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On global identification in structural vector autoregressions
\title{\huge On global identification in structural vector autoregressions\thanks{We thank Thomas Carr, Giuseppe Cavaliere, Luca Fanelli, Michele Piffer, Majid Al-Sadoon and Matthew Read for beneficial discussions and comments. Financial support from the ESRC through the ESRC Centre for Microdata Methods and Practice (CeMMAP)
(grant number RES-589-28-0001) and the European Research Council (Starting grant No. 715940) is gratefully acknowledged. Emanuele Bacchiocchi gratefully acknowledges financial support from Italian Ministry of University and Research (PRIN 2022, Grant 20229PFAX5) and the University of Bologna (RFO grants).}}
\author{
\begin{tabular}{ccc}
Emanuele Bacchiocchi\thanks{University of Bologna, Department of Economics. Email: [email removed]} & &
Toru Kitagawa\thanks{Brown University, Department of Economics. Email: toru\[email removed]} \\
University of Bologna & & Brown University \\
\end{tabular}
}
\date{This draft: 14 October 2024}
\maketitle
\begin{abstract}
In a landmark contribution to the structural vector autoregression (SVARs) literature, Rubio-Ram\'{i}rez, Waggoner, and Zha (2010, `Structural Vector Autoregressions: Theory of Identification and Algorithms for Inference,' \textit{Review of Economic Studies}) show necessary and sufficient conditions for equality restrictions to globally identify the structural parameters of an SVAR. Among them, the sufficient condition shown in their Theorem 7 is the simplest and most attractive for practitioners, reducing the check for global identification to a counting exercise about the number of zero restrictions imposed.
However, their findings build on a set of technical assumptions that should be satisfied for the nice result to hold. Specifically, one of these assumptions states that
the first derivative of the function transforming the parameters one is interested to constrain must have full rank. We show, through a reasonable example, to
what extent the failure of this assumption can invalidate the necessary and sufficient condition provided in their Theorem 7, allowing the possibility that restrictions are redundant, in the sense that one or more restrictions may be implied by other restrictions. In this case, the implied restrictions do not add any identifying information, leading to failure of global identification. We derive a modified, and more general, necessary and sufficient condition for SVAR global identification and show how it can be easily assessed in practice.
\end{abstract}
\begin{flushleft}
\textit{Keywords}: Structural Vector Autoregression, exclusion restrictions, redundant restrictions. \newline
\bigskip
\textit{JEL codes}: C01,C13,C30,C51.
\end{flushleft}
\newpage
\section{Introduction}
\label{sec:intro}
\cite{RWZ10} (henceforth RWZ) provide necessary and sufficient conditions for the global identification of structural parameters in Structural Vector Autoregressions (SVARs) under a general class of zero restrictions imposed on the structural parameters and their (non-)linear transformations, including impulse responses. Exploiting the insights of their global identification analysis, RWZ also develop efficient and practical algorithms to perform estimation and inference for structural parameters and impulse responses. Their analytical and computational innovations have been instrumental to recent developments in the literature, including set-identified SVARs (\citet{ARW18}, \citet{GK18}, \citet{GKR19}, \citet{Volpicella20}, \citet{AD21}), locally-identified SVARs (\citet{BK20}), and SVARs with narrative restrictions (\citet{AR18}, \citet{GKR21}), to list a few. RWZ provide several different versions of the necessary and sufficient conditions for global identification. The one given in Theorem 7 is the simplest and most attractive for practitioners, which reduces the check for global identification to a counting exercise about the number and pattern of imposed zero restrictions without requiring knowledge of the true value of the structural or reduced-form parameters. For instance, in \citet{ACR19} and \citet{Zviadadze17}, the authors apply Theorem 7 of RWZ to judge whether the imposed identifying restrictions deliver global identification or not.
However, their rank condition builds on some technical assumptions that should be checked in advance. Through a reasonable example, we show the importance of such
assumptions and to what extent their failure can lead to missleading results, making the rank condition no longer sufficient. An analytical investigation of this example reveals why it does not guarantee global identification. We find that the condition of Theorem 7 of RWZ cannot detect what we refer to as \textit{redundancy} of imposed identifying restrictions. In this phenomenon, a set of equality restrictions on the structural parameters or impulse responses implicitly forces other structural parameters or impulse responses to zero. If it is present, some (redundant) zero restrictions are already implied by other imposed equality restrictions, so they do not contribute any further identifying information to the system. The condition of Theorem 7 of RWZ, without a prior check of the basic technical assumptions, incorrectly counts the redundant identifying restrictions as if they reduced the dimension of the admissible structural 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.
In the present paper we provide a new necessary and sufficient condition for (exact) global identification that correctly discounts redundant identifying restrictions, and can be used under weaker assumptions than in RWZ. RWZ propose a useful algorithm that sequentially constructs an orthonormal matrix for structural parameter identification that satisfies the identifying restrictions. Building on and modifying their algorithm, our proposed necessary and sufficient condition for global identification checks for the existence of redundant restrictions by verifying whether the orthonormal matrix generated by this sequential algorithm is unique. Verifying uniqueness boils down to checking the rank of a sequence of matrices constraining each column of the orthonormal matrix. Our algorithm only requires values of the reduced-form parameters as an input and check for global identification at that particular point of the parameter space. Our approach is more convenient than analytically checking a priori the theoretical assumptions in RWZ, and, furthermore, provides correct results even under certain failures of their assumptions.
