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Identifiability of Structural Singular Vector Autoregressive Models
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We generalize well-known results on structural identifiability of vector autoregressive (VAR) models to the case where the innovation covariance matrix has reduced rank. Singular structural 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 we provide a rank condition for identifiability of the noise parameters. Since the Yule-Walker (YW) 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.
Keywords: Stochastic singularity, structural vector autoregressive models, identifiability
JEL classification: C32, C50
Singular structural VAR (SVAR) models play an important role in macroeconomic modeling. To introduce the subject, we succinctly discuss Generalized Dynamic Factor Models (GDFM) and Dynamic Stochastic General Equilibrium (DSGE) models, and their relation to singular SVAR models.
In the literature on GDFMs FHLR00,FHLR05,BaiNg07,EJC10, singular VAR models are the essential building block connecting static factors (a static transformation of the denoised observables) to the uncorrelated lower-dimensional shocks. chenandersondeistlerfiller2010 and festschriftbdoanderson2010 treat canonical forms of singular VAR models, i.e. they focus on the reduced form. In FGLR05, it is demonstrated that dynamic factor models (and consequently singular VAR models) are useful for structural modeling. In this article, we provide results regarding identifiability of singular SVAR models and thus analyze Step C in FGLR05 in more detail.
A key issue in the econometric treatment of DSGE models is caused by the fact that the number of exogenous shocks driving the system is often strictly smaller than the number of endogenous variables. This is known as the stochastic singularity problem DeJongDave11 and investigated in, e.g., RugeMurcia07singularDSGE. The relationship between DSGE and SVAR models is analyzed in DeJongDave11, Giacomini13, KilianLut17, and most recently by lippi19. It has been acknowledged KilianLut17 that the usual strategies\footnote{KilianLut17 enumerate 1) adding measurement noise as, e.g. in Sargent89 or Ireland04, and discussed in lippi19, 2) reducing the number of observables bouakezRugemurcia2005habit and 3) augmenting the number of economically interpretable shocks IngramKocherlotkaSaven94,LeeperSims94.} for solving this rank deficiency problem are not satisfactory. Thus, one way forward would be estimation of singular SVAR models.
Singularity of the innovation covariance matrix has two possible consequences for the restrictions imposed by the modeler\footnote{A similar problem appears in KilianLut17 where it is emphasized that the reduced rank of a certain matrix appearing in cointegration analysis must be “taken into account when determining the number of restrictions that have to be imposed for full identification of the structural shocks”.}. 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 SVAR models. Moreover, restrictions on the system parameters are not necessarily over-identifying when the innovation covariance matrix is singular because the YW 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 we define notions which will be necessary later. As preparation for the main results, we collect well-known facts on singular VAR models in reduced form (in particular regarding the possible non-singularity of the Toeplitz matrix appearing in the YW equations) in section 3. In section 4, we analyze restrictions on the noise and system parameters and provide results as to how a singular innovation covariance matrix needs to be taken into account for identifiability analysis. We illustrate that the usual order condition may be misleading in the singular case with a (stochastically singular) DSGE model and provide easily verifiable conditions for under- and over-identification when the YW equations have multiple solutions.\textcolor{red}{ }All proofs are deferred to the Appendix.
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. $z\left(y_{t}\right)_{t\in\mathbb{Z}}=\left(y_{t-1}\right)_{t\in\mathbb{Z}}$. For a (matrix) polynomial $p(z)$, we denote by $\deg(p(z))$ the highest degree of $p(z)$. The transpose of an $\left(m\times n\right)$ dimensional matrix $A$ is represented by $A'$. We use ${\rm vec}\left(A\right)\in\mathbb{R}^{nm\times1}$ to stack the columns of $A$ into a column vector and ${\rm vech}\left(A\right)\in\mathbb{R}^{\frac{n(n+1)}{2}\times1}$ to stack the lower-triangular elements of an $n$-dimensional square matrix $A$ analogously. The $n$-dimensional identity matrix is denoted by $I_{n}$. 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 $span_{R}\left(A\right)$ and $span_{C}\left(A\right)$, respectively, and the projection of $A$ on $span_{R}\left(B\right)$, $B\in\mathbb{R}^{r\times n}$, is $Proj_{R}\left(A|B\right)$ and the projection of $A$ on $span_{C}\left(D\right)$, $D\in\mathbb{R}^{m\times s}$, is $Proj_{C}\left(A|D\right)$. We use $\mathbb{E}\left(\cdot\right)$ for the expectation of a random variable with respect to a given probability space.
