A note on global identification in structural vector autoregressions
aa r X i v : . [ ec on . E M ] F e b A note on global identification in structural vectorautoregressions ∗ Emanuele Bacchiocchi † Toru Kitagawa ‡ University of Bologna University College LondonThis draft: 8 February 2021
Abstract
In a landmark contribution to the structural vector autoregression (SVARs) literature, Rubio-Ram´ırez, Waggoner, and Zha (2010, ‘Structural Vector Autoregressions: Theory of Identificationand Algorithms for Inference,’
Review of Economic Studies ) shows a necessary and sufficientcondition for equality restrictions to globally identify the structural parameters of a SVAR. Thesimplest form of the necessary and sufficient condition shown in Theorem 7 of Rubio-Ram´ırezet al (2010) checks the number of zero restrictions and the ranks of particular matrices withoutrequiring knowledge of the true value of the structural or reduced-form parameters. However,this note shows by counterexample that this condition is not sufficient for global identification.Analytical investigation of the counterexample clarifies why their sufficiency claim breaks down.The problem with the rank condition is that it allows for the possibility that restrictions areredundant, in the sense that one or more restrictions may be implied by other restrictions, inwhich case the implied restriction contains no identifying information. We derive a modifiednecessary and sufficient condition for SVAR global identification and clarify how it can beassessed in practice.
Keywords : Simultaneous equation model, exclusion restrictions, redundant restrictions.
JEL codes : C01,C13,C30,C51. ∗ We thank Thomas Carr, Luca Fanelli, Michele Piffer, and Matthew Read for beneficial discussions and comments.Financial support from the ESRC through the ESRC Centre for Microdata Methods and Practice (CeMMAP) (grantnumber RES-589-28-0001) and the European Research Council (Starting grant No. 715940) is gratefully acknow-ledged. † University of Bologna, Department of Economics. Email: [email protected] ‡ University College London, Department of Economics. Email: [email protected] Introduction
Rubio-Ram´ırez et al. (2010) (henceforth RWZ) provide necessary and sufficient conditions for theglobal identification of structural parameters in Structural Vector Autoregressions (SVARs) undera general class of zero restrictions imposed on the structural parameters and their (non-)lineartransformations, including impulse responses. Exploiting the insights of their global identificationanalysis, RWZ also develop efficient and practical algorithms to perform estimation and inferencefor structural parameters and impulse responses. Their analytical and computational innovationshave been instrumental to recent developments in the literature, including set-identified SVARs(Arias et al. (2018, 2021)), Giacomini and Kitagawa (2020), Giacomini et al. (2021b), Volpicella(2020), Amir-Ahmadi and Drautzburg (2021)), locally-identified SVARs (Bacchiocchi and Kitagawa(2020)), and SVARs with narrative restrictions (Antol´ın-D´ıaz and Rubio-Ram´ırez (2018), Giacomini et al.(2021a)), to list a few. RWZ provide several different versions of the necessary and sufficient con-ditions for global identification. Those given in Theorem 7 are particularly attractive in termsof ease of implementation. Theorem 7 of RWZ characterizes a necessary and sufficient conditionfor (exact) global identification through a set of rank conditions that depends only on the choiceof identifying restrictions and does not require knowledge of the true value of the structural orreduced-form parameters.This note presents a counterexample refuting the sufficiency of the rank conditions of Theorem7 of RWZ, i.e., the rank conditions of Theorem 7 in RWZ are met but global identification fails.An analytical investigation of this counterexample reveals why these rank conditions do not guar-antee global identification. We find that the rank conditions of Theorem 7 of RWZ cannot detectwhat we refer to as redundancy of imposed identifying restrictions. In this phenomenon, a set ofequality restrictions on the structural parameters or impulse responses implicitly forces other struc-tural parameters or impulse responses to zero. If it is present, some (redundant) zero restrictionsare already implied by other imposed equality restrictions, so they do not contribute any furtheridentifying information to the system. The rank conditions of Theorem 7 of RWZ, however, incor-rectly count the redundant identifying restrictions as if they reduce the dimension of the admissiblestructural parameters, resulting in an erroneous conclusion that the model is globally identified.We argue that the redundancy of the identifying restrictions is relevant for empirical applications,rather than being of pure theoretical interest.To modify the sufficiency claim