As an alternative to their Theorem 7, Theorem 1 in RWZ presents a different form of necessary and sufficient condition for global identification. As we illustrate in this note, its proper implementation requires a complete understanding of how the imposed identifying restrictions analytically constrain the impulse responses and the set of structural parameters. For instance, if redundant identifying restrictions are present but one is not aware which zero restrictions can be implied by others, naive implementation of the rank conditions in Theorem 1 of RWZ may also overlook a lack of global identification. To prevent this, it is important to analytically ascertain how a set of equality restrictions translate to zero restrictions for other structural objects. This is feasible for small 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 moderate to large scale SVARs.
The rest of the paper is organized as follows. We first introduce the model and notation in Section II. In Section \ref{sec:ex}, we present an example that
sheds light on Theorem 7 of RWZ. In Section \ref{sec:id} we define the notion of redundant identifying restrictions and provide a modified necessary and sufficient condition for (exact) global identification. Section \ref{sec:conclusion} concludes.
\section{Model}
\label{sec:def}
We maintain the notation used in RWZ. Let $y_t$ be a $n\times 1$ vector of variables observed over the sample $t=1,\ldots, T$. The specification of the SVAR model is
\begin{equation}
\label{eq:SVAR}
y_t^\prime A_0 = \sum_{l=1}^{p}y_{t-l}^\prime A_l + c +\varepsilon_t^\prime,
\end{equation}
where $\varepsilon_t$ is a $n\times 1$ multivariate normal white noise process with null expected value and covariance matrix equal to the identity matrix $I_n$. The $n\times n$ matrices $A_0,\,A_1,\ldots,\,A_p$ are the structural parameters and $c$ is a $1\times n$ vector of constant terms. The structural parameters are $(A_0,A_+)$, where $A_+^\prime\equiv (A_1^\prime,\ldots,\,A_l^\prime,\,c^\prime)$ is a $n\times m$ matrix with $m\equiv np+1$. We also assume that the initial conditions $y_1,\,\ldots,\, y_p$ are given and that $A_0$ is invertible. The set of structural parameters is denoted by $\ensuremath\mathbb{P}^S$, an open dense set of $\ensuremath\mathbb{R}^{(n+m)n}$. The structural form can be written compactly as
\begin{equation}
\label{eq:SVARc}
y_t^\prime A_0 = x_{t}^\prime A_+ + \varepsilon_t^\prime
\end{equation}
where $x_t^\prime=\left(y_{t-1}^\prime,\ldots,\,y_{t-p}^\prime,\,1\right)$.
The reduced-form representation of (\ref{eq:SVARc}) is the standard VAR model,
\begin{equation}
\label{eq:VARc}
y_t^\prime = x_{t}^\prime B + u_t^\prime,
\end{equation}
where $B=A_+A_0^{-1}$, $u_t^\prime=\varepsilon_t^\prime A_0^{-1}$, and $E(u_t\,u_t^\prime)=\Sigma=(A_0A_0^{\prime})^{-1}$. The reduced-form parameters are $(B,\,\Sigma)$, where $\Sigma$ is a symmetric and positive definite matrix. We denote the set of reduced-form parameters by
$\ensuremath\mathbb{P}^R \subset \mathbb{R}^{nm+n(n+1)/2}$.
The relationship between the structural and reduced-form parameters is defined by the function
$g:\ensuremath\mathbb{P}^S\rightarrow\ensuremath\mathbb{P}^R$, where $g\left(A_0,A_+\right)=(A_+A_0^{-1},(A_0A_0^\prime)^{-1})$.
The definition of global identification is the standard one provided by \cite{Rothenberg71ECTA}; the absence of
observationally equivalent parameters in the parametric space. We consider identification of the structural parameters by imposing zero restrictions on
a transformation $f(\cdot)$ of the structural parameter space into the set of $k \times n$ matrices, $k \geq 1$, with domain $U\subset \ensuremath\mathbb{P}^S$. Such linear restrictions are represented by
\begin{equation}
\label{eq:restr}
Q_jf(A_0,A_+) e_j=0,\hspace{1cm}\text{for } j=1,\ldots,n.
\end{equation}
where $Q_j$ is a $k\times k$ selection matrix for $j=1,\ldots,n$, and $e_j$ is the \textit{j}-th column of the $n\times n$ identity matrix $I_n$.
The rank of $Q_j$ is denoted by $q_j$, which also represents the number of restrictions in the \textit{j}-th column of the transformed space
$f(A_0,A_+)$. As in RWZ, we order the columns of $f(A_0,A_+)$ according to
\begin{equation}
\label{eq:ordering}
q_1\geq q_2\geq\ldots\geq q_n.
\end{equation}
We denote the set of orthonormal matrices by $\mathcal{O}(n)$ with generic element $P$.
According to RWZ, the transformation is admissible when the following condition holds.
\begin{condition}
\label{def:adm}
The transformation $f(\cdot)$, with the domain $U$, is admissible if and only if for any $P\in\ensuremath\mathcal{O}\left(n\right)$ and $\left(A_0,A_+\right)\in U$, $f(A_0 P,A_+ P)=f(A_0,A_+) P$.
\end{condition}
Moreover, RWZ impose the following two conditions when proving some of their results, the former of which is at the heart of the present paper.
\begin{condition}
\label{def:reg}
The transformation $f(\cdot)$, with the domain $U$, is regular if and only if $U$ is open and $f$ is continuously differentiable with $f^\prime \left(A_0,A_+\right)$
of rank $kn$ for all $\left(A_0,A_+\right) \in U$.
\end{condition}
\begin{condition}
\label{def:streg}
The transformation $f(\cdot)$, with the domain $U$, is strongly regular if and only if it is regular and $f(U)$ is dense in the set of
$k\times n$ matrices.
