VARMA Models with Single- or Mixed-Frequency Data: New Conditions for Extended Yule–Walker Identification

Abstract

This paper deals with the identifiability of VARMA models with VAR order greater than or equal to the MA order, in the context of mixed-frequency data (MFD) using extended Yule–Walker equations. The main contribution is that necessary and sufficient conditions for identifiability in the single-frequency data case are expressed in an original way and yield new results in the MFD case. We also provide two counterexamples that answer an open question in this topic about whether certain sufficient conditions are necessary for identifiability. Therefore, this paper expands the set of models that can be identified with MFD using extended Yule–Walker equations. The main idea is that with MFD, some autocovariance blocks are not available from observed variables and, in some cases, the new conditions in this paper can be used to reconstruct all the non-available covariance blocks from available covariance blocks.

1. Introduction

In time series models, developing statistically efficient and computationally quick methods for identifying and estimating VAR or VARMA models with single-frequency data (SFD) or mixed-frequency data (MFD) is an important task. In this paper, “SFD means that all the variables of a model are observed at the same discrete-time frequency at which the model operates, and MFD means that some of the variables are observed at the same discrete-time frequency at which the model operates, and others are observed at one or more lower frequencies” [1].

Linear algebra tools are extensively used to study transfer functions, Yule–Walker equations and Hankel matrices associated with identifying and estimating VAR or VARMA models (see, for instance, [1,2,3,4,5,6,7,8,9,10,11,12,13]). In particular, the extended Yule–Walker method (XYW) was proposed by [5], and it is considered in several later papers, in particular in [2,3,8] for estimating VAR models from available covariance matrices of MFD. Two of the principal and parallel strands of this literature which extend the method to the case of VARMA models are [1,2]. The first considers both exact and generic identification and the second considers only exact identification. They make both common and differing assumptions involving the parameters of a VARMA model and prove that their assumptions are, as a whole, sufficient to identify all of the parameters of a VARMA model with MFD. In particular, [1] questions whether its “conditions as a whole are necessary for identification”. This question has motivated our work, and our aim is to expand the methodology to more identifiable models.

This paper is focused on identification. In econometrics and statistics, identification means the coefficients (parameters) of a model are determined uniquely from data covariances (and higher moments, depending on the data distribution) under assumed conditions for the coefficients of a model.

Our aim is to provide conditions for the identifiability of VARMA models that cannot be identified by following the procedure in [1] and, in this regard, to complement it. We use a subset of conditions in [1] and some more, and we prove with two counterexamples that the whole set of conditions in [1] is not necessary. The main results are obtained from subsystems of extended Yule–Walker equations. Therefore, we expand the set of models identified with extended Yule–Walker methods.

Because with MFD some autocovariance blocks are not available from observed variables, the main idea of our work is to provide conditions that, in addition to ensuring that the model with SFD is identifiable, allow rebuilding the unknown blocks from available covariance blocks. This thus yields all the complete autocovariances. As is well known, there is a bijection between the covariance and the corresponding spectral density of the process (see, for instance, [6]), meaning we can ensure the identifiability of the model.

Section 2 summarizes and comments on the sufficient conditions used in [1] to identify VARMA models with SFD and MFD. In Section 3, we prove our main theoretical results. With a suitable change, our paper expands the applicability of XYW methods to more identifiable models. Section 4 tests and illustrates our main insight with two counterexamples. We close this paper with Conclusions, References and an Appendix with a MATLAB subroutine used in a counterexample.

2. The Six Sufficient Conditions for Identification in [1]

Zadrozny, in [1], considers the VARMA(r, q) model

y_t = A₁y_t−1 + … + A_ry_t−r + B₀ε_t + B₁ε_t−1 + … + B_qε_t−q

(1)

where y_t denotes an n × 1 vector of observed variables; p = max{r, q + 1}; A_i for i = 1, …, p and B_j for j = 0, 1, …, p − 1 are n × n matrices, A_r ≠ 0, B_q ≠ 0, A_i = 0 if i = r + 1, …, p; B_i = 0 if i = q + 1, …, p − 1; and ${\epsilon }_t$ denotes an n × 1 white noise vector.

