## 4 Counterexamples  
In Counterexample 1, the conditions in Theorem 1 hold, and thus the VARMA model is identified with MFD. However, Condition IV in [1] does not hold. Therefore, it is not necessary for identifiability in the SFD case. We remark that rank \(A_{r}<n\), but \(\text{rank}(A_{r}\overset{\cdot}{:}B_{q})=n\)  
**Counterexample 1**. : _Consider the VARMA(3, 1) model with \(A_{0}=B_{0}=I\),_  
\[A_{1}= \begin{pmatrix}0&-1/2\\ -1/2&0\end{pmatrix} ,A_{2}= \begin{pmatrix}-1/4&0\\ 0&-1/4\end{pmatrix} ,A_{3}= \begin{pmatrix}-1/2&-1/4\\ -1/4&-1/8\end{pmatrix} ,B_{1}= \begin{pmatrix}1/2&1/2\\ 1/2&0\end{pmatrix} \]  
_and \(E(\varepsilon_{t}^{\,t})=I_{n}\)._  
We have \(q=1\), \(r=3\), \(nr=6\) and we consider \(n_{1}=1\). The model is identifiable in the SFD case because  
\[\text{rank} \begin{pmatrix}K_{q-r+1}&K_{q-r+2}&\cdots K_{q+(n-1)r}\\ \vdots&\vdots&\vdots\\ K_{q}&K_{q+1}&\cdots&K_{q+nr-1}\end{pmatrix} =\text{rank} \begin{pmatrix}K_{-1 }&K_{0}&\cdots K_{4}\\ K_{0}&K_{1}&\ldots K_{5}\\ K_{1}&K_{2}&\cdots K_{6}\end{pmatrix} =6.\]  
Note that Conditions I, ii, III, iv.1 hold. However, Condition IV is not satisfied because rank \([G\,\ldots\,F^{mp}G]=5\
eq 6\) and, as a consequence, rank \(\text{C}_{L}(F,V\,H_{1}^{t})<np\) and Condition V do not hold. Therefore, \(A_{1}\), \(A_{2}\) and \(A_{3}\) could not be uniquely calculated using the procedure in [1]. We prove that Condition v.1 holds (with MATLAB, see Appendix A) as follows:  
1. First, we computed \(C_{i}\) (\(i=0,\,1,\,2,\,3\)) by solving the Yule-Walker equations: \[C_{0}-A_{1}C_{-1}-A_{2}C_{-2}-A_{3}C_{-3}=I+B_{1}{K_{1}}^{\,t}\] \[C_{1}-A_{1}C_{0}-A_{2}C_{-1}-A_{3}C_{-2}=B_{1}\] \[C_{2}-A_{1}C_{1}-A_{2}C_{0}-A_{3}C_{-1}=0\] \[C_{3}-A_{1}C_{2}-A_{2}C_{1}-A_{3}C_{0}=0\] where \(K_{1}=B_{1}+A_{1}\). We then computed \(C_{i}=A_{1}C_{i-1}+A_{2}C_{i-2}+A_{3}C_{i-3}\). for \(i>4\). 2. Second, we obtained that rank \(\widetilde{Q}_{q-r+1,r,nr}=6\). 3. Taking into account that \(\widetilde{Q}_{q-r+1,r,L}=\text{O}_{r}(F,H)\,\,\text{C}_{L}(F,\,V^{*}(F)^{q- q+1}H_{1}^{t})\), \(\text{O}_{r}(F,H)\) is full column rank \(nr\), \(C_{L}(F,\,V^{*}(F)^{q-q+1}H_{1}^{t})\) has \(nr\) rows and rank\(\widetilde{Q}_{q-r+1,r,nr}=6=nr\), and we can affirm that rank \(C_{L}(F,\,V^{*}(F)^{r-q+1}H_{1}^{t})=nr=6\); i.e., Condition v.1 holds. Therefore, \(A_{1}\), \(A_{2}\) and \(A_{3}\) can be uniquely determined by solving (5). Regarding Condition vi, in this example, \(\theta\) has 6 columns, where \(\theta_{b}\) is the submatrix with the 2 nd and 6 th columns of \(\theta\) and rank \(\theta_{b}=2\), i.e., Condition vi holds. Taking into account that \(X= \begin{pmatrix}C_{1}\\ C_{0}\\ C_{-1}\end{pmatrix} \), from (6), we can identify \(C_{1}^{22}\text{and}C_{-1}^{22}\). As a consequence, \(C_{0}\) and \(C_{1}\) are complete. Finally, the unknown \(C_{i}^{22}\) for \(i>1\) can be identified considering that \(C_{i}=A_{1}C_{i-1}+A_{2}C_{i-2}+A_{3}C_{i-3}\) for \(i>1\). Therefore, this model is identified in the MFD case. In the following example, Condition V does not hold because \(r=q\) (Lemma 1). However, the conditions in Theorem 2 hold and therefore the VARMA model is identified with MFD. **Counterexample 2**. : _Consider the VARMA(1, 1) model where \(A_{0}=B_{0}=I\), \(A_{1}= \begin{pmatrix}-1/2&-1/4\\ 1&1/2\end{pmatrix} \), \(B_{1}= \begin{pmatrix}1&4\\ -1/4&1\end{pmatrix} \) and \(E(\varepsilon_{t}\varepsilon_{t}^{t})=I_{n}\)._We have \(r=q=1\), \(nr=2\) and we consider \(n_{1}=1\). The autocovariance matrices are  
\[C_{0}= \begin{pmatrix}4753/256&-1025/128\\ -1025/128&949/64\end{pmatrix} ,C_{1}= \begin{pmatrix}-201/32&275/64\\ 229/16&-51/32\end{pmatrix} ,C_{2}= \begin{pmatrix}-7/16&-7/4\\ 7/8&7/2\end{pmatrix} ,\] \[C_{i}=0\]  
for \(i>2\). Note that Conditions I, ii, III and iv.2 hold. Regarding Condition v.2, note that  
\[\widetilde{Q}_{1,r,nr}=\mathrm{O}_{r}(F,H)\mathrm{C}_{nr}(F,(G^{**}\sum\widetilde {B}_{0}^{\widetilde{\prime}}+FV^{**}H_{1}^{t})=\left(\widetilde{C}_{1}\quad \widetilde{C}_{2}\right)= \begin{pmatrix}-201/32&-7/16\\ 229/16&7/8\end{pmatrix}