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Identifiability of Structural Singular Vector Autoregressive Models
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\title{Identifiability of Structural Singular Vector Autoregressive Models}
\author{Bernd Funovits$^{a,b}$ and Alexander Braumann$^{c}$}
\maketitle
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\vspace{-2cm}
\section*{}
\section*{Affiliations}
\begin{singlespace}
\textbf{$^{a}$University of Helsinki}\\
Faculty of Social Sciences\\
Discipline of Economics\\
P. O. Box 17 (Arkadiankatu 7)\\
FIN-00014 University of Helsinki\\
\end{singlespace}
and
\begin{singlespace}
\textbf{$^{b}$TU Wien}\\
Institute of Statistics and Mathematical Methods in Economics\\
Econometrics and System Theory\\
Wiedner Hauptstr. 8\\
A-1040 Vienna
\end{singlespace}
and
\begin{singlespace}
\textbf{$^{c}$TU Braunschweig}\\
Institute for Mathematical Stochastics\\
Universitätsplatz 2\\
D-38106 Braunschweig\\
\end{singlespace}
\section*{E-mail of Corresponding Author}
[email removed]
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\section*{Abstract}
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.
\textbf{Keywords:} Stochastic singularity, structural vector autoregressive
models, identifiability
\textbf{JEL classification: }C32, C50
\pagebreak{}
\section{Introduction}
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 \citep{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. \citet{chenandersondeistlerfiller2010}
and \citet{festschriftbdoanderson2010} treat canonical forms of singular
VAR models, i.e. they focus on the reduced form. In \citet{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 \citet[page 1332]{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 \citep[page 184f.]{DeJongDave11}
and investigated in, e.g., \citet{RugeMurcia07singularDSGE}. The
relationship between DSGE and SVAR models is analyzed in \citet{DeJongDave11},
\citet{Giacomini13}, \citet[Chapter 6.2]{KilianLut17}, and most
recently by \citet{lippi19}. It has been acknowledged \citep[page 177]{KilianLut17}
that the usual strategies\footnote{\citet{KilianLut17} enumerate 1) adding measurement noise as, e.g.
in \citet{Sargent89} or \citet{Ireland04}, and discussed in \citet{lippi19},
2) reducing the number of observables \citep{bouakezRugemurcia2005habit}
and 3) augmenting the number of economically interpretable shocks
\citep{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 \citet[Chapter 10.2]{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.
\section{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 SVAR\footnote{Most work on SVAR models is performed in the parametrization where
$A_{0}=I_{n}$ \citep[Chapter 8]{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 \citet{luet05}). Except for section 4.2,
it is sufficient to set $A_{0}=I_{n}$. } system
\begin{align}
A_{0}y_{t} & =A_{1}y_{t-1}+\cdots+A_{p}y_{t-p}+B\varepsilon_{t},\label{eq:model}\\
& =\underbrace{\left(A_{1},\ldots,A_{p}\right)}_{=A_{+}}x_{t-1}+B\varepsilon_{t}\nonumber
\end{align}
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
\begin{equation}
\det\left(a(z)\right)\neq0,\ \left|z\right|\leq1,\label{eq:restr_stability}
\end{equation}
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
\begin{equation}
C_{S}{\rm vec}\left(A_{+}'\right)=c_{S}\text{ and }C_{N}{\rm vec}\left(\begin{pmatrix}A_{0} & B\end{pmatrix}\right)=c_{N}\label{eq:restr_modeler}
\end{equation}
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{eq:model}) which satisfy the restrictions imposed
by (\ref{eq:restr_stability}) and (\ref{eq:restr_modeler}).
Next, we discuss the external characteristics which are observed by
the econometrician. The stationary solution of the system (\ref{eq:model})
(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 \citet{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 $\left(y_{t}\right)$.
