Non-Identiability of VMA and VARMA Systems in the
Mixed Frequency Case
1
2
3
Manfred Deistler , Lukas Koelbl , Brian D.O. Anderson
Abstract
This paper is concerned with results on non-identiability of VMA and VARMA
systems from mixed frequency data.
Recently, identiability results for VAR
systems have been shown in a number of papers.
These results have been
extended to VARMA systems, where the MA order is smaller than or equal to
the AR order. In this paper we show that in the VMA case and in the VARMA
case, where the MA order exceeds the AR order, results are completely dierent.
Then, for the case, where the innovation covariance matrix is non-singular, we
typically have non-identiability - not even local identiability. This is due to
the fact that, e.g., in the VMA case, as opposed to the VAR case, the not directly
observed autocovariances of the output can vary freely. In the singular case,
i.e., when the innovation covariance matrix is singular, things may be dierent.
Keywords:
VARMA, VMA, mixed frequency, non-identiability
1 Corresponding
author. Manfred Deistler is with the Institute of Statistics and Mathemat-
ical Methods in Economics, Vienna University of Technology, Vienna, Austria, and Institute
for Advanced Studies, Vienna, Austria, e-mail: [email protected].
2 Lukas
Koelbl is with the Institute of Statistics and Mathematical Methods in Economics,
Vienna University of Technology, Vienna, Austria, e-mail: [email protected]
3 Brian
D.O. Anderson is with the Research School of Engineering, Australian National
University, Canberra, Australia, and Canberra Research Laboratory, National ICT Australia
Ltd., e-mail: [email protected]
Preprint submitted to Elsevier
July 8, 2016
1. Introduction
In many applications we encounter mixed frequency (MF) data in multivariate
time series, i.e., the univariate component processes often are available only at
dierent sampling frequencies.
Problems associated with MF data have been
considered, e.g., in Nijman [15], Kohn and Ansley [13], Chen and Zadrozny
[5], Ghysels et al. [8], Wohlrabe [17], Marcellino and Schumacher [14], Foroni and
Marcellino [6], Ghysels [7]. Here, we are interested in the problem of retrieving
the parameters of the underlying high frequency system, i.e., the system generating the output at the highest sampling frequency available in the data, from MF
data and, in particular, in identiability. Many papers dealing with mixed frequency data use the framework of VAR systems. G(eneric)-identiability from
MF data has been shown in Anderson et al. [3, 2] for the case of VAR systems
and in Anderson et al. [4], Zadrozny [18] for the case of VARMA systems, where
the MA order is smaller than or equal to the AR order. Here, we show that the
situation is completely dierent in case of VMA systems and of VARMA systems with MA order exceeding the AR order. This is a consequence of the fact
that non-observed autocovariances of the VMA output process can freely vary
in a certain neighborhood without violating the assumptions. For the VARMA
case considered here, this is true for special non-observed autocovariances.
In Section 2 we give a detailed analysis of non-identiability of VMA systems
from MF data for the case of stock as well as for the case of ow variables. In
Section 3 we show non-identiability for the VARMA case, where the MA order
exceeds the AR order. Whereas Section 2 and 3 deal with the case, where the
innovation variance matrix is non-singular, Section 4 gives generic identiability
results for the case, where the innovation variance matrix is singular with rank
being suciently small.
2
2. High Frequency VMA Systems and MF Data: Non-Identiability
Results
We consider (high frequency) multivariate MA(r) systems of the form
t∈Z
yt = νt + B1 νt−1 + ... + Br νt−r ,
where
r
is a specied integer,
T
Σ = E νt νt
.
Bi ∈ Rn×n
and
(νt |t ∈ Z)
(2.1)
is white noise with
In addition, we impose the strict miniphase condition
det b(z) 6= 0,
|z| ≤ 1
(2.2)
where
b(z)
= In + B1 z + ... + Br z r
is the corresponding VMA transfer function. We use
well as for the backward shift on
Z.
z
(2.3)
for a complex variable as
In this and in the next section we assume
that
q = rank (Σ) = n
(2.4)
holds. The VMA parameter space considered is of the form
ΘMA
=
n
o
2
∈ Rn r | det b(z) 6= 0, |z| ≤ 1 ×
|
{z
}
SMA
n
o
n(n+1)
2
vech (Σ) ∈ R
| Σ = ΣT , Σ > 0 .
|
{z
}
vec (B1 , . . . , Br )
Σ