\end{condition}
To fix the sign of structural shocks, we need to impose sign normalization rules. Following RWZ, we define them as follows:
\begin{defin}[Normalization rule]
\label{def:norm}
A normalization rule can be characterized by a set $N\subset \ensuremath\mathbb{P}^S$ such that for any structural parameter point
$\left(A_0,A_+\right)\subset \ensuremath\mathbb{P}^S$, there exists a unique $n\times n$ diagonal matrix $D$ with plus or minus ones along the diagonal
such that $(A_{0}D,A_+D)\in N$.
\end{defin}
We are now able to define the set of restricted structural parameters as
\begin{equation}
\label{eq:R}
R = \Big\{\left(A_0,A_+\right)\in U\cap N\,\Big|\,Q_jf(A_0,A_+) e_j=0\text{ for }j=1,\ldots,n\Big\}.
\end{equation}
Following RWZ, we consider the following definition of identification when discussing whether or not the imposed restrictions can globally identify the structural parameters.
\begin{defin}[Exact identification]
\label{def:exact}
Consider an SVAR with restrictions represented by $R$. The SVAR is exactly identified if and only if, for almost any reduced-form parameter
point $(B,\Sigma)$, there exists a unique structural parameter point $\left(A_0,A_+\right)\in R$ such that $g\left(A_0,A_+\right)=(B,\Sigma)$.
\end{defin}
In this definition, if the set of structural parameters under the restrictions $R$ constrains the reduced-form parameters, the domain of the reduced-form parameters for which the almost-sure property is required is restricted to $\tilde{\ensuremath\mathbb{P}}^R \subset \ensuremath\mathbb{P}^R$, where $\tilde{\mathbb{P}}^R$ is the set of reduced-form parameters generated by the structural
parameters satisfying $R$. For instance, if $f(\cdot)$ maps the structural parameters to long-run impulse responses, its domain $U$ restricts the reduced-form VARs to being invertible. Then, $\tilde{\mathbb{P}}^R$ corresponds to the set of reduced-form parameters constrained to invertible VARs.
\section{An illustrative example}
\label{sec:ex}
In the setting described in the previous section, RWZ show a variety of necessary and sufficient conditions for the identifying restrictions $R$ with admissible $f(\cdot)$ to globally identify the structural parameters. Among those, the necessary and sufficient condition for exact identification presented in Theorem 7 of RWZ is the simplest and most attractive in practice, as it reduces verification of exact identification to a simple exercise of computing the ranks of the matrices $Q_j$, $1 \leq j \leq n$. So that our exposition is self-contained, we present Theorem 7 of RWZ here:
\bigskip
\noindent \textbf{Theorem 7 in RWZ}: \textit{Consider an SVAR with admissible and strongly regular restrictions represented by $R$.\footnote{Admissible and strongly regular restrictions represented by $R$ mean $f(\cdot)$ in (\ref{eq:R}) is admissible and strongly regular.} The SVAR is exactly identified if and only if $q_j = n - j$ for $1 \leq j \leq n$. }
\bigskip
The first result in this note is that the assumption in Condition \ref{def:reg}, that looks like a rather technical one, and thus off consideration by practitioners,
can invalidate the ``if'' statement of Theorem 7 in RWZ, as shown by the following example.
\subsection{An example}
\label{sec:contr}
Consider a trivariate SVAR characterized by the following restrictions
\begin{equation}
\label{eq:ExRestr}
\begin{array}{ccc}
A_0 = \left(\begin{array}{ccc}
a_{11} & a_{12} & a_{13}\\
0 & a_{22} & a_{23}\\
0 & a_{32} & a_{33}
\end{array}\right) & \text{ and } &
IR_0 = \left(\begin{array}{ccc}
\times & 0 & \times\\
\times & \times & \times\\
\times & \times & \times
\end{array}\right)
\end{array}
\end{equation}
where $IR_0=(A_0^{-1})^{\prime}$ is the contemporaneous impulse response matrix, the symbol `$\times$' indicates that no restriction is imposed, and `0' represents
a zero (or exclusion) restriction.
The function $f(A_0,A_+)$ will be
\begin{equation}
\label{eq:Exf}
f(A_0,A_+)=\left(\begin{array}{c}A_0\\IR_0\end{array}\right)=
\left(\begin{array}{ccc}
a_{11} & a_{12} & a_{13}\\
0 & a_{22} & a_{23}\\
0 & a_{32} & a_{33}\\
\times & 0 & \times\\
\times & \times & \times\\
\times & \times & \times
\end{array}\right).
\end{equation}
The matrices of restrictions defined in (\ref{eq:restr}) can be specified as
\begin{equation}
\label{eq:QEx}
\begin{array}{lcr}
Q_1 = \left(\begin{array}{cccccc}
0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0\\
\hdashline[2pt/2pt]
0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0
\end{array}\right), & \hspace{2cm} &
Q_2 = \left(\begin{array}{cccccc}
0 & 0 & 0 & 1 & 0 & 0\\
\hdashline[2pt/2pt]
0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0
\end{array}\right).
\end{array}
\end{equation}
According to Theorem 7 in RWZ, the SVAR is exactly (globally) identified, as the rank of the restriction matrices follows $q_1=n-1=2$, $q_2=n-2=1$ and $q_3=n-3=0$. However, analytical investigation shows the current set of identifying restrictions fails to achieve global identification.
