1. INTRODUCTION
The aim of this paper is to shed some light on a problem that arises
in models that impose some sort of structure on covariance matrices. This
is the case with several popular models commonly employed in econometrics
and multivariate statistics: the most prominent in econometrics is
probably the structural vector autoregression (VAR) model, and much of the
present paper will focus on it. Possible extensions of the results
presented here to other contexts will be briefly discussed in the final
section.
The methodology of structural VARs (SVARs) was pioneered by Sims
(1980) and then made popular by countless
applications, some of which were highly influential (Blanchard and Quah,
1989, comes to mind). The issues involved in the
identification and estimation of such models were thoroughly investigated
in Giannini (1992) and Amisano and Giannini
(1997), and the present paper builds heavily on
that work.
In these models, it is assumed that a valid representation for a
vector of n observable variables yt
is a finite-order VAR1
For a complete and
detailed survey paper on VARs, see Canova (1995).
where P(L) is a matrix polynomial in the lag
operator, εt is a vector white noise sequence
with covariance matrix Σε, and the function
μt may depend on deterministic terms and a given
set of exogenous variables. In SVARs, εt can be
thought of as some function of a vector of n unobservable
variables ut (called the structural
shocks), which can be assumed independent, or at least uncorrelated,
and to have unit variance. The researcher's interest is normally
centered on recovering the effect of the
ut's on the
yt's.
In most models of this kind the link between the structural shocks and
the system innovations can be written as
where A and B are square nonsingular matrices of
order n, which are a function of the parameters to be estimated,
θ. In some other cases, the relevant covariance matrix is the
so-called long-run covariance matrix of yt,
which satisfies ΣLR =
C(1)Σε C(1)′, with
C(L) = P(L)−1. In the
rest of the paper, we will use the generic notation Σ for the
covariance matrix of interest.
We may define a covariance structure as a parametric model
featuring two matrices A and B such that
To avoid cumbersome notation, the reference to θ henceforth will
be dropped, and A and B will be implicitly assumed to be
functions of θ.
In this paper, we will consider cases when estimation of the
parameters θ is carried out by optimizing some objective
function:
where σ = vech(Σ) and v(θ) is the vector
function
Estimation of these models amounts to finding the vector θ such
that
A−1BB′(A′)−1
is “closest” (in a sense to be specified) to Σ.
The identification question arises because there is no guarantee that
is unique, because the objective function is defined in terms of
v(θ) and there may be some
(and a corresponding, different pair of matrices
) that give rise to the same value of
.
If the objective function f (σ,v(θ))
possesses derivatives up to the second order, identification can be
investigated through definiteness of the Hessian matrix at the optimum. As
a rule, however, the Hessian matrix is itself a function of the parameters
to be estimated, so it cannot be computed before estimation is carried
out, which renders this type of check nonoperational.2
A procedure is suggested in Giannini (1992) where identification is checked by computing the
information matrix at a random point in the parameter space. This
procedure hinges on the conjecture that in an identified model the
probability of making an incorrect decision is 0. This insight is made
more precise in Section 3.2, where it is shown that in such a case
underidentification occurs on an area with zero Lebesgue measure in the
parameter space.
As done by Johansen (
1995) in the context of linear models, the present
paper provides necessary and sufficient conditions for identification that
depend only on the constraints and can therefore be checked independently
of estimated parameters: this checking procedure can be made automatic, so
that it can be applied to systems of any dimension. Moreover, as a side
benefit, the analytical framework put forward here makes it possible to
give the researcher an intuitive insight as to
why a particular
model fails to meet the requirements for identification; an example will
be given in Section 5.3.
The plan of the paper is as follows. Section 2 provides a brief
reminder of some mathematical concepts that will be used in the rest of
the paper. Section 3 explores a special case in which the A
matrix is fully restricted to establish some concepts and exemplify them
more clearly, whereas Section 4 deals with the general case; some examples
are given in Section 5. Finally, Section 6 summarizes the results and
briefly discusses applications other than those presented here.
