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Large structural VARs with multiple linear shock and impact inequality restrictions
\date{\today}
\maketitle
\begin{abstract}
\begin{singlespace}
\noindent
We propose a high-dimensional structural vector autoregression framework that features a factor structure in the error terms and accommodates a large number of linear inequality restrictions on impact impulse responses, structural shocks, and their element-wise products. In particular, we demonstrate that narrative restrictions can be imposed via constraints on the structural shocks, which can be used to sharpen inference and disentangle structurally interpretable shocks. To estimate the model, we develop a highly efficient sampling algorithm that scales well with both the model dimension and the number of inequality restrictions on impact responses and structural shocks. It remains computationally feasible even in settings where existing algorithms may break down. To illustrate the practical utility of our approach, we identify five structural shocks and examine the dynamic responses of thirty macroeconomic variables, highlighting the model's flexibility and feasibility in complex empirical applications. We provide empirical evidence that financial shocks are the most important driver of business cycle dynamics.
\end{singlespace}
\end{abstract}
\bigskip
\noindent \textbf{Keywords:} Structural Identification, Sign Restrictions, Narrative Restrictions, Large BVARs
\noindent \textbf{JEL classification:} C11, C32, C55, E50 \bigskip
\thispagestyle{empty}
\newpage
\pagenumbering{arabic}
\section{Introduction}
\cite{uhlig2005effects} popularizes the use of sign restrictions on impulse response functions, offering a flexible and theory-consistent approach to identifying structural shocks in Vector Autoregression (VAR) models. Instead of relying on often implausible zero restrictions, sign restrictions constrain the signs of impulse responses based on economic intuition. Building on this framework, \cite{antolin2018narrative} introduce narrative restrictions, which incorporate external prior information about specific historical episodes-such as well-documented policy interventions or economic events-to further sharpen set identification.\footnote{The sign restrictions on shocks over distinct time periods may be interpreted as a more agnostic framework than impact sign restrictions, as noted in \cite{antolin2018narrative}. Nonetheless, it is emphasized that sign restrictions must be grounded in a plausible narrative foundation. } By conditioning structural shocks on particular time periods, narrative restrictions enable researchers to embed economically meaningful prior knowledge directly into the model, enhancing interpretability and credibility of the identified shocks. Using narrative restrictions have become popular recently.\footnote{Important examples that use narrative restrictions include \cite{zeev2018can}, \cite{altavilla2019loan}, \cite{furlanetto2019immigration}, \cite{cheng2020revisiting},
\cite{kilian2020does}, \cite{kilian2022oil}, \cite{laumer2020government}, \cite{redl2020uncertainty}, \cite{zhou2020refining},
\cite{antolin2021structural}, \cite{caggiano2021financial}, \cite{larsen2021components}, \cite{ludvigson2021uncertainty},
\cite{maffei2021does}, \cite{berger2022unified}, \cite{fanelli2022sovereign}, \cite{inoue2022joint}, \cite{badinger2023measuring},
\cite{berthold2023macroeconomic}, \cite{caggiano2023global}, \cite{conti2023bank}, \cite{harrison2023structural}, \cite{herwartz2023point}
\cite{neri2023long}, \cite{reichlin2023monetary}, \cite{ascari2023endogenous}, \cite{boer2024energy} and \cite{ruth2023monetary}.} However, in most applications researcher use rather small dimensional VARs with narrative restrictions. Furthermore, the implementation of narrative restriction rely on accept-and-reject algorithms (see, e.g., \cite{antolin2018narrative} or \cite{giacomini2021identification}), which may become computationally burdensome as the number of restrictions increases. Therefore, applications with narrative restriction are limited to VARs with a small number of narrative restrictions, typically for just a single structural shock.
Since the influential work of \cite{banbura2010large}, large VARs have become a cornerstone of modern structural analysis and forecasting. By incorporating dozens-or even hundreds-of variables, large VARs offer a powerful solution to a key limitation of smaller models: omitted variable bias. This bias can lead to distorted forecasts, unreliable policy conclusions, and incorrect identification of structural shocks. Expanding the information set helps capture the economic environment more comprehensively and reduces the risk of excluding relevant dynamics. As emphasized by \cite{hansen2019two} and \cite{lippi1993dynamic,lippi1994var}, using a narrower information set than that available to economic agents can make a model non-fundamental, thus preventing accurate recovery of structural shocks. Building on this foundation, numerous studies-including \cite{carriero2009forecasting, giannone2015prior, jarocinski2017granger, huber2019adaptive, chan2024large}-have explored various methodological and empirical extensions. Overall, large VARs provide a flexible framework for structural inference in high-dimensional settings to address informational deficiencies.
In this paper, we propose a high-dimensional Structural Vector Autoregression (SVAR) framework capable of accommodating a large number of linear inequality constraints on impact impulse responses, structural shocks, and their element-wise products. While previous literature has primarily focused on small narrative SVAR models, our approach extends these methods by enabling the inclusion of a much larger set of restrictions as well as a greater number of variables. The ability to incorporate a broader set of inequality restrictions enhances the structural set identification, allowing for more precise inference, see \cite{antolin2018narrative}. Moreover, in cases where impact sign restrictions alone cannot easily distinguish between different structural shocks, additional sign restrictions directly on the structural shocks provide a flexible way to separate them.
Recently, the assumption that VAR errors follow a factor structure has gained popularity, with latent factors being interpreted as structural shocks (see, for instance, \cite{Korobilis2022}, \cite{chan2022large}, \cite{hauzenberger2022bayesian}, \cite{banbura2023drives}, \cite{gambetti2023agreed}, \cite{pfarrhofer2024high}, \cite{pruser2024large}, \cite{korobilis2025exploring} and \cite{chan2025large}). We build on this literature, leveraging the factor structure for several key advantages. First, from a computational perspective, the factor structure enables equation-by-equation estimation, simplifying the estimation process and allowing for the inclusion of a rich set of variables in the model. Second, as the number of variables increases (including different measures of prices and output), it is reasonable to assume that the number of structural shocks is smaller than the number of variables, making a factor-based approach well-suited for high-dimensional settings. Third, and particularly important for our paper, the factors have a clear interpretation as structural shocks. This provides a natural framework for incorporating narrative restrictions in the form of prior distributions on the structural shocks, which allows to develop a simple yet efficient algorithm.
Our approach deviates from existing methods such as those in \cite{antolin2018narrative} and \cite{giacomini2021identification}, which typically incorporate narrative restrictions through a pseudo-likelihood. Instead, imposing shock sign restrictions via prior distributions allows us to introduce a new sampling algorithm that offers significant computational efficiency by ensuring that each draw is accepted with certainty. This feature enables us to handle a large number of restrictions on both impact responses and structural shocks, overcoming computational challenges often encountered with accept-and-reject algorithms. Our approach therefore represents a major advancement in the scalability and feasibility of SVAR models, particularly in high-dimensional settings where existing algorithms may be very slow or even infeasible.
We compare our proposed algorithm with the approach of \cite{antolin2018narrative} in a synthetic exercise in which we estimate several ten-variable VARs with different sets of impact and shock sign restrictions. The results indicate that our algorithm achieves substantially faster computation times than the method proposed by \citet{antolin2018narrative} across the different identification schemes.