Let us express the reduced-form covariance matrix and its Cholesky decomposition as
\begin{equation}
\label{eq:ExSigma}
\begin{array}{ccc}
\Sigma = \left(\begin{array}{ccc}
\sigma_{11} & \sigma_{21} & \sigma_{31}\\
\sigma_{21} & \sigma_{22} & \sigma_{32}\\
\sigma_{31} & \sigma_{32} & \sigma_{33}
\end{array}\right) & \Rightarrow &
\Sigma_{tr} = \left(\begin{array}{ccc}
l_{11} & 0 & 0\\
l_{21} & l_{22} & 0\\
l_{31} & l_{32} & l_{33}
\end{array}\right).
\end{array}
\end{equation}
Imposing triangularity on $A_0$, we can obtain $A_0$ and $IR_0=A_0^{-1\prime}$ as
\begin{equation}
\label{eq:ExPar}
\begin{array}{ccc}
A_0^\prime = \Sigma_{tr}^{-1} = \left(\begin{array}{ccc}
\frac{1}{l_{11}} & 0 & 0\\
-\frac{l_{21}}{l_{11}l_{22}} & \frac{1}{l_{22}} & 0\\
\frac{l_{21}l_{32}-l_{22}l_{31}}{l_{11}l_{22}l_{33}} & -\frac{l_{32}}{l_{22}l_{33}} & \frac{1}{l_{33}}
\end{array}\right) & \Rightarrow &
IR_{0}= A_0^{-1\prime} = \left(\begin{array}{ccc}
l_{11} & 0 & 0\\
l_{21} & l_{22} & 0\\
l_{31} & l_{32} & l_{33}
\end{array}\right).
\end{array}
\end{equation}
Consider applying Algorithm 1 in RWZ to determine an orthogonal matrix $P$ that maps the $\left(A_0,A_+\right)$ parameters under triangularity to the one satisfying the imposed restrictions.
First, $f(A_0,A_+)$ is
\begin{equation}
\label{eq:Exfnum}
f(A_0,A_+)=\left(\begin{array}{c}A_0\\IR_0\end{array}\right)=
\left(\begin{array}{ccc}
\frac{1}{l_{11}} & -\frac{l_{21}}{l_{11}l_{22}} & \frac{l_{21}l_{32}-l_{22}l_{31}}{l_{11}l_{22}l_{33}}\\
0 & \frac{1}{l_{22}} & -\frac{l_{32}}{l_{22}l_{33}}\\
0 & 0 & \frac{1}{l_{33}}\\
l_{11} & 0 & 0\\
l_{21} & l_{22} & 0\\
l_{31} & l_{32} & l_{33}
\end{array}\right).
\end{equation}
As in RWZ, let $\bar{Q}_1$ and $\bar{Q}_2$ be the matrices of indicators for the restricted elements of $f(A_0,A_+)$ obtained by removing the row vectors of zeros from $Q_1$ and $Q_2$. Algorithm 1 in RWZ suggests calculating
\begin{equation}
\label{eq:Q1t}
\tilde{Q}_1=\bar{Q}_1f(A_0,A_+)=\left(\begin{array}{ccc}0 & \frac{1}{l_{22}} & -\frac{l_{32}}{l_{22}l_{33}}\\
0 & 0 & \frac{1}{l_{33}}\end{array}\right),
\end{equation}
and finding a unit-length vector that is orthogonal to the row vectors of $\tilde{Q}_1$. The QR decomposition of $\tilde{Q}_1$ and a sign normalization lead to $p_1 = (1,0,0)'$ as a unique unit vector satisfying $\tilde{Q}_1 p_1 = 0$, so we can pin down the first column vector of $P$.
Next, to find the second column vector $p_2$ of $P$, we form the matrix
\begin{equation}
\label{eq:Q2t}
\tilde{Q}_2=\left(\begin{array}{c}\bar{Q}_2f(A_0,A_+)\\p_1^\prime\end{array}\right)
=\left(\begin{array}{ccc}l_{11} & 0 & 0\\ \hdashline[2pt/2pt] 1 & 0 & 0\end{array}\right)
\end{equation}
and search for a unit vector $p_2$ satisfying $\tilde{Q}_2\,p_2=0$. Since the rank of $\tilde{Q}_2$ is one for any value of $l_{11}$, we cannot pin down a unique $p_2$ (up to the sign normalization). From a geometric point of view, any vector belonging to the unit circle
in $\ensuremath\mathbb{R}^3$ orthogonal to the unit vector $p_1=\left(\begin{array}{ccc}1 & 0 & 0\end{array}\right)^\prime$ is admissible as $p_2$. This implies that given any reduced-form parameter value of $\Sigma$, the imposed restrictions fail to pin down a unique orthogonal matrix $P$, implying that, contrary to the claim in Theorem 7 of RWZ, global identification does not hold in this example.
Some packaged algorithms for the QR decomposition, including the Matlab function $qr(\cdot)$, yield an orthogonal vector $p_2$ irrespective of whether it is unique or not. That is, if $\tilde{Q}_{2}$ is not full-rank, these algorithms implicitly select one unit vector $p_2$ from infinitely many admissible ones. As a result, an application of the ``if'' statement of Theorem 7 and naive implementation of Algorithm 1 in RWZ may fail to detect the failure of global identification and mislead subsequent impulse response analysis.
\subsection{Analytical investigation}
\label{sec:analytic}
A correct implementation of the RWZ procedure for checking global identification would have suggested an immediate warning, being the transformation function
$f(A_0,A_+)$ not regular. In fact, the domain of $f(A_0,A_+)$ is the set of $n\times n$ matrices $A_0$ (being $IR_0=A_0^{-1}$), while its domain has dimension $2n\times n$.
