# Multi-Frequency Information Flows between Global Commodities and Uncertainties: Evidence from COVID-19 Pandemic

Authors: Emmanuel Asafo-Adjei; Siaw Frimpong; Peterson Owusu Junior; Anokye Mohammed Adam; Ebenezer Boateng; Robert Ofori Abosompim
Journal: Complexity
 (2022)

---

## Abstract

Owing to the adverse impact of the COVID-19 pandemic on world economies, it is expected that information flows between commodities and uncertainties have been transformed. Accordingly, the resulting twisted risk among commodities and related uncertainties is presumed to rise during stressed market conditions. Therefore, investors feel pressured to find safe haven investments during the pandemic. For this reason, we model a mixture of asymmetric and non-linear bi-directional causality between global commodities and uncertainties at different frequencies through the information flow theory. Consequently, we utilise the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and the Rényi effective transfer entropy techniques to establish the dynamic flow of information. The intrinsic mode functions (IMFs) from the CEEMDAN are carefully extracted into multi-frequencies through cluster analysis to reconstruct the series into high, medium, and low frequencies in addition to the residue. We utilise daily data from December 31st, 2019, to March 31st, 2021, to provide insights into the COVID-19 pandemic. The correlation coefficients and variances demonstrate that the high frequency (IMFs 1–4) which measures the short-term dynamics is the dominant frequency, suggesting short-lived market fluctuations relative to real economic growth for institutional investors. Moreover, outcomes from the multi-frequency entropy indicate a negative bi-directional causality of information flow between global commodities and uncertainties, especially in the long term. Generally, the findings present pertinent inferences for portfolio diversification, policy decisions, and risk management schemes for global commodities and markets volatilities. We, therefore, advocate that market volatilities act as effective hedges for global commodities, and they clearly act as balancing assets rather than substitutes in the long-term dynamics of the COVID-19 pandemic. Investors who delayed in investing within financial markets of commodities and market volatilities are likely to minimise their portfolio risks.