of the rank conditions of Theorem 7 of RWZ, we provide anew necessary and sufficient condition for (exact) global identification that correctly discountsredundant identifying restrictions. RWZ propose a useful algorithm that sequentially constructs anorthonormal matrix for structural parameter identification that satisfies the identifying restrictions.Building on and modifying their algorithm, our proposed necessary and sufficient condition for2lobal identification checks for the existence of redundant restrictions by verifying whether theorthonormal matrix generated by this sequential algorithm is unique. Verifying uniqueness boilsdown to checking the rank of a sequence of matrices constraining each column of the orthonormalmatrix. Although this algorithm requires values of the reduced-form parameters as an input, weshow that it can detect a lack of global identification even with redundant restrictions at almostany values of the reduced-form parameters. This almost-sure property is a key to facilitating theimplementation of the algorithm in practice, as it justifies running the algorithm at one or a fewpoints in the reduced-form parameter space drawn from a prior or posterior distribution or obtainedas a maximum likelihood estimate.As an alternative to their Theorem 7, Theorem 1 in RWZ presents a different form of necessaryand sufficient conditions for global identification. As we illustrate in this note, its proper implement-ation requires a complete understanding of how the imposed identifying restrictions analyticallyconstrain the impulse responses and the set of structural parameters. For instance, if redundantidentifying restrictions are present but one is not aware which zero restrictions can be implied byothers, naive implementation of the rank conditions in Theorem 1 of RWZ may also overlook alack of global identification. To prevent this, it is important to analytically ascertain how a set ofequality restrictions translate to zero restrictions for other structural objects. This is feasible forsmall scale SVARs, but can be less straightforward for medium or large scale SVARs. In contrast,checking our necessary and sufficient condition remains tractable and attractive even for moderateto large scale SVARs.The rest of the paper is organized as follows. We first introduce the model and notation inSection II. In Section III, we present an example that contradicts Theorem 7 of RWZ. In SectionIV we define the notion of redundant identifying restrictions and provide a modified necessary andsufficient condition for (exact) global identification. Section V concludes.
II Model
We maintain the notation used in RWZ. Let y t be a n × t = 1 , . . . , T . The specification of the SVAR model is y ′ t A = p X l =1 y ′ t − l A l + c + ε ′ t , (1)where ε t is a n × I n . The n × n matrices A , A , . . . , A p are the structuralparameters and c is a 1 × n vector of constant terms. The structural parameters are ( A , A + ),3here A ′ + ≡ ( A ′ , . . . , A ′ l , c ′ ) is a n × m matrix with m ≡ np + 1. We also assume that the initialconditions y , . . . , y p are given and that A is invertible. The set of structural parameters is denotedby P S , an open dense set of R ( n + m ) n . The structural form can be written compactly as y ′ t A = x ′ t A + + ε ′ t (2)where x ′ t = (cid:0) y ′ t − , . . . , y ′ t − p , (cid:1) .The reduced-form representation of (2) is the standard VAR model, y ′ t = x ′ t B + u ′ t , (3)where B j = A + A − , u ′ t = ε ′ t A − , and E ( u t u ′ t ) = Σ = ( A − A ′ ) − . The reduced-form parametersare ( B, Σ ), where Σ is a symmetric and positive definite matrix. We denote the set of reduced-form parameters by P R ⊂ R nm + n ( n +1) / . The relationship between the structural and reduced-formparameters is defined by the function g : P S → P R , where g ( A , A + ) = ( A + A − , ( A A ′ ) − ).The definition of global identification is the standard one provided by Rothenberg (1971); theabsence of observationally equivalent parameters in the parametric space. We consider identificationof the structural parameters by imposing zero restrictions on a transformation f ( · ) of the structuralparameter space into the set of k × n matrices, k ≥
1, with domain U ⊂ P S . Such linear restrictionsare represented by Q j f ( A , A + ) e j = 0 , for j = 1 , . . . , n. (4)where Q j is a k × k selection matrix for j = 1 , . . . , n , and e j is the j -th column of the n × n identitymatrix I n . The rank of Q j is denoted by q j , which also represents the number of restrictions in the j -th column of the transformed space f ( A , A + ). As in RWZ, we order the columns of f ( A , A + )according to q ≥ q ≥ . . . ≥ q n . (5)We denote the set of orthonormal matrices by O ( n ) with generic element P .Following RWZ, we say that this transformation is admissible when the following conditionholds. Condition 1.
