Identifiability of Structural Singular Vector Autoregressive Models

We generalize well-known results on structural identifiability of vector autoregressive models (VAR) to the case where the innovation covariance matrix has reduced rank. Structural singular VAR models appear, for example, as solutions of rational expectation models where the number of shocks is usually smaller than the number of endogenous variables, and as an essential building block in dynamic factor models. We show that order conditions for identifiability are misleading in the singular case and provide a rank condition for identifiability of the noise parameters. Since the Yule-Walker equations may have multiple solutions, we analyze the effect of restrictions on the system parameters on over- and underidentification in detail and provide easily verifiable conditions.


Introduction
Singular VAR models play an important role in macroeconomic modeling. As introduction to the subject we succinctly discuss Generalized Dynamic Factor Models (GDFM) and Dynamic Stochastic General Equilibrium (DSGE) models, and their relation to structural singular VAR models.
In the literature on GDFMs (Forni et al., 2000(Forni et al., , 2005Bai and Ng, 2007;Deistler et al., 2010), singular VAR models are the essential building block connecting static factors (a static transformation of the denoised observables) to the uncorrelated lower-dimensional shocks. Chen et al. (2011) andDeistler et al. (2011) treat canonical forms of singular VAR models, i.e. they focus on the reduced form. In Forni et al. (2009), it is shown that dynamic factor models (and consequently singular VAR models) are useful for structural modeling. In this article, we build on Step C in Forni et al. (2009, page 1332) by providing results regarding identifiability of structural singular VAR models.
The econometric treatment of DSGE models often involves representing DSGE models as structural VAR models. The relationship between DSGE and structural VAR models, i.e. all simplifying assumptions necessary to obtain a structural VAR representation from a DSGE model, is analyzed in Giacomini (2013), Kilian and Lütkepohl (2017, Chapter 6.2), and most recently by Lippi (2019). In particular, it is usually required that the number of endogenous variables coincides with the number of exogenous shocks driving the system. It is, however, recognized (Kilian and Lütkepohl, 2017, page 177) that the usual strategies 1 for solving this rank deficiency problem are not satisfactory and very unrealistic. Lippi (2019) analyses the effects of adding measurement noise in more detail.
Singularity of the innovation covariance matrix has two possible consequences for the restrictions imposed by the modeler. On the one hand, the restrictions imposed by the modeler might contradict the restrictions that are implicit due to the singularity structure of the innovation covariance matrix. On the other hand, the restrictions imposed by the modeler might already be contained in the restrictions that are implicit due to the singularity structure of the innovation covariance matrix and are therefore redundant. These cases must be taken into account when analyzing identifiability properties of singular structural VAR models. Moreover, restrictions on the system parameters are not necessarily overidentifying when the innovation covariance matrix is singular because the Yule-Walker equations might have multiple solutions.
The rest of this article is structured as follows. In section 2, we specify the model, we introduce restrictions on model parameters in a general fashion, and define notions which will be needed later. In section 3, we discuss identifiability of the reduced form parameters in singular VAR models. In particular, the possible singularity of the Toeplitz matrix appearing in the Yule-Walker equations is analyzed. In section 4, we discuss restrictions on the noise and system parameters which occur in the literature, and how a singular innovation covariance matrix needs to be taken into account for iden-1 Kilian and Lütkepohl (2017) enumerate 1) adding measurement noise (or rather model approximation errors), 2) reducing the number of observables and 3) augmenting the number of economically interpretable shocks. tifiability analysis. In particular, we discuss the case where the Yule-Walker equations have multiple solutions and provide easily verifiable conditions for under-and overidentification.
The following notation is used in the article. We use z as a complex variable as well as the backward shift operator on a stochastic process, i.e. z (y t ) t∈Z = (y t−1 ) t∈Z . For a (matrix) polynomial p(z), we denote by deg(p(z)) the highest degree of p(z). The transpose of an (m × n) dimensional matrix A is represented by A ′ . For the submatrix of A consisting of rows m 1 to m 2 , 0 ≤ m 1 ≤ m 2 ≤ m, we write A [m 1 :m 2 ,•] and analogously A [•,n 1 :n 2 ] for the submatrix of A consisting of columns n 1 to n 2 , 0 ≤ n 1 ≤ n 2 ≤ n.
We use vec (A) ∈ R nm×1 to stack the columns of A into a column vector and vech (A) ∈ R n(n+1) 2 ×1 to stack the lower-triangular elements of an ndimensional square matrix A analogously. The n-dimensional identity matrix is denoted by I n . An n-dimensional diagonal matrix with diagonal elements (a 1 , . . . , a n ) is represented by diag (a 1 , . . . , a n ). The inequality " > 0" refers to positive definiteness in the context of matrices. For the span of the row space and the column space of A, we write span R (A) and span C (A), respectively, and the projection of A on span R (B), B ∈ R r×n , is P roj R (A|B) and the projection of A on span C (D), D ∈ R m×s , is P roj C (A|D). We use E (·) for the expectation of a random variable with respect to a given probability space.