2. DEFINITIONS AND NOTATION
In this section, we define some concepts and terms that will be used
in the rest of the paper.
2.1. Identification—Generalities
We assume that estimation of the parameters θ is carried out by
optimizing an objective function f (σ,v(θ))
over a parameter space Θ.
It will be assumed that the objective function has derivatives up to
the second order and has at least one optimum
in the interior of Θ. At
,
we obtain an estimate of Σ that obeys
Obvious examples for f (σ,v(θ)) include
- concentrated Gaussian likelihood:
;
- minimum distance estimation (or generalized method of moments
[GMM]): f (σ,v(θ)) =
δ′Λδ where δ = v(θ) − vech Σ
and Λ is some positive definite matrix.
As stated in the introduction,
satisfying equation (5) are not necessarily unique: in fact, (5) is
satisfied by any pair of matrices A1 and
B1 such that
where Q is invertible and H is orthogonal; it is
clear that the matrices A1 and B1
can be considered an equivalent reparametrization of the original model.
The simplest case that can be taken as an example is when H is a
permutation matrix. In this case, B1 is simply
with its columns reordered, that is to say, the ordering of the
structural shocks ut in (2) is changed. The
case when H or Q is arbitrary is nevertheless
uninteresting, because the practically relevant issues arise when
considering whether such matrices Q and H can exist in a
neighborhood of the optimum. In the terminology of Rothenberg (1971), therefore, we are dealing with local,
rather than global identification.
3 In a globally identified model, the maximum of the objective
function is unique. Local identification, on the contrary, stipulates that
multiple maxima can exist, but each of these is an isolated point (see
also Gourieroux and Monfort, 1995, pp.
88–89). From a statistical viewpoint, the most important requirement
is local identification, because it makes it possible to resort to
standard proofs for asymptotic properties of extremum estimators. On the
other hand, local identification can often be made equivalent to global
identification by suitably reshaping the parameter space.
To achieve identification, some constraints on A and
B must be imposed. In this paper, we will follow a long-standing
tradition of imposing a system of ra +
rb = r linear constraints:4
Linear constraints are not only easier to manage
but are also often the most interesting because they lend themselves very
naturally to the representation of some economic theory: a typical example
is zero restrictions on some parameters.
(lowercase symbols will be used throughout to indicate vectorizations
of matrices, i.e., a ≡ vec A). Alternatively, the
constraints can be written in explicit form as follows:
where θ are the unconstrained parameters. The number of elements
in θ is p = pa +
pb, where pa =
n2 − ra and
pb = n2 −
rb. The p columns of the matrix
Sa and Sb form
bases for the null space of the rows of Ra
and Rb, respectively, so that
Ra Sa =
Rb Sb = 0 and
the matrices
[Sa|Ra′]
and
[Sb|Rb′]
have column rank n2 (the symbol | will be used
throughout the paper to indicate horizontal stacking of matrices).
If a pair of matrices A and B satisfies the system
of constraints (6)–(7), we call them admissible. The aim of
this paper is to establish conditions under which an optimum of the
objective function (4) corresponds to the only admissible pair within a
neighborhood.
The standard way of analyzing the constrained optimization problem
would be to start from the first-order conditions for an optimum in terms
of Lagrange multipliers, as in
where G(v(θ)) is the system of constraints
induced on v(θ) by the restrictions (6) and (7). For our
purposes, however, it is best to consider an equivalent way of expressing
these conditions, involving differentials:
if the Jacobian matrix Jv
fJθv has full
column rank, then we have identification at the optimum. This, in turn,
requires that the column rank of
Jθv is full, too. Then, the
system is identified if, inside an arbitrarily small neighborhood of
,
no other vector θ1 (and thus, no corresponding pair of
admissible matrices A1 and B1)
exists such that
Obviously, identification imposes an order condition: because the
number of elements in σ is n(n + 1)/2, the
column rank of the Jacobian matrix
Jθv cannot be full if θ
has more than n(n + 1)/2 elements. As is well known,
however, this is only a necessary condition.5
In fact, this point is sometimes overlooked in the applied
literature: e.g., the statement “In the structural VAR approach,
B can be any structure as long as it has sufficient
restrictions” was taken verbatim from a recent working paper. This
sentence seems to imply that the order condition is considered sufficient,
not just necessary.
