The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
73,907 characters
A Structural Matrix Autoregression Framework for International Spillovers
{\setstretch{.8}
\maketitle
\begin{abstract}
Understanding how macroeconomic shocks propagate across countries requires structural models that can jointly identify country-specific shocks and their international transmission. Yet extending structural vector autoregressions (SVARs) to large multi-country systems is challenging due to rapidly increasing dimensionality, computational costs, and the proliferation of identifying restrictions. This paper develops a Bayesian Structural Matrix Autoregression (BSMAR) framework that exploits the natural matrix structure of international macroeconomic data. By separating dependence across economic variables from dependence across countries, the framework provides a parsimonious representation that substantially reduces the dimensionality of large structural systems. We develop a Bayesian sampling algorithm for posterior inference that accommodates zero, sign, and ranking (magnitude) restrictions, allowing established SVAR identification schemes to be combined with a novel approach to identifying contemporaneous international spillovers. Applying the model to quarterly data for 15 economies, we find substantial heterogeneity in international shock transmission, with demand shocks playing a more prominent role than supply shocks in generating cross-country spillovers.
\noindent
\textit{\textbf{Keywords: }
SVAR; Identification; Spillover; High Dimension.} \\
\noindent
\textit{\textbf{JEL Classification: }C11, C32, C55, F00
}
\end{abstract}
}
\section{Introduction}
Structural vector autoregressions (SVARs) have become a central tool for identifying macroeconomic shocks and studying their dynamic effects on the economy, see, e.g., \textcite{sims1980macroeconomics}, \textcite{blanchard_quah_1989}, and \textcite{kilian2017structural}. Recent applications of sign-restricted SVARs have, in particular, revisited the role of supply and demand disturbances in explaining the post-pandemic inflation episode, see, e.g., \textcite{giannone_primiceri_2025}, \textcite{bergholt2026inflation}, and \textcite{della_chang_jansen_pagliacci_2023}. Yet, despite the inherently international nature of many macroeconomic questions, most structural VAR applications continue to study countries in isolation. This is increasingly difficult to reconcile with the structure of modern economies, where deep trade integration, globally interconnected financial markets, and complex production networks allow shocks originating in one country to propagate rapidly across borders. Understanding these spillovers requires structural models that can simultaneously identify country-specific shocks and characterize their transmission across economies.
Extending structural VARs to a large panel of countries, however, presents a fundamental challenge. As the number of countries increases, the dimension of the system and the number of unknown parameters grow rapidly, resulting in severe overparameterization, high computational costs, and a proliferation of identifying restrictions. Consequently, unrestricted multi-country SVARs quickly become infeasible, even for a moderate number of economies.
Existing approaches address this curse of dimensionality by imposing structure on the cross-country dependence. Dynamic factor models summarize large datasets using a small number of latent factors, see, e.g., \textcite{forni2001generalized}, \textcite{charnavoki2014effects}, \textcite{mumtaz2009transmission}, and \textcite{stock2005implications}. These models are highly effective for studying common business-cycle fluctuations but are not designed to identify a large number of country-specific structural shocks, as they typically assume a relatively small number of structural shocks. Global vector autoregressive (GVAR) models provide another influential framework for studying international macroeconomic linkages, see, e.g., \textcite{pesaran2004modeling}, \textcite{dees2007exploring}, \textcite{crespo2019spillovers}, and \textcite{feldkircher2016international}. By connecting country-specific models through predetermined foreign aggregates, GVARs remain computationally tractable for large panels, but at the cost of imposing uncorrelated contemporaneous cross-country innovations. More generally, the existing literature faces a fundamental trade-off: models that remain computationally feasible impose restrictions on the cross-country structure that limit the identification of country-specific structural spillovers, whereas unrestricted structural models become infeasible as the number of countries increases.
This paper addresses this trade-off by proposing a Bayesian Structural Matrix Autoregression (BSMAR), together with a Bayesian sampling algorithm for drawing from its joint posterior distribution. The key insight is that international macroeconomic data naturally have a matrix structure, with one dimension representing economic variables and the other representing countries. Rather than vectorizing the data into a single high-dimensional system, we exploit this two-way structure directly. The resulting framework separates dependence across economic variables from dependence across countries, yielding a parsimonious representation that preserves the multivariate structure of the data while substantially reducing the dimensionality of the model.
The matrix formulation resolves the three principal obstacles that have prevented the estimation of large structural international systems. First, it drastically reduces the number of unknown parameters relative to an unrestricted SVAR, mitigating overparameterization and improving statistical efficiency. Second, the resulting parsimonious representation makes Bayesian estimation computationally feasible even in systems containing many countries. Third, and most importantly, it fundamentally simplifies the structural identification problem. Under our framework, the contemporaneous impact matrix admits a Kronecker product decomposition into a variable-specific component and a country-specific component. Rather than identifying one prohibitively large structural impact matrix, identification is carried out separately for economic relationships among variables and for contemporaneous spillovers across countries. This decomposition substantially reduces the number of structural parameters and identifying restrictions while preserving a transparent economic interpretation.
Our Bayesian framework further allows researchers to incorporate a broad class of identifying restrictions, including zero, sign, ranking, and more general nonlinear restrictions. This flexibility makes it possible to combine established SVAR identification schemes for the variable-specific component with a novel identification framework for the country-specific component. In particular, we use economically motivated restrictions on the country-specific impact matrix to identify contemporaneous structural spillovers across countries. A central assumption is that a country-specific shock has its largest contemporaneous impact in the country where it originates. We also allow for soft restrictions that incorporate prior beliefs about the direction and relative magnitude of spillovers without ruling out deviations from these beliefs.
We demonstrate the empirical relevance of the framework using quarterly data for 15 economies. Motivated by the recent literature on the role of supply and demand shocks in explaining inflation dynamics, see, e.g., \textcite{giannone_primiceri_2025}, \textcite{bergholt2026inflation}, and \textcite{della_chang_jansen_pagliacci_2023}, we simultaneously identify 30 country-specific structural shocks---one supply shock and one demand shock for each economy---and trace their contemporaneous transmission across countries. To the best of our knowledge, this is the first study to identify contemporaneous structural spillovers among such a large group of countries within a unified structural framework. We find substantial heterogeneity in international shock transmission, with the United States emerging as an important transmitter of shocks and demand shocks playing a more prominent role than supply shocks in generating cross-country spillovers. Furthermore, our results highlight the central role of demand shocks in driving international macroeconomic fluctuations and inflation dynamics. In particular, demand shocks emerge as the main driver of the increase in inflation following the COVID-19 pandemic, underscoring their importance for understanding the post-pandemic inflation surge.
Our paper contributes to two strands of the literature. First, we contribute to the growing literature on matrix-valued time-series models. Matrix autoregressive (MAR) models exploit the two-way structure of matrix-valued data to achieve substantial dimension reduction relative to vectorized VARs, see \textcite{chen2021autoregressive}. Recent developments include Bayesian estimation, variable selection, nonlinear and factor extensions, see \textcite{celani2024matrix}, \textcite{celani2024bayesian}, \textcite{bucci2024smooth}, \textcite{chan2024bayesian}, and \textcite{yu2024twoway}. We extend this literature by developing a structural matrix autoregression that allows researchers to identify contemporaneous structural relationships within countries as well as structural spillovers across countries. Second, we contribute to the literature on international macroeconomic spillovers. Existing approaches, including GVAR and dynamic factor models, either focus on a relatively small number of common shocks or impose substantial structure on the contemporaneous interactions among country-specific shocks, see, e.g., \textcite{charnavoki2014effects}, \textcite{pfarrhofer2025high}, and \textcite{camehl2026explains}. Our framework provides a computationally feasible alternative that simultaneously identifies a large number of country-specific structural shocks and their contemporaneous cross-country transmission.
The remainder of the paper is organized as follows. Section 2 introduces the structural matrix framework, discusses general identification restrictions, and develops the Bayesian sampling algorithm. Section 3 describes the data, presents our empirical identification strategy, and discusses the main empirical findings. Section 4 concludes.
\section{A Large Multicountry SMAR}
\label{sec:model}
In this section, we first present the reduced-form Bayesian Matrix Autoregressive (MAR) model. We then formulate its structural representation and discuss the identification of the structural parameters. Finally, we develop a sampling algorithm for drawing from the joint posterior distribution of the model.
\subsection{Reduced Form MAR}
Let $\mathbf{Y}_t=(y_{t,ij})$ denote an $n \times k$ matrix of endogenous time series, where $i=1,\ldots,n$ indexes the macroeconomic variables for each country and $j=1,\ldots,k$ indexes the countries, for $t=1,\ldots,T$. In our application, the rows of $\mathbf{Y}_t$ correspond to economic variables, while the columns correspond to countries. Hence, the matrix structure reflects the two dimensions that characterize international macroeconomic data: interactions among economic variables and interactions across countries. In total, there are $r=nk$ time series.
The Matrix Autoregressive model, as presented in \textcite{chan2025large,samadi2025matrix}, exploits this structure directly and is given by
\begin{equation}
\label{eq:MAR}
\mathbf{Y}_t
=
\mathbf{A}_1\mathbf{Y}_{t-1}\mathbf{B}_1'
+\ldots+
\mathbf{A}_p\mathbf{Y}_{t-p}\mathbf{B}_p'
+\mathbf{U}_t,
\end{equation}
where $\mathbf{A}_1,\ldots,\mathbf{A}_p$ are $n\times n$ coefficient matrices and $\mathbf{B}_1,\ldots,\mathbf{B}_p$ are $k\times k$ coefficient matrices. The matrices $\mathbf{A}_\ell$ capture dynamic interactions among economic variables, whereas $\mathbf{B}_\ell$ capture dynamic interactions across countries. The model allows for a rich but parsimonious representation of the dynamics of the data, making it suitable for high-dimensional settings where the number of variables is large relative to the number of observations \parencite{chen2021autoregressive}.