Before going into the details let us outline the basic ideas in the ow of arguments behind Lemma 1, Theorem 1, Theorem 2 and Theorem 3. Consider
3
ΓMA
gure 2.1 below. In this gure
to
ΘMA , ιF
is the set of autocovariances corresponding
is the mapping describing this correspondence,
ιM
relates autoco-
variances to spectra and ιP MA parameters to transfer functions. The basic idea
is as follows: First we show for the MA(r) case, for every corresponding spectrum there is a neighborhood, which is contained in the set of MA(r) spectra,
not violating the strict miniphase assumption. In the next step we demonstrate
that by the bijective nature of the mapping between parameters and spectral
densities in the high frequency case, we obtain that varying a particular autocovariance, that cannot be directly observed from mixed frequency data, leads
to a corresponding variation of parameters; thus giving a (non-singleton) subset of a class of observationally equivalent parameters under mixed frequency
observations.
Let
ΘMA
be endowed with the usual Euclidean norm.
(as det is a continuous mapping) the set
and
Σ
is open in
ιP (τ ) = b(e−iλ )
R
n(n+1)
2
.
For every
τ =
SMA
As can be seen easily
dened above is open in
vec (B1 , . . . , Br )
∈ SMA ,
is uniquely given by (2.3). Conversely, for given
2
Rn
r
of course,
b(e−iλ ), τ
is
uniquely given by
1
2π
Bj =
Thus, ιP is a bijection between
SMA
and
ιP (SMA ).
Lemma 1. SMA
Z
π
b
−iλ
e
iλj
e
dλ.
−π
SMA
and ιP
(SMA )
and in this sense we identify
The following lemma is straightforward to show:
(endowed with the Euclidean metric) and
ιP (SMA )
endowed
with the supremum norm
b
−iλ
e
are homeomorphic. Here,
sup
k.kmax
=
sup
λ∈[−π,π]
b
−iλ
e
max
(2.5)
denotes the max norm for matrices with com-
plex entries.
4
In other words, the above lemma says that
SMA ,
when endowed with the Eu-
clidean metric and then endowed with the sup norm (2.5) has the same open
sets, i.e., the two metrics are topologically equivalent.
Now consider the autocovariance sequence
γ : Z → Rn×n of the VMA(r) process
(2.1), where
γ (j) = E yj y0T .
(2.6)
The corresponding spectral density is given by
f (λ)
where
∗
r
1 X
γ (j) e−iλj
2π j=−r
= b e−iλ Σb∗ e−iλ
=
(2.7)
denotes the conjugate transpose. As is well known, there is a bijection
between the covariances and the corresponding spectral density; let us write
ιM (γ) = f .
The set of all autocovariance sequences has to be non-negative
denite; in addition, for the MA case considered here
Moreover,
γ (0)
γ (j) = 0, ∀|j| > r.
is symmetric and positive denite and due to (2.2) we have
f (λ) > 0, ∀λ ∈ [−π, π].
Thus, for the VMA(r) case, we can parametrize the set
of all autocovariances by
ΓMA
=
γ=
vech (γ
T
(0)) , vec (γ (1) , . . . , γ (r))
f (λ) = ιM (γ) (λ) > 0, ∀λ ∈ [−π, π] .
Here, we have identied
above, we identify
vech (γ
T
T
T
∈R
T
(0)) , vec (γ (1) , . . . , γ (r))
n(n+1)
+n2 r
2
T
with
γ.
|
As
ΓMA and ιM (ΓMA ) and we can endow ΓMA with the Euclidean
norm as well as with the sup norm for spectral densities and these norms are
topologically equivalent. We have:
5
Theorem 1.
Proof. Let
Under the assumptions (2.2) and (2.4)
γ ∈ ΓMA .
principal minors of
f
tinuous functions in
neighborhood of
f >0
f
Then,
f (λ) > 0 ∀λ ∈ [−π, π]
being positive on
γ,
[−π, π].
ΓMA
R
n(n+1)
+n2 r
2
.
is equivalent to all leading
Since these minors are con-
they are bounded away from zero.
Thus, there is a
(in the sup and therefore also in the Euclidean norm), where
and thus a neighborhood of
γ
which is contained in
In the next step, let us consider the relation between
denoted by ιF
is open in
(θ) = γ , dened by (2.7).
ΓMA .
θ ∈ ΘMA
and
γ ∈ ΓMA ,
Clearly ιF is continuous. By the spectral
factorization theorem (see Rozanov [16], Hannan [9], Hannan and Deistler [11]),
ιF
is bijective. In Anderson [1], the continuity of the spectral factorization with
the sup norm for spectra and the
ΘMA
As can be seen easily, on
L2
the
norm for transfer functions has been stated.
L2
and the Euclidean norm are the same up
to normalization. Thus, we have:
Theorem 2.
ιF : ΘMA → ΓMA
The mapping
is a homeomorphism.
Summarizing, this can be depicted in the following diagram: Note that, strictly
speaking,
ιP
maps
τ
onto
−iλ
b
e
, but the extension of
ιP
to
ΘMA
is evident.
ιF
θ ∈ ΘMA
γ ∈ ΓMA
ιP
ιM
(b(e−iλ ), Σ)
f
Figure 2.1: Diagram of the homeomorphisms
In this diagram the index
and
P
M
stands for moments,
for parameters.
6
F
stands for the factorization
Now consider the MF case: Let
 
