Conditional projection methods for large-scale Bayesian VARs

bibunit\section{Introduction} Conditional projection methods have been a key component of macroeconometrics since at least doan1984forecasting and underpin the modern practice of counterfactual policy and scenario analysis crump2021large,adrian2026risks. Fully specified structural models commit the analysis to a single model by design and often restrict the size of the usable information set. An alternative is to use structural VARs, in the spirit of the structural scenario analysis (SSA) of antolin2021structural, in which scenarios are implemented as distributional restrictions on observables and structural shocks. Restricting subsets of the latter to their unconditional distribution defines the structural sources of the scenario, attributing it to identified economic forces.\footnote{The Lucas-robust policy counterfactuals of mckay2023can can be obtained as a special case of SSA breitenlechner2024fiscal; we return to this point in Section (ref).} Our paper contributes to this literature by developing a high-dimensional Bayesian VAR for (semi-)structural analysis to obtain conditional predictions with fast estimation and simulation algorithms. Rich information sets have proven successful in both forecasting and structural settings when combined with appropriate regularization banbura2010large. Our framework is a BVAR with a global--local shrinkage prior on the coefficients and a factor structure on the reduced-form errors. This setup serves two purposes. First, the prior and factor structure enable regularization alongside quick and order-invariant estimation of VARs with a large cross section of $n$ endogenous variables and long lag orders kastner2020sparse.\footnote{The latter can reduce bias in dynamic multiplier estimates at longer horizons Baumeister2025comment, which matters here because our conditional projections are built from the same moving-average matrices as impulse response functions (IRFs).} Second, following korobilis2022new, a small number $q\,(\ll n)$ of “primitive” shocks --- the factors --- drive the common dynamics, while cross-sectionally independent components capture idiosyncratic shocks; when the factors are orthogonal and identified, they admit an interpretation as structural economic shocks.\footnote{arias2026large and chan2025largesign discuss related high-dimensional SVAR methods for unrestricted covariance matrices, without the reduced-rank plus idiosyncratic structure. When the number of factors is equal to the number of observables ($q = n$), our framework is close to a conventional SVAR; irrespective of the underlying model, the conditional projection algorithm we develop is directly applicable. The factor assumption about the reduced-form errors is used, e.g., in banbura2026drives,chan2022large,gambetti2023agreed,pruser2024large.} Earlier conditional projection algorithms waggoner1999conditional can be computationally intensive and thus are often limited to smaller dynamic systems; scalability to richer information sets is possible through filtering-based methods banbura2015conditional or the closed-form solutions of antolin2021structural and chan2023conditional for joint predictive distributions. We build on the latter two, with an extension to separate restrictions on observables, structural shocks and idiosyncratic components. Further, we derive a fast and exact algorithm whose cost is cubic only in the number of restrictions, while the size of the forecasted system enters only linearly, which offers speed improvements relative to earlier implementations. We make three contributions: (i) a framework with separate distributional restrictions on observables, structural shocks, and idiosyncratic components; (ii) an exact and computationally fast sampler; and (iii) an empirical demonstration that the structural attribution of a scenario can drive vastly different counterfactual predictions. We illustrate these aspects using a quarterly dataset of $33$ US macroeconomic and financial variables, in which sign restrictions set-identify ten structural shocks. The scenarios are taken from kilian2026impact