---

## Body

## 1. Introduction

Commodities have developed as one of the most important categories of assets in the global economy [1]. From the beginning of the late 10th century, commodity investing has attracted the attention of researchers, policy makers, and investors. Commodities are mainly categorised into metals, energy, and agricultural products [2]. Despite the differences in their categorisations, they are likely to offer diversification, safe haven, or hedge benefits [3–5]. Specifically, among the family of metals, precious metals have induced the attention of academicians, specifically, gold [6]. Gold acts as a dominant safe haven asset [6, 7], but very little is known about the safe haven properties of other metals in times of severe economic and financial havoc [6]. The same can be said of agricultural and energy commodities. Notwithstanding, Ahmed and Sarkodie [8] indicated that commodities depict high probability to remain in low volatility regime than in high volatility regime. They demonstrated that precious metals and agricultural commodities display less inverse response to uncertainties than energy. It can, therefore, be inferred that commodities display high likelihoods of asymmetric dynamics among themselves [3, 9, 10].Commodities are not only considered for their fundamental physical usage but also considered as a potential hedging instrument during uncertainties. Nonetheless, commodities are likely to experience similar chaotic risk as other assets, thereby plummeting their prices. With reference to the 2007-2008 global financial crisis, there have been major uncertainties in the financial and economic setting, including financial contagion risk [11, 12], economic downturns [13, 14], political turmoil [15, 16], Brexit [17, 18], and cryptocurrency crashes which in turn spread contagion and weaken financial stability [19]. These disturbances have an adverse impact on commodities, thereby plunging their hedging or safe haven potentials [1, 6, 8, 11–18].In addition to these disturbances, the emergence of the COVID-19 pandemic has induced many empirical studies on the globe, including its impact on health [20–23], food prices [24], the nexus between equities and cryptocurrencies [25], commodities [8, 26], and major world markets [27, 28]. It can be analysed from these studies that the impact of the COVID-19 pandemic has disrupted most economic activities including financial markets of which commodities are no exception.The contagious impact of COVID-19 on markets does not substantially deviate from the efficient market hypothesis (EMH) [29, 30]. Thus, relevant information about the COVID-19 pandemic may largely reflect in ascertaining current prices of financial assets. The rippling effect would be a fall in market prices over time. On the other hand, the heterogeneous and adaptiveness of market participants across time, market overreaction, mean reversion, excessive volatility, and all other market anomalies provide support to investigating the commodities and uncertainties’ information flow dynamics during the COVID-19 pandemic. Hence, we are curious to ascertain the flow of information between important commodities and uncertainties during the COVID-19 pandemic. We do this to assess the extent of immunity of the commodities to uncertainties in times of economic shocks.The exegesis of the COVID-19 pandemic on commodities has substantially been provided by prior studies with varying outcomes [8, 26–28]. This is partly due to the behavioural intentions of investors which is not the same across time, with fluctuating economic conditions which contradict the EMH. As a result, the asymmetric and time-based varying behaviours of investors are carefully supported by the heterogeneous market hypothesis (HMH) [31] and the adaptive market hypothesis (AMH) [32]. The HMH provides that the numerous market participants take their investment decisions on dissimilar time horizons in line with their risk and return preferences by making reference to their past and current news. Again, the AMH states that markets evolve–due to events and structural changes, adapt–and market efficiency varies in degree at different times. In this study, we employ intrinsic time which concurs with time scales of short, medium, and long term to account for time horizons.Also, we consider the hypothetical advancement of Owusu Junior et al. [28, p. 2] on the competitive market hypothesis (CMH) which argues that “in part, the intensity of information flows and spillover between markets of the same and differing asset classes are exacerbated by rational, albeit irrational investors’ relentless search for competing rewards and risks to satisfy the portfolio goals.” As a result, the intensity of information flows between markets (similar and dissimilar) may lead to high uncertainties to which individual financial markets are susceptible. This is in line with the financial instability of Minsky [33–35] and the asymmetric volatility dynamics.The theoretical exposition of Minsky [33–35] on financial instability provides a linkage between financial market fragility and endogenous speculative investment bubbles. This hypothesis claims that economic tranquillity and stability are not self-sustaining. Accordingly, stability could lead to more optimism which eventually leads to more borrowing in financial markets. Over time, there is a transformation from a stable financial system to a fragile system. Thus, consequently, stable and booming markets drive blindness for increasing risks [36]. The impact may however not be instantaneous but differed even to periods of existing markets stress such as the COVID-19 pandemic which may aggravate the overall market risks. We take cognisance from the dynamics of the pre-COVID-19 pandemic shock transmissions from stable and booming markets which may drive blindness for increasing future risks and the impact of the current COVID-19 pandemic shocks. We institute that asset price response to information may heed from past (due to time delay mechanism and accumulation effect) as well as current information regarding market inefficiency.In addition, arguments on asymmetric volatility [37] in the discussion of uncertainties and commodities markets cannot go unnoticed. This is because fluctuations in market volatilities influence investors’ portfolio choices either by altering the trade-off between risk and return or their predictions of future market performance. According to Chen [38], investors’ desire to hedge against market volatility because rising volatility does not incentivize investment opportunities. Consequently, phases of high volatility tend to correspond to drawdowns in markets which may minimise investors’ confidence [39].Furthermore, Benthall [40] advocates that information flows are causal flows located in the context of other causal linkages. The information flow theory builds on the philosophy of Dretske [41] and the statistics of Pearl [42]. The mathematics of probability and statistics has made it possible to quantify the information flow between variables. Intuitively, reciprocal information exists when two random variables are linked in such a way that one variable can learn about the state of the other from observation of the other [40].We draw insights from the aforesaid theoretical and hypothetical advancements to provide that the competitiveness of markets intensifies the flow of information among them and the stable and booming markets drive blindness for increasing risks and eventually result in an asymmetric volatility information flow to markets. The dynamics of price response to shocks may not be instantaneous due to market friction [43]. In frictionless capital markets with complete information and rational investors, asset prices adjust to new information immediately and completely [44]. However, information imperfections potentially hinder timely price discovery and are associated with delayed stock price adjustment to information [44, 45]. According to Hou and Moskowitz [43], the most delayed firms grasp a large return premium not expounded by size, liquidity, or microstructure effects. They indicate further that delay captures part of the size effect, idiosyncratic risk is priced only among the most delayed firms, and earning drift is monotonically increasing in delay.As a result, the relentlessness of market frictions affecting an asset contingent on the delay with which its price responds to information during stressed conditions as well as shocks from past markets dynamics may offer diversification potentials in the long term. The time delay mechanism has been illustrated within commodity markets by prior studies [46, 47]. The delayed effect of prices response to shocks through uncertainties on individual markets as a result of intensity of information flows between markets which amplifies market risks, and the impact of other external shocks place the empirical analysis in perspective. We expect diversification potentials to increase monotonically from the short to long terms (delay in market dynamics) in times of stressed conditions. These complex analogies are reflected in what we call “the delayed volatility of market competitiveness and external shocks (DVMCES)” phenomenon. The market competitiveness can be analysed from either same or different class of assets.We therefore employ five uncertainty indices which include Economic Policy Uncertainty (EPU), Global Volatility Index (Gvolatility), Cryptocurrency Volatility Index (VCRIX), Chicago Board Options Exchange (CBOE), Volatility Index (VIX), and CBOE Crude Oil Volatility Index (OVX). Gvolatility represents volatility in all financial markets, whereas the VIX is a proxy for investor fear and expectations in the equity markets. Also, the VCRIX is utilised in this study due to the cryptocurrency crashes which in turn spread contagion and weaken financial stability [19]. In addition, we consider the OVX which is one of the most important volatility indices in the commodity markets [48–50]. These four uncertainty indices are employed to investigate whether the competitiveness of markets would either transmit volatility from alternative markets to other markets or shocks from the other markets transmit significant information to the volatilities of alternative markets.Due to the world uncertainty and cotemporary conflicts, there is an increasing body of uncertainty-generating policies that influence economic policy and financial decisions [51]. The EPU index includes a range of concerns, for instance, conflicts in regulations, conflicts over inequality of income distribution, and fluctuations in global prices, to mention a few, that occur around the globe. The creation of the EPU index has therefore induced several empirical studies [6, 52, 53]. From the points so far discussed, we rigorously examine the DVMCES phenomenon in the context of global commodities and uncertainties. This can, however, be extended to other markets.Empirical studies on the flow of information between uncertainties and commodities during the COVID-19 pandemic at diverse intrinsic time are underdeveloped. This investigation is necessary because different commodities are distinguished by different time responses to shocks in the market [54]. An avalanche of studies have examined the spillover effects or nexus between commodities and uncertainties [8, 55, 56]. Studies have been conducted on the volatility transmission in precious metal markets [57, 58], energy markets [59, 60], agricultural commodities [61], regime switching effect of the COVID-19 pandemic, and EPU on three commodity categories—metals, energy, and agriculture [8]. However, these studies did not consider the flow of information at diverse intrinsic times during the COVID-19 pandemic. Also, little is known about the flow of information between global commodities and the five important uncertainties—EPU, Gvolatility, VCRIX, VIX, and OVX, during the COVID-19 pandemic. On the other hand, the few empirical studies that come close to investigating the information flow between commodities and uncertainties [59] did not employ the Rényi transfer entropy at various intrinsic times which corresponds to the stylised facts of financial returns. In addition, most studies have not considered similarities between commodities or uncertainty indices (except GEPU) which is necessary for revealing the level of efficiency or heterogeneity within the markets at specific or diverse intrinsic times.Accordingly, empirical studies on the commodities and uncertainties nexus have warranted methodologies such as linear and non-parametric causality tests [62], bootstrap causality test and the time-varying approach [57], feasible quasi-generalized least squares estimator [55], causality in variance test and impulse response functions [61], entropy-based wavelet analysis [59], time-varying parameter vector autoregression model [60], and GAS and GARCH modelling of precious metals in addition to the Hansen et al. model confidence set in ranking superior set model [58]. None of these methods considered multi-frequency-dependent entropy approach which quantifies information from a probability density function in the short, medium, and long terms.Our study departs from extant literature by first employing the complete ensemble empirical mode decomposition with adaptive Noise (CEEMDAN), offered by Torres et al. [63] to solve the problem of mode mixing caused by the empirical mode decomposition (EMD) method as well as the inability of the EEMD to completely eliminate Gaussian white noise after signal reconstruction [64]. Mode mixing, according to Wu and Huang [65], is defined as any IMF consisting of oscillations of intensely disparate amplitude, mostly caused by intermittency of the driving mechanism. Thus, the physical meaning of an IMF can cease by itself, indicating falsely that there may be diverse physical processes embodied in a mode.The CEEMDAN is a suitable method for sampling and dealing with the noise of signals and significantly attenuates the frequency aliasing problem that may occur with EMD and EEMD as developed by Huang et al. [66] and Wu and Huang [67], respectively. The CEEMDAN method realizes the continuity in frequency between adjacent scales by adding a certain white noise [68]. It uses the original series as the goal of IMF sifting and completely solves the two constraints—inconsistency in the number of decomposition scales and some inevitable error which exists between the reconstructed and original signals [63, 68]. Through the CEEMDAN, modes are reasonably extracted non-recursively, which makes it a fully intrinsic, adaptive, and quasi-orthogonal decomposition method [63].The CEEMDAN meticulously decomposes input signals into their major modes, known as intrinsic mode functions (IMFs), which reproduce the input signal but with varying sparsity qualities. Specifically, in the context of this study, the IMFs represent short, medium, and long-term periods [28]. The CEEMDAN is purposely employed in this study to minimise noise in the data. Information that misrepresents unpretentious core patterns is referred to as noise. Small price corrections in the market, as well as price variations that distort the broader trend, are examples of noise in the financial markets. This suggests that market noise might make it difficult for investors to tell what is driving a trend and whether it is fundamentally changing or simply experiencing short-term volatility. This is in line with the HMH [31] and the AMH [32].Corollary to the CEEMDAN, we present the effective transfer entropy that occurs from the formulation of conditional related information [69]. Transfer entropy quantifies the reduction in uncertainty especially when conditioned on past values in forecasting variables and thus makes it easier to model statistical causality between variables in a natural phenomenon [28, 40, 70]. Consequently, the amount of information that flows between commodities and volatilities can be quantified using transfer entropy. Therefore, the CEEMDAN-based entropy would provide an asymmetric method to measure the flow of information, after accurately decomposing the time series data into their IMFs. This approach is lacking in prior studies on commodities and uncertainties.Our study offers contributions to literature in many ways. First, we adopt the multi-frequency CEEMDAN-based entropy to examine the information flow between commodities and uncertainties. We follow Adam et al. [71] to capture the IMFs into high, medium, and low frequencies, in addition to the residue using cluster analysis to account for only four multi-frequencies. This is done to carefully extract the four dynamics of frequencies in financial time series [71, 72]. This is enriched with detailed information for decision making while aggregating similar IMFs based on mean periods to minimise cumbersome analysis. Second, we assess the similarities among the commodities and uncertainty indices through the Pearson correlation and Kendall tau-b coefficients from the outcome of the cluster analysis [71, 72]. Third, we consider the Rényian transfer entropy (RTE) to deal with issues of non-linearity, non-stationarity, and asymmetry which may occasion a deterministic system to chaos [73]. In addition, the RTE is a log-likelihood ratio transfer entropy which quantifies information from a probability density function. The RTE is specifically used in this study instead of the Shannon entropy to account for tail events associated with pricing relevant financial information in times of COVID-19. Consequently, it is extreme event rather than observation in the centre that comes to light when information flow is utilised [25, 28, 74].Fourth, since multi-frequency analysis is pertinent in this study, we utilise a relatively long time period of COVID-19, which has caused havoc on financial markets [28, 75]. This would offer better discernments and indulgence about the diversification potentials of commodities in times of shocks [76]. Fifth, we utilise five uncertainties relevant to commodities—EPU, Gvolatility, VCRIX, VIX, and OVX. Fifth, we provide a rigorous analysis of 20 commodities categorised into metals, energy, and agriculture. Seventh, the extent to which uncertainty indices can hedge against fluctuations in commodities markets is adequately investigated in this study. We further examine the DVMCES phenomenon which has never been considered by prior studies on information flows [25, 27, 28, 69, 73]. The outcome of the study will bolster confidence in existing investors within these markets to either give up part or all of their investments or ensure their effective management in times of shocks. The study will assist investors in making optimal portfolios, considering the overwhelming global financial influence of the COVID-19 pandemic [56]. Accordingly, the current study, to the best of our knowledge, is among the very few empirical studies that extensively assess information flow between global commodities and uncertainties, while drawing insights from the COVID-19 pandemic.Our empirical analysis revealed that the correlation coefficients and variances demonstrate that the high frequency (IMFs 1–4) which measures the short-term dynamics is the dominant frequency. Findings from the multi-frequency entropy indicate a negative bi-directional causality information transfer between global commodities and uncertainties, especially in the long term. Generally, the findings present pertinent inferences for portfolio diversification, policy decisions, investing risk, and risk management schemes for global commodities and market volatilities. We advocate that investors who delayed in investing within financial markets of commodities and market volatilities during the COVID-19 pandemic are most likely to minimise their portfolio risks.The rest of the study is well thought out as follows. Section2 contains the study’s methodologies on the CEEMDAN-based Rényi transfer entropy, and data sources and description are presented. Section 3 provides analysis of the main results, and theoretical and practical underpinnings are found in Section 4, and the main conclusions are drawn in Section 5.