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 SVAR\footnote{Most work on SVAR models is performed in the parametrization where $A_{0}=I_{n}$ KilianLut17 and investigates how to estimate $B$. Since we also treat structural restrictions on $A_{+}$, rather than on the reduced form parameters $A_{0}^{-1}A_{+}$, we allow here for additional generality and treat the so-called $AB$-model (in the nomenclature of luet05). Except for section 4.2, it is sufficient to set $A_{0}=I_{n}$. } system
where $x_{t-1}=\left(y'_{t-1},\ldots,y'_{t-p}\right)'$ and where the dimension $q$ of the white noise process $(\varepsilon_{t})$ of (economically) fundamental shocks with covariance matrix $I_{q}$ is strictly smaller than $n$, the number of observed variables of $y_{t}$. The matrix $B\in\mathbb{R}^{n\times q}$ has full column rank. This implies that the covariance matrix $\Sigma_{u}$ of the innovations $u_{t}=B\varepsilon_{t}$ is of rank $q<n$.
Furthermore, we assume that the matrices $A_{i}\in\mathbb{R}^{n\times n}$ are such that the stability condition
holds, where $a(z)=A_{0}-A_{1}z-\cdots-A_{p}z^{p}$, and that $\det\left(A_{0}\right)\neq0$. Last, we assume that the system and noise parameters satisfy the restrictions
where $C_{S}$ and $C_{N}$ are of dimensions $\left(r_{S}\times n^{2}p\right)$ and $\left(r_{N}\times\left(n^{2}+nq\right)\right)$, 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 $\left(A_{+},\left(A_{0},B\right)\right)$ in system ((ref)) which satisfy the restrictions imposed by ((ref)) and ((ref)).
Next, we discuss the external characteristics which are observed by the econometrician. The stationary solution of the system ((ref)) (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 $\gamma(s)=\mathbb{E}\left(y_{t}y_{t-s}'\right)$ or equivalently the spectral density $f\left(e^{-i\lambda}\right)=\frac{1}{2\pi}\sum_{s=-\infty}^{\infty}\gamma(s)e^{-is\lambda}$.
We follow Rothenberg71 to define identifiability of parametric models. Two internal characteristics $\left(A_{+}^{(1)},\left(A_{0}^{(1)},B^{(1)}\right)\right)$ and $\left(A_{+}^{(2)},\left(A_{0}^{(2)},B^{(2)}\right)\right)$ are called observationally equivalent if they imply the same external characteristics. An internal characteristic is globally identifiable if there is no other observationally equivalent internal characteristic. Likewise, an internal characteristic $\left(A_{+},\left(A_{0},B\right)\right)$ is locally identifiable if there exists a neighborhood around the parameter $\left(A_{+},\left(A_{0},B\right)\right)$ 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 $\left(y_{t}\right)$.
In order to prepare for the structural case where we will connect to external characteristics uniquely to the deep parameters, we review identifiability of the reduced form of singular VAR models, see also andersondeistlerchenfiller2012. In particular, we discuss the rank of finite sections of the covariance of the observed process and its relation to the rank of the innovation covariance matrix $\Sigma_{u}$. Moreover, we show how $p$, $q$ and the left-kernel $L\in\mathbb{R}^{(n-q)\times n}$ of $\Sigma_{u}$, which are assumed to be known in the identifiability analysis in Section (ref), can be obtained from the external characteristics.