and impose a path for the nominal price of oil in the context of the 2026 Iran War and the closure of the Strait of Hormuz. Successively adding restrictions on the idiosyncratic components and non-driving structural shocks, we trace how the shocks required to deliver the same oil price path change and how this choice shapes the counterfactual predictions for the unrestricted observables. The macroeconomic implications differ markedly. The same scenario is consistent with a benign, mildly inflationary episode when its structural origins are unrestricted, and with pronounced stagflation alongside substantially wider predictive intervals when it is attributed to the structural oil price shock alone. \section{Econometric Framework} Consider a reduced-form VAR for $n$ variables $\bm{y}_t = (y_{1t},\hdots,y_{nt})'$ with $P$ lags in $\bm{x}_t = (\bm{y}_{t-1}',\hdots,\bm{y}_{t-P}')'$: \begin{equation} \bm{y}_t = \bm{a} + \sum_{p=1}^P \bm{A}_p \bm{y}_{t-p} + \bm{\eta}_t = \bm{a} + \bm{A}\bm{x}_t + \bm{\eta}_t, \quad \bm{\eta}_t \sim \mathcal{N}(\bm{0}_n,\bm{\Sigma}); \end{equation} $\bm{a}$ is an intercept, $\bm{A} = [\bm{A}_1,\hdots,\bm{A}_P]$ comprises the dynamic coefficient matrices $\bm{A}_p$, and $\bm{\eta}_t$ is a zero-mean reduced-form error with covariance matrix $\bm{\Sigma}$. Since an infinite number of structural models is consistent with a reduced-form representation as in ((ref)), structural economic interpretations require further restrictions, which are commonly imposed via decompositions of the covariance matrix $\bm{\Sigma}$. We use a factor model for the reduced-form error, because this specification is computationally attractive and encompasses a range of possible avenues for structural identification: \begin{equation} \bm{\eta}_t = \bm{L}\bm{\varepsilon}_t + \bm{S}^{1/2}\bm{u}_t, \quad (\bm{\varepsilon}_t', \bm{u}_t')'\overset{iid}{\sim}\mathcal{N}(\bm{0}_{q+n},\bm{I}_{q+n}), \end{equation} where $\bm{L}$ is an $n \times q$-matrix of loadings associated with $q$ factors $\bm{\varepsilon}_t$, and $\bm{S} = \text{diag}(s_1^2,\hdots,s_n^2) = \bm{S}^{1/2}\bm{S}^{1/2}$ is a diagonal matrix collecting the variances of the idiosyncratic component. This yields the decomposition $\bm{\Sigma} = \bm{L}\bm{L}' + \bm{S}$ anderson1956statistical. Any co-movement in the reduced-form errors is due to the common factors $\bm{\varepsilon}_t$ while the idiosyncratic component is cross-sectionally uncorrelated. For $n = q$, $\bm{S} = \bm{0}_{n\times n}$, and $\bm{L}$ invertible, premultiplying by $\bm{A}_0 \equiv \bm{L}^{-1}$ results in the standard SVAR $\bm{A}_0\bm{y}_t = \bm{A}_0(\bm{a} + \bm{A}\bm{x}_t) + \bm{\varepsilon}_t$ and $\bm{\varepsilon}_t$ admits a structural interpretation. In canonical SVARs, there are as many shocks as observables. Factor models are a dimension-reduction technique with $q \ll n$, based on the idea that only a few economic driving forces are responsible for the majority of the observed business cycle fluctuations. In this case, assuming $\bm{L}$ has full column rank, obtaining an approximate structural form of the model involves the pseudoinverse $\bm{A}_0 = (\bm{L}'\bm{L})^{-1}\bm{L}'$, to yield $\bm{A}_0\bm{y}_t = \bm{A}_0(\bm{a} + \bm{A}\bm{x}_t) + \bm{\varepsilon}_t + (\bm{L}'\bm{L})^{-1}\bm{L}'\bm{S}^{1/2}\bm{u}_t$. Invoking a central limit argument, korobilis2022new argues that the last term vanishes asymptotically for each $t$ as $n\rightarrow\infty$, and the interpretation of the factors as structural shocks follows. {Identifying assumptions}. It is well known that for $q \leq n$, the decomposition $\bm{\Sigma} = \bm{L}\bm{L}' + \bm{S}$ is in general not unique; multiple pairs $(\bm{L}, \bm{S})$ can be consistent with the same reduced-form. More generally, even if $\bm{L}\bm{L}'$ is uniquely identified, $\bm{L}$ itself is not, since for any $q\times q$ orthogonal matrix $\bm{Q}$, the matrix $\tilde{\bm{L}} = \bm{L}\bm{Q}$ yields the same product $\bm{L}\bm{L}' = \tilde{\bm{L}}\tilde{\bm{L}}'$. A solution to the latter issue is to impose zero restrictions directly on $\bm{L}$. This can achieve point identification at the cost of potentially arbitrary exclusion