The transformation f ( · ), with the domain U , is admissible if and only if for any P ∈ Ø and ( A , A + ) ∈ U , f ( A P, A + P ) = f ( A , A + ) P .Moreover, RWZ impose the following two conditions when proving some of their results. Condition 2.
The transformation f ( · ), with the domain U , is regular if and only if U is open and f is continuously differentiable with f ′ ( A , A + ) of rank kn for all ( A , A + ) ∈ U .4 ondition 3. The transformation f ( · ), with the domain U , is strongly regular if and only if it isregular and f ( U ) is dense in the set of k × n matrices.To fix the sign of structural shocks, we need to impose sign normalization rules. FollowingRWZ, we define them as follows: Definition 1 (Normalization rule) . A normalization rule can be characterized by a set N ⊂ P S such that for any structural parameter point ( A , A + ) ⊂ P S , there exists a unique n × n diagonalmatrix D with plus or minus ones along the diagonal such that ( A D, A + D ) ∈ N .We are now able to define the set of restricted structural parameters as R = { ( A , A + ) ∈ U ∩ N | Q j f ( A , A + ) e j = 0 for j = 1 , . . . , n } . (6)Following RWZ, we consider the following definition of identification when discussing whether ornot the imposed restrictions can globally identify the structural parameters. Definition 2 (Exact identification) . Consider an SVAR with restrictions represented by R . TheSVAR is exactly identified if and only if, for almost any reduced-form parameter point ( B, Σ ), thereexists a unique structural parameter point ( A , A + ) ∈ R such that g ( A , A + ) = ( B, Σ ).In this definition, if the set of structural parameters under the restrictions R constrains thereduced-form parameters, the domain of the reduced-form parameters for which the almost-sureproperty is required is restricted to ˜ P R ⊂ P R , where ˜ P R is the set of reduced-form parametersgenerated by the structural parameters satisfying R . For instance, if f ( · ) maps the structuralparameters to long-run impulse responses, its domain U restricts the reduced-form VARs to beinginvertible. Then, ˜ P R corresponds to the set of reduced-form parameters constrained to invertibleVARs. III An Illustrative Counterexample
In the setting described in the previous section, RWZ shows a variety of necessary and sufficientconditions for the identifying restrictions R with admissible f ( · ) to globally identify the structuralparameters. Among those, the necessary and sufficient condition for exact identification presentedin Theorem 7 of RWZ is particularly attractive, as it reduces verification of exact identification toa simple exercise of computing the ranks of the matrices Q j , 1 ≤ j ≤ n . So that our exposition isself-contained, we present Theorem 7 of RWZ here:5 heorem 7 in RWZ : Consider an SVAR with admissible and strongly regular restrictions repres-ented by R . The SVAR is exactly identified if and only if q j = n − j for ≤ j ≤ n . The first result in this note is that the “if” statement of this theorem is false, as shown by thefollowing counterexample.
