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The Anatomy of Commodity Risk: Micro, Market, and Economy-Wide Sources
\title{The Anatomy of Commodity Risk: Micro, Market, and Economy-Wide Sources}
\author{
Nektarios Aslanidis$^{a}$,
Aurelio F. Bariviera$^{a}$,
George Kapetanios$^{b}$,
Vasilis Sarafidis$^{c}$,
Alexia Ventouri$^{b}$\\[0.5em]
\small
$^{a}$Universitat Rovira i Virgili, ECO-SOS, Department of Economics, Spain\\
\small
$^{b}$King's College London, Department of Banking and Finance, UK\\
\small
$^{c}$Brunel University London, Brunel Business School, UK\\[0.3em]
}
\date{}
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\begin{abstract}
We study the anatomy of commodity risk by distinguishing micro, market-level, and economy-wide sources. We develop a two-stage ``divide-and-conquer'' framework that allows sensitivities to these risk sources to vary across commodities while treating economy-wide risk as latent. The first stage uses defactored instrumental-variable estimation to recover commodity-specific sensitivities to micro and market conditions. The second combines principal components with high-dimensional variable selection to identify an observable representation of macro-financial risk. We then construct Risk Intensity Indices (RIIs), which combine estimated sensitivities with prevailing risk conditions to quantify the relative importance of each risk source on a common scale. Market risk is the largest component on average, accounting for about two fifths of total risk intensity and more than half for energy commodities. Risk intensity is also highly concentrated across individual commodities: the top 20\% account for approximately half of micro and market risk intensity, whereas macro risk is more broadly dispersed. The composition of risk varies substantially across sectors and over time, with market risk becoming particularly prominent during episodes of commodity-market stress. Micro and market RIIs also contain information about future volatility and absolute returns. These findings provide investors, risk managers, and policymakers with a diagnostic of where commodity risk is concentrated, which risk layers are most important, and how their importance changes over time. More broadly, our divide-and-conquer framework provides a flexible approach to decomposing layered risk in settings where common risk is latent.
\end{abstract}
\noindent \textbf{Keywords:} Commodity risk, risk intensity, heterogeneous sensitivities, latent common factors, high-dimensional variable selection
\noindent \textbf{JEL Classification:} C23, C38, C55, C58, G13
{\small
\noindent\textit{Acknowledgements:} The authors would like to thank T. Diasakos, T. Panagiotidis, D. Robertson, C. Savva, A. Urquhart, as well as participants at the 31st IPDC (Exeter, 6--7 July 2026), 29th IPDC (Orléans, 4--5 July 2024), and the Macro-Finance Cluster Seminar of the University of York (22 October 2024) for useful comments and suggestions. Aslanidis acknowledges financial support by the Spanish Government, Ministry of Science and Innovation under Project reference PID2022-137382NB-I00.
}
\doublespacing
\section{Introduction \label{sec:intro}}
Commodity markets occupy a distinctive position in the global economy, linking financial markets to the production, consumption, storage, and trade of essential inputs ranging from energy and metals to agricultural goods. Commodity prices are influenced by physical supply and demand conditions, inventories, and other commodity-specific fundamentals
\citep{GortonRouwenhorst2006,GortonHayashiRouwenhorst2013}. At the same time, the increasing participation of financial investors has strengthened the integration of commodity futures with broader financial markets and increased comovement across commodities \citep{TangXiong2012,BasakPavlova2016,Baker2021}. In particular, index investment can generate common movements across commodities that are not fully explained by commodity-specific fundamentals, while recent evidence suggests that index trading can also facilitate the transmission of nonfundamental shocks across commodities included in an index
\citep{DaTangTaoYang2024}. Commodity returns therefore reflect both highly commodity-specific conditions and forces operating more broadly across commodity and financial markets.
Taken together, these considerations suggest that commodity risk has an inherently layered structure. At the micro level, each commodity is exposed to idiosyncratic risks arising from its own price volatility, liquidity, trading activity, inventories, and supply-demand fundamentals. At the market level, commodities are exposed to common movements generated by developments elsewhere in commodity and financial markets, including market returns, volatility, and speculative activity. The importance of such common sources is supported by evidence that common commodity factors help explain the cross-sectional variation in commodity returns \citep{BakshiGaoRossi2019}, as well as by the broader literature on commodity-market financialisation \citep{TangXiong2012,BasakPavlova2016,Baker2021,DaTangTaoYang2024}. In particular, financialisation can increase comovement across commodities and strengthen the transmission of financial shocks, while index trading can propagate nonfundamental shocks across commodities within the same index. Beyond these two layers, commodities are exposed to economy-wide conditions, including global economic activity, monetary and financial conditions, inflation, exchange rates, and economic uncertainty, which can affect many commodity prices simultaneously. Commodity risk can therefore be viewed as arising from three conceptually distinct layers: \emph{micro}, \emph{market}, and \emph{economy-wide} sources.
Identifying sensitivity to different risk sources is informative, but sensitivities alone do not reveal their relative importance to a commodity's overall risk. A large sensitivity to a specific risk source may yield minimal risk intensity if the underlying variable rarely moves. Conversely, a moderate sensitivity to a highly volatile variable can be far more consequential. Furthermore, a collection of disparate sensitivity coefficients lacks a common metric to compare risk contributions across commodities or track how the composition of risk evolves over time. We address this limitation by constructing \emph{Risk Intensity Indices} (RIIs) that combine heterogeneous sensitivities with prevailing risk conditions and measure the magnitude of their contribution on a common scale. Separate indices are constructed for idiosyncratic, market, and economy-wide sources. The RIIs therefore shift the focus from asking simply \emph{how sensitive is a commodity to a particular source of risk?} to asking \emph{where does commodity risk come from, how important are its different sources, and how does its anatomy differ across commodities and over time?} In this sense, the indices provide a parsimonious mapping from a potentially large and heterogeneous set of risk sources into an economically interpretable decomposition of commodity risk intensity.
Our approach differs from existing commodity factor and risk-premium decompositions in both its objective and its unit of analysis. \citet{SzymanowskaDeRoonNijmanVanDenGoorbergh2014} decompose commodity futures risk premia into spot and term premia and study the factors explaining their cross-sectional variation, while \citet{BakshiGaoRossi2019} show that average commodity returns, carry, and momentum describe the cross section of commodity returns and investigate the economic sources underlying these factors. Related work examines how financialisation changes commodity prices, risk premia, volatility, and return dependence \citep{Baker2021,DaTangTaoYang2024}. Our objective is different. Rather than explaining the cross section of expected returns or decomposing risk premia, we identify heterogeneous sensitivities of realised commodity returns to micro, market-level, and economy-wide conditions and use these sensitivities, together with prevailing risk conditions, to measure the intensity of each risk source. The model does not impose an asset-pricing restriction linking these sensitivities to expected returns and does not
estimate the price of any source of risk. Whether the exposures identified here are priced in the cross section of expected commodity returns is a distinct question from the one studied in this paper. Our analysis instead uses the resulting risk intensities to characterise the anatomy of commodity risk across individual commodities, sectors, and time.
Building on this distinction, our contribution is fourfold. First, we develop a two-stage ``divide-and-conquer''estimation framework that enables the consistent identification of heterogeneous sensitivities across all three risk layers in the presence of latent macro risk. Second, we map these sensitivities into Risk Intensity Indices (RIIs), which provide comparable measures of risk intensity at the micro, market, and economy-wide levels. Third, we employ the resulting indices to characterize the anatomy of commodity risk. Specifically, we quantify the relative importance of the three sources of risk for the commodity market as a whole, examine differences in their composition across agricultural, energy, and metal commodities as well as across individual commodities, and trace their evolution over time by relating changes in the structure of risk to major economic and financial episodes. Finally, we establish the forward-looking predictive relevance of the RIIs and find that micro- and market-level risk intensity predict future volatility and absolute returns over horizons of up to twelve periods.
A central methodological challenge arises from the inherently latent nature of macro risk. The set of macrofinancial indicators that may proxy for this unobservable risk factor is typically high-dimensional, heterogeneous, and highly collinear, with limited guidance on their empirical relevance. Consequently, the selection of informative proxies entails a fundamental trade-off between omitted-variable bias, which may arise from excluding relevant information, and overfitting, which may result from incorporating an excessive number of noisy and highly correlated indicators.
To address this challenge, we develop a sequential ``divide-and-conquer'' estimation framework that partitions risk into observed micro and market components and latent macro risk. In Stage 1, heterogeneous sensitivities to the observed micro- and market-level risk components are estimated using commodity-specific instrumental-variable (IV) regressions. IV identification exploits a latent common-factor structure in the commodity-specific risk characteristics that is sufficiently rich to span the latent macro component responsible for endogeneity. Despite the endogeneity induced by the latent component, the resulting IV estimator delivers consistent estimates of heterogeneous sensitivities to the observed risk layers. In Stage 2, PCA is applied to the residuals across commodities to recover the dominant directions of common macro risk. These principal components are mapped to a large set of candidate macro-financial proxies, from which relevant predictors are selected using the Boosting with Multiple Testing (BMT) procedure of \citet{KapetaniosEtal2026}. This yields a sparse and observable representation of macro risk while mitigating overfitting. Commodity-specific macro sensitivities are then obtained by regressing each commodity's residual component on the selected macro risk proxies.
From a methodological perspective, the paper contributes along several dimensions. First, before extracting the latent factors, we partial out the observed sector-return space from the commodity-specific risk characteristics. This ensures that factor extraction targets latent common variation that is incremental to observed market-return movements, thereby separating the two sources of common variation. Second, in contrast to standard factor-augmented models, which approximate the macro component using a small and fixed number of latent factors (e.g., \citet{Bai2009,JuodisSarafidis2018,NorkuteEtal2021,JuodisSarafidis2022b}), we search over a high-dimensional set of observable macro-financial indicators, while allowing only a sparse subset to be relevant. Our candidate set comprises 36 indicators spanning uncertainty, sentiment, market attention, and global financial conditions, assembled from a broad range of studies.\footnote{To the best of our knowledge this is
the first study to employ such a diverse set of indicators within a single framework.} This yields an observable representation of macro risk without requiring the relevant indicators to be specified a priori. Third, to our knowledge, integrating PCA with high-dimensional selection in this way is new to the econometric and broader statistical literature: rather than applying boosting directly to the commodity-specific residuals, it is applied to principal components extracted by PCA. Selection therefore targets the common residual variation and produces a common set of observable macro-financial proxies. We examine the finite-sample performance of this procedure through a Monte Carlo study reported in the Appendix, which shows that the proposed approach performs well in recovering the relevant macro-financial proxies and the associated sensitivities. Fourth, we derive a commodity-specific over-identifying restrictions test for the defactored moment conditions, which provides the basis for a panel-level diagnostic obtained by combining individual test statistics across commodities. Finally, with respect to the RIIs, we formally establish conditions under which the three risk layers are non-reducible: in general, micro, market, and economy-wide risk intensity cannot be collapsed into a single aggregate risk component without losing information about the underlying structure of commodity risk.
The empirical analysis reveals a clear anatomy of commodity risk. For the full sample, market risk is the dominant source, accounting for 42.1\% of total risk intensity, followed by economy-wide and micro risk at 29.7\% and 28.2\%, respectively. The composition of risk nevertheless varies substantially across commodity sectors. Market risk is particularly important for energy commodities, where it accounts for 52.4\% of total risk intensity, compared with 35.4\% for agricultural commodities and 38.1\% for metals. Economy-wide risk, by contrast, is considerably more prominent in agriculture and metals than in energy. The time-varying RIIs also exhibit pronounced increases around major episodes of commodity-market stress, including the COVID-19 pandemic and associated oil-market collapse in 2020, the global energy-market disruptions of late 2021, Russia’s invasion of Ukraine in 2022, the escalation of global trade tensions in April 2025, and the renewed commodity-market turmoil in early 2026. Finally, the RIIs contain forward-looking information about subsequent commodity risk: micro- and market-level risk intensity significantly predict future volatility and absolute returns over horizons of up to twelve periods.
The distribution of risk intensity across commodities reveals substantial differences in concentration across the three risk layers. Micro and market-level risk are relatively concentrated among a subset of commodities: the top 20\% account for approximately half of total intensity in both cases, and the top 10\% account for around one-third. Economy-wide risk is more dispersed, with the corresponding shares falling to 40.6\% and 26.6\%, respectively. Aggregate risk intensity is more evenly distributed still, indicating that concentration within individual risk layers does not translate mechanically into concentration of overall commodity risk.
These findings have implications for both market participants and policymakers. Commodity risk reflects the interaction of commodity-specific conditions, common market forces, and broader macro-financial conditions, with their relative importance varying substantially across commodities and over time. For investors and risk managers, the RIIs provide information on both the concentration of risk across commodities and the sources underlying prevailing risk conditions. Their ability to predict future volatility and absolute returns further gives the indices a forward-looking dimension, suggesting their potential usefulness for monitoring emerging risk conditions and supporting portfolio and risk-management decisions. For policymakers, distinguishing market-level from economy-wide risk can help assess whether periods of elevated commodity risk are primarily associated with conditions within commodity markets or with broader economic and financial developments. More broadly, the “divide-and-conquer” framework can be applied to other settings in which risk reflects multiple sources and a common economy-wide component is latent, particularly when a large set of observable macro-financial indicators is available to proxy the underlying latent component.
\section{Model and Measurement of Layered Risk \label{sec:specification}}
\subsection{Model Specification and Risk Measurement}
We consider the following heterogeneous panel data model of commodity log-returns:
\begin{equation}
r_{i,t}
=
\rho_{i} r_{-i,t} + \boldsymbol{\alpha}_{i}^{\prime}\mathbf{x}_{i,t}
+
\boldsymbol{\beta}_{i}^{\prime}\mathbf{y}_{i,t}
+
\boldsymbol{\gamma}_{i}^{\prime}\mathbf{g}_{t}
+
\varepsilon_{i,t},
\quad
i=1,\dots,N; \quad t=1,\dots,T,
\label{model}
\end{equation}
where $r_{i,t}$ denotes the log-return of commodity $i$ in week $t$.