The error term is assumed to follow a matrix normal distribution,
\begin{equation}
\label{eq:error}
\mathbf{U}_t
\sim
\mathcal{MN}_{n \times k}
(\mathbf{0},\boldsymbol{\Sigma}_c,\boldsymbol{\Sigma}_r),
\end{equation}
where $\boldsymbol{\Sigma}_c$ is a $k\times k$ positive definite column covariance matrix and $\boldsymbol{\Sigma}_r$ is an $n\times n$ positive definite row covariance matrix.\footnote{Note that under the matrix normal distribution, $\operatorname{Cov}(\operatorname{vec}(\mathbf{U}_t))=\boldsymbol{\Sigma}_c\otimes\boldsymbol{\Sigma}_r.$ Thus, the error structure mirrors the two-dimensional organization of the data by separately capturing dependence across countries and across economic variables.} For a more compact representation, define $\mathbf{X}_{t}=\operatorname{diag}(\mathbf{Y}_{t-1},\ldots,\mathbf{Y}_{t-p})$, $
\mathbf{A}
=
(\mathbf{A}_1,\ldots,\mathbf{A}_p)'
\in\mathbb{R}^{np\times n}$ and $
\mathbf{B}
=
(\mathbf{B}_1,\ldots,\mathbf{B}_p)'
\in\mathbb{R}^{kp\times k}
$
denote the stacked autoregressive coefficient matrices. The MAR model can then be written as
\begin{equation*}
\mathbf{Y}_t
=
\mathbf{A}'\mathbf{X}_t\mathbf{B}
+
\mathbf{U}_t.
\end{equation*}
The advantages of this structure become particularly clear when considering the equivalent vector representation. Applying the vectorizing operator yields
\begin{equation}
\label{eq:VAR}
\mathbf{y}_t
=
\boldsymbol{\Phi}_1\mathbf{y}_{t-1}
+\ldots+
\boldsymbol{\Phi}_p\mathbf{y}_{t-p}
+\mathbf{u}_t,
\end{equation}
where $\mathbf{y}_t=\operatorname{vec}(\mathbf{Y}_t)$, $\mathbf{u}_t=\operatorname{vec}(\mathbf{U}_t)$, $\mathbf{u}_t \sim N(\mathbf{0}, \mathbf{\Sigma}_u)$,
$
\boldsymbol{\Phi}_\ell
=
\mathbf{B}_\ell\otimes\mathbf{A}_\ell,
$ and $
\boldsymbol{\Sigma}_u
=
\boldsymbol{\Sigma}_c\otimes\boldsymbol{\Sigma}_r.
$ Hence, the MAR model can be viewed as a VAR in which the large $nk\times nk$ coefficient matrices and covariance matrix are restricted to have a Kronecker product structure. Besides casting the model in a familiar framework, the VAR representation of the MAR allows us to make use of theoretical results in structural identification \parencite{kilian2017structural}, provided that the conditions of stability of the MAR representation are fulfilled \parencite{samadi2025matrix}.
An unrestricted VAR with $nk$ variables and $p$ lags requires $(nk)^2p$ autoregressive parameters. In contrast, the MAR representation requires only $(n^2+k^2)p$ parameters. For example, with $n=2$ economic variables and $k=15$ countries, an unrestricted VAR requires $900p$ autoregressive coefficients, compared with only $229p$ in the MAR model. The reduction becomes increasingly important as the number of countries or variables grows. The MAR model therefore provides a parsimonious representation that is particularly well suited to high-dimensional settings where estimating an unrestricted VAR would be computationally demanding and statistically inefficient \parencite{chen2021autoregressive}.
\subsection{Structural Matrix Autoregression}
\label{sec:smar}
To recover economically meaningful shocks, the reduced-form innovations must be mapped into structural shocks:
\begin{equation}
\label{eq:SVAR}
\mathbf{u}_t=\mathbf{B}_0\mathbf{e}_t,
\qquad
\mathbf{e}_t\sim\mathcal{N}(\mathbf{0},\mathbf{I}_r),
\end{equation}
where $\mathbf{B}_0$ is the structural impact matrix. However, the covariance matrix of the reduced-form innovations only identifies $\mathbf{B}_0$ up to an orthogonal rotation. Specifically, $ \boldsymbol{\Sigma}_u=\mathbf{B}_0\mathbf{B}_0'$, and any orthogonal matrix
$\mathbf{Q} \in \mathcal{O}(r)$ of the orthogonal group $\mathcal{O}(r)=\{\mathbf{Q} \in \mathbb{R}^{r\times r}: \mathbf{Q} \mathbf{Q}'=\mathbf{I}_r \} $
yields an observationally equivalent model
\[
(\mathbf{B}_0\mathbf{Q}')
(\mathbf{B}_0\mathbf{Q}')'
=
\mathbf{B}_0\mathbf{Q}'\mathbf{Q}\mathbf{B}_0'
=
\mathbf{B}_0\mathbf{B}_0'.
\]
Therefore, an unrestricted SVAR requires enough economic restrictions to select a unique rotation among infinitely many observationally equivalent structural representations. This identification problem becomes particularly challenging in large systems. An unrestricted SVAR requires $r^2$ structural parameters, while the reduced-form covariance matrix only pins down $r(r+1)/2$ of them. Point identification therefore requires at least $r(r-1)/2$ additional restrictions on $\mathbf{B}_0$. When the number of variables or units increases, the number of restrictions required quickly becomes impractical.
To overcome this limitation, we exploit the matrix structure of the VAR and impose a parsimonious decomposition of the structural impact matrix. Specifically, we assume that the vectorized system admits the following representation:
\begin{equation}
\label{eq:BSMAR}
\mathbf{Y}_t =
\mathbf{A}_1\mathbf{Y}_{t-1}\mathbf{B}_1'
+\ldots+
\mathbf{A}_p\mathbf{Y}_{t-p}\mathbf{B}_p'
+
\mathbf{B}_r\mathbf{E}_t\mathbf{B}_c',
\qquad
\mathbf{E}_t\sim\mathcal{MN}_{n\times k}(\mathbf{0},\mathbf{I}_k,\mathbf{I}_n),
\end{equation}
where $\mathbf{E}_t$ collects the structural shocks in matrix form, so that $\mathbf{e}_t=\operatorname{vec}(\mathbf{E}_t)$ and $\mathbf{U}_t=\mathbf{B}_r\mathbf{E}_t\mathbf{B}_c'$, and the structural impact matrix is restricted to have the Kronecker form
\[
\mathbf{B}_0=\mathbf{B}_c\otimes\mathbf{B}_r .
\]
This restriction provides two important advantages. First, it substantially reduces the number of unknown structural parameters. Instead of estimating $r^2=n^2k^2$ elements of $\mathbf{B}_0$, the researcher only needs to estimate the $n^2+k^2-1$ free elements contained in $\mathbf{B}_r$ and $\mathbf{B}_c$, one element being fixed by the scale normalisation discussed below.
Second, the Kronecker decomposition provides a more transparent economic interpretation of the identifying restrictions. The matrix $\mathbf{B}_r$ captures contemporaneous relationships among variables within a unit and therefore allows researchers to exploit existing identification schemes from the SVAR literature. It is common to all units: every country shares the same contemporaneous variable structure, merely rescaled across units by the corresponding element of $\mathbf{B}_c$. The same separability applies to the dynamics, where a single set of $\mathbf{A}_\ell$ matrices governs the variable dynamics of every country and a single set of $\mathbf{B}_\ell$ matrices the cross-country dynamics of every variable. This is what buys the dimension reduction, and it is the substantive cost of the specification; the multiple-component extension in the Online Appendix relaxes it for the dynamics by writing the conditional mean as a sum of $g$ bilinear terms. In contrast, $\mathbf{B}_c$ captures cross-unit transmission mechanisms. For $j_1,j_2\in\{1,\ldots,k\}$, we write $b_c^{j_1j_2}$ for the $(j_1,j_2)$ element of $\mathbf{B}_c$; it represents the relative strength of the contemporaneous spillover from country $j_2$ to country $j_1$, and $b_c^{jj}$ is the own effect of country $j$. Consequently, restrictions on $\mathbf{B}_c$ can be directly interpreted as assumptions about economic linkages, such as international spillovers or network effects. Hence, the identification problem is transformed from selecting a rotation of a large unrestricted matrix $\mathbf{B}_0$ to identifying two lower-dimensional matrices with economically meaningful interpretations. An additional implication of this decomposition is that the covariance structure can be separated into row and column components as $\boldsymbol{\Sigma}_r=\mathbf{B}_r\mathbf{B}_r'$ and $
\boldsymbol{\Sigma}_c=\mathbf{B}_c\mathbf{B}_c'$.
An important direction for future research is the development of matrix models that relax some separability restrictions while preserving the computational and statistical advantages of the matrix representation. One promising approach is to approximate the autoregressive VAR coefficients as a sum of Kronecker products, thereby allowing for multiple channels of interaction between countries and indicators while still achieving substantial dimension reduction. Another possibility is to combine a Kronecker-structured component with an unrestricted component in the structural impact matrix $\mathbf{B}_0$. Such specifications would provide a flexible continuum between the fully separable Matrix SVAR model and the unrestricted SVAR model discussed in the Online Appendix. We believe these extensions could be particularly useful in applications involving a relatively small number of countries, where the additional flexibility can be estimated with reasonable precision. However, as the cross-sectional dimension grows, the number of parameters increases rapidly, leading to a substantial rise in estimation uncertainty and diminishing the practical advantages of these more flexible specifications.
