# An Algorithm for the Fisher Information Matrix of a VARMAX Process
## 1 Introduction  
In this paper, we present an algorithm for the symbolic software package Mathematica to compute the asymptotic Fisher information matrix for a multivariate time series, more precisely a controlled vector autoregressive moving average stationary process, or VARMAX process. There are several ways to obtain the Fisher information matrix for time series models. The most used procedure is computing the Hessian of the Gaussian log-likelihood when a model is fitted to the time series, usually after non-linear optimization (since the model is non-linear with respect to the parameters except for the pure VAR case), e.g., [1]. Another much less used method consists of determining the exact (Gaussian) Fisher information, see [2; 3; 4; 5; 6; 7; 8]. This is nice when the data are available but, as [9] indicates, it can be useful to know the matrix at a tentative parameter point before obtaining the data, in order to determine the series length necessary to reach some specific accuracy for the estimators of the parameters. Then, an algorithm for the computation of the so-called asymptotic Fisher information matrix at a given parameter point is what is needed. It was provided in the form of an algebraic expression for a handful of small-order (1 or 2) ARMA models by [PERSON] and [PERSON] in 1970, see [10]. Otherwise, there is a vast literature for scalar models, i.e., simple ARMA models, e.g., [11; 12], seasonal ARMA models [13], ARMAX models [14], single input single output models [15], and for vector models, i.e., VARMA models [16; 17] and VARMAX models [18; 19]. We introduce a new algorithm well-suited to symbolic computation software packages (here Mathematica) and we present an illustrative example. Then, we discuss the invertibility of the Fisher information matrices and provide other examples. The proposed algorithm differs from the works of [PERSON] [3; 20], see also [PERSON] and [PERSON] [21], who developed their methods for VARMAX models, in the sense that they are based on an approximation of the exact Fisher information. Here, we follow [PERSON]'s approach [22] of expressing the asymptotic information matrix by computing acircular integral of a rational function, related to the autoregressive and moving average operators. That was the way used by [PERSON] for VARMA models, see [16], with the difference that we make the effective computation and that we consider VARMAX models. The more fundamental source is [19], who obtained an explicit expression for the Fisher information matrix of a VARMAX model under the form of a sum of two integrals while [18] contains expressions for each element of that matrix. As a matter of fact, we first need to generalize [19], which is developed for the special case where the number of explanatory variables equals the number of variables of interest and, moreover, the coefficient matrix at lag 0 is the identity matrix. These restrictions were explained by the revealed objective of the paper which was to exploit tensor Sylvester matrices and try to extend to VARMAX models the resultant property of the Fisher information matrix wrongly claimed for VARMA models by [17]. Indeed, in [17] (p. 1986), it is written _\"[a] representation [based on tensor Sylvester matrices] is more appropriate...to prove a possible resultant property of the Fisher information matrix of a VARMAX process\"_. Such a tensor Sylvester matrix is associated with two matrix polynomials and it becomes singular if, and only if, the two matrix polynomials have at least one common eigenvalue, see [23]. In [17], it is said that the Fisher information matrix of a VARMA process becomes invertible if, and only if, the tensor Sylvester matrix is invertible, in other words, if, and only if, the autoregressive and moving average matrix polynomials of the VARMA process have no common eigenvalue. This is called the Sylvester resultant property. In [17], the Sylvester resultant property is shown for a scalar ARMAX process but it is no longer true, in general, for other than scalar ARMA or ARMAX processes.