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Nonlinear Drivers of Macroeconomic Tail Risk: A Threshold Stochastic Volatility-in-Mean VAR with Regime-Dependent Leverage
\pagenumbering{arabic}
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\author[1]{Haroon Mumtaz}
\affil[1]{Queen Mary University of London}
\author[2]{Sofia Velasco}
\affil[2]{Banco de España}
\title{Nonlinear Drivers of Macroeconomic Tail Risk:\\ A Threshold Stochastic Volatility-in-Mean VAR with Regime-Dependent Leverage
\thanks{\footnotesize Corresponding author: Haroon Mumtaz ([email removed]). The views expressed in this paper are those of the authors and do not represent those of Banco de España or the ESCB.}
}
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{\LARGE\bf Nonlinear Drivers of Macroeconomic Tail Risk: A Threshold Stochastic Volatility-in-Mean VAR with Regime-Dependent Leverage}
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\begin{abstract}
\setlength{\parskip}{2pt}
The tails of macroeconomic outcomes can respond differently from the centre of their distribution: shocks with modest effects on median growth or inflation can shift downside growth or upside inflation risk. We develop a threshold stochastic-volatility-in-mean VAR with regime-dependent leverage to study their structural drivers. The model allows endogenous interactions between outcomes and volatility, contemporaneous level--volatility dependence, and regime-specific propagation. In nearly 150 years of U.S.\ data, predictive model selection supports three inflation-defined regimes. We identify business-cycle, financial, macroeconomic-uncertainty, and financial-uncertainty shocks and decompose their contributions to growth- and inflation-at-risk. The structural composition of tail risk differs from that of the predictive median. Business-cycle shocks dominate the median response of GNP growth but account for a substantially smaller share of growth-at-risk. Macroeconomic uncertainty makes a material contribution to both growth- and inflation-at-risk, with its share of growth-at-risk increasing with the magnitude of a positive macroeconomic-uncertainty impulse, despite its limited role at the median. In high-inflation states, the contribution of financial uncertainty to inflation-at-risk rises with the magnitude of positive financial-uncertainty impulses.
\end{abstract}
\vspace{-1ex}
\noindent \textbf{JEL Classification}: C11, C32, C53, E32, E44\\
\noindent\textbf{Keywords}: Threshold effects; Stochastic volatility in mean; Leverage; Growth-at-risk; Inflation-at-risk
\clearpage
\spacingset{1.8}
\section{\label{intro}Introduction}
Recent contributions in macroeconomics and finance document pronounced asymmetries in the U.S.\ predictive distributions of economic growth and inflation. Financial conditions are particularly informative about downside risks to future activity \citep{Giglio2016,adrian2019,DelleMonacheDePolisPetrella2024}, while the determinants of inflation can affect different parts of its predictive distribution differently, generating variation in upside inflation risk that is not captured by the conditional mean \citep{LopezSalidoLoria}. Moreover, \citet{loria-matthes-zhang-2025} show that monetary, financial,
uncertainty, and oil-price shocks can generate larger responses at the tenth
percentile of future activity than at the conditional median. The recurrence of this asymmetry across different shocks points toward a role for nonlinear propagation, rather than a mechanism specific to a single disturbance. This raises two related questions: do the shocks that drive movements in the centre of the predictive distribution also drive growth- and inflation-at-risk, and does their propagation through the predictive distribution depend on the state of the economy?
We address these questions by developing a threshold stochastic-volatility-in-mean VAR for the joint dynamics of GNP growth, inflation, the credit spread, and their endogenous stochastic volatilities. The model contains two interacting mean--volatility channels. First, past macroeconomic outcomes enter subsequent volatility dynamics, and innovations to macroeconomic outcomes and volatility can be contemporaneously correlated. Together, these features allow adverse innovations to be associated with higher subsequent volatility, a macroeconomic analogue of the leverage mechanism documented in financial markets \citep{Black1976,Christie1982,Schwert1989}. Second, stochastic volatility can feed back into expected macroeconomic
outcomes through a volatility-in-mean channel \citep{mumtaz2018}, consistent
with the broader real effects of uncertainty emphasised by \citet{Bloom2009}. The threshold structure allows these mean--volatility interactions, as well as the broader dynamics of the system, to change across inflation regimes. As a result, the same shock can reshape the joint predictive distribution of activity, inflation, and financial conditions differently depending on the prevailing inflation state.
The model combines the endogenous mean--volatility interactions of \citet{mumtaz2018} with the state-dependent transmission of uncertainty emphasised by \citet{alessandri2019financial}. Most closely, \citet{CaldaraSVOL} use an SV-in-mean VAR to trace the
structural sources of macroeconomic and financial tail risk. Their model
generates state-dependent distributional responses through endogenous
mean--volatility interactions, even though the dynamic coefficients and the
correlation matrix of the standardised innovations are constant. We allow
these coefficients and innovation correlations to change across
inflation-defined threshold regimes. The model can therefore distinguish
changes in the prevailing volatility state from changes in the parameters
governing mean--volatility feedback and shock transmission. Complementary data-rich evidence relates heterogeneous tail risk across variables to common mean--volatility interactions \citep{caldara2024risk}.\footnote{Related evidence separates common level and volatility factors in large macroeconomic panels and finds economically meaningful second-moment dynamics \citep{GorodnichenkoNg2017}. \citet{MumtazVelasco2026} model the joint evolution of level and volatility factors in a dynamic factor framework.}
Relatedly, \citet{ccm2024} show that BVARs with stochastic volatility can
capture more variable downside than upside growth risk even when their
one-step-ahead conditional predictive distributions are symmetric. Our focus
is on the structural drivers of these risks and how the parameters governing
their transmission vary across inflation regimes.
We apply the model to quarterly U.S.\ data from 1875 to 2024. The long sample permits the high-inflation state to be characterized from repeated episodes under distinct historical circumstances, rather than primarily from the Great Inflation. Predictive model selection based on recursive joint density forecasts of GNP growth and inflation supports a three-regime specification defined by inflation. Inflation therefore indexes the macroeconomic environment in which shocks propagate; it does not identify the source of the structural disturbances.
The estimated inflation regimes are not simply volatility states. The Great Depression and the Global Financial Crisis feature exceptionally high innovation volatility but belong predominantly to the low-inflation regime. The reduced-form estimates further suggest that the relationship between adverse output news and subsequent output volatility weakens as inflation rises.
The structural analysis delivers the paper's main result: the shocks that dominate movements in the centre of the predictive distribution need not dominate movements in its tails. Conditional on the model and the max-share identification, the structural composition of tail risk differs from that of predictive-median responses. Business-cycle shocks dominate the median response of GNP growth but account for a substantially smaller share of growth-at-risk. Macroeconomic uncertainty makes a material contribution to both growth- and inflation-at-risk despite its limited role at the median. Its contribution to growth-at-risk rises with the magnitude of positive macroeconomic-uncertainty shocks, while the contribution of financial uncertainty to inflation-at-risk rises with the magnitude of positive financial-uncertainty shocks in high-inflation histories.
These are model-implied structural decompositions, conditional on the estimated model and identifying restrictions. They do not map an individual reduced-form leverage correlation one-for-one into a structural response. Rather, the results reflect the joint operation of regime-specific mean--volatility interactions, dynamic propagation, and endogenous regime transitions.
Section~\ref{model} presents the model and estimation. Section~\ref{dms} describes the data, model selection, risk measurement, and the estimated inflation regimes. Section~\ref{SVAR} identifies structural shocks, studies their nonlinear tail-risk responses, and decomposes the drivers of those responses. Section~\ref{conclu} concludes.