In an empirical application, we demonstrate the practical utility of our approach by identifying five structural shocks in a large VAR. In this respect, we follow \cite{stock2012disentangling} and estimate 30 variable VAR and disentangle structurally interpretable shocks by means of impact and shock sign restrictions.\footnote{Specifically, we use the same set of 30 variables and identify the same structural shocks as in \cite{stock2012disentangling}. \cite{h24} estimates the same 30-variables VAR and identifies the same 5 structural shocks.} Importantly, we show that shock sign restrictions can help to separate structural shocks and to overall sharpen identification.
In our empirical illustration, we provide evidence suggesting that financial risk shocks are a key driver of business cycle dynamics.
This example underscores the flexibility and scalability of our methodology, which is capable of handling high-dimensional data and a large number of inequality restrictions on the impact responses as well as on the structural shocks.
The remainder of this paper is organized as follows. Section 2 lays out and discusses the econometric framework. Section 3 studies the computational efficiency. Section 4 applies the model to the study of \cite{stock2012disentangling} to identify and investigate the effects of five structural shocks on the economy. Section 4 concludes.
\section{ A large structural VAR with factor structure}
Let $\boldsymbol{y}_t = (y_{1,t}, \dots, y_{n,t})'$ represent an $n \times 1$ vector of endogenous variables at time $t$. The model can be expressed as:
\begin{equation}
\boldsymbol{y}_t = \boldsymbol{b}_0 + \boldsymbol{B}_1 \boldsymbol{y}_{t-1} + \cdots + \boldsymbol{B}_p \boldsymbol{y}_{t-p} + \boldsymbol{u}_t,
\end{equation}
where the error term $\boldsymbol{u}_t$ is decomposed as
\begin{equation}
\boldsymbol{u}_t = \boldsymbol{L} \boldsymbol{f}_t + \boldsymbol{v}_t,
\end{equation}
with $\boldsymbol{v}_t \sim N(\boldsymbol{0}, \boldsymbol{\Sigma})$, where $\boldsymbol{\Sigma} = \text{diag}(\sigma_1^2, \dots, \sigma_n^2)$, and $\boldsymbol{f}_t$ is an $r \times 1$ vector of factors such that $\boldsymbol{f}_t \sim N(\boldsymbol{0}, \boldsymbol{I})$. Concisely, the system can be reformulated as:
\begin{equation}
\boldsymbol{y}_t = (\boldsymbol{I}_n \otimes \boldsymbol{x}'_t) \boldsymbol{\beta} + \boldsymbol{L} \boldsymbol{f}_t + \boldsymbol{v}_t,
\end{equation}
where $\boldsymbol{I}_n$ is the identity matrix of dimension $n$, $\otimes$ represents the Kronecker product, and $\boldsymbol{\beta} = \text{vec}([\boldsymbol{b}_0, \boldsymbol{B}_1, \dots, \boldsymbol{B}_p]')$. The vector $\boldsymbol{x}_t = (1, \boldsymbol{y}'_{t-1}, \dots, \boldsymbol{y}'_{t-p})'$ of dimension $k = 1 + np$ contains an intercept and lagged values. The idiosyncratic component $\boldsymbol{v}_t$ accounts for measurement error or other idiosyncratic noise. In contrast, the $r$ latent factors, $\boldsymbol{f}_t$ impact multiple variables in the system and hence have the interpretation as structural shocks.
More formally we multiply the model with the generalized inverse of $\boldsymbol{L}$ to obtain:
\begin{equation}
\boldsymbol{A} \boldsymbol{y}_t = \boldsymbol{B} \boldsymbol{x}_t + \boldsymbol{f}_t + \boldsymbol{A} \boldsymbol{v}_t,
\end{equation}
where $\boldsymbol{A} = (\boldsymbol{L}' \boldsymbol{L})^{-1} \boldsymbol{L}'$ and $\boldsymbol{B} = (\boldsymbol{A} \boldsymbol{b}_0, \boldsymbol{A} \boldsymbol{B}_1, \dots, \boldsymbol{A} \boldsymbol{B}_p)$. Given that $\boldsymbol{v}_t$ is uncorrelated noise, the Central Limit Theorem (see \cite{bai2003}) implies that $\boldsymbol{A} \boldsymbol{v}_t \to 0$ as $n \to \infty$. Hence, we can write
\begin{equation}
\boldsymbol{f}_t \approx \boldsymbol{A} \boldsymbol{y}_t - \boldsymbol{B} \boldsymbol{x}_t.
\end{equation}
Thus, $\boldsymbol{v}_t$ is treated as noise shocks without structural meaning, while $\boldsymbol{f}_t$ provides a projection of structural shocks in $\mathbb{R}^r$. Consequently, this formulation supports structural analysis using standard methods, including impulse response functions (e.g., \cite{Froni2019}, \cite{Korobilis2022}, \cite{chan2022large}).
Under the assumption of uncorrelated $\boldsymbol{f}_t$ and $\boldsymbol{v}_t$, the conditional variance of $\boldsymbol{u}_t$ is given by:
\begin{equation}
\text{Var}(\boldsymbol{u}_t | \boldsymbol{\Sigma}, \boldsymbol{L}) = \boldsymbol{L} \boldsymbol{L}' + \boldsymbol{\Sigma}.
\end{equation}
To separately identify the common and idiosyncratic components, we adopt the condition $r \leq (n-1)/2$ from \cite{anderson1956statistical}. Under this restriction, for any observationally equivalent model $(\boldsymbol{L},\boldsymbol{\Sigma})$, it holds that $\boldsymbol{L} \boldsymbol{L}' = \boldsymbol{L}^* \boldsymbol{L}^{*'}$ and $\boldsymbol{\Sigma} = \boldsymbol{\Sigma}^*$.
