# An Algorithm for the Fisher Information Matrix of a VARMAX Process
## 6 Conclusions  
In this paper, we have proposed for the first time an exact method to compute the asymptotic Fisher information matrix of VARMAX processes, an algorithm for symbolic computation software, and an implementation for Mathematica. It is based on an extension of the method proposed in [19].  
Therefore, we could not compare it with other algorithms in the general case. Nevertheless, we could compare it to the special case of an ARMAX process where there exists alternative algorithms.  
The open-source packages Octave and Maxima should be able to do the job since their symbolic integration procedure can work on matrices of functions, but they do not work presently. More developments would be needed to circumvent their failing integration.  
We also provide a simple sufficient condition of invertibility of the asymptotic Fisher information matrix and Example 3 shows that the condition is not necessary.  
We did not examine the complexity of our algorithm because the concept is not well-established for symbolic computation. Let us just say that, at this stage, it will take much more running time, as seen in the special case discussed in Section 3. For instance, the timings given for the ARMAX model of Section 3 show that the numerical method is considerably faster. Also, the program implementing the algorithm relies on the capabilities of the symbolic computation package and possible hardware and software limitations. This is the price to pay for the ability to provide exact results, at least when the matrix entries are given as fractions, not as decimal numbers. We believe, however, that the fact that, when it works at least, the notebook that we have produced the result for an arbitrary VARMAX model.  
We should conclude with possible extensions of our results. There have been papers on the asymptotic information matrix for extended models like ARMA models with periodic coefficients [45; 46], state-space models [46], Markov switching VARMA models [47], or VARFIMA models [48]. It is possible that the approach described in this paper can be extended to these extended models.  
Figure 7: Output of the Fisher information matrix for Example 4.  
**Author Contributions:** Conceptualization, A.K. and G.M.; Methodology, A.K. and G.M.; Software, G.M.; Validation, A.K. and G.M.; Formal Analysis, G.M.; Investigation, G.M.; Resources, G.M.; Data Curation, A.K. and G.M.; Writing--Original Draft Preparation, G.M.; Writing--Review and Editing, A.K. and G.M.; Visualization, A.K. and G.M.; Supervision, G.M.; Project Administration, G.M.; Funding Acquisition, nobody. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Data Availability Statement:** Not applicable.  
**Acknowledgments:** We thank [PERSON] for his comments on the errors in [17] and [PERSON] for his comments on possible generalizations, and the two referees for their very useful remarks.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**Abbreviations**  
The following abbreviations are used in this manuscript:  
AR autoregressive ARMA autoregressive moving average ARMAX autoregressive moving average with exogenous variables PDF portable document format SISO single input single output VAR vector autoregressive VARMA vector autoregressive moving average VARMAX vector autoregressive moving average with exogenous variables