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Identification and estimation of structural vector autoregressive models via LU decomposition
\numberwithin{equation}{section}
\theoremstyle{plain}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\theoremstyle{definition}
\newtheorem{defi}[thm]{Definition}
\newtheorem{assumption}[thm]{Assumption}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{rem}[thm]{Remark}
\newtheorem{eg}[thm]{Example}
\title{Identification and estimation
of structural vector autoregressive models
via LU decomposition}
\author[1]{Masato Shimokawa}
\author[2]{Kou Fujimori}
\affil[1, 2]{Faculty of Economics and Law,
Shinshu University.}
\date{}
\maketitle
\begin{abstract}
Structural vector autoregressive (SVAR)
models are
widely used to analyze the simultaneous relationships
between multiple time-dependent data.
Various statistical inference methods
have been studied to
overcome the identification problems of SVAR models.
However, most of these methods impose
strong assumptions for innovation processes
such as the uncorrelation of components.
In this study, we relax the assumptions for
innovation processes and propose an identification method for SVAR models
under the zero-restrictions
on the coefficient matrices,
which correspond to sufficient conditions for
LU decomposition
of the coefficient matrices of the reduced form
of the SVAR models.
Moreover, we establish asymptotically normal estimators for the coefficient matrices and
impulse responses, which enable us to
construct test statistics for the
simultaneous relationships of
time-dependent data.
The finite-sample performance of the proposed method
is elucidated by numerical simulations.
We also present an example of an empirical study
that analyzes the impact of policy rates on unemployment and prices.
\end{abstract}
\section{Introduction}\label{sec:introduction}
Structural vector autoregressive (SVAR) models
studied by \cite{Sims1980}, \cite{Bernanke1986}, and \cite{Blanchard1988} are widely used to
analyze time-dependent macroeconomic data.
A basic SVAR model is expressed as follows:
\begin{equation}\label{eq:intro SVAR}
\bm{Y}_t = \bm{\mu} + \bm{A}_0 \bm{Y}_t + \sum_{s=1}^{p} \bm{A}_{s} \bm{Y}_{t-s} + \bm{v}_t,\quad
t \in \mathbb{Z},
\end{equation}
where $\bm{\mu} \in \mathbb{R}^k$ is an intercept, $\bm{A}_s, s=0,1,\ldots,p$ are $k \times k$
coefficient matrices, and $\{\bm{v}_t\}_{t \in \mathbb{Z}}$ is an innovation process.
The term $\bm{A}_0 \bm{Y}_t$ in the right-hand side represents the
simultaneous relationships between
the components of $\bm{Y}_t$.
When $\bm{I}_k - \bm{A}_0$ is non-singular,
we have the following ordinal vector autoregressive
(VAR) representation:
\begin{equation}\label{eq:intro reduced form}
\bm{Y}_t = \bm{\eta} + \sum_{s=1}^p \bm{B}_s \bm{Y}_{t-s} +
\bm{e}_t,\quad t \in \mathbb{Z},
\end{equation}
where
\[
\bm{\eta} = \bm{Q} \bm{\mu},\quad
\bm{B}_s = \bm{Q} \bm{A}_s,\quad
s=1,\ldots,p,
\]
with $\bm{Q} = (\bm{I}_k - \bm{A}_0)^{-1}$, and
$\bm{e}_t = \bm{Q} \bm{v}_t$.
We call \eqref{eq:intro reduced form}
the reduced form of \eqref{eq:intro SVAR}.
Under some assumptions such as the stationarity
of the process $\{\bm{Y}_t\}_{t \in \mathbb{Z}}$,
we can construct an asymptotically normal
estimator for $\bm{B}_s, s=1,\ldots,p$ using methods
such as
ordinary least squares (OLS).
Meanwhile, some restrictions are required
to recover
$\bm{A}_s, s=0,\ldots,p$
from observable structures.
Such identification problems
have been discussed by several researchers.
Typically, we consider zero restrictions,
e.g., some specific components of $\bm{A}_s, s=0,1,\ldots,p$ are fixed to zero.
Under some additional assumptions, a general identification method based on zero
restrictions
was
proposed by
\cite{Rubio-Ramirez2010}.
\cite{Canova2002} and \cite{Uhlig2005}
proposed identification methods for SVAR models
under the
sign restriction of the coefficient matrices, i.e.,
the signs of some specific impulse
responses
are known.
\cite{Hyvarinen2010} and \cite{Lanne2017}
relaxed the restrictions on the
coefficient matrices
and developed other methods for non-Gaussian processes to allow flexible identification of structures.
To correctly interpret the
analysis of SVAR models,
it is necessary to impose appropriate restrictions
based on prior knowledge and data background.
Regarding this point, previous studies
have focused on
the restrictions
on the coefficient matrices.
Meanwhile, it is often assumed that
$\mathop{\rm Var}\nolimits[\bm{v}_t]$ is a diagonal matrix.
However, the components of $\bm{v}_t$ might be correlated if $\bm{v}_t$ includes some unobservable exogenous variables and does not correspond to the unique shocks of $\bm{Y}_t$.
Because an invalid assumption may
lead to misunderstandings in causal interpretations,
it is imperative to
consider other identification methods under
less restrictive conditions on the innovation process.
In this study, under mild conditions on the innovation process, where
$\mathop{\rm Var}\nolimits[\bm{v}_t]$ is allowed to be a
non-diagonal matrix, we propose an identification method based on zero restrictions on the coefficient matrices
and LU decomposition for
a sub-matrix of $\bm{B} = (\bm{\eta}, \bm{B}_1,\ldots,\bm{B}_p)$.
Moreover, we establish
asymptotically normal estimators for
the coefficient matrices $\bm{A}_s, s=0,\ldots,p$
and impulse responses, enabling us
to construct test statistics for
the hypothesis testing
whose null hypothesis
is that $\mathcal{H}_0: \bm{A}_0 = \bm{O}$,
which can be used to
verify whether the simultaneous relationships
should be considered for the data $\{\bm{Y}_t\}_{t \in \mathbb{Z}}$.
The remainder of this article is organized as follows.
In Section \ref{sec:SVAR model setups},
we describe the model setup of SVAR models
and present a motivational example.
In Section \ref{sec:estimations},
we propose the identification and estimation
methods for $\bm{A}_s, s=0,1,\ldots,p$
via LU decomposition
under appropriate restrictions.
We also consider
the impulse response estimation and
hypothesis testing
for $\bm{A}_0$ in this section.
The numerical simulations used to verify the
asymptotic behavior of the estimators and test statistics are presented in
Section \ref{sec:numerical}.
We further apply the proposed methods to
analyze the impact of policy rates on employment
and prices in this section.
The proofs of the main theoretical results are presented in Section \ref{sec:proofs}.
We discuss the causal interpretations
of the statistical inference for SVAR models
in the Appendix.
\section{SVAR models}\label{sec:SVAR model setups}
\subsection{Model setup and a motivational example}
We consider the following model:
\begin{equation}\label{eq:SVAR}
\bm{Y}_t = \bm{\mu} + \bm{A}_0 \bm{Y}_t + \sum_{s=1}^{p} \bm{A}_{s} \bm{Y}_{t-s} + \bm{v}_t,\quad
t \in \mathbb{Z},
\end{equation}
where $\bm{A}_s \in \mathbb{R}^{k \times k}, s=0,1,\ldots, p$ are the coefficient matrices and
$\{\bm{v}_t\}_{t \in \mathbb{Z}}$ is an i.i.d. innovation
process such that $\,\mathrm{E}[\bm{v}_t] = \bm{0}$.
We suppose that
$\bm{A}_0$ is a $k \times k$
lower-triangular matrix
with all diagonal components being zero,
which means that
$Y_{i, t}$ cannot be the direct cause
of $Y_{j, t}$ for $i >j$.
\if0
The innovation process
$\{\bm{v}_t\}_{t \in \mathbb{Z}}$
can be regarded as direct causes of
$\bm{Y}_t$ which cannot be written by
components of $\bm{Y}_{t-s}, s=0,\ldots,p$.
\fi
Several researchers have assumed that
$\mathop{\rm Var}\nolimits[\bm{v}_t]$ is a diagonal matrix.
Meanwhile,
we consider the existence of
contemporaneous confounding,
indicating that
$\bm{v}_{i, t}$ and $\bm{v}_{j, t}, i \neq j$
are correlated.
We now consider the following motivational example.
\begin{eg}\label{eg:SVAR-ex}
Let $\{{W}_t\}_{t \in \mathbb{Z}}$
be an $\mathbb{R}$-valued unobservable
i.i.d. sequence.