However, without additional restrictions, $\boldsymbol{L}$ is not identified. Any orthogonal matrix $\boldsymbol{Q} \in \mathcal{O}(r)$, where $\mathcal{O}(r) = \{ \boldsymbol{Q} \in \mathbb{R}^{r \times r} : \boldsymbol{Q} \boldsymbol{Q}' = \boldsymbol{I}_m\}$, generates an equivalent model via the transformation $\tilde{\boldsymbol{L}} \tilde{\boldsymbol{f}}_t = \boldsymbol{L} \boldsymbol{Q} \boldsymbol{Q}' \boldsymbol{f}_t$. In this paper, we achieve set identification by placing inequality restrictions on $\boldsymbol{L}$ and $\boldsymbol{f}_{t}$.\footnote{As shown below, our model also allows for inequality restrictions on the product of $\boldsymbol{L}$ and $\boldsymbol{f}$.}
\subsection{Linear inequality restrictions via prior distributions}
We consider inequality restriction on the $\boldsymbol{L}$, $\boldsymbol{f}_{t}$, and on their product.\footnote{In many cases we have a strong consensus in economic theory about the
signs of impulse responses at impact but not at longer horizons (see, e.g. \cite{canova2011business}).} We can express these inequality restrictions as follows:
\begin{align}
\ell_i^{L} &< \boldsymbol{R}_{i}^{L} \boldsymbol{l}_{i'}< \upsilon_i^{L},\\
\ell_{\tilde{t}}^{f} &< \boldsymbol{R}_{\tilde{t}}^{f} \boldsymbol{f}_{\tilde{t}}< \upsilon_{\tilde{t}}^{f},\\
\ell_{i\tilde{t}}^{Lf} &< \boldsymbol{R}_{i\tilde{t}}^{Lf} (\boldsymbol{l}_i' \odot \boldsymbol{f}_{\tilde{t}})< \upsilon_{i\tilde{t}}^{Lf},
\end{align}
where $\boldsymbol{l}_i$ denote the elements of $\boldsymbol{L}$ in the $i-$th equation, $\boldsymbol{R}_{i}^{L}$ is $r_i^L\times r$,
$\boldsymbol{R}_{\tilde{t}}^{f}$ is $r_{\tilde{t}}^f\times r$, $\boldsymbol{R}_{i\tilde{t}}^{Lf} $ is $r_{i\tilde{t}}^{Lf}\times r$ and $\tilde{t}\in t=1,\dots,T$. We assume that $\ell_i^{L} <\upsilon_i^{L}$, $\ell_{\tilde{t}}^{f}< \upsilon_{\tilde{t}}^{f}$ and $\ell_{i\tilde{t}}^{Lf}< \upsilon_{i\tilde{t}}^{Lf}$. Furthermore, we allow elements in the lower and upper bound to be $-\infty$ or $\infty$ to indicate that some inequity restrictions are single-sided. In many applications $\boldsymbol{R}_i^{L}$, $\boldsymbol{R}_{\tilde{t}}^{f}$ and $\boldsymbol{R}_{i\tilde{t}}^{Lf}$ will be equal to $\boldsymbol{I}_r$. The first set of inequality restrictions can be used to implement sign restrictions on $\boldsymbol{L}$ as in \cite{Korobilis2022}. The second set of inequality restriction can be used to implement sign restrictions on the structural shocks $\boldsymbol{f}_{t}$ at specific time periods as in \cite{antolin2018narrative}. Finally, the last set of inequality restrictions can be used to implement a magnitude restrictions on the product of both factors loadings and structural shocks as in \cite{Badinger2023}. In particular, \cite{Badinger2023} impose that one particular shock explains more then half of the unexplained movement of a certain variable at a specific period. This provides a flexible framework for the identification of structural shocks. Instead of only relaying on impact sign restrictions, researchers can also use shock sign restrictions to distinguish between different structural shocks. Or even use a single magnitude restriction as in \cite{Badinger2023} to separate one shock from the others.\footnote{It should be noted that non-linear restrictions can be incorporated via an accept-rejection step. However, this may significantly increase the computational burden or even render sampling infeasible. If such non-linear restrictions can be well approximated by linear restrictions that fit within our framework, the accept-rejection step may become feasible and the additional computational cost can be kept minimal.}
We implement our inequality restrictions using truncated Gaussian prior distributions. In particular we assume
\begin{align}
\boldsymbol{l}_i &\sim N(\boldsymbol{l}_{0,i},\boldsymbol{V}_{\boldsymbol{l}_{i}})\boldsymbol{1} (\ell_i^{L} < \boldsymbol{R}_i^{L} \boldsymbol{l}_i'< \upsilon_i^{L})\bigcup_{\tilde{t}\in \tilde{T}} \boldsymbol{1} (\ell_{i\tilde{t}}^{Lf} < \boldsymbol{R}_{i\tilde{t}}^{Lf} (\boldsymbol{l}_i' \odot \boldsymbol{f}_{\tilde{t}})< \upsilon_{i\tilde{t}}^{Lf}),\\
\boldsymbol{f}_t &\sim N(\boldsymbol{0}, \boldsymbol{I})\boldsymbol{1} (\ell_{t}^{f} < \boldsymbol{R}_{t}^{f} \boldsymbol{f}_{t}< \upsilon_{t}^{f}) \boldsymbol{1} (\ell_{it}^{Lf} < \boldsymbol{R}_{it}^{Lf} (\boldsymbol{l}_i' \odot \boldsymbol{f}_{t})< \upsilon_{it}^{Lf},
\end{align}
where $\tilde{T}$ denotes the set of periods in which a restrictions is specified.
We complete our model specification by describing the further prior distributions.
For the variance terms of the noise we assume $\sigma_{j}^2 \sim \mathcal{IG}(\alpha_0,\beta_0)$. In our empirical application, we set $\boldsymbol{l}_{0,i}= \boldsymbol{0}$, $ \boldsymbol{V}_{\boldsymbol{l}_i}=10 \times \boldsymbol{I}_r$ and $\alpha_0=\beta_0=0$. In high-dimensional settings such as large VARs, it is important to impose shrinkage prior to mitigate the risk of overfitting. We follow \cite{Korobilis2022} and use the horseshoe prior proposed by \cite{carvalho2010horseshoe}. Let $\boldsymbol{\beta}_i$ the VAR coefficients in the $i-$th equation, $i=1,\dots,n$ and $\beta_{i,j}$, the $j-$th coefficient in the $i-$th equation. Consider the prior for $\beta_{i,j}$, $i=1,\dots,n$ and $j=2,\dots,k$:
\begin{align}
\beta_{i,j}|\lambda_i, \psi_{i,j} &\sim N(0,\lambda_{i}\psi_{i,j}),\\
\sqrt{\psi_{i,j}}&\sim C^+(0,1),\\
\sqrt{\lambda_i}&\sim C^+(0,1),
\end{align}
where $C^+(0,1)$ denotes the standard half-Cauchy distribution. The hyperparameter $\lambda_i$ is the global variance components that are common to all elements in $\boldsymbol{\beta}_i$ , whereas each $\psi_{i,j}$ is a local variance component specific to the coefficients $\beta_{i,j}$. To facilitate sampling, we follow \cite{makalic2015simple} and use the following latent variables representations of the half-Cauchy distributions:
\begin{align}
(\psi_{i,j}|z_{\psi_{i,j}})&\sim \mathcal{IG}(1/2,1/z_{\psi_{i,j}}),\quad z_{\psi_{i,j}}\sim \mathcal{IG}(1/2,1),\\
(\lambda_i|z_{\lambda_i})&\sim \mathcal{IG}(1/2,1/z_{\lambda_i}),\quad z_{\lambda_i}\sim \mathcal{IG}(1/2,1),
\end{align}
for $i=1,\dots,n$ and $j=2,\dots,k$.