\section{\label{model}A Stochastic Volatility-in-Mean Threshold VAR with Leverage}
This section develops an empirical framework that combines the endogenous mean--volatility interaction of \citet{mumtaz2018} with the threshold-dependent transmission of uncertainty shocks in \citet{alessandri2019financial}. The former allows level and volatility innovations to be contemporaneously correlated and volatility to feed back into macroeconomic outcomes; the latter shows that the transmission of uncertainty can vary across financial regimes. We allow these two dimensions to interact: both the coefficients governing macroeconomic and volatility dynamics and the correlations between level and volatility shocks can change across regimes. Macroeconomic risk can therefore vary not only with the level of volatility, but also with how level and volatility shocks co-move and propagate. The regimes are determined by an observed lagged variable, while the number of regimes and the variable defining them are selected by out-of-sample predictive performance in Section~\ref{dms}.
\subsection{Model specification}
A regime-switching VAR model with stochastic volatility describes the log-volatility processes and the associated reduced-form dynamics of the VAR system:
\begin{align}
\tilde h_{t+1}&=\sum_{m=1}^{M}\Big(\alpha_m+\theta_m \tilde h_t+\sum_{j=1}^{Q}d_{j,m}\,Y_{t-j}\Big)\mathcal{I}(S_t=m)+\sum_{m=1}^{M}\tilde S_m^{1/2}\,\mathcal{I}(S_t=m)\,\eta_t, \label{eq:volh}\\
Y_t&=\sum_{m=1}^{M}\Big(c_m+\sum_{j=1}^{P}\beta_{j,m}\,Y_{t-j}+\sum_{k=1}^{K}b_{k,m}\,\tilde h_{t-k}\Big)\mathcal{I}(S_t=m)+H_t^{1/2}e_t. \label{eq:obs}
\end{align}
Equation~(\ref{eq:volh}) describes the evolution of the stochastic volatilities, collected in the $N\times1$ vector of unobserved log-volatility processes $\tilde h_t$. Their dynamics depend on lagged values of $\tilde h_t$ through $\theta_m$, which is allowed to be a full matrix and therefore permits the volatility processes to interact dynamically with one another. Volatility is also endogenous: through $d_{j,m}$, past values of the observable variables affect the evolution of $\tilde h_{t+1}$. The innovations to the volatility equation are scaled by $\tilde S_m^{1/2}$, where $\tilde S_m=\mathrm{diag}(\tilde s_m)$ and $\tilde s_m=[s_{1,m},s_{2,m},\dots,s_{N,m}]'$. Thus, both the coefficients governing volatility dynamics and the variances of the volatility shocks are regime specific, with $S_t\in\{1,\dots,M\}$ denoting the regime prevailing at time $t$ and $\mathcal{I}(S_t=m)$ the corresponding indicator.
\noindent Equation~(\ref{eq:obs}) describes the evolution of $Y_t$, the $N\times1$ vector of observable variables. Its conditional mean depends on lags of $Y_t$ through the coefficients $\beta_{j,m}$ and on lags of $\tilde h_t$ through $b_{k,m}$. The latter introduces the stochastic-volatility-in-mean channel: changes in volatility can therefore affect expected macroeconomic outcomes rather than only their dispersion. The reduced-form innovation is $H_t^{1/2}e_t$, where $e_t$ is an $N\times1$ vector of standardised residuals and $H_t=\mathrm{diag}(\exp(\tilde h_t))$ collects their time-varying variances. As in the volatility-transition equation, all reduced-form coefficients are allowed to vary across regimes.
The disturbances are jointly distributed as
\begin{equation}
\varepsilon_t=
\begin{pmatrix}
\underbrace{\eta_t}_{N\times1}\\[10pt]
\underbrace{e_t}_{N\times1}
\end{pmatrix}
\sim N(0,\Sigma_m),
\qquad
\underbrace{\Sigma_m}_{2N\times2N}=
\begin{pmatrix}
\Sigma_{\eta,(m)} & \Sigma_{\eta e,(m)}\\[2pt]
\Sigma_{\eta e,(m)}' & \Sigma_{e,(m)}
\end{pmatrix},
\label{eq:sigmar}
\end{equation}
where each block $\Sigma_{\eta,(m)}$, $\Sigma_{e,(m)}$, and $\Sigma_{\eta e,(m)}$ is $N\times N$. The diagonal elements of $\Sigma_m$ are restricted to one. The off-diagonal block $\Sigma_{\eta e,(m)}$ captures the contemporaneous correlation between shocks to the stochastic volatilities and shocks to the endogenous variables: its $(i,j)$ element is $\operatorname{corr}(\eta_{i,t},e_{j,t})$. Because $\Sigma_{\eta e,(m)}$ is regime specific, both the sign and the strength of this leverage relationship can vary across regimes. The same draw $\varepsilon_t=(\eta_t',e_t')'$ supplies $\eta_t$ to the transition for $\tilde h_{t+1}$ and $e_t$ to the observation equation for $Y_t$. Thus, $\Sigma_{\eta e,(m)}$ describes the contemporaneous correlation between the level and volatility innovations, while the volatility innovation affects the subsequent log-volatility state. Restricting the diagonal of $\Sigma_m$ to unity makes it a correlation matrix, so that all scale is carried by $\tilde S_m^{1/2}$ and $H_t^{1/2}$. The resulting time-varying covariance matrix of the reduced-form disturbances in equations~\eqref{eq:volh}--\eqref{eq:obs} is
\begin{equation}
\underbrace{\Omega_t}_{2N\times2N}=
\sum_{m=1}^{M}\mathcal{I}(S_t=m)
\begin{pmatrix}
\tilde S_m^{1/2} & 0\\
0 & H_t^{1/2}
\end{pmatrix}
\Sigma_m
\begin{pmatrix}
\tilde S_m^{1/2} & 0\\
0 & H_t^{1/2}
\end{pmatrix}',
\label{eq:leverage}
\end{equation}
which combines regime-specific shock correlations and volatility-shock variances with the continuously evolving stochastic volatilities in $H_t$.
The regime process is defined by
\begin{align}
S_t&=1 &&\text{if } Y^{*}_{t-\mathrm{delay}}\le \mathrm{tar}_1,\notag\\
S_t&=m &&\text{if } \mathrm{tar}_{m-1}< Y^{*}_{t-\mathrm{delay}}\le \mathrm{tar}_m, \quad m=2,\dots,M-1,\notag\\
S_t&=M &&\text{if } Y^{*}_{t-\mathrm{delay}}> \mathrm{tar}_{M-1}, \label{eq:regime}
\end{align}
where $Y^{*}_t$ is a threshold series, a possibly transformed lag of one of the endogenous variables; $\mathrm{delay}$ is the lag at which it enters; and $\mathrm{tar}_1\le\dots\le\mathrm{tar}_{M-1}$ are the $M-1$ ordered thresholds. Because the regime is determined by a lagged endogenous variable, regime switches are endogenous rather than governed by a latent transition process.\footnote{The threshold variable and the number of regimes $M$ are selected across candidate specifications in the out-of-sample exercise of Section~\ref{dms}. Conditional on a specification, the threshold values and delay are estimated within the model.}
Following \citet{CaldaraSVOL}, the model contains an endogenous-volatility, or leverage, mechanism and a distinct volatility-in-mean mechanism. The leverage mechanism has a dynamic component, through which past macroeconomic outcomes affect subsequent volatility via $d_{j,m}$, and an innovation component, through which level innovations are correlated with innovations to subsequent volatility via $\Sigma_{\eta e,(m)}$ \citep{Black1976,Christie1982,Schwert1989}. The latter is the regime-specific leverage correlation reported below. Separately, volatility feeds back into expected macroeconomic outcomes through $b_{k,m}$ \citep{mumtaz2018}, consistent with the broader real effects of uncertainty emphasised by \citet{Bloom2009}. The threshold structure allows all three components, together with the remaining macroeconomic and volatility dynamics, to vary across regimes.