restrictions; by contrast, imposing sign restrictions leads to set-identification. Indeed, there is a long-standing tradition to set-identify structural models via sensible sign and zero restrictions baumeister2015sign,arias2018inference which we adopt. {Prior choices and Bayesian estimation}. Before turning to the conditional predictions we briefly summarize prior choices and the MCMC algorithm. We mostly follow earlier papers here, and details about the posteriors and the standard Gibbs sampling algorithm are provided in Appendix (ref). The number of parameters in the conditional mean is $k + n$ where $k = n^2P$, which quickly exceeds $T$ even for moderate $n$. This typically results in overfitting if estimated without any regularization, and can be computationally slow. We follow huber2019adaptive and shrink via a global--local (horseshoe) prior; for $nP \gg T$, fast sampling algorithms are used. Moreover, full-system estimation is often computationally infeasible, which is why recent approaches commonly rely on equation-by-equation estimation carriero2022corrigendum. The factor structure on the reduced-form errors in ((ref)) enables fast and order-invariant equation-by-equation estimation, since ((ref)) reduces to $n$ independent equations conditional on the factors. To impose the sign restrictions on $\bm{L}$, following korobilis2022new, each loading receives a pre-specified restriction $\mathcal{R}_{ij} \in \{+, -, c_{ij}, \text{NA}\}$, collected in the $n\times q$-matrix $\bm{\mathcal{R}}$: sign restrictions ($+$, $-$) are imposed via truncated normal priors on the respective loadings (i.e., informative priors in the spirit of baumeister2015sign), constants $c_{ij} \in \mathbb{R}$ via point masses (with $c_{ij} = 0$ yielding zero restrictions), and unrestricted loadings ($\text{NA}$) receive normal priors. This baseline approach can be combined with additional identifying restrictions and/or model features.\footnote{chan2022large use factor stochastic volatility to exploit heteroskedasticity for point identification; pruser2024large relies on a similar framework and uses non-Gaussian features of the structural shocks for identification. Another option is to replace the normal prior on the structural shocks in ((ref)) with, e.g., a truncated normal prior for specific shocks in specific periods berend2025large. External and internal instruments may be introduced by considering these as observed or partially latent factors korobilis2025exploring,pfarrhofer2025high. } We abstract from these options, noting that most such extensions require only minor adjustments for use with our proposed SSA algorithm. \subsection{Structural Scenario Analysis} The joint vector of forecasts is $\bm{y}_{T+(1:H)} = (\bm{y}_{T+1}',\hdots,\bm{y}_{T+H}')'$, where $H$ refers to the maximum forecast horizon. In the same vein, we stack $\bm{\varepsilon}_{T+(1:H)} = (\bm{\varepsilon}_{T+1}',\hdots,\bm{\varepsilon}_{T+H}')'$ and idiosyncratic errors $\bm{u}_{T+(1:H)} = (\bm{u}_{T+1}',\hdots,\bm{u}_{T+H}')'$ in a joint vector $\bm{\xi}_{T+(1:H)} = (\bm{\varepsilon}_{T+(1:H)}',\bm{u}_{T+(1:H)}')' \sim\mathcal{N}(\bm{0},\bm{I}_{d})$ with $d = (q+n)H$, which is standard normal based on ((ref)). Precise definitions and additional details about the vectors and matrices below are provided in Appendix (ref). Using $\bm{\mathcal{L}}_{(H)} = (\bm{I}_H \otimes \bm{L})$, $\bm{\mathcal{S}}^{1/2}_{(H)} = (\bm{I}_H \otimes \bm{S}^{1/2})$ and $\bm{\mathcal{V}} = [\bm{\mathcal{L}}_{(H)}, \bm{\mathcal{S}}_{(H)}^{1/2}]$ of size $nH \times d$, this allows us to write: $\bm{H}\bm{y}_{T+(1:H)} = \bm{h} + \bm{\mathcal{L}}_{(H)}\bm{\varepsilon}_{T+(1:H)} + \bm{\mathcal{S}}^{1/2}_{(H)}\bm{u}_{T+(1:H)} = \bm{h} + \bm{\mathcal{V}}\bm{\xi}_{T+(1:H)}$. Thus: \begin{equation} \bm{y}_{T+(1:H)} = \bm{H}^{-1}\bm{h} + \bm{H}^{-1}\bm{\mathcal{V}}\bm{\xi}_{T+(1:H)}, \end{equation} where we sometimes also write $\bm{m} = \bm{H}^{-1}\bm{h}$ and $\bm{M}' = \bm{H}^{-1}\bm{\mathcal{V}}$, so $\bm{y}_{T+(1:H)}\sim\mathcal{N}(\bm{m}, \bm{M}'\bm{M})$. The vector $\bm{h}$ collects initial conditions and any deterministic terms, and $\bm{H}$ encodes the dynamic VAR