According to Condition \ref{def:reg}, the transformation $f(\cdot)$, in order to be regular, must have the first derivative of rank $kn$. In the specific example,
the rank should be $nk = 2n^2$. However, the first derivative $f^\prime$ will be of dimension $2n^2\times n^2$, and thus the rank will be at most $n^2$, that is
clearly less than $2n^2$. The identification scheme proposed in the example, thus, cannot be investigated by the RWZ procedure, although it is not so unrealistic from
an empirical point of view. In fact, combining zero restrictions on the contemporaneous relations among the endogenous variables and on the response on impact to
structural shocks could be a quite standard strategy for practitioners. In the following sections we will introduce a strategy to check for global identification even for models not admissible according to Condition \ref{def:reg}.
However, to understand why in this particular case the ``if'' statement of Theorem 7 of RWZ breaks down and how it can be modified, it is useful to determine analytically the special feature of the identifying restrictions specified in Eq. (\ref{eq:ExRestr}).
We begin with the inversion of the $A_0$ matrix; the determinant of $A_0$ is
\begin{equation}
\label{eq:det}
|A_0| = a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}
\end{equation}
and the adjunct matrix is
\begin{equation}
\label{eq:Adj}
\text{Adj}(A_0) = \left(\begin{array}{ccc}
a_{22}a_{33}-a_{32}a_{23} & -(a_{12}a_{33}-a_{32}a_{13}) & a_{12}a_{23}-a_{22}a_{13}\\
-(a_{21}a_{33}-a_{31}a_{23}) & a_{11}a_{33}-a_{31}a_{13} & -(a_{11}a_{23}-a_{21}a_{13})\\
a_{21}a_{32}-a_{31}a_{22} & -(a_{11}a_{32}-a_{31}a_{12}) & a_{11}a_{22}-a_{21}a_{12}
\end{array}\right).
\end{equation}
The inverse is $A_0^{-1}=|A_0|^{-1}\,\text{Adj}(A_0)$. Substituting the two zero restrictions on $A_0$, $a_{21}=0$ and $a_{31}=0$, into $A_0^{-1\prime}$ leads to
\begin{equation}
\label{eq:AinvR}
A_0^{-1\prime} = \frac{1}{a_{11}(a_{22}a_{33}-a_{23}a_{32})}\left(\begin{array}{ccc}
a_{22}a_{33}-a_{32}a_{23} & 0 & 0\\
-(a_{12}a_{33}-a_{32}a_{13}) & a_{11}a_{33} & -(a_{11}a_{32}-a_{31}a_{12})\\
a_{12}a_{23}-a_{22}a_{13} & -a_{11}a_{23} & a_{11}a_{22}
\end{array}\right)=IR_0.
\end{equation}
It is evident that the two restrictions on $A_0$ imply two zero restrictions on $IR_0$, $(A_{0}^{-1 \prime})_{[1,2]} = (A_{0}^{-1 \prime})_{[1,3]}=0$. One of these, $(A_{0}^{-1\prime})_{[1,2]}=0$, is exactly the zero restriction specified for $IR_0$ in (\ref{eq:ExRestr}).
In other words, we intended to impose the three restrictions, but the two imposed on $A_0$ imply the third imposed on $IR_0$,
so this third restriction was redundant. Due to this redundancy, the third restriction does not further constrain the admissible orthonormal matrix $P$, which translates into rank deficiency of $\tilde{Q}_2$.
Although this redundancy phenomenon can occur in some realistic applications,\footnote{Many
influential empirical papers combine restrictions on both contemporaneous relationships among the endogenous variables
and the contemporaneous impulse responses. Examples include \cite{Blanchard89}, \cite{BlanPerotti02}, \cite{Bernanke86}.}
whether or not any of the imposed set of restrictions are redundant cannot be directly assessed by the simple necessary and sufficient condition in Theorem 7 of RWZ. As a way to uncover such redundancy, one may want to examine how a set of zero restrictions imposed on one structural object translates to zero restrictions on other objects. In Section IV below, we modify the necessary and sufficient condition of Theorem 7 of RWZ by offering a systematic way to detect redundancy of the imposed identifying restrictions.
\subsection{Detecting the failure of global identification}
\label{sec:DetectFail}
In their Theorem 6, RWZ provides an alternative necessary and sufficient condition for exact identification of SVARs. If we properly take into account that the imposed zero restrictions imply zero restrictions on other objects, this alternative approach can correctly detect a lack of global identification. We illustrate how in our example.
For $1 \leq j \leq n$ and any $k \times n$ matrix $X$, let $M_j(X)$ be a $(k+j) \times n$ matrix defined by
\begin{equation*}
M_j(X) = \begin{pmatrix} Q_j X \\ I_{j \times j} \mspace{20mu} O_{j \times (n-j)} \end{pmatrix},
\end{equation*}
where $Q_j$ is a $k \times k$ matrix defined in (\ref{eq:restr}). Theorem 6 of RWZ provides a necessary and sufficient condition for exact identification through the rank conditions for $M_j\big(f(A_0,A_+)\big)$.
\bigskip
\noindent \textbf{Theorem 6 in RWZ}: \textit{Consider an SVAR with admissible and strongly regular restrictions represented by $R$. The SVAR is exactly identified if and only if the total number of restrictions is equal to $n(n-1)/2$ and for some $(A_0,A_+) \in R$, $M_j\big(f(A_0,A_+)\big)$ is of rank $n$ for $1 \leq j \leq n$.}
\bigskip
As for the necessary and sufficient condition in Theorem 7, the condition in Theorem 6 is correct, but is limited to admissible and strongly regular transformations
$f(\cdot)$. As we have just seen here before, the analysis could be extended to a broader class of transformations, where the technical assumption on the full rank
of the first derivative of $f(\cdot)$ can be substituted by the less stringent and more practical condition of non-redundancy. Although the utilization of Theorem 6
in RWZ is precluded by the failure of the regularity Condition \ref{def:reg}, it is useful to understand what happens in our example and whether the condition can
be recovered even in the case of non regular transformations.