\subsection{\label{mc}Estimation}
We estimate the model using a Gibbs sampling algorithm. Several conditional posterior distributions are non-standard, including those
of the thresholds, the regime-specific correlation matrices, and the
stochastic volatilities. This section describes the prior distributions and summarizes the main blocks of the sampler; full derivations and implementation details are provided in the Appendix.\footnote{We assess finite-sample recovery in calibrated two- and three-regime simulations with regime-specific dynamics and leverage correlations. In both designs, the sampler accurately recovers the thresholds and delay, tracks the latent log-volatilities closely, and produces coefficient credible intervals that cover the true values. The posterior estimates of the error correlations generally have the same sign as their true values, although some correlations involving latent volatility shocks are attenuated towards zero. The full design and results are reported in the Appendix.}
\subsubsection{Priors and starting values}
\paragraph{VAR and volatility-transition coefficients} Let $\Gamma_m$ collect the coefficients of the observation equation~\eqref{eq:obs} in regime $m$, and $\tilde\Gamma_m$ those of the volatility-transition equation~\eqref{eq:volh}. For both, we follow \citet{Banbura-Giannone-Reichlin-10Paper} and implement a Minnesota-type prior through dummy observations, with a common overall tightness of $0.2$ and prior means for each variable's own first-lag coefficient obtained from individual AR(1) regressions on a pre-sample training period. The coefficients on lagged stochastic volatilities in
equation~\eqref{eq:obs} use a separate prior scale,
$\tilde c=1$, in the dummy-observation notation of the
Appendix, while the prior on every intercept is set effectively flat. Because these dummy observations discipline only the coefficients and not the regime- and time-varying error covariance, the remaining primitives are assigned priors separately below.
\paragraph{Initial log-volatility} Following \citet{cogley-sargent-05}, we use a pre-sample training period to set a Gaussian prior for the initial state $\tilde h_0$, centred on the logarithm of the diagonal elements of the OLS residual covariance matrix estimated on that pre-sample, with prior variance $0.1I$.
\paragraph{Volatility-shock variances and the correlation matrix $\Sigma_m$} The diagonal elements of $\tilde S_m$ are assigned independent inverse-Gamma priors. For $\Sigma_m$, the unit-diagonal restriction already imposed in equation~\eqref{eq:sigmar} leaves only its off-diagonal correlations free; for these we assume a flat prior over the region in which $\Sigma_m$ is positive definite, the natural counterpart of the ``separation strategy'' of \citet{44b58bef-fe61-3589-95e9-7e6440f4511a}, which factors a covariance matrix into standard deviations and a correlation matrix. Unlike a prior placed on the elements of a Cholesky factorisation of $\Sigma_m$, this flat prior is invariant to how the shocks $\varepsilon_t$ are ordered.
\paragraph{Thresholds and delay} Each threshold is assigned a Gaussian prior centred on an economically meaningful percentile of the candidate threshold variable: for the benchmark inflation-threshold specification, the lower threshold is centred on the $50$th percentile and the upper on the $80$th, with a common prior variance of $0.1$, tight relative to the range of every candidate threshold variable. Draws that violate the ordering $\mathrm{tar}_1\le\dots\le\mathrm{tar}_{M-1}$ or leave any regime with too few observations receive zero prior mass. The delay has a discrete uniform prior over its admissible values; as noted above, the threshold variable itself is chosen across candidate specifications by the out-of-sample exercise of Section~\ref{dms}, rather than sampled within a given specification.
\subsubsection{Posterior simulation}
Let $\Psi$ denote the remaining parameters and latent states. The Gibbs sampler cycles through the following blocks:
\begin{enumerate}[itemsep=3pt,parsep=0pt,topsep=4pt]
\item \textbf{Thresholds and delay.}
Conditional on the remaining parameters, the threshold posterior is proportional to the likelihood times the prior, subject to the ordering and minimum-regime-size restrictions. Because the likelihood is a step function of the thresholds, we sample them using a shrinkage slice sampler. The delay is drawn from its discrete conditional posterior, with probabilities proportional to the likelihood at each admissible value.
\item \textbf{Regime-specific coefficients.}
Given the thresholds and delay, the regime allocation $S_t$ is known. Conditional on the stochastic volatilities and the remaining covariance parameters, the coefficients in equations~\eqref{eq:volh} and~\eqref{eq:obs} are sampled regime by regime from conditionally Gaussian regression posteriors. Draws implying explosive dynamics are rejected.
\item \textbf{Volatility-shock variances.}
Conditional on the level and volatility innovations, the diagonal elements of $\tilde S_m$ have non-standard conditional posteriors because the level and volatility disturbances are correlated. We sample these parameters using an independence Metropolis--Hastings step based on an inverse-Gamma approximation to the conditional posterior.
\item \textbf{Regime-specific correlation matrices.}
Given standardized level and volatility innovations, each $\Sigma_m$ is sampled over the space of positive-definite correlation matrices. We update its free correlations one at a time, in random order, using slice sampling; the positive-definiteness restriction implies an admissible interval for each conditional update \citep{44b58bef-fe61-3589-95e9-7e6440f4511a}. This avoids tuning a random-walk proposal and enforces the
positive-definiteness restriction within each scalar update.
\item \textbf{Stochastic volatilities.}
Conditional on the remaining parameters, equations~\eqref{eq:volh}--\eqref{eq:obs} define a nonlinear state-space system because the latent volatilities enter both the conditional mean and variance of $Y_t$. We draw the full path $\{\tilde h_t\}_{t=1}^{T}$ using particle Gibbs with ancestor sampling \citep{JMLR:v15:lindsten14a}, with the state-space matrices switching according to the current regime allocation.
\end{enumerate}
\section{\label{dms}Regime-Dependent Leverage and Macroeconomic Tail Risk}
Our empirical analysis uses a parsimonious three-variable model and nearly 150 years of U.S.\ quarterly data. We compare candidate threshold variables and numbers of regimes by the fit of their recursive joint predictive densities to realised GNP growth and inflation. On this criterion, the data favour the specification with three regimes defined by four-quarter inflation, which we use as the benchmark in the reduced-form and structural analyses below. Throughout the empirical analysis, we define growth-at-risk as the 5th percentile of the predictive distribution of GNP growth and inflation-at-risk as the 95th percentile of the predictive distribution of inflation.
\subsection{Data and variables}
The raw quarterly data span 1875Q2--2024Q1. After transformations, lags, and the pre-sample used to calibrate the priors, the estimation sample begins in 1881Q3. The three variables are real GNP growth, inflation, and the corporate credit spread. Real activity is measured by the quarterly growth rate of real GNP, inflation by the quarterly growth rate of the GNP deflator, and the credit spread by the difference between a corporate bond yield and the 10-year U.S.\ government bond yield. This three-variable macro-financial set-up closely follows \citet{mumtaz2018} and \citet{CaldaraSVOL}, facilitating comparison with their single-regime frameworks.
The historical series combine pre-1947 data from the NBER tables accompanying \emph{The American Business Cycle} with postwar data from FRED and the Global Financial Database. Real GNP and the GNP deflator are each spliced at 1947, while the corporate bond yield combines the NBER series with Moody's Baa yield. The 10-year government bond yield is obtained from the Global Financial Database over the full sample. The Appendix lists the underlying series and their identifiers and provides details on the transformations and splicing.
The length of the sample is particularly important for the threshold exercise because, unlike postwar samples, it contains repeated high-inflation episodes. These include World War I, the 1946--48 inflation surge, the Great Inflation, and the 2022--23 post-pandemic episode. This repeated variation allows us to assess whether changes in mean--volatility interactions recur systematically across high-inflation environments rather than reflecting a particular historical episode. This distinguishes our application from \citet{CaldaraSVOL}, whose estimation sample, 1954Q2--2019Q4, contains the Great Inflation but neither the earlier inflation episodes nor the post-pandemic surge.
