The Dynamics of Commodity Return Comovements

This paper studies comovements in commodity futures markets. We compare factor models with respect to their fit of commodity return comovements. A model based on traded long-short portfolio returns outperforms a macroeconomic model, and explains 96% of the realised comovement. Dissecting the evidence, we find that comovements are driven by the variation of factor covariances as opposed to sensitivities. Intersectoral correlations are more affected than intrasectoral correlations. Volatility comovements only react following the global financial crisis. Our results cast doubt on the persistence of the effects of financialization and emphasize the importance of the dynamics of factor covariances.


Introduction
Since the beginning of the century commodity markets have undergone a massive transformation. Surging demand from emerging countries, deregulation, and an increase of index investment by financial players has affected the correlations of commodity market returns. 1 Average pairwise correlations of commodity market returns increased from 9% before 2004 to 22% after 2004 (Figure 1, Panel A). This increase in correlations is even stronger for return volatilities which rose from 12% to 33% during the same period ( Figure 1, Panel B). This shift has not only occurred to a specific group of commodities but to the whole cross-section ( Figure 2). 2 Our paper aims to shed light on the dynamics of these comovements. Is it the correlation (1) within sectors or (2) between different sectors that drive the time variation in comovements? How much of the comovements can be explained by factor models and thus what proportion was unexpected? Is this increase due to the time variation in (1) the factor sensitivities or (2) the factor covariances?
The main results of our study are threefold. First, we find that a simple factor model can explain 96% of the realized comovements in commodity returns. Second, we document that the high comovements observed during and after financialization are mostly driven by comovements between different commodity sectors rather than within sectors. Third, the time variation in factor covariances as opposed to the time variation in factor sensitivities is the main contributor to the dynamics of comovements during this period.
We begin by comparing two recently proposed empirical models for commodity futures returns: (1) a four-factor model, that includes the market, carry, momentum and basis-momentum factors (Bakshi et al., 2019;Boons and Prado, 2019) and (2) a macro factor model that builds on the information content of 184 macroeconomic variables (Le Pen and Sévi, 2017). We compare the models' ability to fit the covariance structure of commodity returns and find that the model with tradable factors outperforms the macro factor model.
Dissecting the comovement into the part driven by intrasectoral correlations and the part driven by intersectoral correlations, we find that while both parts have increased during and after financialization, it is the comovement between commodities of different sectors that drives the time variation in the total comovement. Next, we decompose the return comovements into a model-implied component and a surprise component. In the data, we find that the model-implied component accounts for virtually all of the realized comovements.
Pushing the analysis further, we show that the model-implied component is the product of factor sensitivities (β) and the covariance matrix of factors (Σ). By fixing one of the two parts to its time-series average, all variation is induced by the other part. Comparing the results shows that while the time-series with fixed factor coefficients is still able to reproduce the main features of the comovement, with fixed factor covariances a meaningful interpretation of commodity return comovements is not possible.
Our work relates to the literature on excess return comovements. Pindyck and Rotemberg (1990) regress commodity returns on 6 selected US macroeconomic variables. Their analysis is extended by Le Pen and Sévi (2017) who consider the information content of 184 macroeconomic variables related to the US, but also international markets. Filtering out the effects of these macro variables on commodity returns, both studies interpret the correlation of the filtered returns as evidence of excess comovement. We find that a simple four-factor model (Bakshi et al., 2019;Boons and Prado, 2019) provides a better fit to the comovement of commodity returns, casting doubt on the amount of excess comovement prevailing in the market once accounted for these factors.
Our work also relates to the broader literature on the modelling of commodity return comovements. Several studies use GARCH-type models to directly model the correlation structure of returns (Deb et al., 1996;Berben and Jansen, 2005;Silvennoinen and Thorp, 2013;Ohashi and Okimoto, 2016). Our approach is different. We use a factor model for commodity returns and explore its implications for comovements. Thus, our methodology enables us to derive the covariance as a product of betas and factor variances and assess their contribution to the model-implied comovements.
Our study also contributes to the growing literature on commonalities of return volatilities. Christoffersen et al. (2019) document an interesting result. They show that while the increase in return comovements of commodities has been temporary, volatility comovements have increased during the crisis and there is no evidence of a decline in the more recent period. One key question that directly follows from this result is: How can volatility comovements persist, while return comovements are temporary? Our results show that volatility comovements are in fact lower during the financialization period, before they jump to a persistently higher regime after the financial crisis of 2008/2009. The increase in return comovements on the other hand is marked by a gradual increase starting before the financial crisis. Thus, it is important to draw a clear distinction between the effects of financialization and the financial crisis on commodity market comovements.
We contribute to the strand of literature on the integration of commodity markets. 3 Tang and Xiong (2012) show that correlations of commodity futures returns have increased substantially and this effect is especially pronounced for commodities that are part of commodity indices. Cheng and Xiong (2014) discuss the impact of financial investors on commodity markets, arguing that they mitigate hedging pressure and improve risk sharing, but also induce shocks due to risk constraints and financial distress. Henderson et al. (2015) use a novel dataset of commodity-linked notes allowing them to identify the relation between price movements and hedging trades. Our findings shed light on how financialization has affected commodity markets. We define financialization as the increase of institutional 3 We concentrate on the financial aspects, but acknowledge that there is also a large macroeconomic literature on this topic, well summarized by Fattouh et al. (2013). Further literature is focussing on spillover effects and contagion, especially during the financial crisis and the boom and bust of commodities (Singleton, 2014).
investors that follow momentum strategies (Bhardwaj et al., 2014), or invest in different generations of commodity indices, incorporating carry and momentum strategies, as well as long-short strategies (Miffre et al., 2012). While it is undisputed that correlations between commodity markets have risen during financialization, the fact that the time variation of factor covariances is the main contributor to the increase in return comovements, helps to explain the long-term effects of financialization. As Bekaert et al. (2009) argue, returns moving together because of fluctuations in the common risk factors, rather than the exposure towards them, casts doubt on a persistent higher degree of market integration since factor covariances exhibit substantial time variation in the short term, but permanent trend changes in comovements are more likely to be induced by changes in betas. Therefore, financialization has affected commodity futures returns in the short-term, but has limited effects on the long-term integration of commodity markets.
Whether changes in factor sensitivities or factor variances drive the increased comovement is of fundamental importance as it calls for different reactions of regulators and practitioners.
From a regulatory perspective, our findings question the view that a stricter regulation of financial trading in commodity markets will reduce comovements. 4 From an asset and risk management perspective, we emphasize the necessity to model the correlation dynamics using tradable factors rather than macroeconomic factors. Our results indicate that failing to model the change in factor variances leads to erroneous risk assessment. For example, using the empirical distribution of the returns with fixed factor covariances results in an estimated 3% decrease of Value-at-Risk (VaR) within the period of financialization, while the realized or model-implied returns show that VaR has increased by 21%.
The remainder of this paper is organised as follows. Section 2 describes the data and methodology. Section 3 introduces and compares the factor models. Section 4 dissects the commodity comovements. Section 5 discusses the implications of our findings, Section 6 documents several robustness checks, and Section 7 concludes. We roll the contract closest to maturity over at the end of the month preceding the month prior to delivery. This approach is analogous to Szymanowska et al. (2014) and enables us to avoid liquidity concerns of futures contracts close to maturity. We roll over all nearbys at the same time, i.e., if t is a roll over date, for any n the (n + 1) th nearby becomes the n th nearby. This has further implications on the computation of the return series, as on the day before a roll over day, we have to account for the fact, that the (n + 1) th nearby will be the n th nearby on the following day. By doing this, we guarantee that the computed return is realizable as based on two prices of the same contract (Singleton, 2014). In formulas, we can write the return on the n th nearby on day t as 5 This choice of sectors is equivalent to Szymanowska et al. (2014), except for the metal sector, which we split into precious and industrial metals. 6 The whole sample comprises data dating back to July 1959. Within this early period the composition of the sample changes drastically, when new commodities are introduced, affecting comovements. Hence, we concentrate on the later period, during which the composition of the sample is constant.
where P (n) t is the price of the n th nearby on day t. Summary statistics for the first nearby returns of all commodities are provided in Table 1 and show common characteristics of commodity markets. Annualized mean returns differ strongly between commodities (7.8% for copper, but −5.2% for corn) and within sectors (13.9% for gasoline, but −9.0% for natural gas), and volatility ranges from 13.9% for live cattle to 44.8% for natural gas.

