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STRUCTURE-BASED SVAR IDENTIFICATION

EMANUELE BACCHIOCCHI RICCARDO “JACK” LUCCHETTI

Working Paper n. 2015-11

LUGLIO 2015

FRANCESCO GUALA

Working Paper n. 2011-18

SETTEMBRE 2011

ARE PREFERENCES FOR REAL?

CHOICE THEORY, FOLK PSYCHOLOGY,

AND THE HARD CASE FOR COMMONSENSIBLE REALISM

FRANCESCO GUALA

Working Paper n. 2011-18

SETTEMBRE 2011

Structure-based SVAR identification

Emanuele Bacchiocchi⇤ Riccardo (Jack) Lucchetti†.

July 10, 2015

Abstract

It may be desirable, for various reasons, to establish criteria for SVARidentification that do not depend on unknown parameters, but only on theset of restrictions that are imposed on the system a priori on theoreticalgrounds.

In the context of linear systems, this was accomplished in Johansen(1995). This paper extends and amends the approach proposed by Luc-chetti (2006); we introduce a set of criteria which ensure identification inde-pendently of unknown parameters for a reasonably general class of modelsand discuss its possible generalization.

Keywords: SVAR, identification, Rado condition.JEL codes: C01, C30, C32.

1 Introduction

As is well known, Structural VARs have been an essential tool for macroecono-metrics for over 30 years, since they were introduced by Sims (1980). Identi-fication issues have been a thorny issue ever since, and were first analyzed ina systematic way a decade later in the 1992 edition of Amisano and Giannini(1997).

It may be desirable, for various reasons, to establish criteria for SVAR iden-tification that do not depend on unknown parameters, but only on the set ofrestrictions that are imposed on the system a priori on theoretical grounds. Inother words, our focus will be on what we consider a very important issue:is it possible to assess identification of a SVAR on the basis of its constraintsalone? Because if it were so, identification would cease to be a property of

⇤Dipartimento di Economia, Management e Metodi Quantitativi (DEMM), Universita degliStudi di Milano.

†Dipartimento di Scienze Economiche e Sociali (DISES), Facolta di Economia ”Giorgio Fua”,Universita Politecnica delle Marche

1

the DGP. Instead, it would be possible to consider identification an inherentcharacteristic of the structure that we superimpose onto the data, independentof the observable evidence. In this case, we would call the model structurallyidentified.

In the context of linear systems, this was accomplished in Johansen (1995),who mostly focused on cointegrated systems, but introduced into the economicliterature a few results from matroid theory which enabled him to draw nec-essary and sufficient conditions for identification in linear systems in a generaland elegant way.

A similar approach was used in Lucchetti (2006) to analyze identification ofmodels, such as Structural VARs, in which linear constraints are imposed on acovariance system. As we will show, the “structure condition” defined in thatarticle is not sufficient, as claimed. In this paper, we extend and amend thesame approach and introduce a set of criteria which ensure identification in-dependently of unknown parameters for the most widely used class of models(the C-model, in Amisano and Giannini’s taxonomy).

In doing so, we also achieve the goal of generalising the approach by Rubio-Ramirez, Waggoner, and Zha (2010), which has gained a notable popularity inrecent years.

The structure of the article is as follows: in the next section, we examine theissue of identification and spell out our main result; then, in Section 3, we givea few examples of application of our criterion. Section 4 extends the resultsto the more general AB-model, with some examples discussed in Section 5.Section 6 concludes and outlines desirable extensions and generalizations ofour present work.

2 Identification in the C-model

2.1 Basic concepts and notation

In this section, we review the identification issue in the C-model variety ofStructural VARs, mainly to establish notation. Following Amisano and Gian-nini (1997), we call a C model a SVAR of the kind

F(L)yt = µt + #t (1)#t = Cut (2)

in which the n-vector of one-step-ahead prediction errors #t is assumed to be alinear function of orthogonal structural shocks ut, so that V(#t) = S = CC0. InAmisano and Giannini’s original classification, this can be seen as a special caseof the so-called AB model A#t = But which is, however, much more seldomused in its full generality. The vast majority of empirical applications relies ona structure such as the one displayed in equation (2).

Under normality, the average log-likelihood can be written as

L = const � ln |C|� 0.5 · trhS(CC0)�1

i

and first-order conditions for maximization are equivalent to

S = CC0 (3)

2

where S is the sample covariance matrix of the VAR residuals; greater gen-erality can easily be achieved by considering equation (3) simply as a set oforthogonality relationships of the kind

E

"#it# jt �

n

Âk=1

cikcjk

���Ft�1

#= 0,

which makes it possible to extend the present setup to QMLE or GMM estima-tion if necessary.

As is well known, this problem is underidentified, since if (3) holds, thenalso S = C⇤C0⇤ holds, where C⇤ = CP and P is an arbitrary orthogonal matrix.