\subsection{Model selection}
We compare seven specifications to determine which variable defines the regimes and how many regimes are required. The no-threshold model serves as the reference specification. The remaining specifications combine two or three regimes with thresholds in real GNP growth, inflation, or the credit spread. Four-quarter inflation defines the regimes in the inflation specifications; the level of the spread does so in the spread specifications. Within each specification, we estimate the threshold values and delay using the priors in Section~\ref{mc}.
Following \citet{GEWEKE2010216}, we compare specifications using joint one-quarter-ahead log predictive scores. At each forecast origin, we re-estimate each model on the expanding information set $\mathcal{D}_t$ and score the posterior predictive density of $Z_{t+1}$, the realised pair of GNP growth and inflation. The criterion rewards models that assign greater probability to the realised joint outcome while accounting for predictive dependence and uncertainty about parameters and latent states. For specification $\mathcal{M}_j$, the cumulative log predictive score is
\begin{equation}
\mathrm{LS}_j=\sum_{t=t_0}^{t_1}\log p\!\left(Z_{t+1}\mid\mathcal{D}_t,\mathcal{M}_j\right),
\label{eq:modelselection_ls}
\end{equation}
where $t_0=1975\mathrm{Q}1$ and $t_1=2022\mathrm{Q}1$, giving 189 common forecast origins. The joint predictive density is approximated by a bivariate kernel density estimate of the posterior predictive simulations. Summing its logarithm over the forecast origins yields a measure of each specification's sequential predictive fit to the sequence of realised GNP-growth--inflation pairs. Differences in cumulative scores measure relative sequential predictive fit for the target pair over the evaluation sample.
Table~\ref{tab:modelselection} reports the resulting scores. Among the seven specifications considered, the specification with three regimes defined by four-quarter inflation attains the highest cumulative score. Under this one-quarter-ahead joint criterion, we select it as the benchmark specification for the reduced-form and structural analyses below.
\begin{table}[!htbp]
\centering
\footnotesize
\caption{\label{tab:modelselection}One-step-ahead joint log predictive scores}
\setlength{\tabcolsep}{8pt}
\begin{tabular}{@{}lc@{}}
\toprule
\rowcolor{gray!10}\textbf{Threshold variable} & \textbf{Log predictive score} \\
\midrule
\rowcolor{gray!10}\multicolumn{2}{@{}l}{\textbf{No threshold regimes}} \\
No-threshold model & $-273.1$ \\
\midrule
\rowcolor{gray!10}\multicolumn{2}{@{}l}{\textbf{Two regimes}} \\
GNP growth & $-303.2$ \\
Inflation & $-281.9$ \\
Credit spread & $-293.3$ \\
\midrule
\rowcolor{gray!10}\multicolumn{2}{@{}l}{\textbf{Three regimes}} \\
GNP growth & $-282.9$ \\
Inflation & \textcolor{blue}{\textbf{$-268.4$}} \\
Credit spread & $-306.6$ \\
\bottomrule
\end{tabular}
\caption*{\footnotesize \textit{Notes:} Sum of joint one-step-ahead log predictive scores for GNP growth and inflation over 189 recursive origins, 1975Q1--2022Q1. Higher values indicate a better one-quarter-ahead joint predictive fit for the realised pair. The realizations scored are 1975Q2--2022Q2. Regimes are defined by four-quarter inflation in the inflation specifications and by the level of the spread in the spread specifications. Bold blue marks the highest score.}
\end{table}
\subsection{\label{post}Inflation regimes}
We re-estimate the selected three-regime specification, in which four-quarter inflation defines the regimes, over the full estimation sample. Figure~\ref{fig:regimeprob} reports the threshold variable, posterior estimates of the two thresholds, and the associated regime probabilities. The posterior median thresholds are $2.86$ and $4.84$ percent, with 90\% credible intervals of $[2.47,3.05]$ and $[4.54,5.37]$, respectively. On average, the posterior regime probabilities assign $61\%$, $17\%$, and $21\%$ of the sample to the low-, moderate-, and high-inflation regimes.
The estimated periodisation has a clear historical interpretation. The high-inflation regime encompasses World War I, the wartime and postwar inflations of the 1940s and early 1950s, the late-1960s run-up and the Great Inflation, and the post-pandemic surge. The moderate regime captures intermediate episodes, including the late 1950s, the post-disinflation period from the mid-1980s to the early 1990s, and 2005--07. The long sample therefore identifies regime-specific propagation from several distinct historical episodes rather than from the Great Inflation alone.
The threshold locations show that the regimes are not symmetric partitions of the historical inflation distribution. Four-quarter inflation has a median of $2.07$ percent and a twenty-fifth percentile of $0.98$ percent over the estimation sample, while almost two-fifths of the pre-1950 observations are at or below zero. The lower and upper thresholds lie at approximately the $62$nd and $78$th percentiles, respectively. Thus, the estimated thresholds place the regime boundaries above the centre of the historical inflation distribution rather than mechanically partitioning the sample into equally populated states. The high-inflation regime covers roughly the upper fifth of the historical distribution, while the moderate regime occupies a comparatively narrow interval above its centre.
The labels low, moderate, and high inflation describe observed threshold states rather than conditional quantiles of future inflation. Conditional on a parameter draw, lagged four-quarter inflation crossing an estimated threshold selects the coefficients, volatility dynamics, and level--volatility interactions governing subsequent propagation. \citet{LopezSalidoLoria} study inflation tails primarily using quantile
regressions. In their complementary Markov-switching regression in
Section~2.3, they compare latent-regime-specific fitted values with
conditional-quantile estimates. Our regime allocation instead depends on
observed lagged inflation and the estimated thresholds. The distinction is
therefore between an observed threshold rule and a latent Markov process,
rather than an exact identification of regimes with conditional quantiles.
\begin{figure}[!htbp]
\centering
\includegraphics[width=\textwidth]{figures/regime_prob3_uncond.pdf}
\caption{\label{fig:regimeprob}Inflation regimes and posterior threshold estimates, 1881Q3--2024Q1.}
\caption*{\footnotesize \textbf{Notes:} The dark line is the threshold variable, four-quarter inflation, at the posterior modal delay (left axis). Solid horizontal lines report posterior median estimates of the lower and upper thresholds; dashed lines show the corresponding 90\% credible intervals. The dotted horizontal lines are the unconditional 5th and 95th percentiles of four-quarter inflation over the estimation sample, $-5.05$ and $9.61$ percent. Shaded areas report the posterior probabilities of the moderate- and high-inflation regimes (right axis); quarters in the low-inflation regime are left unshaded. Grey bars in the lane below the panel mark NBER recessions, from the peak to the trough quarter.}
\end{figure}
\subsection{\label{tailrisk}Macroeconomic Tail Risk}
We next use the estimated model to characterize state-dependent macroeconomic tail risk. Because the model jointly determines activity, inflation, financial conditions, and their volatilities, common shocks can alter the location, dispersion, and dependence of future outcomes. In addition to growth- and inflation-at-risk as defined above, we measure upside credit-spread risk by the 95th percentile of its predictive distribution.
We construct recursive predictive distributions at horizons from one to eight quarters. At each forecast origin, the threshold and no-threshold models are estimated recursively using observations through that origin and simulated forward from the estimated state. Volatility and, in the threshold model, the inflation regime evolve endogenously.