We consider the following model:
\begin{equation}\label{eq:SVAR-ex}
\bm{Y}_t = \bm{A}_0 \bm{Y}_t + \sum_{s=1}^{p} \bm{A}_{s} \bm{Y}_{t-s} + \bm{A}_W{W}_t + \bm{u}_t,\quad
t \in \mathbb{Z},
\end{equation}
where $\bm{A}_s, s=1,\ldots,p \in \mathbb{R}^{3 \times 3}$,
$\bm{A}_0 \in \mathbb{R}^{3 \times 3}$ is a
lower-triangular matrix
with diagonal elements being zero,
and
$\{\bm{u}_t\}$ is an $\mathbb{R}^3$-valued
i.i.d. sequence
independent of ${W}_t$ with $\mathop{\rm Var}\nolimits[\bm{u}_t] = \sigma_u^2 \bm{I}_3$, and
$\bm{A}_W \in \mathbb{R}^{3 \times 1}$.
Thus, we can rewrite the model \eqref{eq:SVAR-ex} as follows:
\[
\bm{Y}_t = \bm{A}_0 \bm{Y}_t + \sum_{s=1}^{p} \bm{A}_{s} \bm{Y}_{t-s} + \bm{v}_t,\quad
t \in \mathbb{Z},
\]
where
\[
\bm{v}_t =\bm{A}_W{W}_t + \bm{u}_t.
\]
Therefore,
$\mathop{\rm Var}\nolimits[\bm{v}_t]$ is not diagonal
unless $\bm{A}_W \bm{A}_W^\top$ is diagonal.
The SVAR model can be
regarded as a special case of
linear structural equation models, as described in the Appendix.
For a linear structural equation model,
we often consider a graphical representation.
Suppose that we observe
$\bm{Y}_{1-p},\ldots,\bm{Y}_T$ for some $T \in \mathbb{N}$ and
consider graph $\bm{G}$ with
the vertex set
$\tilde{\bm{V}} = \{Y_{i,s} : i =1,\ldots,k, s =1-p,\ldots,T\} \cup \{W_{s} : s =1-p,\ldots,T \}$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{fig2.png}
\caption{A graphical representation for the model \eqref{eq:SVAR-ex} with $p=1$}
\label{fig:SVAR-ex}
\end{figure}
Figure \ref{fig:SVAR-ex} shows graph $\bm{G}$,
where the edges represent the
nonzero components
of $\bm{A}_0, \bm{A}_1$ and
$\bm{A}_W$.
Since the model structure does not depend on time, Figure \ref{fig:SVAR-ex} only shows
the components of $\bm{Y}_t$, their direct causes, $W_t$, and $W_{t-1}$
included in the vertex set.
Observe that $W_t, t=1-p,\ldots,T$ are (unobservable)
contemporaneous confounding factors, which
implies correlations between the components of the
innovation process $\{\bm{v}_t\}_{t \in \mathbb{Z}}$.
If we suppose that the
covariance matrix of
the innovation process is
non-diagonal,
we cannot deal with such data
using most of the conventional SVAR
models.
\end{eg}
If the matrix $\bm{I}_k - \bm{A}_0$ is non-singular, then
we have the reduced form of
\eqref{eq:SVAR} as follows:
\begin{equation}\label{eq:SVAR-reduced}
\bm{Y}_t = \bm{Q} \bm{\mu} + \sum_{s=1}^{p} \bm{Q} \bm{A}_{s} \bm{Y}_{t-s} + \bm{Q}\bm{v}_t,
\end{equation}
where
\[
\bm{Q} = (\bm{I}_k - \bm{A}_0)^{-1}.
\]
This reduced form is a typical VAR model.
Thus, to ensure
the unique existence of
a stationary and ergodic solution to
\eqref{eq:SVAR},
it is sufficient to assume the following conditions.
\begin{assumption}\label{assump:stationary}
\begin{itemize}
\item[(i)]
$\bm{A}_0$ is a
lower-triangle matrix with the diagonal elements being zero.
\item[(ii)]
The following polynomial
\[
\left| \bm{I}_{k} - \bm{Q}\bm{A}_1z - \bm{Q}\bm{A}_2z^2 - \cdots - \bm{Q}\bm{A}_pz^p \right|,\quad
z \in \mathbb{C}
\]
is nonzero for all $z \in \{\zeta \in \mathbb{C} : |\zeta| \leq 1\}$.
\end{itemize}
\end{assumption}
Condition (i) of Assumption \ref{assump:stationary}
guarantees that the model
\eqref{eq:SVAR}
has a directed acyclic graph (DAG) structure.
Next, we consider the impulse response functions.
Consider the following VAR$(1)$ representation
of \eqref{eq:SVAR-reduced}:
\[
\bm{\xi}_t = \bm{\Lambda} \bm{\xi}_{t-1} + \bm{\epsilon}_t,
\]
where
$\bm{\xi}_t = (\bm{Y}_t^\top,\ldots,\bm{Y}_{t-p+1}^\top)^\top$,
$\bm{\epsilon}_t = (\bm{e}_t^\top,\bm{0}^\top,\ldots,\bm{0}^\top)^\top$,
\[
\bm{\Lambda}
= \left(\begin{array}{ccccc}
\bm{B}_1 & \bm{B}_2 & \cdots & \bm{B}_{p-1} & \bm{B}_p \\
\bm{I}_k & \bm{O} & \cdots & \bm{O} & \bm{O} \\
\bm{O} & \bm{I}_k & \cdots & \bm{O} & \bm{O} \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
\bm{O} & \bm{O} & \cdots & \bm{I}_k & \bm{O} \\
\end{array}
\right),
\]
and $\bm{B}_s = \bm{Q}\bm{A}_s$ for $s=1,\ldots,p$.
We omit $\bm{\mu}$ here for simplicity because impulse responses do not depend on it.
Under Assumption \ref{assump:stationary},
we have the following moving average (MA) representation:
\begin{equation}\label{eq:SVAR-MA}
\bm{\xi}_t
= \sum_{s=0}^\infty \bm{\Lambda}^s \bm{\epsilon}_{t-s}.
\end{equation}
For the top $k$ rows in
\eqref{eq:SVAR-MA},
there exist
$\bm{\Psi}_0 = \bm{I}_k$ and
$\bm{\Psi}_s \in \mathbb{R}^{k \times k}, s=1,2,\ldots$ such that
\[
\bm{Y}_t
= \sum_{s=0}^\infty \bm{\Psi}_s \bm{e}_{t-s}
= \sum_{s=0}^\infty \bm{\Psi}_s \tilde{\bm{L}} \tilde{\bm{u}}_{t-s},
\]
where
$\tilde{\bm{L}} \in \mathbb{R}^{k \times k}$
is a
lower-unitriangular
matrix obtained by
the LU decomposition of $\mathop{\rm Var}\nolimits[\bm{e}_t]$
and
$\tilde{\bm{u}}_t = \tilde{\bm{L}}^{-1}\bm{e}_t$.
Notably, $\bm{\Psi}_s$ coincides with the
$k \times k$ sub-matrix of $\bm{\Lambda}^s$
corresponding to the first $k$ columns and rows.
Therefore, the orthogonalized impulse response
$\mathrm{OIRF}_{ij}(s)$ and
non-orthogonalized impulse response $\mathrm{IRF}_{ij}(s)$ are, respectively, given by
\[
\mathrm{OIRF}_{ij}(s) = \left(
\bm{\Psi}_s \tilde{\bm{L}}
\right)_{ij},\quad
\mathrm{IRF}_{ij}(s) = \left(
\bm{\Psi}_s
\right)_{ij}.
\]
By considering the SVAR model as a
special case of linear structural equation models,
the coefficient matrices and
impulse responses
can be regarded as causal effects
(see the Appendix for the detail of such interpretations).
\section{Identification and statistical inference of SVAR models}\label{sec:estimations}
\subsection{Estimations of the coefficient matrices}\label{subsec:coef}
In this section, we establish asymptotically normal estimators for
the coefficient matrices of
model \eqref{eq:SVAR}.
We begin by considering
the estimation method for the
reduced form \eqref{eq:SVAR-reduced} of
\eqref{eq:SVAR}.
We rewrite the model as follows:
\begin{equation}\label{eq:SVAR-structure sec4}
\bm{Y}_t = \bm{A}_0 \bm{Y}_t + \bm{A} \bm{X}_{t-1}
+ \bm{v}_t,
\end{equation}
where
$\bm{A} = (\bm{\mu}, \bm{A}_1,\ldots, \bm{A}_p)$
and
$\bm{X}_{t-1} = (1, \bm{Y}_{t-1}^\top,\ldots,\bm{Y}_{t-p}^\top)^\top$.
