# An Algorithm for the Fisher Information Matrix of a VARMAX Process
## 1 Introduction
Indeed, in [24] [PERSON] indicated the error in the claim and explained why the example in [17] was wrong. In the present paper, we correct the assertion stated in [17] for VARMA processes, and we extend the \"if\" part to a class of VARMAX processes. As will be shown, the \"only if\" part of that result is, however, wrong when the dimension of the process is larger than 1. We will also compare our algorithms with other algorithms proposed for special cases in the literature. Although the results are less powerful, they are what is needed in practice: a sufficient condition of invertibility of the Fisher information matrix, indicating a possible lack of identifiability so that parameter estimation is risky. A necessary condition that should involve more information about the matrix polynomials is not as useful. Consider the vector stochastic difference equation representation of a linear system of order \((p,r,q)\) of the process \(\{y_{t},t\in\mathbb{Z}\}\), \(\mathbb{Z}\) is the set of integers  
\[\sum_{j=0}^{p}A_{j}y_{t-j}=\sum_{j=0}^{r-1}C_{j}x_{t-j}+\sum_{j=0}^{q}B_{j} \varepsilon_{t-j},\quad t\in\mathbb{Z} \tag{1}\]  
where \(y_{t},x_{t},\) and \(\varepsilon_{t}\) are, respectively, the \(n\)-dimensional stochastic observed output, the \(m\)-dimensional observed input, and the \(n\)-dimensional unobserved errors, and where \(A_{j}\in\mathbb{R}^{n\times n}\), \(j=1,\ldots,p\), \(C_{j}\in\mathbb{R}^{n\times m}\), \(j=0,\ldots,r-1\), and \(B_{j}\in\mathbb{R}^{n\times n}\), \(j=1,\ldots,q\), are associated parameter matrices. We additionally assume \(A_{0}=B_{0}=I_{n}\), the \(n\times n\) identity matrix. We also assume that \(C_{0}\) is an invertible matrix. In the examples, we will sometimes take \(C_{0}=I_{n}\) fixed instead of being a matrix of parameters so that the maximum lag \(r-1\) is replaced by \(r\). Note that the absence of lag induced by (1) is purely conventional. For example, a lagged effect of \(x_{t}^{\prime}\) on \(y_{t}\) can be produced by defining \(x_{t}=x_{t-1}^{\prime}\). More precisely, we suppose that \(x_{t}\) is stochastic and that \((y_{t},x_{t})\) is a Gaussian stochastic process. The error \(\{\varepsilon_{t},t\in\mathbb{Z}\}\) is a sequence of independent zero-mean \(n\)-dimensional Gaussian random variables with a strictly positive definite covariance matrix \(\Sigma\). We denote this by \(\Sigma\succ 0\). We denote the transposition by \({}^{T}\), the complex conjugate transposition by \({}^{*}\), and the mathematical expectation is \(\mathbb{E}\). We assume \(\mathbb{E}\{x_{t}\varepsilon_{t^{\prime}}^{T}\}=0\), for all \(t,t^{\prime}\). We denote \(z\) as the backward shift operator, for example, \(zx_{t}=x_{t-1}\). Then (1) can be written as  
\[A(z)y_{t}=C(z)x_{t}+B(z)\varepsilon_{t}, \tag{2}\]where  
\[A(z)=\sum_{j=0}^{p}A_{j}z^{j},\quad C(z)=\sum_{j=0}^{r-1}C_{j}z^{j},\quad B(z)= \sum_{j=0}^{q}B_{j}z^{j} \tag{3}\]  
are the associated matrix polynomials, where \(z\in\mathbb{C}\) (the duplicate use of \(z\) as an operator and as a complex variable is usual in the signal and time series literature, e.g., [25; 26; 27]). The assumptions \(\det(A(z))\
eq 0\) for all \(|z|\leq 1\) (causality) and \(\det(B(z))\
eq 0\) for all \(|z|\leq 1\) (invertibility) are imposed so that the eigenvalues of the matrix polynomials \(A(z)\) and \(B(z)\) will be outside the unit circle, see [25]. Remind that the eigenvalues of an \(n\times n\) matrix polynomial \(D(z)\) are the roots of the equation \(\det(D(z))=0\) (see [28], p. 14). We will also use the reciprocal polynomial of \(D(z)\), with degree \(d\), say. It is \(\tilde{D}(z)=z^{d}D(z^{-1})\). If \(D(0)\) is full-rank, there are \(dn\) eigenvalues for \(\tilde{D}(z)\). The eigenvalues of \(D(z)\) are the inverse of those of \(\tilde{D}(z)\), provided the missing eigenvalues of the former are treated as being at \(\infty\). Note that there are restrictions in the VARMAX model being considered. The dimensions of the unobserved errors \(\varepsilon_{t}\) and the observed output \(y_{t}\) are the same.