## 2. Methodology

### 2.1. CEEMDAN

The empirical mode decomposition techniques have gained rapid attention by researchers due to their purely data-driven algorithm to separate scales which are exclusive of predefined basis functions, disparate to wavelet decomposition [77]. Thus, in the wavelet decomposition (for instance, the maximal overlap discrete wavelet transform and discrete wavelet transform), a predefined mother wavelet is needed to decompose a signal, and the selection of the mother wavelet is subjective and influential [77]. Nonetheless, the EMD method resorts to scale mixing problem. This problem was solved with the ensemble empirical mode decomposition method (EEMD) developed by Wu and Huang [67] to incorporate a randomly generated white noise series to the original signal. Thankfully, Torres et al. [63] developed the CEEMDAN to solve the residual noise in the reconstructed signals within the EEMD by appending the noise to the residual of prior iteration instead of the original signal [77].In CEEMDAN, compared to EMD, EEMD, and possibly CEEMD, irrespective of the number of decompositions, the reconstruction error of the signal approaches zero, and the completeness is better. Further, it solves the problem of low decomposition efficiency and saves a great deal of processing power. Again, the output of CEEMDAN follows a Gaussian distribution, so that each IMF followsNθi,1[78]. This is important because the observed data often describe a set of phenomena which may be of different kinds, i.e., which may include phenomena of different quality [79], and these different qualities presents themselves in quantitative discrepancies in financial time series. The global commodity and uncertainty variables were decomposed into seven IMFs and a residual. This was implemented with the libeemd R package [80]. The application of the algorithm is summarised as follows.Begin the number of realizationsN, noise parameters, index for IMF j = 1.Perform the EMD forN realizations; Smt=St+δoWnt,i=1,2,3,…,N, where n refers to the index for realizations; Wnt is the white noise series added to the candidate signal; and δo is the noise parameter for the initial step.The ensemble mean intrinsic mode functions (IMF) are calculated as(1)IMFnt¯=1N∑n=1NIMFnt.The exclusive first residue can be determined as(2)r1t=st−IMFnt¯.EvolveN number of realizations; then, the operator Ej. produces Jth mode obtained by EMD.(3)rjnt=rjt+δjEjWnt,n=1,2,3,…,N,IMFj+1t¯=1N∑n=1NE1rjt+δjEjWnt.The final step is to calculate thejth residue, where j=j+1:(4)rjt=rj−1t−IMFjt¯.


### 2.2. Cluster Analysis

The cluster analysis enables us to group the IMFs which we consider in this study as high, medium, and low frequencies. We consider the high, medium, and low frequencies in this study as multi-frequencies [71, 72]. The multi-frequencies are obtained by observing the mean periods of each IMF. The mean period is the average frequency of each IMF. It is measured as the ratio of the total number of points to the number of peaks [71, 72, 81]. Specifically from the CEEMDAN, the mean period is calculated as(5)total observationsnumber of maxima,where the number of peaks (maxima) is obtained through the extrema function [82]. We then sum up the IMFs based on the mean period obtained to categorise them into their respective multi-frequencies [81].