Let $\mathcal{S}_i$ denote the set of commodities belonging to the same sector as commodity $i$, and let $N_i=|\mathcal{S}_i|$, the cardinality of the $i$th sector. We define the leave-one-out sector return as
\begin{equation}
r_{-i,t}
=
\frac{1}{N_i-1}
\sum_{\substack{j\in\mathcal{S}_i\\j\neq i}}
r_{j,t}.
\label{eq:loo_return}
\end{equation}
Thus, $r_{-i,t}$ captures contemporaneous return movements elsewhere in commodity $i$'s sector while excluding commodity $i$ itself, and $\rho_i$ measures commodity $i$'s heterogeneous sensitivity to these market-level return movements.
The vector
\[
\mathbf{x}_{i,t}
=
(\mathrm{VLT}_{i,t},\mathrm{SKW}_{i,t},
\mathrm{KTS}_{i,t},\mathrm{OI}_{i,t})'
\in\mathbb{R}^{K_x}
\]
collects time-varying commodity-specific risk characteristics: volatility, realised skewness, realised kurtosis, and logged open interest (see Section~\ref{sec:data} for more details). These variables capture distinct dimensions of commodity-specific conditions. Volatility measures the dispersion of returns, while skewness and kurtosis capture asymmetry and tail behaviour, respectively. Open interest captures the scale of outstanding futures positions and provides information about market participation and trading activity specific to each commodity.
The vector
\[
\mathbf{y}_{i,t}
=
(\mathrm{CVLT}_{i,t},\mathrm{CSKW}_{i,t},
\mathrm{CKTS}_{i,t},\mathrm{COI}_{i,t})'
\in\mathbb{R}^{K_y}
\]
collects the leave-one-out sector-level counterparts of the commodity-specific risk characteristics in $\mathbf{x}_{i,t}$. Specifically, $\mathrm{CVLT}_{i,t}$, $\mathrm{CSKW}_{i,t}$, $\mathrm{CKTS}_{i,t}$, and $\mathrm{COI}_{i,t}$ measure volatility, realised skewness, realised kurtosis, and open interest, respectively, among the remaining commodities in commodity $i$'s sector. Each element is constructed as a leave-one-out average, so that $\mathbf{y}_{i,t}$ varies across both $i$ and $t$. The leave-one-out construction captures risk conditions prevailing elsewhere in the same sector while excluding commodity $i$'s own contribution. It therefore distinguishes commodity-specific risk conditions, $\mathbf{x}_{i,t}$, from corresponding sector-level risk conditions, $\mathbf{y}_{i,t}$.
Finally, $\mathbf{g}_{t}\in\mathbb{R}^{K_g}$ represents broader macro-financial risk, including uncertainty, sentiment, market attention, and global financial conditions. Unlike the micro and market layers, the relevant macro-financial variables are not specified a priori and are treated as latent at the estimation stage. Section~\ref{sec:Estimation} describes how an observable sparse representation of this layer is recovered from a
high-dimensional set of candidate macro-financial indicators. The disturbance $\varepsilon_{i,t}$ captures the remaining commodity-specific variation.
The slope coefficients $\boldsymbol{\alpha}_{i}$, $\boldsymbol{\beta}_{i}$, and $\boldsymbol{\gamma}_{i}$ are commodity-specific and capture heterogeneous sensitivities to the three risk layers. The coefficients $\boldsymbol{\alpha}_{i}$ measure sensitivity to commodity-specific risk characteristics, while $\boldsymbol{\beta}_{i}$ capture sensitivity to conditions
prevailing elsewhere in the same sector, which we refer to as market risk. The coefficients $\boldsymbol{\gamma}_{i}$ measure sensitivity to pervasive economy-wide macro-financial conditions. Throughout the paper, we refer to the three sources as the micro, market, and macro risk layers, respectively. Importantly, these sensitivities should be distinguished from prices of risk. They are estimated from conditional equations for realised commodity returns, whereas a risk-pricing interpretation would require an additional asset-pricing restriction linking exposures to expected returns and the estimation of the associated risk premia. Neither is imposed here.
Although the micro, market, and macro risk layers may be correlated, the coefficients represent partial effects conditional on the remaining sources of risk. This distinction is important because commodity-specific conditions, common movements elsewhere in the commodity market, and broader macro-financial conditions may coexist and need not affect individual commodities in the same way. The decomposition is therefore not intended to imply that the three layers originate from mutually exclusive primitive shocks; rather, it distinguishes their separate contributions to commodity returns conditional on the remaining risk sources.
Heterogeneous sensitivities alone do not reveal the relative importance of the different sources of commodity risk. We therefore map the estimated sensitivities into Risk Intensity Indices (RIIs). Throughout, $\widetilde{x}_{k,i,t}$, $\widetilde{y}_{k,i,t}$, and $\widetilde{g}_{k,t}$ denote standardized variables with zero mean and unit variance. Commodity-specific risk intensity is defined as the time-average
magnitude of the sensitivity-weighted impact associated with each risk layer:
\begin{equation}
\mathrm{RII}^{\mathrm{micro}}_{i}
=
\frac{1}{T}\sum_{t=1}^{T}
\left|
\sum_{k=1}^{K_x}
\alpha_{k,i}\widetilde{x}_{k,i,t}
\right|;
\quad
\mathrm{RII}^{\mathrm{market}}_{i}
=
\frac{1}{T}\sum_{t=1}^{T}
\left|
\sum_{k=1}^{K_y}
\beta_{k,i}\widetilde{y}_{k,i,t}
\right|;
\quad
\mathrm{RII}^{\mathrm{macro}}_{i}
=
\frac{1}{T}\sum_{t=1}^{T}
\left|
\sum_{k=1}^{K_g}
\gamma_{k,i}\widetilde{g}_{k,t}
\right|.
\label{eq:RII_i_all}
\end{equation}
Total commodity-level risk intensity is defined as
\begin{equation}
\mathrm{RII}^{\mathrm{total}}_{i}
=
\mathrm{RII}^{\mathrm{micro}}_{i}
+
\mathrm{RII}^{\mathrm{market}}_{i}
+
\mathrm{RII}^{\mathrm{macro}}_{i}.
\label{eq:RII_i_total}
\end{equation}
The RIIs combine the magnitude of estimated sensitivities with prevailing risk conditions. Thus, a large sensitivity need not imply high risk intensity when the corresponding risk condition is close to its typical level, while a more moderate sensitivity can be consequential when the associated risk condition is unusually pronounced. Standardisation places the variables on a common scale, allowing risk intensity to be compared across variables and layers.\footnote{Sensitivities are estimated using variables in their original economic units, preserving their economic interpretation. The RIIs are constructed using standardised variables to remove differences in measurement units and unconditional variability across risk measures.}
Absolute values are used in Eq. \eqref{eq:RII_i_all} because the RIIs measure the intensity rather than the direction of the sensitivity-weighted impact. The sign of an estimated coefficient determines the direction in which commodity returns are associated with a given risk variable, whereas the RII is concerned with the magnitude of that association under prevailing conditions. For example, our estimates indicate that, on average, commodity returns are positively associated with commodity-specific skewness, which captures asymmetry in the commodity's own return distribution, but negatively to market skewness, which captures asymmetry in returns elsewhere in the commodity's sector. These sensitivities operate in opposite directions, but both can represent important sources of risk intensity when the corresponding risk conditions are pronounced. Importantly, the absolute value is applied only after aggregating across variables within each risk layer. This allows individual variables within the same layer to reinforce or offset one another and ensures that the index measures the magnitude of the net sensitivity-weighted impact of that layer.
To examine how the anatomy of commodity risk evolves over time, we construct
cross-sectional average risk intensities for each layer:
\begin{align}
\mathrm{RII}^{\mathrm{micro}}_{t}
&=
\frac{1}{N}\sum_{i=1}^{N}
\left|
\sum_{k=1}^{K_x}
\alpha_{k,i}\widetilde{x}_{k,i,t}
\right|,
\nonumber\\
\mathrm{RII}^{\mathrm{market}}_{t}
&=
\frac{1}{N}\sum_{i=1}^{N}
\left|
\sum_{k=1}^{K_y}
\beta_{k,i}\widetilde{y}_{k,i,t}
\right|,
\nonumber\\
\mathrm{RII}^{\mathrm{macro}}_{t}
&=
\frac{1}{N}\sum_{i=1}^{N}
\left|
\sum_{k=1}^{K_g}
\gamma_{k,i}\widetilde{g}_{k,t}
\right|.
\label{eq:RII_t_layers}
\end{align}
The corresponding total risk intensity is
\begin{equation}
\mathrm{RII}^{\mathrm{total}}_{t}
=
\mathrm{RII}^{\mathrm{micro}}_{t}
+
\mathrm{RII}^{\mathrm{market}}_{t}
+
\mathrm{RII}^{\mathrm{macro}}_{t}.
\label{eq:RII_t_total}
\end{equation}
\subsection{Conceptual Framework and Motivation}
The specification in Eq.~\eqref{model} distinguishes three sources of commodity risk: conditions specific to commodity $i$, conditions prevailing elsewhere in the commodity market, and broader macro-financial conditions. The distinction between the first two layers deserves particular attention because the market variables are constructed as leave-one-out aggregates of
the corresponding commodity-specific characteristics. Although this creates an explicit link between the micro and market layers, it does not make them equivalent. The micro layer captures the conditions of commodity $i$ itself, whereas the market layer captures its sensitivity to conditions prevailing among the remaining commodities.
More generally, correlation across the three risk layers does not imply that one can be recovered from another. Aggregating commodity-specific risk characteristics does not reproduce the sensitivity-weighted market contribution for a given commodity, while correlation between market and macro-financial conditions does not make their respective contributions equivalent. The
following theorem formalises this non-reducibility and establishes the conditions under which such equivalence could arise.
\begin{theorem}[Non-reducibility of layered risk]
\label{theorem:non_reducibility}
For expositional simplicity, consider one scalar characteristic from each risk layer and define the model-implied layered contribution for commodity $i$ as $\ell_{i,t}=\alpha_i x_{i,t}+\beta_i y_{i,t}+\gamma_i g_t$. Let $\mathcal S_i$ denote the sector containing commodity $i$, with $N_i=|\mathcal S_i|$, and suppose that the market characteristic is constructed as the leave-one-out sector average $y_{i,t}=(N_i-1)^{-1}\sum_{j\in\mathcal S_i,j\neq i}x_{j,t}$. Suppose further that market and macro conditions are related according to $y_{i,t}=\lambda_i g_t+\eta_{i,t}$, where $\lambda_i\neq0$ and $g_t$ is not identically zero. Then:
\begin{enumerate}
\item The aggregated micro contribution of the remaining commodities in sector $\mathcal S_i$ and the market contribution are distinct unless
\[
\sum_{\substack{j\in\mathcal S_i\\j\neq i}}
\left(\alpha_j-\frac{\beta_i}{N_i-1}\right)x_{j,t}=0
\qquad\text{for every }t.
\]
A sufficient condition for the two contributions to coincide is
$\alpha_j=\beta_i/(N_i-1)$ for $j\in\mathcal S_i$, $j\neq i$.
This condition is also necessary if the sample second-moment matrix of the characteristics $\{x_{j,t}:j\in\mathcal S_i,\ j\neq i\}$ is nonsingular, or if equality is required as an identity for every possible realization of these characteristics.
\item The macro contribution $\gamma_i g_t$ and the market contribution $\beta_i y_{i,t}$ are distinct unless
\[
\beta_i\eta_{i,t}=(\gamma_i-\beta_i\lambda_i)g_t
\qquad\text{for every }t.
\]
Thus, when $\beta_i\neq0$, equality between the two contributions requires $\eta_{i,t}=(\gamma_i/\beta_i-\lambda_i)g_t$ for every $t$. In other words, the market-specific component $\eta_{i,t}$ must be exactly proportional to the macro factor $g_t$, with a proportionality coefficient
determined by the market and macro sensitivities. If $g_t$ and $\eta_{i,t}$ are linearly independent over the sample, this condition cannot hold for $\beta_i\neq0$. In that case, equality between the two contributions is possible only in the trivial case $\beta_i=\gamma_i=0$.
\end{enumerate}
\end{theorem}
\begin{proof}
See Online~\ref{reg_conditions_lemmas}.
\end{proof}
The first statement of Theorem~\ref{theorem:non_reducibility} shows that aggregation of micro contributions does not, in general, recover the market contribution. Exact equality requires the coefficient discrepancies between the two contributions to generate a linear combination of the underlying commodity characteristics that vanishes at every sample date. A sufficient condition is $\alpha_j=\beta_i/(N_i-1)$ for all $j\in\mathcal S_i$, $j\neq i$, which is also necessary when the corresponding sample second-moment matrix is nonsingular. Thus, absent exact linear dependence among the underlying characteristics, equality requires the micro sensitivities to satisfy a restrictive proportionality condition determined by commodity $i$'s market sensitivity and the number of commodities in its sector.
The second statement establishes an analogous result for market and macro risk. Given $y_{i,t}=\lambda_i g_t+\eta_{i,t}$, equality between the market
and macro contributions requires the market-specific component
$\eta_{i,t}$ to be exactly proportional to the macro factor $g_t$, with the
proportionality coefficient determined by the market and macro sensitivities.