In order to express the conditions in [1], and others in this paper, we need the following notation:

a(z) = I − A₁z −…− A_pz^p, b(z) = B₀ + B₁z + … + B_qz^q, where z ∈ ℂ,

$$\mathit{F}=\left(\begin{array}{c}\begin{array}{cc}{A}_1& I_n\\ ⋮& 0\end{array}\begin{array}{cc}\cdots & 0\\ \ddots & ⋮\end{array}\\ \begin{array}{cc}⋮& ⋮\\ A_p& 0\end{array}\begin{array}{cc}\ddots & I_n\\ \cdots & 0\end{array}\end{array}\right)\mathrm{i}\mathrm{s} np\times np,\mathit{G}=\left(\begin{array}{c}\begin{array}{c}{B}_0\\ ⋮\end{array}\\ B_{p-1}\end{array}\right)\mathrm{i}\mathrm{s} np n,\mathit{H}=(I_n,0_{n\times n},\dots ,0_{n\times n})\mathrm{i}\mathrm{s} n\times np,$$

C_L(F, G) = [G FG F²G … F^L−1G] is np × nL and

O_L(F, H)= [H^t (HF)^t(HF²)^t … (HF^L−1)^t]^t is nL×np, for L = 1, 2, …

Assuming that the VARMA(r, q) model is stationary and that ∑ = E(${\epsilon }_t{\epsilon }_t^t$) is positive definite:

K(z) = a⁻¹(z)b(z) = $\sum _{i=0}^{\infty }{K}_i{z}^i$ is the transfer function,

C_i = E(${y_{t}y}_{t-i}^T$) = $\sum _{j=0}^{\infty }{K}_{i+j}{\sum K}_j^t$ for i$\in$ℤ$,$ is the i-th population covariance matrix.

To study mixed-frequency data cases (with stock variables), we consider n = n₁ + n₂ variables; the first n₁ variables are high-frequency variables observed in every period and, given N$\in \mathbb{N}$, the last n₂ variables are low-frequency variables observed only for t$\in${0, N, 2N, 3N, …}. Furthermore, we consider the following partition:

$${\mathit{C}}_{\mathit{i}}=\left(\begin{array}{cc}{\mathit{C}}_{\mathit{i}}^{\mathit{f}\mathit{f}}& {\mathit{C}}_{\mathit{i}}^{\mathit{f}\mathit{s}}\\ {\mathit{C}}_{\mathit{i}}^{\mathit{s}\mathit{f}}& {\mathit{C}}_{\mathit{i}}^{\mathit{s}\mathit{s}}\end{array}\right),$$

where $C_i^{ff}$, $C_i^{fs},C_i^{sf} and C_i^{ss}$ are n₁ × n₁, n₁ × n₂, n₂ × n₁ and n₂ × n₂ blocks, respectively.

Note that all the covariance blocks are available from variables observed with MFD, except $C_i^{ss}$ if I ≠ kN, for any integer k. However, the XYW method in [1] only considers the first n₁ columns in each C_i. Therefore, let us denote:

${\tilde{\mathit{C}}}_{\mathit{i}}$ as the first n₁ columns of C_i, for i$\in$ℤ;

H₁ as the matrix with the first n₁ rows of H;

H₂ as the matrix with the last n₂ rows of H.

Zadrozny, 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) ≠ 0 if |z| ≤ 1.

Condition II: VARMA model (1) is regular with B₀ lower triangular and non-singular and ∑ = I_n.

Condition III: VARMA model (1) is miniphase, i.e., det b(z) ≠ 0 if |z| < 1.

Condition IV: rank C_np(F, G) = np.

Condition V: rank C_L(F, V$H_1^t)$ = np and rank O_L(F, H₁) = np, for sufficiently large L, where V = $\sum _{k=0}^{\infty }{F}^k{G\sum G}^t{\left(F^t\right)}^k$ = [C_np(F, G)…] [C_np(F, G)…]^t 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”. Condition V written above can be read in [1] (p. 445): “Section 3 and Section 4 proved that parameters of VARMA model (2.1) are identified (…) for MFD if C_L(F, V$H_1^t)$ and O_L(F, H₁) have full rank, (…) for sufficiently large L”, because to identify VAR parameters with the specific procedure in [1] (pp. 441–442), rank D₁ = np is necessary, and therefore the full rank of C_L(F, V$H_1^t)$. (We have taken into account that matrices D₁ and E₁ in [1] (p. 442) must be written without a subscript in the second$\tilde{H})$. Moreover, to identify VMA parameters with the specific procedure in [1] (pp. 442–444), the full rank of O_L(F, H₁) is necessary.

Condition VI: The nq × nq matrix $\left(\begin{array}{cc}\begin{array}{cc}-B_1{B}_0^{-1}& I_n\\ ⋮& 0\end{array}& \begin{array}{cc}& 0\\ \ddots & \end{array}\\ \begin{array}{cc}& \mathrm{⁝}\\ -B_q{B}_0^{-1}& 0\end{array}& \begin{array}{cc}& I_n\\ \cdots & 0\end{array}\end{array}\right)$ is diagonalizable, i.e., it has a linearly independent set of eigenvectors.

Remark 1. 