Methodology
Having constructed the return series of commodity futures, we now aim to decompose the covariance of returns. Suppose that for a commodity i the return on the first nearby, R i , emerges from its linear exposure to K factors. For every commodity market i, we run the time-series regression: where α i is the intercept, F is the T × K matrix of factors, β i is the K × 1 vector of slope coefficients and i denotes the residual. The covariance of two commodity returns R i and R j is then given by where Σ F denotes the K × K covariance matrix of the factors. Equation (3)

Model Selection
In this section, we introduce two factor models. The first model uses tradable factors, and the second model uses macroeconomic variables to explain commodity futures returns.
After describing different ways to estimate both models, we compare their ability to describe commodity market comovements.

Commodity Factor Model
The first empirical model we consider is an extension of Bakshi et al. (2019), who find that the market, basis and momentum factors can describe the time-series and cross-sectional variation in commodity futures returns. We augment this model with the recently proposed basis-momentum factor (Boons and Prado, 2019). Hence, we can write the model in form of Equation (2) as: where R MRKT , R BAS , R MOM and R BASMOM are returns on the market, basis, momentum and basis-momentum portfolio, respectively, are the respective slope coefficients and i denotes the error term.
The market portfolio is an equally-weighted long-only portfolio of all commodity markets. 7 For the basis portfolio, we compute the basis B i for each commodity market i as: where P (1) i ) is the price of the first (second) nearby contract, and M (1) i ) is the time to maturity in days of the first (second) nearby contract. Note that by compounding the basis, we correct for differences in maturities. After sorting the cross-section of commodities by their basis, we construct the basis portfolio as an equally-weighted long-short portfolio, that opens a long position in the 17 commodities with the highest basis and opens a short position in the 17 commodities with the lowest basis.
For the momentum factor, we consider the past performance of each market over the last 12 months and construct an equally-weighted portfolio opening a long position in the 17 commodities with the highest historical returns and opening a short position in the 17 commodities with the lowest historical returns.
The basis-momentum factor is based on the difference between first nearby momentum and second nearby momentum in each market. The portfolio consists of a long position in the 17 commodities with the highest difference in historical returns and a short position in the 17 commodities with the lowest difference in historical returns. All portfolios are rebalanced each month.