coefficients in a banded structure. The inverse $\bm{H}^{-1}$ by construction then collects the coefficient matrices of the vector moving average (VMA) representation of the underlying VAR. Specifically, these VMA coefficient matrices are $\bm{\Phi}_j = \bm{J}\bm{\mathcal{A}}^{j}\bm{J}'$ with $\bm{\Phi}_0 = \bm{I}_{n}$, defined using $\bm{J} = [\bm{I}_n,\bm{0}_{n\times n(P-1)}]$ and the companion form $\bm{\mathcal{A}} = [\bm{A}; \bm{I}_{n(P-1)}, \bm{0}_{n(P-1)\times n}]$. $\bm{H}^{-1}$ features copies of $\bm{\Phi}_0$ on its main diagonal, and the $h$th subdiagonal for $h = 1,2,\hdots,H-1,$ consists of copies of $\bm{\Phi}_h$. Intuitively, consider the IRF of the endogenous variables to a structural shock. The IRF to a structural shock in $\bm{\varepsilon}_{T+1}$ is defined as $\partial \bm{y}_{T+h+1} / \partial \bm{\varepsilon}_{T+1}' = \bm{\Phi}_h\bm{L}$ for $h = 0, 1, \hdots, H - 1$, with the timing convention that the impact response $h = 0$ refers to period $T+1$. The subdiagonal structure of $\bm{H}^{-1}$ propagates forward any vector it multiplies according to the dynamics of the model, e.g., structural shock impacts at each horizon, via $\bm{H}^{-1}\bm{\mathcal{L}}_{(H)}$ in ((ref)). The example showcases that IRFs and the associated VMA matrices are the fundamental building blocks of any conditional projection and scenario analysis, which can be implemented via specific restrictions, to which we turn next. Stochastic restrictions as in andersson2010density are imposed on observed variables, where there are $r_y$ restrictions, and $\bm{R}^{(y)}$ is $r_y \times nH$: \begin{equation} \bm{R}^{(y)}\bm{y}_{T+(1:H)} \sim \mathcal{N}(\bm{r}^{(y)}, \bm{\Omega}^{(y)}), \end{equation} on the structural shocks: $\bm{R}^{(\varepsilon)}\bm{\varepsilon}_{T+(1:H)} \sim \mathcal{N}(\bm{r}^{(\varepsilon)},\bm{\Omega}^{(\varepsilon)})$ with $r_{\varepsilon}$ restrictions and $\bm{R}^{(\varepsilon)}$ is $r_{\varepsilon} \times qH$; and, generalizing earlier papers, on the idiosyncratic component: $\bm{R}^{(u)}\bm{u}_{T+(1:H)} \sim \mathcal{N}(\bm{r}^{(u)},\bm{\Omega}^{(u)})$ with $r_{u}$ restrictions and $\bm{R}^{(u)}$ is $r_{u} \times nH$. The latter are stacked using $\bm{R}^{(\xi)} = \text{bdiag}(\bm{R}^{(\varepsilon)}, \bm{R}^{(u)})$, $\bm{r}^{(\xi)} = [\bm{r}^{(\varepsilon)}; \bm{r}^{(u)}]$ and $\bm{\Omega}^{(\xi)} = \text{bdiag}(\bm{\Omega}^{(\varepsilon)},\bm{\Omega}^{(u)})$, and written as: \begin{equation} \bm{R}^{(\xi)}\bm{\xi}_{T+(1:H)} \sim \mathcal{N}(\bm{r}^{(\xi)}, \bm{\Omega}^{(\xi)}). \end{equation} Plugging ((ref)) into ((ref)) yields $\bm{R}^{(y)}\left(\bm{m} + \bm{M}'\bm{\xi}_{T+(1:H)}\right) \sim \mathcal{N}(\bm{r}^{(y)}, \bm{\Omega}^{(y)})$; combined with ((ref)) we have: \begin{equation*} \underbrace{\begin{bmatrix} \bm{R}^{(y)}\bm{M}'\\ \bm{R}^{(\xi)} \end{bmatrix}}_{\bm{R}} \bm{\xi}_{T+(1:H)} \sim \mathcal{N}\big( \underbrace{\begin{bmatrix} \bm{r}^{(y)} - \bm{R}^{(y)}\bm{m}\\ \bm{r}^{(\xi)} \end{bmatrix}}_{\bm{r}}, \underbrace{\begin{bmatrix} \bm{\Omega}^{(y)} &\\ & \bm{\Omega}^{(\xi)} \end{bmatrix}}_{\bm{\Omega}} \big). \end{equation*} There are a total of $r = r_y + r_\varepsilon + r_u$ restrictions, $\bm{R}$ is $r \times (n+q)H$ and assumed to have full row rank (no redundancies), $\bm{r}$ is $r \times 1$ and $\bm{\Omega}$ is $r \times r$. These restrictions specify the distribution that $\bm{R}\bm{\xi}_{T+(1:H)}$ is required to have under the conditional forecast distribution. The restricted distribution of the innovations is the revision of their unconditional standard normal distribution that satisfies the restrictions while minimizing the Kullback--Leibler divergence to the unconditional distribution; antolin2021structural prove that in the Gaussian case this is equivalent to entropic tilting as in robertson2005forecasting. This revision replaces the implied marginal of $\bm{R}\bm{\xi}_{T+(1:H)}$ with $\mathcal{N}(\bm{r},\bm{\Omega})$, while leaving the conditional $\bm{\xi}_{T+(1:H)}$ given $\bm{R}\bm{\xi}_{T+(1:H)}$ unchanged. This yields the restricted shocks $\bm{\xi}_{T+(1:H)}^{\ast}\sim\mathcal{N}(\bm{\mu}_\xi, \bm{V}_\xi)$, with moments: \begin{equation} \bm{V}_\xi = (\bm{I}_{d} - \bm{R}^{+}\bm{R}) + \bm{R}^{+}\bm{\Omega}\bm{R}^{+\prime}, \quad \bm{\mu}_\xi = \bm{R}^{+}\bm{r}, \end{equation} where $\bm{R}^{+} = \bm{R}'(\bm{R}\bm{R}')^{-1}$ denotes the Moore--Penrose inverse of $\bm{R}$. This approach draws the full sequence of innovations $\bm{\xi}_{T+(1:H)}^{\ast}$ compatible with the restrictions. Inserting the restricted $\bm{\xi}_{T+(1:H)}^{\ast}$ into ((ref)) yields the final conditional projection, a draw from $\bm{y}_{T+(1:H)} \:\vert\:\bm{R},\bm{r},\bm{\Omega} \sim \mathcal{N}(\bm{\mu}_y, \bm{V}_y)$ with $\bm{\mu}_y = \bm{m} + \bm{M}'\bm{\mu}_{\xi}$ and $\bm{V}_y = \bm{M}'\bm{V}_{\xi}\bm{M}$ in terms of the observables.\footnote{These moments coincide with antolin2021structural; Appendix (ref) provides derivations, alongside an alternative that interprets the restrictions as noisy measurements of $\bm{R}\bm{\xi}_{T+(1:H)}$ and updates via Bayes' rule (combining rather than replacing the model-based forecast), which coincides with the above for $\bm{\Omega} = \bm{0}$ but differs otherwise.} Draws from the distribution of $\bm{\xi}_{T+(1:H)}^{\ast}$ can be obtained quickly via Algorithm (ref); we drop the time subscripts for notational convenience. The algorithm is related to cong2017fast and draws from the desired distribution, with $\mathbb{E}(\bm{\xi}^{\ast}) = \bm{\mu}_\xi$ and $\mathrm{Var}(\bm{\xi}^{\ast}) = \bm{V}_\xi$. Based on the auxiliary draws $\bm{\xi}_{0} \sim \mathcal{N}(\bm{0}_{d},\bm{I}_{d})$ and $\bm{v}_0 \sim \mathcal{N}(\bm{0}_r,\bm{\Omega})$, the restrictions hold by construction, $\bm{R}\bm{\xi}^{\ast} = (\bm{r} + \bm{v}_0)\sim\mathcal{N}(\bm{r},\bm{\Omega})$; hard constraints of the form $\bm{R}\bm{\xi}_{T+(1:H)} = \bm{r}$ are nested trivially for $\bm{v}_0 = \bm{0}_r$. More formal statements and a proof are provided in Appendix (ref). \begin{algorithm}[ht] \caption{Conditional forecast sampling algorithm.} \begin{enumerate}[label=(\arabic*)] • Sample $\bm{\xi}_{0} \sim \mathcal{N}(\bm{0}_{d},\bm{I}_{d})$ and $\bm{v}_0 \sim \mathcal{N}(\bm{0}_r,\bm{\Omega})$ independently; • Return $\bm{\xi}^{\ast} = \bm{\xi}_{0} + \bm{R}'(\bm{R}\bm{R}')^{-1}(\bm{r} + \bm{v}_0 - \bm{R}\bm{\xi}_{0})$, using \begin{itemize}[leftmargin = *,label={--}] • Solve for $\bm{x}$ from $(\bm{R}\bm{R}')\bm{x} = (\bm{r} + \bm{v}_0 - \bm{R}\bm{\xi}_{0})$; • Return $\bm{\xi}^{\ast} = \bm{\xi}_{0} + \bm{R}'\bm{x}$. \end{itemize} \end{enumerate} \end{algorithm} Algorithm (ref) factorizes only the $r \times r$ matrix $\bm{R}\bm{R}'$, so the cost of each draw is cubic only in the number of restrictions, while the size of the forecasted system enters linearly ($\mathcal{O}(r^2nH + r^3)$). By contrast, precision-based samplers factorize the complementary block of dimension $nH - r$ and gain the advantage once most of the system is restricted, while approaches that sample from the full $d$-dimensional Gaussian (or the $nH$-dimensional distribution of the observables) are never the fastest option in our comparisons and remain within a small factor of it only in small systems. In typical conditioning exercises with sparse to moderately dense restrictions, our algorithm is thus the fastest option, with an advantage that widens with the dimension of the stacked forecast vector; a detailed comparison of computation times is provided in Appendix (ref). \section{Empirical Application} We use a dataset patterned after crump2021large in our empirical illustration. The quarterly dataset includes $n = 33$ series: $31$ macroeconomic and financial variables for the US, plus two auxiliary variables to impose ranking restrictions; after transformations, the data span 1965Q1 to 2025Q4. Linked to the maximum forecast horizon $H = 20$ (i.e., $5$ years until 2030Q4), we use $P = (H - 1) = 19$ lags, which leaves the lag polynomial unrestricted at every horizon of the forecast window, so that the effective estimation sample starts in 1969Q4.