In the current example, the total number of restrictions imposed is 3 and it meets the condition for the total number of restrictions with $n=3$. We hence focus on checking the rank condition for $M_j\big(f(A_0,A_+)\big)$, $j=1,2,3$.
In this check, we substitute the following matrices into $f(A_0,A_+)$:
\begin{equation}
\label{eq:ExRestrA}
\begin{array}{ccc}
A_0 = \left(\begin{array}{ccc}
a_{11} & a_{12} & a_{13}\\
0 & a_{22} & a_{23}\\
0 & a_{32} & a_{33}
\end{array}\right) & \text{ and } &
IR_0 = \left(\begin{array}{ccc}
\times & 0 & 0\\
\times & \times & \times\\
\times & \times & \times
\end{array}\right),
\end{array}
\end{equation}
where the symbol `$\times$' denotes the parameters in Eq. (\ref{eq:AinvR}). We obtain, if $a_{22}a_{33}-a_{32}a_{23} \neq 0$,
\begin{equation}
\label{eq:RkExA}
\begin{array}{lcl}
M_1\big(f(A_0,A_+)\big)=\left(\begin{array}{ccc}
0 & a_{22} & a_{23}\\
0 & a_{32} & a_{33}\\
\hdashline[2pt/2pt]
1 & 0 & 0\end{array}\right) & \hspace{2cm} & \mathrm{rank}\,(M_1)=3\\&&\\
M_2\big(f(A_0,A_+)\big)=\left(\begin{array}{ccc}
a_{22}a_{33}-a_{32}a_{23} & 0 & 0\\
\hdashline[2pt/2pt]
1 & 0 & 0\\
0 & 1 & 0\end{array}\right) & \hspace{2cm} & \mathrm{rank}\,(M_2)=2<3.
\end{array}
\end{equation}
Hence, the rank condition of Theorem 6 in RWZ fails. This is consistent with the conclusion in our analysis above; the imposed restrictions uniquely pin down the first column vector of $P$, but not the second column vector of $P$. Thus, plugging in the expression of $f(A_0,A_+)$ obtained analytically under the imposed restrictions, the rank condition of Theorem 6 of RWZ correctly detects the failure 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 by the imposed restrictions is crucial to correctly performing the check of the rank condition in Theorem 6 of RWZ. For instance, in the current example, if we were not aware of the redundancy issue of the identifying restrictions and incorrectly let the $(1,3)$-element of $M_2\big(f(A_0,A_+)\big)$ be an unknown potentially nonzero free parameter, we would have erroneously claimed that $M_2\big(f(A_0, A_+)\big)$ were of rank 3 and concluded that the exact identification holds. If the dimension of the SVAR is large, exhaustively investigating and figuring out the entire set of constraints implied by the zero restrictions on $f(A_0,A_+)$ is challenging. In such a case, immediate implementation of the rank conditions of Theorem 6 of RWZ is limited.
\subsection{Other examples of transformation}
\label{sec:OtherEx}
In the previous illuminating example we have considered zero restrictions on $A_0$ and $IR_0$. The next two examples introduce VARs, of potential interest in empirical applications, where the transformation functions are not regular, and thus not contemplated by the RWZ's machinery.
\begin{ex}[Short-run zero restrictions on a monetary policy shock]
\label{ex:Lags}
In practically all triangular quarterly SVARs,\footnote{See, among many others, \cite{CEE05JPE} and \cite{BG06RESTATS}.} it is assumed a monetary policy shock to have no impact within the quarter it hits
the economy. In a monthly VAR, it does correspond to zero restrictions on $IR_0$, $IR_1$, $IR_2$ and $IR_3$ of the monetary policy shock on indicators of the real economy, like industrial production.
If the DGP is, for instance, a VAR with just two lags, the regularity assumption in Condition \ref{def:reg} is no longer valid. To see this, it is sufficient to write the
transformation function as
\begin{equation}
\label{eq:ExLags}
f(A_0,A_1,A_2) = \Big(IR_0^\prime,IR_1^\prime,IR_2^\prime,IR_3^\prime\Big)^\prime,\nonumber
\end{equation}
with $IR_h = \Big(A_0^{-1}J^\prime F^h J\Big)^\prime$, where
\begin{equation}
\label{eq:ExLagsMat}
F = \left(\begin{array}{cc}A_1A_0^{-1} & I_n \\ A_2A_0^{-1} & 0\end{array}\right)
\hspace{1cm}\mathrm{and} \hspace{1cm}
J = \left(\begin{array}{c}I_n \\ 0\end{array}\right).\nonumber
\end{equation}
The dimension of $f(A_0,A_1,A_2)$ is $4n\times n$. Then, the rank of $f^\prime(A_0,A_1,A_2)$ is at most $3n$, that is strictly less than $4m$.
Condition \ref{def:reg}, thus, is not satisfied.
\end{ex}
\begin{ex}[Short-run, long-run, cumulated long-run restrictions]
\label{ex:Cumul}
Consider an SVAR model mixing aggregate demand and supply shocks \'{a} la \cite{BlanQuah1}, as well as permanent productivity shocks as in \cite{KPSW91}.