\subsection{Gibbs Sampler}
In this section, we develop an efficient posterior sampler to estimate the model. To impose the inequality restrictions on the factors or factor loadings we use the sampler from \cite{botev2017normal}. In contrast to existing rejecting sampling approaches (see, e.g. \cite{antolin2018narrative} or \cite{giacomini2021identification}) each draw is accepted with certainty. Posterior draws can be obtained by sampling sequentially from the conditional distributions:
\begin{enumerate}
\item $p(\boldsymbol{f}|\boldsymbol{y},\boldsymbol{\beta}, \boldsymbol{L},\boldsymbol{\Sigma}, \boldsymbol{\lambda}, \boldsymbol{\psi}, \boldsymbol{z}_{\lambda},\boldsymbol{z}_{\boldsymbol{\psi}})=p(\boldsymbol{f}|\boldsymbol{y}, \boldsymbol{\beta}, \boldsymbol{L}, \boldsymbol{W}, \boldsymbol{\Sigma})$;
\item $p( \boldsymbol{L}|\boldsymbol{y}\boldsymbol{\beta},\boldsymbol{f},\boldsymbol{\Sigma}, \boldsymbol{\lambda}, \boldsymbol{\psi}, \boldsymbol{z}_{\lambda},\boldsymbol{z}_{\boldsymbol{\psi}})=\prod_{i=1}^n=p(\boldsymbol{l}_i|\boldsymbol{y}_i, \boldsymbol{\beta}_i,\boldsymbol{f}, \boldsymbol{\sigma}^2_i)$
\item $p(\boldsymbol{\beta}|\boldsymbol{y}, \boldsymbol{L}, \boldsymbol{f},\boldsymbol{\Sigma}, \boldsymbol{\lambda}, \boldsymbol{\psi}, \boldsymbol{z}_{\lambda},\boldsymbol{z}_{\boldsymbol{\psi}})=\prod_{i=1}^n=p(\boldsymbol{\beta}_i|\boldsymbol{y}_i,\boldsymbol{l}_i,\boldsymbol{f}, \boldsymbol{\sigma}^2_i)$
\item $p(\boldsymbol{\Sigma}|\boldsymbol{y},\boldsymbol{\beta}, \boldsymbol{L},\boldsymbol{f}, \boldsymbol{\lambda}, \boldsymbol{\psi}, \boldsymbol{z}_{\lambda},\boldsymbol{z}_{\boldsymbol{\psi}})=\prod_i^n p(\sigma_i^2|\boldsymbol{y}_i,\boldsymbol{f}_i,\boldsymbol{l}_i,\boldsymbol{\beta}_i) $;
\item $p(\boldsymbol{\lambda}|\boldsymbol{y},\boldsymbol{\beta}, \boldsymbol{L},\boldsymbol{f},\boldsymbol{v}, \boldsymbol{\psi}, \boldsymbol{z}_{\lambda},\boldsymbol{z}_{\boldsymbol{\psi}})=\prod_{l=1}^n p(\lambda_i|\boldsymbol{\beta}, \boldsymbol{\psi}, z_{\lambda_i})$;
\item $p(\boldsymbol{\psi}|\boldsymbol{y},\boldsymbol{\beta}, \boldsymbol{L},\boldsymbol{f},\boldsymbol{\Sigma}, \boldsymbol{\lambda}, \boldsymbol{z}_{\lambda},\boldsymbol{z}_{\boldsymbol{\psi}})=\prod_{i=1}^n\prod_{j=2}^k p(\psi_{i,j}|\beta_{i,j}, \lambda_i,z_{\psi_{i,j}})$;
\item $p( \boldsymbol{z}_{\lambda}|\boldsymbol{y},\boldsymbol{\beta}, \boldsymbol{L},\boldsymbol{f},\boldsymbol{\Sigma}, \boldsymbol{\lambda}, \boldsymbol{\psi},\boldsymbol{z}_{\boldsymbol{\psi}})=\prod_{i=1}^n p(z_{\lambda_i}|\lambda_i)$;
\item $p(\boldsymbol{z}_{\boldsymbol{\psi}}|\boldsymbol{y},\boldsymbol{\beta}, \boldsymbol{L},\boldsymbol{f},\boldsymbol{\Sigma}, \boldsymbol{\lambda}, \boldsymbol{\psi}, \boldsymbol{z}_{\lambda})=\prod_{i=1}^n\prod_{j=2}^k p(z_{ \psi_{i,j}}| \psi_{i,j})$,
\end{enumerate}
with $\boldsymbol{y}_i=(y_{i,1},\dots,y_{i,T})'$ be a $T\times 1$ vector of observations of the $i-$th variable.
\textbf{Step 1} First, we sample $\boldsymbol{f}_t$ for $t=1,\dots,T$. We can use standard regression results (see, e.g., \cite{chan2019bayesian}) to obtain
\begin{equation}
(\boldsymbol{f}_t|\boldsymbol{y},\boldsymbol{\beta}, \boldsymbol{L})\sim N(\hat{\boldsymbol{f}}_t,\boldsymbol{K}_{\boldsymbol{f}}^{-1}),
\end{equation}
where
\begin{equation}
\boldsymbol{K}_{\boldsymbol{f}}^{-1}=(\boldsymbol{I}_r+\boldsymbol{L}' \boldsymbol{\Sigma} \boldsymbol{L})^{-1}, \quad \hat{\boldsymbol{f}}_t=\boldsymbol{K}_{\boldsymbol{f}}(\boldsymbol{L} \boldsymbol{\Sigma}^{-1}(\boldsymbol{y}_t- (\boldsymbol{I}_n \otimes \boldsymbol{x}'_t)\boldsymbol{\beta} )).
\end{equation}
We use the efficient truncated Normal generator provided by \cite{botev2017normal} in order to sample the factors.
\textbf{Step 2} Second, we sample $\boldsymbol{L}$. Given the latent factors $\boldsymbol{f}$, the VAR becomes $n$ unrelated regressions and we can sample $\boldsymbol{L}$ equation by equation. Remember that $\boldsymbol{\beta}_i$ and $\boldsymbol{l}_i$ denote, respectively, the VAR coefficients and the factor loadings in the $i-$th equation. Then, the $i-$th equation of the VAR can be expressed as
\begin{equation}
\boldsymbol{y}_i=\boldsymbol{X}_i \boldsymbol{\beta}_i+\boldsymbol{F} \boldsymbol{l}_i +\boldsymbol{v}
\label{eqbyeq}
\end{equation}
where $\boldsymbol{F}=(\boldsymbol{f}_1,\dots,\boldsymbol{f}_r)$ the $ T\times r$ matrix of factors with $ \boldsymbol{f}_i=(f_{i,1},\dots,f_{i,T})'$. The vector of noise $\boldsymbol{v}=(v_{i,1},\dots,v_{i,T})'$ is distributed as $N(\boldsymbol{0},\boldsymbol{I}_T \sigma_i^2)$.
Then using standard linear regression results, we get
\begin{equation}
(\boldsymbol{l}_i|\boldsymbol{y}_i,\boldsymbol{f}, \boldsymbol{\beta}_i, \sigma_i^2)\sim N( \hat{\boldsymbol{l}_i},\boldsymbol{K}^{-1}_{\boldsymbol{l}_i})\boldsymbol{1} (\ell_{t}^{f} < \boldsymbol{R}_{t}^{f} \boldsymbol{f}_{t}< \upsilon_{t}^{f})\boldsymbol{1} (\ell_{it}^{Lf} < \boldsymbol{R}_{it}^{Lf} (\boldsymbol{l}_i' \odot \boldsymbol{f}_{t})< \upsilon_{it}^{Lf},
\end{equation}
where
\begin{align*}
\boldsymbol{K}_{\boldsymbol{l}_i}=\boldsymbol{V}^{-1}_{\boldsymbol{l}_i}+\boldsymbol{\sigma}^{-2}_i \boldsymbol{F}'\boldsymbol{F}, \quad \hat{\boldsymbol{l}}_i=\boldsymbol{K}^{-1}_{\boldsymbol{l}_i}(\boldsymbol{V}^{-1}_{\boldsymbol{l}_i}\boldsymbol{l}_{0,i}+\sigma^{-2}_i \boldsymbol{F} (\boldsymbol{y}_i-\boldsymbol{X}_i \boldsymbol{\beta}_i)).