\subsection{General Structural Identification Restrictions}
Because the structural-form representation is not identified, additional identifying restrictions are required to narrow the identified set and recover the structural shocks. In this section, we present a general framework for imposing such restrictions within our sampling algorithm presented in Section~\ref{sec:Estimation}. The particular restrictions adopted in the empirical application are described in Section~\ref{sec:emp_id_restrictions}.
\subsubsection{Hard Parameter Restrictions}
Define $\boldsymbol{\theta}
=
\left(
\operatorname{vec}(\mathbf{B}_r)',
\operatorname{vec}(\mathbf{B}_c)'
\right)'
\in \mathbb{R}^{d\times 1},$ which stacks all structural parameters of the model. Following the notation commonly used in the literature \parencite{hou2024large, read2025fast}, we impose linear restrictions on $\boldsymbol{\theta}$ by organizing them into three classes: \emph{zero} restrictions (linear equality constraints), \emph{sign} restrictions (one-sided linear inequalities), and \emph{magnitude} restrictions (two-sided linear inequalities).
\paragraph{Zero restrictions.}
Zero restrictions impose linear equality constraints that fix selected elements of $\boldsymbol{\theta}$ at zero. Let $r_E$ denote the number of such restrictions, let $\mathbf{R}_{E}$ be an $r_E\times d$ selection matrix, and let $\mathbf{d}_E$ be an $r_E\times1$ vector. The restrictions are written as
\begin{equation*}
\mathbf{R}_{E}\boldsymbol{\theta}=\mathbf{d}_E,
\qquad
\mathbf{d}_E=\mathbf{0}.
\end{equation*}
This corresponds to the special case $\mathbf{d}_i=\mathbf{0}$ of the column-specific equality restriction
$\mathbf{R}^{E}_i\mathbf{b}_i=\mathbf{d}_i$ considered in \textcite{hou2024large}. Such exclusion restrictions rule out specific contemporaneous relationships, for example by setting the contemporaneous effect of one unit on another to zero.
\paragraph{Sign restrictions.}
Sign restrictions impose one-sided linear inequalities on selected elements or linear combinations of $\boldsymbol{\theta}$. Collecting the $s$ individual restrictions into a vector-valued function, we write \parencite{read2025fast}
\begin{equation*}
\mathbf{S}(\boldsymbol{\theta})\geq \mathbf{0}_{s\times1},
\qquad
\mathbf{S}(\boldsymbol{\theta})
=
\big(
S^{(1)}(\boldsymbol{\theta}),
\ldots,
S^{(s)}(\boldsymbol{\theta})
\big)' .
\end{equation*}
This is a special case of the general inequality restriction
\[
\mathbf{l}<\mathbf{R}_{I}\boldsymbol{\theta}<\mathbf{u},
\]
where one of the bounds is set to $\pm\infty$ \parencite{hou2024large}. Typical applications include restricting the direction of a contemporaneous response to a structural shock or imposing a sign restriction on cross-unit spillover effects.
\paragraph{Magnitude restrictions.}
Magnitude restrictions constrain the size of linear combinations of structural parameters through two-sided inequalities:
\begin{equation*}
\mathbf{l}<\mathbf{R}_{I}\boldsymbol{\theta}<\mathbf{u},
\end{equation*}
where $\mathbf{R}_{I}$ is an $r_I\times d$ matrix and $\mathbf{l}$ and $\mathbf{u}$ are $r_I\times1$ vectors of lower and upper bounds, respectively. These restrictions can be used to bound the magnitude of contemporaneous effects or to impose relative magnitude relationships among structural parameters. A common example is a dominance restriction, which requires a particular structural coefficient to be larger in absolute value than competing coefficients, such as
\[
|b_c^{j_2j_2}|\geq |b_c^{j_1j_2}|,
\qquad
\forall j_1\neq j_2.
\]
Although this condition appears nonlinear because of the absolute values, it can be interpreted as a set of linear inequalities conditional on the sign of the relevant coefficients. Consequently, the orientation of the inequalities depends on the current parameter draw, implying that the corresponding bounds must be updated at each iteration. In practice, such restrictions are typically imposed through an accept--reject step, retaining only normalized draws that satisfy the dominance condition.
Collecting the three classes of restrictions into
\[
\mathcal{S}
=
\{\mathbf{R}_{E},\,\mathbf{S},\,(\mathbf{R}_{I},\mathbf{l},\mathbf{u})\},
\]
the identified set is given by \parencite[cf.][]{arias_rubio_ramirez_waggoner_2018,read2025fast}
\[
\mathcal{D}(\boldsymbol{\theta}\mid\mathcal{S})
:=
\left\{
\boldsymbol{\theta}\in\mathbb{R}^{d}:
\mathbf{R}_{E}\boldsymbol{\theta}=\mathbf{0},
\;
\mathbf{S}(\boldsymbol{\theta})\geq\mathbf{0},
\;
\mathbf{l}<\mathbf{R}_{I}\boldsymbol{\theta}<\mathbf{u}
\right\}.
\]
This set is the analogue, in the present $B$-type parameterization, of the rotation-based identified set
\[
\mathcal{Q}(\boldsymbol{\phi}\mid S)
=
\{
\mathbf{Q}\in\mathcal{O}(r):
S(\boldsymbol{\phi},\mathbf{Q})\geq\mathbf{0}
\}
\]
defined in \textcite{read2025fast}. Because the imposed restrictions generally identify a set rather than a unique structural parameter vector, all elements of $\mathcal{D}(\boldsymbol{\theta}\mid\mathcal{S})$ are observationally equivalent: they generate the same likelihood value and cannot be distinguished using the sample information alone.
\subsubsection{Soft Parameter Restrictions}
\label{sec:soft_sign}
We follow \textcite{baumeister2015sign} by placing the prior directly on the structural parameters $\boldsymbol{\theta}$, so that the identifying information is stated on economically meaningful objects — contemporaneous effects and spillovers, rather than on rotations. The conditional posterior is
\[
p(\boldsymbol{\theta} \mid \mathbf{Y}, \mathbf{A}, \mathbf{B}) \propto \mathcal{L}(\mathbf{Y} \mid \boldsymbol{\theta}, \mathbf{A}, \mathbf{B})\, p(\boldsymbol{\theta}),
\]
with $\mathcal{L}(\cdot)$ the matrix-Gaussian likelihood and $p(\boldsymbol{\theta})$ the prior. A \emph{soft} sign restriction on $\theta_i$ is a continuous prior that favours, but does not impose, a sign: a Gaussian $p(\theta_i) = \mathcal{N}(\theta_i;\, \mu_{\theta_i}, \sigma_{\theta_i}^2)$ with $\mu_{\theta_i} > 0$ and $\sigma_{\theta_i}^2$ chosen so that $P(\theta_i > 0)$ is high (say $0.95$) places most, but not all, of the prior mass on the restricted region. If the likelihood contradicts the restriction sharply enough, the posterior still assigns probability to sign violations, so identification operates through probabilistic weighting of structural models rather than through deterministic exclusion. The interpretation of this approach is closely related to the smooth regularisation approach by \textcite{read2025fast}. \\
\subsubsection{Normalisation Restrictions}
The Kronecker product is unique only up to multiplication by a non-zero scalar $\gamma$. The likelihood function is unchanged if, for $\ell = 1,\ldots,p$, the pair $
(\mathbf{A}_\ell,\mathbf{B}_\ell)
$ is replaced by $
(\gamma\, \mathbf{A}_\ell,\gamma^{-1} \, \mathbf{B}_\ell)
$, and likewise if $
(\mathbf{B}_c,\mathbf{B}_r)
$ is replaced by $
(\gamma \, \mathbf{B}_c,\gamma^{-1}\, \mathbf{B}_r)
$.
We follow \textcite{chan2025large} and remove this indeterminacy by fixing the $(1,1)$ element of $\mathbf{B}_\ell$ at one for $\ell = 1, \ldots, p$, and, for the structural matrices, the $(1,1)$ element of $\mathbf{B}_c$ at one. Note that this normalisation fixes the scale and changes the how the prior mean for this element is set.
\subsection{Bayesian Estimation}\label{sec:Estimation}
In this section we develop our Gibbs-Sampler to draw from the joint posterior distribution.
\subsubsection{Prior Specification}
Before deriving approaches to obtain draws from conditional posterior distributions, we describe the prior distributions. To simplify computations we follow \textcite{chan2025large} and assume natural conjugate prior on the pairs $\mathbf{A}$ and $\mathbf{B}$. For the unrestricted elements in $\boldsymbol{B}_r$ and $\boldsymbol{B}_c$ we assume independent Gaussian distributions.