The VAR model, as in \cite{KPSW91}, can be made of a) non-stationary $I(1)$ (potentially cointegrated) variables, where the long-run dynamics is governed by a
combination of both permanent and transitory structural shocks; b) first differences of non-stationary $I(1)$ variables, and c) stationary variables.
The set of constraints, thus, could be represented by zero restrictions on the response of impact $IR_0$, on the long-run responses $IR_{\infty}$, defined as
\begin{equation}
\label{eq:IRinfty}
IR_{\infty} = \Bigg(A_0^\prime-\sum_{l=1}^{p} A_l^\prime\Bigg)^{-1},\nonumber
\end{equation}
and on the cumulated impulse responses ($L_{\infty}^c=\sum_{h=0}^{\infty}IR_h$).
In the same vain as in the Example \ref{ex:Lags}, if the estimated reduced-form specification is characterized by just one lag, Condition \ref{def:reg} is no longer valid
and the RWZ's machinery cannot be implemented.
\end{ex}
\section{Extending the necessary and sufficient condition for exact identification}
\label{sec:id}
In this section we provide a modified necessary and sufficient condition for exact identification that eliminates the redundancy issue erroneously invalidating Theorem 7 of RWZ and extend the condition to a broader class of transformation functions. Our proposal relies on the sequential feature of Algorithm 1 in RWZ and checks the rank condition for uniqueness of the $j$-th column vector $p_j$ for each $j = 1, \dots, n$.
Given the reduced-form parameters $(B, \Sigma)$, choose $(A_0, A_+)$ to be an unrestricted set of structural parameters satisfying $\Sigma = (A_0^{\prime})^{-1} (A_0)^{-1}$ and $B=A_+ A_0^{-1}$, such as $A_0^{\prime} = \Sigma_{tr}^{-1}$ and $A_+ = B (\Sigma_{tr}^{-1})'$. Let
\begin{equation}
\label{eq:Qt}
\tilde{Q}_1 = Q_1 f(A_0, A_+), \ \text{and} \ \tilde{Q}_j=\left(\begin{array}{c}
Q_jf(A_0,A_+)\\p_1^\prime\\\vdots\\p_{j-1}^\prime
\end{array}\right) \ \text{for $j=2,\ldots,n$}.
\end{equation}
By Theorem 5 and Algorithm 1 of RWZ, the exact identification of SVARs follows if and only if, for almost every reduced-form parameters $(B, \Sigma)$, the orthogonality conditions $\tilde{Q}_jp_j=0$ combined with 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 $1 \leq j \leq n$. This is, however, not a sufficient condition, because if any of the orthogonal vectors $(p_1, \dots, p_{j-1})$ is linearly dependent on the row vectors of $Q_j f(A_0,A_+)$, a rank-deficient $\tilde{Q}_j$ fails to pin down a unique
$p_j$. This is exactly the mechanism that caused the systematic failure of global identification in our illustrative counterexample. To rule out such rank-deficiency in the characterization of the global identification condition, we introduce the following concept:
\begin{defin}[Non-redundant restrictions]
\label{def:RedRes}
Given reduced-form parameter $(B,\Sigma)$, let $A_0^{\prime} = \Sigma_{tr}^{-1}$ and $A_+ = B (\Sigma_{tr}^{-1})'$. Identifying restrictions for an SVAR that
are represented by zero restrictions $Q_jf(A_0,A_+) e_j=0$, $j=1, \dots, n$, are \textit{non-redundant} at given reduced-form parameter point, $(B,\Sigma)$ if for
every $j=2, \dots, n$, orthogonal vectors $(p_1, \dots, p_{j-1})$ are linearly independent of the row vectors of $Q_jf(A_0,A_+)$, i.e., $\tilde{Q}_j$ defined in
Eq. (\ref{eq:Qt}) is full row-rank for all $j=2, \dots, n$.
\end{defin}
If the imposed zero restrictions are non-redundant and the rank condition of Theorem 7 in RWZ holds, we can guarantee
\begin{equation}
\label{eq:RedRes}
\mathrm{rank}\,(\tilde{Q}_j) = \mathrm{rank}\, \left(\begin{array}{c}Q_jf(A_0,A_+)\\p_1^\prime\\\vdots\\p_{j-1}^\prime\end{array}\right)=n-1
\end{equation}
for all $j=1, \dots, n$. We can therefore solve for an orthonormal matrix $P$ uniquely by sequentially solving $\tilde{Q}_j p_j = 0$, for $j=1, \dots, n$. If non-redundancy of the imposed restrictions holds for almost any reduced-form parameter point $(B,\Sigma)$, we can achieve exact identification. We hence obtain the following theorem that modifies Theorem 7 of RWZ.
\begin{theo}[A necessary and sufficient condition for exact identification]
\label{theo:NecSuffCond}
Consider an SVAR with admissible restrictions represented by $R$. The SVAR is exactly identified at the point $(A_0,A_+)\in R$ if and only if
$q_j=n-j$ for $j=1,\ldots,n$ and the restrictions are non-redundant at $(A_0,A_+)$.
\end{theo}
\begin{proof}
Let the first $j-1$ shocks be identified. It means that the quantity
\begin{equation}
\label{eq:Pj}
\setlength{\dashlinegap}{2pt}
P_{j-1} = \left[\begin{array}{c:c:c:c}
p_1 & p_2 & \cdots & p_{j-1}
\end{array}\right]\nonumber
\end{equation}
is uniquely determined. If the \textit{j}-th shock is not identified, instead, then $p_j$ and $p_j^*$, by construction orthogonal to $P_{j-1}$,
are both admissible, with $p_j^*$ any unit-length rotation of $p_j$.