Macro Factor Model
The second model builds on the work of Pindyck and Rotemberg (1990 macro factor model. The total variation of the first nine components jointly explains 34.4% of the total variation in all 184 dimensions. 9 We can write the model as a specification of Equation (2) and estimate the following time-series regression for every commodity market i as where α i is the intercept, PC 1 to PC 9 are the first to ninth principal component, respectively, i are the respective slope coefficients, and i is the error term.

Estimation of Factor Sensitivities
In the following, we will introduce three different ways to obtain a series of parameter estimates for the described models.

Constant Beta
Up to this point, we assumed the exposure of commodity returns to the pricing factors is constant, i.e., β it = β i for every t. In practice, this means we use the full sample period to estimate the parameters of interest.

Re-Estimated Beta
However, it is possible that these factor exposures change over time. The simplest way to introduce time variation is by re-estimating the models in Equation (4) and (6). We do this using a rolling window of the past 36 months, and obtain a monthly series of coefficients. 10

Parametric Beta
Another approach is to model the time variation explicitly. We follow Bekaert et al. (2009) and use several financial variables to capture the dynamics of β. More formally, we model β it using K M factors as: where β k it is the k th entry of the coefficient vector β it , γ 0 ik is the intercept and γ ik is the and compute the coefficients, β it , by plugging the estimated coefficients, γ 0 i and γ ik , into Equation (7). Note that effectively the 'Constant Beta' approach is nested within Equation (8), if we set the financial variables M t to zero.

Model Comparison
As we are interested in how comovements of commodity returns change over time, we will look at the decomposition from Equation (3) in a rolling window manner. Let τ be a rolling window of 36 months, then the the correlation of two commodity returns can be written as where R i and R j are the returns on the respective commodities i and j, β i and β j are the respective slope coefficients, Σ F,τ is the covariance of pricing factor within the window, and i and j are the respective error terms. As for the covariance in Equation (3) the correlation decomposes into a realized, model-implied and residual part.
Since correlations are conditional on volatilities, heteroskedasticity can bias the conditional correlation coefficients. Therefore, we follow Le Pen and Sévi (2017) and adjust the correlation coefficient as proposed by Forbes and Rigobon (2002). We denote the heteroskedasticity-adjusted correlation coefficient as where ρ ij is the non-adjusted correlation coefficient, Var short (R i ) is the variance of R i over half the observations compared to Var long (R i ). In our baseline model this refers to 36 months for Var long (R i ) and 18 months for Var short (R i ). Applying this adjustment to the left and righthand side of Equation (9), we obtain the heteroskedasticity-adjusted correlations, ρ real ij (τ ) * , ρ model ij (τ ) * and ρ resid ij (τ ) * for the realized, model-implied and residual part, respectively. Now, we can define the comovement measure (CM) as the weighted average of the off-diagonal correlation coefficients. More precisely, we define where w ij are weights such that i,j,i =j w ij = 1. 11 Following from Equation (3), the realized comovement, CM real , can be decomposed into a model-implied and a residual part.
To assess the performance of the different models, we measure mean absolute error (MAE) and root mean squared error (RMSE) of the model as: where T denotes the number of time-series observations. Bekaert et al. (2009) use these measures for equity comovement as well as Anderson (2017) for credit default swaps.
Before we compare the models with respect to the comovements, we take a look at the results of the time-series regressions for the commodity factor model in Table 2 and the macro factor model in Table 3. The reported coefficients equal the coefficients for the whole sample period, i.e., the 'Constant Beta' approach. As a first indication we look at the R 2 and find that the commodity factor model can explain 25.0% of the return variation over all commodities on average, while the macro factor model can only explain 12.2%. Separating the single commodities into sectors also shows that the commodity factor model performs reasonably well in all sectors apart from live stock. The macro factor model however, can only explain more than 10% of the variation for energy and metal commodities, which is in line with expectations as energy and metal markets are closer related to the economic 11 Note that we sum over all non-diagonal elements instead of only the above-diagonal part because the transformation in Equation (10) is not symmetric. The weights are calculated as the average market value of traded contracts over the rolling window τ . The necessary multipliers to obtain the value of a contract are listed in Table A.1 of the Online Appendix.