Zadrozny, in [1], proves that Conditions I, II, III and IV are sufficient for identifiability with SFD. However, if Conditions I, II and III hold, Condition IV is sufficient but not necessary in the SFD case (see Counterexamples 1 and 2).

Deistler et al. in [6] prove that if q > r, a VARMA(r, q) model with MFD is not identifiable. Next, we will show that rank O_L (F, H₁) = np in Condition V excludes not only the case q > r, but also the case q = r.

Lemma 1. 

If q ≥ r, then rank O_L(F, H₁) < np.

Proof. 

Considering that if q ≥ r then p = q + 1 and A_r+1 = … = A_p = 0.

On the one hand, the first n(p − r) columns in the n(p − r) × np matrix

$${(H}_{}^t{\left(H_{}F\right)}^t{\left(H_{}{F}^2\right)}^t\cdots {\left(H_{}{F}^{p-r-1}\right)}^t{)}^t$$

form a lower triangular matrix with ones on the diagonal.

Therefore, its rank is exactly n(p − r). As a result

$$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}{(H}_1^t{\left(H_{1}F\right)}^t{\left(H_1{F}^2\right)}^t\cdots {\left(H_1{F}^{p-r-1}\right)}^t{)}^t=n_1(p-r).$$

(2)

On the other hand, considering the nr × np matrix$({\left(H_{}{F}^{p-r}\right)}^t\cdots$ ${\left(H_{}{F}^{p-1}\right)}^t{)}^t$, it is easy to see that the submatrix formed by its last nr columns is lower triangular with ones on the diagonal, and therefore,

$$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}({\left(H_{}{F}^{p-r}\right)}^t\cdots {\left(H_{}{F}^{p-1}\right)}^t{)}^t=nr.$$

(3)

Moreover, if i ≥ p, HFⁱ = A₁HFⁱ⁻¹ + … + A_rHF^i−r and, from (3), rank $({\left(H_{}{F}^{p-r}\right)}^t\cdots$ ${\left(H_{}{F}^{L-1}\right)}^t{)}^t$ = nr.

As a consequence:

$$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}({\left(H_1{F}^{p-r}\right)}^t\cdots {\left(H_1{F}^{L-1}\right)}^t{)}^t\le nr \mathrm{i}\mathrm{f} L>p$$

(4)

From (2) and (4), rank O_L(F, H₁) ≤ n₁(p − r) + nr = n₁p − n₁r + (n₁ + n₂)r = n₁p + n₂r < np and Lemma 1 has been proven. □

Note that if we do not have the hypothesis from Lemma 1, i.e., if q < r, then p = r and we do not have (2). Therefore, rank O_L(F, H₁) could be equal to np.

3. Reconstructing Missing Blocks in Autocovariance Matrices

Our goal in this work is to extend the set of models that can be identified with Yule–Walker methods in the MFD case. We provide new conditions to identify the VARMA model in the MFD case from an original form of rewriting necessary and sufficient conditions in the SFD case. These conditions are expressed based on the parameters of the model. We will demonstrate with two counterexamples that the sufficient conditions in [1] are not necessary to identify MFD models. Thus, we consider one of the questions opened in [1] to be resolved.

In our proposal, we treat the cases r > q and r = q separately, giving new conditions to replace the corresponding ones in [1]. In addition, the fourth and fifth conditions are different in each of the two cases.

3.1. Case I: r > q

In this case, our main result will be Theorem 1. We previously introduced the necessary notation, the new conditions and some previous results (Hanzon’s Theorem and Lemma 2).

Let us denote the following matrices:

$$\mathit{G}\mathit{*}=\left(\begin{array}{c}\begin{array}{c}0\\ ⋮\\ 0\end{array}\\ \begin{array}{c}{B}_0\\ ⋮\\ B_q\end{array}\end{array}\right)\mathrm{a}\mathrm{n}\mathrm{d} \mathit{X}=\left(\begin{array}{c}\begin{array}{c}{C}_q\\ C_{q-1}\end{array}\\ \begin{array}{c}⋮\\ C_{q-r+1}\end{array}\end{array}\right) \mathrm{a}\mathrm{r}\mathrm{e} nr\times n,\mathit{F}\mathit{*}=\left(\begin{array}{c}\begin{array}{cc}{A}_1& \begin{array}{cc}{A}_2& \end{array}\\ I_n& \begin{array}{cc}0& \end{array}\end{array}\begin{array}{cc}\cdots & A_r\\ \cdots & 0\end{array}\\ \begin{array}{cc}0& \begin{array}{cc}\ddots & \ddots \end{array}\\ ⋮& \begin{array}{cc}& \end{array}\end{array}\begin{array}{cc}\ddots & ⋮\\ 0& 0\end{array}\\ \begin{array}{cc}\begin{array}{cc}0& \begin{array}{cc}\cdots & 0\end{array}\end{array}& \begin{array}{cc}{I}_n& 0\end{array}\end{array}\end{array}\right)\mathrm{i}\mathrm{s} nr\times nr,$$