If $p_j$ is admissible, then
\[
Q_jf(A_0,A_+)\,p_j=0
\]
and, similarly, if $p_j^*$ is admissible, then
\[
Q_jf(A_0,A_+)\,p_j^*=0.
\]
In other words, both $p_j$ and $p_j^*$ belong to the null space of $Q_jf(A_0,A_+)$, that, by assumption, is of dimension $j$.
Moreover, by construction, they are orthogonal to $P_{j-1}$, too. The two vectors, thus, must be contained in the intersection of the two
null spaces of $Q_jf(A_0,A_+)$ and $P_{j-1}$, that is equivalent to the null space of the $(n-j)+(j-1)\times n$ matrix
\[
\tilde{Q}_j = \left(\begin{array}{c}
Q_jf(A_0,A_+)\\ P_{j-1}^\prime
\end{array}\right).
\]
Being $p_j$ and $p_j^*$ linearly independent, the null space of $\tilde{Q}_j$ must be at least of dimension two.
Now, using the rank-nullity theorem, we can say that $\mathrm{rank}\,\,\tilde{Q}_j\leq n-2$, and, thus, cannot be full.
This proves the sufficiency of the rank condition in the Theorem \ref{theo:NecSuffCond}.
Proving the other direction of the condition is very simple. In fact, if the rank of $\tilde{Q}_j$ is equal to $n-1$ (that means that the
restrictions are not redundant), then the orthogonal complement of $\tilde{Q}_j$ will have dimension equal to one. Thus, there will be just two
candidates for the vector $p_j$, but being one opposite to the other, only one will be retained. If this happens for every $j=\{1,\ldots,n\}$,
there will be just one orthogonal matrix $P$ transforming the parameter point $(A_0,A_+)\in U$ into the parameter point $(A_0P,A_+P)\in R$,
that satisfies the restrictions. Finally, the fact that this last result does correspond to global identification can be proved by using
Theorem 5 in RWZ, that does not require the restrictions to be neither regular nor strongly regular.
\end{proof}
In comparison to Theorem 7 of RWZ, our Theorem \ref{theo:NecSuffCond} adds the non-redundancy condition of the imposed restrictions as a part of necessary and sufficient condition.
Importantly, removing Conditions \ref{def:reg} and \ref{def:streg} from the assumptions precludes the nice results in Theorem 3 in RWZ, saying that if the model is identified
in one point of $R$, it is identified almost everywhere in $R$. Our result, instead, is specific to the single point we are interested in. Building on and modifying Algorithm 1 of RWZ, the next algorithm correctly judges if exact identification
holds or not at the specific point $\big(B,\,\Sigma\big)\in\,R$, that could be the ML estimation, or any draw from the posterior distribution of the reduced-form parameters
in a Bayesian VAR.
\begin{algo}
\label{algo:Exact}
Consider an SVAR with admissible restrictions represented by $R$ that satisfy $q_j=n-j$, for $j=1,\ldots,n$.
Let $\big(B,\,\Sigma\big)$, be any reduced-form parameters.
Perform the following steps:
\begin{enumerate}
\item Let $A_0^{\prime} = \Sigma_{tr}^{-1}$ and $A_+ = B (\Sigma_{tr}^{-1})'$, where $\Sigma_{tr}$ is the lower-triangular Cholesky factor of $\Sigma$.
\item For each $j=1, \dots, n$, sequentially check the rank conditions for non-redundancy, i.e., check if $\mathrm{rank}\,(\tilde{Q}_j) = n-1$ holds,
where $\tilde{Q}_1=Q_1f(A_0,A_+)$ and
\begin{equation}
\label{eq:QtAlgo}
\tilde{Q}_j=\left(\begin{array}{c}Q_jf(A_0,A_+)\\p_1^\prime\\\vdots\\p_{j-1}^\prime\end{array}\right)
\end{equation}
for $j = 2, \dots, n$, and $p_j$ is an $n \times 1$ unit-length vector satisfying $\tilde{Q}_j p_j = 0$ which is unique (up to sign normalization) if
$\mathrm{rank}(\tilde{Q}_{j})=n-1$ holds for all preceding $j=1, \dots, j-1$.
\end{enumerate}
If the reduced-form parameter point passes Step 2 of the current algorithm, we conclude that the imposed identifying restrictions $R$ achieve exact identification.
If not, we conclude that the imposed identifying restrictions do not achieve exact identification.
\end{algo}
The constructions of the orthonormal vectors $p_1, \dots, p_n$ by solving $\tilde{Q}_j p_j = 0$ sequentially for $j=1, \dots, n$, as incorporated in Step 2 of Algorithm \ref{algo:Exact}, is proposed in Algorithm 1 of RWZ. For the purpose of checking exact identification, the important feature of our algorithm is the step of checking $\mathrm{rank}(\tilde{Q}_j) = n-1$ for all $j=1, \dots, n$. This extra step, which is absent in Algorithm 1 of RWZ, is necessary to detect failure of exact identification due to redundancy of the identifying restrictions.
\section{Conclusion}
\label{sec:conclusion}
Based on a counterexample, this note demonstrates the importance of some technical assumptions to be met for the sufficiency claim in Theorem 7 of RWZ, commonly used by applied macro-economists because of its simplicity, to be correctly implemented. Analytical investigation of this counterexample reveals the issue of redundancy among the identifying restrictions, which the rank conditions of Theorem 7 of RWZ, in the absence of such regulatory assumptions, overlook. To rectify this, and to open to larger sets of admissible parameter restrictions, we present, firstly, a new necessary and sufficient condition for exact identification and, secondly, a computational algorithm that 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 SVARs under their choice of equality identifying restrictions.
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