The reduced form
can be represented as follows.
\begin{equation}\label{eq:SVAR-reduced sec4}
\bm{Y}_t = \bm{B} \bm{X}_{t-1} + \bm{e}_t,
\end{equation}
where
$\bm{B} = \bm{Q} \bm{A}$ and
$\bm{e}_t = \bm{Q}\bm{v}_t$.
Suppose that we observe $(\bm{Y}_{1-p},\ldots,\bm{Y}_T)$.
We define the data matrix
$\bm{Y}$ and the design matrix
$\bm{X}$ as follows:
\[
\bm{Y} = (\bm{Y}_1,\ldots,\bm{Y}_T)^\top\quad
\mbox{and}\quad
\bm{X} = (\bm{X}_0,\ldots,\bm{X}_{T-1})^\top,
\]
respectively.
Let $r=1+kp$ and
$\bm{\Theta} \subset \mathbb{R}^{k \times r}$
be a compact parameter space of $\bm{B}$
and $\bm{B}_*$ be the true value of $\bm{B}$.
Then, we consider the following least squares estimators $\hat{\bm{B}}_T$ and $\hat{\bm{b}}_T$
for $\bm{B}$ and $\,\mathrm{vec}(\bm{B})$:
\[
\hat{\bm{B}}_T = \bm{Y}^\top \bm{X} (\bm{X}^\top \bm{X})^{-1}
= \left(
\sum_{t=1}^T \bm{Y}_t \bm{X}_{t-1}^\top
\right)
\left(
\sum_{t=1}^T \bm{X}_{t-1} \bm{X}_{t-1}^\top
\right)^{-1},
\]
and
\[
\hat{\bm{b}}_T
= \,\mathrm{vec}(\hat{\bm{B}}_T)
= \bm{b}_* + \left\{\left(
\frac{1}{T}
\sum_{t=1}^T \bm{X}_{t-1} \bm{X}_{t-1}^\top
\right)^{-1} \otimes \bm{I}_k\right\}
\left(
\frac{1}{T} \sum_{t=1}^T \bm{X}_{t-1} \otimes \bm{e}_t
\right),
\]
where $\bm{b}_* = \,\mathrm{vec}(\bm{B}_*)$.
To establish the asymptotic behavior of the
estimator $\hat{\bm{b}}_T$, we assume the following conditions.
\begin{assumption}\label{assump:regularity LSE}
\begin{itemize}
\item[(i)]
It holds that
\[
\,\mathrm{E}[\|\bm{v}_t\|_2^4] < \infty,\quad
t \in \mathbb{Z}.
\]
\item[(ii)]
$\mathop{\rm Var}\nolimits[\bm{v}_t]$ and
$\,\mathrm{E}[\bm{X}_{t-1} \bm{X}_{t-1}^\top]$ are positive definite.
\item[(iii)]
The true value $\bm{B}_*$ is an interior point in
the parameter space $\bm{\Theta} \subset \mathbb{R}^{k \times r}$.
\end{itemize}
\end{assumption}
Then, we have the following asymptotic normality
of $\hat{\bm{b}}_T$.
\begin{prop}\label{prop:normality LSE B}
Under Assumptions \ref{assump:stationary} and \ref{assump:regularity LSE}, it holds that
\[
\sqrt{T}(\hat{\bm{b}}_T - \bm{b}_*)
\to^d N(\bm{0}, \bm{\Gamma}^{-1} \otimes \bm{\Sigma}),\quad
T \to \infty,
\]
where
$\bm{\Sigma} = \,\mathrm{E}[\bm{e}_t \bm{e}_t^\top]$
and $\bm{\Gamma} = \,\mathrm{E}[\bm{X}_{t-1} \bm{X}_{t-1}^\top]$.
\end{prop}
The proof can be found in, e.g.,
\cite{Lutkepohl2013} or \cite{Hamilton1994},
therefore, we omit it.
Next, we introduce
estimators for $\bm{Q}$, $\bm{A}_0$, and $\bm{A}$
using LU decomposition.
Denote
\[
\bm{A} = (\bm{a}_1,\bm{a}_2,\ldots,\bm{a}_r).
\]
where $\bm{a}_1= \bm{\mu}$.
The following condition is
sufficient to construct
estimators for $\bm{Q}$, $\bm{A}_0$, and $\bm{A}$
via LU decomposition
and derive its asymptotic behavior.
\begin{assumption}\label{assump:regularity A}
There exists a (known) $k$-tuple of distinct
positive numbers $(j_1,\ldots,j_k)$ such that
$(\bm{a}_{j_1 *},\ldots,\bm{a}_{j_k *})$
is a non-singular
upper-triangular matrix,
where
$\bm{A}_*$ is the true value of $\bm{A}$.
\end{assumption}
Let $g: \mathbb{R}^{k \times r} \to \mathbb{R}^{k \times k}$ be the map
defined as follows
\[
g(\bm{C}) = (\bm{c}_{j_1},\ldots,\bm{c}_{j_k}),
\]
where $\bm{c}_m$ is the $m$-th column of the
matrix $\bm{C}$.
Similarly to
\cite{Rubio-Ramirez2010},
we should impose zero restrictions for
lower-triangular
part of $g(\bm{A})$ based on
the data background
(see Section \ref{subsec:real data}
for a concrete example).
Note that
$g(\bm{B}) = \bm{Q} g(\bm{A})$.
For a nonsingular matrix
$\bm{C} \in \mathbb{R}^{k \times k}$
which allows
an LU-decomposition with
a lower-unitriangular matrix,
we introduce the following notation:
\[
\bm{C} = \bm{L}(\bm{C}) \bm{U}(\bm{C}).
\]
Let $\bm{q}, \bm{a}_0$, and $\bm{a}$ be the vectorizations
of $\bm{Q}, \bm{A}_0$, and $\bm{A}$, respectively,
i.e.,
\[
\bm{q} = \,\mathrm{vec}(\bm{Q}),\quad
\bm{a}_0 = \,\mathrm{vec}(\bm{A}_0),\quad
\mbox{and}
\quad
\bm{a} = \,\mathrm{vec}(\bm{A}).
\]
We define the estimators for $\bm{q}, \bm{a}_0$, and $\bm{a}$ as follows.
\[
\hat{\bm{q}}_T
= f_1(\hat{\bm{b}}_T) := \,\mathrm{vec}(\bm{L}_g(\hat{\bm{B}}_T)),
\]
\[
\hat{\bm{a}}_{0T}
= f_2(\hat{\bm{b}}_T) := \,\mathrm{vec}(\bm{I}_k - \bm{L}_g(\hat{\bm{B}}_T)^{-1}),
\]
and
\[
\hat{\bm{a}}_{T}
= f_3(\hat{\bm{b}}_T) := \left(\bm{I}_r \otimes
\bm{L}_g(\hat{\bm{B}}_T)\right)^{-1}\hat{\bm{b}}_T,
\]
where $\bm{L}_g := \bm{L} \circ g$.
The asymptotic normality of
the estimators follows from the delta method.
\begin{thm}\label{thm:normality A}
Suppose that Assumptions \ref{assump:stationary},
\ref{assump:regularity LSE}, and \ref{assump:regularity A} hold.
Then, the following conditions hold:
\begin{equation}\label{eq:normality q}
\sqrt{T}(\hat{\bm{q}}_T - \bm{q}_*)
\to^d N(\bm{0}, \bm{\Sigma}_1),
\end{equation}
\begin{equation}\label{eq:normality ao}
\sqrt{T}(\hat{\bm{a}}_{0T} - \bm{a}_{0*})
\to^d N(\bm{0}, \bm{\Sigma}_2),
\end{equation}
\begin{equation}\label{eq:normality a}
\sqrt{T}(\hat{\bm{a}}_T - \bm{a}_*)
\to^d N(\bm{0}, \bm{\Sigma}_3),
\end{equation}
as $T \to \infty$,
where
\[
\bm{\Sigma}_l = \bm{J}_l \bm{\Sigma}_b \bm{J}_l^\top
\]
with
$\bm{\Sigma}_b = \bm{\Gamma}^{-1} \otimes \bm{\Sigma}$ and
\[
\bm{J}_l = \left.\frac{\partial f_l(\bm{b})}{\partial \bm{b}}\right|_{\bm{b} = \bm{b}_*},\quad
l=1,2,3.
\]
\end{thm}
The asymptotic covariance matrices
are singular because some components are fixed at $0$ or $1$ by LU decomposition.
\subsection{Estimation of impulse responses}\label{subsec:impulse}
Next, we construct estimators for the impulse response $\mathrm{IRF}_{ij}(h)$ for
$i, j = 1,2,\ldots,k$ and $h>0$.