Figure~\ref{fig:tailcompare} provides an illustrative comparison at the four-quarter horizon. It contrasts the percentile paths implied by the threshold model with those from a model that retains stochastic volatility but imposes a single propagation mechanism throughout the sample. The two specifications generate similar risk measures outside high-inflation periods. During high-inflation periods, however, the threshold model generally implies more adverse tail quantiles. The comparison therefore illustrates how allowing for state-dependent propagation changes the model-implied predictive distribution of activity and inflation. Because the specifications differ jointly in their coefficients, volatility dynamics, and level--volatility interactions, the comparison does not isolate a single channel.
\begin{figure}[!htbp]
\centering
\includegraphics[width=\textwidth]{figures/tailrisk_compare_oos_h4_uncond.pdf}
\caption{\label{fig:tailcompare}Tail risk with and without inflation regimes.}
\caption*{\footnotesize \textbf{Notes:} Four-quarter-ahead 5th and 95th predictive percentiles from recursively estimated threshold (navy) and no-threshold (green) models; grey lines are realised outcomes. GNP growth and inflation are cumulated over the forecast horizon, that is summed over the $h$ quarters following the origin, and the credit spread is measured at its target date. The heavier line denotes the adverse tail. The horizontal red lines are the unconditional 5th and 95th percentiles of the realised series over the same targets, so the distance between a predictive percentile and its reference line measures risk relative to a typical quarter. Background shading reports the modal threshold-model regime at the origin, moderate inflation in amber and high inflation in red; quarters in the low-inflation regime are left unshaded. Grey bars in the lane at the foot of each panel mark NBER recessions, from the peak to the trough quarter. Forecast origins: 1975Q1--2022Q1.}
\end{figure}
The comparison is illustrative. We next evaluate the relative predictive performance of the threshold and no-threshold specifications over horizons from one to eight quarters. Table~\ref{tab:tailscores} reports two complementary diagnostics. Panel A reports RMSE ratios for the posterior predictive means and therefore assesses central forecast accuracy. Panel B reports outcome-weighted log-score differences in the sense of \citet{Amisano-Giacomini-07}, which place greater weight on realised outcomes far from the centre of the recursive estimation sample and hence assess relative density performance when outcomes are unusually distant from typical conditions.\footnote{The weight rises with the distance of the realised outcome from the centre of each recursive estimation sample. We report 100 times the average score difference. Outcome-weighted log scores are diagnostic rather than proper scoring rules \citep{gneiting-ranjan-11}.}
The forecast evidence is favourable to the threshold specification, but not uniformly so. Posterior-mean accuracy is broadly comparable across models: the threshold specification improves the credit spread at short horizons and inflation at longer horizons, whereas output-growth point forecasts are similar. The outcome-weighted log score favours the threshold specification in most variable--horizon comparisons, particularly at short horizons for GNP growth. This pattern warrants caution. The largest positive differences reflect individual pandemic target quarters rather than typical relative performance, and the gains are not uniform across variables or horizons. The results therefore provide suggestive evidence that inflation-regime dependence can improve selected central forecasts and tail-outcome density diagnostics, rather than establish general density dominance.
\begin{table}[!htbp]
\centering
\footnotesize
\caption{\label{tab:tailscores}Out-of-sample forecast comparison}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}l*{8}{c}@{}}
\toprule
\textbf{Variable} & \multicolumn{8}{c}{\textbf{Forecast horizon}} \\
\cmidrule(l){2-9}
& \textbf{H1} & \textbf{H2} & \textbf{H3} & \textbf{H4} & \textbf{H5} & \textbf{H6} & \textbf{H7} & \textbf{H8} \\
\midrule
\rowcolor{gray!10}\multicolumn{9}{@{}l}{\textbf{A. Posterior-mean accuracy: RMSE ratio}} \\
Output growth & \textcolor{blue}{\textbf{0.996}} & \textcolor{blue}{\textbf{0.992}} & \textcolor{blue}{\textbf{0.992}} & 1.004 & 1.002 & 1.005 & 1.010 & 1.007 \\
Inflation & 1.053 & 1.024 & \textcolor{blue}{\textbf{0.999}} & \textcolor{blue}{\textbf{0.971}} & \textcolor{blue}{\textbf{0.954}} & \textcolor{blue}{\textbf{0.939}} & \textcolor{blue}{\textbf{0.930}} & \textcolor{blue}{\textbf{0.920}} \\
Credit spread & \textcolor{blue}{\textbf{0.937}} & \textcolor{blue}{\textbf{0.930}} & \textcolor{blue}{\textbf{0.939}} & \textcolor{blue}{\textbf{0.903}} & \textcolor{blue}{\textbf{0.999}} & 1.003 & 1.003 & 1.006 \\
\midrule
\rowcolor{gray!10}\multicolumn{9}{@{}l}{\textbf{B. Tail-outcome density diagnostic: weighted log-score difference}} \\
Output growth & \textcolor{blue}{\textbf{27.1}} & \textcolor{blue}{\textbf{43.8}} & \textcolor{blue}{\textbf{102.8}} & $-$11.5 & \textcolor{blue}{\textbf{0.5}} & \textcolor{blue}{\textbf{0.0}} & \textcolor{blue}{\textbf{0.9}} & \textcolor{blue}{\textbf{0.5}} \\
Inflation & $-$0.1 & \textcolor{blue}{\textbf{0.0}} & \textcolor{blue}{\textbf{0.1}} & \textcolor{blue}{\textbf{0.1}} & \textcolor{blue}{\textbf{0.0}} & \textcolor{blue}{\textbf{0.1}} & \textcolor{blue}{\textbf{0.3}} & \textcolor{blue}{\textbf{0.2}} \\
Credit spread & \textcolor{blue}{\textbf{124.9}} & $-$6.6 & $-$0.8 & $-$3.3 & $-$47.3 & $-$16.2 & $-$0.3 & $-$29.1 \\
\bottomrule
\end{tabular}
\caption*{\footnotesize \textit{Notes:} Based on 189 recursive forecast origins. GNP growth and inflation are evaluated as average growth over the horizon, the cumulated change divided by $h$; the spread is the target-date level. Panel A reports threshold/no-threshold RMSE ratios; values below one favour the threshold model. Panel B reports 100 times threshold-minus-no-threshold outcome-weighted log-score differences; positive values favour the threshold model. Bold blue marks values that favour the threshold model. Panel B differences are averages across origins and are dominated by a few extreme quarters, chiefly 2020Q1--2020Q2, when both models assign very little density to the realised outcomes.}
\end{table}
Complementing the relative score comparisons, we assess the absolute calibration of each one-quarter-ahead predictive density using the probability integral transform (PIT), defined as the predictive cumulative distribution function evaluated at the realised outcome \citep{diebold-gunther-tay-98}. Under calibration, PITs are uniformly distributed. We test uniformity using the Kolmogorov--Smirnov (KS) and Cram\'er--von Mises (CvM) statistics of \citet{rossi-sekhposyan-19}.\footnote{We use the one-step-ahead critical values in \citet{rossi-sekhposyan-19}. As the models are recursively re-estimated, the test outcomes are indicative.} Unlike the outcome-weighted score, which compares models at realised outcomes, the PIT diagnostics assess each predictive density separately; the two exercises therefore need not rank the models identically.