$$\mathit{\theta }=\left(\begin{array}{c}\begin{array}{c}{H}_1{F}^{*}\\ ⋮\\ H_1{\left(F^{*}\right)}^{nr}\end{array}\\ H_2{\left(F^{*}\right)}^{kN-q}\\ \begin{array}{c}{H}_2{\left(F^{*}\right)}^{kN-q}{\left(F^{*}\right)}^N\\ ⋮\\ H_2{\left(F^{*}\right)}^{kN-q}{\left(F^{*}\right)}^{(nr-1)N}\end{array}\end{array}\right) \mathrm{i}\mathrm{s} n^{2}r\times nr,J=\left(\begin{array}{c}\begin{array}{c}\begin{array}{cc}{C}_{q+1}^{ff}& C_{q+1}^{fs}\end{array}\\ ⋮\\ \begin{array}{cc}{C}_{q+nr}^{ff}& C_{q+nr}^{fs}\end{array}\end{array}\\ \begin{array}{cc}{C}_{kN}^{sf}& C_{kN}^{ss}\end{array}\\ \begin{array}{c}\begin{array}{cc}{C}_{(k+1)N}^{sf}& C_{\left(k+1\right)N}^{ss}\end{array}\\ ⋮\\ \begin{array}{cc}{C}_{(k+nr-1)N}^{sf}& C_{\left(k+nr-1\right)N}^{ss}\end{array}\end{array}\end{array}\right)\mathrm{i}\mathrm{s} n^{2}r\times n,$$

where k is an integer such that kN > q.

θ_b is the submatrix formed with columns of θ, such that the ith column of θ is inθ_b, if I ≠ 1, …, n₁, n + 1, …, n + n₁, 2n + 1, …, 2n + n₁, …, (r − 1)n + 1, …, (r − 1)n + n₁ and also the ith row of X $=\left(\begin{array}{c}\begin{array}{c}{C}_q\\ C_{q-1}\end{array}\\ \begin{array}{c}⋮\\ C_{q-r+1}\end{array}\end{array}\right)$ is not a row of C_j or C_−j with j$\in${0, N, 2N, 3N…}.

θ_a is the submatrix of θ with the columns that are not in θ_b.

Below we set out the new conditions. Note that although [1] refers to a state space model associated with the VARMA model, we only need to note the algebraic properties that hold in certain matrices constructed from the parameters of the VARMA model.

Condition ii: The VARMA model is regular with B₀ = I and ∑ positive definite.

Condition iv.1: rankC_nr(F, G^*) = nr.

Condition v.1: rank C_nr(F, ${{V^{*}\left(F^t\right)}^{r-q-1}H}_1^t)$ = nr where

V^* = $\sum _{k=0}^{\infty }{F}^k{{G^{*}\sum (G}^{*})}^t{\left(F^t\right)}^k$ = [C_np(F, G^*$\sum$^1/2)…] [C_np(F, G^*$\sum$^1/2)…]^t, which exists because the model is stationary.

Condition vi:θ_b is full column rank.

We will also make use of [11] (Theorem 3.1.3.2–1 (iii), Theorem 3.1.2.3–29 (iii) and Corollary 2.4.3–25), which we summarize in the following theorem that we call Hanzon’s Theorem.

Hanzon’s Theorem: Considering K_−i = 0 if i > 0, for i, j, h$\in \mathbb{N}$, we denote

$$\begin{gathered} {\mathit{M}}_{\mathit{i},\mathit{j},\mathit{h}}=\left(\begin{array}{c}\begin{array}{cc}{K}_i& K_{i+1 }\\ K_{i+1}& K_{i+2}\end{array}\begin{array}{cc}\cdots & K_{i+h-1}\\ \ddots & K_{i+h}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ K_{i+j-1}& \end{array}\begin{array}{cc}\ddots & \mathrm{⁝}\\ \cdots & { K}_{i+j+h-2}\end{array}\end{array}\right)\mathrm{a}\mathrm{n}\mathrm{d} \\ {\mathit{Q}}_{\mathit{i},\mathit{j},\mathit{h}}=\left(\begin{array}{c}\begin{array}{cc}{C}_i& C_{i+1 }\\ C_{i+1}& C_{i+2}\end{array}\begin{array}{cc}\cdots & C_{i+h-1}\\ \ddots & C_{i+h}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ C_{i+j-1}& \end{array}\begin{array}{cc}\ddots & ⋮ \\ \cdots & { C}_{j+j+h-2}\end{array}\end{array}\right) \end{gathered}$$

If model (1) is stationary, regular and miniphase, the following conditions are satisfied for any orders r and q:

• rank M_{q−r+1,∞,∞} = rank M_q−r+1,r,nr
• rank Q_{q−r+1,r,∞} = rank Q_q−r+1,r,nr
• rank M_q−r+1,r,nr = rank Q_q−r+1,r,nr

As a consequence, we can deduce the following Lemma.