In order to achieve identification, it is necessary to make more qualificationsto the above setup: the traditional solution has been to impose a system of plinear constraints on C of the kind

Rvec C = d. (4)

A special case of linear restrictions occurs when C is assumed to be lower-triangular. This was Sims’s (1980) original proposal, and is sometimes calleda “recursive” identification scheme. It has a number of interesting properties,among which the fact that the ML estimator of C is just the Cholesky decom-position of S, the sample covariance matrix of VAR residuals. This has beenthe most frequently used variant of a SVAR model, partly for its ease of inter-pretation, partly for its ease of estimation.

The general case (4), first analyzed in Amisano and Giannini (1997), hasbeen recently re-assessed and somewhat extended in a recent contribution byRubio-Ramirez, Waggoner, and Zha (2010), who provide conditions for globalidentification based on linear and some nonlinear restrictions of the parame-ters. However, their general results are all obtained by considering equation-by-equation restrictions (which are less general than those expressible via equa-tion 4); moreover, the rank condition they obtain rely on the availability of theunknown parameters, which is precisely what we want to avoid.

An alternative identification strategy was introduced by Rigobon (2003),who suggested to exploit heteroskedasticity to identify simultaneous equationmodels without having to impose restrictions on the parametric space.1 Thisapproach has been extended to SVAR models by Lanne and Lutkepohl (2008)and generalized in Bacchiocchi (2014) and Bacchiocchi and Fanelli (2014), whocombine heteroskedasticity and structural breaks with the traditional approachof imposing restrictions on the parameters of the model.

In this paper, we focus on the traditional identification strategy and weleave generalizations for future work; as a consequence of equation (4), thevector c = vec C can be written as

c = Sq + s (5)

where q is the vector contianing the q = n2 � p free parameters (assumedvariation-free). This setup is more general than it looks, since it can also han-dle long-run constraints of the kind proposed by Blanchard and Quah (1989)

1The idea that heteroskedasticity can be helpful in identifying econometric models has beenoriginally proposed by Sentana and Fiorentini (2001) in a factor model context, that could be ex-tended to SVAR models too.

3

or King, Plosser, Stock, and Watson (1991). However, it cannot be used withmore exotic identification schemes such as inequality-based ones like in Uhlig(2005).

As argued in Amisano and Giannini (1997) and Lucchetti (2006), local iden-tification is tied to the rank of the matrix

H = R(I ⌦ C) eDn (6)

being full; in this case, eDn is a n2 ⇥ n(n�1)2 matrix whose columns span the null

space of the duplication matrix Dn (see Magnus, 1988).2 Note that, in the caseof exact identification, this matrix is square; introducing extra restrictions addsextra rows, but the number of columns, which only depends on n, remainsunchanged.

This introduces a necessary and rather obvious order condition by whichthe number of linearly independent restrictions p must be at least n(n�1)

2 . How-ever, as is well known, the order condition is necessary but not sufficient. Ev-idently, the elements of H depend on the unknown parameters, since the Cmatrix is a function of q, so it would seem that its rank cannot be establishedin general, but only on a case-by-case basis for each point in the parameterspace Rq. Lucchetti (2006) introduces an additional condition, called the struc-ture condition, which is independent of the matrix C and should ensure (to-gether with the order condition) sufficiency for identification apart from a zero-Lebesgue-measure set in the parameter space. Unfortunately, there are caseswhen this does not work. In the next section, we explain why and provide acounter-example.

2.2 An example when the structure condition fails

As an illustrative example, consider a case with 4 variables, in which the con-straints are

c41 = c32 = c23 = c12 = c13 = c14 = 0

Since n = 4, n(n�1)2 = 6 so the order condition is satisfied. In this particular

case the H matrix in equation 6 is square but singular for any choice of C. Asan example, consider this particular choice for C (non-zero entries are randomuniforms, just for the sake of the example):

C =

2

664

0.74 0 0 00.40 0.40 0 0.130.31 0 0.23 0.86

0 0.65 0.25 0.82

3

775

in which we take R to be

R =

2

6666664

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

3

7777775.

2The matrix eDn can also be defined as a basis for the space of the vectorization of n ⇥ nhemisymmetric matrices, so that any hemisymmetric matrix H has a vectorized form which satis-fies h = eDn j for some n (n � 1) /2 ⇥ i vector j.

4

and d = 0. Clearly, the ordering of the rows of R is immaterial for the purposesof identification. The matrix H, whose rank ought to be full if the model wereidentified, is

H =

2

6666664

0.65 0.25 0.82 0 0 0�0.31 0 0 0.23 0.86 0

0 �0.40 0 �0.40 0 0.13�0.74 0 0 0 0 0

0 �0.74 0 0 0 00 0 �0.74 0 0 0

3

7777775

which is, instead, singular: the set of its three rightmost columns has evidentlyrank 2. Following the arguments in Lucchetti (2006), it can be shown that thisimplies the existence of an infinitesimal rotation matrix of the form

2

664

0 0 0 00 0 a b0 �a 0 c0 �b �c 0

3

775

which, in turn, implies the existence of an orthogonal matrix which rotates,in a neighborhood of C, the space of structural shocks in an observationallyequivalent way, so to make the system under-identified.