\paragraph{Prior information on $\mathbf{A}$}
We assume a Gaussian prior on $\mathbf{A}$. Namely, $p(\operatorname{vec}(\mathbf{A})\mid\boldsymbol{\Sigma}_r) = \mathcal{N}\!\big(\operatorname{vec}(\underline{\mathbf{A}}), \boldsymbol{\Sigma}_r \otimes \mathbf{V}_{\mathbf{A}}\big)$ with density function:
\begin{equation}
p(\mathbf{A}) \propto |\mathbf{V}_{\mathbf{A}}|^{-\frac{n}{2}} |\boldsymbol{\Sigma}_r|^{-\frac{np}{2}} \operatorname{e}^{-\frac{1}{2} \operatorname{tr}\left(\boldsymbol{\Sigma}_r^{-1} (\mathbf{A} - \underline{\mathbf{A}})' \mathbf{V}_\mathbf{A}^{-1} (\mathbf{A} - \underline{\mathbf{A}}) \right)},
\end{equation}
where $\underline{\mathbf{A}}$ is the prior mean, $\mathbf{V}_{\mathbf{A}}$ is the $np \times np$ prior covariance matrix, and $\operatorname{tr}(\cdot)$ denotes the trace operator. We specify both to encode Minnesota-type beliefs \parencite{litterman1986} adapted to the matrix setting by \textcite{chan2025large}. Because all series enter in growth rates, we shrink towards white noise and set $\underline{\mathbf{A}} = \mathbf{0}$. The matrix $\mathbf{V}_{\mathbf{A}}$ is diagonal and is the object through which the shrinkage hyperparameter $\kappa_{\mathbf{A}}$ acts. Throughout, $i \in \{1,\dots,n\}$ indexes the rows of $\mathbf{Y}_t$, that is the economic variables, and $j \in \{1,\dots,k\}$ indexes its columns, that is the countries. Recall that $\mathbf{A} = (\mathbf{A}_1, \ldots, \mathbf{A}_p)'$. Let $m_A=(\ell-1)n+i$ denote the row of $\mathbf{A}$ that carries the coefficients on row variable $i$ at lag $\ell \in \{1,\dots,p\}$. To fix the scale, let $\hat{s}^2_{i,j}$ be the residual variance of an AR(4) regression fitted to the single series $y_{t,ij}$, and let $\hat{s}^2_{i,\bullet} \;=\; \frac{1}{k}\sum_{j=1}^{k} \hat{s}^2_{i,j}$
denote its average across the $k$ countries in row $i$. The corresponding diagonal element of $\mathbf{V}_{\mathbf{A}}$ is then
\begin{equation}
v_{\mathbf{A},m_A m_A} \;=\; \frac{\kappa_{\mathbf{A}}}{\ell^{2}\, \hat{s}^{2}_{i,\bullet}},
\qquad m_A = (\ell-1)n + i.
\label{eq:VA}
\end{equation}
The prior variance thus factors into three pieces. The term $\ell^{-2}$ shrinks higher-order lags towards zero more aggressively, reflecting the belief that distant lags carry little information. The term $\hat{s}^{-2}_{i,\bullet}$ makes the prior invariant to the units in which each row of $\mathbf{Y}_t$ is measured. Finally, $\kappa_{\mathbf{A}}$ scales the whole matrix and therefore governs the \emph{overall} shrinkage strength: as $\kappa_{\mathbf{A}} \to 0$ the prior collapses onto $\underline{\mathbf{A}}$, while a large $\kappa_{\mathbf{A}}$ leaves $\mathbf{A}$ essentially unrestricted. Equivalently, $\mathbf{V}_{\mathbf{A}} = \kappa_{\mathbf{A}} \mathbf{C}_{\mathbf{A}}$, where $\mathbf{C}_{\mathbf{A}}$ is a fixed diagonal matrix determined by the data and the lag structure alone.\\
Rather than fixing $\kappa_{\mathbf{A}}$ at a conventional value, we treat it as unknown and let the data determine its value, following \textcite{giannone_lenza_primiceri_2015}. We place a hierarchical gamma prior $\kappa_{\mathbf{A}} \sim \mathcal{G}(c_{\mathbf{A},1}, c_{\mathbf{A},2})$ and we follow \textcite{chan2025large} by setting $c_{\mathbf{A},1} = c_{\mathbf{A},2} = 5$, a weakly informative choice centred at one.
\paragraph{Prior information for $\mathbf{B}$ }
The prior on $\mathbf{B}$ is close in spirit, the density function is given by:
\begin{equation}
p(\mathbf{B}) \propto |\mathbf{V}_{\mathbf{B}}|^{-\frac{k}{2}} |\boldsymbol{\Sigma}_c|^{-\frac{kp}{2}} \operatorname{e}^{-\frac{1}{2} \operatorname{tr}\left(\boldsymbol{\Sigma}_c^{-1} (\mathbf{B} - \underline{\mathbf{B}})' \mathbf{V}_\mathbf{B}^{-1} (\mathbf{B} - \underline{\mathbf{B}}) \right)}.
\end{equation}
We set the prior mean $\underline{\mathbf{B}} = (\mathbf{I}_k, \ldots, \mathbf{I}_k)'$ to represent no-spillovers and country-homogeneous beliefs on $\mathbf{B}$. The matrix $\mathbf{V}_{\mathbf{B}}$ is built symmetrically to \eqref{eq:VA}: let $m_B=(\ell-1)k+j$ denote the row of $\mathbf{B}$ that carries the coefficients on column country $j$ at lag $\ell$, and let $\hat{s}^{2}_{\bullet,j} = n^{-1}\sum_{i=1}^{n} \hat{s}^{2}_{i,j}$ denote the average AR(4) residual variance down column $j$. We set
\[
v_{\mathbf{B},m_B m_B} \;=\; \frac{\kappa_{\mathbf{B}}}{\ell^{2}\, \hat{s}^{2}_{\bullet,j}},
\qquad m_B = (\ell-1)k + j ,
\]
so that $\kappa_{\mathbf{B}}$ determines how informative the prior is: a small $\kappa_{\mathbf{B}}$ pins $\mathbf{B}_\ell$ at the identity, a large one results in an uninformative prior, and again $\mathbf{V}_{\mathbf{B}} = \kappa_{\mathbf{B}} \mathbf{C}_{\mathbf{B}}$ with $\mathbf{C}_{\mathbf{B}}$ fixed and diagonal. Again we follow \textcite{chan2025large} by using a hierarchical gamma prior $\kappa_{\mathbf{B}} \sim \mathcal{G}(c_{\mathbf{B},1}, c_{\mathbf{B},2})$ with $c_{\mathbf{B},1} = c_{\mathbf{B},2} = 5$ and estimate it from the data.
\paragraph{Priors on $\mathbf{B}_r$ and $\mathbf{B}_c$}
We specify \emph{Gaussian} priors on the free elements of $\mathbf{B}_r$ and $\mathbf{B}_c$,
\[
\theta_i \sim \mathcal{N}\!\left(\mu_{\theta_i},\, \sigma^2_{\theta_i}\right), \qquad \theta_i \in \boldsymbol{\theta},
\]
with $\boldsymbol{\mu}_{\theta}$ and the diagonal $\boldsymbol{\Sigma}_{\theta}$ collecting the prior means and variances discussed in Section \ref{sec:soft_sign}. The prior on the covariance blocks is therefore \emph{induced} by the prior on the structural parameters rather than elicited separately, which is what allows the identifying restrictions to be stated on economically interpretable objects.
\subsubsection{Conditional posteriors}
\paragraph{Sampling $\mathbf{A}$ and $\mathbf{B}$.}
Conditional on $\mathbf{B}_r$ and $\mathbf{B}_c$, the two covariance matrices are known, and the coefficient blocks retain their conjugate form. The conditional posteriors for $\mathbf{A}$ and $\mathbf{B}$ are
\begin{equation}
\operatorname{p}\big(\operatorname{vec}(\mathbf{A}) \mid \boldsymbol{\Sigma}_r, \mathbf{B}, \boldsymbol{\Sigma}_c, \kappa_{\mathbf{A}}, \kappa_{\mathbf{B}}\big) \sim \mathcal{N}\!\big(\operatorname{vec}(\widehat{\mathbf{A}}), \boldsymbol{\Sigma}_r \otimes \mathbf{K}_{\mathbf{A}}^{-1}\big),
\end{equation}
with
\begin{equation*}
\mathbf{K}_\mathbf{A}
= \mathbf{V}_\mathbf{A}^{-1}
+ \sum_{t=1}^T\mathbf{X}_t \mathbf{B} \mathbf{\Sigma}_c^{-1} \mathbf{B}' \mathbf{X}_t',
\qquad
\widehat{\mathbf{A}}
= \mathbf{K}_A^{-1} \left( \mathbf{V}_A^{-1} \underline{\mathbf{A}}
+ \sum_{t=1}^T \, \mathbf{X}_t \mathbf{B} \mathbf{\Sigma}_c^{-1} \mathbf{Y}_t' \right).
\end{equation*}
Similarly, the conditional posterior of $\mathbf{B}$ is given by
\begin{equation}
\operatorname{p}\big(\operatorname{vec}(\mathbf{B}) \mid \boldsymbol{\Sigma}_c, \mathbf{A}, \boldsymbol{\Sigma}_r, \kappa_{\mathbf{A}}, \kappa_{\mathbf{B}}\big) \sim \mathcal{N}\!\big(\operatorname{vec}(\widehat{\mathbf{B}}), \boldsymbol{\Sigma}_c \otimes \mathbf{K}_{\mathbf{B}}^{-1}\big),
\end{equation}
where
\begin{equation*}
\mathbf{K}_\mathbf{B}
= \mathbf{V}_\mathbf{B}^{-1}
+ \sum_{t=1}^T \, \mathbf{X}_t' \mathbf{A} \mathbf{\Sigma}_r^{-1} \mathbf{A}' \mathbf{X}_t,
\qquad
\widehat{\mathbf{B}}
= \mathbf{K}_\mathbf{B}^{-1} \left( \mathbf{V}_\mathbf{B}^{-1} \underline{\mathbf{B}}
+ \sum_{t=1}^T \, \mathbf{X}_t' \mathbf{A} \mathbf{\Sigma}_r^{-1} \mathbf{Y}_t \right).
\end{equation*}
One of the key advantages of this approach is its computational efficiency. In particular, drawing the autoregressive component of the model costs $\mathcal{O}(n^3p^3)$ for $\mathbf{A}$ and $\mathcal{O}(k^3p^3)$ for $\mathbf{B}$, rather than the $\mathcal{O}\big((nk)^3p^3\big)$ required by the corresponding unrestricted VAR; see \textcite{chan2025large} for a detailed discussion. For the normalisation restrictions we implement the algorithm by \textcite{nobile2000comment}.