In the next section, another necessary
condition, called the
structure condition, will be developed; it
covers some cases of interest where the order condition holds yet the
model is underidentified. It will be argued that a sufficient condition
for identification can be obtained by requiring that both order and
structure conditions hold.
2.2. Decomposition of Square Matrices
As is well known (see, e.g., Lütkepohl,
1996), any n × n matrix X can be
written as
where X+ is symmetric and
X− is hemisymmetric6
A square matrix X for which X =
−X′ holds is called hemisymmetric, or
sometimes skew-symmetric.
and they are defined as
Let us now consider the space
:
any element of this space can be considered the vectorization of an
(n × n) matrix. In a parallel fashion, the space
Ω can be subdivided into two orthogonal subspaces Ω+
and Ω−: any vector x ∈ Ω can be
written as x = x+ +
x−, where x+ =
vec(X+) ∈ Ω+ and
x− = vec(X−) ∈
Ω−. It can be shown that Ω+ has
dimension n(n + 1)/2 and Ω−
has dimension n(n − 1)/2. We call
matrix whose columns form a basis for Ω−, so that
any hemisymmetric matrix H has a vectorized form that satisfies
for some vector φ.
One useful operator7
An alternate
notation for Knn, which is sometimes used,
e.g., in Pollock (1979), is
.
in this context is the
n2 ×
n2 matrix
Knn, which is
defined by the property
Knn vec(
A) =
vec(
A′). It can be shown (see, e.g., Magnus and Neudecker,
1988, p. 46) that
Knn is symmetric and orthogonal, i.e.,
2.3. The Infinitesimal Rotation Operator
A matrix operator that will be useful in the rest of the paper is the
so-called infinitesimal rotation operator.8
For the properties of the infinitesimal rotation operator, see
Weisstein (2004).
Consider a vector
x0: we are interested in an infinitesimal displacement
of
x0 that preserves its norm, i.e., a vector
x1 =
x0 + d
x such that
d∥
x∥ = 0. Then it is possible to define a matrix
H as the matrix such that
x1 = (
I +
H)
x0. Two properties of
H will be of
interest.
(1) (I + H) must be orthogonal because
x1′x1 =
x0′(I + H)′(I +
H)x0 =
x0′x0 must hold for any
x0, and therefore (I +
H)′(I + H) = I;
(2) H is hemisymmetric because
d(x′x) = x0′dx
+ (dx′)x0 = 0 and therefore H
= −H′.
In this case, we say that the transformation is an infinitesimal
rotation and H is the corresponding infinitesimal rotation
operator.
Note that conditions 1 and 2 imply H′H = 0;
this result is a consequence of H being infinitesimal. An example
can be easily given when
:
in this case, we have
so
from a Taylor expansion we get
as δ → 0, Δx becomes dx and the
remainder term R(δ) disappears; therefore, when δ is
infinitesimal, dx = Hx, where H is
infinitesimal and hemisymmetric.
3. IDENTIFICATION IN THE
C-MODEL
We begin by examining the special case in which A =
I. In the terminology introduced in Giannini (1992), which will be adopted from here on, this
corresponds to the so-called C-model. In this section, therefore,
we analyze the special case where all the elements of A are
constrained with Ra = I and
da = vec I; because
Ra is full rank,
Sa does not exist and
sa = da. In
practice, we will only be concerned with constraints on the B
matrix: therefore, to avoid clutter, in this section the constraint
matrices Rb and
Sb will be referred to simply as R
and S.
The structure of interest here is
.