\begin{table}[!htbp]
\centering
\footnotesize
\caption{\label{tab:pit}Calibration of the one-step-ahead predictive densities}
\setlength{\tabcolsep}{8pt}
\begin{tabular}{@{}lcccc@{}}
\toprule
& \multicolumn{2}{c}{\textbf{Kolmogorov--Smirnov}} & \multicolumn{2}{c}{\textbf{Cram\'er--von Mises}} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}
\rowcolor{gray!10}\textbf{Variable} & \textbf{Threshold} & \textbf{No threshold} & \textbf{Threshold} & \textbf{No threshold} \\
\midrule
GNP growth & \textcolor{blue}{\textbf{1.70}}$^{***}$ & 1.90$^{***}$ & \textcolor{blue}{\textbf{0.66}}$^{**}$ & 0.85$^{***}$ \\
Inflation & \textcolor{blue}{\textbf{1.21}}$^{*}$ & 1.44$^{**}$ & \textcolor{blue}{\textbf{0.46}}$^{*}$ & 0.66$^{**}$ \\
Credit spread & \textcolor{blue}{\textbf{1.03}} & 1.18 & \textcolor{blue}{\textbf{0.27}} & 0.31 \\
\bottomrule
\end{tabular}
\caption*{\footnotesize \textit{Notes:} Rossi--Sekhposyan KS and CvM statistics for PIT uniformity from 189 recursive one-step-ahead origins, 1975Q1--2022Q1. Lower values are better; bold blue marks the lower statistic. $^{***}$, $^{**}$ and $^{*}$ denote rejection at the 1, 5 and 10 percent levels, respectively.}
\end{table}
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.9\textwidth]{figures/pit_calibration.pdf}
\caption{\label{fig:pit}PIT calibration of the one-step-ahead predictive densities.}
\caption*{\footnotesize \textit{Notes:} The top and middle rows show PIT histograms for the threshold and no-threshold models, respectively; the solid and dashed lines give the uniform density and its pointwise 95 percent interval. The bottom row plots the empirical PIT CDFs against the 45-degree line; the shaded region is the 5 percent KS band. Recursive one-step-ahead forecasts from origins 1975Q1--2022Q1.}
\end{figure}
Table~\ref{tab:pit} and Figure~\ref{fig:pit} show a relative calibration improvement for the threshold specification: both diagnostics are lower for all variables, especially inflation. Nevertheless, both models reject PIT uniformity for GNP growth and inflation, whereas the credit-spread densities are compatible with uniformity under either specification. The threshold specification therefore yields smaller empirical departures from PIT uniformity, although departures remain for GNP growth and inflation at the reported significance levels.
\FloatBarrier
\subsection{\label{leverage}State-Dependent Leverage and Volatility-in-Mean Feedback}
The model contains three related channels linking the first and second moments: lagged macroeconomic outcomes affect subsequent volatility through $d_{j,m}$; level and volatility innovations are contemporaneously correlated through $\Sigma_{\eta e,(m)}$; and volatility feeds back into expected macroeconomic outcomes through the volatility in-mean coefficients $b_{k,m}$. The threshold structure allows these channels, together with the remaining macroeconomic and volatility dynamics, to differ between inflation regimes.
Figure~\ref{fig:volleverage} distinguishes two reduced-form objects. Its left-hand side plots each observed variable with the posterior median of its innovation volatility, $\exp(h_{i,t}/2)$, and the inflation-regime chronology. Appendix Table~\ref{tab:volobscorr} summarizes this descriptive comovement over both the full historical sample and the postwar window used by \citet{CaldaraSVOL}. Its right-hand side reports the model's leverage estimates, the contemporaneous correlations $\operatorname{corr}(\eta_{i,t},e_{j,t})$ between first- and second-moment innovations.
Both panels report reduced-form features of the fitted model. The historical co-movements on the left are descriptive, whereas the right-hand correlations summarize the contemporaneous dependence between level and volatility innovations. Neither identifies the response to an individual structural shock or a causal transmission channel. The figure also shows that inflation regimes are not simply volatility states: the Great Depression and the Global Financial Crisis lie predominantly in the low-inflation regime despite exceptionally high innovation volatility.
\begin{figure}[!htbp]
\centering
\includegraphics[width=\textwidth]{figures/combined_vol_leverage.png}
\caption{\label{fig:volleverage}Innovation volatility and leverage correlations across inflation regimes.}
\caption*{\footnotesize \textbf{Notes:} \emph{Left:} dotted lines show observed values (left axis); solid lines show posterior median innovation standard deviations, $\exp(h_{i,t}/2)$, with 68\% credible intervals (right axis). Moderate- and high-inflation periods are shaded yellow and red, respectively. \emph{Right:} posterior medians with 68\%/90\% credible intervals for the regime-specific leverage block $\Sigma_{\eta e,(m)}$ of equation~\eqref{eq:sigmar}, whose $(i,j)$ element is $\operatorname{corr}(\eta_{i,t},e_{j,t})$, the correlation between the innovation to the volatility of variable $i$ and the level innovation of variable $j$; solid intervals denote own-variable correlations and lighter intervals cross-variable correlations. The vertical line denotes zero.}
\end{figure}
\citet{CaldaraSVOL} document three descriptive patterns: weak GDP growth coincides with higher GDP-growth innovation volatility; inflation and its innovation volatility co-move positively on average, but inflation falls as its volatility rises during the Global Financial Crisis; and financial stress co-moves with financial volatility. Appendix Table~\ref{tab:volobscorr} recovers the broad signs of these patterns in the shared postwar sample, but the longer historical sample shows that they vary across inflation environments.
Output provides the clearest connection between the descriptive evidence and the innovation-level estimates. Weak GNP growth tends to coincide with elevated output innovation volatility, a pattern evident in major recessions such as the Great Depression and the Global Financial Crisis, which are predominantly classified in the low-inflation regime. The innovation-level evidence is consistent with this pattern: in the low-inflation regime, negative output innovations co-occur with positive innovations to subsequent output volatility. This own-output leverage correlation attenuates in the moderate regime and is centred near zero when inflation is high. The estimates therefore suggest that, in the low-inflation state, adverse output news is associated with greater uncertainty about future output, whereas this reduced-form association weakens as inflation rises. This pattern is potentially relevant for downside growth risk, but does not identify the structural disturbance responsible for it.
For inflation, the positive postwar association with innovation volatility is concentrated in moderate- and high-inflation observations. It reverses in the low-inflation state and in the full historical sample. Own-inflation leverage is imprecisely estimated across regimes, so this descriptive evidence does not establish a stable relation between inflation innovations and innovations to subsequent inflation volatility.
Finally, the credit-spread evidence is the closest counterpart to the EBP result in \citet{CaldaraSVOL}: financial stress co-moves positively with its innovation volatility in the shared postwar sample. In the full history, this relation is concentrated in the low-inflation state, which includes the Great Depression and the Global Financial Crisis, and does not extend uniformly to the other regimes. The posterior medians of own-spread leverage are positive but imprecisely estimated. Relatedly, the cross-variable leverage correlation between inflation innovations and innovations to subsequent credit-spread volatility changes sign across regimes, but is imprecisely estimated and should be interpreted only as suggestive evidence of a state-dependent reduced-form link between inflation and financial volatility. Given this posterior uncertainty, we do not interpret these reduced-form correlations as evidence about particular economic shocks. Section~\ref{SVAR} instead examines how identified shocks propagate through the full estimated system.
\subsection{Predictive risk of entering the high-inflation regime}
The model also provides a measure of near-term transition risk (regime-entry
risk) that complements inflation-at-risk. At each historical origin classified
outside the high-inflation regime, we use the full-sample posterior
and smoothed volatility states to compute the probability of entering
that regime at least once over the following four quarters.\footnote{These
probabilities are historical risk assessments conditional on
full-sample information, rather than forecasts constructed using
only the information available at each origin.}
Whereas inflation-at-risk describes the upper tail of the predictive
inflation distribution, the transition probability quantifies the
likelihood of moving into an environment in which the dynamics of
macroeconomic outcomes and volatility, and hence shock propagation,
differ.
Figure~\ref{fig:regimerisk} shows that the probabilities rise ahead
of the highlighted transitions, indicating that the estimated model
associates the histories preceding these episodes with elevated
near-term transition risk. In both highlighted episodes, the
probability reaches $0.91$ two quarters before entry, compared with
a pre-1965 average of $0.25$. Most of the increase occurs in the
preceding quarter. The exercise thus provides a probabilistic
characterization of the build-up to high-inflation episodes, going
beyond the classification of realised inflation into regimes.