\end{align*}
We use the efficient truncated Normal generator provided by \cite{botev2017normal} in order to sample $\boldsymbol{l}_i$.
\textbf{Step 3} Third, we sample $\boldsymbol{\beta}$ equation by equation based on (\ref{eqbyeq}). Equation by equation estimation simplifies the estimation and allows for the estimation with a large number of variables. Again,
using standard linear regression results, we get
\begin{equation}
(\boldsymbol{\beta}_i|\boldsymbol{y}_i,\boldsymbol{f}, \boldsymbol{l}_i, \sigma_i^2)\sim N( \hat{\boldsymbol{l}_i},\boldsymbol{K}^{-1}_{\boldsymbol{\beta}_i}),
\end{equation}
where
\begin{align*}
\boldsymbol{K}_{\boldsymbol{\beta}_i}=\boldsymbol{V}^{-1}_{\boldsymbol{\beta}_i}+\boldsymbol{\sigma}^{-2}_i \boldsymbol{X}_i'\boldsymbol{X}_i, \quad \hat{\boldsymbol{\beta}}_i=\boldsymbol{K}^{-1}_{\boldsymbol{\beta}_i}(\boldsymbol{V}^{-1}_{\boldsymbol{l}_i}\boldsymbol{\beta}_{0,i}+\sigma^{-2}_i \boldsymbol{X}_i (\boldsymbol{y}_i-\boldsymbol{F}\boldsymbol{l}_i)).
\end{align*}
\textbf{Step 4} Next, we sample $\sigma^2_{i}$ for $i=1,\dots,n$. Given $\boldsymbol{f}$ the model reduces to $n$ independent linear regressions. Therefore, we can use standard regression results (see, e.g., \cite{chan2019bayesian}) to obtain
\begin{equation}
(\sigma_{i}^2|\boldsymbol{y}, \boldsymbol{f}, \boldsymbol{L})\sim \mathcal{IG}\left(\alpha_0+\frac{T}{2},\beta_0+0.5\sum_{t=1}^T(y_{it}-\boldsymbol{X}_{it}\boldsymbol{\beta}_i -\boldsymbol{l}_i\boldsymbol{f}_t)^2\right).
\end{equation}
\textbf{Step 5} Lastly, we sample the hyperparameter $\lambda_{i}$ and $\psi_{i,j}$ from our shrinkage prior for the VAR coefficients as well as the mixing variables $z_{\lambda_{i}}$ and $z_{\psi_{i,j}}$. Using the latent variable representation of the half Cauchy distribution, we obtain
\begin{align*}
p(\psi_{i,j}|\beta_{i,j}, \lambda_{i}, z_{\psi_{i,j}})&\propto \psi_{i,j}^{\frac{1}{2}} \text{e}^{-\frac{1}{2\lambda_{i}\psi_{i,j}}(\beta_{i,j})^2 }\times \psi^{-\frac{3}{2}}\text{e}^{-\frac{1}{\psi_{i,j}z_{\psi_{i,j}}}}\\
&=\psi_{i,j}^{-2}\text{e}^{-\frac{1}{\psi_{i,j}}\left(\frac{1}{z_{\psi_{i,j}}}+\frac{(\beta_{i,j})^2}{2\lambda_{i}}\right)},
\end{align*}
which is the kernel of the following inverse-gamma distribution:
\begin{equation}
(\psi_{i,j}|\beta_{i,j}, \lambda_{i}, z_{\psi_{i,j}})\sim \mathcal{IG}\left(1, \frac{1}{z_{\psi_{i,j}}}+\frac{(\beta_{i,j})^2}{2\lambda_{i}}\right).
\end{equation}
Furthermore,
\begin{align*}
p(\lambda_i|\boldsymbol{\beta}, \boldsymbol{\psi}, z_{\lambda_i})&\propto \prod_{i} \lambda_i^{-\frac{1}{2}} \text{e}^{-\frac{1}{2\lambda_i \psi_{i,j}}(\beta_{i,j})^2 }\times \lambda_i^{-\frac{3}{2}} \text{e}^{-\frac{1}{\lambda_i z_{\lambda_i}} },\\
&=\lambda_i^{-\left(\frac{k}{2}+1 \right)}
\text{e}^{-\frac{1}{\lambda_i}\left(\frac{1}{z_{\lambda_i}} +\sum_{j=2}^{k}\frac{(\beta_{i,j})^2}{2\psi_{i,j}} \right) },
\end{align*}
which is the kernel of the following inverse-gamma distribution:
\begin{equation}
(\lambda_i|\boldsymbol{\beta}, \boldsymbol{\psi}, z_{\psi_{\lambda_i}})\sim \mathcal{IG}\left(\frac{k}{2}, \frac{1}{z_{\lambda_i} }+\sum_{j=2}^{k}\frac{(\beta_{i,j})^2}{2\psi_{i,j}}\right).
\end{equation}
Finally, we sample the latent variables $\boldsymbol{z}_{\boldsymbol{\psi}}$ and $\boldsymbol{z}_{\boldsymbol{\lambda}}$. In particular, $z_{\psi_{i,j}}\sim \mathcal{IG}(1,1+\psi_{i,j}^{-1})$ for $i=1,\dots,n$ and $j=2,\dots,k$. Similarly, we have $z_{\lambda_{l}}\sim \mathcal{IG}(1,1+\lambda_i^{-1})$ for $i=1,\dots,n$.
\section{Computational Efficiency} \label{section_compute_efficiency}
This section evaluates the computational performance of our proposed algorithm in comparison to the approach developed by \citet{antolin2018narrative} (henceforth DDR18). To assess the relative efficiency, we consider three different structural VAR specifications with $n=10$ variables, lag length $p=4$, $k=5$ structural shocks, and a sample size of $T=148$ observations.\footnote{In our empirical application in Section \ref{section_emp_application}, we use quarterly data from 1983Q1 to 2019Q4, corresponding to $T = 148$. We identify $k=5$ structural shocks and set the lag length to $p=4$, consistent with the simulation design here.}
Each VAR specification differs in the number of shock sign restrictions imposed: (i) 15 impact sign restrictions and no shock sign restrictions, (ii) 15 impact sign restrictions and 3 shock sign restrictions, and (iii) 15 impact sign restrictions and 6 shock sign restrictions. For each specification, we simulate 10 synthetic data sets and report the average runtime required by both algorithms to generate 100 admissible draws.\footnote{\label{foot:comparison}For our algorithm, we discard the first 1,000 draws as burn-in and retain every \nth{10} draw thereafter, yielding an effective sample of 1,000 draws. For the DDR18 algorithm, we adhere to the specifications outlined by \citet{antolin2018narrative}, employing 1,000 importance-weighted draws and capping the number of attempted draws of the reduced-form parameters and the $\mathbf{Q}$ matrix at 100 each. We use flat priors as \citet{antolin2018narrative}. It is important to note that \citet{antolin2018narrative} estimate a three-variable VAR, which naturally requires less prior shrinkage than the ten-variable VAR used in our simulations. This difference affects computational burden and must be considered when interpreting runtime comparisons. Additionally, \citet{antolin2018narrative} generate independent posterior draws for the reduced-form coefficients.} Details of the data-generating process are provided in Appendix \ref{app_data_gen}.
\begin{table}[t]
\centering
\caption{Comparison of DDR18 algorithm and proposed algorithm}
\label{tab:comparison}
\begin{tabular}{lllccc}
\toprule
\textbf{Restrictions} & \textbf{\# Restrictions} & \textbf{Code} & Mean \\
\midrule
Impact Sign & 15 & DRR18 & 32.09 \\
Shock Sign & 0 & Proposed & 0.18 \\
\hline
Impact Sign & 15 & DRR18 & 33.32 \\
Shock Sign & 3 & Proposed & 0.23 \\
\hline
Impact Sign & 15 & DRR18 & 34.58 \\
Shock Sign & 6 & Proposed & 0.23 \\
\bottomrule
\end{tabular}
\vspace{0.75em}
\begin{minipage}{\textwidth}
\footnotesize
\footnotesize
\textit{Notes:} The table reports the average runtime (in minutes) required by the DDR18 algorithm and our proposed algorithm to obtain 100 admissible draws across three different structural VAR specifications. The \# Restriction column indicates the number of imposed impact and shock sign restrictions.