The matrix $\bm{\Psi}_h = (\mathrm{IRF}_{ij}(h))_{i, j = 1,\ldots,k}$ satisfies
\[
\bm{\Psi}_h = [\bm{\Lambda}^h]_k^k,
\]
where $[\bm{\Lambda}^h]_k^k$ is the $k \times k$ sub-matrix of $\bm{\Lambda}^h$ corresponding to the top $k$ columns and rows.
Therefore, there exists a differentiable map
$f_{4, h}$ such that
$f_{4, h}(\bm{b}) = \bm{\psi}_h$, where
$\bm{\psi}_h = \,\mathrm{vec}(\bm{\Psi}_h)$.
We define an estimator for $\bm{\psi}_h$
by $\hat{\bm{\psi}}_{h T} = f_{4, h}(\hat{\bm{b}}_T)$.
The following proposition was proved by
\cite{lutkepohl1990}.
\begin{prop}\label{prop:estimator psi}
Suppose that assumptions \ref{assump:stationary},
\ref{assump:regularity LSE}, and \ref{assump:regularity A} hold.
For every $h >0$, it holds that
\begin{equation}\label{eq:normality psi}
\sqrt{T}(\hat{\bm{\psi}}_{hT} - \bm{\psi}_{h*})
\to^d N(\bm{0}, \bm{\Sigma}_{4, h}),\quad
T \to \infty,
\end{equation}
where
$\bm{\psi}_{h*} = \,\mathrm{vec}([\bm{\Lambda}_*^h]_k^k)$,
\[
\bm{\Lambda}_*
=\left(\begin{array}{ccccc}
\bm{B}_{1*} & \bm{B}_{2*} & \cdots & \bm{B}_{p-1*} & \bm{B}_{p*} \\
\bm{I}_k & \bm{O} & \cdots & \bm{O} & \bm{O} \\
\bm{O} & \bm{I}_k & \cdots & \bm{O} & \bm{O} \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
\bm{O} & \bm{O} & \cdots & \bm{I}_k & \bm{O} \\
\end{array}
\right),
\]
and $\bm{\Sigma}_{4, h} = \bm{J}_{4, h}\bm{\Sigma}_{\bm{b}} \bm{J}_{4, h}^\top$ with
\[
\bm{J}_{4, h} = \left.\frac{\partial f_{4, h}(\bm{b})}{\partial \bm{b}}\right|_{\bm{b} = \bm{b}_*}.
\]
\end{prop}
The general expression of
each component of $\bm{J}_{4, h}$ can be found in
\cite{lutkepohl1990}.
Let $\bm{\Psi}_{h*}^{\mathrm{o}}, h>0$
be a matrix defined by
\[
\bm{\Psi}_{h*}^{\mathrm{o}}
= \bm{\Psi}_{h*}\bm{Q}_{*},
\]
which corresponds to a ``total effect'' for a
linear structural equation model as described in
the Appendix.
We define the estimator for
$\bm{\psi}_{h*, T}^{\mathrm{o}}= \,\mathrm{vec}(\bm{\Psi}_{h*}\bm{Q}_{*})$
as follows.
\[
\hat{\bm{\psi}}_{hT}^{\mathrm{o}}
= f_{5, h}(\hat{\bm{b}}_T)
:= \left(
\bm{L}_g(\hat{\bm{B}}_T)^\top \otimes \bm{I}_k
\right) f_{4, h}(\hat{\bm{b}}_T).
\]
Because the map $f_{5, h}$ for every $h$
is differentiable,
we obtain the following proposition based on the delta method.
\begin{prop}\label{prop:estimator total effect}
Suppose that Assumptions \ref{assump:stationary},
\ref{assump:regularity LSE}, and \ref{assump:regularity A} hold.
Then, it holds that
\begin{equation}\label{eq:normality total effect}
\sqrt{T}(\hat{\bm{\psi}}_{hT}^{\mathrm{o}} - \bm{\psi}_{h *}^{\mathrm{o}})
\to^d N(\bm{0}, \bm{\Sigma}_{5, h})
,\quad T \to \infty,
\end{equation}
where
$\bm{\psi}_{h *}^{\mathrm{o}} = \,\mathrm{vec}(\bm{\Psi}_{h *} \bm{Q}_*)$,
$\bm{Q}_*$ is the true value of $\bm{Q}$, and
$\bm{\Sigma}_{5, h} = \bm{J}_{5, h}\bm{\Sigma}_{\bm{b}} \bm{J}_{5, h}^\top$ with
\[
\bm{J}_{5, h} = \left.\frac{\partial f_5(\bm{b})}{\partial \bm{b}}\right|_{\bm{b} = \bm{b}_*}.
\]
\end{prop}
\subsection{Tests for simultaneous relationships}\label{subsec:test}
In this section, we consider the following
test:
\begin{equation}\label{eq:test A0}
\mathcal{H}_0:\
\bm{A}_0 = O,\quad
\mathcal{H}_1:\ \bm{A}_0 \neq O.
\end{equation}
Let $\hat{\bm{e}}_t = \bm{Y}_t - \hat{\bm{B}}_T \bm{X}_{t-1}$ and
\[
\hat{\bm{\Sigma}}_{bT}
= \left(
\frac{1}{T} \sum_{t=1}^T \bm{X}_{t-1} \bm{X}_{t-1}^\top
\right)^{-1}
\otimes \left(
\frac{1}{T} \sum_{t=1}^T \hat{\bm{e}}_t \hat{\bm{e}}_t^\top
\right).
\]
The estimators
for $\bm{\Sigma}_l, l=1,2,3$ and $\bm{\Sigma}_{l, h}, l=4, 5$ can be constructed as follows:
\[
\hat{\bm{\Sigma}}_l
= \hat{\bm{J}}_l \hat{\bm{\Sigma}}_{\bm{b}}\hat{\bm{J}}_l^\top,\quad l=1,2,3,
\]
and
\[
\hat{\bm{\Sigma}}_{l, h}
= \hat{\bm{J}}_{l, h} \hat{\bm{\Sigma}}_{\bm{b}}\hat{\bm{J}}_{l, h}^\top,\quad l=4, 5
\]
with
\[
\hat{\bm{J}}_{l T}=\left.\frac{\partial f_l(\bm{b})}{\partial \bm{b}}\right|_{\bm{b} = \hat{\bm{b}}_{lT}},\quad
l=1,2,3
\]
and
\[
\hat{\bm{J}}_{l, h T}=\left.\frac{\partial f_{l, h}(\bm{b})}{\partial \bm{b}}\right|_{\bm{b} = \hat{\bm{b}}_{lT}},\quad
l=4, 5.
\]
The following lemma is obtained from
the asymptotic normality of the estimators.
\begin{lem}\label{lem:estimation sub-vector}
Suppose that Assumptions \ref{assump:stationary},
\ref{assump:regularity LSE}, and \ref{assump:regularity A} hold.
For every
$\bm{w}_1, \bm{w}_2 \in \mathbb{R}^{k^2}$,
and $\bm{w}_3 \in \mathbb{R}^{k(1+kp)}$,
let
\[
z_{1T}(\bm{w}_{1})
:= \sqrt{T} (\bm{w}_{1}^\top \hat{\bm{\Sigma}}_{1} \bm{w}_{1})^{-1/2} \bm{w}_{1}^\top
(\hat{\bm{q}}_T - \bm{q}_*),
\]
\[
z_{2T}(\bm{w}_{2})
:= \sqrt{T} (\bm{w}_{2}^\top \hat{\bm{\Sigma}}_{2} \bm{w}_{2})^{-1/2} \bm{w}_{2}^\top
(\hat{\bm{a}}_{0T} - \bm{a}_{0*}).
\]
and
\[
z_{3T}(\bm{w}_{3}) := \sqrt{T} (\bm{w}_{3}^\top \hat{\bm{\Sigma}}_{\bm{b}} \bm{w}_{3})^{-1/2} \bm{w}_{3}^\top
(\hat{\bm{b}}_T - \bm{b}_*).
\]
Then, it holds that
\begin{equation}\label{eq:normality comp b}
z_{lT}(\bm{w}_{l})
\to^d N(0, 1), \quad T \to \infty,\quad
l=1,2,3.
\end{equation}
\end{lem}
Under the null hypothesis
$\mathcal{H}_0$,
we have
$
\bm{Q}_* = \bm{I}_k$, $\bm{A}_{0*} = \bm{O}$,
and $g(\bm{B}_*) = g(\bm{A}_*)$ is a
non-singular
upper-triangular
matrix.