Model
Here, we start by defining the model, i. e. the system and noise parameters as well as the stability, singularity, and researcher imposed restrictions. Next, we describe the observed quantities that are available to the econometrician; in our case the second moments. Last, we discuss the notion of identifiability, i. e. the connection between the internal and external characteristics.
We consider a singular structural VAR system where the white noise process(ε t ) of (economically) fundamental shocks is uncorrelated across time with covariance matrix I q , and the first q rows of B ∈ R n×q , q ≤ n, are w.l.o.g. linearly independent. We denote the covariance matrix of the innovations u t = Bε t by Σ u . It follows that we can express the remaining (n − q) rows of B as Furthermore, we assume that the matrices A i ∈ R n×n are such that the stability condition det (a(z)) = 0, |z| ≤ 1, holds, where a(z) = I n − A 1 z − · · · − A p z p . Last, we assume that the parameters satisfy the restrictions where C A and C B are of dimensions r A × n 2 p and (r B × nq), respectively, describing the (a-priori known) restrictions imposed by the modeler. To summarize, we define the internal characteristics that we would like to identify as the parameters (A + , B) in system (1) which satisfy the restrictions imposed by (2), (3), and (4).
Next, we discuss the external characteristics which are observed by the econometrician. The stationary solution of the system (1) (together with the restrictions imposed on the parameters) is called a singular VAR process.
Having available all finite joint distributions of the singular VAR process corresponds to the maximal information we could possibly obtain regarding external characteristics. Another commonly used set of external characteristics is the second moment information contained in the singular VAR process, i.e. the autocovariance function γ(s) = E y t y ′ t−s or equivalently the spec- We follow Rothenberg (1971) to define identifiability of parametric models. corresponding to the internal characteristic such that there is no other observationally equivalent internal characteristic in this neighborhood. In this article, we focus on identifiability from second moment information, i.e. the external characteristics correspond to the spectral density of the observed process (y t ).

VAR Models
In order to fix ideas and prepare for the structural case, we review some identifiability properties of singular VAR models in reduced form. First, we characterize the non-identifiable case in terms of external and internal characteristics. Subsequently, we show that the transfer function k(z) = a(z) −1 B is not affected by possible non-identifiability of A + .
One way to connect the observable characteristics to the internal characteristics is by using the Yule-Walker equations 2 , i.e.
invertible, there is a unique internal characteristic (A + , Σ) for a given external characteristic (γ(0), . . . , γ(p)). While for VAR models with non-singular innovation covariance matrix it can be shown that Γ p is non-singular, this is not the case for VAR models with singular innovation covariance matrix.
For VAR models with a singular innovation covariance matrix, the dimension it follows from reordering of the Yule-Walker equations as holds thatā(0) = I n , deg (ā(z)) = p, andB = B and therefore the term in brackets in is another solution of the Yule-Walker equations which implies that Γ p is singular.
Last, consider a matrix R(z) whose entries are rational functions and for This implies that it is not possible to identify the parameters in a(z) by re-stricting, e.g., k(0) or k(1).

Imposing Structural Restrictions
In this section, we discuss identifiability of noise and system parameters in the case of singular SVAR systems. First, we derive a condition which ensures that the modeler imposed restrictions on the noise parameters are not in contradiction with the singularity of the innovation covariance matrix.
Subsequently, we derive a rank condition similar to the previous literature and note that the order condition does not provide useful information in the stochastically singular case. Second, we discuss the consequences of researcher imposed restrictions on system parameters on their identifiability properties in terms of under-and overidentification. In particular, we show that it is uncommon that researcher imposed restrictions do not solve the underidentification problem (if the number of restrictions is at least as large as the rank deficiency of Γ p ).
We start by focusing on the case Γ p > 0 and affine restrictions on the elements in B which appear in short-run restrictions, see Kilian and Lütkepohl (2017, Chapter 8) for the non-singular case. The conditions we derive are local in nature. Next, we deal with the case where Γ p may be singular and where affine restrictions on the elements in A + are imposed in order to connect our results to the identifiability analysis of the reduced form of singular VAR models in Deistler et al. (2011);Chen et al. (2011). These results concern global identifiability.