\section{Identifiability Issues in Reduced Form Singular VAR Models \label{sec:YuleWalker}}
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
\citet{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{sec:structural_identifiability}, 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{eq:model}) 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, $\Gamma_{p}=\left(\begin{smallmatrix}\gamma(0) & \gamma(1) & \cdots & \gamma(p-1)\\
\gamma(-1) & \gamma(0)\\
\vdots & & \ddots\\
\gamma(-p+1) & & & \gamma(0)
\end{smallmatrix}\right)$ 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 $\Sigma_{u}$ \citep{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).
\section{\label{sec:structural_identifiability}Imposing Structural Restrictions}
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 \citet[Chapter 8]{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.
\subsection{Affine Restrictions on the Noise Parameters}
In the light of the discussion in Section \ref{sec:YuleWalker}, we
start with a singular $\Sigma_{u}$ and with researcher imposed restrictions
given by
\begin{equation}
C_{N}{\rm vec}\left(\left(A_{0},B\right)\right)=c_{N}.\label{eq:restr_B_affine}
\end{equation}
Here, $C_{N}=\left(\begin{smallmatrix}C_{A_{0}} & 0_{r_{A_{0}}\times nq}\\
0_{r_{B}\times n^{2}} & C_{B}
\end{smallmatrix}\right)$ 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{eq:restr_B_affine}) (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{eq:restr_B_affine}) 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
\begin{equation}
\left(I_{q}\otimes L\right){\rm vec}\left(A_{0}^{-1}B\right)=0.\label{eq:restr_B_singul}
\end{equation}
The condition for when the solution set of the joint system of restrictions
given in (\ref{eq:restr_B_affine}) and (\ref{eq:restr_B_singul})
is non-empty is given in the following
\begin{lem}
\label{lem:rest_affine_compatible}Let $L\in\mathbb{R}^{(n-q)\times n}$
be a basis of the left-kernel of $\Sigma_{u}$, define $\mathcal{N}:=\left\{ \left[\left(A_{0}^{-1}B\right)',I_{q}\right]\otimes LA_{0}^{-1}\right\} $,
and let $M:=C_{N}-Proj_{R}(C_{N}|\mathcal{N})$ be the perpendicular
of the projection of $C_{N}$ on the row-span of $\mathcal{N}$. The
restrictions $C_{N}{\rm vec}\left(A_{0},B\right)=c_{N}$ are consistent
with the singularity of $\Sigma_{u}$ if and only if ${\rm rk}(M)={\rm rk}\begin{pmatrix}M & c_{N}\end{pmatrix}$,
i.e. if and only if $c_{N}$ is in the image of $M$.
\end{lem}
\begin{rem}
When we consider the SVAR setting in which $A_{0}=I_{n}$, we only
need to check whether $c_{B}$ is contained in the column space of
$C_{B}-Proj\left(C_{B}|\left(I_{q}\otimes L\right)\right)$.
\end{rem}
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 ${\rm rk}(M)={\rm rk}\begin{pmatrix}M & c_{N}\end{pmatrix}$
is satisfied.
\begin{prop}
\label{prop:BmodelLocalIdent}Let $A_{0}$ and $B$ be $\left(n\times n\right)$
and $\left(n\times q\right)$-dimensional matrices of full column
rank, let $n>q$, and let $C_{N}{\rm vec}\left(A_{0},B\right)=c_{N}$
hold. For given $\Sigma_{u}$, the matrix $\left(A_{0},B\right)$
is the unique solution of $\Sigma_{u}=A_{0}^{-1}B\left(A_{0}^{-1}B\right)'$
if and only if $c_{N}$ is in the image of $M=C_{N}-Proj_{R}\left(C_{N}|\mathcal{N}\right)$
and the matrix $\left(\begin{smallmatrix}-2D_{n}^{+}(\Sigma_{u}\otimes A_{0}^{-1}) & 2D_{n}^{+}(A_{0}^{-1}B\otimes A_{0}^{-1})\\
C_{A_{0}} & 0\\
0 & C_{B}
\end{smallmatrix}\right)$ is of (full column) rank $n^{2}+nq$.