One way to connect the observable characteristics to the internal characteristics is by using the YW equations\footnote{They are obtained by right-multiplying $\left(y_{t}',\ldots,y_{t-p}'\right)$ on ((ref)) and taking expectations.}, i.e. \[ \bar{A}_{+}\Gamma_{p}=\gamma_{p}\ \text{and}\ \Sigma_{u}=\gamma(0)-\bar{A}_{+}\gamma_{p}', \] where $\bar{A}_{+}=\left(\bar{A}_{1},\ldots,\bar{A}_{p}\right)=A_{0}^{-1}A_{+}$ are the reduced form system parameters, $\Gamma_{p}=\left(
\right)$ and $\gamma_{p}=\left(\gamma(1),\ldots,\gamma(p)\right)$. If $\Gamma_{p}$ is invertible, there is a unique internal characteristic $\left(\bar{A}_{+},\Sigma_{u}\right)$ for a given external characteristic $\left(\gamma(0),\ldots,\gamma(p)\right)$. While for VAR models with non-singular innovation covariance matrix it can be shown \citep[page 112]{HannanDeistler12} that $\Gamma_{r}$ is non-singular for all $r\in\mathbb{N}$, this is not the case for VAR models that have a singular innovation covariance matrix. Indeed, it is easy to see \citep[Theorem 7]{andersondeistler09} that, for $s>0$, $rk\left(\Gamma_{p+s}\right)=rk\left(\Gamma_{p}\right)+s\cdot q$ holds. Even $\Gamma_{p}$ might be rank deficient: Consider a solution $\left(\bar{A}_{+}^{(1)},\Sigma_{u}\right)$ of the YW equations and the polynomial matrix $U(z)=I_{n}+cc'z$ where $c'\in\mathbb{R}^{1\times n}$ is non-trivial and in the left-kernel of both $\bar{A}_{p}^{(1)}$ and $\bar{B}=A_{0}^{-1}B$. One can verify that $U(z)\bar{a}^{(1)}(z)$, where $\bar{a}^{(1)}(z)$ is the polynomial corresponding to $\bar{A}_{+}^{(1)}$, is also a polynomial matrix of degree $p$ and solves the YW equations which implies that $\Gamma_{p}$ has a non-trivial left-kernel. Note that the perpendicular of the projection is unique irrespective of how the projection itself is parametrized. More formally, $\Sigma_{u}^{(1)}=\gamma(0)-\bar{A}_{+}^{(1)}\gamma_{p}'=\gamma(0)-\bar{A}_{+}^{(2)}\gamma_{p}'=\Sigma_{u}^{(2)}$ holds even if $\bar{A}_{+}^{(1)}\neq\bar{A}_{+}^{(2)}$ for two solutions $(\bar{A}_{+}^{(1)},\Sigma_{u}^{(1)})$ and $(\bar{A}_{+}^{(2)},\Sigma_{u}^{(2)})$ of the Yule-Walker equations.
Examining the ranks of $\Gamma_{r}$ for some consecutive values of $r$, the integer-valued parameters $q$ and $p$ can be obtained. Having the rank $q$ of the innovation covariance $\Sigma_{u}$ available, it is straightforward to obtain (a basis of) the left-kernel $L\in\mathbb{R}^{(n-q)\times n}$ of $\Sigma_{u}$ AlSadoon17ranktests.
For the remainder of this article, we will assume that $p,q,$ and $L$ are known by the practitioner (in addition to the other external characteristics).
In this section, we discuss identifiability of noise and system parameters in the case of singular SVAR models. First, we derive a condition which ensures that the modeler imposed restrictions on the noise parameters are not in contradiction to the singularity of the innovation covariance matrix. Subsequently, we derive a rank condition similar to the previous literature and illustrate with a New-Keynesian DSGE model that the order condition does not provide useful information in the stochastically singular case. Secondly, we discuss whether researcher imposed restrictions on system parameters are under-, just- or over-identifying. 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 $\left(I_{n}\otimes\Gamma_{p}\right)$).
We start with affine restrictions on the noise parameters $\left(A_{0},B\right)$ which appear in short-run restrictions, see KilianLut17 for the non-singular case. The conditions that we derive are local in nature. Next, we deal with the case where $\Gamma_{p}$ may be singular and where affine restrictions on the elements in $A_{+}$ are imposed. These results concern global identifiability.