Thus, the presence of market-specific variation does not by itself establish
non-reducibility; what matters is whether that variation contains a component
that is linearly independent of the macro factor. In particular, if $g_t$
and $\eta_{i,t}$ are linearly independent, the market and macro contributions
cannot coincide except in the trivial case $\beta_i=\gamma_i=0$.
\begin{remark}
Theorem~\ref{theorem:non_reducibility} characterizes non-reducibility in terms of the restrictions required for distinct risk layers to generate identical contributions. Correlation across layers is not sufficient for one layer to subsume another: exact reducibility requires specific restrictions on either the sensitivities and aggregation structure or the dependence between the underlying risk drivers.
\end{remark}
If the macro-risk variables $\mathbf{g}_t$ were directly observed, the model in Eq.~\eqref{model} could, under standard exogeneity conditions, be estimated commodity by commodity using least squares. In practice, however, $\mathbf{g}_t$ is latent. Because latent macro risk may be correlated with the observed micro- and market-level variables, omitting it induces endogeneity and renders ordinary least squares estimation inconsistent.
Since the macro-risk variables are unobserved, they need to be proxied by observable macro-financial indicators. The difficulty is that the set of potentially relevant indicators is large, diverse, and highly collinear, with limited guidance as to which indicators are empirically relevant.\footnote{Macro risk itself is not assumed to be high-dimensional. High dimensionality arises from the set of observable macro-financial indicators considered as potential proxies for the latent macro-risk process.} Uncertainty, sentiment, and financial conditions, for example, can each be represented by several competing measures, often available across different dimensions and horizons.\footnote{A case in point is ``economic uncertainty'', which can be proxied by several alternative indices \citep{BakerBloomDavis2016,CaldaraIacoviello2022,AhirBloomFurceri2022,JuradoLudvigsonNg2015}, often available at different forecast horizons and levels of geographic aggregation.} Identifying the informative subset therefore involves a trade-off between omitted-variable bias, if relevant indicators are excluded, and overfitting, if too many noisy and collinear proxies are retained.\footnote{This issue parallels the broader problem of variable proliferation in empirical finance, where a growing set of candidate predictors complicates model selection and increases the risk of overfitting and spurious inference \citep{Cochrane2011,HarveyLiuZhu2016}.}
Our ``divide-and-conquer'' procedure addresses these challenges sequentially. In Stage~1, heterogeneous sensitivities to observed micro- and market-level risk are estimated using commodity-specific instrumental-variable regressions. IV identification exploits a latent common-factor structure in the commodity-specific risk characteristics that is sufficiently rich to span the latent macro component responsible for endogeneity. Instruments are constructed from variation in the observed risk characteristics after removing this latent common component. Despite the endogeneity induced by the latent macro component, the resulting estimator consistently recovers the heterogeneous sensitivities to the observed risk layers.
In Stage~2, we extract the $r$ leading principal components from the Stage~1 residuals, where $r$ is determined from the data using eigenvalue-based procedures or information criteria
\citep{AhnHorenstein2013,BaiNg2002}. These components span the dominant directions of common residual variation. We then map each component to a high-dimensional set of observable macro-financial indicators using the Boosting with Multiple Testing procedure of \citet{KapetaniosEtal2026}. Collecting the indicators selected across components yields a sparse set of observable proxies for macro risk.\footnote{Online~\ref{sec:Appendix_MC} examines the finite-sample performance of the PCA--BMT procedure.}
The commodity-level RIIs in Eqs.~\eqref{eq:RII_i_all}--\eqref{eq:RII_i_total} characterise the cross-sectional distribution of risk intensity and allow us to assess its concentration across individual commodities. The time-varying
indices in Eqs.~\eqref{eq:RII_t_layers}--\eqref{eq:RII_t_total} instead track the evolution and changing composition of commodity risk over time. Together, these measures show where risk intensity is concentrated across commodities and how the relative importance of its micro, market, and macro sources evolves over time.
It is useful at this point to distinguish the RIIs from measures used to quantify downside financial risk, such as Value-at-Risk (VaR) and Expected Shortfall (ES). VaR and ES characterise the tail of the return distribution at a specified horizon and confidence level, addressing the magnitude of potential losses under adverse outcomes. The RIIs address a different question: what is the relative importance of the micro, market, and macro sources? Thus, while VaR and ES characterise the distribution and severity of downside losses, the RIIs characterise the relative intensity and composition of the different sources of commodity risk.
\begin{comment}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{Figures/Divide_and_Conquer_V4.png}
\caption{Sequential Estimation Framework}
\label{fig:Div_Conq}
\end{figure}
\end{comment}
\section{Divide-and-Conquer Estimation Framework \label{sec:Estimation}}
This section provides the analytical formulation of the ``divide-and-conquer'' estimation framework for model~\eqref{model}. We begin by making explicit that the macro-risk component is latent:
\begin{equation}
r_{i,t}
=\rho_{i} r_{-i,t} +
\boldsymbol{\alpha}_{i}^{\prime}\mathbf{x}_{i,t}
+
\boldsymbol{\beta}_{i}^{\prime}\mathbf{y}_{i,t}
+
u_{i,t},
\qquad
u_{i,t}
=
\boldsymbol{\gamma}_{i}^{\prime}\mathbf{g}_{t}
+
\varepsilon_{i,t},
\quad
i=1,\dots,N;\quad t=1,\dots,T.
\label{model_with_u}
\end{equation}
The macro-risk vector $\mathbf g_t$ is initially unobserved and is therefore absorbed into the composite error $u_{i,t}$.
Stacking the $T$ observations for commodity $i$ gives
\begin{equation}
\mathbf r_i
=\rho_{i} \mathbf{r}_{-i} +
\mathbf X_i\boldsymbol{\alpha}_i
+
\mathbf Y_i\boldsymbol{\beta}_i
+
\mathbf u_i,
\qquad
\mathbf u_i
=
\mathbf G\boldsymbol{\gamma}_i
+
\boldsymbol{\varepsilon}_i,
\label{model_vectori}
\end{equation}
where $\mathbf r_i=(r_{i,1},\ldots,r_{i,T})'$,
$\mathbf X_i=(\mathbf x_{i,1},\ldots,\mathbf x_{i,T})'$
is $T\times K_x$, $\mathbf Y_i=(\mathbf y_{i,1},\ldots,\mathbf y_{i,T})'$ is $T\times K_y$, $\mathbf G=(\mathbf g_1,\ldots,\mathbf g_T)'$, and $\boldsymbol{\varepsilon}_i
=(\varepsilon_{i,1},\ldots,\varepsilon_{i,T})'$.
A first challenge in estimating Eq.~\eqref{model_vectori} is that the commodity-specific and market-level risk characteristics may be correlated with the latent macro-financial conditions that also enter the commodity
return equation, $\mathbf G\boldsymbol{\gamma}_i$, giving rise to endogeneity.
A separate issue is that the commodity-specific characteristics may co-move with the observed leave-one-out sector return, $r_{-i,t}$. We distinguish this observed source of common variation from the latent common variation in the commodity-specific characteristics through the representation
\begin{equation}
\mathbf{x}_{i,t}
=
\boldsymbol{\phi}_{i} r_{-i,t}
+
\boldsymbol{\Lambda}_{i}\mathbf f_t
+
\mathbf v_{i,t},
\label{x_factor}
\end{equation}
where $\mathbf f_t\in\mathbb R^{K_f}$ denotes a vector of latent common
factors, $\boldsymbol{\phi}_i$ and $\boldsymbol{\Lambda}_i$ contain the
corresponding commodity-specific loadings, and $\mathbf v_{i,t}$ is
idiosyncratic. The first term captures common variation in the
commodity-specific characteristics associated with observed sector-return
movements, while the second captures latent common variation that is
incremental to the observed sector-return space. The precise identifying
separation between these two components is stated below.
Note that because the market-level risk variables $\mathbf{y}_{i,t}$ are constructed as leave-one-out averages of the corresponding commodity-specific characteristics, they inherit both the sector-return-related and latent common variation present in the cross-section of $\mathbf{x}_{j,t}$. In particular, the latent factor structure represented by $\mathbf{f}_t$ is also embedded in $\mathbf{y}_{i,t}$. Hence, $\mathbf{y}_{i,t}$ does not provide an independent source from which the latent factor space needs to be estimated.
For compactness, define
\begin{equation}
\mathbf C_i
=
(\mathbf r_{-i},\mathbf X_i,\mathbf Y_i),
\qquad
\boldsymbol{\theta}_i
=
(\rho_i,\boldsymbol{\alpha}_i',
\boldsymbol{\beta}_i')',
\end{equation}
so that
\begin{align}
\mathbf r_i
&=
\mathbf C_i\boldsymbol{\theta}_i+\mathbf u_i,
\label{compact}\\
\mathbf X_i
&=
\mathbf r_{-i}\boldsymbol{\phi}_i'
+
\mathbf F^0\boldsymbol{\Lambda}_i'
+
\mathbf V_i,
\label{pca_vectori}
\end{align}
where
$\mathbf F^0=(\mathbf f_1,\ldots,\mathbf f_T)'$
denotes the $T\times K_f$ population factor matrix.
Our two-stage estimation proceeds as follows.
\noindent
\textbf{Stage 1: Micro and Market Risk}
We first partial out the observed sector-return variation from the commodity-specific risk characteristics. Let $\mathcal S_1,\ldots,\mathcal S_J$ denote the $J$ commodity sectors, with $N_s=|\mathcal S_s|$ and $\sum_{s=1}^J N_s=N$, and define the sector-average return
$$
\bar{\mathbf r}_s
=
\frac{1}{N_s}
\sum_{j\in\mathcal S_s}\mathbf r_j,
\qquad s=1,\ldots,J.
$$
Collect the sector-average returns in the $T\times J$ matrix $\mathbf R = \left(
\bar{\mathbf r}_1,\ldots,\bar{\mathbf r}_J
\right)$, and define the associated residual-maker matrix $\mathbf M_{\mathbf R} = \mathbf I_T
- \mathbf R(\mathbf R'\mathbf R)^{-1}\mathbf R'$.
Rather than partialling out a different leave-one-out sector return for each commodity before factor extraction, we project all commodity-specific risk characteristics onto the orthogonal complement of the common space spanned by the observed sector returns. This removes the pervasive observed sector-return component in Eq.~\eqref{pca_vectori} while applying the same transformation to every commodity. In particular, for $i\in\mathcal S_s$,
$$
\mathbf r_{-i}
=
\frac{N_s}{N_s-1}\bar{\mathbf r}_s
-
\frac{1}{N_s-1}\mathbf r_i,
$$
and therefore
$$
\mathbf M_{\mathbf R}\mathbf r_{-i}
=
-\frac{1}{N_s-1}
\mathbf M_{\mathbf R}\mathbf r_i.
$$
Hence, under the regularity conditions stated below, the appropriately normalised magnitude of the sector-return component remaining after the common projection is of order $N_s^{-1}$ and is asymptotically negligible as the sector size increases.
For lagged characteristics, the projection is defined using the correspondingly lagged sector-return space. Specifically, for $\tau=0,\ldots,\max\{\zeta_x,\zeta_y\}$, let
\[
\mathbf R_{-\tau}
=
\left(
\bar{\mathbf r}_{1,-\tau},\ldots,
\bar{\mathbf r}_{J,-\tau}
\right),
\]
where $\bar{\mathbf r}_{s,-\tau}$ denotes the sector-average return
aligned with $\mathbf X_{i,-\tau}$, and define
\[
\mathbf M_{\mathbf R_{-\tau}}
=
\mathbf I_T
-
\mathbf R_{-\tau}
\left(
\mathbf R_{-\tau}'\mathbf R_{-\tau}
\right)^{-1}
\mathbf R_{-\tau}'.
\]
For $\tau=0$, $\mathbf R_{-0}=\mathbf R$ and
$\mathbf M_{\mathbf R_{-0}}=\mathbf M_{\mathbf R}$.
We define $\mathbf F^0_{-\tau}$ as the latent pervasive component of the commodity-specific risk characteristics that remains after accounting for the observed sector-return space. Accordingly, the decomposition is identified through the normalization
$\mathbf R_{-\tau}'\mathbf F^0_{-\tau}
= \mathbf 0$, $\tau=0,\ldots,\max\{\zeta_x,\zeta_y\}$. This normalization distinguishes the latent common component from the observed sector-return component in the reduced-form representation of the commodity-specific characteristics; it does not imply that the economic forces underlying the two components are mutually independent.
It follows that, for $i\in\mathcal S_s$,
$$
\mathbf M_{\mathbf R_{-\tau}}\mathbf X_{i,-\tau}
=
\mathbf F^0_{-\tau}\boldsymbol{\Lambda}_i'
+
\mathbf M_{\mathbf R_{-\tau}}\mathbf V_{i,-\tau}
+
\mathbf D_{i,-\tau},
$$
where
$$
\mathbf D_{i,-\tau}
=
\mathbf M_{\mathbf R_{-\tau}}
\mathbf r_{-i,-\tau}\boldsymbol{\phi}_i',
$$
and the contribution of $\mathbf D_{i,-\tau}$ is asymptotically negligible as $N_s\rightarrow\infty$.
We therefore estimate the latent factor space separately for each $\tau=0,\ldots,\max\{\zeta_x,\zeta_y\}$ by applying PCA to the projected characteristics. Denote the resulting estimated factor matrices by $\widehat{\mathbf F}_{-\tau}$. Up to rotation, they are obtained from the leading eigenvectors of
\begin{equation}
\frac{1}{NT}
\sum_{i=1}^{N}
\mathbf M_{\mathbf R_{-\tau}}
\mathbf X_{i,-\tau}
\mathbf X_{i,-\tau}'
\mathbf M_{\mathbf R_{-\tau}}.
\label{factor_covariance}
\end{equation}
The number of factors, $K_f$, is determined using standard eigenvalue-based criteria.
\begin{remark}
The projection on the observed sector-return space is required because sector returns themselves constitute a pervasive source of common variation in the commodity-specific risk characteristics. If PCA were applied directly to the unprojected characteristics, the extracted components could combine common variation associated with observed sector returns with the latent
common variation of interest. By first removing the finite-dimensional sector-return space, PCA is instead applied to the remaining common variation. Because the same sector-return space is removed from every commodity at a given lag, the projected characteristics retain a common factor representation and PCA can exploit information from the full cross-section.