\footnote{We pre-screen each series for outliers: observations farther than five interquartile ranges from the central quartiles are treated as missing and imputed prior to estimation, inspired by similar procedures often used with dynamic factor models carriero2021addressing. The results without outlier-adjustment are qualitatively similar. All series are then standardized to zero mean and unit variance; conditioning targets are mapped into standardized units accordingly, and all results are reported, using an inverse transformation after estimation, in the original units.} A similar dataset has been used in the applications of arias2026large,chan2025largesign, from which we also take the sign restrictions to enable a structural interpretation. Specifically, we set-identify $q = 10$ shocks: demand, investment, financial, monetary policy, government spending, technology, labor supply, wage bargaining, oil price and consumer sentiment. The number of shocks $q$ is thus fixed by the adopted identification scheme. Detailed variable descriptions and sign restrictions, as well as the resulting IRFs for selected key indicators, are provided in Appendix (ref). The estimated IRFs qualitatively match the responses reported in the cited papers. \begin{figure}[htbp] \caption{Oil-price scenarios. Left panel: imposed restriction targets (WTI, in USD per barrel) by forecast horizon; blank cells are unrestricted. Right panel: observed data (solid black line), restriction targets (black circles), and conditional forecast distributions of the oil price under hard and soft constraints (posterior medians alongside 68% credible sets).} \end{figure} We consider counterfactual scenarios from 2026Q1 onwards, based on assumptions about the nominal price of oil (WTI, in USD per barrel) in the context of the 2026 Iran War and the associated closure of the Strait of Hormuz. The \texttt{moderate}, \texttt{adverse} and \texttt{severe} scenarios are provided by kilian2026impact based on a DSGE model described in that paper; they reflect closures of increasing duration, with higher and later peaks of the oil price and a later return to a scenario-specific baseline. The left panel of Figure (ref) lists the timing and numerical magnitudes of these restrictions; at all other horizons the oil price is unrestricted and thus model-determined. The right panel of Figure (ref) shows the resulting predictive distributions when the targets are imposed as hard versus distributional (soft) constraints. For the soft constraints, we impose the restrictions on the mean but retain the model-implied variance, $\bm{\Omega}^{(y)} = \bm{R}^{(y)}\bm{M}'\bm{M}\bm{R}^{(y)\prime}$. For each parameter draw, the conditional means of the two versions coincide exactly, since $\bm{\mu}_\xi$ in ((ref)) does not depend on $\bm{\Omega}$ so the tightness of the constraints matters only for the predictive uncertainty. In the empirical results that follow, we focus only on the distributional version, so that predictive uncertainty is reflected more adequately antolin2021structural. The results in Figures (ref) and (ref) refer to different types of restrictions on the shocks that we combine with the scenario restrictions on oil prices. In addition to the restrictions on observables (\texttt{obs}), which are identical across all conditioning schemes, our framework allows us to also restrict the structural (\texttt{struc}) and idiosyncratic (\texttt{idio}) shocks, which is consequential for the counterfactual paths of the unrestricted observables. For \texttt{idio}, we restrict all idiosyncratic components to follow their unconditional distribution, that is, the scenario must be delivered by the (unrestricted) structural shocks alone. For \texttt{struc}, we restrict all structural shocks but the oil price shock to their unconditional distribution, that is, all but that shock are “non-driving” shocks and only the structural oil price shock may deviate. Combining these options yields the four conditioning schemes labeled \texttt{obs}, \texttt{obs/idio}, \texttt{obs/struc} and \texttt{obs/idio/struc} in the figures.\footnote{Restricting shocks to their unconditional distribution amounts to normally distributed soft constraints with zero mean and unit covariance: for \texttt{idio}, $\bm{R}^{(u)} = \bm{I}_{nH}$; for \texttt{struc}, $\bm{R}^{(\varepsilon)}$ selects all but the oil price shock at every horizon, labeling the latter the driving shock and all others non-driving; in both cases $\bm{r} = \bm{0}$ and $\bm{\Omega} = \bm{I}$ of conformable dimension. Other (sets of) driving shocks can be designated by adjusting the dimensions of the restricted block.