Suppose that another admissible matrix B1 =
B0 + dB exists, where dB is
infinitesimal. The model is underidentified if B1 is
observationally equivalent to B0, i.e., if
If (10) holds, it is possible to define an infinitesimal rotation
matrix H as the matrix such that B1 =
B0(I + H). From its definition, the
matrix B1 can be written in vectorized form as
follows:
If B1 is to be admissible, then we must have
Rb1 = d; however, this implies
Rb0 + R(H′ [otimes ]
I)b0 = d, and therefore,9
This reasoning could have been equivalently, and
certainly more compactly, put by requiring that, for
B0 + dB to be admissible, the condition
Rdb = 0 has to be satisfied, where dB =
BH for some hemisymmetric H; in this case, however,
brevity would have possibly come at the expense of clarity.
using
(9),
Consider now the ith row of R(H′
[otimes ] I). If we call ei the
ith column of the identity matrix, we have
where h = vec H and Ri
is an n × n matrix whose vectorization is the
ith row of R. To be an infinitesimal rotation matrix,
H must be hemisymmetric. Therefore, it is possible to write
h as
and equation (12) as
where
is any basis for Ω− (see Section 2.2). In short, if
a nonnull φ exists that satisfies equation (13) for each i,
the model is underidentified at θ.
Giannini (1992), in his analysis of the
C-model, indicates (p. 28) that identification holds if and only
if the matrix
has full column rank.10
The original
notation was slightly altered to match ours.
This condition was
obtained by considering maximum likelihood estimation, because it ensures
that the information matrix is positive definite. It is easy to see that
this condition is exactly equivalent to requiring that equation (13) has
no nontrivial solutions: if the matrix in (14) has full rank, then
for any nonnull φ.
Condition (13) can be rephrased as
where
and p is the number of constraints (the number of rows in
R). In other words, the system is unidentified at θ if some
φ ≠ 0 exists that satisfies the preceding equations for every
i. It is important to note that, for a given θ, the
existence of nonzero solutions to the system (15) (and therefore
underidentification) depends only on the matrices
Ti and the vectors
ti, which are functions of the
constraints alone. In general, it could be thought that the existence
of a solution to the system (15) for some φ ≠ 0 may depend on
θ. However, it will be shown in Section 3.2 that the choice of θ
is practically immaterial.
For any given θ, the system (15) can be written as
where
The matrix T(θ) has n(n −
1)/2 rows and p columns, so its rank must be less than or
equal to min(p,n(n − 1)/2). Solutions
to (16) do not exist (and hence the model is identified) if and only if
the rank of T(θ) equals n(n −
1)/2.
3.1. The Structure Condition
The existence of solutions to (16), and therefore underidentification,
occurs necessarily whenever the order condition fails, because in this
case p < n(n − 1)/2 and the rank
of T(θ) is, at most, p. However, there might be
cases when (13) holds for any value of θ ∈ Θ even though
the order condition might be met. We will call such a model
structurally underidentified: in these cases, there is at least
one H that satisfies equation (13) whatever the choice of θ.
Therefore, it must also be true that
for some infinitesimal rotation H.
By employing the properties of the Kronecker product and of the
vectorization operator in a manner similar to the one used in the
derivation of equation (15), it is possible to write the ith
column of (H′ [otimes ] I)S as
(H′ [otimes ] I)Sei =
(I [otimes ] Si)h, where
Si is an n × n matrix
whose vectorization is the ith column of S. Similarly,
we can transform (H′ [otimes ] I)s into
(H′ [otimes ] I)s = (I [otimes ]
S)h.
The problem can now be stated as follows: given the matrix
if there is an infinitesimal rotation H such that Th
= 0, then the system is structurally unidentified;
11 Notice that each row of the matrix T can be written
as
vec(Si′Rj)′
for all possible combinations of the columns of R and
[S|s]. This may help computationally.
Another point that may help in a practical software implementation is
that, as a rule, a large number of rows in T are zero and clearly
can be left out without influencing the rank of T.
in
turn, this implies the existence of a nonnull vector φ such that
.
