## 2 The Six Sufficient Conditions for Identification in [1]  
[PERSON], in [1], considers the VARMA(\(r,q\)) model  
\[y_{t}=A_{1}y_{t-1}+\ldots+A_{r}y_{t-r}+B_{0}\varepsilon_{t}+B_{1}\varepsilon_ {t-1}+\ldots+B_{q}\varepsilon_{t-q} \tag{1}\]  
where \(y_{t}\) denotes an \(n\times 1\) vector of observed variables; \(p=\max\{r,\,q+1\}\); \(A_{i}\) for \(i=1,\,\ldots\), \(p\) and \(B_{j}\) for \(j=0,\,1,\ldots,\,p-1\) are \(n\times n\) matrices, \(A_{r}\
eq 0\), \(B_{q}\
eq 0\), \(A_{i}=0\) if \(i=r+1,\,\ldots,\,p\); \(B_{i}=0\) if \(i=q+1,\,\ldots,\,p-1\); and \(\varepsilon_{t}\) denotes an \(n\times 1\) white noise vector. In order to express the conditions in [1], and others in this paper, we need the following notation:  
\[\textbf{{a(z)}}=I-A_{1}z-\ldots-A_{p}z^{p},\textbf{{b(z)}}=B_{0}+B_{1}z+ \ldots+B_{q}z^{q},\,\text{where}\,z\in\mathbb{C},\]  
\[\textbf{{F}}= \begin{pmatrix}A_{1}&I_{n}\cdots&0\\ \vdots&0\ddots&\vdots\\ \vdots&\vdots\ddots&I_{n}\\ A_{p}&0\cdots&0\end{pmatrix} \text{is}\,\,np\times np,\textbf{{G}}= \begin{pmatrix} B_{0}\\ \vdots\\ B_{p-1}\end{pmatrix} \text{is}\,\,np\,\,n,\textbf{{H}}=(I_{n},0_{n\times n}, \ldots,0_{n\times n})\text{is}\,\,n\times np,\]  
\[\textbf{{C}}_{\textbf{{L}(F,G)}}=[G\,\,FG\,F^{2}G\ldots F^{L-1}G]\text{ is}\,\,np\,\times\,nL\text{ and}\]  
\[\textbf{{O}}_{\textbf{{L}(F,H)}}=[H^{t}\,(HF)^{t}(HF^{2})^{t}\ldots(HF^{L-1})^{ t}]\text{ is}\,\,nL\times np,\,\text{for}\,\,L=1,\,2,\,\ldots\]  
Assuming that the VARMA(\(r,\,q\)) model is stationary and that \(\sum=\text{E}(\varepsilon_{t}\varepsilon_{t}^{t})\) is positive definite:  
\(\textbf{{K(z)}}=a^{-1}(z)b(z)=\sum_{i=0}^{\infty}K_{i}z^{i}\) is the transfer function,  
\(\textbf{{C}}_{i}=\text{E}(y_{t}y_{t-i}^{T})=\sum_{j=0}^{\infty}K_{i+j}\sum K_{ i}^{j}\) for \(i\in\mathbb{Z}\), is the \(i\)-th population covariance matrix. To study mixed-frequency data cases (with stock variables), we consider \(n=n_{1}+n_{2}\) variables; the first \(\textbf{{n_{1}}}\) variables are high-frequency variables observed in every period and, given \(N\in\mathbb{N}\), the last \(\textbf{{n_{2}}}\) variables are low-frequency variables observed only for \(t\in\{0,N,\,2N,\,3N,\,\ldots\}\). Furthermore, we consider the following partition:  
\[\textbf{{C}}_{i}= \begin{pmatrix}C_{i}^{\prime\prime}&C_{i}^{\prime\prime}\\ C_{i}^{\prime\prime}&C_{i}^{\prime\prime}\end{pmatrix} ,\]where \(C_{i}^{ff}\), \(C_{i}^{fs}\), \(C_{i}^{sf}\)_and_\(C_{i}^{ss}\) are \(n_{1}\times n_{1}\), \(n_{1}\times n_{2}\), \(n_{2}\times n_{1}\) and \(n_{2}\times n_{2}\) blocks, respectively. Note that all the covariance blocks are available from variables observed with MFD, except \(C_{i}^{ss}\) if \(I\
eq kN\), for any integer \(k\). However, the XYW method in [1] only considers the first \(n_{1}\) columns in each \(C_{i}\). Therefore, let us denote:  
\(\widetilde{C}_{i}\) as the first \(n_{1}\) columns of \(C_{i}\), for \(i\in\mathbb{Z}\);  
\(\mathbf{H_{1}}\) as the matrix with the first \(n_{1}\) rows of \(H\);  
\(\mathbf{H_{2}}\) as the matrix with the last \(n_{2}\) rows of \(H\). [PERSON], in [1], proves that under the following six sufficient conditions, a VARMA model (1) is identified by the population covariance of its variables observed with MFD. **Condition I:** VARMA model (1) is stationary, i.e., \(\det a(z)\
eq 0\) if \(\,|z\,|\,\leq 1\). **Condition II:** VARMA model (1) is regular with \(B_{0}\) lower triangular and non-singular and \(\sum I_{n}\). **Condition III:** VARMA model (1) is miniphas, i.e., \(\det b(z)\
eq 0\) if \(\,|z\,|\,<1\). **Condition IV:** rank \(C_{mp}(F,\,G)=np\). **Condition V:** rank \(C_{L}(F,\,VH_{1}^{t})=np\) and rank \(O_{L}(F,\,H_{1})=np\), for sufficiently large \(L\), where \(V=\sum_{k=0}^{\infty}F^{k}G\sum G^{t}\big{(}F^{t}\big{)}^{k}=[C_{mp}(F,\,G) \ldots]\) [\(C_{mp}(F,\,G)\ldots]\)] which exists because the model is stationary. **Note on Condition V:** Condition V in [1] (p. 441) reads: \"VARMA model (2.1) is observable for sufficiently large \(L\), for the MFD being considered\".