conditions.
These preliminary results are clearly in favor of the commodity factor model, but they are only informative about the model's ability to fit each single commodities variation in the time series. For the comovement of commodity markets instead, the model needs to account for the connections between different commodity markets, which cannot be deducted from time-series regressions that are estimated separately for each commodity market. Table 4 reports MAE and RMSE for both models and all three estimation methods, Constant, Parametric and Re-estimated Beta. The conclusions conveyed by both panels of Table 4 are twofold.
First, for all three estimation methods and both performance measures, the commodity factor model outperforms the macro factor model by a large margin. For the 'Re-estimated Beta' approach the MAE is only 0.0290 for the commodity factor model compared to 0.1452 for the macro factor model. In relative terms, the average mistake in estimating the correlation with the macro factor model as compared to the commodity factor model is more than five times larger. Independent of the measure and estimation technique the error is always at least twice as large for the macro factor model. Second, the results are clearly advocating the re-estimation method over the 'Constant Beta' and the 'Parametric Beta' approach. While the introduction of time-varying betas in general reduces the RMSE by only 5% from 0.0949 to 0.0903 for the commodity factor model, re-estimating the parameters reduces the error by 63% from 0.0903 to 0.0344. These results also hold for the MAE and for the macro factor model, although re-estimating does improve the MAE (RMSE) for the macro factor model by only 11% (14%), suggesting less variation in coefficients for the macro factor model.
Concluding this section, the commodity factor model with re-estimated coefficients is the best model to fit the comovement structure of commodity futures returns and will thus be the benchmark for the dissection of commodity comovements in the following section. 12

Dissecting Commodity Comovements
This section introduces two ways of decomposing the proposed commodity comovement measure to obtain insights over the drivers of the time variation in commodity comovements.
Further, we look at differences between return and volatility comovements in commodity markets. Following the previous section, we use the best model and estimation method to fit the commodity return comovements, i.e., the four-factor model using market, basis, momentum and basis-momentum factors with re-estimated betas. While restricted to the monthly frequency for the model comparison, we can now use the daily frequency. 13

Intra vs. Intersectoral Return Comovements
We can decompose the comovement measure into comovements within the same sector and between different sectors. This is interesting as they indicate different ways of market integration. A change in intrasectoral comovements is more likely caused through sector specific channels, e.g., the shale oil and gas boom for energy markets or weather conditions for agricultural markets. Intersectoral comovements, however, indicate a more general form of market integration as they are more likely caused by financial interconnectedness following financialization.
To disentangle these effects, we simply partition the realized comovement into those correlation coefficients of commodities within the same sector and those of different sectors and obtain two disjoint parts, where S i and S j denote the commodity sector of commodity i and j, such that, if i and j 13 Robustness checks show that the commodity factor model performs even better in matching the comovement at the daily frequency than at the monthly frequency. are in the same sector, S i = S j , while if they are in different sectors S i = S j .
Of course, the decomposition in Equation (14) does not take into account the number of weighted correlation pairs in the disjoint parts. In our sample of 34 commodities covering 8 sectors, there are 561 different commodity market correlations pairs. From those 561 pairs, 62 are intrasectoral pairs and 499 are intersectoral pairs. Therefore, the intersectoral part comprises more than eight times as many correlation pairs than the intrasectoral part.
However, intrasectoral correlations are higher on average since commodities of the same sector are closer related to each other. To adjust for this imbalance, we scale both sums accordingly and obtain the intersectoral and intrasectoral comovement measures, where W intra and W inter are scalars such that i,j,i=j, We split the sample into three subperiods before, during, and after financialization. The results of the above decomposition are presented in Table 5. Panel A of Table 5 confirms that comovements have increased during financialization and remain on a higher level afterwards. The realized comovement increased by 130% from 0.172 before to 0.395 after financialization. This increase is matched in both parts, intrasectoral as well as intersectoral comovements, but is relatively much stronger for intersectoral comovements, which increased by 177% from 0.093 to 0.258, while intra sectoral comovements increased by 17% from 0.663 to 0.775. All changes are significant at the 1% level.
Panel B and C of Table 5 report the standard deviation of the comovement measures and their correlation with the realized comovement for the different periods. The volatility of comovement increases throughout the sample period. However, the volatility of the intrasectoral comovement has decreased after financialization, exhibiting only 34% of the realized comovements volatility in the most recent period. The intersectoral comovement instead, shows the same pattern as the realized comovement with a steady increase of volatility. Together with the high correlation between intersectoral and realized comovements of 0.929 over the whole sample, these results show that the time variation in comovements is driven by intersectoral comovements rather than intrasectoral comovements during and after financialization. Figure 3 shows the time series of comovements and the decomposition into the intraand intersectoral part, confirming the discussed observations. Both, intra-and intersectoral comovements have contributed to the long-term increase in comovements during financialization, but the short-term time variation leading to a peak of comovement shortly after the crisis and a decrease in the most recent period is driven by the intersectoral comovements.