It links the macroeconomic conditions and volatility states
prevailing at each historical origin to the model-implied likelihood
of a subsequent change in the propagation environment.
Entry risk is not simply a volatility alarm. It is essentially zero in 2009Q1 despite the exceptionally high innovation volatility documented in Figure~\ref{fig:volleverage}. The measure therefore distinguishes an extreme-volatility episode from one that carries a high near-term probability of entering a state with different propagation mechanisms.
\begin{figure}[!htbp]
\centering
\includegraphics[width=\textwidth]{figures/regime_entry_risk.pdf}
\caption{\label{fig:regimerisk}Predictive probability of entering the high-inflation regime.}
\caption*{\footnotesize \textbf{Notes:} The filled area is the predictive probability that an origin outside the high-inflation regime enters it at least once over the following four quarters; the measure is undefined within that regime. The thin line is observed four-quarter inflation (right axis), and high-inflation periods are shaded in red. Grey bars in the lane below the horizontal axis mark NBER recessions, from the peak to the trough quarter. Panel (a) plots the full sample and panel (b) the same series postwar. The inflation axis in panel (a) is truncated at $-16$ percent, so the 1921 deflation trough lies below the panel. Open circles mark 1973Q2, 2009Q1, and 2021Q3.}
\end{figure}
Regime-entry risk and upside inflation risk are related but not redundant. Their positive, imperfect association (a correlation of $0.68$ at the four-quarter horizon) reflects their different objects. Inflation-at-risk is an intensive-margin measure of the severity of adverse inflation outcomes, whereas regime-entry risk is an extensive-margin measure of a transition into a state in which the propagation of shocks changes.
\section{\label{SVAR}Decomposing Macroeconomic Tail Risk}
The preceding analysis shows that the predictive distribution of macroeconomic outcomes varies across inflation regimes and provides reduced-form evidence of state dependence in some dimensions of mean--volatility interactions. We now turn to the structural drivers of these distributional changes. Specifically, we ask whether the shocks that account for movements in the centre of the predictive distribution are also the main drivers of growth- and inflation-at-risk, and whether their effects on tail risk depend on the inflation state and on the sign and size of the shock.
\subsection{Identification of shocks}
We identify four structural disturbances using the multi-shock max-share approach of \citet{Carriero02012025}: a \emph{business-cycle} shock targeting the forecast-error variance of GNP growth; a \emph{financial} shock targeting that of the credit spread; a \emph{macroeconomic-uncertainty} shock targeting the joint forecast-error variance of GNP-growth and inflation volatility; and a \emph{financial-uncertainty} shock targeting credit-spread volatility.
Candidate impact matrices take the form $A_0=C_0P_0$, where $C_0$ is the lower Cholesky factor of $\Omega_t$, the time-$t$ covariance matrix of the scaled reduced-form innovations, and $P_0$ is orthonormal. The first four columns of $A_0$ define the identified shocks. This rotation-based construction does not impose an economic recursive ordering. Because the model is nonlinear, with regime switching, volatility feedback, and time-varying shock scales, we compute simulated generalised impulse responses \citep{koop-pesaran-potter-96} and generalised forecast-error variance decompositions \citep{lanne-nyberg-16}, conditional on the initial state and cumulated over one year.
Following \citet{Carriero02012025}, we identify the four shocks jointly. The rotation $P_{0}$ maximises the equally weighted sum of the shares of the four targets explained by their corresponding shocks, subject to the requirement that each shock explains a larger share of its own target than of any other target.\footnote{These are the baseline inequality restrictions of \citet{Carriero02012025}, their equation 3.2. Cumulating the FEV shares over one year follows the variant in their Appendix~G.} We normalise the business-cycle shock to raise GNP growth on impact, the financial shock to raise the credit spread, and each uncertainty shock to raise its target volatility. Identification is repeated at every posterior draw and initial state. Thus, comparisons across inflation regimes hold the economic max-share criterion fixed, while allowing both the identified impact vector and its subsequent propagation to be state dependent.
\subsection{Nonlinear tail-risk responses}
For each posterior draw and initial state, we simulate the model with and without an impulse to an identified shock, using common future innovations in the two paths. The regime evolves endogenously along each simulated path. The response of growth-at-risk or inflation-at-risk is the difference between the relevant conditional quantiles of the shocked and baseline predictive distributions. We average these responses across histories within each inflation regime and report posterior medians and $68\%$ credible intervals. We use these responses to distinguish three forms of nonlinear propagation. State dependence refers to differences in the response to the same impulse across inflation regimes; sign asymmetry to departures from mirror symmetry between the responses to
equally sized positive and negative impulses; and size dependence to departures from proportionality as the magnitude of the impulse changes.
We focus on the low- and high-inflation regimes to provide a sharper contrast between inflation environments. The moderate regime remains fully embedded in the estimated model, entering the identification, simulated dynamics, and endogenous regime transitions. We do not report its responses separately because the smaller number of moderate-inflation histories results in less precise regime-specific estimates, while the two outer regimes provide the clearest comparison across inflation environments.
Figure~\ref{fig:risksize} compares the centre and tails of the predictive distribution over shocks ranging from $-5$ to $5$ standard deviations. Panel~(a) reports the predictive median, the $50^{th}$ percentile, as a centre-of-distribution benchmark; panel~(b) reports growth-at-risk and inflation-at-risk. The predictive median is distinct from the conditional mean used in the twelve-quarter impulse responses in the Appendix, since the predictive distribution can be asymmetric; it also serves as the centre-of-distribution benchmark in the decomposition below. The larger impulses are stress experiments designed to trace departures from proportional propagation, rather than representative shock realisations. The Appendix reports the twelve-quarter dynamics following one-standard-deviation shocks: uncertainty shocks have persistent effects on growth-at-risk at that scale, while state dependence is clearest for inflation-at-risk. Under proportional propagation, scaling an impulse scales its response proportionally, so responses trace a straight line through the origin. Curvature within either the positive or negative branch therefore indicates size dependence, while departures from equal-and-opposite responses to equally sized positive and negative impulses indicate sign asymmetry. The predictive-median responses in panel~(a) are broadly proportional within each regime, whereas the stronger departures in panel~(b) are concentrated in the tails of the predictive distribution.
The tail responses display the strongest nonlinearities for the uncertainty shocks. Positive macroeconomic-uncertainty shocks generate an increasingly large
deterioration in growth-at-risk. Large positive macroeconomic-uncertainty disturbances are therefore not simply scaled-up versions of smaller ones: their effect becomes disproportionately concentrated in the downside of the growth distribution. The low- and high-inflation responses diverge mainly at the largest impulses, where the wide $68\%$ credible intervals leave the magnitude of the state contrast imprecisely estimated. Negative business-cycle shocks also worsen growth-at-risk more in low-inflation histories. This tail-specific asymmetry should not be read as uniformly stronger business-cycle transmission when inflation is low, and the averages are not recession-specific decompositions. The asymmetry across the growth distribution is consistent with evidence of
more variable downside than upside growth risk \citep{adrian2019,ccm2024}.
The model also allows uncertainty to respond endogenously to macroeconomic
disturbances, a distinction emphasised by \citet{LMN2020}.