### 2.3. Rényi Transfer Entropy

Before we discuss the Rényi transfer entropy, we present the idea of Shannon entropy as a measure of uncertainty upon which transfer entropy is embedded in information theory [83]. We consider a probability distribution with diverse results of a given experiment pj. Following Hartley [84], each symbol’s average information is detailed as(6)H=∑j=1nPjlog21Pjbits,where n denotes number of diverse symbols with respect to the probabilities Pj.The concept of Shannon entropy was introduced in 1948 by Shannon [85]. It indicates that for a discrete random variable (J) that has a probability distribution of (Pj), the average number of bits necessary to optimally encode independent draws [83] can be presented as(7)HJ=−∑j=1nPjlog2Pj.With the notion of Markov processes, Shannon entropy is aligned with the concept of Kullback–Leibler distance [86] in order to measure the information flow between two time series. We present I and J as two discrete random variables with corresponding marginal probabilities of Pi and Pj and joint probability Pi,j, with dynamic structures in line with a stationary Markov process of order k (processI) and I (processJ). The Markov property signifies that the probability to detect I at time t+1 in state i conditional on the k previous observations is pit+1it,…,it−k+1=pit+1it,…,it−k. To encode the reflection in t+1, the average bit number required once the ex ante k values are known can be illustrated as(8)hjk=−∑iPit+1,itklogPit+1|itk,where itk=it,…,it−k+1 (similar in the same respect for process J). In a bivariate perspective as well as relying on the Kullback–Leibler distance [86], information flow from process J to process I is measured by computing the deviation from the generalized Markov property Pit+1|itk=Pit+1|itk,jtI. The Shannon transfer entropy can thus be presented as(9)TJ⟶Ik,l=∑Pit+1,itk,jtIlogPit+1|itk,jtIPit+1|itk,where TJ⟶I calculates the information flow from J to I. Analogously, TI⟶J, as a measure for the information flow from I to J, can be derived. The main direction of the information flow can be concluded by calculating the difference between TJ⟶I and TI⟶J.Based on the Shannon entropy so far discussed, we present the Rényi transfer entropy [87] which is contingent on a weighting parameter q and can be calculated as(10)HJq=11−qlog∑jPqj,with q>0. For q⟶1, Rényi entropy converges to Shannon entropy. For 0<q<1, thus, low probability events receive more weight, while for q>1, the weights benefit outcomes j with a higher initial probability. As a result, Rényi entropy permits to emphasize diverse distribution areas, depending on parameter q [70, 83].Applying the escort distribution [88] ∅qj=pqj/∑jpqj with q>0 to normalize the weighted distributions, Rényi transfer entropy [86] is derived as(11)RTJ⟶Ik,l=11−qPit+1,itk,jtIlog∑i∅qitkPqit+1|itk∑i,j∅qitk,jtIPqit+1|itk,jtI.It is worth noting that the Rényi transfer entropy can have negative values. As a result, knowing the history ofJ depicts even greater uncertainty than would otherwise be indicated by only knowing only the history of I.The transfer entropy parameters are biased in small samples [89]. The correction of the bias to calculate the effective transfer entropy is(12)ETEJ⟶Ik,l=TJ⟶Ik,l−TJshuffled⟶Ik,l,where TJshuffled⟶Ik,l depicts the transfer entropy via a shuffled form of the time series J, that is, selecting values at random from the observed time series J and realigning them to form a new time series, destroying the time series dependencies of J, and not forgetting the statistical dependencies between J and I. This enjoins TJshuffled⟶Ik,l to come together to zero with increasing sample size, and any nonzero value of TJshuffled⟶Ik,l is due to small sample effects.As a result, repeated shuffling and the average of the shuffled transfer entropy assessments across all replications can be used as a small sample bias estimator. This is subtracted from the Shannon or Rényi transfer entropy estimate to get a bias-corrected effective transfer entropy estimate.Relying on a Markov block bootstrap, the statistical significance of the transfer entropy estimates, as given by equation (12), can be inspected as provided by Dimpfl and Peter [74]. This preserves the dependencies within the variables J and I but ignores the statistical dependencies between J and I opposing to shuffling. The distribution of the estimates under the null hypothesis of no information movement is then determined by repeated estimation of transfer entropy. The associated pvalue is given by 1−q^T, where q^T signifies the simulated distribution’s quantile, which is defined by the transfer entropy estimate [83].


### 2.4. Data Sources and Description

The study employs 20 daily commodity prices which can be relatively described as aggregated and individual commodities. They include global commodities (Acommodity), global energy (Aenergy), global metals (Ametals), industrial metals (Imetals), Brent, gasoline, heating oil (Htoil), natural gas (Ngas), petroleum, cocoa, coffee, corn, cotton, soybeans, wheat, gold, lead, nickel, palladium, and zinc. We further present five uncertainty indices: the US Economic Policy Uncertainty (EPU), NASDAQ 100 Volatility Target (Gvolatility), Cryptocurrency Volatility Index (VCRIX), Chicago Board Options Exchange (CBOE) Volatility Index (VIX), and CBOE Crude Oil Volatility Index (OVX). The daily data span from December 31st, 2019, to March 31st, 2021, yielding a total of 306 observations after balancing the data. The suggested time frame is chosen to encompass the COVID-19 pandemic.The commodities are carefully selected to include various categorisations of important commodities mostly employed by extant literature in addition to commodities which are less considered in empirical literature. We include both aggregated and individual commodity indices to provide a detailed information flow dynamics with uncertainties during COVID-19 pandemic. The variables are selected based on consistent data availability for the chosen period and their importance for investment decisions. We do this to reveal hidden relationship of multi-frequency information flow between the commodities and uncertainty indices to establish the DVMCES phenomenon analogous to the competitive market hypothesis [28]. The data on the commodities, Gvolatility, VIX, and OVX, were obtained from investing.com, whereas EPU and VCRIX were obtained from the websites https://www.policyuncertainty.com/index.html [51] and https://data.thecrix.de/data/vcrix.csv, respectively, with the US dollars as the currency value where appropriate. The study was based on daily returns of rt=lnPt−lnPt−1, where rt is the incessantly compounded return and Pt and Pt−1 are current index and preceding index correspondingly.