III.1 A counterexample
Consider a trivariate SVAR characterized by the following restrictions A = a a a a a a a and IR = × ×× × ×× × × (7)where IR = ( A − ) ′ is the contemporaneous impulse response matrix, the symbol ‘ × ’ indicatesthat no restriction is imposed, and ‘0’ represents a zero (or exclusion) restriction. The function f ( A , A + ) will be f ( A , A + ) = A IR ! = a a a a a a a × ×× × ×× × × . (8)The matrices of restrictions defined in (4) can be specified as Q = , Q = . (9)According to Theorem 7 in RWZ, the SVAR is exactly (globally) identified, as the ranks of therestriction matrices follow q = n − q = n − q = n − Admissible and strongly regular restrictions represented by R mean f ( · ) in (6) is admissible and strongly regular. Σ = σ σ σ σ σ σ σ σ σ ⇒ Σ tr = l l l l l l . (10)Imposing triangularity on A , we can obtain A and IR = A − ′ as A ′ = Σ − tr = l − l l l l l l − l l l l l − l l l l ⇒ IR = A − ′ = l l l l l l . (11)Consider applying Algorithm 1 in RWZ to determine an orthogonal matrix P that maps the( A , A + ) parameters under triangularity to the one satisfying the imposed restrictions.First, f ( A , A + ) is f ( A , A + ) = A IR ! = l − l l l l l − l l l l l l − l l l l l l l l l l . (12)As in RWZ, let ¯ Q and ¯ Q be the matrices of indicators for the restricted elements of f ( A , A + )obtained by removing the row vectors of zeros from Q and Q . Algorithm 1 in RWZ suggestscalculating ˜ Q = ¯ Q f ( A , A + ) = l − l l l l ! , (13)and finding a unit-length vector that is orthogonal to the row vectors of ˜ Q . The QR decompositionof ˜ Q and a sign normalization lead to p = (1 , , ′ as a unique unit vector satisfying ˜ Q p = 0,so we can pin down the first column vector of P .Next, to find the second column vector p of P , we form the matrix˜ Q = ¯ Q f ( A , A + ) p ′ ! = l ! (14)and search for a unit vector p satisfying ˜ Q p = 0. Since the rank of ˜ Q is one for any value of7 , we cannot pin down a unique p (up to the sign normalization). From a geometric point ofview, any vector belonging to the unit circle in R orthogonal to the unit vector p = (cid:16) (cid:17) ′ is admissible as p . This implies that given any reduced-form parameter value of Σ , the imposedrestrictions fail to pin down a unique orthogonal matrix P , implying that, contrary to the claim inTheorem 7 of RWZ, global identification does not hold in this example.Some packaged algorithms for the QR decomposition, including the Matlab function qr ( · ), yieldan orthogonal vector p irrespective of whether it is unique or not. That is, if ˜ Q is not full-rank,these algorithms implicitly select one unit vector p from infinitely many admissible ones. As aresult, an application of the “if” statement of Theorem 7 and naive implementation of Algorithm1 in RWZ may fail to detect the failure of global identification and mislead subsequent impulseresponse analysis. III.2 Analytical investigation
To understand why the “if” statement of Theorem 7 of RWZ breaks down and how it can bemodified, it is useful to determine analytically the special feature of the identifying restrictionsspecified in (7).We begin with the inversion of the A matrix; the determinant of A is | A | = a a a + a a a + a a a − a a a − a a a − a a a (15)and the adjunct matrix isAdj( A ) = a a − a a − ( a a − a a ) a a − a a − ( a a − a a ) a a − a a − ( a a − a a ) a a − a a − ( a a − a a ) a a − a a . (16)The inverse is A − = | A | − Adj( A ). Substituting the two zero restrictions on A , a = 0 and a = 0, into A − ′ leads to A − ′ = 1 a ( a a − a a ) a a − a a − ( a a − a a ) a a − ( a a − a a ) a a − a a − a a a a = IR . (17)It is evident that the two restrictions on A imply two zero restrictions on IR , ( A − ′ ) [1 , =( A − ′ ) [1 , = 0. One of these, ( A − ′ ) [1 , = 0, is exactly the zero restriction specified for IR in (7).In other words, we intended to impose the three restrictions, but the two imposed on A imply thethird imposed on IR , so this third restriction was redundant. Due to this redundancy, the third8estriction does not further constrain the admissible orthonormal