# An Algorithm for the Fisher Information Matrix of a VARMAX Process
## 5 Numerical Experiments
\begin{split} Row3=\left(\frac{4}{105^{*}},\frac{38}{2625^{*}}, \frac{16}{3},\frac{152}{75},0,0,-4,-\frac{38}{25},0,0,-\frac{4}{3},-\frac{38}{75 }\right)\\ Row4=\left(\frac{152}{2625^{*}},\frac{1444}{65625^{*}},\frac{152}{75 },\frac{13276}{1875},0,0,-\frac{38}{25},-\frac{3319}{625},0,0,-\frac{38}{75}, -\frac{3319}{1875}\right)\\ Row7=\left(-\frac{4}{7},-\frac{38}{175},-4,-\frac{38}{25},0,0,4, \frac{38}{25},0,0,0,0\right)\\ Row8=\left(-\frac{152}{175^{*}},-\frac{1444}{4375^{*}},-\frac{38}{2 5^{*}},-\frac{3319}{625},0,0,\frac{38}{25},\frac{3319}{625},0,0,0\right)\\ Row11=\left(\frac{8}{15},\frac{76}{375^{*}},-\frac{4}{3},-\frac{38} {75},0,0,0,0,0,\frac{4}{3},\frac{38}{75}\right)\\ Row12=\left(\frac{304}{375^{*}},\frac{2888}{9375^{*}},-\frac{38} {75},-\frac{3319}{1875},0,0,0,0,0,\frac{38}{75},\frac{3319}{1875}\right)\end{split} \]  
_and it appears that \(\text{Row}3=-\text{Row}11-\text{Row}7\) and \(\text{Row}4=-\text{Row}8-\text{Row}12\)._  
**Example 3**. : _Let \(A\), \(B\), and \(C\) be defined by_  
\[A=\left( \begin{array}{cc}0.6&0.2\\ 0.4&-0.6\\ \end{array} \right),\quad B=\left( \begin{array}{cc}0.5&0.76\\ 0.25&-0.5\\ \end{array} \right),\quad C=\left( \begin{array}{cc}0.7&0.1\\ -0.5&-0.7\\ \end{array} \right). \tag{11}\]  
_This example is a generalization of the bivariate VARMA(1,1) model in (Example 1, [17]). There, the two matrix polynomials had the same two eigenvalues. In [17], it was concluded, wrongly, that the Fisher information matrix should be singular although the numerical computations in Matlab lead to the smallest eigenvalue equal to \(0.0067\). This incoherency was explained by numerical inaccuracy. In [24], [PERSON] gave a second view to that example and discovered that the necessity and sufficiency of invertibility of the Fisher information matrix is only a necessary condition. For the VARMAX model related to (11), the three matrix polynomials have the same eigenvalues \(\pm 5/\sqrt{11}\). Exact computations with Mathematica indicate, however, that the determinant of the \(12\times 12\) Fisher information matrix is strictly positive and that the smallest eigenvalue is \(0.0919\). The inverse of the asymptotic Fisher information matrix is shown in Figure 6. This example provides a counter-example that the sufficient condition in Theorem 1 is not necessary._  
**Example 4**. : _This example is based on the example in (Section 4.1, [7]) so with \(n=2\), \(m=3\), \(p=q=r=1\), \(\Sigma=l_{2}\), and \(R_{x}(z)=I_{3}\). Let \(A_{1}=O_{2\times 2}\), \(C_{0}=C_{1}=O_{2\times 3}\), and \(B_{1}\) be defined by:_  
\[B_{1}=\left( \begin{array}{cc}6/5&1/2\\ -7/5&-1/5\\ \end{array} \right). \tag{12}\]  
_The total number of parameters is \(4+6+6+4=20\). In (Section 4.1, [7]), the exact Fisher information matrix was computed and compared to those obtained using the E4 Toolbox for Matlab,  
Figure 6: Output of the inverse of the Fisher information matrix for Example 3.  
see [5; 44], but (Appendix B, [7]) contains the results for the asymptotic Fisher information matrix obtained using individual formulas in [18] for each of the 10 different blocks, the computation of the integrals being performed using the algorithm in [33]. Here, we do not bother with complex individual expressions and the computation of integrals. We simply use the notebook described in Figure 1 and adapt the model specifications. See the resulting information matrix in Figure 7.