eigenvalue. Note that [17] has implicitly assumed that the determinants of \(A(z)\) and \(B(z)\) have their maximum degrees. For a discussion of identifiability without co-primeness, see [41]. For VARMA models, Ref. [24] has shown that it is not true to say, as [17] did, that the conditions in Theorem 1 are also necessary for invertibility of the Fisher information matrix. The approach of [17] used tensor Sylvester matrices which are a generalization of Sylvester matrices for the case of matrix polynomials instead of scalar polynomials. Consequently, the expectations of [19] mentioned in the second paragraph of Section 1 are not met for VARMAX models. Moreover, since there are three matrix polynomials and \(C(z)\) does not need to be a square matrix, the tensor Sylvester matrices are not useful. It is also not useful to detail the alternative representations to (6) developed in (Sections 2.4 to 2.7, [19]) using the tensor Sylvester matrices. Let us just say that these representations seem to be correct in the sense that we have checked on all our examples for which \(m=n\) that the Fisher information matrix obtained using them is identical to what is obtained using (7)-(9).