matrix P , which translates intorank deficiency of ˜ Q .Although this redundancy phenomenon can occur in some realistic applications, whether or notany of the imposed set of restrictions are redundant cannot be directly assessed by the simple rankconditions in Theorem 7 of RWZ. As a way to uncover such redundancy, one may want to examinehow a set of zero restrictions imposed on one structural object translates to zero restrictions onother objects. In Section IV below, we modify the necessary and sufficient condition of Theorem 7of RWZ by offering a systematic way to detect redundancy of the imposed identifying restrictions. III.3 Detecting the failure of global identification
In their Theorem 6, RWZ provides an alternative necessary and sufficient condition for exactidentification of SVARs. If we properly take into account that the imposed zero restrictions implyzero restrictions on other objects, this alternative approach can correctly detect a lack of globalidentification. We illustrate how in our example.For 1 ≤ j ≤ n and any k × n matrix X , let M j ( X ) be a ( k + j ) × n matrix defined by M j ( X ) = Q j XI j × j O j × ( n − j ) ! , where Q j is a k × k matrix defined in (4). Theorem 6 of RWZ provides a necessary and sufficientcondition for exact identification through the rank conditions for M j ( f ( A , A + )). Theorem 6 in RWZ : Consider an SVAR with admissible and strongly regular restrictions repres-ented by R . The SVAR is exactly identified if and only if the total number of restrictions is equalto n ( n − / and for some ( A , A + ) ∈ R , M j ( f ( A , A + )) is of rank n for ≤ j ≤ n . In the current example, the total number of restrictions imposed is 3 and it meets the conditionfor the total number of restrictions with n = 3. We hence focus on checking the rank condition for M j ( f ( A , A + )), j = 1 , ,
3. In this check, we substitute the following matrices into f ( A , A + ): A = a a a a a a a and IR = × × × ×× × × , (18) Many influential empirical papers combine restrictions on both contemporaneous relationships amongthe endogenous variables and the contemporaneous impulse responses. Examples include Blanchard (1989),Blanchard and Perotti (2002), Bernanke (1986). × ’ denotes the parameters in Eq. (17). We obtain, if a a − a a = 0, M ( f ( A , A + )) = a a a a rank ( M ) = 3 M ( f ( A , A + )) = a a − a a rank ( M ) = 2 < . (19)Hence, the rank condition of Theorem 6 in RWZ fails. This is consistent with the conclusion in ouranalysis above; the imposed restrictions uniquely pin down the first column vector of P , but not thesecond column vector of P . Thus, plugging in the expression of f ( A , A + ) obtained analyticallyunder the imposed restrictions, the rank condition of Theorem 6 of RWZ correctly detects thefailure of global identification due to the redundancy among the imposed identifying restrictions.It is important to note that understanding analytically the whole set of constraints implied bythe imposed restrictions is crucial to correctly performing the check of the rank condition in Theorem6 of RWZ. For instance, in the current example, if we were not aware of the redundancy issue ofthe identifying restrictions and incorrectly let the (1 , M ( f ( A , A + )) be an unknownpotentially nonzero free parameter, we would have erroneously claimed that M ( f ( A , A + )) wereof rank 3 and concluded that the exact identification holds. If the dimension of the SVAR islarge, exhaustively investigating and figuring out the entire set of constraints implied by the zerorestrictions on f ( A , A + ) is challenging. In such a case, immediate implementation of the rankconditions of Theorem 6 of RWZ is limited. IV Modified necessary and sufficient condition for exact identific-ation