\end{minipage}
\end{table}
Table \ref{tab:comparison} summarizes the computational comparison between the proposed algorithm and DDR18 across the three VAR configurations. It is evident that the proposed algorithm achieves substantially faster runtimes in all cases. As highlighted in Footnote~\ref{foot:comparison}, the DDR18 algorithm was originally developed for small-scale VARs, whereas our approach is explicitly designed for high-dimensional VARs with factor error structures, which requires more sophisticated shrinkage methods. Importantly, in our simulation design, we construct the data-generating process such that the imposed shock sign restrictions align with the true signs of the underlying structural shocks. In the absence of this feature, the DDR18 algorithm fails to find a sufficient number of admissible draws.\footnote{See Appendix \ref{app_data_gen} for a detailed discussion of the data-generating process. Our computations were carried out using MATLAB R2024b on an Intel Core Ultra 7 with a 1.70 GHz base speed and 8 GB of RAM.}
Table \ref{tab:comparison} clearly shows that the proposed algorithm achieves substantially faster runtimes than the DDR18 algorithm. Notably, the runtime of the proposed method remains stable even as the number of shock sign restrictions increases. In contrast, as the number of variables $n$ or the number of identifying restrictions grows - e.g., to $n = 20$ or $n = 30$ with more impact and shock sign restrictions - the DDR18 algorithm struggles or even fails to find admissible draws within a feasible timeframe. These results underscore the computational advantages of our approach and highlight its suitability for estimating large-scale VARs with extensive identifying restrictions. In the empirical application that follows, we apply the proposed algorithm to a thirty-variable VAR identified with 60 impact sign restrictions and 26 shock sign restrictions.
\section{Application to \cite{stock2012disentangling}} \label{section_emp_application}
This section presents an empirical application based on the framework of \cite{stock2012disentangling}, who estimate a 30-variable vector autoregression (VAR) to assess the relevance of five macroeconomic shocks during the Great Financial Crisis. This empirical application serves the purpose to illustrate the effectiveness of our proposed algorithm to combine impact and shock sign restrictions in large a structural VAR.\footnote{\cite{h24} also estimates the same 30-variable VAR and identifies the structural shocks using multiple instruments in a proxy SVAR.} In line with their approach, we estimate a large-scale VAR model comprising the same set of variables, simultaneously identifying five structural shocks: an oil supply (news) shock, a monetary policy shock, a technology shock, a financial risk shock, and a government spending shock. Differing from \cite{stock2012disentangling}, we employ the above-outlined structural VAR with factor error structure. Further, we extend the sample period to span from 1983Q1 to 2019Q4 and employ both impact sign and shock sign restrictions for the identification of structural shocks.\footnote{\cite{bgk24} note that this sample period captures a relatively stable macroeconomic environment, excluding the heightened oil price volatility of the 1970s as well as the COVID-19 pandemic and the geopolitical shock resulting from the Russian invasion of Ukraine.} A detailed overview of the 30 variables included in the VAR is provided in Section \ref{data_set}. Following the transformations employed in \cite{chan2025rankingrestrictions}, most variables are expressed in logarithms. Given the quarterly frequency of the data, the model is estimated using a lag length of
$p=4$. Posterior inference is conducted using the described Gibbs sampler. Specifically, we perform 6,000 iterations, discarding the first 1,000 as burn-in and retaining every 10th of the remaining draws for inference.
\subsection{Impact Sign and Shock Sign Restrictions} \label{section_restrictions}
In this section, we outline and discuss the implementation of impact sign and shock sign restrictions. Particular emphasis is placed on the shock sign restrictions for 2001Q3 and 2008Q4, given their role in disentangling structural shocks. The reasoning of the remaining shock sign restrictions are deferred to Appendix \ref{appendix_restrictions}.
Regarding the impact sign restrictions imposed on the $L$-matrix via $\ell_i^{L} < \boldsymbol{R}_i^{L} \boldsymbol{l}_i' < \upsilon_i^{L}$ in Equation (7), we draw on a well-established literature on shock identification using sign restrictions. For the identification of a positive oil supply (news) shock, as presented in Table \ref{tab:restrictions}, we follow \cite{banbura2023drives} and assume that such a shock leads to an increase in oil prices, as well as in both consumer and producer prices.\footnote{\cite{banbura2023drives} elaborate on the identification of oil supply and demand shocks, referring to \cite{peersman2009oil}, \cite{kilian2008economic}, \cite{kilian2009not}, \cite{kilian2014role}, and \cite{baumeister2019structural}. Notably, our oil supply news is closely related to the shock identified by \cite{kaenzig2021}, where a positive shock implies the expectation of lower oil supply.} In addition, we assume that a positive oil supply shock induces a monetary tightening, depresses stock prices, and worsens labor market conditions. Output and its components are left unrestricted. Concerning monetary policy shocks, we adopt the identification strategy of \cite{chan2025rankingrestrictions}, under which a tightening of monetary policy is associated with increases in interest rates and bond yields, declines in stock prices, and a contraction in economic activity.\footnote{Extensive discussions on the identification of monetary policy shocks in VAR frameworks can be found in \cite{sims1972money}, \cite{sims1980macroeconomics}, \cite{sims1986forecasting}, \cite{faust1998robustness}, \cite{canova2005var}, \cite{canova2002monetary}, \cite{uhlig2005effects}, and \cite{rubio2010structural}.} The technology shock is identified following \cite{chan2025rankingrestrictions}, with the assumption that it increases economic activity, raises labor demand, and exerts downward pressure on prices. In response to such a shock, stock markets are expected to appreciate, while the policy rate declines in reaction to the deflationary environment.\footnote{For additional discussion on the sign identification of technology shocks, see \cite{dedola2007does} and \cite{peersman2009technology}.} To identify a financial risk shock, we assume a widening of the yield spread between corporate and ten year government bonds, a decline in stock market valuations, and a decrease in consumer prices, an approach consistent with \cite{chan2025rankingrestrictions}.\footnote{See also \cite{Korobilis2022} and \cite{furlanetto2019identification} for further discussion on the identification of financial shocks.} Finally, a government spending shock is identified by assuming an increase in government expenditures, a reduction in unemployment, and a rise in (stock) prices. This identification strategy also aligns with \cite{chan2025rankingrestrictions}.\footnote{See \cite{laumer2020government} and \cite{forni2010fiscal} for detailed treatments of fiscal shock identification.}