Factor Sensitivities vs. Factor Covariances
Apart from splitting the comovement into subgroups, there is another way of dissecting the time variation of commodity comovements. Within the commodity factor model, the time variation of comovements has two potential sources, the fluctuations in factor sensitivities (β) and the fluctuation in factor covariances (Σ F ). We follow Bekaert et al. (2009) and set one of the two channels to its time-series average, so all time variation is induced by the other part. We denote the respective correlations as where Σ F is the time-series average of factor covariances and β i is the time-series average of factor sensitivities for asset i. We denote the heteroskedasticity-adjusted version as ρ fixedβ ij (τ ) * and ρ fixedΣ ij (τ ) * , respectively, and obtain the comovement measure with fixed factor exposures, CM fixedβ , and the comovement measure with fixed factor covariances, CM fixedΣ , as  With fixed time variation in factor sensitivities (β), the comovement measure CM fixedβ is still closely related to the realized and model-implied comovements. It also shows an increasing trend for the average comovement as well as for the volatility of comovement.
The correlation with the realized comovement exceeds 0.95 for the financialization and postfinancialization period indicating that we capture the time variation in comovement for this period.
Fixing the time variation in factor covariances (Σ) however, produces a very different picture. Panel A of Table 6 shows that with fixed factor covariances the level of comovement is decreasing throughout the sample, contrary to what we observe during financialization. The volatility of the comovement measure CM fixedΣ is only slightly increasing and the correlation with the realized comovement over the whole sample is −0.716, with 0.197 before, −0.828 during, and −0.527 after financialization.
Finally, the graphs in Figure 4 confirm these results. The time series of comovements is mainly driven by changes in the factor covariance structure rather than the sensitivities of the commodities towards these factors. While factor covariances exhibit substantial time variation in the short term, permanent trend changes in comovements are more likely to come from changes in betas (Bekaert et al., 2009). Therefore, the evidence suggests that financialization has affected the factor covariance structure rather than the intensity to which commodities are exposed to these factors.

Return vs. Volatility Comovements
The previous results are especially interesting with respect to the study of Christoffersen et al.
(2019), who argue that commodity return correlations have returned to their pre-crisis level, but find a persistent higher degree of volatility correlations after financialization, suggesting another way of market integration. To address this point, we also look at comovements between commodity return volatilities, by applying the same framework as before to return volatilities as the underlying variables. Let V i be the series of monthly return volatilities for a commodity market i and the correlation of the monthly volatilities of the return volatilities of commodity market i and j over a rolling window τ of 36 months.
Then, we can define the comovement measure of volatilities (CMV) analogously to returns as the weighted average of all correlation pairs: where w ij are weights such that i,j,i =j w ij = 1.
In Table 7, we present mean and standard deviation of the comovement measure for returns (CM) and volatilities (CMV). In Panel A, we report the statistics as before using the returns or volatilities of the 34 commodities as underlying variables. In Panel B, we use the sector returns or volatilities as the underlying variables, eliminating the intrasectoral part of the comovement as those are aggregated within the sector return.
As for return comovements we find an increase in volatility comovements over the sample The graphs in Figure 5 emphasize the differences between the evolution of return and volatility comovements, with both having increased significantly after financialization. While the shift in commodity return comovements has been gradual over the financialization period, commodity volatility comovements have been effected during the financial crisis jumping to higher regime.
These results advocate a clear differentiation between the gradual effects of financialization on the one hand, and the eruptive effect of the financial crisis on the other hand.

Market Integration and Financialization
Our results add to the discussion on market integration and financialization of commodity markets. We show that a four factor model extension of Bakshi et al. (2019) is able to explain most of the comovement within commodity markets leaving a negligible part of excess comovement. This is in contrast to the literature on excess comovement which usually finds larger parts of unexplained comovement.
This result demonstrates that it is crucial how to define excess comovement. Pindyck and Rotemberg (1990) say excess comovement is anything that cannot be explained by the common effects of inflation, changes in aggregate demand, interest rates or exchange rates.
Even after extending this set of variables to 184 macro variables (Le Pen and Sévi, 2017), these models leave a significant amount of excess comovement which they try to explain with non-economic variables. Instead, we are able to internalise this excess comovement using a model with tradable factors, enabling us to analyse the whole comovement subsequently.
Since we are able to capture the entire comovement with this parsimonious model, we can analyse which part of the model is contributing the most to the time variation and therefore the increase in comovements during the financialization period. This gives insight into whether the change is persistent or not. We find that it is mostly the covariance of the factors that introduces the time variation into comovements. As the factor covariance (Σ) is more affected by short-term changes than the factor sensitivities (β), which relate factor returns to the commodity returns, this is evidence that the effects of financialization are less strong in the long term.
However, that financialization has affected commodity markets as a whole is evident from the dissection of intra-and intersectoral comovements. We document a significant increase in comovements between different commodity sectors as well as within sectors during financialization. This result supports the argument that index investment in commodity markets has increased seemingly unrelated commodity markets through financial channels, rather than only intensifying existing linkages between commodities of the same sector.
Finally, we have shown that return comovements and volatility comovements are effected differently by financialization. While both have increased significantly post financialization, the channels are arguably different. The gradual increase of commodity return comovements during financialization is not matched in volatility comovements. Instead volatilities even comove less during the financialization period, before they jump significantly during the financial crisis. This observation is interesting as it motivates a discussion of the distinct effects of financialization and the financial crisis on commodity markets.