Furthermore, it is also instructive to explore the rank deficiency of H byconsidering its rows as well as its columns. First, it should be noted that, underthe above set of restrictions, S1,4 = S4,1 = 0 identically, since the first and thefourth rows of C are orthogonal by construction. We conjecture that this leadsto underidentification since the corresponding moment condition on E(#1,t ·#4,t) contains no information on any of the elements in C.

This is mirrored by considering that the top row in H is a linear combinationof the bottom three rows. In terms of identification, the meaning of this factis that, given S1,4 = 0, the four restrictions on c41, c12, c13 and c14 could becombined into three linear restrictions on C. Note that this happens despitenone of the linear constraints being linearly dependent on the other ones, asthe constraints matrix R has full row rank.

In Lucchetti (2006), it was argued that checking for the rank of H is equiva-lent to considering the existence of solutions to the system

j0 eD0n(I ⌦ R0

i)[Sq + s] = j0Tiq + j0ti = 0 for i = 1 . . . p (7)

with

Ti ⌘ eD0n(I ⌦ R0

i)S

ti ⌘ eD0n(I ⌦ R0

i)s

and Ri is a n ⇥ n matrix whose vectorization is the ith row of R. In other words,the system is unidentified at q if some j 6= 0 exists which satisfies the aboveequations for every i. While this is true, the check proposed in Lucchetti (2006)is flawed, and it is instructive to see why; construct the matrix

T =

2

6664

R(I ⌦ S1) eDnR(I ⌦ S2) eDn

...R(I ⌦ Sq) eDn

3

7775

5

where Si is an n⇥ n matrix whose vectorisation is the i-th column of S; solutionof the system of equations (7) amounts to finding q and j such that

(q0 ⌦ Ip)Tj = 0;

clearly, this is always possible if the column rank of T is not full, so the structurecondition in Lucchetti (2006) amounts to checking for the rank of T; however,the equation above may also have solutions if a nonzero j exists such that theproduct Tj is non-null, but lies in the kernel space of (I ⌦ q0), that is Sp(q? ⌦Ip).

Since the structure condition fails to capture this particular feature of themodel, Lucchetti (2006)’s setup erroneously labels the above model as identi-fied. In the next section, we propose a theorem that overcomes this deficiency,by defining a necessary and sufficient condition for structural identification.For reasons that will become obvious, we will call this the Rado condition.

2.3 The measure of under-identified points in the parameter

space

The next result, already presented in Johansen (1995) for linear systems ofequations and Rubio-Ramirez, Waggoner, and Zha (2010) for global identifi-cation in SVAR models, helps in practically checking whether a model is effec-tively identified or not.

Theorem 1. Consider the SVAR model in Eq. (2), with restrictions on the parametersas described in Eqs. (4)-(5). If the order condition is met, then either H is singular forall values of q 2 Rq or the set of vectors q that makes it singular has zero Lebesguemeasure.

Proof. H has p rows and n(n�1)2 columns, so if the order condition p � n(n�1)

2is met, then the rank of H0H equals that of H. Since C in equation (6) is anaffine function of q, it follows that the determinant |H0H| is a polynomial offinite order in q. As is well known (see Traynor and Caron (2005)), a finite-order polynomial function Rq 7! R is either identically 0, or non-zero almosteverywhere. Call Q0 the set of q satisfying |H0H| = 0; then either Q0 = Rq, orQ0 ⇢ Rq, where l(Q0) = 0, and the claim follows.3

The most powerful consequence of the above theorem is that if one q existssuch that the model is locally identified at it, then it is locally identified almost ev-erywhere. This result provides theoretical justification to the standard practiceof checking identification by simply calculating rk(H) for randomly selectedvalues for the free parameters q.

Moreover, it should be noted that, by the continuity of the differential func-tion, if Q0 ⇢ Rq, there exists some q⇤ /2 Q0 in every neighborhood of q 2 Q0,so the subset of parameter points for which identification is attained is eitherempty, or dense in Rq.

3We use l(·) for Lebesgue measure, as per standard notation.

6

2.4 The main theorem

As we argued, a necessary and sufficient condition for the model to be struc-turally identified is that the matrix H in equation (6) is full rank for any possiblevalue of the unknown parameter vector q (apart form a zero-measure subset).To analyze the conditions under which this is true, we study the equivalentproblem of the existence of solutions to the system of equations (7).