f
yt 
yt =   ,
yts
where the
the
nf -dimensional
fast component
ns -dimensional slow component yts
ytf
(2.8)
is observed for every
is observed only for
t∈Z
t ∈ N Z (N
and
being an
integer larger than one). Thus, we consider the MF stock case here. Throughout
we assume that
nf ≥ 1
ourselves to the case
and
N =2
ns ≥ 1.
Without loss of generality we restrict
here. Let


ff
fs
γ (j) γ (j)
γ (j) = 

γ sf (j) γ ss (j)
denote the corresponding partitioned autocovariance matrices. Then,
γ f s (j), γ sf (j)
and
γ ss (2j), j ∈ Z
can be directly observed, i.e., directly es-
timated from data, but this is not the case for
instance, the case
r = 1;
ΓMA
due to its openness.
dimension
ns × ns
γ ss (j), j
odd.
Consider, for
then, under the mixed frequency scheme described
above, the autocovariances
neighborhood of the true
γ f f (j),
γ ss (1) ∈ Rns ×ns
vech (γ
T
can be freely varied in a certain
T
(0)) , vec (γ (1))
T
without leaving the set
This variation corresponds to a set
say, and, by what was stated before
ι−1
F (O)
O ⊆ ΓMA
of
is a subset of
the class of observationally equivalent VMA systems of topological dimension
ns × ns .
Thus, we have shown:
Theorem 3.
Under the assumptions (2.2) and (2.4) in the MF stock case, the
MA(r) parameters are not identiable.
As immediately can be seen we do not even have local identiability in this
case. A heuristic argument concerning the non-identiability can be given by the
7
counting condition, e.g., for the case
n = 2, r = 1, N = 2 we can directly observe
6 autocovariances whereas the corresponding VMA(1) system is described by 7
free parameters. As shown in the following example, such an argument may be
misleading:
Example.
Consider the VMA(1) system, where

ff
b
B1 = 
bsf
and
|bf f |, |bss | < 1.