In this section we provide a modified necessary and sufficient condition for exact identification thateliminates the redundancy issue that invalidates Theorem 7 of RWZ. Our proposal relies on thesequential feature of Algorithm 1 in RWZ and checks the rank condition for uniqueness of the j -thcolumn vector p j for each j = 1 , . . . , n .Given the reduced-form parameter ( B, Σ ), set ( A , A + ) to be an unrestricted set of structuralparameters satisfying Σ = ( A ′ ) − ( A ) − and B = A + A − , such as A ′ = Σ − tr and A + = B ( Σ − tr ) ′ .10et ˜ Q = Q f ( A , A + ) , and ˜ Q j = Q j f ( A , A + ) p ′ ... p ′ j − for j = 2 , . . . , n. (20)By Theorem 5 and Algorithm 1 of RWZ, the exact identification of SVARs follows if and only if,for almost every reduced-form parameters ( B, Σ ), the orthogonality conditions ˜ Q j p j = 0 combinedwith the sign normalization restrictions pin down a unique orthogonal matrix P .For P to be uniquely determined, it is necessary to have q j = n − j for all 1 ≤ j ≤ n . This is,however, not a sufficient condition, because if any of the orthogonal vectors ( p , . . . , p j − ) is linearlydependent on the row vectors of Q j f ( A , A + ), a rank-deficient ˜ Q j fails to pin down a unique p j .This is exactly the mechanism that caused the systematic failure of global identification in ourillustrative counterexample. To rule out such rank-deficiency in the characterization of the globalidentification condition, we introduce the following concept: Definition 3 (Non-redundant restrictions) . Given reduced-form parameter (
B, Σ ), let A ′ = Σ − tr and A + = B ( Σ − tr ) ′ . Identifying restrictions for a SVAR that are represented by zero restrictions Q j f ( A , A + ) e j = 0, j = 1 , . . . , n , are non-redundant at given reduced-form parameter point, ( B, Σ )if for every j = 2 , . . . , n , orthogonal vectors ( p , . . . , p j − ) are linearly independent of the row vectorsof Q j f ( A , A + ), i.e., ˜ Q j defined in (20) is full row-rank for all j = 2 , . . . , n .If the imposed zero restrictions are non-redundant and the rank condition of Theorem 7 in RWZholds, we can guarantee rank ( ˜ Q j ) = rank Q j f ( A , A + ) p ′ ... p ′ j − = n − j = 1 , . . . , n . We can therefore solve for an orthonormal matrix P uniquely by sequentiallysolving ˜ Q j p j = 0, for j = 1 , . . . , n . If non-redundancy of the imposed restrictions holds for almostany reduced-form parameter point ( B, Σ ), we can achieve exact identification. We hence obtainthe following theorem that modifies Theorem 7 of RWZ. We provide a proof in the Appendix.
Theorem 1 (A necessary and sufficient condition for exact identification) . Consider an SVARwith admissible and strongly regular restrictions represented by R . The SVAR is exactly identifiedif and only if q j = n − j for j = 1 , . . . , n and the restrictions are non-redundant at almost anyreduced-form parameter ( B, Σ ) .
11n comparison to Theorem 7 of RWZ, our Theorem 1 adds the almost-sure non-redundancycondition of the imposed restrictions as a part of necessary and sufficient condition. Accordingly,the modified necessary and sufficient condition of our Theorem 1 may not appear as simple aschecking the ranks of Q j matrices. However, the next theorem, which extends Theorem 3 of RWZto the current setting, leads to an easy-to-implement procedure for assessing the almost-sure non-redundancy condition: Theorem 2.
Consider an SVAR with admissible and regular restrictions represented by R thatsatisfies q j = n − j for ≤ j ≤ n . Let ˜ P R ⊂ P R be the set of reduced-form parameters ( B, Σ ) generated by the structural parameters satisfying R . Let K be the set of reduced-form parameters ( B, Σ ) ∈ ˜ P R that satisfy the non-redundancy condition, i.e., the rank conditions of Definition 3holds. Either K is empty or the complement of K is of measure zero in ˜ P R . A practical implication of this theorem is that we can assess exact identification of SVARs bychecking the rank conditions of non-redundancy at some finite number of points of (
B, Σ ) ∈ ˜ P R drawn from a probability distribution supporting ˜ P R . Such probability distribution can be a prioror posterior distribution for the reduced-form parameters in a Bayesian VAR. Building on andmodifying Algorithm 1 of RWZ, the next algorithm correctly judges if exact identification holds ornot, almost surely in terms of the sampling probability therein. Algorithm 1.