Lemma 2. 

Suppose r > q and Conditions I, II or ii, and III hold. Therefore, Condition IV implies Condition iv.1.

Proof. 

We can easily see that

K_i = HFⁱG for i ≥ 0

and M_0,r,nr = O_r(F, H) C_nr(F, G). Taking into account that O_r(F, H) is full column rank nr and, from Condition IV, C_nr(F, G) is full row rank nr, then rank M_0,r,nr = nr.

By Hanzon’s Theorem, rank M_0,r,nr = rank M_0,r,∞ = nr, rank M_q−r+1,r,nr = rank M_{q−r+1,r,∞}. Since all the columns of M_0,r,∞ are columns of the matrix M_{q−r+1,r,∞}, rank M_{q−r+1,r,∞} ≥ rank M_0,r,∞. Since the matrix has nr rows, rank M_{q−r+1,r,∞} = nr. By Hanzon’s Theorem, rank M_q−r+1,r,nr = rank M_{q−r+1,r,∞} = nr.

We can easily see that

K_j = HF^r−q−1+jG* for j ≥ q − r + 1, (note that q − r + 1 ≤ 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. □

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:

(a) 

Condition iv.1 is necessary and sufficient for identifiability of the VARMA(r, q) model (1) in the SFD case.

(b) 

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 ≥ 0, C_q−r+1+j = $\sum _{\mathit{i}=0}^{\mathit{\infty }}{\mathit{K}}_{\mathit{i}+\mathit{q}-\mathit{r}+1+\mathit{j}}{\sum \mathit{K}}_{\mathit{i}}^{\mathit{t}}={\sum _{\mathit{i}=0}^{\mathit{\infty }}{\mathit{H}\mathit{F}}^{\mathit{i}+\mathit{j}}\mathit{G}}^{\mathit{*}}\sum {\mathit{G}}^{\mathit{*}\mathit{t}}{\left({\mathit{F}}^{\mathit{i}-\mathit{q}+\mathit{r}-1}\right)}^{\mathit{t}}{\mathit{H}}^{\mathit{t}}$ = HF^j$\left(\sum _{\mathit{i}=0}^{\mathit{\infty }}{\mathit{F}}^{\mathit{i}}{\mathit{G}}^{\mathit{*}}{\sum \mathit{G}}^{\mathit{*}\mathit{t}}{\left({\mathit{F}}^{\mathit{i}}\right)}^{\mathit{t}}\right){{\left({\mathit{F}}^{-\mathit{q}+\mathit{r}-1}\right)}^{\mathit{t}}\mathit{H}}^{\mathit{t}}$.

Therefore, denoting V^* = ${\sum _{i=0}^{\infty }{F}^{i}G}^{*}\sum G^{*t}{\left(F^i\right)}^t$,

C_q−r+1+j = H F^jV*(F^−q+r−1)^tH^t for j ≥ 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 = 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 (A₁, A₂, …, A_r) can be uniquely identified from the autocovariance matrices of the process, solving the following system:

$$\left(A_r\dots A_1\right)Q_{q-r+1,r,nr}^{}=(C_{q+1}{C}_{q+2}\dots C_{q+nr}).$$

Taking into account that Q_q−r+1,r,nr = O_r(F, H) C_nr(F, V^*(F^t)^r−q+1H^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^t)^r−q+1H^t) is full row rank nr.

If we substitute H for H₁ in the autocovariances

$${\tilde{C}}_{j+q-r+1}=H{F}^j{V}^{*}{\left(F^{-q+r-1}\right)}^t{H}_1^t \mathrm{f}\mathrm{o}\mathrm{r} j \ge 0.$$

Denoting

$${\tilde{Q}}_{q-r+1,r,L}=\left(\begin{array}{c}\begin{array}{cc}{\tilde{C}}_{q-r+1}& {\tilde{C}}_{q-r+2}\\ {\tilde{C}}_{q-r+2}& {\tilde{C}}_{q-r+3}\end{array}\begin{array}{cc}\cdots & {\tilde{C}}_{q-r+L}\\ \ddots & {\tilde{C}}_{q-r+L+1}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ {\tilde{C}}_q& {\tilde{C}}_{q+1}\end{array}\begin{array}{cc}\ddots & ⋮\\ \cdots & {\tilde{C}}_{q+L-1}\end{array}\end{array}\right)$$

we have

$${\tilde{Q}}_{q-r+1,r,L}=O_r(F,H) C_L(F,V*{\left(F^t\right)}^{r-q+1}{H}_1^t).$$

Keeping Hamilton–Cayley in mind, rank C_L(F, V^*(F^t)^r−q+1$H_1^t$) does not change when L > nr.