As a consequence, a C-model is structurally identified if and only if
the matrix
has full column rank n(n − 1)/2. This
condition will be henceforth called the structure condition.
The structure condition can be more easily checked via the matrix
The system is structurally identified if and only if
is invertible. If not, the matrix H has a vectorization that
lies in the right null space of
.
Note that this condition is wholly independent of θ, so structural
identification is a property of the model as such. Therefore, this
condition can be checked prior to estimation, like the order
condition.
3.2. Sufficiency
Both the order and structure conditions are, by themselves, necessary
but not sufficient. However, the order and the structure condition form a
quasi-sufficient condition taken together: if equation (16) is
satisfied for some φ ≠ 0 beyond the trivial case of p <
n(n − 1)/2, then the values of θ that
satisfy (16) define a set of measure 0, and therefore the model is
identified almost everywhere in Θ.
This assertion can be proven by a line of reasoning similar to that in
Johansen (1995, p. 130): consider the matrix
T(θ) in (16) as a function of θ. For (16) to have
nonzero solutions, this matrix must have rank less than
n(n − 1)/2, and therefore the determinant of
the following square matrix:
must be 0. Because the function |D(θ)| is
a polynomial in θ, it is either identically 0 or has a set of
solutions that forms a closed set in Θ with zero Lebesgue
measure.
Because D has n(n − 1)/2 rows,
its determinant is clearly zero identically if the number of its columns
is less than n(n − 1)/2: this takes us back to
the order condition. Nevertheless, |D| = 0 may also
happen if the order condition is satisfied yet the rank of D is
less than n(n − 1)/2; but in this case,
equation (16) is satisfied for any θ, and therefore the model fails
to meet the structure condition. As a consequence, when both conditions
hold, (16) has no solutions in Θ but for a set of measure 0.
Moreover, this set is closed, and the set of points in Θ where the
two conditions together are sufficient to ensure identification is open
and dense in Θ.
3.3. An Example
Let us analyze a C-model where n = 2 and B
is lower triangular. As is well known, this case is identified, as
B is simply the Cholesky decomposition of Σ, which we assume
positive definite. In this case, the number of free parameters is 3 and
the parameter space Θ is the subset of
such that B is invertible, namely,
.
The matrix B has the form
so that the constraint matrices can be written as
and the corresponding S matrix is
(the vector s equals 0); as for
,
the only possible choice (up to a scalar) when n = 2 is the
vector
The order condition is evidently satisfied, as the number of free
parameters equals n(n + 1)/2 = 3. The structure
condition can be checked via
and therefore the model is identified almost everywhere in
.
In this case, the dimension of the matrices is small enough to make it
feasible to check identification by developing the argument analytically,
which is rather instructive. Because B must be lower triangular
to be admissible, the identification problem boils down to establishing
whether postmultiplying B by an arbitrary infinitesimal rotation
may result in another lower triangular matrix. In formulas:
The resulting matrix is clearly not admissible (i.e., lower
triangular) unless θ1φ = 0. If φ ≠ 0, then
θ1 must be 0. This implies that the only region in
where the model is underidentified is the plane θ1 = 0,
which has zero Lebesgue measure and is outside Θ by hypothesis
anyway.
The same result can be obtained by considering condition (15)
directly; because we have only one constraint, then we have only one
Ti matrix and one
ti vector, and these equal
whereas t1 = 0. The model is underidentified as
long as
holds for some φ ≠ 0, but in this case, the only solution is
θ1 = 0. Hence, the model is identified almost everywhere
in
and everywhere in Θ.
4. IDENTIFICATION IN THE
AB-MODEL
This case is the most general: contrary to the previous section, we
analyze the situation where neither A nor B is fully
restricted, so the identification question revolves around establishing
the existence of two matrices (A + dA) and (B +
dB) that are still admissible.