In order to do this, define T(q) as

T(q) =⇥

T1q + t1 T2q + t2 · · · Tpq + tp⇤

. (8)

Note that, in the case of d = 0, all the ti-s are zero.4 It is useful, at this point, torestate Rado’s (1942) theorem, in the form employed in Johansen (1995):

Theorem 2. Suppose you have a collection of matrices M1, M2, . . . , Mn; then, inorder to obtain a set of linearly independent vectors v1, v2, . . . , vn, where vi = Mili,a necessary and sufficient condition is that for all k = 1, . . . n and all sets of indices1 i1 < ... < ik n,

rk⇥Mi1 | . . . |Mik

⇤ � k.

Now, define Ti = (Ti | ti) and consider

T =⇥

T1 T2 · · · Tp⇤

,

whose size is n(n�1)2 ⇥ pq with pq > n(n�1)

2 . This matrix collects all the vectorscomposing Ti, i = 1 . . . p. The next theorem, based on the result in Theorem 1,when combined with the order condition p � n(n�1)

2 , provides a necessary andsufficient condition for the T (q) matrix to have full rank n(n�1)

2 , independentlyof the free parameters q.

Put a different way, the theorem provides a necessary and sufficient condi-tion for the model to be identified that is only based on the imposed restric-tions, and that can be easily checked before the estimation process. As in Jo-hansen (1995), it can be said that identification of the SVAR is a characteristicof the model, rather than being tied to the parameter matrix C.

Theorem 3. Consider the specification in equations (2)-(3) and the set of restrictionsgiven by equations (4)-(5), with p � n(n�1)

2 . Given an integer 1 k n(n�1)2 and

a corresponding integer h = p � n(n�1)2 + k, consider all possible sets of h increasing

integers between 1 and p: 1 i1 < i2 < . . . ih p; if

rk�Ti1 | . . . |Tih

� � k, (9)

for any k, then the function rk[T(q)] equals n(n�1)2 almost everywhere in Rq.

We call the Rado condition the condition by which the inequalities (9) are simul-taneously satisfied. Therefore, the SVAR model is structurally identified if and only ifthe Rado condition is met, except for a zero Lebesgue measure set of values for q.

4This, for example, is the case of recursive identification and is by far the most common setupin the C model.

7

Proof. The theorem directly derives from Theorem 1 and Theorem 2. Provingthe necessity condition is trivial: if the model is identified, then T(q) is of fullcolumn rank. As a consequence, it will be possible to choose some vectors qi,for i = 1 . . . p, or equivalently (almost surely) a common vector q, such thatthe n (n � 1) /2 columns in T(q) are linearly independent. This implies that,for each k, the matrix

�Ti1 | . . . |Tik

�must contain at least k linearly independent

vectors. If, instead, the conditions for theorem 2 are not met for a generic setof vectors qi, then, given Theorem 1, they will not be met anywhere in theparametric space, so that Q0 = Rq.

As for sufficiency, consider the just identified case first, in which p = n(n�1)2 .

If the conditions of Theorem 2 are met, then it is possible to choose vectorsTiqi + ti that are linearly independent such that the matrix

⇥T1q1 + t1 | T2q2 + t2 | . . . | Tpqp + tp

⇤

is invertible for such specific qi vectors. But if this is possible, then the possi-bility that Q0 = Rq is ruled out; therefore, even if a specific point q may belongto the zero-measure set Q0, identification is attained at some other (arbitrarilyclose) point q⇤.

Finally, consider the over-identified case, i.e. p � n(n�1)2 . In this case, the

number of columns in T (q) as defined in Eq. (8) is larger than the number ofrows. Thus, it is impossible to check a condition for all columns of the T(q) tobe linearly independent, as at least p � n(n�1)

2 will be obtained as linear com-bination of the others. Therefore, we have to verify that the spaces spanned by�

T1| . . . |Tp�

will generate exactly n(n � 1)/2 linearly independent vectors. Toprove the result it is sufficient to start our rank checks at h =

hp � n(n�1)

2 + 1i

instead of at h = 1.

It is perhaps useful to give an explicit example of the condition on Theorem3: Consider a trivariate system with 4 restrictions so that n = 3, n(n�1)

2 = 3and p = 4. Theorem 3 states that we first we have to check the rank of all thematrices of the type

�Ti1 |Ti2

�, that is for (i1, i2) equal to

(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4);

the rank of all the above combinations should be at least 1. Then, we considermatrices of the kind

�Ti1 |Ti2 |Ti3

�, so we need the combinations below:

(1, 2, 3), (1, 3, 4), (2, 3, 4);

all these matrices must have at least rank 2. Finally, we check for the only com-bination of 4 submatrices: (T1|T2|T3|T4); this matrix must have rank 3.

If p n(n�1)2 the order condition fails, and the model cannot be identifiable

as there are not enough empirical moments in the covariance matrix S to esti-mate the free parameters of the model. If, instead, p � n(n�1)

2 , the number ofimposed restrictions is large enough compared to the number of distinct ele-ments in S, and the necessary condition is met. At this stage, the condition inEq. (9) becomes a necessary and sufficient condition for the identifiability ofthe SVAR model.