ff
0
σ
,
Σ
=


bss
σ sf
n = 2, bf s = 0
is xed

σf s 
>0
σ ss
(2.9)
Now let us dene an equivalent (in terms of directly observ-
able autocovariances) VMA(1) system with

B̄1
σ̄ ss
where



ff
fs
0
0
σ
σ




= B1 + δ  f s

 , Σ̄ = 
σ
sf
ss
− σf f 1
σ
σ̄
sf 2
ss 2
ss
ss
2 (σ )
1 + (b ) σ + 2δb + δ
σf f
=
> 0,
2
1 + (bss + δ)
δ ∈ (−1−bss , 1−bss ) is the free variation.
ΘMA , γ̄(0) = γ(0)
γ̄ ss (1) 6= γ ss (1) for δ 6= 0.
VMA(1) systems satises
varying
δ,
vec
B̄1
T
b̄f s = 0,

0

2
(σ f s ) 
ss
σ̄ − σf f
If we assume that the class of equivalent
it can be shown that the formulas above, by
dene a class of observationally equivalent systems, which is not a
singleton. Note that in this case we have 6 unknown parameters (3 for
3 for
Σ)
, vech Σ̄
but

0
γ̄(1) = B̄1 Σ̄ = γ(1) + δ 
0
and therefore
It follows that
B1
and
and 6 directly observable autocovariances and therefore the counting
condition fails.
8
T T
∈
Until now, only the case where stock variables are observed, has been considered.
Now we assume that the observed slow component
wt
is obtained by the linear
observation scheme
s
s
wt = k0 yts + k1 yt−1
+ · · · + kN −1 yt−N
+1 ,
where
ki ∈ Rns ×ns
are known and
k0
t ∈ NZ
(2.10)
is non-singular. Then, the corresponding
autocovariances which can be directly observed, are
T
γ f f (j) = E yjf y0f
,
= E yjf w0T ,
T
γ wf (j) = E wj y0f
,
γ ww (N j) = E wN j w0T , j ∈ Z.
γ f w (j)
As shown in Anderson et al. [2], Section 5,
structed from
struct
γ ss (j)
γ f f (j)
and
γ f w (j).
γ f s (j)
and
γ sf (j)
can be recon-
Therefore, in a last step, we have to recon-
from
γ ww (N j) =
N
−1
X
ki γ ss (N j + i − h) khT ,
j∈Z
(2.11)
i,h=0
since
γ ww (N j)
γ ss (j).
are the only directly observed autocovariances depending on
Note that
γ ww (N j) = 0
for
largest integer smaller than or equal to
|j| >
γ ss
N
+1
, where
bαc
α, and therefore we only have
1 directly obsered autocovariances from γ ww
from
r−1
and
denotes the
r−1
N
+1 +
r +1 unknown autocovariances
(with non-negative lags) which are unequal to zero. For simplicity, we
only consider the case
ns = 1 ,
generalizations being straightforward.
In this
PN −1
ss
γ ww (N j) =
i,h=0 ki kh γ (N j + i − h) and thus we
r−1
equations with
N + 1 + 1 equations and r + 1 unknown
case (2.11) changes to
obtain a system of
autocovariances
γ ss .
Under the additional assumption that
9
kN −1
is unequal to
zero, we can uniquely reconstruct the
γ ss
obtain identiability. Nevertheless, in the case
autocovariances
For the case
PN −1
i=0
γ ss
r >1
and therefore we
we have more unknown
than equations and thus we do not obtain identiability.
r = 1, ns ≥ 1
ki ⊗ ki ,
r=1
for the case
where
⊗
and
N ∈ N
we have to assume that
kN −1
and
denotes the Kronecker product, are non-singular to
obtain identiability. Note that this excludes the case of stock variables.
If we relax the strict miniphase assumption (2.2), in the sense that we allow
roots on and outside the unit circle, we may have g(eneric)-identiability in
certain extreme cases, e.g., for
r = 1, n = 2
and
b (z)
has zeros at
1
and
−1.
Here, we say that a subset of a given set is generic, if it contains an open and
dense subset of this set.
3. High Frequency VARMA Systems and MF Data: Non-Identiability
Results
In this section we consider VARMA(p, r) systems
yt + A1 yt−1 + ... + Ap yt−p = νt + B1 νt−1 + ... + Br νt−r ,
where
Ai , Bi ∈ Rn×n
(νt |t ∈ Z)
and
are parameter matrices,
is white noise with
T
Σ = E νt νt
b(z) = In + B1 z + ... + Br z r .
.
t∈Z
(3.1)
p, r ∈ N are specied integers and
Let
a(z) = In + A1 z + ... + Ap z p
We impose the usual stability condition
det a(z) 6= 0,
|z| ≤ 1
as well as the strict miniphase condition (2.2).
(3.2)
Throughout this section we
assume that
r > p.
10
(3.3)
The case
(yt )
r≤p
has been treated in Anderson et al. [4]. The spectral density of
is given by
f (λ)
=
=
1 −1 −iλ −iλ
a
e
b(e )Σb∗ (e−iλ )a−∗
2π
∞
1 X
γ (j) e−iλj .
2π j=−∞
−iλ
e
(3.4)
The parameter space considered here is of the form
ΘARMA
= {(A1 , . . . , Ap , B1 , . . . , Br ) | det a(z) 6= 0, |z| ≤ 1, det b(z) 6= 0, |z| ≤ 1
rank (Ap )
= rank (Br ) = n, (a(z), b(z))
Σ | Σ = ΣT , Σ > 0 .
Note that in the high frequency case
ΘARMA
=
rank (Br )
= n
×
(3.5)
is identiable (see Hannan [10],
Hannan and Deistler [11]) and that the conditions
and rank (Ap )
is left co-prime}
(a(z), b(z))
are generically fullled.
is left co-prime
In the next step we
(A1 , . . . , Ap ) are generically identiable from MF
T
f
postmultiply (3.1) by yt−j
, j = r + 1, . . . , r + np, and take
show that the AR parameters
stock data. If we
the expectation, we obtain analogously to Chen and Zadrozny [5]
f
yt−r−1
T T
T f
f
, yt−r−2 , . . . , yt−r−np