Consider an SVAR with admissible and strongly regular restrictions represented by R that satisfies q j = n − j , for j = 1 , . . . , n . Let ( B m , Σ m ) , m = 1 , . . . , M be M number of drawsof the reduced-form parameters from a probability distribution supporting ˜ P R . M does not have tobe large and a small integer M ≥ should suffice.For each m = 1 , . . . , M , perform the following steps:1. Let A ′ = Σ − tr,m and A + = B m ( Σ − tr,m ) ′ , where Σ tr,m is the lower-triangular Cholesky factor of Σ m .2. For each j = 1 , . . . , n , sequentially, check the rank conditions for non-redundancy, i.e., checkif rank ( ˜ Q j ) = n − holds, where ˜ Q = Q f ( A , A + ) and ˜ Q j = Q j f ( A , A + ) p ′ ... p ′ j − (22) for j = 2 , . . . , n , and p j is an n × vector satisfying ˜ Q j p j = 0 which is unique (up to signnormalization) if rank( ˜ Q j ′ ) = n − holds for all preceding j ′ = 1 , . . . , j − . f at least one drawn reduced-form parameter point passes Step 2 of the current algorithm, we con-clude that the imposed identifying restrictions R achieve exact identification. If none of the drawnreduced-form parameter points passes Step 2, we conclude that the imposed identifying restrictionsdo not achieve exact identification. The constructions of the orthonormal vectors p , . . . , p n by solving ˜ Q j p j = 0 sequentially for j = 1 , . . . , n , as incorporated in Step 2 of Algorithm 1, is proposed in Algorithm 1 of RWZ. Forthe purpose of checking exact identification, the important feature of our algorithm is the step ofchecking rank( ˜ Q j ) = n − j = 1 , . . . , n . This extra step, which is absent in Algorithm 1of RWZ, is necessary to detect failure of exact identification due to redundancy of the identifyingrestrictions. V Conclusion
Based on a counterexample, this note demonstrates that the sufficiency claim in Theorem 7 of RWZ,commonly used by applied macro-economists because of its simplicity, is not correct. Analyticalinvestigation of this counterexample reveals the issue of redundancy among the identifying restric-tions, which the rank conditions of Theorem 7 of RWZ overlook. To rectify this, we present a newset of necessary and sufficient conditions for exact identification and a computational algorithmthat can correctly detect redundant identifying restrictions and is easy to implement in practice.We recommend this procedure to any researchers who wish to check global identification of SVARsunder their choice of equality identifying restrictions.
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Appendix: Proofs of Theorems
The proof proceeds via a sequence of lemmas modifying those shown in RWZ.
Lemma 1. If q j = n − j for j = 1 , . . . , n and all the restrictions are non-redundant, then for every ( A , A + ) ∈ U , there exists a P ∈ Ø such that ( A P, AP ) ∈ R .Proof. This lemma modifies Lemma 5 in RWZ by adding the non-redundancy condition. Underthe non-redundancy condition, for every j = 1 , . . . , n , the rank of ˜ Q j in Eq. (22) is equal to n − p j .The next lemma modifies Lemma 6 in RWZ by explicitly assuming non-redundancy. Lemma 2. If q j = n − j for j = 1 , . . . , n and all the restrictions are non-redundant, then thereexists ( A , A + ) ∈ R , such that M j ( f ( A , A + )) is of rank n for j = 1 , . . . , n .Proof. The proof proceeds similarly to the proof of Lemma 6 in RWZ, except for the necessarymodification due to the additional assumption we impose for non-redundancy. If the imposedrestrictions are non-redundant as assumed in the current lemma, there are exactly q j = n − j number of independent restrictions operating for the structural parameters for j = 1 , . . . , n . Let V j be the column space of ( Q j f ( A , A + )) ′ for ( A , A + ) ∈ ( U ∩ N ). Moreover, let V ⊥ j be the linearsubspace of R n that is orthogonal to V j . In case of