The constraints to be put on A + dA and B +
dB to remain admissible can be specified in a way akin to
equation (11) by writing
where (I + Q) is nonsingular and (I +
H) is orthogonal, so that
(in equation (20), there would be a term (H′ [otimes ]
Q), which, however, disappears because both matrices are
infinitesimal).
As in the previous section, H has to be an infinitesimal
rotation for (I + H) to be orthogonal. The only
additional complication with respect to Section 3 is that it is now
necessary to take the matrix Q into consideration also. Moreover,
Q is not necessarily an infinitesimal rotation operator, because
I + Q need not be orthogonal, although it is required
that it is invertible. However, if we only consider what happens in a
neighborhood of the optimum, it suffices to say that |I +
Q| ≠ 0 for any Q, as long as Q is
infinitesimal, by the continuity of the determinant function. Therefore,
if we define q = vec Q, it is sufficient to consider the
condition q ≠ 0 without any further qualifications.
The equation parallel to (12) is the following system:
identification holds at θ unless it is possible to find two
infinitesimal nonzero matrices Q and H (with H
= −H′) satisfying the preceding system.
With a little algebra, similar to that used in Section 3, we may
reexpress some of the matrix products as
because Knn q =
vec(Q′) (see Section 2.2) and
.
The symbol Ra,i indicates the
(n × n) matrix such that
vec(Ra,i)′ is the ith
row of Ra. These rearrangements lead us to
examine the following system of equations:
where
which admits nonzero solutions if and only if the model is
underidentified.
4.1. The Order and Structure Conditions
The order and structure conditions can be stated in a way very similar
to Section 3. Again, the order condition requires that the number of free
parameters pa +
pb does not exceed n(n +
1)/2 or, equivalently, that the number of restrictions put on
A and B is at least n2 +
(n(n − 1)/2). If this requirement were not
met, then there would always be solutions for some nonzero q
and/or φ to the equation
where
The structure condition requires that there are no infinitesimal
matrices Q and H (with H hemisymmetric) that
satisfy
By considering the columns of Sa and
Sb one at a time, we get
and
so that lack of identification implies (and is implied by) the
existence of nontrivial solutions to this system, which evidently
parallels the problem of finding solutions to equation (17). One thing
that must be noted for the index i is that, in general, it runs
from 1 to p, because the number of columns in
Sa and Sb is
p; however, if the parameter vector θ can be split into
θ =
[θa′|θb′]′,
as is usually the case, then the columns of
Sa from the (pa
+ 1)th onward are all zero and so can be omitted without affecting the
existence of solutions to the system (24)–(25); the same applies to
the first pa columns of
Sb.
The matrix T equivalent to the one in equation (18) then
becomes
and the system is structurally identified provided there are no
trivial solutions to
As a consequence, an operational procedure for checking the structure
condition could simply amount to verifying whether the matrix
is singular.12
The quasi-sufficiency property of the order and structure conditions
combined can be assessed by means of an argument similar to that developed
in Section 3.2 by noting that the rank of the matrix T(θ) in
equation (23) is less than n2 + (n(n
− 1)/2) either identically or for a set of measure 0 in
Θ.
4.2. Special Cases
In certain cases, checking the structure condition involves simpler
matrices; the C-model, e.g., clearly emerges as a special case:
if we impose the constraint A = I, then we have
Ra = I and
[Sa|sa]
= vec I, so the matrix T reduces to
which clearly has full column rank if and only if its southeast corner
has. This, in turn, is precisely the matrix T in equation
(18).
Another notable special case arises when it is assumed that
AΣA′ = I. This case is referred to as
the K-model in Giannini (1992), and,
because AΣA′ = I implies
A′A = Σ−1, it could be
conjectured that the requisites for structural identification ought to be
similar to those in the C-model. This is indeed the case, as the
restrictions put on B parallel those previously put on
A, for we have Rb = I and
[Sb|sb]
= vec I; as a consequence, equation (27) simplifies to
which implies
;
therefore, if a nonzero solution exists, the equality
must hold. This happens only if the rank of
is less than n(n − 1)/2.