Risk Management
For practitioners in risk management the covariance and hence comovements are crucial as they determine the riskiness of a portfolio. We therefore look into the effect of the comovement on the Value-at-Risk (VaR), a common risk measure, which we compute for a portfolio P as where α is the confidence level, the notional amount is $1,000,000, Φ −1 is the inverse normal distribution function and σ P is the portfolio's volatility. The covariance of commodity returns enters the VaR through the volatility of the portfolio, since where w i are weights such that N i=1 w i = 1. Hence, changes in the covariance structure will affect the risk measure. Anderson (2017) uses the VaR to illustrate changes in the covariance of credit default swaps. In a similar fashion, we will analyse the change of VaR using different versions for the commodity covariance as before. For a rolling window τ of 36 months, we as the realized covariance, the model-implied covariance, the estimated covariance with fixed betas and the estimated covariance with fixed factor covariances, where β iτ and β jτ are the coefficients from the time-series regression in Equation (4) and Σ F,τ is the covariance of the factors. Again, β i , β j and Σ F denote the time-series averages.
In Table 8 we compare the VaR before, during and after financialization to see whether the model-implied covariance estimates are able to capture the changes in risk. The first column shows the VaR using the covariance of the realized returns, Cov real , for the computation. In the second column, we use the model-implied covariance Cov model . We find that the modelimplied covariance gives a good estimate of the realized VaR, capturing the increasing risk during and after financialization.
In the third (fourth) column of Table 8, we compute VaR without time variation in the factor sensitivities (factor covariances). When not allowing for time variation in the coefficients, VaR estimates are reasonably close to the realized and model-implied estimates.
They are especially able to capture the increased risk post financialization. The VaR based on covariances without time variation in factor covariances instead, cannot capture this change of risk. Instead, 5% VaR is rather stable through the periods from $12,119 per $1,000,000 before financialization to $12,492 per $1,000,000 during financialization and $12,491 per $1,000,000 after financialization. We find a similar pattern for the 1% VaR in Panel B of Table 8.
This result points out the importance of the variation in factor covariances for the comovements. It is crucial for investors to adjust for the changes in factor covariances to be able to assess the risks of a commodity portfolio correctly. However, variations in the sensitivities of a portfolio to certain factors do only affect the VaR negligibly.

Robustness Checks
To eliminate possible concerns about the choice of model, we run several robustness checks and discuss the results in this section organized by potential reasons. The relevant results are presented in the Online Appendix.

Model Settings
To start with, we want to convince the reader that the choice of model is not dependent on some of the parameters chosen within the regression setup. Therefore, we repeat the analysis of Section 3 altering the size of the rolling window to 24 months and 60 months (Table A.3), omitting the adjustment for heteroskedasticity or changing the computation of Var short in Equation (10) to be the 12-month or 6-month variance (Table A.4).
As the referred tables show, the specification of these parameters does not change the general result that the commodity factor model with re-estimated betas is the best model to explain the comovement of commodity futures returns.

Sample Choice
There are several reason, why we decided to concentrate our analysis on the set of 34 com-

Number of Factors
As we have shown that the extension of the model by Bakshi et al. (2019) adding a basismomentum factor is able to explain the comovements between commodity returns, it is natural to ask which of the four factors, market, basis, momentum and basis-momentum contributes the most. To shed light on this, we run the same analysis using different subsets of factors and compare their performance in the same way as before.
It is evident from Panel A of Table A.6 that the market factor alone is able to explain much more of the comovement than the other three factors. Since we use the whole cross section of commodities to compute all factors, this effect is unlikely to originate from the choice of commodities within the portfolios. To rule out the possibility that the performance of the model is driven by the fact that the factors include the commodity returns themselves, we repeat the analysis using unique factor portfolios for each commodity market that exclude the market itself, e.g., the market portfolio for corn includes all commodity markets but the corn market itself. Panel B of

Alternative Comovement Measures
One concern is that our model comparison might be biased by the way we measure comovements. Part of the literature has addressed commodity comovements methodologically using Vector Autoregressive models for the return volatility as in Diebold et al. (2017) or assessing the excess comovement, i.e., the comovement of the error term in Equation (3), with a GARCH framework as Ohashi and Okimoto (2016). 15 Nevertheless, we favor a simple factor model as we are interested in the explainable part of the commodity future returns. We look at the pairwise correlations as they appear to be at the heart of the financialization debate (Bhardwaj et al., 2015).
However, the high explanatory power of the market factor for the comovement measure casts doubts on whether this is induced by how we measure comovement. Let us illustrate this with a simple example. Assume a simple one-factor model only including an equallyweighted market factor, i.e., with α i , β i and i as intercept, slope and residual, respectively. Recall that since the market factor here is an equally-weighted average of all constituents, the average exposure to this factor (β) must equal one and hence the average of all possible products of β i and β j as well, i.e., If we additionally assume the comovement measure to be equally-weighted, i.e. w ij = 1 N 2 for a number of N commodity markets, and we average over all covariances including the 15 Although Forbes and Rigobon (2002) state that using correlation coefficients is the most straight forward framework.