\end{remark}
Last, we eliminate the influence of $\widehat{\mathbf F}$ from the entire model in Eq.~\eqref{compact} using the transformation
\begin{equation}
\mathbf M_{\widehat{\mathbf F}}\mathbf r_i
=
\mathbf M_{\widehat{\mathbf F}}\mathbf r_{-i}\rho_i
+
\mathbf M_{\widehat{\mathbf F}}\mathbf X_i\boldsymbol{\alpha}_i
+
\mathbf M_{\widehat{\mathbf F}}\mathbf Y_i\boldsymbol{\beta}_i
+
\mathbf M_{\widehat{\mathbf F}}\mathbf u_i
=
\mathbf M_{\widehat{\mathbf F}}\mathbf C_i\boldsymbol{\theta}_i
+
\mathbf M_{\widehat{\mathbf F}}\mathbf u_i .
\label{projected_model}
\end{equation}
The factor space recovered in Eq.~\eqref{factor_covariance} is statistical rather than inherently macro-financial. Common variation across commodity risk characteristics may reflect, for example, common trading or hedging activity, financialisation, sectoral conditions, as well as broader macro-financial forces. We therefore do not require the latent factor matrix $\mathbf F^0$ to coincide with the macro-risk matrix $\mathbf G$. For identification, it is sufficient that $\operatorname{col}(\mathbf G) \subseteq \operatorname{col}(\mathbf F^0)$, or equivalently that $\mathbf G=\mathbf F^0\mathbf A'$ for some conformable
matrix $\mathbf A$. This spanning condition requires the latent macro-financial variation affecting commodity returns to be contained in the common factor space of the commodity-specific risk characteristics.
Consequently, $\mathbf M_{\mathbf F^0}\mathbf G=\mathbf 0$, and consistency of the estimated factor space implies that the transformation in Eq.~\eqref{projected_model} asymptotically removes the latent macro component from the estimating equation. Importantly, the spanning condition forms part of the maintained restrictions underlying instrument validity, which can be assessed indirectly using the over-identification tests developed below.
To estimate $\boldsymbol{\theta}_i$ we construct instruments for the observed micro- and market-level variables in the defactored model. The instrument set is given by\footnote{In practice, the lag orders $\zeta_x$ and $\zeta_y$ are selected using the Lasso approach of
\citet{BelloniEtal2012}, as in \citet{ChenEtal2025}.}
\begin{align}
\widehat{\mathbf Z}_i
=
\Big(
&
\mathbf r_{-i},\;
\mathbf r_{-i,-1},\ldots,
\mathbf r_{-i,-\zeta_r},
\nonumber\\
&
\mathbf M_{\widehat{\mathbf F}}\mathbf X_i,\;
\mathbf M_{\widehat{\mathbf F}_{-1}}\mathbf X_{i,-1},
\ldots,
\mathbf M_{\widehat{\mathbf F}_{-\zeta_x}}
\mathbf X_{i,-\zeta_x},
\nonumber\\
&
\mathbf M_{\widehat{\mathbf F}}\mathbf Y_i,\;
\mathbf M_{\widehat{\mathbf F}_{-1}}\mathbf Y_{i,-1},
\ldots,
\mathbf M_{\widehat{\mathbf F}_{-\zeta_y}}
\mathbf Y_{i,-\zeta_y}
\Big).
\label{Instruments_matrix}
\end{align}
Instrument relevance is provided by the idiosyncratic component of the commodity-specific risk characteristics. Defactoring removes their latent common component but preserves idiosyncratic variation that remains informative about the corresponding endogenous regressors. Formally, this relevance requirement is captured by the full-column-rank condition on $\mathbf A_{i}$ in Assumption 5 below.
For the micro- and market-risk blocks, projecting the contemporaneous and lagged variables onto the orthogonal complement of $\widehat{\mathbf F}_{-\tau}$ asymptotically removes the latent common component underlying their endogeneity. The resulting defactored variables and their selected lags therefore provide valid instruments for the
commodity-specific and market-level risk characteristics under the maintained moment conditions.
The leave-one-out sector return $\mathbf r_{-i}$ is treated separately. Unlike the micro- and market-risk characteristics, it is not defactored when constructing the instrument set. The observed sector-return space is
separated from the latent factor space in the factor decomposition above, while the spanning condition ensures that the latent macro component of the composite error is removed by the transformation in
Eq.~\eqref{projected_model}. Subject to the maintained orthogonality conditions with the remaining idiosyncratic disturbance, the contemporaneous leave-one-out sector return and its selected lags can therefore enter the instrument set directly.
Using the instruments in Eq.~\eqref{Instruments_matrix}, the
commodity-specific IV estimator of $\boldsymbol{\theta}_{i}$ is defined as
\begin{equation}
\widehat{\boldsymbol{\theta}}_{i}
=
\left(
\widehat{\mathbf A}_{i,T}^{\prime}
\widehat{\mathbf B}_{i,T}^{-1}
\widehat{\mathbf A}_{i,T}
\right)^{-1}
\widehat{\mathbf A}_{i,T}^{\prime}
\widehat{\mathbf B}_{i,T}^{-1}
\widehat{\mathbf c}_{i,T},
\label{ivi}
\end{equation}
\begin{equation}
\widehat{\mathbf A}_{i,T}
=
\frac{1}{T}
\widehat{\mathbf Z}_{i}^{\prime}
\mathbf M_{\widehat{\mathbf F}}\mathbf C_i,
\qquad
\widehat{\mathbf B}_{i,T}
=
\frac{1}{T}
\widehat{\mathbf Z}_{i}^{\prime}
\mathbf M_{\widehat{\mathbf F}}
\widehat{\mathbf Z}_{i},
\qquad
\widehat{\mathbf c}_{i,T}
=
\frac{1}{T}
\widehat{\mathbf Z}_{i}^{\prime}
\mathbf M_{\widehat{\mathbf F}}\mathbf r_i .
\label{tabgi}
\end{equation}
The IV estimator in Eq.~\eqref{ivi} exploits the population orthogonality condition that, for each $i$,
\begin{equation}
p\!\lim_{T\to\infty} T^{-1}\mathbf{Z}_{i}^{\prime}\mathbf{M}_{\mathbf{F}^{0}}\mathbf{u}_{i}=\mathbf{0}.
\label{moments_pop}
\end{equation}
For a generic parameter vector $\boldsymbol\theta$, define the sample moment vector
\begin{equation}
\overline{\mathbf g}_{iT}(\boldsymbol\theta)
=
\frac{1}{T}
\widehat{\mathbf Z}_i'
\mathbf M_{\widehat{\mathbf F}}
\left(
\mathbf r_i-\mathbf C_i\boldsymbol\theta
\right)
=
\widehat{\mathbf c}_{i,T}
-
\widehat{\mathbf A}_{i,T}\boldsymbol\theta.
\label{sample_moments}
\end{equation}
The estimator in Eq.~\eqref{ivi} satisfies the first-order conditions
\begin{equation}
\widehat{\mathbf A}_{i,T}'
\widehat{\mathbf B}_{i,T}^{-1}
\overline{\mathbf g}_{iT}
\left(
\widehat{\boldsymbol\theta}_i
\right)
=
\mathbf 0.
\label{moments_sample}
\end{equation}
When $q_i>p$, these $p$ first-order conditions do not in general imply that all $q_i$ sample moments in
$\overline{\mathbf g}_{iT}(\widehat{\boldsymbol\theta}_i)$ are individually equal to zero.
Under the regularity conditions stated below and instrument validity, $\widehat{\boldsymbol{\theta}}_i$ is $\sqrt{T}$-consistent and asymptotically normal. Once the latent common component is projected out, the remaining estimation problem takes the form of a standard commodity-specific IV regression. We summarise below the main conditions governing identification, factor structure, heterogeneity, and limiting inference. More primitive moment, dependence, factor-estimation, and projection conditions used to establish these high-level results are collected in Online Appendix D.
\begin{description}
\item[\textbf{Assumption 1 (Idiosyncratic errors).}]
$\varepsilon_{i,t}$ and $\text{v}_{k,i,t}$ have zero mean, finite $(8+\delta)$ moments for some $\delta>0$, and satisfy weak dependence conditions over $t$ and independence across $i$.
\item[\textbf{Assumption 2 (Common component structure).}]
$\mathbf f_t=\boldsymbol C_x(L)\mathbf q_{f,t}$ and
$\mathbf g_t=\boldsymbol C_g(L)\mathbf q_{g,t}$, where
$\boldsymbol C_x(L)$ and $\boldsymbol C_g(L)$ are absolutely summable,
$\mathbf q_{f,t}\sim i.i.d.(\mathbf 0,\boldsymbol\Sigma_f)$ and
$\mathbf q_{g,t}\sim i.i.d.(\mathbf 0,\boldsymbol\Sigma_g)$, with finite
fourth moments. Letting
$\mathbf F^0=(\mathbf f_1,\ldots,\mathbf f_T)'$ and
$\mathbf G=(\mathbf g_1,\ldots,\mathbf g_T)'$, we assume $\mathrm{col}(\mathbf G) \subseteq
\mathrm{col}(\mathbf F^0)$.
\item[\textbf{Assumption 3 (Sector structure and factor loadings).}]
The number of sectors $J$ is fixed and, for each
$s=1,\ldots,J$, $\frac{N_s}{N}\longrightarrow\pi_s$,
$0<\pi_s<1$. The loadings $\boldsymbol{\phi}_i$, $\boldsymbol{\Lambda}_i$, and $\boldsymbol{\gamma}_i$ have finite fourth moments and are independent of $\varepsilon_{i,t}$ and $\mathbf v_{i,t}$. Moreover, for each sector, $\frac{1}{N_s} \sum_{i\in\mathcal S_s}
\boldsymbol{\Lambda}_i \boldsymbol{\Lambda}_i'
\overset{p}{\longrightarrow} \mathbf Q_{\Lambda,s}$,
$\mathbf Q_{\Lambda,s}>0$. The relevant remaining second-moment matrices are positive definite.
\item[\textbf{Assumption 4 (Random coefficients).}]
$\boldsymbol{\theta}_i
=
\boldsymbol{\theta}+\boldsymbol{\eta}_i$,
where
\[
\boldsymbol{\theta}_i
=
(\rho_i,
\boldsymbol{\alpha}_i',
\boldsymbol{\beta}_i')',
\]
and $\boldsymbol{\eta}_i$ is i.i.d. with mean zero, finite fourth moments, and sub-exponential tails.
\item[\textbf{Assumption 5 (Identification and limiting distribution).}]
Let $\mathcal H$ denote the sigma-field generated by the heterogeneous coefficients and loadings. For each fixed $i$, the following conditions hold conditionally on $\mathcal H$, for almost every realization, as $N,T\to\infty$ with $N/T\to c\in(0,\infty)$. The instrument dimension $q_i\geq p$ is fixed. We assume
\[
\widehat{\mathbf A}_{i,T}\overset{p}{\to}\mathbf A_i,
\qquad
\widehat{\mathbf B}_{i,T}\overset{p}{\to}\mathbf B_i,
\]
where $\mathbf A_i$ has full column rank $p$ and $\mathbf B_i$ is positive definite. Moreover,
\[
\frac{1}{\sqrt T}
\mathbf Z_i'\mathbf M_{\mathbf F^0}\mathbf u_i
\overset{d}{\longrightarrow}
N(\mathbf 0,\boldsymbol\Omega_i),
\]
where $\boldsymbol\Omega_i$ is positive definite. For uniform results over $i$, $\mathbf B_i$ and $\boldsymbol\Omega_i$ are uniformly positive definite.
\begin{comment}
Let $\mathbf m_{i,t}^{\circ}\in\mathbb R^{q_i}$ denote a conditionally covariance-stationary, mean-zero population moment process, and define
\[
\mathbf a_{iT}
=
\frac{1}{\sqrt T}\sum_{t=1}^T\mathbf m_{i,t}^{\circ},
\qquad
\boldsymbol\Gamma_i(h)
=
\mathbb E\!\left(
\mathbf m_{i,t}^{\circ}
\mathbf m_{i,t-h}^{\circ\prime}
\mid\mathcal H
\right).
\]
Assume
\[
\sum_{h=-\infty}^{\infty}
\|\boldsymbol\Gamma_i(h)\|<\infty,
\qquad
\boldsymbol\Omega_i
=
\sum_{h=-\infty}^{\infty}\boldsymbol\Gamma_i(h)
>0,
\]
and
\[
\mathbf a_{iT}
\overset{d}{\longrightarrow}
N(\mathbf 0,\boldsymbol\Omega_i).
\]
Finally, assume the population-score representation
\[
\frac{1}{\sqrt T}
\mathbf Z_i'\mathbf M_{\mathbf F^0}\mathbf u_i
=
\mathbf a_{iT}+o_p(1).
\]
For uniform results over $i$, $\mathbf B_i$ and
$\boldsymbol\Omega_i$ are assumed to be uniformly positive definite.
\end{comment}
\end{description}
The full-column-rank condition on $\mathbf A_i$ is the standard IV relevance condition, requiring the instrument set to retain sufficient independent variation to identify the $p$ elements of $\boldsymbol{\theta}_{i}$. In the present setting, this identifying variation is provided by the idiosyncratic component of the observed risk characteristics that remains after their latent common component has been removed.
Additional high-level moment, dependence, pervasiveness, and regularity conditions required for the asymptotic arguments are reported in
Online~\ref{reg_conditions_lemmas}. These conditions ensure the uniform laws of large numbers and central limit results required for PCA-based factor extraction and commodity-specific IV estimation, as well as the
asymptotic negligibility of the residual leave-one-out sector-return component after projection on the observed sector-return space.