For a matrix
$\bm{C} \in \mathbb{R}^{k\times (1+kp)}$,
$g(\bm{C}) \in \mathbb{R}^{k \times k}$ is a sub-matrix of $\bm{C}$.
We define the sub-vectors
$\bm{q}_{\mathrm{sub}}$ and $\bm{a}_{0 \mathrm{sub}}$
of $\bm{q}$ and $\bm{a}_0$,
which comprise the lower-triangular components
of
$\bm{Q}$ and $\bm{A}_0$, except for diagonal components, respectively.
Similarly, let $\bm{\beta}_{\mathrm{sub}}$
be a sub-vector which comprises
the lower-triangular
components of
$g(\bm{B})$, except for diagonal components.
Then, we have
\[
\bm{w}^\top \bm{q}_{*\mathrm{sub}}
= \bm{w}^\top \bm{a}_{0*\mathrm{sub}}
= \bm{w}^\top \bm{\beta}_{*\mathrm{sub}} =0
\]
for every vector $\bm{w} \in \mathbb{R}^{k(k-1)/2}$.
Thus, we obtain the following theorem.
\begin{thm}\label{thm:test}
Suppose that Assumptions \ref{assump:stationary},
\ref{assump:regularity LSE}, and \ref{assump:regularity A} hold.
For $l=1,2,3$ and
$\bm{v} \in \mathbb{R}^{k(k-1)/2}$,
consider the following test statistics
$z_{lT}$
\[
z_{1T} = \sqrt{T} (\bm{v}^\top \hat{\bm{\Sigma}}_{1\mathrm{sub}} \bm{v})^{-1/2} \bm{v}^\top \hat{\bm{q}}_{T\mathrm{sub}},
\]
\[
z_{2T} = \sqrt{T} (\bm{v}^\top \hat{\bm{\Sigma}}_{2\mathrm{sub}} \bm{v})^{-1/2} \bm{v}^\top \hat{\bm{a}}_{0 T\mathrm{sub}},
\]
and
\[
z_{3T} = \sqrt{T} (\bm{v}^\top \hat{\bm{\Sigma}}_{\bm{b}\mathrm{sub}} \bm{v})^{-1/2} \bm{v}^\top \hat{\bm{\beta}}_{T\mathrm{sub}},
\]
where
$\hat{\bm{\Sigma}}_{1\mathrm{sub}}, \hat{\bm{\Sigma}}_{2\mathrm{sub}},$ and
$\hat{\bm{\Sigma}}_{\bm{b}\mathrm{sub}}$ are
sub-matrices of
$\hat{\bm{\Sigma}}_{1}, \hat{\bm{\Sigma}}_{2},$ and
$\hat{\bm{\Sigma}}_{\bm{b}}$
corresponding to
sub-vectors $\hat{\bm{q}}_{T\mathrm{sub}}$,
$\hat{\bm{a}}_{0T\mathrm{sub}}$, and
$\hat{\bm{\beta}}_{T\mathrm{sub}}$, respectively.
\begin{itemize}
\item[(i)]
Under the null hypothesis $\mathcal{H}_0$,
\[
z_{lT} \to^d N(0, 1),\quad T \to \infty,\quad
l=1,2,3.
\]
\item[(ii)]
Assume in addition that
$
\bm{v}_1^\top \bm{q}_{*},
\bm{v}_2^\top \bm{a}_{0*},
\bm{v}_3^\top \bm{b}_*
\neq 0
$
under the alternative hypothesis $\mathcal{H}_1$,
then,
\[
\,\mathrm{P}(|z_{lT}| > c) \to 1,\quad T \to \infty,\quad
l=1,2,3
\]
for every $c>0$.
\end{itemize}
\end{thm}
\begin{rem}\label{rem:asymptotic theory}
Similarly to the example of the empirical study described in the next section,
the number of observations
of time series data that seems to satisfy stationarity tends to be small. However, SVAR models have
at least $k^2p$
parameters,
and we often analyze the data such that the $k^2p/T \not\asymp 0$.
To handle such data, we should consider
central limit theorems or Gaussian approximations under
high-dimensional settings, e.g.,
the case where $k^2p/T \to \kappa \in (0, 1)$ as
$T \to \infty$.
See \cite{Koike2021},
\cite{Chernozhukov2023} for the
recent development of high-dimensional
asymptotic theory,
\cite{Fang2023} for the degenerate case, and
\cite{Belloni2018} for martingale high-dimensional
central limit theorems.
\end{rem}
\section{Numerical studies}\label{sec:numerical}
\subsection{Simulations}
In this section, we verify the finite-sample performances of the proposed estimators and test statistics.
Because each component of
$f_l(\bm{b})$ and $f_{l, h}(\bm{b})$ for $l=1,\ldots,5$
can be represented as a
rational function of the components of $\bm{b}$,
we can calculate their derivatives using, e.g.,
``PyTorch'' in Python.
In the sequel, we consider a stationary model
with $k=p=5$.
The innovation process $\{\bm{v}_t\}_{t \in \mathbb{Z}}$ is generated by the following model:
\[
\bm{v}_t = \bm{A}_W \bm{W}_t + \bm{u}_t,
\]
where $\{\bm{W}_t\}_{t \in \mathbb{Z}}$,
$\{\bm{u}_t\}_{t \in \mathbb{Z}}$ are two and five-dimensional mutually uncorrelated
i.i.d. Laplace distributed with mean $\bm{0}$
and variance $0.5$, respectively, and
\[
\bm{A}_W =
\begin{pmatrix}
0.5 & -0.5 \\
0.5 & 0.5 \\
-0.5 & 0.5 \\
0.4 & 0.6 \\
-0.4 & -0.6 \\
\end{pmatrix}.
\]
The true values of the coefficient matrices are
shown with the heat maps (Figure \ref{fig:parameter simulation}),
where the $x$-axis of the figure for $g(\bm{A})$
represents the column number of $\bm{A}$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.6]{params.jpg}
\caption{Heat maps of the coefficient matrices}
\label{fig:parameter simulation}
\end{figure}
We calculate the proposed
estimators and test statistics
for $1000$ replications.
First, we evaluate the performances of the
estimators $\hat{\bm{b}}_T$, $\hat{\bm{q}}_T$,
$\hat{\bm{a}}_{0T}$, and $\hat{\bm{a}}_T$,
using the mean of empirical bias (MB)
and mean of empirical mean absolute error (MMAE).
Tables \ref{table:est1}--\ref{table:est3}
summarize the results.
\begin{table}[H]
\caption{Estimators of the coefficient matrices $(T=100)$}
\label{table:est1}
\centering
\begin{tabular}{lcccc}
\hline
& $\hat{\bm{b}}_T$ & $\hat{\bm{q}}_T$ & $\hat{\bm{a}}_{0T}$ & $\hat{\bm{a}}_T$ \\
\hline \hline
MB & 0.008 & 0.062 & 0.065 & 0.026 \\
MMAE & 0.11 & 0.155 & 0.16 & 0.138 \\
\hline
\end{tabular}
\end{table}
\begin{table}[H]
\caption{Estimators of the coefficient matrices $(T=200)$}
\label{table:est2}
\centering
\begin{tabular}{lcccc}
\hline
& $\hat{\bm{b}}_T$ & $\hat{\bm{q}}_T$ & $\hat{\bm{a}}_{0T}$ & $\hat{\bm{a}}_T$ \\
\hline \hline
MB & 0.004 & 0.043 & 0.049 & 0.019 \\
MMAE & 0.072 & 0.124 & 0.127 & 0.092 \\
\hline
\end{tabular}
\end{table}
\begin{table}[H]
\caption{Estimators of the coefficient matrices $(T=500)$}
\label{table:est3}
\centering
\begin{tabular}{lcccc}
\hline
& $\hat{\bm{b}}_T$ & $\hat{\bm{q}}_T$ & $\hat{\bm{a}}_{0T}$ & $\hat{\bm{a}}_T$ \\
\hline \hline
MB & 0.002 & 0.019 & 0.019 & 0.008 \\
MMAE & 0.043 & 0.079 & 0.077 & 0.053 \\
\hline
\end{tabular}
\end{table}
We also calculate the estimators for impulse responses for $h=1, 2, 3$.
The results are summarized in
Table \ref{table:est4}--\ref{table:est6}.
