## 3 Reconstructing Missing Blocks in Autocovariance Matrices
### Case 2: \(r=q\)  
In this section, our main result will be Theorem 2.  
For this case, neither \(G\) nor \(G^{*}\) allow us to state sufficient conditions similar to those in Theorem 1. Therefore, we consider the following \(nr\times n\) matrix:  
\[\mathbf{G^{\ast\ast}}=\begin{pmatrix}A_{1}+B_{1}\\ \vdots\\ A_{r}+B_{r}\end{pmatrix},\]  
and, considering \(p=r\), we state the following conditions, which change with respect to Case 1.  
**Condition iv.2**: rank \(C_{nr}(F,\ G^{\ast\ast})\) = \(nr\).
**Condition v.2**: rank \(C_{nr}(F,\ (G^{\ast\ast}\sum\limits_{i=0}^{\sim}+\mathrm{FV}^{\ast\ast}H_{1}^{ \dagger}))\) = \(nr\) where \(\widetilde{B_{0}^{t}}\) denotes the first \(n_{1}\) columns of \(B_{0}^{t}\) and \(V^{\ast\ast}=\sum_{i=0}^{\infty}F^{i}G^{\ast\ast}\sum G^{\ast\ast t}\left(F \right)^{t}\), which exists because the model is stationary.  
We are in a position to state the following theorem.  
Theorem 2: _If \(r=q\) and Conditions I, II or ii, and III hold:_  
1. _Condition iv.2 is necessary and sufficient for identifiability of the VARMA(r, r) model (1) in the SFD case.__._
2. _Conditions iv.2, v.2 and vi are sufficient for identifiability of the VARMA(r, r) model (1) in the MFD case._  
Proof.: The proof of Theorem 2 is similar to that of Theorem 1, except for some specific details, because with \(G^{**}\), we have that  
\[K_{j+1}=H^{j}G^{**}\text{ if }j\geq 0,\]  
\[C_{j}=H^{j-1}G^{**}\sum B_{0}^{t}+H^{j}\sum_{i=0}^{\infty}F^{i}G^{**}\sum G^{** }\left(F^{i}\right)^{t}H^{t}=H^{j-1}\left(G^{**}\sum B_{0}^{t}+FV^{**}H^{t}\right)\]  
for \(j\ \geq\ 1\) with \(V^{**}=\sum_{i=0}^{\infty}F^{i}G^{**}\sum G^{**}\left(F^{i}\right)^{t}\).  
Taking into account that  
\[M_{1,r,nr}=\text{O}_{r}(F,H)\ C_{nr}(F,G^{**})\text{ and }\text{Q}_{1,r,nr}= \text{O}_{r}(F,H)\ C_{nr}(F,(G^{**}{\sum}{\text{B}_{0}}^{t}+FV^{**}H^{t})),\]  
Conditions I, II or ii, III and iv.2 imply the full rank \(nr\) of the matrices \(M_{1,r,nr}\), \(Q_{1,r,nr}\) and \(\text{C}_{nr}(F,(G^{**}{\sum}B_{0}^{t}+FV^{**}H^{t}))\).  
If in the autocovariances we substitute \(H\) for \(H_{1}\) and \(B_{0}^{t}\) for \(\overset{\sim}{B_{0}^{t}}\):  
\[\overset{\sim}{C}_{j+q-r+1}=H^{j-1}(G^{**}{\sum}\overset{\sim}{B_{0}^{t}}+FV^{ **}H_{1}^{t})\text{ for }j\geq 1\]  
\[\overset{\sim}{Q}_{1,r,nr}=\left(\begin{array}{cccc}\overset{\sim}{C}_{1}& \overset{\sim}{C}_{2}\cdots&\overset{\sim}{C}_{nr}\\ \overset{\sim}{C}_{2}&\overset{\sim}{C}_{3}\ddots&\overset{\sim}{C}_{nr+1}\\ \vdots&\vdots&\ddots&\vdots\\ \overset{\sim}{C}_{r}&\overset{\sim}{C}_{r+1}\cdots&\overset{\sim}{C}_{nr+r-1 }\end{array}\right)\text{ and then }\]  
\[\overset{\sim}{Q}_{1,r,nr}=O_{r}(F,H)\ C_{nr}(F,(G^{**}\sum\overset{\sim}{B_{0} ^{t}}+FV^{**}H_{1}^{t})\]  
Condition v.2 implies \(\text{rank}\overset{\sim}{Q}_{1,r,nr}=nr\), and therefore \((A_{1},\)\(A_{2},\)\(\ldots,\)\(A_{r})\) can be uniquely identified from the autocovariance matrices of the process, solving (5).  
The last part of the proof of this theorem is identical to that of Theorem 1. Therefore, Theorem 2 has been proven. \(\square\)  
As a consequence of section (a) of the previous Theorem and of the Theorem in [9], we give the following Corollary 2.  
Corollary 2: _Suppose \(r=q\) and that Conditions I, ii and III hold. In this case, \(\text{rank}(A_{r}\overset{\cdot}{B_{r}})=n\) and \((a(z),\)\(b(z))\) is left coprime iff rank \(C_{nr}(F,G^{**})=nr\)._  
Note that the results in this section consider certain blocks available in the autocovariance matrices that [1] ignores. In particular, if \(r>q+1\), we use \(C_{i}^{ff}\) and \(C_{i}^{sf}\) for \(i=q-r+1\), \(\ldots\), \(1\), in \(\overset{\sim}{Q}_{q-r+1,r,L}\) solving (5) and \(C_{i}^{f_{\text{s}}}\) for \(i=q+1\), \(\ldots\), \(q+nr\) and \(i=\{kN\), \((k+1)N\), \(\ldots\)\((k+nr-1)N\}\) in \(X_{2a}\) solving (6).