Theorem~\ref{TH2} formalises that, despite the endogeneity of the observed micro- and market-level variables induced by latent macro risk, the proposed commodity-specific IV estimator consistently recovers the heterogeneous sensitivities and supports standard inference under the stated conditions.
\begin{theorem}[Asymptotic Distribution of the Commodity-Specific IV Estimator]
\label{TH2}
Consider the model in Eqs.~\eqref{compact} and \eqref{x_factor}, under Assumptions 1--5 and the additional regularity conditions reported in the Online Appendix. Then, as $N,T\to\infty$ such that $N/T\rightarrow c$, with $0<c<\infty$, for each fixed $i$ and almost every realization of $\mathcal H$,
\begin{equation}
\sqrt{T}
\left(
\widehat{\boldsymbol{\theta}}_i-\boldsymbol{\theta}_i
\right)
\overset{d}{\longrightarrow}
N\left(
\mathbf 0,\,
\mathbf V_i
\right)
\qquad\text{conditionally on }\mathcal H,
\end{equation}
where
\begin{equation}
\mathbf V_i
=
(\mathbf A_i'\mathbf B_i^{-1}\mathbf A_i)^{-1}
\mathbf A_i'\mathbf B_i^{-1}
\boldsymbol{\Omega}_i
\mathbf B_i^{-1}\mathbf A_i
(\mathbf A_i'\mathbf B_i^{-1}\mathbf A_i)^{-1}.
\end{equation}
\end{theorem}
\begin{proof}
See Online~\ref{reg_conditions_lemmas}.
\end{proof}
The ``divide-and-conquer'' framework generates over-identifying restrictions that allow the validity of the instrument set to be assessed separately for each commodity. Importantly, these restrictions provide an indirect diagnostic relevant to the spanning condition in Assumption~2, according to which the latent macro-risk component is contained in the common factor space extracted from the commodity-specific risk characteristics after accounting for observed sector-return variation. We therefore derive a commodity-specific over-identification test, which also provides the basis for the panel-level diagnostic reported later in Section~\ref{sec:results}. Because the defactored moment conditions may exhibit serial dependence, the test is based on a heteroskedasticity and autocorrelation consistent (HAC) estimator of the long-run covariance matrix.
Let $\widehat{\mathbf u}_i = \mathbf r_i-\mathbf C_i\widehat{\boldsymbol\theta}_i$ denote the residual vector from the baseline IV estimator in
Eq.~\eqref{ivi}, and define
\[
\widehat{\mathbf m}_{i,t}
=
\widehat{\mathbf z}_{i,t}^{*}\,
\widehat u_{i,t}^{*},
\qquad
t=1,\dots,T,
\]
where $\widehat{\mathbf z}_{i,t}^{*}$ denotes the $t$th row of
$\mathbf M_{\widehat{\mathbf F}}\widehat{\mathbf Z}_i$ and
$\widehat u_{i,t}^{*}$ is the $t$th element of
$\mathbf M_{\widehat{\mathbf F}}\widehat{\mathbf u}_i$.
Define the sample autocovariance matrices
\begin{equation}
\widehat{\boldsymbol\Gamma}_{i,\ell}
=
\frac{1}{T}
\sum_{t=\ell+1}^{T}
\widehat{\mathbf m}_{i,t}
\widehat{\mathbf m}_{i,t-\ell}^{\prime},
\qquad
\ell=0,1,\dots,L_T,
\label{vc1}
\end{equation}
and the HAC estimator of the long-run covariance matrix
\begin{equation}
\widehat{\boldsymbol\Omega}_{iT}
=
\widehat{\boldsymbol\Gamma}_{i,0}
+
\sum_{\ell=1}^{L_T}
k\!\left(\frac{\ell}{L_T+1}\right)
\left(
\widehat{\boldsymbol\Gamma}_{i,\ell}
+
\widehat{\boldsymbol\Gamma}_{i,\ell}^{\prime}
\right),
\label{hac_omega}
\end{equation}
where $k(\cdot)$ is a kernel function and $L_T$ is a bandwidth parameter satisfying $L_T\to\infty$ and $L_T/T\to0$ as $T\to\infty$. Using $\widehat{\boldsymbol\Omega}_{iT}$ as the second-step weighting matrix, define the efficient two-step estimator
\begin{equation}
\widetilde{\boldsymbol\theta}_i
=
\left(
\widehat{\mathbf A}_{i,T}'
\widehat{\boldsymbol\Omega}_{iT}^{-1}
\widehat{\mathbf A}_{i,T}
\right)^{-1}
\widehat{\mathbf A}_{i,T}'
\widehat{\boldsymbol\Omega}_{iT}^{-1}
\widehat{\mathbf c}_{i,T}.
\label{efficient_iv}
\end{equation}
The corresponding residual vector is
\[
\widetilde{\mathbf u}_i
=
\mathbf r_i-\mathbf C_i\widetilde{\boldsymbol\theta}_i.
\]
The estimator $\widetilde{\boldsymbol\theta}_i$ is introduced solely for constructing the over-identification test; the baseline estimator $\widehat{\boldsymbol\theta}_i$ in Eq.~\eqref{ivi} remains the estimator used in the two-stage empirical procedure.
The commodity-specific over-identification statistic is then defined as
\begin{equation}
S_{iT}
=
\frac{1}{T}
\left(
\widetilde{\mathbf u}_i'
\mathbf M_{\widehat{\mathbf F}}
\widehat{\mathbf Z}_i
\right)
\widehat{\boldsymbol\Omega}_{iT}^{-1}
\left(
\widehat{\mathbf Z}_i'
\mathbf M_{\widehat{\mathbf F}}
\widetilde{\mathbf u}_i
\right).
\label{sargan}
\end{equation}
Equivalently,
\[
S_{iT}
=
T\,
\overline{\mathbf g}_{iT}
(\widetilde{\boldsymbol\theta}_i)'
\widehat{\boldsymbol\Omega}_{iT}^{-1}
\overline{\mathbf g}_{iT}
(\widetilde{\boldsymbol\theta}_i),
\]
where $\overline{\mathbf g}_{iT}(\boldsymbol\theta)$ is defined in
Eq.~\eqref{sample_moments}.
The limiting distribution of the statistic, allowing for serial correlation in the moment process, is given in the following theorem.\footnote{Related IV estimators for panel models with latent common components have been studied in the literature (e.g., \citet{NorkuteEtal2021}). To the best of our knowledge, however, the commodity-specific over-identification test for the defactored moment conditions considered
here has not previously been developed. Theorem~\ref{TH3} therefore provides a formal basis for assessing instrument validity within the present divide-and-conquer framework.}
\begin{theorem}[Commodity-Specific Over-Identification Test]
\label{TH3}
Consider the model in Eqs.~\eqref{compact}--\eqref{pca_vectori}, and suppose
that Assumptions 1--5 hold together with the additional regularity conditions
reported in the Online Appendix, including Assumption~A.5 on consistency of
the long-run covariance estimator. Let $S_{iT}$ be defined as in
Eq.~\eqref{sargan}, with the sample moments evaluated at
$\widetilde{\boldsymbol\theta}_i$ defined in Eq.~\eqref{efficient_iv}.
Under the null hypothesis that the $q_i$ population moment conditions are
valid, as $N,T\to\infty$ with $N/T\to c\in(0,\infty)$, for each fixed $i$
and almost every realization of $\mathcal H$,
\[
S_{iT}
\overset{d}{\longrightarrow}
\chi^2_{q_i-p}
\qquad\text{conditionally on }\mathcal H,
\]
where $q_i$ is the fixed number of instruments for commodity $i$ and
$p=\dim(\boldsymbol\theta_i)$.
\end{theorem}
\begin{proof}
See Online~\ref{reg_conditions_lemmas}.
\end{proof}
\noindent
\textbf{Stage 2: Latent Economy-Wide Common Risk Sources}
In Stage~2, we use the commodity-specific IV residuals from Stage~1 to identify observable proxies for latent economy-wide macro risk and estimate the corresponding heterogeneous sensitivities. We begin by forming $\widehat{\mathbf{u}}_{i} = \mathbf{r}_{i} - \mathbf{C}_{i}\widehat{\boldsymbol{\theta}}_{i}$
\begin{comment}
\begin{equation}
\widehat{\mathbf{u}}_{i} = \mathbf{r}_{i} - \mathbf{C}_{i}\widehat{\boldsymbol{\theta}}_{i},\label{IV_res}
\end{equation}
\end{comment}
which consistently estimates $\mathbf u_i
=
\mathbf G\boldsymbol\gamma_i+\boldsymbol\varepsilon_i$ and captures the variation in commodity returns remaining after controlling for the observed sector-return, micro-risk, and market-risk sources. We next extract the $r$ leading principal components from the empirical covariance matrix $(NT)^{-1} \sum_{i=1}^{N} \widehat{\mathbf{u}}_{i} \widehat{\mathbf{u}}_{i}^{\prime}$, where $r$ is determined using standard eigenvalue-based methods \citep{BaiNg2002,AhnHorenstein2013}. Under the maintained factor structure, these principal components recover, up to rotation, the common component of the Stage~1 residuals generated by $\mathbf G$. They therefore provide a statistical representation of the latent macro-risk space remaining after the observed risk sources have been controlled for. We then apply the Boosting with Multiple Testing (BMT) algorithm of \citet{KapetaniosEtal2026} to each extracted principal component using a high-dimensional set of candidate macro-financial variables $\boldsymbol{\mathcal{Z}} \in \mathbb{R}^{T \times n}$, where $\boldsymbol{\mathcal{Z}} = \left(\boldsymbol{\mathcal{Z}}_{1}, \dots, \boldsymbol{\mathcal{Z}}_{n}\right)$. Methodological details of BMT are provided in the Online \ref{subsec:BMT_method}. The high dimensionality arises because macro risk is latent, while observable macro-financial indicators available to proxy it are numerous, highly collinear, and provide limited guidance as to which are truly informative. The procedure selects, for each component $j=1,\dots,r$, a sparse index set $\mathcal{S}_{j} \subset {1,\dots,n}$.
The union of predictors selected across the $r$ components defines the set of observable macro-financial proxies for the latent macro-risk
space. Formally, letting $\mathcal{S} = \bigcup_{j=1}^{r} \mathcal S_{j}$, the corresponding design matrix is denoted by $\boldsymbol{\mathcal Z}_{(\mathcal{S})}$. Finally, we regress $\widehat{\mathbf{u}}_{i}$ on $\boldsymbol{\mathcal Z}_{(\mathcal{S})}$ to obtain commodity-specific sensitivities to the identified macro risk sources, $\widehat{\boldsymbol{\gamma}}_i$.\footnote{\citet{KapetaniosEtal2026} show that the post selection least squares estimator is asymptotically equivalent to the infeasible oracle estimator that would be obtained if the true model were known in advance.}
The PCA--BMT step therefore links the common variation remaining in commodity returns after Stage~1 to observable macro-financial risk sources through a
structured dimension-reduction and selection procedure. PCA recovers the common residual variation associated with latent macro risk, while BMT identifies the macro-financial indicators that are informative about this
variation. The resulting predictor set provides a sparse and interpretable set of observable proxies for the latent macro-risk space and permits estimation of heterogeneous macro-risk sensitivities across commodities. Finite-sample evidence in Online~\ref{sec:Appendix_MC} documents the performance of the procedure in high-dimensional settings and compares it with alternative regularisation methods such as Lasso.
\begin{remark}
\textnormal{The treatment of the latent macro-risk component differs from standard factor-augmented approaches, which typically retain a small number of estimated latent factors in the final specification \citep[e.g.,][]{Bai2009,NorkuteEtal2021}. Here, the extracted factors serve an intermediate role: they recover the common residual variation to which BMT is subsequently applied. The high-dimensionality arises from the candidate set of observable macro-financial indicators, not from the dimension of the underlying macro-risk process. BMT selects a sparse subset of these indicators to provide an observable representation of the latent common component.
}
\end{remark}
\begin{remark}
\textnormal{To the best of our knowledge, applying BMT to principal components extracted from the commodity-specific IV residuals, rather than directly to the residuals themselves, is novel. The distinction is important. Direct commodity-by-commodity selection could produce different sets of macro-financial indicators across commodities, making it difficult to identify a common observable representation of macro risk. Moreover, non-selection of an indicator could reflect either a zero sensitivity or insufficient power to pass the selection threshold, complicating aggregation and subsequent Mean Group inference. The PCA--BMT procedure instead performs selection on the common residual variation and then estimates heterogeneous commodity-specific sensitivities conditional on a common selected indicator set.
}
\end{remark}
Once the commodity-specific coefficients $\widehat{\boldsymbol{\theta}}_i$ and
$\widehat{\boldsymbol{\gamma}}_i$ have been obtained, they can be aggregated to conduct population-level inference while preserving heterogeneity in the underlying commodity-specific sensitivities. In particular, the Mean Group (MG) estimator of the population average $\boldsymbol{\theta}$ is
\begin{equation}
\widehat{\boldsymbol{\theta}}
=
\frac{1}{N}
\sum_{i=1}^{N}
\widehat{\boldsymbol{\theta}}_i .
\label{ivmg}
\end{equation}
The following result establishes that averaging the commodity-specific estimates consistently recovers the population-average sensitivities and permits standard inference on these averages.