In the light of the discussion in Section (ref), we start with a singular $\Sigma_{u}$ and with researcher imposed restrictions given by
Here, $C_{N}=\left(
\right)$ is block-diagonal and has full row rank, and $c'_{N}=\left(c'_{A_{0}},c'_{B}\right)$. In order to show the existence of a unique pair $(A_{0},B)$ for parametrising $\Sigma_{u}=A_{0}^{-1}BB'\left(A_{0}^{'}\right)^{-1}$, one usually calls on the implicit functions theorem. While in the non-singular SVAR case the system of equations to be analyzed always has at least one solution, it might happen in the singular SVAR case that the set of solutions of ((ref)) (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 ((ref)) imposed by the researcher do not contradict the singularity structure of the model. 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 $\Sigma_{u}$ implies $L\left(A_{0}^{-1}B\right)=0$, where the rows of $L\in\mathbb{R}^{(n-q)\times n}$ span the left-kernel of $\Sigma_{u}$, which is equivalent to
The condition for when the solution set of the joint system of restrictions given in ((ref)) and ((ref)) is non-empty is given in the following
The singularity of $\Sigma_{u}$ restricts the set of admissible restrictions on the parameter space. If $C_{N}$ does not “interfere” with the singularity restrictions, i.e. if $C_{N}$ lies in the orthogonal complement of $span_{R}\left(\mathcal{N}\right)$ or expressed differently if $Proj_{R}\left(C_{B}|\mathcal{N}\right)=0$, then $M=C_{N}$ and condition ${\rm rk}(M)={\rm rk}
$ is satisfied.
To illustrate Proposition (ref), we discuss a version of the New-Keynesian monetary business cycle model LubikSchorfheide03,castelnuovo13 featuring a “supply-shifting” shock in the new-Keynesian Phillips Curve (NKPC). We thus consider the model
where $\left(\pi_{t},x_{t},R_{t}\right)$ denote inflation, output gap, and nominal interest rate in log-deviation from a unique steady state. The conditional expectations are to be understood as linear projections on the space spanned by present and past components of the uncorrelated shocks $\varepsilon_{t}^{\pi}$ and $\varepsilon_{t}^{R}$ which are white noise processes (whose variance is normalized to one for the sake of simplicity). The parameters of the model are the subjective time preference factor $\beta\in\left(0,1\right)$, $\phi\geq0$ the elasticity of the interest response of the central bank, and the slope parameters $\kappa$ and $\tau$.
For specific parameter values $(\beta,\phi,\tau,\kappa)=\left(\frac{4}{5},\frac{39}{38},\frac{3}{4},\frac{1}{2}\right)$, we solve this system of equations involving conditional expectations of future endogenous variables Sims01,funo17full and obtain the unique causal stationary solution $\left(
\right)=B\left(
\right)$, where $B=\left(
\right)$, of the DSGE model described above. The innovation covariance matrix is obviously singular. The restrictions on $B$ are described by $C_{B}{\rm vec}(B)=c_{B}$, with $C_{B}=\left(
\right)$ and $c_{B}=\left(
\right)$. In order to apply Proposition \ref{prop:BmodelLocalIdent}, we need to check the condition ${\rm rk}(M)={\rm rk}
$ of Lemma \ref{lem:rest_affine_compatible}. The perpendicular of the projection of $C_{B}$ on the row-span of $\left(I_{2}\otimes L\right)$ for $L=
$, is given by $M=\left(
\right)$. For any value $\tau\in\mathbb{R}\backslash\left\{ 0\right\} $ the relation ${\rm rk}(M)={\rm rk}
$ is satisfied. We can now apply Proposition \ref{prop:BmodelLocalIdent} and check the rank of $\left(
\right)=\left(
\right)$ is equal to 6 for any values $\kappa$ and $\tau$.
We now focus on imposing linear restrictions on the structural parameters $A_{+}$ in the case where $\Gamma_{p}$ is singular. Thus, without restrictions on $A_{+}$, there are multiple observationally equivalent solutions of the YW equations (one particular solution plus the left kernel of $\Gamma_{p}$). We will start by considering the case where $A_{0}=I_{n}$ (such that the reduced form parameters $\bar{A}_{+}$ coincide with $A_{+}$). This simplifies the discussion and allows us to illustrate why the identifiability problem (for $A_{0}$ not necessarily equal to the identity matrix) can “generically” be solved by (the right number of) arbitrary restrictions on $A_{+}$.\footnote{To be more precise, it can be considered uncommon that $s\cdot n$, where $s$ is the dimension of the kernel of $\Gamma_{p}$, “random” restrictions on ${\rm vec}\left(A_{+}'\right)$ do not solve the identifiability problem.}
Two aspects deserve special attention. First, the particular solutions (canonical representatives of the equivalence class of observational equivalence) introduced in festschriftbdoanderson2010 and chenandersondeistlerfiller2010 can be obtained by choosing a particular set of restrictions on ${\rm vec}\left(A_{+}'\right)$. Secondly, singular SVAR models are special in the sense that some researcher imposed restrictions are not over-identifying in the sense that imposing them does not restrict the feasible covariance structures. In Lemma (ref) we provide a condition for checking whether the researcher imposed restrictions on $A_{+}$ are over-identifying.