\paragraph{Sampling $\kappa_{\mathbf{A}}$ and $\kappa_{\mathbf{B}}$.}
Because the shrinkage parameters enter the prior variances multiplicatively through $\mathbf{V}_{\mathbf{A}} = \kappa_{\mathbf{A}}\mathbf{C}_{\mathbf{A}}$ and $\mathbf{V}_{\mathbf{B}} = \kappa_{\mathbf{B}}\mathbf{C}_{\mathbf{B}}$, combining the gamma priors with the coefficient priors yields generalised inverse Gaussian (GIG) conditional posteriors,
\begin{align*}
\kappa_{\mathbf{A}} \mid \mathbf{A}, \boldsymbol{\Sigma}_r
&\sim
\mathcal{GIG}\!\left(
c_{\mathbf{A},1}-\tfrac{n^{2}p}{2},\;
2c_{\mathbf{A},2},\;
\operatorname{tr}\!\big(\boldsymbol{\Sigma}_r^{-1}(\mathbf{A}-\underline{\mathbf{A}})'\mathbf{C}_{\mathbf{A}}^{-1}(\mathbf{A}-\underline{\mathbf{A}})\big)
\right), \\
\kappa_{\mathbf{B}} \mid \mathbf{B}, \boldsymbol{\Sigma}_c
&\sim
\mathcal{GIG}\!\left(
c_{\mathbf{B},1}-\tfrac{k^{2}p}{2},\;
2c_{\mathbf{B},2},\;
\operatorname{tr}\!\big(\boldsymbol{\Sigma}_c^{-1}(\mathbf{B}-\underline{\mathbf{B}})'\mathbf{C}_{\mathbf{B}}^{-1}(\mathbf{B}-\underline{\mathbf{B}})\big)
\right),
\end{align*}
where $\mathcal{GIG}(\lambda,a,b)$ denotes the density proportional to $x^{\lambda-1}\operatorname{e}^{-(ax+b/x)/2}$ on $x>0$; we draw from it with the algorithm of \textcite{devroye2014random}. The prior covariance matrices are then rebuilt from the new draws at each sweep.
\paragraph{Sampling $\mathbf{B}_r$ and $\mathbf{B}_c$.}
The structural factors do not inherit the conjugacy of the coefficient blocks. Conditional on $(\mathbf{A}, \mathbf{B})$ the reduced-form innovations are known,
\[
\mathbf{U}_t(\mathbf{A},\mathbf{B}) \;=\; \mathbf{Y}_t - \sum_{\ell=1}^{p} \mathbf{A}_\ell \mathbf{Y}_{t-\ell} \mathbf{B}_\ell' ,
\qquad
\mathbf{u}_t \;=\; \operatorname{vec}\big(\mathbf{U}_t\big) \;=\; \mathbf{B}_0\, \mathbf{e}_t,
\qquad
\mathbf{B}_0 = \mathbf{B}_c \otimes \mathbf{B}_r ,
\]
with $\mathbf{e}_t = \operatorname{vec}(\mathbf{E}_t) \sim \mathcal{N}(\mathbf{0}, \mathbf{I}_{nk})$. Gathering the free elements of $\mathbf{B}_r$ and $\mathbf{B}_c$ in the vector $\boldsymbol{\theta}$, the conditional likelihood is
\begin{equation}
\log \mathcal{L}(\mathbf{Y} \mid \boldsymbol{\theta}, \mathbf{A}, \mathbf{B})
= \text{const}
- T\Big( k \log\big|\det \mathbf{B}_r\big| + n \log\big|\det \mathbf{B}_c\big| \Big)
- \frac{1}{2} \sum_{t=1}^{T} \operatorname{tr}\!\left( \boldsymbol{\Sigma}_r^{-1} \mathbf{U}_t \boldsymbol{\Sigma}_c^{-1} \mathbf{U}_t' \right),
\label{eq:lik_theta}
\end{equation}
where the log-determinant term uses $\log|\det \mathbf{B}_0| = k \log|\det \mathbf{B}_r| + n \log|\det \mathbf{B}_c|$, the Jacobian of the map from shocks to innovations. Combining \eqref{eq:lik_theta} with the Gaussian prior $\boldsymbol{\theta} \sim \mathcal{N}(\boldsymbol{\mu}_{\theta}, \boldsymbol{\Sigma}_{\theta})$ of the previous subsection and the identified set $\mathcal{D}(\boldsymbol{\theta}\mid\mathcal{S})$ gives the conditional posterior
\begin{equation}
p(\boldsymbol{\theta} \mid \mathbf{Y}, \mathbf{A}, \mathbf{B})
\;\propto\;
\mathcal{L}(\mathbf{Y} \mid \boldsymbol{\theta}, \mathbf{A}, \mathbf{B})\;
p(\boldsymbol{\theta} \mid \boldsymbol{\mu}_{\theta}, \boldsymbol{\Sigma}_{\theta})\;
\mathbf{1}\left\{ \boldsymbol{\theta} \in \mathcal{D}(\boldsymbol{\theta}\mid\mathcal{S}) \right\}.
\label{eq:post_theta}
\end{equation}
Because $\boldsymbol{\theta}$ enters the likelihood through both the determinant and the inverse of $\mathbf{B}_0$, \eqref{eq:post_theta} belongs to no known family: there is no closed-form draw and no conjugate update. We sample it with an elliptical slice sampler, which we describe next.
\subsubsection{Elliptical slice sampling for $\boldsymbol{\theta}$}
\label{sec:ess}
To sample from the conditional posterior \eqref{eq:post_theta} of the structural parameters we implement the elliptical slice sampling algorithm \parencite{murray2010elliptical}, following the approach of \textcite{read2025fast} for soft-sign-restricted SVARs. The method is tailored to targets of exactly this shape — a Gaussian prior $\boldsymbol{\theta} \sim \mathcal{N}(\boldsymbol{\mu}_{\theta},\boldsymbol{\Sigma}_{\theta})$ multiplied by an arbitrary likelihood — and it requires neither a Metropolis accept-reject step nor any stepsize tuning.\footnote{ We have also experimented with the DREAM algorithm by \textcite{Vrugt2016_DREAM}, an efficient Metropolis--Hastings scheme that, unlike the elliptical slice sampler, accommodates arbitrary (non-Gaussian) priors on the structural parameters. The algorithm is well suited to high-dimensional targets and reduces the correlation in the draws; the cost is that it requires running multiple chains and a larger number of simulations. }
\paragraph{The proposal.} Let $\boldsymbol{\theta}^{(m)}$ denote the current state. The sampler draws an auxiliary direction from the prior, $\boldsymbol{\nu}\sim \mathcal{N}(\mathbf{0},\boldsymbol{\Sigma}_{\theta})$, and moves along the ellipse through $\boldsymbol{\theta}^{(m)}$ and $\boldsymbol{\nu}$:
\begin{equation}
\boldsymbol{\theta}^{\ast}(\alpha)
=
\boldsymbol{\mu}_{\theta} + \big(\boldsymbol{\theta}^{(m)}-\boldsymbol{\mu}_{\theta}\big)\cos\alpha + \boldsymbol{\nu}\sin\alpha,
\qquad \alpha \in [0,2\pi).
\label{eq:ess_proposal}
\end{equation}
The construction is such that $\boldsymbol{\theta}^{\ast}(\alpha)$ is marginally distributed as $\mathcal{N}(\boldsymbol{\mu}_{\theta},\boldsymbol{\Sigma}_{\theta})$ for \emph{every} angle: the prior is invariant along the ellipse. Two special cases are worth noting. At $\alpha = 0$ we recover the current state, $\boldsymbol{\theta}^{\ast}(0) = \boldsymbol{\theta}^{(m)}$; at $\alpha = \pi/2$ we obtain an independent prior draw, $\boldsymbol{\theta}^{\ast}(\pi/2) = \boldsymbol{\mu}_{\theta} + \boldsymbol{\nu}$. The angle therefore interpolates between a null move and a bold, prior-sized move, and it is the only quantity that has to be chosen.
\paragraph{The slice.} Because the prior is constant along \eqref{eq:ess_proposal}, it cancels from the acceptance condition and the slice
threshold depends only on the likelihood:
\begin{equation}
\log y
=
\log \mathcal{L}(\mathbf{Y}|\boldsymbol{\theta}^{(m)},\mathbf{A},\mathbf{B}) + \log w,
\qquad w\sim\mathrm{Unif}(0,1).
\label{eq:ess_slice}
\end{equation}
A proposal is accepted whenever it lies above this threshold. The angle is located by a shrinking bracket rather than by tuning:
\begin{enumerate}
\item Draw $\alpha \sim \mathrm{Unif}[0,2\pi)$ and initialise the bracket $[\alpha_{\min}, \alpha_{\max}] = [\alpha - 2\pi,\; \alpha]$.
\item Form $\boldsymbol{\theta}^{\ast}(\alpha)$ from \eqref{eq:ess_proposal} and evaluate the likelihood \eqref{eq:lik_theta}.
\item If $\log \mathcal{L}(\mathbf{Y}\mid\boldsymbol{\theta}^{\ast}(\alpha),\mathbf{A},\mathbf{B}) \ge \log y$ and $\boldsymbol{\theta}^{\ast}(\alpha)$ is feasible, accept: set $\boldsymbol{\theta}^{(m+1)} = \boldsymbol{\theta}^{\ast}(\alpha)$ and stop.
\item Otherwise shrink the bracket towards the current state, setting $\alpha_{\min} = \alpha$ if $\alpha < 0$ and $\alpha_{\max} = \alpha$ otherwise, draw a new $\alpha \sim \mathrm{Unif}(\alpha_{\min}, \alpha_{\max})$, and return to step 2.