As previously noted, we impose shock sign restrictions on the oil supply, financial risk, and government spending shocks in 2001Q3 and 2008Q4 through $\ell_{\tilde{t}}^{f} < \boldsymbol{R}_{\tilde{t}}^{f} \boldsymbol{f}_{\tilde{t}} < \upsilon_{\tilde{t}}^{f}$ in Equation (8), as these restrictions are essential for the separation of the respective shocks. In 2001Q3, the terrorist attacks on September 11th resulted in a sudden and pronounced increase in financial risk, reflecting heightened uncertainty across financial markets. The temporary closure of the New York Stock Exchange and the flight of investors toward safe-haven assets such as Treasury bonds substantiate this assessment.\footnote{This interpretation is further supported by the Financial Stress Index; see \cite{stlfsi4}.} In terms of the oil shock, we assume it to be positive, as uncertainty surrounding oil supply rose significantly following the attacks, with market participants anticipating potential production cuts and hence, higher oil prices. Additionally, we assume a positive government spending shock in response to the enactment of the Emergency Supplemental Appropriations Act on September 18th, 2001.\footnote{For further details, see \cite{publiclaw107-38}, \cite{ramey2018government}, and \cite{blanchard2002empirical}.} For the fourth quarter of 2008, we impose a negative oil supply news shock, a positive financial risk shock, and a negative government spending shock. The bankruptcy of Lehman Brothers in late September 2008 precipitated widespread turmoil in global financial markets, leading to a significant increase in financial uncertainty during the subsequent quarter. Consequently, we posit a positive financial risk shock. \footnote{See again \cite{stlfsi4} for empirical evidence.} Regarding the oil supply shock, the financial crisis contributed to a downward revision of oil price expectations, thereby alleviating supply constraints. \footnote{In fact, the Organization of the Petroleum Exporting Countries (OPEC) held two meetings in 2008Q4 to address the decline in oil prices \citep{opecmeetings2008}.} In terms of government spending, the election of President Obama resulted in a reduced expectation of government expenditure due to announcements of cuts in military spending. \footnote{Additionally, the American Recovery and Reinvestment Act, aimed at stimulating the economy, was announced by President Obama in January 2009 and signed into law in February 2009 \citep{ARRA2009}.} As illustrated in Table \ref{tab:restrictions}, the narrative shock sign restrictions for both the financial risk and government spending shocks are critical for disentangling the two shocks. The additional shock sign restrictions further refine the identification process.
It is noted that, for the monetary policy and financial risk shocks, in principle, it would be possible for the two shocks to exhibit the same sign patterns.
However, given our set of restrictions, the data do not support the same sign patterns for both shocks, and thus our restrictions are sufficient to disentangle the shocks. In particular, we verify the uniqueness of the sign patterns for every posterior draw of the impact matrix $L$ and the structural shocks $f_{t}$.
\begin{table}[t]
\centering
\caption{Impact Sign and Shock Sign Restrictions to Identify Structural Shocks}
\label{tab:restrictions}
\vspace{0.5em}
\textbf{Impact Sign Restrictions}
\vspace{0.5em}
\resizebox{\textwidth}{!}{
\begin{tabular}{lccccc}
\toprule
\textbf{Variable} & \textbf{Oil Supply} & \textbf{Monetary Policy} & \textbf{Technology} & \textbf{Financial Risk} & \textbf{Government Spending} \\
\midrule
GDP & & $-1$ & $+1$ & & \\
Consumption & & $-1$ & $+1$ & & \\
Investment & & $-1$ & & & \\
Gov. Spending & & & & & $+1$ \\
Hours & & & $-1$ & & \\
Real Compensation & & & $+1$ & & $-1$ \\
Output per Hour & & & $+1$ & & \\
Unit Labor Cost & & & & $+1$ & $+1$ \\
Industrial Production & & $-1$ & & & \\
Capacity Utilization & & & & & \\
Employment & $-1$ & $-1$ & $+1$ & & \\
Unemployment & $+1$ & $+1$ & $-1$ & & $-1$ \\
Housing Starts & & & & & \\
M\&T Sales & & & & & \\
M\&T Inventories & & & & & \\
PCE Index & $+1$ & $-1$ & $-1$ & $-1$ & $+1$ \\
GDP Deflator & $+1$ & $-1$ & $-1$ & $-1$ & $+1$ \\
CPI & $+1$ & $-1$ & $-1$ & $-1$ & $+1$ \\
PPI & $+1$ & & $-1$ & & \\
FED Funds Rates & $+1$ & $+1$ & $-1$ & & \\
3-Month TB & & $+1$ & & & \\
1-Year TB & & $+1$ & & & \\
10-Year TB & & $+1$ & & & \\
BAA-GS10 Spread & & & $-1$ & $+1$ & \\
Monetary Base & & $-1$ & & & \\
M2 & & $-1$ & & & \\
S\&P500 & $-1$ & $-1$ & $+1$ & $-1$ & $+1$ \\
Dow Jones & $-1$ & $-1$ & $+1$ & $-1$ & $+1$ \\
US Dollar Index & & $+1$ & & & \\
Oil Price & $+1$ & & $+1$ & & \\
\bottomrule
\end{tabular}
}
\vspace{0.1em}
\textbf{Narrative Shock Sign Restrictions}
\vspace{0.5em}
\resizebox{\textwidth}{!}{
\begin{tabular}{lccccc}
\toprule
& \textbf{Oil Supply} & \textbf{Monetary Policy} & \textbf{Technology} & \textbf{Financial Risk} & \textbf{Government Spending} \\
\midrule
$>0$ &
\begin{tabular}{@{}l@{}}1986Q3, 1988Q4\\\textbf{2001Q3}, 2016Q4\end{tabular} &
&
&
\begin{tabular}{@{}l@{}}\textbf{2001Q3}, \textbf{2008Q4}\end{tabular} &
\begin{tabular}{@{}l@{}}1990Q4, \textbf{2001Q3}, 2001Q4,\\2002Q1, 2002Q3, 2003Q1,\\2006Q2, 2007Q4\end{tabular} \\
\addlinespace
$<0$ &
\begin{tabular}{@{}l@{}}2001Q4, \textbf{2008Q4} \\ 2014Q4\end{tabular} &
&
&
\begin{tabular}{@{}l@{}}1998Q4, 1999Q4\end{tabular} &
\begin{tabular}{@{}l@{}}1986Q3, 1988Q4, 1989Q4,\\1991Q4, \textbf{2008Q4}, 2011Q3,\\2013Q1\end{tabular}
\\
\bottomrule
\end{tabular}
}
\vspace{0.5em}
\begin{minipage}{\textwidth}
\footnotesize
\textit{Notes:} This table summarizes the impact sign restrictions and narrative shock sign restrictions to identify the five structural shocks. In the upper table, $+1$ ($-1$) implies a positive (negative) on impact response of the respective variable in the left column. In the lower table, $>0$ ($<0$) restricts the respective shock in the column to be positive (negative).