\end{prop}
\begin{rem}
Considering for simplicity the case where $A_{0}=I_{n}$ and following
\citet{Rothenberg71}, the restrictions imposed on the structural
parameter $B$ are $C_{B}{\rm vec}(B)=c_{B}$ as well as $\left(I_{q}\otimes L\right)vec(B)=0$
which suggests that the matrix $\frac{\partial}{\partial\left(vec\left(B\right)\right)'}\left(\begin{smallmatrix}{\rm vec}h\left(BB'\right)-{\rm vech}\left(\Sigma_{u}\right)\\
C_{B}{\rm vec}(B)-c_{B}\\
\left(I_{q}\otimes L\right){\rm vec}(B)
\end{smallmatrix}\right)$ needs to be of rank $nq$. However, it is not necessary to include
$\left(I_{q}\otimes L\right)$ in Proposition \ref{prop:BmodelLocalIdent}
because $\left(I_{q}\otimes L\right){\rm vec}(B)=0$ is already implied
by the fact that $BB'=\Sigma_{u}$. Put differently, the inequality
$rk\left(\begin{smallmatrix}2D_{n}^{+}(B\otimes I_{n})\\
C_{B}\\
\left(I_{q}\otimes L\right)
\end{smallmatrix}\right)\leq rk\left(\begin{smallmatrix}2D_{n}^{+}(B\otimes I_{n})\\
C_{B}
\end{smallmatrix}\right)$ holds.
\end{rem}
\begin{rem}
If $q<n$, the usual order condition requiring that the number of
rows in $\left(\begin{smallmatrix}2D_{n}^{+}\left(B\otimes I_{n}\right)\\
C_{B}
\end{smallmatrix}\right)$ be larger than or equal to the number of columns is not useful. Consider
the case where there are no researcher imposed restrictions. While
the order condition is satisfied for $q\leq\frac{n+1}{2}$, the matrix
$D_{n}^{+}\left(B\otimes I_{n}\right)$ of dimension $\left(\frac{n(n+1)}{2}\times nq\right)$
is of course rank deficient with co-rank $\frac{q(q-1)}{2}$.
\end{rem}
\begin{rem}
The rank of the matrix $\left(\begin{smallmatrix}2D_{n}^{+}\left(B\otimes I_{n}\right)\\
C_{B}
\end{smallmatrix}\right)$ drops if some restrictions in $C_{B}$ are already implied by the
singularity structure of $\Sigma_{u}$, i.e. if for the $r$-th row
$\left[C_{B}\right]_{\left[r,\bullet\right]}\subseteq{\rm span_{R}}\left(I_{q}\otimes L\right)$
holds. Thus, the $\frac{q(q-1)}{2}$ additional restrictions which
are necessary to obtain a matrix $\left(\begin{smallmatrix}2D_{n}^{+}\left(B\otimes I_{n}\right)\\
C_{B}
\end{smallmatrix}\right)$ of full column rank must not be contained in the row space of $D_{n}^{+}\left(B\otimes I_{n}\right)$.
\end{rem}
\subsubsection{Illustration}
To illustrate Proposition \ref{prop:BmodelLocalIdent}, we discuss
a version of the New-Keynesian monetary business cycle model \citep{LubikSchorfheide03,castelnuovo13}
featuring a ``supply-shifting'' shock in the new-Keynesian Phillips
Curve (NKPC). We thus consider the model
\begin{align*}
\pi_{t} & =\beta\mathbb{E}_{t}\left(\pi_{t+1}\right)+\kappa x_{t}+\varepsilon_{t}^{\pi}\\
x_{t} & =\mathbb{E}_{t}\left(x_{t+1}\right)-\tau\left(R_{t}-\mathbb{E}_{t}\left(\pi_{t+1}\right)\right)\\
R_{t} & =\phi\mathbb{E}_{t}\left(\pi_{t+1}\right)+\varepsilon_{t}^{R}.