\noindent
\begin{proposition}[Asymptotic Distribution of the Mean Group Estimator]
\label{prop:MG_distribution}
Consider the model in Eqs.~(\ref{compact})--(\ref{pca_vectori}), under Assumptions 1--5. Then, as $N,T\to\infty$ such that $N/T\to c$, with $0<c<\infty$,
\begin{equation}
\sqrt{N}
\left(
\widehat{\boldsymbol{\theta}}-\boldsymbol{\theta}
\right)
\overset{d}{\rightarrow}
N\left(
\mathbf{0},\boldsymbol{\Sigma}_{\eta}
\right),
\end{equation}
and
\begin{equation}
\widehat{\boldsymbol{\Sigma}}_{\eta}
-
\boldsymbol{\Sigma}_{\eta}
\overset{p}{\rightarrow}
\mathbf{0},
\end{equation}
where
\begin{equation}
\widehat{\boldsymbol{\Sigma}}_{\eta}
=
\frac{1}{N-1}
\sum_{i=1}^{N}
\left(
\widehat{\boldsymbol{\theta}}_i-\widehat{\boldsymbol{\theta}}
\right)
\left(
\widehat{\boldsymbol{\theta}}_i-\widehat{\boldsymbol{\theta}}
\right)' .
\label{MG_varcov}
\end{equation}
\end{proposition}
\begin{proof}
See Online~\ref{reg_conditions_lemmas}.
\end{proof}
\noindent
We also summarize the heterogeneous macro-risk sensitivities using the Mean Group estimator
\begin{equation}
\widehat{\boldsymbol{\gamma}}
=
\frac{1}{N}
\sum_{i=1}^{N}
\widehat{\boldsymbol{\gamma}}_i,
\end{equation}
with its variance-covariance matrix estimated from the cross-sectional dispersion of the commodity-specific estimates, analogously to Eq.~\eqref{MG_varcov}. The post-selection oracle property established by \citet{KapetaniosEtal2026} provides the basis for inference following the BMT selection step.
\begin{remark}
\textnormal{
In the empirical implementation, we also consider a Median Group estimator as a robustness check. This replaces the cross-sectional mean with the cross-sectional median of the commodity-specific coefficients and is less sensitive to outliers and heavy-tailed heterogeneity. Results are reported in Online~\ref{sec:Appendix_Robustness}.
}
\end{remark}
\section{Data \label{sec:data}}
Our sample consists of $N=52$ exchange-traded commodity futures contracts obtained from LSEG Datastream (Eikon). For each commodity we retrieve the daily settlement price of the nearest-maturity series (Datastream mnemonic suffix \texttt{TRc1}) together with the corresponding open interest, defined as the total number of contracts entered into and not yet liquidated, that is, the aggregate purchase or sale commitment outstanding at the close of trading.
The cross-section is deliberately broad along three dimensions. First, it spans the three conventional sectors of the commodity complex: agriculture and livestock (24 contracts), energy and petrochemicals (14 contracts), and metals (14 contracts). Second, it spans venues in North America (CBOT, CME, CSCE, KCBT, NYMEX, COMEX, NYCE), Europe (ICE, LIFFE, LME), and Asia (DCE, ZCE, MCX, NCDEX, KLSE, SICOM, GME). Third, and most important for our purposes, it deliberately mixes contracts that are constituents of the major investable commodity indices, such as WTI and Brent crude, gold, copper, and the LME base metals, with contracts that lie outside the index universe and are traded predominantly by local commercial participants, such as cardamom, kapas, mentha oil, barley and maize on the Indian exchanges, eggs on the Dalian Commodity Exchange, and the Chinese ferroalloys. This design is central to the identification of heterogeneous exposures: the financialisation literature predicts that index membership, rather than physical characteristics alone, governs the strength of an asset's comovement with the commodity market as a whole \citep{TangXiong2012,BasakPavlova2016}, and a sample restricted to index constituents cannot test that prediction. Table~\ref{tab:commodity_list} lists the full cross-section by sector, settlement currency, and Datastream series.
Our classification assigns the petrochemical contracts, namely linear low-density polyethylene, polypropylene, polyvinyl chloride, and methanol, to the energy sector, since their production costs and price dynamics are driven principally by crude oil, naphtha, and coal feedstocks rather than by agricultural or metallurgical fundamentals. We assign the ferroalloys (ferrosilicon, silicon manganese) and the steel complex (iron ore, hot-rolled coil steel) to metals. Livestock and dairy contracts are grouped with agriculture.
The sample runs at weekly frequency from August 2014 to April 2026, yielding a balanced panel of $T=611$ weekly observations per commodity and $NT=31,772$ observations in total. The start date is dictated by the availability of continuous price and open-interest histories for the Asian contracts, which are the most recently listed in the cross-section.
Thirteen of the fifty-two contracts settle in currencies other than the US dollar: the MCX and NCDEX contracts in Indian rupees, the DCE and ZCE contracts in Chinese renminbi, and the KLSE palm oil contract in Malaysian ringgit. We convert all settlement prices to US dollars at the contemporaneous spot exchange rate before computing returns, realised moments, and market aggregates, so that the panel is expressed throughout in a single numeraire and the cross-section is directly comparable. This is the convention of the commodity asset-pricing literature \citep{GortonRouwenhorst2006,TangXiong2012}, and it has the natural interpretation that returns are those realised by an unhedged dollar-based investor.
{color{red} Based on daily and weekly data, returns and higher moments of both individual contracts and market measures are computed, as detailed in Online Appendix A. }
Stage 2 of our sequential approach involves projecting dominant principal components onto a high-dimensional set of candidate macro-financial risk indicators. To this end, we collect 36 variables that serve as proxies for a range of risk channels, including uncertainty, sentiment, market attention, and global financial conditions. These variables have been carefully assembled from a wide array of influential papers, and to the best of our knowledge this is the first study to employ such a diverse set of indicators within a single framework.
The full set of these variables and their definitions are listed in Table~\ref{tab:meta-variables}.
This list is by no means exhaustive. Additional indices not considered here include the global financial uncertainty index by \cite{Caggiano2023} (only available up to 2020); and the global financial cycle index by \cite{Miranda-Agrippino2020} (only recently extended to include recent years).
\section{Results \label{sec:results}}
\subsection{Micro, Market, and Macro Risk Sensitivities \label{sec:risk_sensitivities}}
Table~\ref{tab:MG_sensitivities} reports Mean Group estimates of the commodity return sensitivities to the micro, market, and macro risk sources identified by our two-stage procedure. Results are shown for the full sample and separately for agriculture, energy, and metals.
At the micro level, in the full sample, commodity-specific volatility is negatively associated with returns (\(-0.047\)). This negative contemporaneous association is consistent with related evidence linking volatility and commodity returns \citep{FuertesMiffreFernandezPerez2015} and, more broadly, with the negative contemporaneous relationship documented between unexpected stock returns and unexpected changes in market volatility \citep{FrenchSchwertStambaugh1987}. This relationship is strongest for energy (\(-0.110\)), weaker for agriculture (\(-0.052\)), and absent for metals. Commodity-specific skewness is positively and highly significantly associated with returns in all three sectors, with remarkably similar coefficients ranging from \(0.019\) to \(0.025\). The positive coefficient indicates that, conditional on market-level distributional characteristics, weeks characterised by an increase in the right-skewness of an individual commodity’s return distribution are associated with higher contemporaneous returns.\footnote{This contemporaneous sensitivity should not be interpreted as evidence on a skewness risk premium. Existing asset-pricing studies, such as \citet{FernandezPerezFrijnsFuertesMiffre2018}, examine whether skewness predicts subsequent futures returns. The two objects address different economic questions and need not have the same sign.} Thus, while asymmetry in the commodity's own return distribution is systematically related to contemporaneous returns, the corresponding average sensitivity to own tail thickness is considerably weaker. Open interest has no statistically significant Mean Group coefficient in any sector. This is an average cross-commodity result and does not imply that open interest is irrelevant for every individual commodity.
Market conditions have a strong association with individual commodity returns. The leave-one-out market return enters positively and significantly in every sector, with a full-sample sensitivity of \(0.509\). The association is particularly strong for energy, where the coefficient exceeds one (\(1.038\)), compared with \(0.489\) for metals and \(0.213\) for agriculture. This strong comovement is consistent with evidence that commodity returns contain an important common component \citep{ChristoffersenLundeOlesen2019,DieboldLiuYilmaz2018}. The sectoral estimates further show that this common market component is especially important for energy returns.
Market volatility is significantly positively associated with returns in the energy sector (\(0.198\)), but significantly negatively associated with returns in the metals sector (\(-0.067\)). The difference in signs is consistent with the distinct exposure of these sectors to commodity-market stress. Energy markets may be particularly sensitive to episodes in which supply disruptions simultaneously increase uncertainty and prices, whereas metals may be more closely exposed to the deterioration in global demand conditions that often accompanies broader periods of market stress. This interpretation is also consistent with the commodity-market connectedness evidence of \citet{DieboldLiuYilmaz2018}, who identify energy as an important transmitter of volatility shocks within commodity markets. By contrast, the corresponding coefficient for agriculture is small and statistically insignificant. At the aggregate level, the opposing sensitivities of energy and metals may therefore contribute to the lack of statistical significance of the pooled market-volatility coefficient. The opposing signs can also be interpreted in the context of the layered specification. Conditional on market volatility, higher commodity-specific volatility represents an increase in risk that is relatively concentrated in the individual commodity. Conversely, conditional on a commodity’s own volatility, higher market volatility captures a change in broader risk conditions affecting other commodities in the relevant market. Thus, although the two volatility measures may be related, they capture different dimensions of volatility exposure once each is conditioned on the other. Consequently, there is no theoretical requirement for them to exhibit the same association with contemporaneous returns.
This distinction between commodity-specific and market-level risk is even more systematic when considering higher moments. Market skewness enters negatively and highly significantly in all three sectors, in contrast to the positive coefficient on commodity-specific skewness. Conditional on market skewness, an increase in a commodity’s own skewness captures a change in the asymmetry of its return distribution that is specific to that commodity. Conversely, conditional on its own skewness, an increase in market skewness reflects a change in the asymmetry of return distributions elsewhere in the sector. The opposing coefficients therefore indicate that the association between distributional asymmetry and returns depends on the level at which the change in skewness occurs. In other words, the economic relevance of higher-moment variation is not determined solely by whether returns become more or less skewed, but also by whether the change in skewness is commodity-specific or occurs at the broader market level. A similar, although less pronounced, distinction emerges for kurtosis: commodity-specific kurtosis is weakly negative, whereas market kurtosis is positive and highly significant across all three sectors. Taken together, these results provide empirical evidence that the level at which higher-moment variation occurs matters for its association with returns, supporting a distinction between commodity-specific and market-level higher-moment risk rather than treating skewness and kurtosis as undifferentiated measures of risk.
Before turning to the latent macro component, we assess the empirical validity of the Stage~1 identification strategy using standard IV diagnostics. Based on the stacked specification across commodities, the Kleibergen--Paap test rejects the null of underidentification ($LM=50.602$, $p=0.004$), while the Kleibergen--Paap rk Wald $F$-statistic is 276.612, providing no indication of weak identification. The panel over-identification test does not reject the null of valid moment conditions ($J_{NT}=28.610$, $p=0.329$).\footnote{The test builds directly on the commodity-specific over-identification test developed in Theorem 3: letting $p_{iT}$ denote the p-value of the test for commodity $i$, the individual p-values are combined using the Fisher statistic ($J_{NT}=-2\sum_{i=1}^{N}\ln p_{iT}$), with inference obtained using a cross-sectional bootstrap to accommodate dependence across commodities.} This result is particularly relevant in our setting because instrument validity relies on the spanning condition in Assumption~2: if the extracted common factor space failed to absorb the latent macro component, the defactored moment conditions would generally be violated.
PCA identifies a single dominant component in the first-stage residuals, accounting for 91.4\% of total residual variation. BMT selects six observable macro-financial indicators: the broad U.S. dollar index (DTWEXBGS), Treasury-market volatility (MOVE), consumer sentiment (UMCSENT), five-year expected inflation (EXPINF5YR), equity-market volatility (VIX), and oil-related geopolitical risk. The bottom panel of Table~\ref{tab:MG_sensitivities} reports Mean Group estimates of the corresponding sensitivities. All six indicators exhibit statistically significant average sensitivities, with consistent signs across the agriculture, energy, and metals sectors. This degree of cross-sector consistency is particularly noteworthy given the substantially greater heterogeneity observed in some of the market-level sensitivities.
The positive sensitivity to the dollar index warrants a conditional interpretation. The conventional negative relationship between the U.S. dollar and commodity prices is generally associated with pricing and invoicing effects, as well as global demand channels, that operate through broad movements in commodity markets. In our Stage 2 specification, however, these market-level effects have already been removed. Consequently, the estimated dollar coefficient should not be interpreted as capturing the conventional unconditional dollar–commodity relationship. Rather, it measures the association between movements in the dollar and the residual common component of commodity returns, conditional on—and after accounting for—the observed market-level effects. Under this interpretation, the dollar may also reflect its role as a global financial risk factor, operating through financial conditions and the risk-taking capacity of market participants. This interpretation is consistent with the evidence in \citet{AvdjievEtal2019}, who emphasize the broader role of the U.S. dollar in global financial conditions.
This interpretation is reinforced by a robustness exercise that addresses a potential measurement concern. Because all settlement prices are converted to U.S. dollars prior to the computation of returns, one might suspect that the estimated dollar exposure reflects
currency translation rather than an economic channel, since for contracts denominated in foreign currency the dollar return embeds the movement of the local currency against the dollar. We therefore re-estimate Stage 2 separately for the thirty-nine dollar-denominated
contracts and for the thirteen contracts settling in Chinese renminbi, Indian rupees, or Malaysian ringgit. The dollar exposure remains positive and highly significant in both subsamples (see Table \ref{tab:robustness_dollar}, at $0.0026$ ($z=10.50$) and $0.0033$ ($z=11.38$) respectively. The former
estimate is decisive for the concern at hand: for contracts that already settle in dollars no translation term can arise by construction, so the positive loading cannot be a measurement artefact. The modest difference between the two estimates is not specific to the dollar. All six macro coefficients, together with the constant, are larger in absolute value in the foreign-currency subsample by a nearly uniform factor of approximately $1.29$
(standard deviation $0.03$ across the seven parameters), and the dollar coefficient relative to the VIX coefficient is virtually identical in the two groups ($1.07$ and $1.04$). This pattern is consistent with a difference in the scale of the residual component rather than with a currency-specific channel, as these locally traded contracts exhibit weak market-level exposures in Stage 1 and therefore retain more residual variation.