5. A FEW EXAMPLES
5.1. An Unidentified 2 × 2 Case
This is an example of a model reported in Giannini (1992), where it is shown that the order condition is
not sufficient, by itself, to ensure identification. In the context of the
present paper, it provides a simple yet enlightening example of failure of
the structure condition. We have n = 2, and A has the
following structure:
in other words, any (2 × 2) admissible matrix has the same
elements on the diagonal, whereas the off-diagonal elements have the
opposite sign. In this case, the requirement that A be invertible
is accommodated by setting
,
as the only vector (θ1,θ2) that makes
A singular is the zero vector.
The fact that B = I by hypothesis allows us to
classify this model as a K-model and thus focus on (29) for
checking the structure condition, rather than on the more complex equation
(28). The constraint matrices can be written as
and both da and
sa are suitably shaped zero vectors.
The order condition is obviously met, because the number of free
parameters (two) is smaller
13 Or
equivalently, the number of constraints (two) is greater than
n(n − 1)/2 = 1.
than
n(
n + 1)/2 = 3. As far as the structure condition
goes, we can use equation (29); using the same
as in Section 3.3, we have
which has rank 0, and therefore the model is structurally
unidentified.
In this case, the matrices are small enough to analyze in some detail
the effect of an infinitesimal rotation on the matrix A
symbolically. Because any hemisymmetric matrix of order 2 can be written
as
then the product A1 = A(I +
H) equals
which is clearly admissible. Therefore, there is an infinity of
admissible matrices A1 that satisfy
.
5.2. The “Standard” AB Model
The model analyzed here corresponds to the most common setup in
structural VARs: the one where A is lower triangular with ones on
the diagonal and B is diagonal.
This example is admittedly rather contrived, as this model could be
easily reparametrized into a K-model, but here it just serves the
purpose of providing a simple example of constraints put on both
A and B. The constraint matrices are
The matrix
,
computed via equation (28), equals
Because
is invertible, the system is structurally identified. Therefore, both
order and structure conditions hold, and identification is attained.
5.3. The Blanchard (1989) Model
The model put forward in Blanchard (1989) is
possibly one of the most influential applications of structural VARs in
the applied macroeconomic literature. Five variables are modeled jointly:
gross national product (GNP), unemployment, prices, wages, and money
supply. The model can be described as an AB model with the following
structure:
where names for the individual parameters are chosen to be consistent
with Blanchard's. As can be easily seen, the model is underidentified
because the number of free parameters is 17 > n(n +
1)/2 = 15 and therefore the order condition is not satisfied. In the
original paper, identification is achieved by setting the two parameters
a34 and c to fixed values, thereby ensuring
that the order condition is met. It can be shown, via the approach
presented here, that those restrictions are indeed sufficient for the
structure condition to hold and, as a consequence, for the model to be
identified.
It may be argued, however, that identification is not necessarily
guaranteed by adding any pair of constraints, because the order condition
would be met but the structure condition might not be. In fact, an
alternative restriction strategy shows such a case:
14 The economic rationale of the choice presented here is
obviously irrelevant in the present context.
keep
a34 fixed (as in the original paper) at a value
but set the parameter c free and fix a21
instead to 1.
Because n = 5, the matrix
has 35 rows and columns and is too big to be reproduced here. However, it
can be checked that it is nonsingular and thus that the structure
condition holds. Moreover, the matrix
is block-diagonal and its north-west 9 × 9 block equals
The matrix
is nonsingular, and the model is identified. On the contrary, it is
interesting to consider what happens if, instead of setting
a21 to a fixed nonzero value, we set it at 0: in this
case, the relevant block of the matrix
becomes
As can be seen, the ninth column is a zero vector, so that
is singular and the structure condition is not met; as a consequence, in
this case the two restrictions
do not suffice to ensure identification.