\end{enumerate}
Since the bracket always contains $\alpha = 0$, at which the proposal coincides with the current state -- feasible and above the threshold by construction -- the loop is guaranteed to terminate. The update is therefore rejection-free in the sense that every sweep returns a draw, and it needs no stepsize: the bracket adapts itself. In practice we work in the whitened coordinates $\mathbf{z} = \mathbf{D}^{-1}(\boldsymbol{\theta}-\boldsymbol{\mu}_{\theta})$, with $\mathbf{D} = \boldsymbol{\Sigma}_{\theta}^{1/2}$ diagonal, so that $\boldsymbol{\nu}$ is a standard normal draw; this is equivalent to \eqref{eq:ess_proposal} and numerically more stable. The scheme scales well with the dimension of $\boldsymbol{\theta}$, which matters here because the structural block grows with both the number of countries and the number of variables.
\paragraph{Enforcing the identified set.}
The three classes of restrictions defining $\mathcal{D}(\boldsymbol{\theta}\mid\mathcal{S})$ are enforced at different stages of the sampling update. Zero (equality) and normalization restrictions are imposed by construction: the slice sampler operates only on the unrestricted elements of $\boldsymbol{\theta}$, while the constrained elements are reinstated when reconstructing the structural matrices. This is equivalent to sampling in the null space of the equality constraint matrix $\mathbf{R}_{E}$. Sign restrictions are incorporated into the feasibility check in Step~3. Consequently, any proposal that violates a sign restriction is treated in the same manner as one falling below the slice level, causing the sampling bracket to shrink until a feasible proposal is found. Finally, magnitude (two-sided inequality) restrictions are enforced through an accept--reject step applied to the normalized draw. A proposal is retained only if all inequality constraints are satisfied; otherwise, the previous draw is preserved\footnote{One can also impose the magnitude restrictions in the ESS step. In our implementation, we only implement sign restrictions in the feasibility check of Step~3 given that the number of strict sign restrictions is smaller, leading to fewer tries of the ESS sampler. Rejections at this step are rare; the sampler records the magnitude acceptance rate for every run.}. Together, these steps ensure that every retained draw belongs to the identified set $\mathcal{D}(\boldsymbol{\theta}\mid\mathcal{S})$.
\subsubsection{Summary of the Gibbs-Sampler}
\label{sec:gibbs}
Collecting the results above, the posterior is explored by cycling through four blocks, each conditioning on the most recent draw of all the others:
\begin{enumerate}
\item $p\big(\mathbf{A} \mid \mathbf{Y}, \mathbf{B}, \mathbf{B}_r, \mathbf{B}_c, \kappa_{\mathbf{A}}\big)$
\item $p\big(\mathbf{B} \mid \mathbf{Y}, \mathbf{A}, \mathbf{B}_r, \mathbf{B}_c, \kappa_{\mathbf{B}}\big)$
\item $p\big(\mathbf{B}_r, \mathbf{B}_c \mid \mathbf{Y}, \mathbf{A}, \mathbf{B}\big)$
\item $p\big(\kappa_{\mathbf{A}} \mid \mathbf{A}, \mathbf{B}_r\big)$ and $p\big(\kappa_{\mathbf{B}} \mid \mathbf{B}, \mathbf{B}_c\big)$
\end{enumerate}
Only the third block is non-standard; the other three are available in closed form. Algorithm \ref{alg:mcmc_sampler} states the full sweep.
\begin{algorithm}[t]
\caption{MCMC Sampler for the BSMAR Model}
\label{alg:mcmc_sampler}
\begin{algorithmic}[1]
\State \textbf{Initialize} $\mathbf{A}^{(0)}, \mathbf{B}^{(0)}, \kappa_{\mathbf{A}}^{(0)}, \kappa_{\mathbf{B}}^{(0)}, \boldsymbol{\theta}^{(0)}$
\State \textbf{Compute} initial structural matrices $\mathbf{B}_r^{(0)}, \mathbf{B}_c^{(0)}$ from $\boldsymbol{\theta}^{(0)}$ and covariance components $\boldsymbol{\Sigma}_r^{(0)} = \mathbf{B}_r^{(0)} {\mathbf{B}_r^{(0)}}', \boldsymbol{\Sigma}_c^{(0)} = \mathbf{B}_c^{(0)} {\mathbf{B}_c^{(0)}}'$
\For{$m = 1$ \textbf{to} $M$}
\State \textbf{Step 1: Autoregressive Dynamics}
\State Compute $\mathbf{K}_{\mathbf{A}}$ and $\widehat{\mathbf{A}}$ using $\mathbf{B}^{(m-1)}$ and $\boldsymbol{\Sigma}_c^{(m-1)}$
\State Draw $\operatorname{vec}(\mathbf{A}^{(m)}) \sim \mathcal{N}(\operatorname{vec}(\widehat{\mathbf{A}}), \boldsymbol{\Sigma}_r^{(m-1)} \otimes \mathbf{K}_{\mathbf{A}}^{-1})$
\State \textbf{Step 2: Cross-Sectional Dependencies}
\State Compute $\mathbf{K}_{\mathbf{B}}$ and $\widehat{\mathbf{B}}$ using $\mathbf{A}^{(m)}$ and $\boldsymbol{\Sigma}_r^{(m-1)}$
\State Draw $\operatorname{vec}(\mathbf{B}^{(m)}) \sim \mathcal{N}(\operatorname{vec}(\widehat{\mathbf{B}}), \boldsymbol{\Sigma}_c^{(m-1)} \otimes \mathbf{K}_{\mathbf{B}}^{-1})$
\State \textbf{Step 3: Structural Parameters (Elliptical Slice Sampling)}
\State Draw candidate auxiliary direction $\boldsymbol{\nu} \sim \mathcal{N}(0, \boldsymbol{\Sigma}_{\theta})$
\State Generate slice log-threshold $\log y = \log \mathcal{L}(\mathbf{Y} \mid \boldsymbol{\theta}^{(m-1)}, \mathbf{A}^{(m)}, \mathbf{B}^{(m)}) + \log \omega$, $\omega \sim \mathcal{U}(0,1)$
\State \textbf{Repeat} until acceptable $\alpha^\star$ is found:
\State \hspace{1cm} Propose $\boldsymbol{\theta}^{\ast}(\alpha) = \boldsymbol{\mu}_{\theta} + (\boldsymbol{\theta}^{(m-1)}-\boldsymbol{\mu}_{\theta})\cos\alpha + \boldsymbol{\nu}\sin\alpha$ \Comment{free elements only; zero-restricted elements reinstated}
\State \hspace{1cm} Evaluate log-likelihood $\log \mathcal{L}^{\ast} = \log \mathcal{L}(\mathbf{Y} \mid \boldsymbol{\theta}^{\ast}(\alpha), \mathbf{A}^{(m)}, \mathbf{B}^{(m)})$
\State \hspace{1cm} Shrink bracket $[\alpha_{\min}, \alpha_{\max}]$ if $\log \mathcal{L}^{\ast} < \log y$ or a sign restriction is violated
\State Set $\boldsymbol{\theta}^{(m)} = \boldsymbol{\theta}^{\ast}(\alpha^\star)$
\State Reshape $\boldsymbol{\theta}^{(m)}$ to obtain new $\mathbf{B}_r^{(m)}$ and $\mathbf{B}_c^{(m)}$ (after normalization)
\State Accept $\boldsymbol{\theta}^{(m)}$ only if the magnitude restrictions hold; otherwise set $\boldsymbol{\theta}^{(m)} = \boldsymbol{\theta}^{(m-1)}$
\State Update $\boldsymbol{\Sigma}_r^{(m)} = \mathbf{B}_r^{(m)} {\mathbf{B}_r^{(m)}}'$ and $\boldsymbol{\Sigma}_c^{(m)} = \mathbf{B}_c^{(m)} {\mathbf{B}_c^{(m)}}'$
\State \textbf{Step 4: Shrinkage Parameters}
\State Draw $\kappa_{\mathbf{A}}^{(m)}$ and $\kappa_{\mathbf{B}}^{(m)}$ from their $\mathcal{GIG}$ conditional posteriors
\State Update prior covariance matrices $\mathbf{V}_{\mathbf{A}}$ and $\mathbf{V}_{\mathbf{B}}$
\EndFor
\end{algorithmic}
\end{algorithm}
\subsection{Illustration of the Sampler}
In this subsection, we illustrate the behaviour of the sampler on a single simulated DGP that deliberately misspecifies part of the identifying restrictions, so that we can examine how the method responds when a sign restriction is imposed incorrectly. The settings match the theory of supply/demand shocks from the next section. Assume the model in equation \eqref{eq:BSMAR} with $n=2$, $k=3$ and $p=2$. We simulate a sample of $T = 230$ observations. We generate the coefficient dynamics $\mathbf{A}_\ell$ and $\mathbf{B}_\ell$ for $\ell = 1, \ldots, p$ following:
\begin{itemize}
\item For the first lag $\mathbf{A}_1$: diagonal elements are drawn from $\mathcal{U}(0.5, 0.9)$ to induce persistence, while off-diagonal elements are drawn from $\mathcal{U}(-0.5, 0.5)$.
\item For higher lags $\ell = 2, \ldots, p$: all elements of $\mathbf{A}_\ell$ are drawn from $\mathcal{N}(0, 0.05^2)$, reflecting the prior belief that higher-order dynamics are less important.
\item For all lags $\ell = 1, \ldots, p$: the cross-sectional matrices $\mathbf{B}_\ell$ have diagonal elements drawn from $\mathcal{U}(0.8, 1)$ and off-diagonal elements from $\mathcal{U}(0.15, 0.2)$.