To simplify discussion, we note that vectorizing the (transposed) YW equations leads to $\left(I_{n}\otimes\Gamma_{p}\right){\rm vec}\left(A_{+}'\right)={\rm vec}\left(\gamma_{p}'\right)$. In festschriftbdoanderson2010, the authors choose the first linearly independent rows of $\Gamma_{p}$ as a basis of the row space (or equivalently column space) of $\Gamma_{p}$ to define a particular solution of the YW equations. To fix ideas, consider a $\Gamma_{p}$ whose first $\left(np-s\right)$ linearly independent rows are selected by premultiplying $S_{1}'$ of dimension $\left(\left(np-s\right)\times np\right)$, containing only zeros and ones, and denote by $S_{2}'$ the $\left(s\times np\right)$-dimensional matrix containing zeros and ones such that $S_{1}'S_{2}=0$. A basis of the column space thus consists of the columns of $\Gamma_{p}S_{1}$, i.e. the elements $S_{2}'A_{+}'$ are restricted to zero. Restricting each column of $A_{+}'$ to be orthogonal to the columns of $S_{2}$ therefore results in a unique solution of the YW equations, i.e. the matrix in brackets in $\left[
\right]vec\left(A_{+}'\right)=\left(
\right)$ is of full rank. We denote the unique solution of the equation above by $\widehat{{\rm vec}\left(A_{+}'\right)}$.
In chenandersondeistlerfiller2010, the authors choose the minimum norm solution of the YW equations as the particular solution. Let $I_{n}\otimes\left[\left(
\right)\left(
\right)\left(
\right)\right]$ be the singular value decomposition (SVD)\footnote{$\left(V_{1},V_{2}\right)$ are an orthonormal eigenbasis describing the image and the kernel of $\Gamma_{p}$ respectively, and $D_{11}$ is a diagonal matrix with positive diagonal elements.} of $\left(I_{n}\otimes\Gamma_{p}\right)$ of rank $n^{2}p-ns=n\cdot rk\left(\Gamma_{p}\right)$. The particular solution is such that coordinates corresponding to the basis vectors $V_{2}$ are set equal to zero. Put differently, ${\rm vec}\left(A_{+}'\right)$ is required to be orthogonal to the columns of $\left(I_{n}\otimes V_{2}\right)$, i.e. $\left[
\right]{\rm vec}\left(A_{+}'\right)=\left(
\right)$. We denote the unique solution of the equation above by $\widetilde{{\rm vec}\left(A_{+}'\right)}$.