Financial-market uncertainty provides a particularly informative interpretation of the latent macro-financial component. Both the VIX and MOVE enter with negative and statistically significant sensitivities, indicating that heightened uncertainty in equity and Treasury markets is associated with lower commodity returns, even after conditioning on the observed commodity-market component. This pattern is consistent with the latent factor captures a dimension of financial stress that is not fully reflected in broad commodity-market movements. The result is consistent with the broader literature documenting stronger linkages between commodity and financial markets during periods of heightened uncertainty. \citet{SilvennoinenThorp2013}, for example, show that increases in the VIX are associated with higher commodity-return volatility and stronger commodity–equity correlations, particularly for a substantial fraction of commodity–equity pairs. \citet{ChengKirilenkoXiong2015} provide complementary evidence from investor behaviour, documenting reductions in financial traders’ net-long positions in commodity futures in response to heightened market distress. \citet{AdamsGluck2015} likewise document persistent comovement between commodity and equity markets in the post-financialization period. Finally, \citet{ChristoffersenLundeOlesen2019} identify a pronounced common factor in commodity futures volatility that is related to stock-market volatility and the business cycle. Our estimates complement this evidence by showing that the financial-stress channel extends to Treasury-market volatility and remains present after the observed commodity-market component has been removed. The negative sensitivities to both VIX and MOVE therefore suggest that the latent factor captures a residual dimension of financial conditions that operates across commodity sectors beyond the variation explained by the common observed market layer.
Consumer sentiment also exhibits a negative and statistically significant sensitivity across all three commodity sectors. This coefficient warrants a similarly conditional interpretation. Because broad commodity-market movements have already been accounted for in Stage 1, the Stage 2 estimate should not be interpreted as the unconditional effect of changes in consumer sentiment on commodity returns. Rather, it captures the association between consumer sentiment and the residual common component of commodity returns, conditional on the variation already explained by the observed commodity-market component. Economically, this residual relationship may reflect changes in risk appetite, expectations, or broader perceptions of economic conditions that are not fully captured by movements in the commodity market as a whole.
Expected inflation provides a distinct perspective on commodity price dynamics. The literature on the inflation-hedging properties of commodities has traditionally focused on the relationship between commodity returns and realised or unexpected inflation, with evidence suggesting that the strength of this relationship varies substantially across commodity classes and over time. By contrast, EXPINF5YR is a forward-looking measure of medium-term inflation expectations and therefore captures a different dimension of the inflation environment. Its coefficient captures the association between changes in medium-term inflation expectations and the residual common component of commodity returns, conditional on the observed commodity-market layer. One possible interpretation is that higher inflation expectations convey information about anticipated monetary and financial conditions. To the extent that stronger inflation expectations are accompanied by expectations of tighter monetary policy and higher real interest rates, they may exert downward pressure on commodity prices through several channels, including the opportunity cost of holding inventories, expectations of future commodity demand, and portfolio reallocation. This interpretation is consistent with the literature emphasizing the role of real interest rates and monetary policy in commodity-price determination. Frankel (2008), for example, highlights the role of real interest rates in determining real commodity prices, while more recent evidence documents cost-of-carry, expected-demand, and financial-market channels through which monetary policy affects commodity prices.
Finally, oil-related geopolitical risk enters positively in all sectors, with the largest coefficient for energy. This is consistent with supply-disruption and precautionary-demand channels through which geopolitical tensions raise commodity prices, particularly for energy-intensive markets; see, for example, \citet{CaldaraIacoviello2022}. The larger energy sensitivity is economically intuitive given the direct exposure of oil markets to geopolitical disruptions.
Taken together, the Stage~2 estimates point to a common macro-financial component that is broad in scope but remarkably consistent across sectors. The signs should be interpreted jointly and conditionally, because the selected indicators form an observable representation of residual common variation after the micro and market components have been removed.
\subsection{Risk Intensity Indices \label{sec:risk_indices}}
The coefficient estimates establish economically distinct sensitivities to micro, market, and macro risk, but they do not by themselves reveal the realised importance of each risk layer. A large sensitivity may be associated with a risk condition that is close to its typical level, whereas a more moderate sensitivity may be consequential when the corresponding risk condition is unusually pronounced. We therefore turn to the Risk Intensity Indices (RIIs), which combine the estimated heterogeneous sensitivities with prevailing risk conditions and place the three risk layers on a common scale. The RIIs should be interpreted as measures of realised risk intensity conditional on the estimated sensitivities and observed risk conditions, rather than as structural variance decompositions or causal attributions of commodity returns. Their shares therefore describe the composition of the RII itself.
Figure~\ref{fig:component_rii} plots the aggregate Micro, Market, and Macro RIIs over time. The three indices display markedly different patterns of time variation. Most strikingly, the Market RII exhibits pronounced short-lived spikes around episodes of severe commodity-market disruption. The first major episode occurs in April 2020, during the COVID-19 shock and the extraordinary dislocation in oil markets. In particular, the sharp increase around 22 April closely follows the collapse of the May WTI futures contract into negative territory on 20 April, amid the severe demand and storage pressures prevailing at the time. The Market RII rises again sharply in late 2021, with the spike around 1 December coinciding with the emergence of the Omicron variant and a period of already elevated stress in global energy markets. The largest movements around 2022 are also consistent with the index capturing periods of broad commodity-market stress. The exceptionally large spike in early March occurs immediately following Russia’s invasion of Ukraine, which generated substantial uncertainty and supply concerns across energy, agricultural commodities, and industrial metals. Its timing also coincides closely with the extraordinary disruption in the nickel market, culminating in the suspension of nickel trading by the London Metal Exchange on 8 March. A further increase around November 2022 occurred amid aggressive global monetary tightening, a strong U.S. dollar, weakening global growth expectations, and uncertainty surrounding Chinese commodity demand, providing a different macro-financial context for elevated commodity-market risk. More recently, the pronounced spike in April 2025 coincides with the sharp escalation in U.S. tariff and trade-policy uncertainty and the associated adjustment in oil and industrial-metal markets. The renewed elevation toward the end of the sample appears broader in nature and is therefore more appropriately interpreted as a period of heightened commodity-market turbulence rather than attributed to a single event.
These episodes are informative about the economic content of the Market RII. Its largest movements coincide with shocks that affected several segments of the commodity complex, including energy, metals, and agricultural markets. The resulting pattern is consistent with the Market RII assigning greater intensity to periods in which market-wide commodity risk conditions become unusually pronounced. The Micro RII also increases during some of these episodes, but its movements are generally less extreme, consistent with a more stable contribution from commodity-specific risk alongside occasional episodes of heightened idiosyncratic stress. The differences across the indices should nevertheless be interpreted as differences in the realised intensity of the estimated risk layers rather than as direct evidence that any particular event caused the movements in a given index.
The Macro RII displays a distinctly different temporal profile. Rather than the sharp spikes characterising the Market RII, it evolves more smoothly over the sample. It is relatively subdued around 2019–20, increases noticeably through 2021, and remains at a higher level over much of the subsequent period. This pattern is consistent with the macro-financial component varying more gradually than the market layer over the sample. Because the Macro RII is constructed from the selected macro-financial representation of the residual common component, however, its interpretation is conditional on the information set used to calculate that component. Micro risk lies between the market and macro patterns, displaying a relatively stable underlying intensity punctuated by occasional increases. Taken together, the three indices indicate that commodity risk varies not only in magnitude but also in the relative intensity of its micro, market, and macro-financial components.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.90\textwidth]{Figures/Component_RII_t.png}
\caption{Total Risk Intensity Index (RII)}
\label{fig:component_rii}
\end{figure}
\noindent
The changing composition of risk becomes clearer when the
three RIIs are expressed as shares of total RII, as in
Figure~2. Around the COVID-19 and oil-market
collapse,\footnote{This is consistent with evidence of
heightened connectedness and risk transmission across
commodity markets during the COVID-19 pandemic
\citep{FaridEtAl2022,QiaoHan2023}.} and again during the commodity-market disruptions surrounding Russia's invasion of Ukraine, the market layer accounts for a very large share of total risk intensity. These episodes therefore involve not only elevated commodity risk but also a pronounced shift in its composition toward market risk. Outside such episodes, the market share declines, while the relative contributions
of the micro and, particularly, macro RIIs become larger. The macro share becomes more prominent after 2021 and remains substantial throughout much of the latter part of the sample, whereas the micro share is comparatively stable. The composition of commodity risk intensity is therefore strongly state dependent. Importantly, these shifts occur although the estimated sensitivities are time-invariant, illustrating why sensitivities alone are insufficient to assess the relative importance of the three risk layers under prevailing market conditions.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.90\textwidth]{Figures/Share_of_Total_RII.png}
\caption{Share of Total RII(\%)}
\label{fig:RII_shares}
\end{figure}
\noindent
Average risk composition also differs substantially across sectors. For the full sample, market risk accounts for 42.1\% of total RII, while macro and micro risk account for 29.7\% and 28.2\%, respectively. Thus, although the market layer is the largest component on average, the remaining RII is almost evenly divided between the macro and micro layers. These percentages should be interpreted as shares of total RII, rather than as percentages of return variance or as structural causal contributions.
The sectoral estimates reveal important differences behind these aggregate shares.
For energy, market risk accounts for 52.4\% of total risk intensity, compared with 26.2\% for micro risk and 21.4\% for macro risk. Energy is the only sector in which a single source accounts for more than half of total intensity. This strong market orientation is consistent with the pronounced market sensitivities documented in the previous subsection and reinforces the distinctive role of energy within commodity markets. Agriculture has a more balanced risk composition. Macro risk accounts for 36.1\% of total intensity, closely followed by market risk at 35.4\%, with micro risk accounting for 28.5\%. Agriculture is the only sector in which macro risk is the largest source on average, although its share is only marginally higher than that of market risk. For metals, market risk is again the largest source at 38.1\%, followed by micro risk at 33.0\% and macro risk at 28.9\%. The relatively narrow range across the three shares contrasts with the strong concentration of energy risk in the market layer. Sectoral differences therefore extend beyond the magnitude of risk intensity to its composition: energy is strongly market-oriented, agriculture assigns nearly equal importance to market and macro risk, while metals show the most even distribution across the three layers.
The level of total risk intensity also differs across sectors. The aggregate Total RII is 4.990 for energy, compared with 3.617 for agriculture and 2.740 for metals. These differences reflect the interaction between the prevailing risk conditions and the heterogeneous sensitivities estimated for individual commodities. In particular, a sector can exhibit a relatively high total RII because its commodities have stronger estimated sensitivities to one or more risk layers, because the associated risk conditions are more pronounced, or because both occur simultaneously. The RII framework therefore provides information that cannot be obtained from the coefficient estimates alone.
\begin{table}[htbp]
\centering
\caption{Risk Intensity Across Asset Groups: Micro, Market, Macro and Total Decomposition}
\label{tab:rii_groups}
\begin{tabular}{lccccccc}
\toprule
& Full Sample & Green & Non-Green & Stable & Non-Stable & DeFi & Non-DeFi \\
\midrule
\multicolumn{8}{c}{\textbf{Micro Risk Intensity Index (RII)}} \\
& 1.229 & 1.107 & 1.302 & 0.074 & 1.433 & 1.097 & 1.317 \\
& 16.8\% & 13.8\% & 18.8\% & 2.9\% & 17.6\% & 16.0\% & 17.2\% \\
\addlinespace
\multicolumn{8}{c}{\textbf{Market Risk Intensity Index (RII)}} \\
& 3.754 & 3.888 & 3.674 & 0.669 & 4.298 & 3.998 & 3.591 \\
& 51.2\% & 48.6\% & 53.1\% & 26.1\% & 52.6\% & 58.5\% & 46.9\% \\
\addlinespace
\multicolumn{8}{c}{\textbf{Macro Risk Intensity Index (RII)}} \\
& 2.343 & 3.002 & 1.947 & 1.819 & 2.435 & 1.743 & 2.742 \\
& 32.0\% & 37.5\% & 28.1\% & 71.0\% & 29.8\% & 25.5\% & 35.8\% \\
\addlinespace
\multicolumn{8}{c}{\textbf{Total Risk Intensity Index (RII)}} \\
& 7.326 & 7.997 & 6.923 & 2.562 & 8.167 & 6.838 & 7.651 \\
& 100.0\% & 100.0\% & 100.0\% & 100.0\% & 100.0\% & 100.0\% & 100.0\% \\
\bottomrule
\end{tabular}
\end{table}
Cross-sectional concentration provides a further dimension of the RII results. Table~\ref{tab:rii_concentration} shows that risk intensity is concentrated among a relatively small share of commodities, particularly in the micro and market layers. For micro risk, the top 10\% of commodities account for 30.3\% of aggregate intensity and the top 20\% for almost half (48.8\%). At the other end of the distribution, the lowest 20\% account for only 4.8\% and the lowest 10\% for 2.2\%. Micro risk intensity is therefore highly concentrated across commodities.
Market risk exhibits a similar degree of concentration. The top 10\% of commodities account for 33.4\% of aggregate Market RII (the largest top-decile share among the three layers) and the top 20\% account for 45.9\%, compared with only 7.6\% for the lowest 20\%. This finding is particularly informative because the underlying market variables are common to commodities within each sector, yet their contribution to risk intensity differs substantially across commodities. Exposure to a common market environment therefore generates highly heterogeneous risk intensity through differences in commodity-specific sensitivities.