8

3 A few example C-models

The next examples provide some details on the implementation of the identifi-cation condition developed in Theorem 3 presented in Section 2.

3.1 Unidentified bivariate SVAR

This example is analyzed in detail both in Amisano and Giannini (1997) andin Lucchetti (2006). Consider the following bivariate SVAR model where thematrix of contemporaneous relations is defined as

C =

✓q1 q2

�q2 q1

◆.

The structure of the C matrix presents a problem similar to the one dis-cussed in Section 2.2, i.e. it implicitly imposes a covariance matrix of the re-duced form as follows

CC0 =✓

q21 + q2

2 00 q2

1 + q22

◆

where the covariance or the residuals is restricted to be zero and the two vari-ances are equal. The model thus presents two parameters to be estimated butonly one empirical moment. The matrices of restrictions are

R =

✓1 0 0 �10 1 1 0

◆S =

0

BB@

1 00 �10 11 0

1

CCA eDn =

0

BB@

01�10

1

CCA .

The T1 and T2 matrices become

T1 = D02�

I2 ⌦ R01�

S =

=�

0 1 �1 0�

0

BB@

1 0 0 00 �1 0 00 0 1 00 0 0 �1

1

CCA

0

BB@

1 00 �10 11 0

1

CCA =�

0 0�

T2 = D02�

I2 ⌦ R02�

S =

=�

0 1 �1 0�

0

BB@

0 1 0 01 0 0 00 0 0 10 0 1 0

1

CCA

0

BB@

1 00 �10 11 0

1

CCA =�

0 0�

.

The order condition is clearly met ( n(n�1)2 = 1 and p = 2). Using Theorem

3, however, gives the following rank conditions

rk (T1) = 0 < 1 , rk (T2) = 0 < 1 , rk�

T1 | T2�= 0 < 2 .

The model, thus, cannot be identifiable.

9

3.2 Trivariate Cholesky SVAR

Consider the triangular trivariate SVAR model. Rubio-Ramirez, Waggoner,and Zha (2010) prove that this model is not simply locally identified, but alsoglobally. The matrix of simultaneous relations is defined as

C =

0

@⇤ 0 0⇤ ⇤ 0⇤ ⇤ ⇤

1

A

where asterisks (‘⇤’) denote unrestricted coefficients. The matrices of restric-tions can be written as

R =

0

@0 0 0 1 0 0 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 0

1

A , S =

0

BBBBBBBBBBBB@

�1 0 0 0 0 00 0 0 �1 0 00 0 0 0 �1 00 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 1

1

CCCCCCCCCCCCA

.

The order condition is clearly satisfied. The T1, T2 and T3 matrices are

T1 =

0

@�1 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

1

A

T2 =

0

@0 0 0 0 0 0�1 0 0 0 0 00 0 0 0 0 0

1

A

T1 =

0

@0 0 0 0 0 00 0 0 �1 0 00 1 0 0 0 0

1

A .

The rank condition provided by Theorem 3 gives the following results

rk (T1) = 1 � 1, rk (T2) = 1 � 1, rk (T3) = 2 � 1

rk (T1|T2) = 2 � 2 rk (T1|T3) = 2 � 2 rk (T2|T3) = 2 � 2

rk�

T1 | T2 | T3�= 3 � 3

indicating that the model is identifiable.

3.3 Unidentified four-variable SVAR

In this section we use the new results in Theorem 3 to show that the unidenti-fied SVAR model described in Section 2.2 can now be correctly detected. Thenecessary and sufficient condition for identification in Eq. (9) suggests to cal-culate the rank of all possible combinations of the T1, . . . , T6 matrices, where

10

p = 6 indicates the number of restrictions (rows of R). This is rather boring todo by hand, but is also easy to check automatically, via appropriate software.Considering the example described in Section 2.2, the rank condition is verifiedfor singles, pairs and triples of Ti, i = 1, . . . , 6, but fails when we check

rk (T1 | T4 | T5 | T6) = 3 4

indicating that the model is under-identified, in accordance to the previousanalysis reported in Section 2.2.

4 Identification in the AB-model

Following the terminology of Amisano and Giannini (1997) and Lutkepohl(2006), the AB-model considers possible simultaneous relationships among theobservable variables and the structural shocks. Classic empirical applicationsusing this specification can be found, among many others, in Bernanke (1986),Blanchard (1989) and Blanchard and Perotti (2002). Concentrating out the dy-namic part of the VAR, the simultaneous relations are defined by

A#t = But (10)

where, as before, #t is the one-step-ahead prediction errors, with E (#t#0t) =S, and ut collects the standardized and uncorrelated structural shocks. Thespecification in Eq. (10) naturally induces a set of non-linear restrictions on theparameter space given by

ASA0 = BB0. (11)

The number of parameters, here, is clearly 2n2, while that of empirical mo-ments remains ‘at most’ n(n + 1)/2, i.e. the number of elements in the es-timable covariance matrix of the residuals S. As a consequence, at least p =2n2 � n(n + 1)/2 = n2 + n(n � 1)/2 restrictions on the parameters must beincluded for identification.