E yt

y
 t−1  T
T 
T  . 

f
f
 ..  y f

(−A1 , . . . , −Ap ) E 
,
y
,
.
.
.
,
y
t−r−np
t−r−1
t−r−2






yt−p
{z
}
|
ZARMA

where
ZARMA ∈ Rnp×npnf .
[2, 4], it can be shown that
Theorem 4.
=
(3.6)
With the same techniques as in Anderson et al.
ZARMA
has generically full row rank. Thus, we get:
Under the assumptions (3.2), (2.2) and (2.4) the AR parameters
11
of (3.1) are g-identiable.
Let
G = (In , 0, . . . , 0)
and

−A1
···


 In

A=



..
−Ap−1
.
be the companion form of the system parameters
a (z).
For
equal to
k=
α,
, where
N
dαe









In
r−p+1 −Ap
0
(A1 , . . . , Ap ) corresponding to
denotes the smallest integer larger than or
we obtain the following system of equations


γ (kN )




 γ ((k + 1) N ) 






 γ ((k + 2) N ) 




.
.


.




γ ((k + np − 1)N )
|
{z
}

=
O1
kN −r

GA





 GAkN −r AN 
γ (r)






.


.
.
 GAkN −r A2N  
.







.
.

 γ (r − p + 1)
.




GAkN −r A(np−1)N
|
{z
}
O
Analogously to Koelbl [12], Anderson et al. [4] it can be shown that
generically full column rank. Thus, we can uniquely obtain
O
has
γ ss (j), r ≥ |j| ≥
r − p + 1 ≥ 2 from (A1 , . . . , Ap ) and the second moments contained in O1 , which
can be directly observed in the case of stock data, by using the formula


γ (r)




.