\end{itemize}
To ensure stationarity of the model, the spectral radius of the full $nkp \times nkp$ companion matrix of the vectorized VAR must be less than one \parencite{samadi2025matrix}. In practice, since our DGP sets $p=2$ with small higher-lag coefficients, we use the first-lag approximation $\rho = \max|\lambda(\mathbf{B}_1 \otimes \mathbf{A}_1)|$ as a conservative check. If $\rho > 1$, all coefficient matrices are rescaled by the factor $1/(1.1 \cdot \rho)$. For identification purposes, we impose the normalization $[\mathbf{B}_\ell]_{11} = 1$ for all lags $\ell$. Given the generated coefficient matrices and the structural impact matrices $\mathbf{B}_r$ and $\mathbf{B}_c$, we compute the true structural impulse response functions (SIRFs) as:
\[
\text{SIRF}_h = \mathbf{J} \boldsymbol{\Phi}^h \mathbf{J}' (\mathbf{B}_c \otimes \mathbf{B}_r), \quad h = 0, 1, \ldots, H,
\]
where $\boldsymbol{\Phi}$ is the companion matrix of the vectorized model, $\mathbf{J} = [\mathbf{I}_{nk} \; \mathbf{0}]$ is a selector matrix, and $H$ denotes the forecast horizon. These true SIRFs serve as the benchmark for evaluating the estimation performance. For the simulations we set:
\begin{align*}
\mathbf{B}_r &= \begin{bmatrix} 1 & 0.8 \\ -0.8 & 1 \end{bmatrix}, \\
\mathbf{B}_c &= \begin{bmatrix}
1 & 0.8 & 0.6 \\
-0.4 & 1 & 0.8 \\
-0.2 & -0.4 & 1
\end{bmatrix}.
\end{align*}
We impose the correct identifying restrictions on all elements of $\mathbf{B}_r$. For $\mathbf{B}_c$, by contrast, we deliberately impose the positive sign restriction $b_c^{j_1j_2}>0 \; \forall j_1,j_2$, even though the data-generating process places genuinely negative spillovers in the lower triangle ($b_c^{21}, b_c^{31}, b_c^{32} < 0$). These elements in $\mathbf{B}_c$ are therefore misspecified: a hard positive sign restriction would force the corresponding contemporaneous responses onto the wrong side of zero and distort the identified set. Instead, we encode the sign as \emph{soft} prior information: the Gaussian prior on $\boldsymbol{\theta}$ is centred on positive values but given enough variance that, where the likelihood strongly contradicts the imposed sign, the posterior can place mass on negative values. This setup lets us check whether the sampler recovers the true responses despite the misspecified restriction.
\begin{figure}[H]
\centering
\includegraphics[width=0.85\textwidth]{figures/simulation/identification_soft_sign.pdf}
\caption{\footnotesize Response of all variables to the structural shock $e_{(1,2)}$ under a misspecified positive sign restriction on $\mathbf{B}_c$, estimated with soft prior information. The posterior median (solid) and $68\%$ credible set (shaded) track the true SIRF (dashed) even for the responses governed by the negative, wrongly-restricted spillovers in the first column of $\mathbf{B}_c$.}
\end{figure}
\section{Studying High Dimensional International Spillovers}
\label{sec:empirical}
There has been renewed interest in understanding the drivers of recent inflation, particularly whether demand or supply shocks have played the dominant role. This question has been addressed using single-country SVAR models identified through sign restrictions \parencite[e.g.,][]{giannone_primiceri_2025, bergholt2026inflation, della_chang_jansen_pagliacci_2023}. Spillovers during the pandemic period have been analyzed theoretically by \textcite{di2023pandemic}.
We contribute to this literature by providing a multicountry perspective that allows for spillovers across countries. In this section, we apply our proposed framework to a panel of 15 economies. We identify country-specific supply and demand shocks and study their spillovers to other countries over time. We begin by describing the data, followed by discussing our identification strategy, and conclude by presenting our empirical findings.
\subsection{Data and Specification}
We use quarterly data for a panel of 15 OECD economies spanning North America, Europe, and the Asia--Pacific region. For each country, we observe real gross domestic product (GDP) and the consumer price index (CPI). The data are obtained from Macrobond. Two further economies, China and Norway, were considered but dropped because their series begin too late to cover the estimation sample. For the identification strategy described in the next section, the countries are partitioned into eight \emph{large} and seven \emph{small} economies (see Table~\ref{tab:oecd_countries}).
The estimation sample covers the period from 1998Q1 to 2019Q4. To construct the historical decompositions, we use an out-of-sample evaluation period from 2020Q1 to 2024Q4, encompassing both the COVID-19 pandemic and the subsequent global inflation episode. Prior to estimation, GDP and CPI are transformed into yearly growth rates and standardized. After the estimation we undo the standardization to facilitate interpretation. We estimate the model using $p=4$ quarterly lags. Details about the sampler are presented in the appendix.
\begin{table}[t]
\centering
\caption{Countries in the empirical application}
\label{tab:oecd_countries}
\begin{threeparttable}
\begin{tabular}{llll}
\toprule
\multicolumn{2}{c}{\textbf{Large economies}} & \multicolumn{2}{c}{\textbf{Small economies}} \\
\cmidrule(lr){1-2} \cmidrule(lr){3-4}
\textbf{ISO} & \textbf{Country} & \textbf{ISO} & \textbf{Country} \\
\midrule
USA & United States & KOR & South Korea \\
CAN & Canada & NLD & Netherlands \\
DEU & Germany & SWE & Sweden \\
FRA & France & FIN & Finland \\
GBR & United Kingdom & BEL & Belgium \\
ESP & Spain & AUS & Australia \\
ITA & Italy & PRT & Portugal \\
JPN & Japan & & \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\footnotesize
\item Quarterly real GDP and consumer price index data sourced from Macrobond, transformed to yearly growth rates. The United States is ordered first, as required by the contemporaneous exogeneity restriction; the ordering of the remaining economies does not affect the identifying restrictions. Estimation sample: 1998Q1 to 2019Q4.
\end{tablenotes}
\end{threeparttable}
\end{table}
\subsection{Identification of Supply and Demand in a Multi-Country Setup}
\label{sec:emp_id_restrictions}
We now turn to the identification of supply and demand shocks in the multicountry BSMAR model and present the full set of restrictions used in our baseline empirical specification. Recall that the contemporaneous structural impact matrix is given by the bilinear decomposition $\mathbf{B}_0 = \mathbf{B}_c \otimes \mathbf{B}_r$. With output and inflation as the variables in each country block, this structure allows us to identify country-specific supply and demand shocks while accounting for contemporaneous international spillovers.
Our identifying restrictions are summarized in Table~\ref{tab:emp_restrictions}. We motivate these assumptions as follows. First, the sign restrictions imposed on $\mathbf{B}_r$ follow the standard economic interpretation of supply and demand shocks (see, e.g., \cite{giannone_primiceri_2025}). Both shocks are signed as expansionary for economic activity: a positive demand shock raises activity and inflation, whereas a positive (favourable) supply shock raises activity while lowering inflation. The two shocks are therefore distinguished by the sign of the inflation response. We further impose positive diagonal elements in $\mathbf{B}_c$ to ensure that own-country effects are consistent with the expected direction of domestic shocks. For the off-diagonal elements, we assume that international spillovers are likely to operate in the same direction as domestic effects, which we implement using soft sign restrictions.
To separate country specific shocks from each other, we impose magnitude restrictions based on the assumption that country-specific shocks have their largest impact on the country in which they originate. The zero restrictions impose that shocks originating in smaller economies do not generate contemporaneous responses in larger economies. We relax these restrictions in Section~\ref{sec:altind} and examine the robustness of our results. Finally, given the prominent role of the US in the global economy, we assume in our baseline specification that the US is not contemporaneously affected by shocks originating in other countries. In Section~\ref{sec:altind}, we relax this assumption as well and assess the robustness of our findings.
\begin{table}[t]
\centering
\caption{Identifying restrictions in the baseline empirical specification, using the notation $b_c^{j_1j_2}$ of Section~\ref{sec:smar}. Rows of $\mathbf{B}_r$ are GDP and CPI, columns the supply and demand shocks.}
\label{tab:emp_restrictions}
\begin{tabular}{@{}p{2.3cm} p{2.0cm} p{4.6cm} p{4.4cm}@{}}
\toprule
\textbf{Family} & \textbf{Target} & \textbf{Restriction} & \textbf{Imposed via} \\
\midrule
Sign & $\mathbf{B}_r$ &
$\operatorname{sign}(\mathbf{B}_r) = \left(\begin{smallmatrix} + & + \\ - & + \end{smallmatrix}\right)$ &
Hard restriction; violating draws rejected in the elliptical slice sampler. \\
Sign (prior) & $\operatorname{diag}(\mathbf{B}_c)$ &
$b_c^{jj} > 0$ &
Positive own effects via a tight prior ($b_{c,0}^{jj}=1$, $v_{c,0}^{jj}=0.1$). \\
Soft sign & $\mathbf{B}_c$ off-diag &
$P(b_c^{j_1j_2}>0) = 0.95$ a priori &
Gaussian prior $b_{c,0}^{j_1j_2}=0.5$, $v_{c,0}^{j_1j_2}\approx0.0924$, so $0.5/\sqrt{v}=z_{0.95}$; data can override. \\
Magnitude & $\mathbf{B}_c$ columns &
$|b_c^{j_2j_2}| \ge |b_c^{j_1j_2}|\ \ \forall\, j_1 \neq j_2$ &
Accept--reject step on the normalized draw of $\mathbf{B}_c$. \\
Zero & $\mathbf{B}_c$ (small$\to$large) &
$b_c^{j_1j_2} = 0$ if $j_1$ large, $j_2$ small &
Exact equality $\mathbf{R}_E\boldsymbol{\theta}=\mathbf{0}$; only free elements sampled. \\
Separation & $\mathbf{B}_c$ first row &
$\mathbf{B}_c(1,:) = (1, 0, \dots, 0)$ &
Contemporaneous exogeneity of the U.S.\ ordered first, fixing the scale. \\
\bottomrule
\end{tabular}
\end{table}
Taken together, these restrictions define the identified set $\mathcal{D}(\boldsymbol{\theta}\mid\mathcal{S})$ of the baseline specification and allow us to recover, for each country, its supply and demand shocks as well as the cross-border spillovers between countries.