Condition v.1 implies $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}{\tilde{Q}}_{q-r+1,r,nr}=nr$, and therefore (A₁, A₂, …, A_r) can be uniquely identified from available autocovariance matrices of the process in the MFD case, solving the system $\left(A_r\dots A_1\right){\tilde{Q}}_{q-r+1,r,nr}^{}=({\tilde{C}}_{q+1}{\tilde{C}}_{q+2}\dots {\tilde{C}}_{q+nr})$ as follows:

$$\left(A_r\dots A_1\right)=\left({\tilde{C}}_{q+1}{\tilde{C}}_{q+2}\dots {\tilde{C}}_{q+nr}\right){\tilde{Q}}_{q-r+1,r,nr}^T{\left[{\tilde{Q}}_{q-r+1,r,nr}{\tilde{Q}}_{q-r+1,r,nr}^T\right]}^{-1}$$

(5)

The last part of the proof of this theorem aims to show that, once (A₁, A₂, ..., A_r) are calculated, if Condition vi holds, we can uniquely reconstruct the unknown blocks of the autocovariance matrices.

Note that H(F^*)ⁱX = C_q+i for i > 0, i.e., H₁(F^*)ⁱX = $\left(\begin{array}{cc}{C}_{q+i}^{ff}& C_{q+i}^{fs}\end{array}\right)$ and H₂(F*)ⁱX = $\left(\begin{array}{cc}{C}_{q+i}^{sf}& C_{q+i}^{ss}\end{array}\right)$. Therefore, J = θX and, in particular, J₂ = θX₂, where J₂ and X₂ are the submatrices formed by the last n₂ columns of J and X, respectively.

Note that, keeping Hamilton–Cayley in mind, θ has been defined such that its rank does not change when some rows of H(F^*)ⁱare added to θ, for some i > nr. Moreover, J has been defined only with available autocovariance blocks. If we rearrange the rows of X₂ such that θX₂ = (θ_a θ_b) $\left(\begin{array}{c}{X}_{2a}\\ X_{2b}\end{array}\right)$, we make it such that X_2b is the submatrix containing the unknown blocks. To calculate X_2b, we solve the following system of linear equations:

J₂−θ_aX_2a = θ_bX_2b,

(6)

for Condition vi, (6) has as a unique solution

$$X_{2b}={\left({\theta }_b^t{\theta }_b\right)}^{-1}{\theta }_b^t(J_2-{\theta }_a{X}_{2a}).$$

As a consequence, we have $C_i^{ss}$ for i = q − r + 1, …, q and we can calculate $C_i^{ss}$ for i > q, considering that C_i = A₁C_i−1 + … + A_rC_i−r for i > q.

Therefore, Theorem 1 has been proven. □

The following Corollary 1 is a consequence of section (a) of the previous Theorem and of the Theorem in [9].

Corollary 1. 

Suppose r > q and that Conditions I, ii and III hold. In this case: rank(A_r⁝B_q) = n and (a(z), b(z)) is left coprime iff rankC_nr(F, G^*) = nr.

Note that, unlike [1], we can consider rank A_r < n or rank B_q < n.

3.2. 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 × n matrix:

$$\mathit{G}\mathit{*}\mathit{*}=\left(\begin{array}{c}\begin{array}{c}{A_1+B}_1\\ ⋮\end{array}\\ {A_r+B}_r\end{array}\right),$$

and, considering p = r, we state the following conditions, which change with respect to Case 1.

Condition iv.2: rank C_nr(F, G^**) = nr.

Condition v.2: rank C_nr(F, ${(G^{**}\sum \widetilde{B_0^t}+\mathrm{F}{\mathrm{V}}^{**}H}_1^t\left)\right)$ = nr where $\widetilde{B_0^t}$ denotes the first n₁ columns of $B_0^t$ and V^** = ${\sum _{i=0}^{\infty }{F}^{i}G}^{**}{\sum G}^{**t}{\left(F^i\right)}^t$, 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:

(a) 

Condition iv.2 is necessary and sufficient for identifiability of the VARMA(r, r) model (1) in the SFD case.