Again, it is worthwhile analyzing the situation symbolically: because
the ninth column of
is zero, it follows that equation (27) is satisfied by a 10-element
vector φ = 0 and a 25-element vector q whose elements are
all 0 except for the ninth one. Hence, the corresponding Q matrix
has one nonnull element on the fourth row, second column. By considering
the product
it is easy to see that admissibility of the resulting matrix depends
on the value λa21: for a21 =
0, the matrix is admissible for any λ ≠ 0. The same goes for the
product (I + Q)B:
again, the matrix (I + Q)B is admissible
whatever the value of λ. This means that, despite the fact that the
number of free parameters equals n(n + 1)/2 = 15,
the model is underidentified, because there are infinite pairs of
admissible matrices A and B such that
A−1B is invariant.
It is interesting to note that an economic interpretation can be
given. If the level of economic activity does not have an instantaneous
impact on unemployment (a21 = 0), then it becomes
impossible to tell what the instantaneous impact of unemployment on wages
(λ + a42) is: this happens because the restriction
a41 = 0 (output does not affect wages instantaneously)
becomes uninformative.
6. CONCLUSIONS AND DIRECTIONS FOR FUTURE
RESEARCH
The main object of this paper is to put forth a method for assessing
identification of a covariance structure. The introduction of the
structure condition makes it possible to state a sufficient condition for
identification solely in terms of the set of constraints.
The ideas were presented here in the context of structural VARs
estimation but may prove useful for other multivariate models. A closely
related class of models is the structural vector correction models (see,
e.g., King, Plosser, Stock, and Watson, 1991),
where identification and estimation also take into account the possibility
of having to deal with a cointegrated system. This makes it possible to
identify a covariance structure by differentiating between permanent and
transitory innovations.
Another set of models that lend themselves quite naturally to be
analyzed in the present framework is that of LISREL (linear structural
relationships) models (or structural equation models). These models are
widely used in applied sociology15
For a
survey, see Mueller (1996).
and can be
briefly described as models where a vector of observable variables
yt is thought to depend on some exogenous
variables
xt through a known number of latent
variables λ
t. The assumed data generating process
can be summarized as
or graphically:
Estimation of these models is carried out by expressing the covariance
matrix of
(yt|xt)
(called the implied covariance matrix) in terms of the unknown
parameters, so it seems plausible that the concepts in this paper should
readily apply.
The identification issue in multivariate generalized autoregressive
conditional heteroskedasticity (GARCH) models could also be analyzed along
these lines: factor GARCH models have been put forward rather often to
overcome problems arising from the massive number of parameters necessary
for medium-scale models, and they essentially reproduce equation (2) with
conditionally heteroskedastic ut's; some
of the latest examples in this vein are Vrontos, Dellaportas, and Politis
(2003), van der Weide (2002), and Lanne and Saikkonen (2005). In these models, however, the very presence of
heteroskedasticity makes it impossible to use the same analytical
framework presented here, because it is not possible to write the
objective function as in (4). It can be said that, in general,
heteroskedasticity is actually helpful when it comes to identification:
the GO-GARCH (generalized orthogonal GARCH) model in van der Weide (2002), for example, could be considered a
C-model with unrestricted B. Another interesting
application of this idea was recently proposed by Rigobon (2003). Research by the author is currently under way
in this direction.
Some points may deserve further investigation. First, it might be
interesting to extend the present approach to make it feasible not only to
check whether the model is identified or not but also to compute its
overidentification rank. Furthermore, a theoretical issue that may be
possible to tackle within the present framework could be to analyze the
relationship between the set of points in the parameter space that give
rise to singular A and/or B matrices and the set
where identification fails, despite the fact that both the order and
structure conditions are satisfied. We do know that both are sets with 0
Lebesgue measure; it is natural to ask if anything more stringent can be
said in general. Finally, the possibility of generalizing the present
approach to nonlinear constraints should be mentioned. Because all the
arguments presented applied to an arbitrarily small neighborhood of the
parameter space, perhaps some kind of linearization of the constraints
might be attempted.