Affine Restrictions on the Noise Parameters
Assume that Γ p > 0, such that A + is identified from the Yule-Walker equations, and that the restrictions imposed by the researcher are given by where C B has full row rank. In order to prove local identifiability, one usually calls on the implicit functions theorem. While in the non-singular SVAR case the system of equations to be analyzed has always at least one solution, it might happen in the singular SVAR case that the set of solutions of (5) (for which the restrictions imposed by the researcher are satisfied) is the empty set. Since the premises of the implicit function theorem are such that there must be at least one solution, one needs to make sure that the affine restrictions (5) imposed by the researcher do not contradict the implicit singularity restrictions (2). In the following, we will provide an analytical condition which implies and is implied by a non-empty solution set.
The linear dependence structure induced by the singularity of Σ u , see equa- which is equivalent to The condition for when the solution set of the joint system of restrictions given in (5) and (6) is non-empty is given in the following Proof. We write C B as orthogonal sum, i.e.
and substitute it into equation (5): In order to fulfill the singularity restrictions of Σ u , equation (6)  is of (full column) rank nq.
Remark 3. Following Rothenberg (1971), the restrictions imposed on the structural parameter B are C B vec(B) = c B as well as Lvec(B) = 0 which suggests that the matrix needs to be of rank nq. However, it is not necessary to include L in Proposition 1 because Lvec(B) = 0 is already implied by the fact that BB ′ = Σ u .
Put differently, the inequality rk

holds.
Proof. Consider the following system of equations: The matrix ∂ϕ ∂vec(B) ′ can be calculated using standard rules for matrix differentiation (Lütkepohl, 1996) as and ∂ϕ 2 ∂vec(B) ′ (vec(B)) = C B . Here, D + n is the pseudo-inverse of the duplication matrix D n which fulfills D n vech(A) = vec(A) for a matrix A ∈ R n×n , and K nn ∈ R n 2 ×n 2 is a commutation matrix such that vec(A ′ ) = K nn vec(A). While the order condition is satisfied for q ≤ n+1 2 , the matrix D + n (B ⊗ I n ) of dimension n(n+1) 2 × nq is of course rank deficient with co-rank q(q−1) 2 .