(b) 

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 = HF^jG** if j ≥ 0,

$$\begin{gathered} C_j={\mathit{HF}}^{j-1}{G}^{**}\sum B_0^t+{\mathit{HF}}^j{\sum }_{i=0}^{\infty }{F}^i{G}^{**}\sum {G^{**}}^t{\left(F^i\right)}^t{H}^t={\mathit{HF}}^{j-1}(G^{**}\sum B_0^t+{\mathit{FV}}^{**}{H}^t) \\ \mathrm{for} j \ge 1 \mathrm{with} {\mathrm{V}}^{**}={\sum }_{i=0}^{\infty }{F}^i{G}^{**}\sum {G^{**}}^t{\left(F^i\right)}^t. \end{gathered}$$

Taking into account that

M_1,r,nr = O_r(F, H) C_nr(F, G**) and Q_1,r,nr = O_r(F, H) C_nr(F, (G**∑B₀^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 C_nr(F, (G^**$\sum B_0^t$ + FV^**H^t)).

If in the autocovariances we substitute H for H₁ and $B_0^t$ for $\widetilde{B_0^t}$:

${\tilde{C}}_{j+q-r+1}$ = HF^j−1(G^**$\sum \widetilde{B_0^t}$ + FV^**$H_1^t)$ for j ≥ 1

$${\tilde{Q}}_{1,r,nr}=\left(\begin{array}{c}\begin{array}{cc}{\tilde{C}}_1& {\tilde{C}}_2\\ {\tilde{C}}_2& {\tilde{C}}_3\end{array}\begin{array}{cc}\cdots & {\tilde{C}}_{nr}\\ \ddots & {\tilde{C}}_{nr+1}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ {\tilde{C}}_r& {\tilde{C}}_{r+1}\end{array}\begin{array}{cc}\ddots & ⋮\\ \cdots & {\tilde{C}}_{nr+r-1}\end{array}\end{array}\right)\mathrm{a}\mathrm{n}\mathrm{d} \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}$$

$${\tilde{Q}}_{1,r,nr}=O_r(F,H) C_{nr}(F,(G^{**}\sum \widetilde{B_0^t}+{\mathit{FV}}^{**}{H}_1^t)$$

Condition v.2 implies $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}{\tilde{Q}}_{1,r,nr}=nr$, and therefore (A₁, A₂, …, 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. □

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, rank(A_r⁝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 $C_i^{ff}\mathrm{a}\mathrm{n}\mathrm{d}$ $C_i^{sf}$ for i = q – r + 1, …, 1, in ${\tilde{Q}}_{q-r+1,r,L}$ solving (5) and $C_i^{fs}$ for i = q + 1, …, q + nr and i = {kN, (k + 1)N, … (k + nr−1)N} in X_2a solving (6).

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 rank(A_r⁝B_q) = n.

Counterexample 1. 

Consider the VARMA(3, 1) model with A₀ = B₀ = I,

$$A_1=\left(\begin{array}{cc}0& -1/2\\ -1/2& 0\end{array}\right),A_2=\left(\begin{array}{cc}-1/4& 0\\ 0& -1/4\end{array}\right),A_3=\left(\begin{array}{cc}-1/2& -1/4\\ -1/4& -1/8\end{array}\right),B_1=\left(\begin{array}{cc}1/2& 1/2\\ 1/2& 0\end{array}\right)$$

and E(${{\epsilon }_t}_t^t$ ) = I_n.

We have q = 1, r = 3, nr = 6 and we consider n₁ = 1.

The model is identifiable in the SFD case because

$$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\begin{array}{ccc}{K}_{q-r+1}& K_{q-r+2}& \cdots \\ ⋮& ⋮& \\ K_q& K_{q+1}& \cdots \end{array}\begin{array}{c}{K}_{q+\left(n-1\right)r}\\ ⋮\\ K_{q+nr-1}\end{array}\right)=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(\begin{array}{ccc}{K}_{-1}& K_0& \cdots \\ K_0& K_1& \dots \\ K_1& K_2& \cdots \end{array}\begin{array}{c}{K}_4\\ K_5\\ K_6\end{array}\right)=6.$$

Note that Conditions I, ii, III, iv.1 hold.

However, Condition IV is not satisfied because rank [G … F^npG] = 5 ≠ 6 and, as a consequence, rank C_L(F, V$H_1^t)$ < np and Condition V do not hold. Therefore, A₁, A₂ and A₃ could not be uniquely calculated using the procedure in [1].