 = OT O −1 OT O1 .
.
.




γ (r − p + 1)
12
(3.7)
γ ss (j), |j| > r
We obtain
γ ss (1)
cannot reconstruct
n2s r − p −
r−p N
Theorem 5.
case, for
from
γ (j) = −
from (3.7).
Pp
i=1
Ai γ (j − i).
Obviously, we
To be more specic, there are still
autocovariances missing.
Under the assumptions (3.2), (2.2) and (2.4) in the MF stock
r > p, ΘARMA
is not identiable.
Proof. Let the VARMA spectral density
f (λ)
of notation, we restrict ourselves to the case

0
f¯ (λ, δ) = f (λ) + 
0
be given by (3.4). For simplicity
ns = 1.
We dene

0
δ
−iλ
e
iλ
e
+
(3.8)


and
a
−iλ
e
f¯ (λ, δ) a∗
−iλ
e
=
f (λ) a∗

0
+a e−iλ 
0 δ
a
−iλ
e
−iλ
e
(3.9)

0
−iλ
e
 ∗
 a
iλ
−iλ
e
.
+e
Note that the left hand side of (3.9) is a trigonometric polynomial matrix of
degree
r.
As can be seen easily, Theorem 1 implies that for
δ
in a suitable
neighborhood of zero this left hand side is a spectral density of an MA(r) system.
Thus, (3.8) corresponds to a VARMA(p, r) spectral density where all
(yt )
second moments of
course
γ ss (−1)),
stay the same with the exception of
which has been disturbed by
δ,
where
neighborhood of zero. Because left co-primeness of
conditions rank (Ap )
sponding to
f¯ (λ, δ)
=
rank (Br )
= n
δ
γ ss (1)
(and of
varies in a certain
(a (z) , b (z))
and the rank
are open properties, the system corre-
can be described in the parameter space
ΘARMA .
Since the
mapping between the second moments and the parameters is homeomorphic,
the variation of
δ
gives the corresponding variation in the parameter space.
13
Note that the basic idea of the proof is not applicable for the case
p ≥ r.
Now
we consider the more general observation scheme (2.10). First of all, observe
that again
γ f s (j)
and
γ sf (j)
can be reconstructed from
γ f f (j)
and
Therefore, we can g-identify the system parameters, see (3.6).
step, we want to reconstruct the missing autocovariances
(2.11). For simplicity, we just investigate the case
straightforward.
For
k =
r−p
N
+1
γ
ns = 1,
ss
γ f w (j).
In a second
(j), j ∈ N
from
generalizations are
we can arrange the following system of
equations


γ ww (αN )














γ ww ((α + 1) N ) 


γ ww ((α + 2) N ) 



.

.
.


ww
γ
((α + np − 1)N )
|
{z
}

=
F
O1
where
O1F



γ f s (r)






ss (r)


γ




.


.


.






.


.
f
s
.
  γ (r − p + 1) 





P
N −1
ss (r − p + 1)
i−h+kN −r A(np−1)N
γ
(0, 1) G
k
k
A
i
h
i,h=0
|
{z
}
OF
and
OF
can be constructed from the observed autocovariances and the
system parameters. Again it can be shown that
rank and therefore we obtain
OF
has generically full column
γ ss (j) , j ≥ r − p + 1 ≥ 2.
to reconstruct the remaining unknown autocovariances
with the help of
with
r−p N

P
N −1
i−h+kN −r
(0, 1) G
i,h=0 ki kh A
P
N −1
i−h+kN −r AN
(0, 1) G
i,h=0 ki kh A
P
N −1
i−h+kN −r A2N
(0, 1) G
i,h=0 ki kh A