Macro risk is less concentrated, although sizeable differences remain across the distribution. The top 10\% and 20\% of commodities account for 26.6\% and 40.6\% of aggregate Macro RII, respectively, compared with 7.6\% for the lowest 20\%. Total risk intensity is less concentrated still: the top 20\% account for 40.1\% of total intensity, whereas the lowest 20\% account for 8.9\%. The concentration of macro risk is particularly noteworthy given the relatively similar Mean Group macro sensitivities across sectors reported in Table~\ref{tab:MG_sensitivities}. Similar sector-average sensitivities do not imply similar risk intensity across individual commodities, because sector averages can conceal substantial heterogeneity in the underlying commodity-specific sensitivities. When these heterogeneous sensitivities interact with common macro-financial conditions, they generate substantial cross-sectional differences in Macro RII. Common macro-financial risk can therefore be broad in scope while contributing differently to the realised risk intensity of individual commodities. This distinction illustrates why the RII measures add information beyond average sensitivity estimates: sensitivities describe exposure, whereas the RIIs quantify the resulting intensity conditional on the prevailing risk conditions.
Taken together, the RII results provide a decomposition of the composition and magnitude of realised commodity risk intensity across risk layers, sectors, individual commodities, and time. Market risk accounts for the largest share of aggregate RII and becomes particularly prominent during major commodity-market disruptions, while the relative contributions of the micro and macro layers are more stable outside such episodes. At the same time, substantial cross-sectional concentration remains even within a common risk environment, reflecting heterogeneous commodity-specific sensitivities. These findings support the interpretation of the three RIIs as complementary measures of distinct risk layers rather than interchangeable components of a single aggregate risk measure. The predictive implications of these measures are examined separately in Section 5.3.
\begin{table}[htbp]
\centering
\caption{Risk Intensity Concentration Across Upper and Lower Percentiles}
\label{tab:rii_concentration}
\begin{tabular}{lcccc}
\toprule
& Top 10\% & Top 20\% & Lowest 20\% & Lowest 10\% \\
\midrule
\textbf{Micro RII} & 29.1\% & 50.1\% & 98.4\% & 99.8\% \\
\textbf{Market RII} & 20.5\% & 37.9\% & 97.9\% & 99.9\% \\
\textbf{Macro RII} & 30.4\% & 49.1\% & 98.1\% & 99.7\% \\
\textbf{Total RII} & 20.6\% & 36.6\% & 97.1\% & 99.8\% \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Predictive Content of the Risk Intensity Indices}
\label{sec:RII_prediction}
We next ask a distinct question from the contemporaneous decomposition in Section~5.2: whether the RIIs contain information about subsequent commodity risk. The purpose of this exercise is not to provide an alternative decomposition of the relative importance of the three risk layers, but to examine whether the risk conditions summarised by the RIIs have forward-looking content. We consider two measures of subsequent risk, volatility and absolute returns, at horizons of one week ($h=1$), four weeks ($h=4$), and twelve weeks ($h=12$). For \(h=1\), the dependent variable is the risk measure in the following week. For \(h=4\) and \(h=12\), we use the average risk measure over non-overlapping subsequent four- and twelve-week intervals, respectively, thereby avoiding mechanically induced serial correlation from overlapping observations. Each specification controls for the corresponding contemporaneous risk measure to account for persistence, includes commodity fixed effects, and uses standardized RIIs, so that the coefficients measure the association between a one-standard-deviation increase in current risk intensity and subsequent risk.
Table~\ref{tab:RII_predictive} reports the in-sample predictive regressions. The main result is that both the Micro and Market RIIs contain information about subsequent commodity risk beyond that contained in its current level. Their coefficients are positive and statistically significant for both volatility and absolute returns at all three horizons. By contrast, the coefficient on the Macro RII is small and statistically insignificant throughout. Thus, risk intensity originating at the commodity-specific and market levels has forward-looking content, whereas the economy-wide component, despite accounting for a substantial share of contemporaneous total risk intensity in Section~5.2, does not provide incremental information about subsequent volatility or absolute returns.
The relative magnitudes of the Micro and Market coefficients should not be over-interpreted. Although their point estimates display somewhat different patterns across horizons, equality of the two coefficients cannot be rejected in any of the six specifications, with $p$-values ranging from 0.121 to 0.845. Table~\ref{tab:RII_predictive} therefore does not establish that one of these risk layers is systematically more important than the other for predicting future risk. Its more robust implication is that both contain incremental forward-looking information after conditioning on current risk.
We assess out-of-sample predictive content using an expanding-window forecasting exercise with the Heterogeneous Autoregressive (HAR) model of \citet{Corsi2009} as the benchmark. The HAR model is well suited to this setting because it parsimoniously captures the strong persistence of volatility through components measured over different horizons. It therefore provides a demanding benchmark for assessing whether the RIIs contain predictive information beyond that already embedded in past volatility. Given the strong short-horizon predictive content of the Market RII in the in-sample volatility regressions, we augment the HAR specification with this index.
\begin{table}[!htbp]
\centering
\caption{In-Sample Predictive Content of the Risk Intensity Indices}
\label{tab:RII_predictive}
\begin{tabular}{lcccccc}
\toprule
& \multicolumn{3}{c}{Volatility}
& \multicolumn{3}{c}{Absolute Return} \\
\cmidrule(lr){2-4}\cmidrule(lr){5-7}
& $h=1$ & $h=4$ & $h=12$
& $h=1$ & $h=4$ & $h=12$ \\
\midrule
Micro RII
& 0.083$^{*}$
& 0.083$^{***}$
& 0.102$^{***}$
& 0.178$^{***}$
& 0.125$^{***}$
& 0.107$^{***}$ \\
& (0.047)
& (0.027)
& (0.030)
& (0.050)
& (0.033)
& (0.028) \\[0.3em]
Market RII
& 0.212$^{***}$
& 0.149$^{***}$
& 0.047$^{**}$
& 0.163$^{***}$
& 0.143$^{***}$
& 0.058$^{***}$ \\
& (0.046)
& (0.031)
& (0.021)
& (0.045)
& (0.027)
& (0.021) \\[0.3em]
Macro RII
& -0.016
& -0.014
& -0.020
& -0.031
& -0.026
& -0.022 \\
& (0.043)
& (0.045)
& (0.048)
& (0.049)
& (0.050)
& (0.051) \\
\midrule
Current risk & Yes & Yes & Yes & Yes & Yes & Yes \\
Commodity fixed effects & Yes & Yes & Yes & Yes & Yes & Yes \\
\midrule
$p$-val.: Market = Micro
& 0.121 & 0.127 & 0.183
& 0.845 & 0.632 & 0.187 \\
Commodities
& 52 & 52 & 52 & 52 & 52 & 52 \\
Observations & 31,720 & 7,904 & 2,600 & 31,720 & 7,904 & 2,600 \\
\bottomrule
\end{tabular}
\vspace{0.5em}
\begin{minipage}{0.98\textwidth}
\footnotesize
\textit{\textbf{Notes}:} The table reports fixed-effects regressions of future commodity risk on the standardized Micro, Market, and Macro Risk Intensity Indices (RIIs). Coefficients and standard errors are multiplied by 100 and therefore express changes in future risk in percentage points. For $h=1$, the dependent variable is risk in the following week. For $h=4$ and $h=12$, it is average risk over non-overlapping subsequent four- and twelve-week intervals, respectively. The use of non-overlapping intervals avoids mechanically inducing serial correlation through overlapping multi-period outcomes. All specifications control for the corresponding contemporaneous risk measure and commodity fixed effects. Standard errors, clustered at the commodity level, are reported in parentheses. The reported $p$-values correspond to tests of equality between the coefficients on the Market and Micro RIIs. Significance levels: $^{*}$ 10\%, $^{**}$ 5\%, $^{***}$ 1\%.
\end{minipage}
\end{table}
Table~\ref{tab:oos_forecasts} reports the out-of-sample forecasting results. The Clark--West test indicates that the Market RII contains incremental predictive information at short horizons. Using the conventional one-sided test, the null of no incremental predictive ability is rejected at the 5\% level for $h=1$ ($p=0.029$) and at the 10\% level for $h=4$ ($p=0.075$), but not at $h=12$ ($p=0.539$). Thus, the out-of-sample evidence supports the in-sample finding that the Market RII contains forward-looking information, particularly at short horizons.\footnote{The positive Clark--West statistics at the shorter horizons are compatible with the RMSE ratios being close to, but slightly above, one and the out-of-sample $R^2$ values being slightly negative. Because HAR is nested within HAR+RII, estimation of the additional RII coefficient introduces estimation noise under the null that its population coefficient is zero, which can raise the raw mean squared forecast error of the larger model. The Clark--West statistic adjusts the forecast-error differential for this effect and therefore tests a different null from an unadjusted comparison of forecast losses.}
\begin{table}[htbp]
\centering
\caption{Out-of-Sample Forecast Performance: HAR versus HAR + Market RII}
\label{tab:oos_forecasts}
\small
\begin{tabular}{l@{\hspace{1.2cm}}c@{\hspace{1.4cm}}c@{\hspace{1.4cm}}c}
\hline\hline
& $h=1$ & $h=4$ & $h=12$ \\
\hline
HAR RMSE & 0.0254 & 0.0157 & 0.0125 \\
HAR + RII RMSE & 0.0254 & 0.0157 & 0.0126 \\
RMSE ratio & 1.0000 & 1.0008 & 1.0017 \\
& & & \\
HAR MAE & 0.0149 & 0.0099 & 0.0084 \\
HAR + RII MAE & 0.0149 & 0.0100 & 0.0084 \\
MAE ratio & 1.0006 & 1.0026 & 0.9993 \\
& & & \\
OOS $R^2$ (\%) & -0.007 & -0.157 & -0.332 \\
& & & \\
Clark--West $t$-statistic & 1.911 & 1.446 & -0.098 \\
Clark--West $p$-value & 0.029 & 0.075 & 0.539 \\
\hline\hline
\multicolumn{4}{p{0.82\textwidth}}{\footnotesize
\textit{Notes:} The benchmark model is the HAR specification. The augmented
model adds the Market Risk Intensity Index (RII). RMSE and MAE ratios
are defined as the loss of the HAR+RII model relative to the HAR model, so
values below one favour HAR+RII. OOS $R^2$ is measured relative to the HAR
benchmark. Clark--West $p$-values are one-sided.} \\
\end{tabular}
\end{table}
The magnitude of the associated gains in point-forecast accuracy is nevertheless small. The RMSE ratios are 1.0000, 1.0008, and 1.0017 at the one-, four-, and twelve-week horizons, respectively, while the MAE ratios are similarly close to one. The out-of-sample $R^2$ values are also close to zero. Hence, although the Clark--West test identifies incremental predictive information in the Market RII at the shorter horizons, incorporating this information produces little change in conventional point-forecast accuracy relative to the HAR benchmark.
\section{Concluding Remarks \label{sec:conclusion}}
This paper studies the sources of commodity risk and their relative importance across commodities and over time. We distinguish three layers: commodity-specific conditions, conditions prevailing elsewhere in the commodity market, and broader macro-financial conditions. The empirical challenge is that sensitivities to these sources are heterogeneous, while the relevant macro-financial component is not directly observed.
We address this challenge using a two-stage ``divide-and-conquer'' framework that accommodates heterogeneous sensitivities to observed micro and market risk and identifies the latent macro-financial component from a high-dimensional set of candidate indicators.
The results show that the three layers play economically different roles. Market sensitivities vary substantially across sectors, with particularly strong market exposure in energy, whereas sensitivities to the selected macro-financial sources are much more similar across agriculture, energy, and metals. These coefficient estimates, however, provide only part of the picture. Once sensitivities are combined with prevailing risk conditions through the Risk Intensity Indices, substantial differences emerge in both the magnitude and composition of commodity risk across sectors, individual commodities, and time.
Market risk is the largest source of commodity risk intensity on average, accounting for roughly 42\% of the total, compared with approximately 30\% for macro risk and 28\% for micro risk. These averages conceal pronounced sectoral differences. Market risk accounts for more than half of total risk intensity in energy, whereas agriculture assigns almost equal importance to market and macro risk, and metals exhibit a more balanced composition across the three layers. Risk composition also changes markedly over time, with the market layer becoming dominant during major commodity-market disruptions.
Risk intensity is also highly concentrated across commodities. The top 20\% of commodities account for almost half of aggregate micro and market risk intensity and about 40\% of macro risk intensity. Importantly, this concentration persists even though average macro sensitivities are relatively similar across sectors. Common macro-financial conditions can therefore affect commodities very differently once heterogeneous commodity-specific sensitivities are taken into account.
The RIIs also contain information about future commodity risk. In-sample regressions show that both the Micro and Market RIIs predict future volatility and absolute returns after controlling for current risk, whereas the Macro RII has no incremental predictive content. Out of sample, the Market RII contains additional forecasting information at short horizons according to the Clark--West test, although any associated gains in point-forecast accuracy appear economically small. The predictive evidence therefore complements the contemporaneous decomposition: the RIIs characterize the composition of current commodity risk and, for the micro and market layers, also contain information about its future evolution.
Several extensions appear particularly useful. The observable representation of macro risk is necessarily conditional on the candidate macro-financial information set, and expanding this information set may uncover additional dimensions of common risk. The market layer could also be generalized beyond sector-based leave-one-out measures to incorporate economic or financial networks linking commodities. Finally, while the RIIs contain short-horizon predictive information, their value for portfolio allocation, hedging, and tail-risk forecasting remains to be explored.
More broadly, the divide-and-conquer approach offers a promising framework for studying layered risk beyond commodity markets. Many economic and financial settings involve risk generated at multiple levels, heterogeneous sensitivities across units, and common components that are latent rather than directly observed. Separating observed local and market sources from latent common risk provides a way to measure their relative importance without imposing homogeneous responses across units.
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