In summary, we can observe the consistency of the proposed estimators.
\begin{table}[H]
\caption{Estimators for (controlled) total effects $(T=100)$}
\label{table:est4}
\centering
\begin{tabular}{lcccccc}
\hline
& $\hat{\bm{\psi}}_{1T}$ & $\hat{\bm{\psi}}_{2T}$ & $\hat{\bm{\psi}}_{3T}$ & $\hat{\bm{\psi}}_{1T}^{\mathrm{o}}$ & $\hat{\bm{\psi}}_{2T}^{\mathrm{o}}$ & $\hat{\bm{\psi}}_{3T}^{\mathrm{o}}$ \\
\hline \hline
MB & 0.008 & 0.009 & 0.006 & 0.030 & 0.031 & 0.032 \\
MMAE & 0.118 & 0.105 & 0.102 & 0.145 & 0.138 & 0.143 \\
\hline
\end{tabular}
\end{table}
\begin{table}[H]
\caption{Estimators for (controlled) total effects $(T=200)$}
\label{table:est5}
\centering
\begin{tabular}{lcccccc}
\hline
& $\hat{\bm{\psi}}_{1T}$ & $\hat{\bm{\psi}}_{2T}$ & $\hat{\bm{\psi}}_{3T}$ & $\hat{\bm{\psi}}_{1T}^{\mathrm{o}}$ & $\hat{\bm{\psi}}_{2T}^{\mathrm{o}}$ & $\hat{\bm{\psi}}_{3T}^{\mathrm{o}}$ \\
\hline \hline
MB & 0.004 & 0.003 & 0.004 & 0.019 & 0.023 & 0.022 \\
MMAE & 0.076 & 0.069 & 0.066 & 0.095 & 0.094 & 0.099 \\
\hline
\end{tabular}
\end{table}
\begin{table}[H]
\caption{Estimators for (controlled) total effects $(T=500)$}
\label{table:est6}
\centering
\begin{tabular}{lcccccc}
\hline
& $\hat{\bm{\psi}}_{1T}$ & $\hat{\bm{\psi}}_{2T}$ & $\hat{\bm{\psi}}_{3T}$ & $\hat{\bm{\psi}}_{1T}^{\mathrm{o}}$ & $\hat{\bm{\psi}}_{2T}^{\mathrm{o}}$ & $\hat{\bm{\psi}}_{3T}^{\mathrm{o}}$ \\
\hline \hline
MB & 0.002 & 0.002 & 0.002 & 0.007 & 0.008 & 0.008 \\
MMAE & 0.047 & 0.042 & 0.040 & 0.057 & 0.055 & 0.062 \\
\hline
\end{tabular}
\end{table}
Next, we show the asymptotic normality
of the proposed estimators
$\hat{\bm{q}}_T, \hat{\bm{a}}_{0T}, \hat{\bm{a}}_T$,
and $\hat{\bm{\psi}}_{h, T}$ for
$h=1,2,3$.
To do this, we introduce the following
random variables.
\begin{align*}
s_{1T} &= \sqrt{T}(\bm{1}^{\top}_{k^2}\hat{\bm{\Sigma}}_{1}\bm{1}_{k^2})^{-\frac{1}{2}}\bm{1}^{\top}_{k^2}(\hat{\bm{q}}_T-\bm{q}_{*}) \\
s_{2T} &= \sqrt{T}(\bm{1}^{\top}_{k^2}\hat{\bm{\Sigma}}_{2}\bm{1}_{k^2})^{-\frac{1}{2}}\bm{1}^{\top}_{k^2}(\hat{\bm{a}}_{0T}-\bm{a}_{0*}) \\
s_{3T} &= \sqrt{T}(\bm{1}^{\top}_{kr}\hat{\bm{\Sigma}}_{3}\bm{1}_{kr})^{-\frac{1}{2}}\bm{1}^{\top}_{kr}(\hat{\bm{a}}_{T}-\bm{a}_{*}) \\
s_{5,hT} &= \sqrt{T}(\bm{1}^{\top}_{k^2}\hat{\bm{\Sigma}}_{5,h}\bm{1}_{k^2})^{-\frac{1}{2}}\bm{1}^{\top}_{k^2}(\hat{\bm{\psi}}_{hT}^{\mathrm{o}}-\bm{\psi}_{h*}^{\mathrm{o}}) ,
\end{align*}
where $\bm{1}_{k^2}=(1,1,\ldots,1)^\top \in \mathbb{R}^{k^2}$ and $\bm{1}_{kr}=(1,1,\ldots,1)^\top \in \mathbb{R}^{kr}$.
Figures \ref{fig:hist1}--\ref{fig:hist6}
show the histograms of the random variables over
1000 replications;
the $y$-axis indicates the relative frequency of the
statistics over the replications
and the dashed line is the density function of the
standard normal distribution.
Moreover, Tables \ref{table:size1} and
\ref{table:size2} show the
empirical tail probabilities of the
random variables.
Even for the relatively small observations $T=100$,
the estimators are well approximated by
the normal distribution.
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{coef100.png}
\caption{Histograms of $s_{iT}$ for $i=1,2,3$. $(T=100)$}
\label{fig:hist1}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{coef200.png}
\caption{Histograms of $s_{iT}$ for $i=1,2,3$. $(T=200)$}
\label{fig:hist2}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{coef500.png}
\caption{Histograms of $s_{iT}$ for $i=1,2,3$. $(T=500)$}
\label{fig:hist3}
\end{figure}
\begin{table}[H]
\caption{Empirical tail probabilities}
\label{table:size1}
\centering
\begin{tabular}{lccc}
\hline
& $|s_{1T}|>1.96$ & $|s_{2T}|>1.96$ & $|s_{3T}|>1.96$ \\
\hline \hline
$T=100$ & 0.100 & 0.048 & 0.030 \\
$T=200$ & 0.056 & 0.036 & 0.027 \\
$T=500$ & 0.061 & 0.047 & 0.039 \\
\hline
\end{tabular}
\end{table}
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{effect100.png}
\caption{Histograms of $s_{5, h T}$ for $h=1,2,3$. $(T=100)$}
\label{fig:hist4}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{effect200.png}
\caption{Histograms of $s_{5, h T}$ for $h=1,2,3$. $(T=200)$}
\label{fig:hist5}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{effect500.png}
\caption{Histograms of $s_{5, h T}$ for $h=1,2,3$. $(T=500)$}
\label{fig:hist6}
\end{figure}
\begin{table}[H]
\caption{Empirical tail probabilities}
\label{table:size2}
\centering
\begin{tabular}{lcccc}
\hline
& $|s_{5,1T}|>1.96$ & $|s_{5,2T}|>1.96$ & $|s_{5,3T}|>1.96$ \\
\hline \hline
$T=100$ & 0.048 & 0.056 & 0.035 \\
$T=200$ & 0.051 & 0.075 & 0.050 \\
$T=500$ & 0.044 & 0.078 & 0.069 \\
\hline
\end{tabular}
\end{table}
Finally, we check the behaviors of the test
statistics $z_{iT}, i=1,2,3$ with $\bm{v} = \bm{1}=(1,\ldots,1)^\top \in \mathbb{R}^{k(k-1)/2}$
under the alternative hypothesis
$\mathcal{H}_1$.
Figures \ref{fig:hist alt1}--\ref{fig:hist alt3} show
the histograms of $z_{iT}, i=1,2,3$
under $\mathcal{H}_1$ and
the dashed line indicates the density function
of the standard normal distribution.
Moreover, the powers of the test statistics
are summarized in Table \ref{table:power}.
The consistency of the test also seems valid.
In terms of the power,
it may be better to use $z_{3T}$ than $z_{1T}$ and
$z_{2T}$ for the test.
\begin{table}[H]
\caption{Power of the test statistics}
\label{table:power}
\centering
\begin{tabular}{lccc}
\hline
& $z_{1T}$ & $z_{2T}$ & $z_{3T}$ \\
\hline \hline
$T=100$ & 0.350 & 0.225 & 0.668 \\
$T=200$ & 0.554 & 0.459 & 0.925 \\
$T=500$ & 0.877 & 0.846 & 1.000 \\
\hline
\end{tabular}
\end{table}
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{power100.png}
\caption{Distribution of test statistics under $\mathcal{H}_1$ $(T=100)$}
\label{fig:hist alt1}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{power200.png}
\caption{Distribution of the test statistics under $\mathcal{H}_1$ $(T=200)$}
\label{fig:hist alt2}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{power500.png}
\caption{Distribution of the test statistics under $\mathcal{H}_1$ $(T=500)$}
\label{fig:hist alt3}
\end{figure}
\subsection{Empirical studies}\label{subsec:real data}
We analyze the impact of policy rates on unemployment
and prices in U.S. from 1992 to 2018, which are
quarterly data.