### 2.5. Preliminary Analysis

Figure1 provides the time-varying prices and returns of both global commodities (black trends) and uncertainties (red trends). The different colour trends within the plots are presented to enhance identification. It can be observed from the plots that at the early sections of 2020, the prices for all markets trend upwards, after a plunging spike between February, 2020, and May, 2021. That is, the prices of global commodities are experiencing a rapid increase which concur with the assertion made by Zhang et al. [90] of markets rebound later in the COVID-19 pandemic since most busineses and economies had learnt how to survive. Thus, the dynamics of most markets have begun to return to normal. Generally, it can be observed from the plots that fluctuations in global commodities are similar which makes analysis of this study during COVID-19 highly comparable. Furthermore, the uncertainties display inverse relationships with the global commodities, except the Global Volatility Index, which depicts increasing trend, and possibly the Cryptocurrency Volatility Index. The log-return plots exhibit volatility clustering as anticipated due to the stylised facts of financial time series [91].Figure 1
Time series plots of prices and returns.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Table1 shows the preliminary statistical analysis for the returns series. The negative mean returns indicate the poor performance of financial assets during COVID-19 while the positive returns depict markets able to withstand shocks. The negative skewness specifically for the aggregated, agricultural, and metal commodities suggests that investing in these assets should be done with caution since there is a likelihood for lower returns in a going concern. Also, it can be observed from the Jarque–Bera statistic that all the series are non-normally distributed. However, we found that most of the returns series are stationary as shown by the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test, except for series such as the aggregated and agricultural commodities. In Table 2, the non-linearity tests indicate a mixture of linear and non-linear relationships in the variables at various significance levels. As a result, analysis with the CEEMDAN-based RTE is robust for dealing with issues of non-stationarity, non-linearity, and asymmetric relationships [28].Table 1
Preliminary statistical analysis.
ReturnsMeanStd. dev.SkewnessKurtosisJarque−BeraKPSSAggregated commoditiesAcommodity0.0010.014−1.38910.158749.234∗∗∗0.495∗∗Aenergy−0.0060.107−14.046226.878646987.1∗∗∗0.534Ametals0.0010.010−0.5574.37839.941∗∗∗0.153∗∗Imetals−0.0010.028−13.336212.749568139.4∗∗∗0.589∗∗EnergyBrent0.0130.23616.710287.8111045063∗∗∗0.229Gasoline0.0140.24516.516283.5051013796∗∗∗0.220Htoil0.0130.24316.917292.4391079185∗∗∗0.238Ngas0.0010.0410.6705.524103.723∗∗∗0.093Petroleum0.0000.044−1.99319.6083707.138∗∗∗0.374∗AgriculturalCocoa−0.0040.061−15.390257.984838297.3∗∗∗0.301Coffee−0.0140.247−16.879291.5531072618∗∗∗0.366∗Corn−0.0080.137−16.804289.8691060174∗∗∗0.434∗Cotton−0.0150.250−17.249300.0131136212∗∗∗0.386∗Soybeans−0.0200.344−17.311301.4481147183∗∗∗0.372∗Wheat−0.0130.223−17.078296.0831106443∗∗∗0.390∗MetalsGold0.0030.04715.254255.126819662.2∗∗∗0.411∗Lead0.0000.015−0.2154.10817.942∗∗∗0.128Nickel0.0000.017−0.5405.04868.091∗∗∗0.155Palladium0.0010.033−0.43620.5823938.329∗∗∗0.021Zinc0.0040.05415.626263.341873746.4∗∗∗0.198UncertaintiesEPU0.0080.5130.5925.49496.917∗∗∗0.118Gvolatility0.0080.12217.116296.9881113263∗∗∗0.319VCRIX0.0140.23612.827200.573504434.8∗∗∗0.290VIX−0.0010.104−0.43018.0192875.934∗∗∗0.050OVX0.0100.18710.009142.305251709.3∗∗∗0.407∗Note. ∗, ∗∗, and ∗∗∗ indicate significance at 10%, 5%, and 1% levels, respectively.Table 2
Non-linearity tests for global commodities and uncertainties.
ReturnsTeraesvirta’s neural network testWhite neural network testKeenan’s one-degree test for non-linearityTsay’s test for non-linearityLikelihood ratio test for threshold non-linearityAggregated commoditiesAcommodity17.319∗∗∗19.247∗∗∗0.0364.395∗∗∗3.624Aenergy9.803∗∗∗8.238∗∗0.0140.0864.960Ametals0.1130.0911.3962.09010.698Imetals0.3130.3530.0180.2442.736EnergyBrent9.058∗∗7.8070.3930.1342.170Gasoline7.287∗∗6.471∗∗0.0010.0501.582Htoil0.7920.5860.0850.1842.060Ngas0.8560.9162.940∗1.748∗∗17.061Petroleum12.271∗∗∗9.671∗∗∗1.0582.351∗∗18.076∗∗AgriculturalCocoa3.0022.8790.8390.9121.800Coffee0.6190.5190.0011.3781.114Corn0.7670.7920.0030.0284.294Cotton1.2821.1740.0010.3061.069Soybeans2.0631.9890.0010.1563.164Wheat0.3860.3903.137e − 072.1672.123MetalsGold2.4783.1095.4930.3042.053Lead6.485∗∗4.5590.1260.12313.472∗∗Nickel3.4502.8681.6201.9334.078Palladium0.8364.28515.793∗∗∗3.55∗∗∗31.494∗∗∗Zinc1.7631.8430.0130.1631.238UncertaintiesEPU3.6912.5962.4133.142∗∗7.467Gvolatility5.631∗5.623∗0.0040.2733.469OVX2.5661.7460.0500.2802.883VCRIX1.0661.3084.893e − 0557.75∗∗∗1.609VIX1.1992.8190.1500.2094.980Note. ∗, ∗∗, and ∗∗∗ indicate significance at 10%, 5%, and 1% levels, respectively.


## 2.1. CEEMDAN

The empirical mode decomposition techniques have gained rapid attention by researchers due to their purely data-driven algorithm to separate scales which are exclusive of predefined basis functions, disparate to wavelet decomposition [77]. Thus, in the wavelet decomposition (for instance, the maximal overlap discrete wavelet transform and discrete wavelet transform), a predefined mother wavelet is needed to decompose a signal, and the selection of the mother wavelet is subjective and influential [77]. Nonetheless, the EMD method resorts to scale mixing problem. This problem was solved with the ensemble empirical mode decomposition method (EEMD) developed by Wu and Huang [67] to incorporate a randomly generated white noise series to the original signal. Thankfully, Torres et al. [63] developed the CEEMDAN to solve the residual noise in the reconstructed signals within the EEMD by appending the noise to the residual of prior iteration instead of the original signal [77].In CEEMDAN, compared to EMD, EEMD, and possibly CEEMD, irrespective of the number of decompositions, the reconstruction error of the signal approaches zero, and the completeness is better. Further, it solves the problem of low decomposition efficiency and saves a great deal of processing power. Again, the output of CEEMDAN follows a Gaussian distribution, so that each IMF followsNθi,1[78]. This is important because the observed data often describe a set of phenomena which may be of different kinds, i.e., which may include phenomena of different quality [79], and these different qualities presents themselves in quantitative discrepancies in financial time series. The global commodity and uncertainty variables were decomposed into seven IMFs and a residual. This was implemented with the libeemd R package [80]. The application of the algorithm is summarised as follows.Begin the number of realizationsN, noise parameters, index for IMF j = 1.Perform the EMD forN realizations; Smt=St+δoWnt,i=1,2,3,…,N, where n refers to the index for realizations; Wnt is the white noise series added to the candidate signal; and δo is the noise parameter for the initial step.The ensemble mean intrinsic mode functions (IMF) are calculated as(1)IMFnt¯=1N∑n=1NIMFnt.The exclusive first residue can be determined as(2)r1t=st−IMFnt¯.EvolveN number of realizations; then, the operator Ej. produces Jth mode obtained by EMD.(3)rjnt=rjt+δjEjWnt,n=1,2,3,…,N,IMFj+1t¯=1N∑n=1NE1rjt+δjEjWnt.The final step is to calculate thejth residue, where j=j+1:(4)rjt=rj−1t−IMFjt¯.