} \begin{figure}[ht] \caption{Posterior means of the counterfactual structural-shock distributions across forecast horizons, by conditioning scheme (column blocks) and scenario (rows within blocks).} \end{figure} Figure (ref) shows the means of the counterfactual shock distributions across horizons under the four conditioning schemes, that is, the realizations of the structural shocks required to deliver the stipulated path of the observed oil price. As antolin2021structural emphasize, restricting the structural origins of a scenario is pertinent: the required shocks differ markedly across conditioning schemes, both in their composition and in their magnitudes. By construction, if the structural shocks are left unrestricted (\texttt{obs}), we obtain the most likely combination of shocks under the model's unconditional shock distribution (the minimal Kullback--Leibler tilt) that generates the stipulated shift in nominal oil prices. Notably, the largest contribution on impact does not come from the oil price shock itself (mean deviations of $0.5$ to $0.8$ standard deviations, depending on the scenario), but from positive financial shocks (about $1$); negative technology shocks (around $-0.6$) also contribute meaningfully. After the oil-price-increasing impact, the signs of these deviations reverse in subsequent periods, providing the offsetting forces that restore the baseline; beyond the final restricted horizon, none of the structural shocks deviate meaningfully from their unconditional distribution. The deviations also increase in size and persistence with the severity of the scenario. Under \texttt{obs/idio}, the composition of the required shocks is virtually unchanged, but the deviations roughly double (e.g., up to more than $2$ standard deviations for the financial shock): with the idiosyncratic margin shut down, the structural shocks alone must deliver the scenario. The picture changes fundamentally once the non-driving structural shocks are restricted. Under \texttt{obs/struc}, only the oil price shock deviates, with a moderate mean of about $1$ standard deviation on impact, while all other structural shocks are forced to their unconditional distribution (up to Monte Carlo noise). In this case, the bulk of the adjustment is absorbed by the idiosyncratic component of the oil price equation (a mean deviation of about $4$ standard deviations on impact; not shown in Figure (ref)), whereas all other idiosyncratic components remain at their unconditional distribution. When both margins are closed (\texttt{obs/idio/struc}), the required magnitude of the structural oil price shock increases by an order of magnitude, offset by large negative realizations in subsequent quarters. Shock sequences of this size are no “modest interventions” in the wording of leeper2003modest, and thus not robust to the Lucas critique. Only the special case of imposing the target for the first forecast period alone and attributing it to the structural oil price shock would amount to a scenario constructed purely from an identified shock, which is robust to the Lucas critique in the sense of mckay2023can, under the assumption that the transmission of the identified shock is invariant to the intervention. A single shock, however, may fail to deliver the stipulated multi-period path of the oil price; considering a richer menu of driving shocks could address this and might yield more realistic (smaller) required shock sequences. This also relates to the approach of caravello2024evaluating, whose counterfactuals build on two complementary inputs, namely conditional forecasts and the causal effects of policy shocks, which correspond to $\bm{m}$ and blocks of $\bm{M}'$ in our framework. \begin{figure}[ht] \caption{Conditional forecasts of real GDP growth and GDP deflator inflation (year-over-year, in percent) under the alternative conditioning schemes (rows) and scenarios (columns). The solid black line shows the observed data, the dashed black line the median unconditional forecast. The colored line and shaded area are the posterior median and 68% credible set of the conditioning scheme; the thin grey lines repeat the median conditional forecasts of the other three schemes; y-axis clipped for readability.