The first systematic contribution in terms of the identification has been pro-vided by Amisano and Giannini (1997). Write the pa and pb independent re-strictions on A and B as

Ravec A = ra and Rbvec B = rb, (12)

or equivalently, in explicit form,

vec A = Saqa + sa and vec B = Sbqb + sb; (13)

where qa and qb are of dimension qa ⇥ 1 and qb ⇥ 1, respectively, and containthe q = qa + qb free parameters to estimate; Amisano and Giannini (1997) intro-duce a necessary and sufficient identification condition based on the followingpb ⇥ [n(n � 1)/2 + qa] matrix

Rb�

B ⌦ BB0�eDn

���� �h

A�1 ⌦ �BB0��1

iSa

�(14)

that must have full column rank for the model to be identified. As for the C-model previously discussed, this matrix depends on the true values of A andB, that are unknown.

11

In what follows, we use the intuition in Lucchetti (2006) and check for theexistence of a pair of matrices

A⇤ = A + dA = (I + Q) AB⇤ = B + dB = (I + Q) B (I + H)

that are observationally equivalent to A and B in a neighborhood of the twotrue matrices A and B.

The two infinitesimal transformation matrices (I + Q) and (I + H) mustobey different requisites: (I + H) should be orthogonal, while invertibility issufficient for (I + Q). This implies that, while it is sufficient to consider a sim-ple infinitesimal non-zero matrix Q 6= 0, H must be an infinitesimal rotationwith H = �H0. The two new matrices of parameters A⇤ and B⇤ will be admis-sible if

Rada = Ra (I ⌦ Q) (Saqa + sa) = 0Rbdb = Rb

⇥(I ⌦ Q) +

�H0 ⌦ I

�⇤(Sbqb + sb) = 0.

where a = vec A and b = vec B. With a little algebra, the identification condi-tion can be shown to be equivalent to the following system of equations5

⇥q0 | f0 ⇤

V(q) = 0 (15)

where

V(q) =

"Ua

1q + ua1 · · · Ua

pa q + uapa Ub

1q + ub1 · · · Ub

pbq + ub

pb

0 · · · 0 Tb1 q + tb

1 · · · Tbpb

q + tbpb

#(16)

with

Uai = Knn (Ra,i ⌦ In) Sa ua

i = Knn (Ra,i ⌦ In) sa i = 1, . . . , paUb

i = Knn (Rb,i ⌦ In) Sb uai = Knn (Rb,i ⌦ In) sb i = 1, . . . , pb

Tbi = eD0

n

⇣In ⌦ R0

b,i

⌘Sb ub

i = eD0n

⇣In ⌦ R0

b,i

⌘sb i = 1, . . . , pb.

in which Ra,i and Rb,i, similarly to Ri defined in Section 2.1, are defined suchthat their vectorization is equal to the i-th row of A and B, respectively. The ma-trix Knn is the n2 ⇥ n2 matrix defined by the property Knnvec (A) = vec (A0).6

If the system admits as its unique solution the null vector [ q0 | f0 ] = [ 0 ],or equivalently, V(q) has full row rank, then the model is identified. Obviously,V(q) is a function of the free parameters q, but then we can employ Theorem2 and obtain a necessary and sufficient condition independent of q like we didfor the C-model, discussed in Theorem 3.

For convenience, define

V(q) =⇥

V1 · · · Vpa Vpa+1 · · · Vpa+pb

⇤ �Ip ⌦ q

�=

= V

�Ip ⌦ q

�

5See Lucchetti (2006), pag 247, Eq. (23).6It can be shown that Knn is symmetric and orthogonal; see for instance Magnus and Neudecker

(1988, p. 46).

12

where q =�q0a | 1 | q0b | 1

�0, and

Vi =

8>>>>>><

>>>>>>:

0

@ Ua1 | ua

1... 0

0... 0

1

A for 1 i pa

0

@ Ub1 | ub

1... 0

0... Tb

1 | tb1

1

A for pa + 1 i pa + pb

The theorem can now be stated as:

Theorem 4. Consider the specification in Eqs. (10)-(11) and the set of p restrictionsgiven by Eqs. (12)-(13), with p = pa + pb � n2 + n(n�1)

2 . Given an integer 1 k ⇣n2 + n(n�1)

2

⌘and a corresponding integer h = p�

⇣n2 + n(n�1)

2

⌘+ k, consider all

possible sets of h increasing integers between 1 and p: 1 i1 < . . . < ih p, if

rk�Vi1 | . . . |Vih

� � k, (17)

for any k, then the function rk[V(q)] equals n2 + n(n�1)2 almost everywhere in Rq,

where q = 2n2 � p is the number of free parameters.We call the Rado condition the condition by which the inequalities (17) are simul-

taneously satisfied. Therefore, the SVAR model is structurally identified if and only ifthe Rado condition is met, except for a zero Lebesgue measure set of values for q.