+1
γ ww (N j), j = 0, . . . ,
equations and
r−p N
r−p+1
In a last step, we have
γ ss (j), j = 0, . . . , r − p
. Thus, we get a system of equations
unknown autocovariances and therefore
we cannot expect to uniquely reconstruct the remaining autocovariances except
for the case
r = p+1
(under the additional assumption that
summarize, VARMA(p, p
not for
+ 1)
kN −1 6= 0).
To
systems are g-identiable in the ow case, but
r > p + 1.
4. High Frequency VARMA Systems and MF Data: The Singular
Case
Here, we consider the singular case, i.e., the case rank (Σ)
= q < n.
as is shown below, as opposed to the regular case rank (Σ)
14
= n,
In this case,
we may have
identiability for VMA and VARMA systems with
r > p,
if
q
is smaller than or
nf .
equal to
In an evident notation we write the spectral density
as
f (λ) of the VARMA system


ff
fs
f (λ) f (λ)
f (λ) = 
.
f sf (λ) f ss (λ)
Clearly,
f
has normal rank equal to
q.
(4.1)
Let

ff
Σ
Σ=
Σsf

Σf s 

Σss
it follows that
f f f (λ)
be partitioned accordingly.
Lemma 2.
For
Σf f > 0
is non-singular almost every-
where.
Proof. Let
H(t) =
span {yj
: j ≤ t}
and
Hf (t) =
span
n
o
yjf : j ≤ t
be the
Hilbert spaces spanned by all univariate components of the
yj , j ≤ t
f
the yj ,
be the projec-
j ≤ t,
f
tion of yt onto
ytf
=
f
yt|t−1
+νtf
and thus
respectively.
f
Furthermore, let y
t|t−1 and
H(t − 1)
Hf (t − 1),
= zt +ηt .
Σf f > 0 ,
and
Now
respectively.
f
Then, it follows that
ηt ηtT
H (t) ⊆ H(t) implies E
which is generically fullled, implies
zt
and
≥E
f f f (λ) > 0
νtf
T νtf
almost ev-
erywhere.
If
q = nf
holds, then generically (in the parameter space of singular VARMA
systems, where
rank equal to
q =
q.
rank (Σ) has been prescribed) already
In the mixed frequency case
obtained from mixed frequency data.
f f f (λ)
f f f (λ), f f s (λ)
and
has normal
f sf (λ)
are
This is true for the stock, but also for
the more general linear observation scheme (2.10).
15
Then, we obtain
f ss (λ)
(according to the Wiener ltering formula) as
−1
f ss (λ) = f sf (λ) f f f (λ)
and thus, in this case, i.e., if
f f s (λ)
(4.2)
f f f (λ) has already normal rank equal to q , f (λ) is
generically unique from the second moments which can be directly observed or
reconstructed in the case of more general linear observation schemes. As is easily
seen, this argument can also be applied to the case
q.
q < nf ,
if
f f f (λ)
has rank
In this way we have directly reconstructed, generically, the high frequency
spectral density and therefore the well known high frequency identiability result
holds.
Note that this result hold for all
between
p
and
Theorem 6.
p, r,
not depending on the relation
r.
For
nf ≥ q ,
the VARMA(p, r) classes are g-identiable.
5. Conclusions
This paper deals with the non-identiability of VMA and VARMA(p, r)
r>p
systems in the case of mixed frequency observations. The main result is that
opposed to the classes of VAR and VARMA systems, where the MA order is
smaller than or equal to the AR order, generic identiability cannot be obtained
in case of regular systems and stock variables. With some modications, these
results hold through for more general observation schemes. For the singular case
q = rank (Σ) ≤ nf
generic identiability can be obtained.
6. Acknowledgments
Support by the Oesterreichische Nationalbank (Oesterreichische Nationalbank,
Anniversary Fund, project number: 16546), the FWF (Austrian Science Fund
16
under contract P24198/N18) and NICTA is gratefully acknowledged. NICTA is
funded by the Australian Government.
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