## 2.2. Cluster Analysis

The cluster analysis enables us to group the IMFs which we consider in this study as high, medium, and low frequencies. We consider the high, medium, and low frequencies in this study as multi-frequencies [71, 72]. The multi-frequencies are obtained by observing the mean periods of each IMF. The mean period is the average frequency of each IMF. It is measured as the ratio of the total number of points to the number of peaks [71, 72, 81]. Specifically from the CEEMDAN, the mean period is calculated as(5)total observationsnumber of maxima,where the number of peaks (maxima) is obtained through the extrema function [82]. We then sum up the IMFs based on the mean period obtained to categorise them into their respective multi-frequencies [81].


## 2.3. Rényi Transfer Entropy

Before we discuss the Rényi transfer entropy, we present the idea of Shannon entropy as a measure of uncertainty upon which transfer entropy is embedded in information theory [83]. We consider a probability distribution with diverse results of a given experiment pj. Following Hartley [84], each symbol’s average information is detailed as(6)H=∑j=1nPjlog21Pjbits,where n denotes number of diverse symbols with respect to the probabilities Pj.The concept of Shannon entropy was introduced in 1948 by Shannon [85]. It indicates that for a discrete random variable (J) that has a probability distribution of (Pj), the average number of bits necessary to optimally encode independent draws [83] can be presented as(7)HJ=−∑j=1nPjlog2Pj.With the notion of Markov processes, Shannon entropy is aligned with the concept of Kullback–Leibler distance [86] in order to measure the information flow between two time series. We present I and J as two discrete random variables with corresponding marginal probabilities of Pi and Pj and joint probability Pi,j, with dynamic structures in line with a stationary Markov process of order k (processI) and I (processJ). The Markov property signifies that the probability to detect I at time t+1 in state i conditional on the k previous observations is pit+1it,…,it−k+1=pit+1it,…,it−k. To encode the reflection in t+1, the average bit number required once the ex ante k values are known can be illustrated as(8)hjk=−∑iPit+1,itklogPit+1|itk,where itk=it,…,it−k+1 (similar in the same respect for process J). In a bivariate perspective as well as relying on the Kullback–Leibler distance [86], information flow from process J to process I is measured by computing the deviation from the generalized Markov property Pit+1|itk=Pit+1|itk,jtI. The Shannon transfer entropy can thus be presented as(9)TJ⟶Ik,l=∑Pit+1,itk,jtIlogPit+1|itk,jtIPit+1|itk,where TJ⟶I calculates the information flow from J to I. Analogously, TI⟶J, as a measure for the information flow from I to J, can be derived. The main direction of the information flow can be concluded by calculating the difference between TJ⟶I and TI⟶J.Based on the Shannon entropy so far discussed, we present the Rényi transfer entropy [87] which is contingent on a weighting parameter q and can be calculated as(10)HJq=11−qlog∑jPqj,with q>0. For q⟶1, Rényi entropy converges to Shannon entropy. For 0<q<1, thus, low probability events receive more weight, while for q>1, the weights benefit outcomes j with a higher initial probability. As a result, Rényi entropy permits to emphasize diverse distribution areas, depending on parameter q [70, 83].Applying the escort distribution [88] ∅qj=pqj/∑jpqj with q>0 to normalize the weighted distributions, Rényi transfer entropy [86] is derived as(11)RTJ⟶Ik,l=11−qPit+1,itk,jtIlog∑i∅qitkPqit+1|itk∑i,j∅qitk,jtIPqit+1|itk,jtI.It is worth noting that the Rényi transfer entropy can have negative values. As a result, knowing the history ofJ depicts even greater uncertainty than would otherwise be indicated by only knowing only the history of I.The transfer entropy parameters are biased in small samples [89]. The correction of the bias to calculate the effective transfer entropy is(12)ETEJ⟶Ik,l=TJ⟶Ik,l−TJshuffled⟶Ik,l,where TJshuffled⟶Ik,l depicts the transfer entropy via a shuffled form of the time series J, that is, selecting values at random from the observed time series J and realigning them to form a new time series, destroying the time series dependencies of J, and not forgetting the statistical dependencies between J and I. This enjoins TJshuffled⟶Ik,l to come together to zero with increasing sample size, and any nonzero value of TJshuffled⟶Ik,l is due to small sample effects.As a result, repeated shuffling and the average of the shuffled transfer entropy assessments across all replications can be used as a small sample bias estimator. This is subtracted from the Shannon or Rényi transfer entropy estimate to get a bias-corrected effective transfer entropy estimate.Relying on a Markov block bootstrap, the statistical significance of the transfer entropy estimates, as given by equation (12), can be inspected as provided by Dimpfl and Peter [74]. This preserves the dependencies within the variables J and I but ignores the statistical dependencies between J and I opposing to shuffling. The distribution of the estimates under the null hypothesis of no information movement is then determined by repeated estimation of transfer entropy. The associated pvalue is given by 1−q^T, where q^T signifies the simulated distribution’s quantile, which is defined by the transfer entropy estimate [83].