Proof. The proof of this theorem follows the same reasoning of the proof ofTheorem 3. Here, the Vi matrices substitute the Ti matrices, while the numberof rows of V is n2 + n(n�1)

2 rather than n(n�1)2 . When the number of restrictions

is strictly equal to what required by the necessary condition for identification,then the set of indices for which the rank is checked in Eq. (17), goes from1 to p. If, instead, the number of restrictions exceeds the minimum required(the model is potentially overidentified) the rank in Eq. (17) is checked forall indices from p �

⇣n2 + n(n�1)

2

⌘+ 1 to p, where p �

⇣n2 + n(n�1)

2

⌘is the

degree of overidentification. The reason is that, when the model is potentiallyoveridentified, the number of columns of V(q), that must be of full-row rank,is larger than the number of rows. What is important, however, is that theremust be n2 + n(n�1)

2 linearly independent column among the p > n2 + n(n�1)2

columns composing the V(q) matrix.

5 AB-models: two examples

In this section we present two examples on the implementation of the previ-ously developed strategy to study identification in the general AB-model. Theformer is an artificial bivariate model of some theoretical interest; the latter isa trivariate SVAR model presented by Blanchard and Perotti (2002) in a well-known empirical application.

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5.1 Bivariate AB-model

Consider the bivariate AB-SVAR model in Eq. (10) with

A =

✓1 q1

�q1 1

◆B =

✓q2 00 q3

◆. (18)

The non-linear restrictions implicitly imposed are given by

S = A�1BB0(A0)�1

=1

�1 + q2

1�2

0

@q2

2 + q21q2

3 �q1q22 + q1q2

3

�q1q22 + q1q2

3 q23 + q2

1q22

1

A .

Global identification dictates that a unique solution exists to the non-linearsystem of three equations in three unknowns connecting the empirical mo-ments in S to the parameters q1, q2 and q3. It is interesting to note that theRubio-Ramirez, Waggoner, and Zha (2010) approach for checking for globalidentification cannot be applied to AB-SVAR models. It is true that this modelcould be turned into a C-model, but then identification would rely on a non-linear, cross-equation constraint, which is also impossible to handle in Rubio-Ramirez, Waggoner, and Zha’s approach.

Local identification, instead, can be checked by using the result in Theorem4, when considering the set of restrictions given by

Ra =

0

@1 0 0 00 0 0 10 1 1 0

1

A ra =

0

@110

1

A Rb =

✓0 1 0 00 0 1 0

◆ra =

✓00

◆

Sa =

0

BB@

01

�10

1

CCA sa =

0

BB@

1001

1

CCA Sb =

0

BB@

1 00 00 00 1

1

CCA sa =

0

BB@

0000

1

CCA .

Checking for the rank of all combinations of indices as discussed in The-orem 4 shows that the model is locally identified. However, if one imposesthe further restriction that q2 = q3, then the relation between the empiricalmoments and the parameters becomes

S =

0

B@q2

21+q2

10

0 q22

1+q21

1

CA

that is similar to the C-model investigated in Section 3.1. As expected, althoughthe number of restrictions is larger than required, the Rado condition fails andthe model is not identified.

5.2 Blanchard-Perotti AB-model

The following model is inspired by the seminal contribution by Blanchard andPerotti (2002) on the effects of fiscal shocks to the real economy. They consider

14

a three-dimensional VAR for taxes, spending and GDP, all in real per capitaterms. Once concentrating out the reduced-form parameters, the model can bewritten as

0

@1 0 �a10 1 �b1

�c1 �c2 1

1

A

0

@ttgtxt

1

A =

0

@a3 a2 0b2 b3 00 0 c3

1

A

0

@#t

t#

gt

#xt

1

A (19)

where #t =⇣

#tt , #

gt , #x

t

⌘0is the vector of reduced-form residuals while ut =

⇣ut

t , ugt , ux

t

⌘0is the vector of uncorrelated unit-variance structural shocks.

The model is clearly not identified as the number of parameters to estimateexceeds the number of distinct elements in the reduced-form covariance ma-trix (9 > n(n�1)

2 = 6). For the order condition to hold, then, at least three morerestrictions must be imposed.

Starting from the specification in Eq. (19), here below we consider alter-native combinations of further restrictions (more or less backed by economicintuition) and, for each of these, check for the identification of the model us-ing the Rado condition developed in Theorem 4. All the results are confirmedby the traditional Amisano and Giannini’s approach by calculating the rank ofthe matrix in Eq. (14) for randomly generated A and B. A horizontal bar overa parameter, as in a1, means that the parameter is calibrated to some non-zerovalue, usually on the basis of previous studies.