We prove that Condition v.1 holds (with MATLAB, see Appendix A) as follows:

(i)

First, we computed C_i (i = 0, 1, 2, 3) by solving the Yule–Walker equations:

C₀ − A₁C₋₁ − A₂C₋₂ − A₃C₋₃ = I + B₁K₁^t

C₁ − A₁C₀ − A₂C₋₁ − A₃C₋₂ = B₁

C₂ − A₁C₁ − A₂C₀ − A₃C₋₁ = 0

C₃ − A₁C₂ − A₂C₁ − A₃C₀ = 0

where K₁ = B₁ + A₁. We then computed C_i = A₁C_i−1 + A₂C_i−2 + A₃C_i−3. for i > 4.

(ii)

Second, we obtained that rank ${\tilde{Q}}_{q-r+1,r,nr}$ = 6.

(iii)

Taking into account that ${\tilde{Q}}_{q-r+1,r,L}$ = O_r(F, H) C_L(F, V^*(F^t)^r−q+1$H_1^t$), O_r(F, H) is full column rank nr, C_L(F, V^*(F^t)^r−q+1$H_1^t$) has nr rows and rank${\tilde{Q}}_{q-r+1,r,nr}$ = 6 = nr, and we can affirm that rank C_L(F, V^*(F^t)^r−q+1$H_1^t$) = nr = 6; i.e., Condition v.1 holds.

Therefore, A₁, A₂ and A₃ can be uniquely determined by solving (5).

Regarding Condition vi, in this example, θ has 6 columns, where θ_b is the submatrix with the 2nd and 6th columns of θ and rank θ_b = 2, i.e., Condition vi holds.

Taking into account that X = $\left(\begin{array}{c}{C}_1\\ C_0\\ C_{-1}\end{array}\right),$ from (6), we can identify $C_1^{22}\mathrm{a}\mathrm{n}\mathrm{d}{C}_{-1}^{22}$. As a consequence, C₀ and C₁ are complete. Finally, the unknown $C_i^{22}$ for i > 1 can be identified considering that C_i = A₁C_i−1 + A₂C_i−2 + A₃C_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 A₀ = B₀ = I, A₁ = $\left(\begin{array}{cc}-1/2& -1/4\\ 1& 1/2\end{array}\right)$, B₁ = $\left(\begin{array}{cc}1& 4\\ -1/4& 1\end{array}\right)$ and E(${\epsilon }_t{\epsilon }_t^t$) = I_n.

We have r = q = 1, nr = 2 and we consider n₁ = 1. The autocovariance matrices are

$$C_0=\left(\begin{array}{cc}4753/256& -1025/128\\ -1025/128& 949/64\end{array}\right),C_1=\left(\begin{array}{cc}-201/32& 275/64\\ 229/16& -51/32\end{array}\right),C_2=\left(\begin{array}{cc}-7/16& -7/4\\ 7/8& 7/2\end{array}\right),C_i=0$$

for i > 2.

Note that Conditions I, ii, III and iv.2 hold.

Regarding Condition v.2, note that

${\tilde{Q}}_{1,r,nr}$ = O_r(F, H))C_nr(F, (G^**$\sum \widetilde{B_0^t}$ + FV^**$H_1^t$) = $\left(\begin{array}{cc}{\tilde{C}}_1& {\tilde{C}}_2\end{array}\right)$ = $\left(\begin{array}{cc}-201/32& -7/16\\ 229/16& 7/8\end{array}\right)$. Due to the fact that rank $\left(\begin{array}{cc}{\tilde{C}}_1& {\tilde{C}}_2\end{array}\right)=$nr = 2, O_r(F, H) has nr columns and C_nr(F, (G^**$\sum \widetilde{B_0^t}$ + FV^**$H_1^t$)) has nr rows, then rank C_nr(F, (G^**$\sum \widetilde{B_0^t}$ + FV^**$H_1^t)$) = nr = 2; i.e., Condition v.2 holds.

Therefore, A₁ is uniquely determined by solving (5).

Regarding Condition vi, taking into account that HF^* = A₁ is a submatrix of θ and θ_b is the second column of θ, then rank θ_b = 1 and Condition vi holds. Taking into account that X = C₁, from (6), we can identify $C_1^{22}$. Since C₀ and C₁ are complete, the unknown $C_i^{22}$ for i > 1 can be identified considering that C_i = A₁C_i−1 for i > 1. Therefore, this model is identified in the MFD case.

5. Conclusions

In this work, we have helped to expand the set of VARMA models identified by extended Yule–Walker methods. It provides new necessary and sufficient conditions in the simple-frequency data case, and sufficient conditions in the mixed-frequency data case. The main results are embodied in two theorems, two corollaries and two counterexamples. The two counterexamples allow us to affirm that models are identifiable for which the sufficient conditions for identifiability in [1] do not hold.