## 3 Reconstructing Missing Blocks in Autocovariance Matrices
### Case I: \(r>q\)
Since all the columns of \(M_{0,r,\omega}\) are columns of the matrix \(M_{q-r+1,r,\omega}\), \(\mathrm{rank}\ M_{q-r+1,r,\omega}\geq\mathrm{rank}\ M_{0,r,\omega}\). Since the matrix has \(nr\) rows, \(\mathrm{rank}\ M_{q-r+1,r,\omega_{0}}=nr\). By Hanzon's Theorem, \(\mathrm{rank}\ M_{q-r+1,r,\omega}=\mathrm{rank}\ M_{q-r+1,r,\omega_{0}}=nr\). We can easily see that  
\[K_{j}=HF^{r-q-1+j}G^{*}\text{ for }j\geq q-r+1,\text{ (note that }q-r+1\leq 0)\]and \(M_{q-r+1,r,nr}=O_{r}(F,H)\ C_{nr}(F,G^{*})\). Considering that \(M_{q-r+1,r,nr}\) is full rank and \(O_{r}(F,H)\) is full column rank \(nr\), then \(C_{nr}(F,G^{*})\) is full row rank \(nr\), i.e., Condition iv.1 holds. \(\square\)  
In light of the above, we can now state the following Theorem. **Theorem 1**. : _If \(r>q\) and Conditions I, II or ii, and III hold:_  
1. _Condition iv.1 is necessary and sufficient for identifiability of the VARMA(r, q) model (1) in the SFD case._
2. _Conditions iv.1, v.1 and vi are sufficient for identifiability of the VARMA(r, q) model (1) in the MFD case._  
Proof. : For \(j\geq 0\), \(C_{q-r+1+j}=\sum_{i=0}^{\infty}K_{i+q-r+1+j}\sum\vec{K}_{i}^{t}=\sum_{i=0}^{ \infty}H\vec{F}^{i+j}\vec{G}^{*}\sum\vec{G}^{*t}(\vec{F}^{i-q+r-1})^{t}H\vec{H} =H\vec{F}(\sum_{i=0}^{\infty}\vec{F}^{i}\vec{G}^{*}\sum\vec{G}^{*t}(\vec{F}^{t }))^{t}(\vec{F}^{-q+r-1})^{t}H\vec{H}^{t}\). Therefore, denoting \(\
abla^{*}=\sum_{i=0}^{\infty}F^{i}\vec{G}^{*}\sum G^{*t}\left(F\right)^{t}\),  
\[C_{q-r+1+j}=H\,F^{j}\
abla^{*}(\vec{F}^{-q+r-1})^{t}H^{t}\ \text{for}\ j\geq 0.\]  
Taking into account that \(M_{q-r+1,r,nr}=O_{r}(F,H)\ C_{nr}(F,\ G^{*})\), \(O_{r}(F,H)\) is full column rank \(nr\) and, from Condition iv.1, we have that \(C_{nr}(F,G^{*})\) is full row rank \(nr\), then rank \(M_{q-r+1,r,nr}=nr\). From Hanzon's Theorem, rank \(M_{q-r+1,r,nr}=\text{rank}\ Q_{q-r+1,r,nr}=nr\), and therefore, under Conditions I, II or ii, III and iv.1, a VARMA model is identified with population covariance of its variables observed with SFD. Note that \((A_{1},A_{2},\ldots,A_{r})\) can be uniquely identified from the autocovariance matrices of the process, solving the following system:  
\[(A_{r},\ldots\ A_{1})Q_{q-r+1,r,nr}=\big{(}C_{q+1}C_{q+2}\ldots C_{q+nr}\big{)}.\]  
Taking into account that \(Q_{q-r+1,r,nr}=O_{r}(F,H)\ C_{nr}(F,V^{*}(F)^{q-q+1}H^{t})\), rank \(Q_{q-r+1,r,nr}=nr\) and \(O_{r}(F,H)\) is full column rank \(nr\) then  
\[C_{nr}(F,V^{*}(F)^{r-q+1}H^{t})\ \text{is full row rank}\ nr.\]  
If we substitute \(H\) for \(H_{1}\) in the autocovariances  
\[\widetilde{C}_{j+q-r+1}=HF^{j}V^{*}(F^{-q+r-1})^{t}H_{1}^{t}\ \text{for}\ j\ \geq 0.\]  
Denoting  
\[\widetilde{Q}_{q-r+1,r,L}= \begin{pmatrix}\widetilde{C}_{q-r+1}&\widetilde{C}_ {q-r+2}\cdots&\widetilde{C}_{q-r+L}\\ \widetilde{C}_{q-r+2}&\widetilde{C}_{q-r+3}\ddots&\widetilde{C}_{q-r+L+1}\\ \vdots&\vdots&\ddots&\vdots\\ \widetilde{C}_{q}&\widetilde{C}_{q+1}\cdots&\widetilde{C}_{q+L-1}\end{pmatrix} \]  
we have  
\[\widetilde{Q}_{q-r+1,r,L}=O_{r}(F,H)\ C_{L}(F,V*(F^{t})^{r-q+1}H_{1}^{t}).\]  
Keeping Hamilton-Cayley in mind, rank \(C_{L}(F,V^{*}(F)^{r-q+1}H_{1}^{t})\) does not change when \(L>nr\). Condition v.1 implies \(\text{rank}\widetilde{Q}_{q-r+1,r,nr}=nr\), and therefore \((A_{1},A_{2},\ldots,A_{r})\) can be uniquely identified from available autocovariance matrices of the process in the MFD case, solving the system \((A_{r}\ldots\ A_{1})\widetilde{Q}_{q-r+1,r,nr}=(\widetilde{C}_{q+1}\widetilde {C}_{q+2}\ldots\widetilde{C}_{q+nr})\) as follows:  
\[(A_{r}\ldots\ A_{1})=(\widetilde{C}_{q+1}\widetilde{C}_{q+2}\ldots\widetilde {C}_{q+nr})\widetilde{Q}_{q-r+1,r,nr}^{\widetilde{T}}[\widetilde{Q}_{q-r+1,r,nr} \widetilde{Q}_{q-r+1,r,nr}]^{T} \tag{5}\]The last part of the proof of this theorem aims to show that, once (\(A_{1}\), \(A_{2}\),..., \(A_{r}\)) are calculated, if Condition vi holds, we can uniquely reconstruct the unknown blocks of the autocovariance matrices.