\end{align*}
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 \citep{Sims01,funo17full} and obtain
the unique causal stationary solution $\left(\begin{smallmatrix}R_{t}\\
\pi_{t}\\
x_{t}
\end{smallmatrix}\right)=B\left(\begin{smallmatrix}\varepsilon_{t}^{R}\\
\varepsilon_{t}^{\pi}
\end{smallmatrix}\right)$, where $B=\left(\begin{smallmatrix}1 & 0\\
-\kappa\tau & 1\\
-\tau & 0
\end{smallmatrix}\right)$, 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 $C_{B}=\left(\begin{smallmatrix}1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right)$ and $c_{B}=\left(\begin{smallmatrix}1\\
0\\
1\\
0
\end{smallmatrix}\right)$. In order to apply Proposition \ref{prop:BmodelLocalIdent}, we need
to check the condition ${\rm rk}(M)={\rm rk}\begin{pmatrix}M & c_{B}\end{pmatrix}$
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 $L=\begin{pmatrix}\tau & 0 & 1\end{pmatrix}$, is given by $M=\left(\begin{smallmatrix}\frac{1}{1+\tau^{2}} & 0 & -\frac{\tau}{1+\tau^{2}} & 0 & 0 & 0\\
0 & 0 & 0 & \frac{1}{1+\tau^{2}} & 0 & -\frac{\tau}{1+\tau^{2}}\\
0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & -\frac{\tau}{1+\tau^{2}} & 0 & \frac{\tau^{2}}{1+\tau^{2}}
\end{smallmatrix}\right)$. For any value $\tau\in\mathbb{R}\backslash\left\{ 0\right\} $ the
relation ${\rm rk}(M)={\rm rk}\begin{pmatrix}M & c_{B}\end{pmatrix}$
is satisfied. We can now apply Proposition \ref{prop:BmodelLocalIdent}
and check the rank of $\left(\begin{smallmatrix}2D_{3}^{+}(B\otimes I_{3})\\
C_{B}
\end{smallmatrix}\right)=\left(\begin{smallmatrix}2 & 0 & 0 & 0 & 0 & 0\\
-\kappa\tau & 1 & 0 & 1 & 0 & 0\\
-\tau & 0 & 1 & 0 & 0 & 0\\
0 & -2\kappa\tau & 0 & 0 & 2 & 0\\
0 & -\tau & -\kappa\tau & 0 & 0 & 1\\
0 & 0 & -2\tau & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right)$ is equal to 6 for any values $\kappa$ and $\tau$.
\subsection{Affine Restrictions on the System Parameters}
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 \citet{festschriftbdoanderson2010} and
\citet{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{lem:overidentifying} 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 \citet{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 $\left[\begin{smallmatrix}\left(I_{n}\otimes\Gamma_{p}\right)\\
\left(I_{n}\otimes S_{2}'\right)
\end{smallmatrix}\right]vec\left(A_{+}'\right)=\left(\begin{smallmatrix}{\rm vec}\left(\gamma_{p}'\right)\\
0_{n\times1}
\end{smallmatrix}\right)$ is of full rank. We denote the unique solution of the equation above
by $\widehat{{\rm vec}\left(A_{+}'\right)}$.
In \citet{chenandersondeistlerfiller2010}, the authors choose the
minimum norm solution of the YW equations as the particular solution.