no restrictions for certain j , let V ⊥ j be the whole R n . Because U is an open dense set, and given the assumption that q j = rank ( Q j ) = n − j , it ispossible to find some values of ( A , A + ) ∈ ( U ∩ N ) such that rank ( Q j f ( A , A + )) = q j = n − j , whichimplies dim ( V j ) = n − j and dim ( V ⊥ j ) = j . For any f ( A , A + ) ∈ ( U ∩ N ), let P i = ( p , p , . . . , p i )be an n × i matrix of orthogonal vectors in R n . Moreover, let P i be the linear subspace of R n generated by the columns of P i and let P ⊥ i be the linear subspace of R n orthogonal to the columnvectors of P i . The dimension of P i is clearly equal to i , while that of P ⊥ i is n − i . Now, given( A , A + ) ∈ ( U ∩ N ), according to Algorithm 1, it is possible to define the elements in P ∈ Ø in thefollowing recursive way: p ∈ H ≡ V ⊥ p ∈ H ≡ V ⊥ ∩ P ⊥ p ∈ H ≡ V ⊥ ∩ P ⊥ ... p j ∈ H j ≡ V ⊥ j ∩ P ⊥ j − ... p n ∈ H n ≡ V ⊥ n ∩ P ⊥ n − where H j ∈ R n is the set of feasible p j given the restrictions on the j -th column of f ( A , A + ) andthe set of previous orthogonal vectors collected in P j − . Given the assumption of non redundantrestrictions, for j = 1 , . . . , n , rank ( ˜ Q j ) = n −
1, and according to Lemma 1, we obtain thatdim ( H j ) = 1. Moreover, being ( p , . . . , p j − ) mutually orthogonal by construction, dim ( P j − ) = j −
1. Thus, because dim ( P j − ) = j − V j ) = n − j , in order for rank ( ˜ Q j ) to be equalto n −
1, the vector spaces V j and P j − must be disjoint. For j = 1 , . . . , n , thus, the number ofrestrictions effectively operating in the columns of f ( A , A + ) is equal to n − j = q j .Having proved this, the remaining part of the proof follows exactly as the proof of Lemma 6 inRWZ. Lemma 3. If q j = n − j for j = 1 , . . . , n and all the restrictions are non-redundant, then the SVARis exactly identified. roof. This lemma modifies Lemma 7 in RWZ by adding the non-redundancy condition. We haveseen that without the further assumption of non-redundant restrictions, the set of all f ( A , A + ) ∈ U such that there exists an orthogonal matrix P = I n with ( A P, A + P ) ∈ R , denoted by G in RWZ,could be of strictly positive measure. According to Lemma 1 and Algorithm 1, the situation of G having a positive measure is precluded. Accordingly, under the assumption of q j = n − j , for j = 1 , . . . , n , and non-redundancy of the restrictions, the claim of exact identification follows as inthe proof of Lemma 7 in RWZ.The previous Lemmas 1 to 3 allow to prove the sufficient part of the condition in Theorem 1.We now move to the other direction and show the following lemma. Lemma 4.
If the SVAR is exactly identified, then q j = n − j for j = 1 , . . . , n and all the restrictionsare non-redundant.Proof. This lemma modifies Lemma 9 in RWZ by explicitly claiming non-redundancy in its con-clusion. The first part of the proof, which consists of proving that an exactly identified SVARpresents a pattern of restrictions of the form q j = n − j , for j = 1 , . . . , n , is essentially the same.What we need to prove is that if the model is exactly identified, then all the restrictions are nonredundant. Using the result in Theorem 5 in RWZ, we can say that if an SVAR is exactly identified,than for almost every structural parameter point ( A , A + ) ∈ U there exists a unique P ∈ Ø suchthat ( A P, A + P ) ∈ R . Moreover, we have seen that such a P matrix can be obtained through ourAlgorithm 1. However, sequentially for each j = 1 , . . . , n , for the algorithm to obtain a unique p j ,the rank of ˜ Q j must be equal to n −
1, proving thus the result.
Proof of Theorem 1.
The claim follows from Lemma 3 and Lemma 4.
Proof of Theorem 2.
This theorem is based on Lemma 2 given above and Theorem 3 in RWZ.In fact, according to Lemma 2, if q j = n − j for j = 1 , . . . , n, . . . , n