variances on the diagonal, then
Because in this case, the market variance, Var(R MRKT ), is equal to the comovement measure, we obtain zero average residual covariance, i.e., There are three reasons, we think our results are not driven by this tautology. First, the comovement measure, we use, differs from the simplified example above as we do not consider the diagonal elements, i.e., the variances, and we do not equally-weight the covariances, more specifically we do not use the same weights for market factor and comovement measure.
Second, since we apply the adjustment for heteroskedasticity by Forbes and Rigobon (2002) the equations for covariances do not hold for correlations. Third, we conduct the following robustness check to address this issue.
To show that our results are not driven by the described averaging effect, we define the partial comovement measure where ρ real ij (τ ) * is the heteroskedasticity-adjusted correlation coefficient between commodity return i and j during the period τ and j =i w j = 1. Analogously to Equation (11), we define the model-implied partial comovement measure CM model i . We find an average MAE (RMSE) of 0.0474 (0.0532) over the 34 partial commodity comovements, while it is 0.0229 (0.0308) for the cumulated measure including all commodity pairs (see Table A.6).

Conclusion
This paper examines the comovements of commodity futures returns and variances. We start from first principles and show that a four factor model as extension of Bakshi et al. (2019) is able to explain a large proportion of the realized comovements. Importantly, this result suggests that there is very little evidence of excess comovements.
We confirm previous evidence of increased comovement during and after financialization and pin its source down to the intersectoral comovements. Dissecting the evidence further, we show that changes in coefficients play a minor role in our understanding of comovements.
This result poses a challenge to the literature on the integration of commodity markets.
The increase of return comovements during financialization is mainly driven by a temporary increase in factor covariances casting doubt on commodity markets becoming more integrated in the long run. Lastly, we find increased comovement of volatilities following the financial crisis, advocating a discussion of the distinct effects of financialization and the financial crisis on commodity comovements.
Economic Journal, 100 (403)   The average correlation rose from 9% to 22% for returns and from 12% to 33% for volatilities between the two periods. The horizontal axis lists the commodity ticker symbols, for details see Table A.1 of the Online Appendix.

Figure 3: Intersectoral and Intrasectoral Commodity Return Comovements
This figure shows the realized commodity return comovement measure, CM real , the intersectoral comovement measure, CM inter and the intrasectoral comovement measure, CM intra , as defined in Equation (15). Comovements are measured as the weighted average of all correlation pairs, and then dissected into those pairs from different and similar markets. The correlations are adjusted for heteroskedasticity (Forbes and Rigobon, 2002) and computed over a rolling window of 36 months. The light grey shaded area marks the period of financialization from December 2000 to July 2009, including dark shaded NBER recession.

Figure 4: Comovement Measure with Fixed Betas or Factor Covariances
This figure shows the realized commodity return comovement measure, CM real , the comovement measure with fixed factor exposures , CM fixedβ , and the comovement measure with fixed factor covariances, CM fixedΣ , as defined in Equation (17). The correlations are adjusted for heteroskedasticity (Forbes and Rigobon, 2002) and computed over a rolling window of 36 months. The light grey shaded area marks the period of financialization from December 2000 to July 2009, including dark shaded NBER recession.

Figure 5: Commodity Comovement Measure for Returns and Volatilities
This figure shows the realized commodity return comovement measure, CM real , and the realized commodity volatility comovement measure, CMV real , as defined in Equation (19). The correlations are adjusted for heteroskedasticity (Forbes and Rigobon, 2002) and computed over a rolling window of 36 months. The light grey shaded area marks the period of financialization from December 2000 to July 2009, including dark shaded NBER recession.
where R i is the monthly return on commodity i, α i is the intercept, R M RKT , R BAS , R M OM and R BASM OM are the returns on market, basis, momentum and basis-momentum portfolios, respectively, and i is the error term. Standard errors are corrected according to Newey and West (1987) with 2 lags and respective t-stats are reported in parenthesis.   (6),