Let $I_{n}\otimes\left[\left(\begin{smallmatrix}V_{1} & V_{2}\end{smallmatrix}\right)\left(\begin{smallmatrix}D_{11} & 0_{(n^{2}p-s)\times s}\\
0_{s\times(n^{2}p-s)} & 0_{s\times s}
\end{smallmatrix}\right)\left(\begin{smallmatrix}V_{1}'\\
V_{2}'
\end{smallmatrix}\right)\right]$ 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. $\left[\begin{smallmatrix}\left(I_{n}\otimes\Gamma_{p}\right)\\
\left(I_{n}\otimes V_{2}'\right)
\end{smallmatrix}\right]{\rm vec}\left(A_{+}'\right)=\left(\begin{smallmatrix}{\rm vec}\left(\gamma_{p}'\right)\\
0_{s\times1}
\end{smallmatrix}\right)$. 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
\begin{equation}
\underbrace{\left(I_{n}\otimes\Gamma_{p}\right)S_{A}}_{=n^{2}p\times\left(n^{2}p-r\right)}=\begin{pmatrix}\tilde{U}_{1} & \tilde{U}_{2}\end{pmatrix}\begin{pmatrix}\tilde{D}_{11} & 0\\
0 & 0_{\tilde{s}\times\tilde{s}}
\end{pmatrix}\begin{pmatrix}\tilde{V}_{1}'\\
\tilde{V}_{2}'
\end{pmatrix}.\label{eq:svd_intersection}
\end{equation}
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
\begin{lem}
\label{lem:overidentifying}In the case $A_{0}=I_{n}$, the restrictions
described by the matrix $C_{S}$ are not over-identifying if and only
if
\begin{equation}
\left[I_{n^{2}p}-\tilde{U}_{1}\tilde{U}_{1}'\right]\left(I_{n}\otimes V_{1}\right)=0,\label{eq:check_overidentifying}
\end{equation}
where $\tilde{U}_{1}$ is obtained from (\ref{eq:svd_intersection}).
There is a unique solution of the YW equations if and only if the
right-kernel of $\left(I_{n}\otimes\Gamma_{p}\right)S_{A}$ is trivial.
\end{lem}
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 $\Gamma_{p}=\left(\begin{smallmatrix}1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 0
\end{smallmatrix}\right)$ and $C_{S}=I_{3}\otimes\left(0,1,0\right)$, such that $S_{A}=\left(I_{3}\otimes\left(\begin{smallmatrix}1 & 0\\
0 & 0\\
0 & 1
\end{smallmatrix}\right)\right)$ and $\left(I_{n}\otimes\Gamma_{p}\right)S_{A}=\left(I_{3}\otimes\left(\begin{smallmatrix}1 & 0\\
0 & 0\\
0 & 0
\end{smallmatrix}\right)\right)$. 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,
$\left[I_{n^{2}p}-\tilde{U}_{1}\tilde{U}_{1}'\right]\left(I_{n}\otimes V_{1}\right)=\left(I_{3}\otimes\left(\begin{smallmatrix}0 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 0
\end{smallmatrix}\right)\right)\left(I_{3}\otimes\left(\begin{smallmatrix}1 & 0\\
0 & 1\\
0 & 0
\end{smallmatrix}\right)\right)\neq0$ and the right-kernel of $\left(I_{n}\otimes\Gamma_{p}\right)S_{A}=\left(I_{3}\otimes\left(\begin{smallmatrix}1 & 0\\
0 & 0\\
0 & 0
\end{smallmatrix}\right)\right)$ is non-empty. The non-generic nature of this example is summarized
in
\begin{prop}
\label{prop:Aplus}Let $C_{S}\in\mathbb{R}^{ns\times n^{2}p}$ be
of full row rank and let $\Gamma_{p}$ be singular with rank deficiency
equal to $s$. The set of restrictions $\left\{ C_{S}\in\mathbb{R}^{ns\times n^{2}p}\,|\,\eqref{eq:check_overidentifying}\text{ does not hold}\right\} $
is of Lebesgue measure zero in $\mathbb{R}^{ns\times n^{2}p}$. A
generic, randomly chosen restriction $C_{S}$ can thus be used to
obtain a unique solution of the system of equations $\left[\begin{smallmatrix}\left(I_{n}\otimes\Gamma_{p}\right)\\
C_{S}
\end{smallmatrix}\right]{\rm vec}\left(A_{+}'\right)=\left(\begin{smallmatrix}{\rm vec}\left(\gamma_{p}'\right)\\
0_{s\times1}
\end{smallmatrix}\right)$ and the system parameters are globally identified.
\end{prop}
\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.
\section{Conclusion}
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.
\section{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.
\section{Data Availability Statement}
There is no data involved in this study.
\pagebreak{}
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\selectlanguage{british}