\subsection{Empirical Results}
We organize the discussion around four sets of objects: structural impulse responses, forecast error variance decompositions (FEVD), out-of-sample historical decompositions, and connectedness measures. Diagnostics are reported in the Appendix.
\paragraph{Structural impulse responses.}
Figure \ref{fig:sirf_cpi_usa} reports the structural impulse responses of GDP and CPI growth across the panel to the identified U.S.\ aggregate demand (AD) shock. The responses are homogeneous in sign and dynamics and respect the sign restrictions imposed through $\mathbf{B}_r$: the demand shock raises both activity and prices on impact. The domestic U.S.\ responses are the largest and the most precisely estimated, but the shock also spills over to the rest of the panel, raising activity and prices abroad. The responses are persistent over the first ten periods and revert to zero at longer horizons. The responses are similar in shape across countries, while their magnitude and posterior uncertainty differ. Given the dominant role of the US, almost all responses exclude zero from their posterior credible bands. For shocks originating in other countries, we find, as expected, wider posterior bands, since smaller economies play a lesser role in driving the system.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figures/Baseline/CPI_USA.pdf}
\caption{Structural impulse responses of GDP and CPI growth across the panel to the identified U.S.\ aggregate demand (AD) shock (baseline specification).}
\label{fig:sirf_cpi_usa}
\end{figure}
\paragraph{Out-of-sample historical decompositions.}
We follow \textcite{giannone_primiceri_2025} by calculating out-of-sample historical decompositions for the period 2020Q1--2024Q4 to answer the question if the recent increase in inflation was demand or supply driven. Figure \ref{fig:HD_oos_CPI_US_GBR} reports the decomposition of U.S.\ and U.K.\ CPI. The figure has four panels: the upper row splits each CPI series into domestic and foreign supply and demand shocks, while the lower row disaggregates the same series by region of origin. In the US, demand shocks clearly dominate the development of CPI during the pandemic. At the onset of the pandemic, domestic demand shocks were the key contributors to inflation; as it unfolded, foreign demand shocks became more prominent. The origin of the shocks is mostly domestic, with minor contributions from Western Europe, East Asia, and the Pacific. The UK, by contrast, exhibits a more varied pattern over the COVID period: the early phase is characterised by opposing domestic and foreign shocks, whereas later periods are dominated by demand forces acting in a common direction.
Placing the two economies side by side reinforces this contrast: U.S.\ inflation is overwhelmingly demand-driven and domestic in origin, whereas the U.K.\ shows a larger foreign contribution with more supply shocks playing a bigger role. Overall, in line with \textcite{giannone_primiceri_2025} these decompositions point to demand shocks as the main explanatory force behind recent inflation developments, with the US playing a central role in transmitting spillovers to the UK.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\textwidth]{figures/Baseline/HD_oos_CPI_compare_legend_supdem_nonUS.pdf}
\vspace{0.4em}
\includegraphics[width=\textwidth]{figures/Baseline/HD_oos_CPI_US_GBR.pdf}
\vspace{0.4em}
\includegraphics[width=0.55\textwidth]{figures/Baseline/HD_oos_CPI_compare_legend_regional.pdf}
\caption{Out-of-sample historical decomposition of CPI for the U.S.\ (right column) and the U.K.\ (left column), baseline specification. Top row: supply/demand decomposition; bottom row: regional decomposition. The supply/demand legend is shown above; the regional legend below.}
\label{fig:HD_oos_CPI_US_GBR}
\end{figure}
\paragraph{Forecast error variance decompositions.}
We present the FEVD in Figure \ref{fig:fevd_supdem}, which decomposes the shock contributions into demand and supply from domestic, foreign, and US sources at each forecast horizon $h=1,\ldots,24$. For most variables, domestic shocks are the major contributors to the forecast-error variance, with a domestic share above 50\% in most countries. Economies such as the US, Japan, and the UK contain local shocks with stronger explanatory power. Focusing on US shocks alone, their contribution to the rest of the world economy is small, with US AD shocks accounting for the larger part of it; a few economies---Canada, Germany, and the UK---load more heavily on these shocks, as expected given that they are major trading partners. In contrast to the out-of-sample historical decompositions, we now see that supply shocks also play a significant role in the behaviour of the system: at least 20\% of the FEVD in every country is explained by supply forces, with varying domestic and foreign contributions. Altogether we see that demand forces have the larger explanatory power and that the domestic shocks still account for large share of the FEVD explanation. The large role of aggregate demand spillovers confirms their importance as a channel of international transmission.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figures/Baseline/FEVD_decomp_supdem.pdf}
\vspace{0.6em}
\includegraphics[width=0.6\textwidth]{figures/Baseline/FEVD_decomp_supdem_legend.pdf}
\caption{Forecast error variance decomposition by shock type (supply vs.\ demand), baseline specification.}
\label{fig:fevd_supdem}
\end{figure}
\paragraph{Connectedness.}
A further way to summarise the spillover effects is the connectedness measure of \textcite{diebold2014network}. Figure \ref{fig:Connectedness_network_firstexo} presents the connectedness network implied by our model, showing the spillovers between countries in both magnitude and direction: the thickness of each edge represents the strength of the directional spillover, while the colour of each node indicates whether the country is a net transmitter (red) or a net receiver (blue) and its size the magnitude of that net position. The US has strong spillover effects on almost all countries, and Spain, the UK, Italy, and Canada are also net transmitters, while the remaining countries are net receivers. The ranking of spillovers changes with the horizon: short horizons display fewer connections, because the contemporaneous exogeneity of the US and the zero restrictions on small-to-large spillovers bind directly on the impact matrix. In the Online Appendix, we report results for more horizons and under the alternative assumptions.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\textwidth]{figures/Baseline/Connectedness_network_h1.pdf}
\vspace{0.4em}
\includegraphics[width=0.55\textwidth]{figures/Baseline/Connectedness_network_legend.pdf}
\caption{Connectedness network of the spillover effects between countries}
\label{fig:Connectedness_network_firstexo}
\end{figure}
The sampler mixes well. The posterior traces show no drift or sticking, and across the alternative specifications every parameter block has a median inefficiency factor below five. The full set of diagnostics, together with the sampler settings, is reported in the Online Appendix.
\subsection{Alternative Identification Assumptions}
\label{sec:altind}
We relax some of the identification assumptions made above. The baseline specification imposes hard zeros on the contemporaneous impact of small economies on large ones, i.e.\ $b_c^{j_1j_2}=0$ whenever country $j_1$ is large and country $j_2$ is small. While economically motivated, this exclusion is a strong assumption, and one may prefer to let the data speak more freely about small-to-large spillovers. We therefore consider two progressively weaker treatments of these entries: a \emph{shrinkage} specification that enforces the zeros softly through the prior, and a \emph{flexible} specification that leaves them entirely unrestricted.
Instead of fixing the small-to-large entries at zero, the shrinkage specification keeps them as free parameters but encodes the exclusion through the prior information. We centre each restricted entry at zero and assign it a tight prior variance,
\[
b_{c,0}^{j_1j_2}=0,\qquad v_{c,0}^{j_1j_2}=0.001 \qquad (j_1\text{ large},\ j_2\text{ small}).
\]
Then, the restriction is enforced softly and the posterior is pulled towards zero. The data, however, may overturn it wherever the evidence is sufficiently strong. This replaces the hard equality restriction $\mathbf{R}_E\boldsymbol{\theta}=\mathbf{0}$ with an informative Gaussian prior on the same coordinates.
The flexible specification removes the distinction between big and small countries altogether, retaining the prior information in Table \ref{tab:emp_restrictions} but excluding the zero restrictions. The Online Appendix additionally reports a multiple-component extension, which replaces the single bilinear term in the conditional mean by a sum of $g=3$ such terms and so relaxes the separability of the dynamics rather than the zero restrictions.
Re-estimating the model under both alternatives leaves the impulse responses, FEVDs and out-of-sample historical decompositions essentially unchanged relative to the baseline, as documented in the Online Appendix. The only noteworthy change is observed in the connectedness network, where some smaller countries have a disproportionate spillover compared to their size and, expectedly,
more connections between units are present.
\section{Summary}
This paper develops a Bayesian Structural Matrix Autoregression (BSMAR) framework for identifying country-specific macroeconomic shocks and their contemporaneous transmission across countries in large international systems. By exploiting the matrix structure of the data and decomposing the structural impact matrix into variable- and country-specific components, the framework substantially reduces the dimensionality of the model while allowing for rich cross-country spillovers. The proposed Bayesian sampler accommodates a broad class of identification restrictions, providing a flexible framework for structural analysis in high-dimensional systems.
Applying the model to 15 economies, we find substantial heterogeneity in international shock transmission, with demand shocks playing a more prominent role than supply shocks in driving cross-country spillovers. The United States emerges as an important transmitter of shocks, generating sizable spillovers to economic activity and inflation across countries. Our results further show that demand shocks were the main driver of the post-COVID increase in inflation, highlighting the importance of demand-side forces in understanding the global inflation surge. Overall, our findings underscore the importance of jointly accounting for country-specific shocks and contemporaneous international spillovers when studying the international transmission of macroeconomic disturbances.
\printbibliography
\medskip
\clearpage