Commodity
where R i is the monthly return on commodity i, α i is the intercept, P C 1 , . . . , P C 9 are the first nine principal components of a set of 184 macro variables following Le Pen and Sévi (2017), and i is the error term. Standard errors are corrected according to Newey and West (1987) with 2 lags and respective t-stats are reported in parenthesis. Commodity sectors are separated by horizontal lines. Returns are in percentage points. Results for the fifth to ninth PC are omitted.    (4) and the macro factor model in Equation (6). The comovement is defined as in Equation (11), where τ is a 36 months rolling window, ρ real ij (τ ) * and ρ model ij (τ ) * are the heteroskedasticityadjusted correlation coefficients of the realized and model-implied commodity returns, and w ij are weights such that i,j,i =j w ij = 1. The MAE and RMSE are computed as in Equation (12) and Equation (13): (12 & 13) In Rows 'Constant Beta', the respective models are estimated once for the whole sample period. In Rows 'Parametric Beta', the coefficients are parametrized using the 3-month US LIBOR rate, the term spread between 10-year and 3-months US Treasury bills, the default spread between Moody's BAA and AAA Corporate Bonds Indices, the TED-spread between 3month LIBOR and the Treasury rate, and the CBOE Volatility Index. In Rows 'Re-estimated Beta', the coefficients are re-estimated for each rolling window.      (20) VaR α (P ) : where α is the confidence level, the notional is $1,000,000, Φ −1 is the inverse normal distribution function and σ P is the standard deviation of portfolio returns R P . The portfolio's volatility is based on the different covariance matrices in Equation (22)   This table reports the root mean squared error (RMSE) for the comovement measure of the model-implied commodity returns with respect to the commodity factor model in Equation (4) and the macro factor model in Equation (6). The comovement is defined as in Equation (11), where τ is a 24 (60) months rolling window, ρ real ij (τ ) * and ρ model ij (τ ) * are the heteroskedasticity-adjusted correlation coefficients of the realized and model-implied commodity returns, and w ij are weights such that i,j,i =j w ij = 1. In Rows 'Constant Beta', the respective models are estimated once for the whole sample period. In Rows 'Parametric Beta', the coefficients are parametrized using the 3-month US LIBOR rate, the termspread between 10-year and 3-months US Treasury bills, the default spread between Moody's BAA and AAA Corporate Bonds Indices, the TED-spread between 3-month LIBOR and the Treasury rate, and the CBOE Volatility Index. In Rows 'Re-estimated Beta', the coefficients are re-estimated for each rolling window.  (4) and the macro factor model in Equation (6). The comovement is defined as in Equation (11), where τ is a 36 months rolling window, ρ real ij (τ ) * and ρ model ij (τ ) * are the heteroskedasticityadjusted correlation coefficients of the realized and model-implied commodity returns, and w ij are weights such that i,j,i =j w ij = 1. The heteroskedasticity adjustment follows Forbes and Rigobon (2002) as described in Equation (10) where ρ ij is the non-adjusted correlation coefficient, Var short (R i ) is the variance of R i over a shorter horizon compared to Var long (R i ), which is computed over 36 months. In Panel A, there is no heteroskedasticity adjustment applied, i.e., Var short (R i ) is equal to Var long (R i ) or δ i = 0. In Panel B, Var short (R i ) uses a third of the observations of Var long (R i ), i.e., 12 months, and in Panel C Var short (R i ) is computed over 6 months. This table reports the root mean squared error (RMSE) for the comovement measure of the model-implied commodity returns with respect to the commodity factor model in Equation (4) and the macro factor model in Equation (6). The comovement is defined as in Equation (11), where τ is a 36 months rolling window, ρ real ij (τ ) and ρ model ij (τ ) are the correlation coefficients of the realized and model-implied commodity returns, and w ij are weights such that i,j,i =j w ij = 1. In Rows 'Constant Beta', the respective models are estimated once for the whole sample period. In Rows 'Parametric Beta', the coefficients are parametrized using the 3-month US LIBOR rate, the termspread between 10-year and 3-months US Treasury bills, the default spread between Moody's BAA and AAA Corporate Bonds Indices, the TED-spread between 3month LIBOR and the Treasury rate, and the CBOE Volatility Index. In Rows 'Re-estimated Beta', the coefficients are re-estimated for each rolling window. Panel A uses the same set of 21 commodities as Szymanowska et al. (2014). Panel B uses the set of 34 commodities excluding the commodities of the sector listed.  (4), where R i is the return on commodity i, α i is the intercept, R M RKT , R BAS , R M OM and R BASM OM are the returns on market, basis, momentum and basis-momentum portfolios and i is the error term. The first column lists the factors included in the model, where the market factor (MRKT) is an equally-weighted average over all commodity returns. The basis factor (BAS) is the return on a long-short portfolio that buys the 17 commodities with the highest basis and sells the 17 commodities with the lowest basis. The momentum factor (MOM) is the return on a long-short portfolio that buys the 17 commodities with the best 12-months performance and sells the 17 commodities with worst 12-month performance. The basis-momentum factor (BASMOM) is the return on a long-short portfolio that buys the 17 commodities with the best 12-months basis performance and sells the 17 commodities with the worst basis performance. The last row augments the model with the first nine PCs of the set of 184 macro variables as in Le Pen and Sévi (2017). MAE and RMSE are computed as in Equation (12) and (13), with the heteroskedasticity-adjusted correlation coefficients re-estimated every month over a rolling window of 36 months. Panel A uses the whole cross section to compute factor returns, Panel B computes factors separately for every commodity markets based on all other 33 markets. This table reports mean absolute error (MAE) and root mean squared error (RMSE) for the model-implied partial commodity return comovement measure of the commodity factor model (4). The comovement is defined as in Equation (27) CM real where τ is a 36 months rolling window, ρ real ij (τ ) * and ρ model ij (τ ) * are the heteroskedasticityadjusted correlation coefficients of the realized and model-implied commodity returns, which are re-estimated over a rolling window of 36 months, and w j are weights such that W i = j =i w j . RMSE and MSE for a certain asset i are computed as follows: The average over all assets is reported in bold at the end.