5.2.1 Further restrictions I

b1 = 0 a1 = a1 b2 = 0

or alternativelyb1 = 0 a1 = a1 a2 = 0

This is the specification used by the authors in their empirical application.In particular, they justify the choice of b1 = 0 by saying that it is not possi-ble to identify any automatic feedback from economic activity to governmentspending. Moreover, the elasticity to output of net taxes (tax minus transfer)can be calibrated by using external information and thus considered as fixedwithin the SVAR framework. Concerning the last restriction, the authors statethat there are no convincing ways to establish a priori, after a simultaneous andunexpected change in taxes and expenditure, which of the two variables, taxesand public spending, reacts first. For this reason they present both the resultswith b2 = 0 or alternatively a2 = 0.

These three further restrictions satisfy the necessary order condition of iden-tification, being p = pa + pb = 7 + 5 = 12 equal to n2 + n(n�1)

2 = 9 + 3 = 12.Moreover, using the Rado condition defined in Eq. (17), it is possible to showthat, independently of the free parameters, the model is almost everywhereidentified. In fact, given the restrictions b2 = 0 and a1 = a1, the cyclically ad-justed tax and public spending, tt = tt � a1xt and gt = gt � b1xt respectively,can be used as instruments to estimate c1 and c2 in a regression of xt on tt andgt. Furthermore, the consistently estimated tt and gt allow to estimate the re-maining two parameters a2 and b1, provided that, in turn, one of the two is

15

restricted to zero. Finally, the diagonal elements of the B matrix are the stan-

dard deviations of the estimated structural shocks ut =⇣

utt , ug

t , uxt

⌘0.

5.2.2 Further restrictions II

b1 = 0 a1 = a1 c1 = 0

or alternativelyb1 = 0 a1 = a1 c2 = 0

This set of restrictions, differently from the previous one, imposes that oneof the two elasticities of output to taxes and public spending is null, i. e. thatoutput does not respond, within the quarter, to changes in the level of taxesor spending. Moreover, we consider that both gt and tt react to unexpectedshocks in taxes and spending ut

t and ugt , respectively.

As before, the number of restrictions satisfies the necessary order condition.Looking at the sufficiency, instead, the Rado condition is now not supportedby the set of restrictions, indicating that the model is not identified. In fact,although b1 = 0 and a1 = a1 help consistently estimating c1 and c2, this doesnot allow to consistently estimate a2 and b2.

5.2.3 Further restrictions III

b1 = 0 a1 = a1 a2 = b3

In this case, the further restrictions, other than, as before b1 = 0 and a1 = a1,impose that gt and tt react to ug

t through coefficients of the same magnitude.This would be a very interesting case to analyse from an economic point ofview because it would describe an economy that, for some reason, follows apolicy of automatic stabilizing certain budget items. For example, during therecent financial crisis, the Italian budget contained certain tax increases (mostlyexcises), that were supposed to be applied automatically and conditionally toa certain deficit target not being met. This was mainly conceived as a way toreassure the markets about the commitment by the Italian government to fiscaldiscipline, and was repeatedly interpreted as such by Italian and EU authori-ties in public statements.

From a strictly econometric point of view, this restriction is of particularinterest because it involves coefficients in two different equations; this type ofconstraint cannot be handled in the framework proposed by Rubio-Ramirez,Waggoner, and Zha (2010), in which cross-equation restrictions are not con-sidered. And yet, in this scenario, a cross-equation restriction such as a2 = b3would be very a natural and direct way to translate an institutional feature intoa parametric restriction.

The necessary order condition continues to hold, but differently from theprevious case, now the Rado condition holds too, so the model is locally iden-tified. This last cross-equation restriction, apart from being interesting from aneconomic point of view is in fact indispensable for identifying all the parame-ters of the model.

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6 Conclusions and future directions

In this paper, we have developed a method for checking whether a given SVARis identified on the pure basis of the restrictions the covariance matrix is sub-ject to, independently of its parameters. Therefore, it is possible to investigateidentification prior to estimation. Our analysis is not limited to C-models, butalso extends to the more general AB model.

A very interesting research avenue lies in investigating qualitatively thepossible under-identification issues that derive from the relationship betweenobservables and parameters; in the example we gave in subsection 2.2, lack ofidentification was directly linked to the absence of information contained inone of the moment conditions (3). It may be possible to derive a formal crite-rion for detecting such cases by considering which one, of the set of inequalities(9), fails to hold. The example given in subsection 3.3 looks quite promising inthat respect.

A more difficult open question is whether our theorem can be combinedwith extra conditions for checking global identification, rather than local. Thiswill be the object of future work, as well as the usage of the results presentedhere for dealing with the heteroskedastic case and make the analysis in Bac-chiocchi (2014) more general and rigorous.

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