Affine Restrictions on the System Parameters
Assuming that B is identified, we now focus on imposing linear restrictions • The minimum norm solution (MNS) in Chen et al. (2011) is such that for the coordinate functions V ′ 1 ∈ R np−s×np pertaining to the basis V 1 of span C (Γ p ), both V ′ 1 V 1 = I np−s and V ′ 1 Γ p V 1 are diagonal. Moreover, for the orthogonal complement V 2 of the column space of Γ p , we have that V ′ 2 Γ p V 2 = 0 and in particular that V ′ 1 Γ p V 2 = 0.
• The selection solution in Deistler et al. (2011) (to be described in more detail below) is special since its coordinate functions, say S ′ 1 of dimension ((np − s) × np), are (row-) vectors containing only one nonzero element (equal to one) and are thus particularly simple. While S ′ 1 S 1 = I np−s holds as well, S ′ 1 Γ p S 1 is not diagonal and, moreover, S ′ 2 Γ p S 2 and S ′ 1 Γ p S 2 are non-zero (where S 2 is the orthogonal complement of S 1 in the np-dimensional Euclidean space ).
Note that for small changes in the parameter A + , the coordinate functions V ′ 1 change, i.e. they are data-dependent, while the rows of S ′ 1 are fixed.
To simplify discussion, we note that vectorizing the (transposed) Yule-Walker equations leads to In Deistler et al. (2011), the authors choose the first linearly independent rows of Γ p as a basis of the row space (or equivalently column space) of Γ p to define a particular solution of the Yule-Walker equations. To fix ideas, consider a Γ p whose first (np − s) linearly independent rows are selected by premultiplying S ′ 1 of dimension ((np − s) × np), containing only zeros and ones, and denote by S ′ 2 the (s × np)-dimensional matrix containing zeros and ones such that S ′ 1 S 2 = 0. A basis of the column space consists thus of the columns of Γ p S 1 , i.e. the elements S ′ 2 A ′ + are restricted to zero.
Restricting each column of A ′ + to be orthogonal to S 2 results thus in a unique solution of the Yule-Walker equations, i.e. the matrix in brackets in is of full rank. We denote the unique solution of the equation above by vec A ′ + .
In Chen et al. (2011), the authors choose the minimum norm solution of the Yule-Walker equations as the particular solution. Let be the singular value decomposition (SVD) 3 of (I n ⊗ Γ p ) of rank n 2 p − ns = n · rk (Γ p ). The particular solution is such that coordinates corresponding to the basis vectors V 2 are set equal to zero. Put differently, vec A ′ + is required to be orthogonal to V 2 , i.e.
We denote the unique solution of the equation above by vec A ′ + .
It is of course possible to represent the particular solution vec A ′ + in terms of the eigenbasis (V 1 , V 2 ) of (I n ⊗ Γ p ) and vice versa. While the coordinate representations vec A ′ + and vec A ′ + usually differ, I n ⊗ x ′ t−1 vec A ′ + and I n ⊗ x ′ t−1 vec A ′ + represent the same projection (component wise on the space spanned by the columns of Γ p or equivalently on the space spanned by the components of x t−1 ). By construction, we have that span C (Γ p ) = span C (V 1 ) = span C (Γ p S 1 ) and, in particular, that the rank of the projection of Γ p S 1 on span C (Γ p ) is of the same rank as Γ p . This projection idea can be used to investigate whether researcher imposed restrictions on the system parameters are "true" restrictions (in the sense that they restrict the possible covariance structures of the model) or whether the restrictions are sufficient to guarantee a unique solution.
Let C A vec A ′ + = 0, where C A ∈ R r×n 2 p is of full row rank, be the researcher imposed restrictions and denote the (right-) kernel of C A by S A ∈ R n 2 p×(n 2 p−r) . If span C ((I n ⊗ Γ p ) S A ) ⊇ span C (I n ⊗ V 1 ), then the researcher imposed restrictions are not overidentifying in the sense that without them the same set of covariance structures are feasible. The validity of this inclusion of spaces can be investigated using projections (calculated with, e.g., SVDs). In order to do so, we define the SVD of If span C ((I n ⊗ Γ p ) S A ) ⊇ span C (I n ⊗ V 1 ) holds, then we can express the column space of (I n ⊗ V 1 ) in terms of the columns of ((I n ⊗ Γ p ) S A ) and, in other words, the projection of (I n ⊗ V 1 ) on the column space of ((I n ⊗ Γ p ) S A ) must coincide with (I n ⊗ V 1 ). Expressed in terms of SVDs, this leads to Proposition 2. The restrictions C A are not overidentifying if and only if There is a unique solution of the Yule-Walker equations if and only if the kernel of (I n ⊗ Γ p ) S A is trivial.
Note that counting restrictions is not enough for concluding on the unique solvability of the Yule-Walker equations. As an example, consider Γ p = Of course, this example is special in the following sense.
Proposition 3. Let C A ∈ R ns×n 2 p be of full row rank and let Γ p be singular with rank deficiency equal to s. The set of restrictions C A ∈ R s×n 2 p | (7) does not hold is of Lebesgue measure zero in R ns×n 2 p . A generic, randomly chosen restric-tion C A can thus be used to obtain a unique solution of the system of equations    and the system parameters are globally identified.
Proof. Let S A of dimension n 2 p × n(np − s) denote the matrix obtained as the orthogonal complement of C A . Since (I n ⊗ Γ p ) = I n ⊗ ( V 1 V 2 ) D 11 0 (n 2 p−s)×s 0 s×(n 2 p−s) 0 s×s it is obvious that (I n ⊗ Γ p ) S A does not have full rank if and only if (I n ⊗ V ′ 1 ) S A is of reduced rank (smaller than n 2 p − ns). For given Γ p , the elements in the matrix of restrictions C A (and therefore also the ones in S A ) are free (up to the requirement that the rows of C A be linearly independent). The determinant det ((I n ⊗ V ′ 1 ) S A ) is thus a multivariate polynomial in the elements of S A . This determinant is either identically zero or zero only on a set of Lebesgue measure zero. Since for S A = (I n ⊗ V 1 ) the determinant is equal to one, the determinant is not identically zero.

Conclusion
In this article, we generalize the well-known identifiability results for structural VAR models to the case of a singular innovation covariance matrix.
The first main difference to the regular case is that the restrictions on the noise parameters B might be in contradiction to the singularity of the innovation covariance matrix. Moreover, the researcher imposed restrictions might already be contained in the restrictions implied by the singularity of the innovation covariance matrix and have therefore no further "identifying effect". The second main difference pertains mainly to restrictions on the system parameters A + . We provide conditions under which the researcher imposed restrictions are overidentifying and show that underidentification can be considered an unusual case when the rank deficiency coincides with the number of restrictions.

Acknowledgements
Financial support by the Research Funds of the University of Helsinki as well as by funds of the Oesterreichische Nationalbank (Austrian Central Bank, Anniversary Fund, project number: 17646) is gratefully acknowledged.

Data Availability Statement
There is no data involved in this study.