\enlargethispage{\baselineskip}
{\looseness=-1
For inflation-at-risk, financial uncertainty provides the clearest state contrast. Positive financial-uncertainty shocks leave the upper tail broadly unchanged in low-inflation histories but generate an increasingly convex rise when inflation is high. In high-inflation histories, large positive financial-uncertainty disturbances are therefore not simply scaled-up versions of smaller ones: their effect on the upper tail of the inflation distribution increases disproportionately with shock size. The corresponding responses of the predictive median are much smaller, indicating a change in tail risk beyond a shift in the centre of the distribution. Again, the magnitude of the contrast at the largest impulses is imprecisely estimated. The inflation-at-risk response to the financial shock is also imprecise at large shock sizes, so we do not attach a separate economic interpretation to it. These results are consistent with, but do not mechanically follow from, the regime-dependent reduced-form evidence in Section~\ref{leverage}: identified shocks propagate through the full system, in which volatility feedback, multivariate dynamics, and endogenous regime transitions jointly affect the location and dispersion of future outcomes.\par}
\clearpage
\begin{landscape}
\thispagestyle{empty}
\vspace*{\fill}
\centering
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/risksize_level_h4.pdf}
\par\smallskip
\textbf{(a)} Predictive-median responses.
\end{minipage}\hfill
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/risksize_all_h4.pdf}
\par\smallskip
\textbf{(b)} Tail-risk responses.
\end{minipage}
\begin{minipage}{\linewidth}
\captionof{figure}{Shock-size responses of the predictive distribution.}\label{fig:risksize}
\captionof*{figure}{\footnotesize \textbf{Notes:} Each panel reports one-year-ahead responses to impulses of $\pm 0.5$, $\pm 1$, $\pm 2$, $\pm 3$, and $\pm 5$ standard deviations. Panel~(a) reports the predictive median of GNP growth (top) and inflation (bottom). Panel~(b) reports growth-at-risk, the response of the $5^{th}$ percentile of GNP growth (top), and inflation-at-risk, the response of the $95^{th}$ percentile of inflation (bottom). Columns correspond to the business-cycle, financial, macroeconomic-uncertainty, and financial-uncertainty shocks. Blue solid lines average responses across low-inflation histories and red dashed lines across high-inflation histories; shaded areas are $68\%$ credible intervals across posterior draws. Under proportional propagation, the points in panel~(a) lie on a straight line through the origin.}
\end{minipage}
\vspace*{\fill}
\end{landscape}
\clearpage
\FloatBarrier
\subsection{Decomposing the drivers of tail risk}
The shocks that dominate movements in the centre of the predictive distribution need not be those that dominate movements in its tails. To quantify this distinction, we construct FEVD-style shares of the model-implied responses of growth- and inflation-at-risk and compare them with the corresponding decompositions of the predictive median. For each target quantile and shock size, the share assigned to an identified direction is its cumulated squared response divided by the corresponding sum over six orthogonal directions. Following the normalisation of \citet{lanne-nyberg-16}, we use these shares as an FEVD-style structural decomposition of model-implied tail-risk responses.
The first two columns of Figure~\ref{fig:qfevd_size} report shares based on responses of the predictive median, the centre-of-distribution benchmark; they decompose the predictive-median responses reported in panel~(a) of Figure~\ref{fig:risksize}. The final two columns report the analogous tail-quantile shares. The exercise concerns hypothetical response paths, not an historical accounting of realised episodes.
\begin{figure}[!htbp]
\centering
\includegraphics[width=\textwidth]{figures/qfevd_size_area_h4.pdf}
\caption{\label{fig:qfevd_size}Structural decomposition of tail-risk responses against shock size.}
\caption*{\footnotesize \textbf{Notes:} Each panel reports a one-year decomposition based on cumulated squared responses to shocks ranging from $-5$ to $5$ standard deviations. A contribution is the squared response to an identified shock divided by the corresponding sum over six orthogonal directions. ``Other'' combines the two orthogonal directions not assigned an economic label by the max-share identification, together with any departure from exact additivity in the nonlinear construction. The top row concerns GNP growth and the bottom row inflation. The first two columns report the benchmark based on responses of the predictive median, the $50^{th}$ percentile; the final two report the tail-quantile shares, for the $5^{th}$ percentile of GNP growth and the $95^{th}$ percentile of inflation. Within each pair, the panels average over low- and high-inflation initial states respectively. For each posterior draw the five plotted shares sum to one by construction, since ``other'' is computed as one minus the sum of the four identified shares. Areas report posterior medians, across draws, of the individual shares; because these componentwise medians need not sum exactly to one, they are renormalised for display.}
\end{figure}
Figure~\ref{fig:qfevd_size} reveals a sharp distinction between the structural composition of the centre of the predictive distribution and that of tail risk. The business-cycle shock accounts for more than $80\%$ of the predictive-median decomposition of GNP growth, but its contribution to growth-at-risk is substantially smaller. Macroeconomic uncertainty fills part of this gap: despite its limited role at the median, it makes a material contribution to both growth- and inflation-at-risk. The sizeable ``other'' component for inflation-at-risk also shows that the four economically labelled shocks do not exhaust the structural directions relevant for inflation tail risk.
These differences are nonlinear. Under proportional linear propagation, changing the sign or scale of an impulse would leave these squared-response shares unchanged. Here the tail shares vary across the negative and positive branches and with impulse size, whereas median-response shares are comparatively flat. The macroeconomic-uncertainty share of growth-at-risk rises with the size of a positive macroeconomic-uncertainty impulse. In high-inflation histories, financial uncertainty likewise gains importance for inflation-at-risk following larger positive financial-uncertainty impulses. Because the shares are based on squared responses, they describe relative importance, not the sign of the quantile response.
The decomposition thus adds information beyond the response functions. It shows that the structural composition of tail risk depends on the state of the economy and on both the sign and size of the disturbance, even where the decomposition of the predictive median changes little. The threshold structure therefore implies state dependence not only in the predictive distribution of macroeconomic risk, but also in the structural composition of its model-implied responses.
\FloatBarrier
\section{\label{conclu}Conclusion}
This paper asks whether the shocks that account for movements in the centre of the predictive distribution also account for macroeconomic tail risk, and whether their propagation through the distribution changes with the state of the economy. We address these questions with a threshold stochastic-volatility-in-mean VAR in which leverage, volatility-in-mean feedback, and the remaining macroeconomic and volatility dynamics vary across regimes. Predictive model selection supports a three-regime specification defined by inflation in a long U.S.\ sample.
The estimated regimes characterize distinct propagation environments rather than simply periods of high or low volatility. The Great Depression and the Global Financial Crisis belong predominantly to the low-inflation regime despite exceptionally high innovation volatility, while the reduced-form estimates suggest that mean--volatility dependence differs across inflation states.
Conditional on the model and the max-share identification, the structural decomposition shows that the composition of tail risk differs from that of the predictive median. Business-cycle shocks dominate the median response of GNP growth but account for a substantially smaller share of growth-at-risk. Macroeconomic uncertainty, by contrast, has a limited role at the median yet makes a material contribution to both growth- and inflation-at-risk. Its contribution to growth-at-risk rises with the magnitude of a positive macroeconomic-uncertainty impulse. Financial uncertainty displays a corresponding pattern for inflation-at-risk in high-inflation histories.
These findings are conditional structural decompositions of model-implied responses, rather than historical shock accountings or causal interpretations of individual reduced-form leverage correlations. The sizeable residual component in the inflation-at-risk decomposition also shows that the four economically labelled shocks do not exhaust the structural directions relevant for inflation tail risk. The central implication is that a decomposition of the centre of the predictive distribution, or a measure of volatility alone, need not reveal the structural composition of macroeconomic tail risk. The shocks that dominate movements in the centre of the predictive distribution need not be those that dominate risks in its tails, and this distinction can itself depend on the state of the economy and on the sign and magnitude of the disturbance.
\bibliographystyle{ecta}
\bibliography{references/references}
\clearpage
\begin{center}
{\large\bfseries Appendix}
\end{center}
\spacingset{1.45}