} \end{figure} To economize on space, we focus on the consequences for two unrestricted observables in Figure (ref), real GDP growth and inflation in the GDP deflator (both year-over-year). Irrespective of the scenario and conditioning scheme, the increase in oil prices is associated with inflationary pressures as measured by the GDP deflator, but the implied magnitudes differ markedly. The pass-through is weakest under \texttt{obs/struc}, where peak median inflation of $3.5$ to $3.6$ percent barely exceeds the unconditional forecast of about $3.3$ percent. It is somewhat stronger under \texttt{obs} (peaks of $4.0$ to $4.8$ percent, depending on the scenario), and most pronounced when the idiosyncratic components are restricted: peak median inflation reaches $4.8$ to $6.5$ percent under \texttt{obs/idio} and $5.0$ to $7.8$ percent under \texttt{obs/idio/struc}. Inflation peaks at or within two quarters of the scenario-specific oil price peak, and the median paths largely return to the unconditional forecast within about two to three years. The scenario does not result in a contraction of GDP growth unless both the idiosyncratic and the non-driving structural shocks are restricted to their unconditional distribution. Under \texttt{obs} and \texttt{obs/struc}, the counterfactual medians stay within $1.4$ percentage points of the unconditional forecast, and under \texttt{obs/idio} the paths are mildly expansionary on impact followed by a modest slowdown. When the scenario is explicitly tied to a structural oil price shock (\texttt{obs/idio/struc}), by contrast, the counterfactual paths produce a pronounced pattern of stagflation. Median GDP growth troughs at $-0.6$, $-2.6$ and $-5.7$ percent in the \texttt{moderate}, \texttt{adverse} and \texttt{severe} scenarios, respectively, with the trough coinciding with the scenario-specific oil price peak. As the oil price returns to its baseline, growth overshoots before gradually converging back to the unconditional forecast. This conditioning scheme also carries substantially higher predictive uncertainty: at the trough of the \texttt{severe} scenario, the 68% band for GDP growth spans roughly $20$ percentage points, compared with $4$ to $5$ percentage points under the other schemes and for the unconditional forecast. Certainty about the structural origin of the scenario thus comes at the price of much more diffuse counterfactual predictions. \section{Conclusions} This paper develops methods for SSA in high-dimensional BVARs with a factor structure on the reduced-form errors. The framework accommodates separate distributional restrictions on observables, structural shocks and idiosyncratic components, and we propose a fast simulation-based algorithm. In an application to oil price scenarios in the context of the 2026 closure of the Strait of Hormuz, we show that restricting the structural origins of a scenario is pertinent. The same stipulated oil price path is consistent with outcomes ranging from a mostly benign, mildly inflationary episode to pronounced stagflation, depending on which types of shocks are permitted to deliver it. Future research avenues include extensions of the model to include stochastic volatility and other forms of parameter time variation, which would allow predictive uncertainty and transmission dynamics to change over time; conditional on the respective parameter paths, the model remains Gaussian and the algorithms developed in this paper apply with minor modifications. The framework also lends itself to explicit manipulations of impulse responses to design policy counterfactuals, e.g., by computing the conditional forecast of the system initialized at its equilibrium, subjecting it to a single structural shock on impact, and restricting its propagation through selected observables. {\setstretch{1.0}\putbib}