## 2.4. Data Sources and Description

The study employs 20 daily commodity prices which can be relatively described as aggregated and individual commodities. They include global commodities (Acommodity), global energy (Aenergy), global metals (Ametals), industrial metals (Imetals), Brent, gasoline, heating oil (Htoil), natural gas (Ngas), petroleum, cocoa, coffee, corn, cotton, soybeans, wheat, gold, lead, nickel, palladium, and zinc. We further present five uncertainty indices: the US Economic Policy Uncertainty (EPU), NASDAQ 100 Volatility Target (Gvolatility), Cryptocurrency Volatility Index (VCRIX), Chicago Board Options Exchange (CBOE) Volatility Index (VIX), and CBOE Crude Oil Volatility Index (OVX). The daily data span from December 31st, 2019, to March 31st, 2021, yielding a total of 306 observations after balancing the data. The suggested time frame is chosen to encompass the COVID-19 pandemic.The commodities are carefully selected to include various categorisations of important commodities mostly employed by extant literature in addition to commodities which are less considered in empirical literature. We include both aggregated and individual commodity indices to provide a detailed information flow dynamics with uncertainties during COVID-19 pandemic. The variables are selected based on consistent data availability for the chosen period and their importance for investment decisions. We do this to reveal hidden relationship of multi-frequency information flow between the commodities and uncertainty indices to establish the DVMCES phenomenon analogous to the competitive market hypothesis [28]. The data on the commodities, Gvolatility, VIX, and OVX, were obtained from investing.com, whereas EPU and VCRIX were obtained from the websites https://www.policyuncertainty.com/index.html [51] and https://data.thecrix.de/data/vcrix.csv, respectively, with the US dollars as the currency value where appropriate. The study was based on daily returns of rt=lnPt−lnPt−1, where rt is the incessantly compounded return and Pt and Pt−1 are current index and preceding index correspondingly.


## 2.5. Preliminary Analysis

Figure1 provides the time-varying prices and returns of both global commodities (black trends) and uncertainties (red trends). The different colour trends within the plots are presented to enhance identification. It can be observed from the plots that at the early sections of 2020, the prices for all markets trend upwards, after a plunging spike between February, 2020, and May, 2021. That is, the prices of global commodities are experiencing a rapid increase which concur with the assertion made by Zhang et al. [90] of markets rebound later in the COVID-19 pandemic since most busineses and economies had learnt how to survive. Thus, the dynamics of most markets have begun to return to normal. Generally, it can be observed from the plots that fluctuations in global commodities are similar which makes analysis of this study during COVID-19 highly comparable. Furthermore, the uncertainties display inverse relationships with the global commodities, except the Global Volatility Index, which depicts increasing trend, and possibly the Cryptocurrency Volatility Index. The log-return plots exhibit volatility clustering as anticipated due to the stylised facts of financial time series [91].Figure 1
Time series plots of prices and returns.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Table1 shows the preliminary statistical analysis for the returns series. The negative mean returns indicate the poor performance of financial assets during COVID-19 while the positive returns depict markets able to withstand shocks. The negative skewness specifically for the aggregated, agricultural, and metal commodities suggests that investing in these assets should be done with caution since there is a likelihood for lower returns in a going concern. Also, it can be observed from the Jarque–Bera statistic that all the series are non-normally distributed. However, we found that most of the returns series are stationary as shown by the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test, except for series such as the aggregated and agricultural commodities. In Table 2, the non-linearity tests indicate a mixture of linear and non-linear relationships in the variables at various significance levels. As a result, analysis with the CEEMDAN-based RTE is robust for dealing with issues of non-stationarity, non-linearity, and asymmetric relationships [28].Table 1
Preliminary statistical analysis.
ReturnsMeanStd. dev.SkewnessKurtosisJarque−BeraKPSSAggregated commoditiesAcommodity0.0010.014−1.38910.158749.234∗∗∗0.495∗∗Aenergy−0.0060.107−14.046226.878646987.1∗∗∗0.534Ametals0.0010.010−0.5574.37839.941∗∗∗0.153∗∗Imetals−0.0010.028−13.336212.749568139.4∗∗∗0.589∗∗EnergyBrent0.0130.23616.710287.8111045063∗∗∗0.229Gasoline0.0140.24516.516283.5051013796∗∗∗0.220Htoil0.0130.24316.917292.4391079185∗∗∗0.238Ngas0.0010.0410.6705.524103.723∗∗∗0.093Petroleum0.0000.044−1.99319.6083707.138∗∗∗0.374∗AgriculturalCocoa−0.0040.061−15.390257.984838297.3∗∗∗0.301Coffee−0.0140.247−16.879291.5531072618∗∗∗0.366∗Corn−0.0080.137−16.804289.8691060174∗∗∗0.434∗Cotton−0.0150.250−17.249300.0131136212∗∗∗0.386∗Soybeans−0.0200.344−17.311301.4481147183∗∗∗0.372∗Wheat−0.0130.223−17.078296.0831106443∗∗∗0.390∗MetalsGold0.0030.04715.254255.126819662.2∗∗∗0.411∗Lead0.0000.015−0.2154.10817.942∗∗∗0.128Nickel0.0000.017−0.5405.04868.091∗∗∗0.155Palladium0.0010.033−0.43620.5823938.329∗∗∗0.021Zinc0.0040.05415.626263.341873746.4∗∗∗0.198UncertaintiesEPU0.0080.5130.5925.49496.917∗∗∗0.118Gvolatility0.0080.12217.116296.9881113263∗∗∗0.319VCRIX0.0140.23612.827200.573504434.8∗∗∗0.290VIX−0.0010.104−0.43018.0192875.934∗∗∗0.050OVX0.0100.18710.009142.305251709.3∗∗∗0.407∗Note. ∗, ∗∗, and ∗∗∗ indicate significance at 10%, 5%, and 1% levels, respectively.Table 2
Non-linearity tests for global commodities and uncertainties.
ReturnsTeraesvirta’s neural network testWhite neural network testKeenan’s one-degree test for non-linearityTsay’s test for non-linearityLikelihood ratio test for threshold non-linearityAggregated commoditiesAcommodity17.319∗∗∗19.247∗∗∗0.0364.395∗∗∗3.624Aenergy9.803∗∗∗8.238∗∗0.0140.0864.960Ametals0.1130.0911.3962.09010.698Imetals0.3130.3530.0180.2442.736EnergyBrent9.058∗∗7.8070.3930.1342.170Gasoline7.287∗∗6.471∗∗0.0010.0501.582Htoil0.7920.5860.0850.1842.060Ngas0.8560.9162.940∗1.748∗∗17.061Petroleum12.271∗∗∗9.671∗∗∗1.0582.351∗∗18.076∗∗AgriculturalCocoa3.0022.8790.8390.9121.800Coffee0.6190.5190.0011.3781.114Corn0.7670.7920.0030.0284.294Cotton1.2821.1740.0010.3061.069Soybeans2.0631.9890.0010.1563.164Wheat0.3860.3903.137e − 072.1672.123MetalsGold2.4783.1095.4930.3042.053Lead6.485∗∗4.5590.1260.12313.472∗∗Nickel3.4502.8681.6201.9334.078Palladium0.8364.28515.793∗∗∗3.55∗∗∗31.494∗∗∗Zinc1.7631.8430.0130.1631.238UncertaintiesEPU3.6912.5962.4133.142∗∗7.467Gvolatility5.631∗5.623∗0.0040.2733.469OVX2.5661.7460.0500.2802.883VCRIX1.0661.3084.893e − 0557.75∗∗∗1.609VIX1.1992.8190.1500.2094.980Note. ∗, ∗∗, and ∗∗∗ indicate significance at 10%, 5%, and 1% levels, respectively.