We use the seasonally adjusted
unemployment level data (U.S. Bureau of Labor Statistics, Unemployment Level [UNEMPLOY], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/UNEMPLOY, February 18, 2025),
the seasonally adjusted core consumer price index data (U.S. Bureau of Labor Statistics, Consumer Price Index for All Urban Consumers: All Items Less Food and Energy in U.S. City Average [CPILFESL], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/CPILFESL, February 18, 2025),
and the federal funds effective rate (Board of Governors of the Federal Reserve System (US), Federal Funds Effective Rate [DFF], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/DFF,
February 18, 2025).
In the sequel, we modify the data and
use the following notations.
\begin{itemize}
\item $Y_{1,t}$ : growth rate of the unemployment level (\%).
\item $Y_{2,t}$ : inflation rate (growth rate of the core consumer price index) (\%).
\item $Y_{3,t}$ : difference of federal funds effective rate (\%).
\end{itemize}
Figure \ref{fig:real data} shows the plot of
the time series $\{Y_{i, t}\}, i=1,2,3$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{realdata.png}
\caption{Plot of time series $Y_{i, t}, i=1,2,3$}
\label{fig:real data}
\end{figure}
We apply the proposed method to the modified data.
We choose the lag $p=4$, which
enables us to consider the correlations
between the three time series
up to one year ago.
The estimated coefficient matrices
of the reduced form are as follows (the number of observations is
$T = 104$).
\begin{align*}
\hat{\bm{B}}_{1T} &=
\begin{pmatrix}
0.061 & 0.634 & -3.973 \\
-0.003 & 0.549 & -0.037 \\
-0.017 & -0.37 & 0.651
\end{pmatrix}, \hspace{6pt}
\hat{\bm{B}}_{2T} =
\begin{pmatrix}
0.368 & 4.763 & -1.284 \\
-0.037 & -0.003 & 0.115 \\
0.651 & 0.005 & 0.258
\end{pmatrix} \\
\hat{\bm{B}}_{3T} &=
\begin{pmatrix}
0.119 & 1.575 & 0.780 \\
-0.001 & 0.201 & 0.054 \\
-0.006 & 0.235 & -0.026
\end{pmatrix}, \hspace{6pt}
\hat{\bm{B}}_{4T} =
\begin{pmatrix}
-0.04 & -1.956 & 1.127 \\
-0.003 & -0.109 & -0.026\\
0.003 & -0.311 & -0.159
\end{pmatrix}
\end{align*}
Figure \ref{fig:total effect real data} shows the
estimated curve of the impulse responses
and their $0.95$ confidence intervals.
\begin{figure}[H]
\centering
\includegraphics[scale=0.6]{irf.png}
\caption{non-orthogonalized impulse response function}
\label{fig:total effect real data}
\end{figure}
Note that lag $s=4$ means one year.
Since we assume that $Y_{3,t}$ does not affect $Y_{1,t}$ and $Y_{2,t}$ (this assumption derives from the ordering of the components of $\bm{Y}_{t}$),
non-orthogonalized impulse responses in Figure \ref{fig:total effect real data} can be regarded as 'total effects' from Proposition \ref{prop:impulse response} in Appendix.
We expect that a policy rate decrease
will cause an improvement in
unemployment.
The upper curve of the
significance interval in the left figure seems consistent
with our intuition.
Meanwhile, the lower curve in the right figure
seems to well describe the
impact of the policy rate on the prices
because the prices may be depressed
if the policy rate increases.
However,
given that the monetary policy from 1992
to 2018 had often been shifted about every three years, the long-term effects shown by estimated values are more reasonable than the edges of significance intervals.
We checked the accuracy of the estimated model
in terms of the standard deviation, root mean squared error, and adjusted R-squared coefficient, which are summarized in
Table \ref{table:prediction real data}.
The results indicate that
the model is applicable to fit the data.
\begin{table}[H]
\caption{Accuracy of the estimated model}
\label{table:prediction real data}
\centering
\begin{tabular}{lccc}
\hline
& $\{\hat{Y}_{1,t}\}$ & $\{\hat{Y}_{2,t}\}$ & $\{\hat{Y}_{3,t}\}$ \\
\hline \hline
Standard deviation & 3.557 & 0.130 & 0.306 \\
Root mean squared error & 3.631 & 0.092 & 0.260 \\
Adjusted R-squared & 0.416 & 0.611 & 0.552 \\
\hline
\end{tabular}
\end{table}
To estimate the coefficient matrices $\bm{A}_s, s=0,1,\ldots,4$ of the SVAR model,
we consider the following assumptions
corresponding to
Assumption \ref{assump:regularity A}.
\begin{assumption}\label{assump:real data}
\begin{itemize}
\item[(1)]
The difference of the policy rate
$\{Y_{3, t}\}_{t}$ does not
depend on $(Y_{1, t-4}, Y_{2, t-4})^\top$ directly.
\item[(2)]
$Y_{3, t}$ does not affect $Y_{2, t+1}$ directly.
\item[(3)]
$Y_{3, t}$ affects $Y_{3, t+1}$ directly.
\item[(4)]
$Y_{1, t}$ affects $Y_{2, t+4}$ directly.
\item[(5)]
$Y_{3, t}$ affects $Y_{1,t+4}$ directly.
\end{itemize}
\end{assumption}
\begin{rem}
Each assumption is considered based on the
following background.
\begin{itemize}
\item[(1)]
The policy rates are determined based on the
behavior of the current economic indicators
by the Federal Open Market Committee per
about six weeks.
\item[(2)]
Prices of commodities are considered to be
directly affected
by production
costs including wages of workers.
In this study, we simply assume that
the policy rates affect prices through unemployment level.
\item[(3)]
The policy rate tends to be gradually changing.
Here, we attempt to approximate the causal structure by simplifying $Y_{3,t}$ as a direct cause of $Y_{3,t+1}$.
\item[(4)]
The condition is a standard assumption
because the relationship between inflation and unemployment is widely observed.
\item[(5)]
Naturally,
the policy rate is considered to affect
unemployment through
various economic activities.
In this study, we simply assume that
$Y_{3, t}$ is a direct cause of $Y_{1, t+4}$.
\end{itemize}
\end{rem}
Under Assumption \ref{assump:real data},
we consider the following
restrictions for $\bm{A}_s, 0 \leq s \leq 4$.
\begin{align*}
\bm{A}_0 &=
\begin{pmatrix}
0 & 0 & 0 \\
* & 0 & 0 \\
* & * & 0
\end{pmatrix}, \hspace{6pt}
\bm{A}_1 =
\begin{pmatrix}
* & * & * \\
* & * & * \\
* & * & \dagger
\end{pmatrix}, \hspace{6pt}
\bm{A}_4 =
\begin{pmatrix}
* & * & \dagger \\
\dagger & * & 0 \\
0 & 0 & 0
\end{pmatrix},
\end{align*}
where
$\dagger$ and $*$ mean
nonzero parameters and arbitrary
$\mathbb{R}$-valued parameters, respectively.
Then, the estimated coefficient
matrices of the SVAR model
are as follows.
\begin{align*}
\hat{\bm{A}}_{0T} &=
\begin{pmatrix}
0.000 & 0.000 & 0.000 \\
-0.023 & 0.000 & 0.000 \\
-0.126 & 0.648 & 0.000
\end{pmatrix}, \hspace{6pt}
\hat{\bm{Q}}_{T} =
\begin{pmatrix}
1.000 & 0.000 & 0.000 \\
-0.023 & 1.000 & 0.000 \\
-0.141 & 0.648 & 1.000
\end{pmatrix} \\
\hat{\bm{A}}_{1T} &=
\begin{pmatrix}
0.061 & 0.634 & -3.973 \\
-0.002 & 0.564 & -0.130 \\
-0.008 & -0.646 & 0.176
\end{pmatrix}, \hspace{6pt}
\hat{\bm{A}}_{2T} =
\begin{pmatrix}
0.368 & 4.763 & -1.284 \\
0.006 & 0.226 & -0.029 \\
0.053 & 0.781 & -0.113
\end{pmatrix} \\
\hat{\bm{A}}_{3T} &=
\begin{pmatrix}
0.119 & 1.575 & 0.780 \\
0.002 & 0.238 & 0.073 \\
0.009 & 0.302 & 0.037
\end{pmatrix}, \hspace{6pt}
\hat{\bm{A}}_{4T} =
\begin{pmatrix}
-0.040 & -1.956 & 1.127 \\
-0.004 & -0.155 & 0.000 \\
0.000 & -0.486 & 0.000
\end{pmatrix}.
\end{align*}
Moreover, we consider the
following test
\[
\mathcal{H}_0: \bm{A}_0 = \bm{O},\quad
\mathcal{H}_1: \bm{A}_0 \neq \bm{O}.