While the coordinate representations $\widehat{{\rm vec}\left(A_{+}'\right)}$ and $\widetilde{{\rm vec}\left(A_{+}'\right)}$ usually differ, $\left(I_{n}\otimes x_{t-1}'\right)\widehat{{\rm vec}\left(A_{+}'\right)}$ and $\left(I_{n}\otimes x_{t-1}'\right)\widetilde{{\rm vec}\left(A_{+}'\right)}$ represent the same projection (component wise on the space spanned by the columns of $\Gamma_{p}$ or equivalently on the space spanned by the components of $x_{t-1}$). By construction, we have that $span_{C}\left(\Gamma_{p}\right)=span_{C}\left(V_{1}\right)=span_{C}\left(\Gamma_{p}S_{1}\right)$ and, in particular, that the rank of the projection of $\Gamma_{p}S_{1}$ on $span_{C}\left(\Gamma_{p}\right)$ is equal to the rank of $\Gamma_{p}$. 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) and whether the restrictions are sufficient to guarantee a unique solution. Let $C_{S}{\rm vec}\left(A_{+}'\right)=0$, where $C_{S}\in\mathbb{R}^{r_{S}\times n^{2}p}$ is of full row rank, be the researcher imposed restrictions and denote the (right-) kernel of $C_{S}$ by $S_{A}\in\mathbb{R}^{n^{2}p\times\left(n^{2}p-r_{S}\right)}$. If $span_{C}\left(\left(I_{n}\otimes\Gamma_{p}\right)S_{A}\right)\supseteq span_{C}\left(I_{n}\otimes V_{1}\right)$, then the researcher imposed restrictions are not over-identifying in the sense that without them the same set of covariance structures are feasible. In order to investigate the validity of this inclusion of spaces, we define the SVD of
If $span_{C}\left(\left(I_{n}\otimes\Gamma_{p}\right)S_{A}\right)\supseteq span_{C}\left(I_{n}\otimes V_{1}\right)$ holds, then we can express the column space of $\left(I_{n}\otimes V_{1}\right)$ in terms of the columns of $\left(\left(I_{n}\otimes\Gamma_{p}\right)S_{A}\right)$ and, in other words, the projection of $\left(I_{n}\otimes V_{1}\right)$ on the column space of $\left(\left(I_{n}\otimes\Gamma_{p}\right)S_{A}\right)$ must coincide with $\left(I_{n}\otimes V_{1}\right)$. Expressed in terms of SVDs, this leads to
Returning to the general case where $A_{0}$ is not necessarily equal to the identity matrix, we will now show that it is in general enough to impose as many restrictions as there are basis vectors in the kernel of $\left(I_{n}\otimes\Gamma_{p}\right)$. Notice that $C_{S}vec\left(A'_{+}\right)=\left[C_{S}\left(A_{0}\otimes I_{np}\right)\right]vec\left(\bar{A}'_{+}\right)$ such that for given $A_{0}$, the restrictions on the parameters $\bar{A}_{+}$ can be obtained straight-forwardly from the ones on $A_{+}$.
In order to provide some intuition for the following result, we consider a quite special example where it is not sufficient to count the number of restrictions for deducing identifiability of the system parameters. Consider $\Gamma_{p}=\left(
\right)$ and $C_{S}=I_{3}\otimes\left(0,1,0\right)$, such that $S_{A}=\left(I_{3}\otimes\left(
\right)\right)$ and $\left(I_{n}\otimes\Gamma_{p}\right)S_{A}=\left(I_{3}\otimes\left(
\right)\right)$. Even though the order condition (that the rank deficiency of $\left(I_{n}\otimes\Gamma_{p}\right)$ is equal to the number of restrictions) is satisfied, they are not sufficient for obtaining a unique solution of the YW equations. Indeed, $\left[I_{n^{2}p}-\tilde{U}_{1}\tilde{U}_{1}'\right]\left(I_{n}\otimes V_{1}\right)=\left(I_{3}\otimes\left(
\right)\right)\left(I_{3}\otimes\left(
\right)\right)\neq0$ and the right-kernel of $\left(I_{n}\otimes\Gamma_{p}\right)S_{A}=\left(I_{3}\otimes\left(
\right)\right)$ is non-empty. The non-generic nature of this example is summarized in
\textcolor{red}Notice, however, that it is not possible to solve the identifiability problem for $A_{+}$ by restricting the transfer function $a(z)^{-1}b$ (e.g., by restricting the long-run coefficients in $k(1)=a(1)^{-1}b$). Since two observationally equivalent pairs $\left(a^{(1)}(z),b^{(1)}\right)$ and $\left(a^{(2)}(z),b^{(2)}\right)$ have by definition the same transfer function, restricting $a(z)^{-1}b$ directly has the effect of either excluding the whole equivalence class or not providing additional information for distinguishing different pairs $\left(a(z),b\right)$ with the same transfer function.
In this article, we generalize the well-known identifiability results for SVAR 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 $(A_{0},B)$ might contradict 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 therefore do not have any further “identifying effect”. The second main difference pertains mainly to restrictions on the structural system parameters $A_{+}$. We provide conditions under which the researcher imposed restrictions on these parameters are over-identifying and show that underidentification can be considered an unusual case when the rank deficiency coincides with the number of restrictions.
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.
There is no data involved in this study.
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