\end{minipage}
\end{table}
\subsection{Empirical Results} \label{emp_app_results}
In this section, we present selected results from our empirical analysis. Figure \ref{fig_selected_irfs} displays the impulse response functions (IRFs) of key macroeconomic and financial variables, including real GDP, industrial production, unemployment, the Consumer Price Index (CPI), and the S\&P 500 index, in response to identified structural shocks. The oil supply news shock, which manifests as an increase in oil prices, leads to a contraction in economic activity, a decline in labor demand, and an increase in consumer prices. As expectations about future economic conditions deteriorate, equity markets also decline. These dynamics are consistent with the empirical findings in the literature (e.g., \cite{kaenzig2021}, \cite{kilian2009impact}, \cite{kilian2009not}, \cite{hamilton2003oil}, \cite{ramey2011oil}). A contractionary monetary policy shock is found to reduce output, raise unemployment, and lower both prices and equity valuations (\cite{christiano2005nominal}, \cite{gertler2015monetary}, \cite{jarocinski2020deconstructing}). This response pattern suggests that the identified monetary policy shock is purged from any information effects or central bank signaling, thereby capturing a pure policy shock (see \cite{jarocinski22}, \cite{bs23a}, \cite{bs23b}). A positive technology shock increases both output and labor demand, while exerting downward pressure on prices due to productivity improvements. The stock market responds favorably. This is largely in line with existing evidence on the expansionary effects of technological innovations (\cite{peersman2009technology}). A shock to financial risk has strongly contractionary effects: output and prices decline persistently, unemployment rises, and stock markets fall. These results corroborate prior findings on the macroeconomic consequences of financial stress (\cite{gilchrist2012credit}, \cite{caldara2016macroeconomic}, \cite{christiano2014risk}, \cite{jurado2015measuring}, \cite{Korobilis2022}). Finally, an increase in government spending yields an expansion in output, a reduction in unemployment, upward pressure on prices, and a short-run increase in equity prices. These responses are consistent with a substantial body of empirical work (\cite{blanchard2002empirical}, \cite{ramey1998costly}, \cite{ramey2011identifying}, \cite{auerbach2012measuring}, \cite{mountford2009effects}, \cite{laumer2020government}). The complete set of impulse responses is provided in Section \ref{app_add_emp_results} of the Online Appendix.
\begin{figure}[t]
\centering
\caption{Selected Impulse Responses}
\includegraphics[width=0.99\textwidth]{pics/XXXnsim=250burnin=1000thin=10Try=1SV=0mag=0r=5Narrative=1narrs1imp2Hshrinkage=6plag4Sign=1SignT1NumRes=1start1983end2019proxyrule0_5x5_IRFs.pdf}
\label{fig_selected_irfs}
\vspace{0.5em}
\begin{minipage}{0.95\textwidth}
\footnotesize
\textit{Notes:} Median impulse responses of selected variables to an oil supply, monetary policy, technology, financial risk and government spending shock. The solid red line depict the medians and the shaded red areas the 68\% credible bands.
\end{minipage}
\vspace{-10pt}
\end{figure}
Table \ref{tab_fevd} presents the forecast error variance decomposition (FEVD) of GDP across various forecast horizons. The FEVD illustrates the proportion of the variance in GDP forecast errors that can be attributed to each of the identified structural shocks. Across all horizons, financial risk shocks emerge as the dominant driver of GDP fluctuations. In contrast, government spending shocks exhibit only a moderate contribution to the forecast error variance of GDP, while monetary policy and technology shocks account for a relatively smaller share of GDP fluctuations across all horizons. Our empirical results are in line with \cite{h24} who estimates the same 30-variable VAR and proxy-identifies the 5 structural shocks. In the Online Appendix, Section \ref{hd_gdp} shows the historical decomposition of GDP corroborating the finding that financial risk shocks are the largest driver of the business cycle. Further, in Section \ref{app_shorter_period} we show that if the sample is shortened to 2012 financial risk shocks take on an even more important role in driving the business cycle.\footnote{Further results on the above-explained shortened sample can be obtained on request. It is noted that our results from the shortened sample are even more similar to \cite{h24} as their baseline sample ends in 2012.}
\begin{table}[t]
\centering
\caption{Forecast error variance decompositions of GDP}
\resizebox{\textwidth}{!}{
\begin{tabular}{lccccc}
\toprule
& \textbf{Oil Supply} & \textbf{Monetary Policy} & \textbf{Technology} & \textbf{Financial Risk} & \textbf{Government Spending} \\
\midrule
$H=1$ & 28\% & 2\% & 4\% & 57\% & 8\% \\
$H=5$ & 28\% & 2\% & 5\% & 57\% & 8\% \\
$H=10$ & 29\% & 3\% & 6\% & 55\% & 9\% \\
$H=20$ & 29\% & 3\% & 7\% & 53\% & 9\% \\
\bottomrule
\end{tabular}
}
\label{tab_fevd}
\vspace{0.5em}
\begin{minipage}{\textwidth}
\footnotesize
\textit{Notes:} Table presents posterior medians of the forecast error variance decompositions of GDP attributed to the oil supply, monetary policy, technology, financial risk, and government spending shocks at horizons $H=1$, $H=5$, $H=10$ and $H=20$.
\end{minipage}
\end{table}
\section{Conclusion}
This paper presents a novel high-dimensional SVAR framework that accommodates a large number of linear inequality restrictions on both impact impulse responses and structural shocks, including their interactions. By incorporating a factor structure in the error terms and introducing a computationally efficient sampling algorithm that avoids rejection-based procedures, our approach addresses key limitations of existing methods - most notably, their limited scalability in high-dimensional settings. The flexibility to combine sign and shock restrictions enhances the precision of structural shock identification and enables economically meaningful inference even in large systems.
In a synthetic data exercise, we demonstrate the advantages of our proposed algorithm in large-scale VAR settings with multiple impact and shock sign restrictions, comparing our results to those of \citet{antolin2018narrative}. In our empirical application, we identify five structural shocks in a VAR with 30 variables, highlighting the practical utility of the method, particularly in revealing the prominent role of financial shocks in driving business cycle fluctuations. Regarding narrative shock sign restrictions, we show their effectiveness in sharpening inference by shrinking the identified set. More importantly, we demonstrate that shock sign restrictions can facilitate the disentanglement of structurally distinct shocks, as exemplified by the separation of financial risk and government risk shocks.
Overall, the proposed framework substantially broadens the scope for credible and computationally tractable structural analysis in high-dimensional VAR environments.
\section*{Disclosure statement}\label{disclosure-statement}
The authors have no conflicts of interest to declare. DeepL (version 1.47.0) was used to assist with grammar and language refinement in selected sections of the manuscript. All final edits and interpretations remain the sole responsibility of the authors. The data used in this study are available from the Federal Reserve Economic Data (FRED) database and from LSEG Data \& Analytics.
\clearpage
\small{\setstretch{0.85}
\addcontentsline{toc}{section}{References}
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\title{Online Appendix - Large structural VARs with multiple linear shock and impact inequality restrictions}
\begin{titlepage}
\setlength{\droptitle}{-3cm}
\maketitle
\begin{abstract}
\begin{singlespace}
\noindent
This is the online appendix for ``Large structural VARs with multiple linear shock and impact inequality restrictions''. The reader is referred to the paper for more detailed information.
\end{singlespace}
\end{abstract}
\bigskip
\noindent \textbf{Keywords:} Structural Identification, Sign Restrictions, Narrative Restrictions, Large BVARs \bigskip
\noindent \textbf{JEL classification:} C11, C32, C55, E50 \bigskip
\end{titlepage}
\pagenumbering{arabic}