\]
Based on the test statistics $z_{3T}$ defined
in the previous section with
$\bm{v} = \bm{1} = (1,\ldots,1)^\top \in \mathbb{R}^{k(k-1)/2}$,
the results are summarized in Table \ref{table:test real data}.
\begin{table}[H]
\caption{Test result}
\label{table:test real data}
\centering
\begin{tabular}{cccc}
\hline
Test statistics & $T$ & $z$-value & $p$-value (two-sided test) \\
\hline \hline
$z_{3T}$ & 104 & -1.79 & 0.0734 \\
\hline
\end{tabular}
\end{table}
We cannot reject the null hypothesis
at a significance level of $\alpha=0.05$.
As described in Proposition \ref{prop:impulse response} in Appendix,
when $\bm{A}_0 = \bm{O}$,
all non-orthogonalized impulse responses
can be regarded
not only as partial effects, but also as ``total effects".
Therefore, if the null hypothesis
is accepted,
we can treat non-orthogonalized impulse responses
from not only $e_{k,t}$ but also $e_{j,t}, j=1,...,k-1$
as indicators of the overall effects.
However,
given that
we obtained a relatively small
$p$-value for a small observations $T=104$,
we continue our discussion
on the estimated value of the SVAR model.
Because one of the purposes of the policy rate is
to stabilize employment and prices,
the signs of the third row
of $\hat{\bm{A}}_{0T}$ may be consistent
with our intuition.
The sign of the $(2,1)$ component
of $\hat{\bm{A}}_{0T}$ also
seems reasonable because
unemployment may cause a decline in
wages, and hence,
have a negative impact on prices.
However, the absolute value of the
$(2, 1)$ component
may be overestimated because
prices are rigid.
Although we simplified the causal structure, obtaining an $\hat{\bm{A}}_{0T}$ that aligns with intuition may serve as evidence that the model can approximate the structure to some extent.
In summary, the proposed model
seems to describe the data structure.
In particular, the non-orthogonalized
impulse responses suggest that
it may take about one year for
the policy rate decrease to
improve unemployment
and policy rates increase
may depress the prices in the short term.
See the Appendix for the causal interpretation
of the non-orthogonalized impulse responses.
\section{Proofs}\label{sec:proofs}
\begin{proof}[Proof of Theorem \ref{thm:normality A}]
It follows from the basic properties of the
$\,\mathrm{vec}$ operator and the
matrix Kronecker product that
\begin{eqnarray*}
\bm{q}_*
&=& \,\mathrm{vec}(\bm{Q}_*)
= \,\mathrm{vec}(\bm{L}_g(\bm{B}_*)) = f_1(\bm{b}_*) \\
\bm{a}_{0*}
&=& \,\mathrm{vec}(\bm{A}_{0*}) = \,\mathrm{vec}(\bm{I}_k - \bm{L}_g(\bm{B}_*)^{-1})
= f_2(\bm{b}_*) \\
\bm{a}_*
&=& \,\mathrm{vec}(\bm{Q}_*^{-1}\bm{B}_*)
= \left(\bm{I}_r \otimes \bm{L}_g(\bm{B}_*)
\right)^{-1} \bm{b}_*
= f_3(\bm{b}_*).
\end{eqnarray*}
We show that the map $\bm{L}$ is differentiable at $g(\bm{B}_*) = g(\,\mathrm{vec}^{-1}(\bm{b}_*))$.
For every matrix $\bm{C}=(c_{ij})_{1 \leq i, j \leq k}$ that admits the
LU decomposition, the corresponding
lower-triangular matrix
$\bm{L}(\bm{C})=(l_{ij})_{1 \leq i, j \leq k}$ and
the upper-triangular matrix
$\bm{U}(\bm{C}) = (u_{ij})_{1 \leq i, j \leq k}$
are solutions to the
following equation:
\[
\begin{pmatrix}
c_{11} & c_{12} & c_{13} & \cdots & c_{1k} \\
c_{21} & c_{22} & c_{23} & \cdots & c_{2k} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
c_{k1} & c_{k2} & c_{k3} & \cdots & c_{kk}
\end{pmatrix}
= \begin{pmatrix}
1 & 0 & 0 & \cdots & 0 \\
l_{21} & 1 & 0 & \cdots & 0 \\
l_{31} & l_{32} & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
l_{k1} & l_{k2} & l_{k3} & \cdots & 1 \\
\end{pmatrix}
\begin{pmatrix}
u_{11} & u_{12} & u_{13} & \cdots & u_{1k} \\
0 & u_{22} & u_{23} & \cdots & u_{2k} \\
0 & 0 & u_{33} & \cdots & u_{3k} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & u_{kk} \\
\end{pmatrix}.
\]
We can observe that for every $i, j =1,2,\ldots,k$,
$l_{ij}$ and $u_{ij}$ can be
written as
a rational function
of some components of $\bm{C}$.
Under our assumptions,
$\bm{Q}_*$ and $g(\bm{A}_*)$ are non-singular
lower and upper-triangular
matrices, respectively,
which implies that
all leading principal sub-matrices
are non-singular, therefore,
$g(\bm{B}_*) = \bm{Q}_* g(\bm{A}_*)$
admits
the LU decomposition.
Combining these facts, we obtain
the differentiability
of $\bm{L}$ at $g(\bm{B}_*)$.
Thus, we can apply the delta method to
derive the conclusion.
\end{proof}
We omit the proof of
Propositions
\ref{prop:estimator psi}, and \ref{prop:estimator total effect} because they are direct consequences of
the delta method.
\begin{proof}[Proof of Theorem \ref{thm:test}]
We provide the proof of the assertion (ii),
because (i) is a direct consequence of
Lemma \ref{lem:estimation sub-vector}.
We consider the case where $l=1$.
Note that
\[
\{|z_{1T}| > c\}
=
\left\{
\left|\frac{z_{1T}}{\sqrt{T}}\right|
> \frac{c}{\sqrt{T}}
\right\}
= \{
A_T^+ > 0
\} \cup
\{
A_T^- <0
\},
\]
where
\[
A_T^+ = \frac{z_{1T}}{\sqrt{T}}-\frac{c}{\sqrt{T}}\quad
\mbox{and}\quad
A_T^- = \frac{z_{1T}}{\sqrt{T}}+\frac{c}{\sqrt{T}}.
\]
Noting that $\hat{\bm{\Sigma}}_1$ and
$\hat{\bm{q}}_T$ are consistent
estimators
for $\bm{\Sigma}_1$ and $\bm{q}_*$, respectively,
we have
\begin{eqnarray*}
\frac{z_{1T}}{\sqrt{T}}
&=& (\bm{v}^\top \hat{\bm{\Sigma}}_{1\mathrm{sub}} \bm{v})^{-1/2} \bm{v}^\top (\hat{\bm{q}}_{\mathrm{sub}} -\bm{q}_{*\mathrm{sub}})
+(\bm{v}^\top \hat{\bm{\Sigma}}_{1\mathrm{sub}} \bm{v})^{-1/2} \bm{v}^\top \bm{q}_{*\mathrm{sub}} \\
&=& o_p(1) + (\bm{v}^\top {\bm{\Sigma}}_{1\mathrm{sub}} \bm{v})^{-1/2} \bm{v}^\top \bm{q}_{*\mathrm{sub}},\quad
T \to \infty.
\end{eqnarray*}
Therefore, it holds that
\[
A_T^+ = A_T^- = o_p(1) + A,
\quad T \to \infty,
\]
with
\[
A=(\bm{v}^\top {\bm{\Sigma}}_{1\mathrm{sub}} \bm{v})^{-1/2} \bm{v}^\top \bm{q}_{*\mathrm{sub}}.
\]
By the continuous mapping theorem, we have
$1_{\{A_T^+ >0\}}
\to^p 1_{\{A >0\}}$ and
$1_{\{A_T^- <0\}}
\to^p 1_{\{A <0\}}$ as $T \to \infty$.
Therefore, it follows from the
dominated convergence theorem that
\begin{eqnarray*}
\lim_{T \to \infty}\,\mathrm{P}(|z_{1T}| > c)
&=& \lim_{T \to \infty} \left\{
\,\mathrm{P} (A_T^+>0) + \,\mathrm{P} (A_T^-<0)
\right\} \\
&=& \lim_{T \to \infty} \left\{
\,\mathrm{E} \left[1_{\{A_T^+>0\}}\right] + \,\mathrm{E}\left[ 1_{\{A_T^-<0\}} \right]
\right\} \\
&=& \,\mathrm{E}\left[1_{\{A^+ >0\}}\right] + \,\mathrm{E}\left[1_{\{A^- <0\}}\right]= 1,
\end{eqnarray*}
which ends the proof for $l=1$.
The assertion for $l=2,3$ can be proved similarly.
\end{proof}