## 3. Results and Discussion

### 3.1. Reconstruction of IMFs

Analysis of the study is performed using 7 IMFs and a residue decomposed through the CEEMDAN technique for the global commodities and uncertainties. The mean period of each IMF expressed as a ratio of total number of points to the number of peaks, Pearson product moment correlation between each IMF and the original series, the variance of each IMF as a percentage of the original series, and the sum of the entire IMFs and residue are presented in Tables3 and 4. Specifically, the mean frequency depicts the average frequency of each IMF, and the correlation illustrates the degree of connectedness of each IMF to the original series, whereas the variance of each IMF as a percentage of the original series elucidates the influence of each IMF to the total volatility of the original series [71, 72, 81]. We do these to provide a substantive information about the global commodity and uncertainty dynamics.Table 3
Measures of IMFs and residue obtained through CEEMDAN.
VariablesIMF1IMF2IMF3IMF4IMF5IMF6IMF7ResidueAcommodityμ2.775.006.9312.7125.4233.8976.25ρ0.73∗∗∗0.38∗∗∗0.40∗∗∗0.35∗∗∗0.27∗∗∗0.030.20∗∗∗0.16∗∗∗σ1260.37%7.52%11.18%5.78`%3.85%2.52%1.75%2.90%σ2260.37%7.52%11.18%5.78%3.85%2.52%1.75%2.90%Aenergyμ2.684.844.369.2416.0530.561.00ρ0.71∗∗∗0.68∗∗∗0.59∗∗∗0.49∗∗∗0.39∗∗∗0.28∗∗∗0.16∗∗∗0.23∗∗∗σ1222.56%6.22%3.83%4.63%3.02%6.45%5.87%17.57%σ2222.56%6.22%3.83%4.63%3.02%6.45%5.87%17.57%Ametalsμ2.635.007.6311.7320.3350.8350.83ρ0.72∗∗∗0.48∗∗∗0.49∗∗∗0.24∗∗∗0.23∗∗∗0.11∗∗0.16∗∗∗0.08∗σ1254.47%11.29%12.79%3.79%3.80%1.02%1.71%0.60%σ2254.47%11.29%12.79%3.79%3.80%1.02%1.71%0.60%Imetalsμ2.684.425.1710.1721.7927.7361.00ρ0.70∗∗∗0.67∗∗∗0.60∗∗∗0.50∗∗∗0.37∗∗∗0.26∗∗∗0.12∗∗0.26∗∗∗σ1222.55%3.63%7.32%4.89%2.66%5.23%4.15%21.44%σ2222.55%3.63%7.32%4.89%2.66%5.23%4.15%21.44%Brentμ2.774.493.727.8212.7127.7376.25ρ0.71∗∗∗0.71∗∗∗0.61∗∗∗0.53∗∗∗0.39∗∗∗0.30∗∗∗0.17∗∗∗0.19∗∗∗σ1218.34%5.08%4.67%3.54%5.52%5.16%9.18%22.52%σ2218.34%5.08%4.67%3.54%5.52%5.16%9.18%22.52%Gasolineμ2.684.693.397.2613.2630.576.25ρ0.70∗∗∗0.72∗∗∗0.61∗∗∗0.53∗∗∗0.41∗∗∗0.30∗∗∗0.19∗∗∗0.19∗∗∗σ1219.17%6.55%4.27%2.83%4.63%4.99%8.97%20.69%σ2219.17%6.55%4.27%2.83%4.63%4.99%8.97%20.69%Htoilμ2.804.553.597.2612.7130.576.25ρ0.70∗∗∗0.72∗∗∗0.62∗∗∗0.54∗∗∗0.41∗∗∗0.31∗∗∗0.19∗∗∗0.20∗∗∗σ1217.55%7.36%3.78%3.13%4.40%5.11%7.51%19.16%σ2217.55%7.36%3.78%3.13%4.40%5.11%7.51%19.16%Ngasμ2.805.657.0912.7123.4643.5776.25ρ0.77∗∗∗0.45∗∗∗0.43∗∗∗0.27∗∗∗0.20∗∗∗0.10∗∗0.050.07σ1266.18%9.77%8.70%3.83%1.62%3.40%1.89%0.82%σ2266.18%9.77%8.70%3.83%1.62%3.40%1.89%0.82%Petroleumμ2.754.776.2210.8920.3338.1376.25ρ0.72∗∗∗0.35∗∗∗0.38∗∗∗0.33∗∗∗0.23∗∗∗0.020.17∗∗∗0.12∗∗σ1266.64%9.63%10.30%9.39%1.84%5.06%1.96%3.66%σ2266.64%9.63%10.30%9.39%1.84%5.06%1.96%3.66%Cocoaμ2.824.494.629.5316.9427.7350.83ρ0.73∗∗∗0.69∗∗∗0.62∗∗∗0.53∗∗∗0.44∗∗∗0.31∗∗∗0.20∗∗∗0.22∗∗∗σ1213.52%6.98%3.77%5.76%3.07%7.06%5.95%16.01%σ2213.52%6.98%3.77%5.76%3.07%7.06%5.95%16.01%Coffeeμ2.824.553.727.4415.5227.7350.83ρ0.72∗∗∗0.71∗∗∗0.62∗∗∗0.52∗∗∗0.42∗∗∗0.30∗∗∗0.17∗∗∗0.23∗∗∗σ1215.47%8.26%3.29%4.08%2.23%6.40%8.30%22.10%σ2215.47%8.26%3.29%4.08%2.23%6.40%8.30%22.10%Cornμ2.804.693.968.9714.5238.1350.83ρ0.70∗∗∗0.70∗∗∗0.63∗∗∗0.51∗∗∗0.41∗∗∗0.29∗∗∗0.18∗∗∗0.24∗∗∗σ1217.18%7.60%3.43%3.96%4.11%5.49%6.28%18.63%σ2217.18%7.60%3.43%3.96%4.11%5.49%6.28%18.63%Cottonμ2.774.423.396.2212.2027.7361.00ρ0.71∗∗∗0.72∗∗∗0.63∗∗∗0.51∗∗∗0.41∗∗∗0.29∗∗∗0.19∗∗∗0.23∗∗∗σ1216.67%8.34%3.25%4.14%3.49%4.88%6.87%17.91%σ2216.67%8.34%3.25%4.14%3.49%4.88%6.87%17.91%Soybeansμ2.804.923.356.112.225.4261.00ρ0.70∗∗∗0.72∗∗∗0.63∗∗∗0.52∗∗∗0.40∗∗∗0.29∗∗∗0.21∗∗∗0.23∗∗∗σ1216.21%9.12%3.07%3.74%3.74%5.40%6.54%16.56%σ2216.21%9.12%3.07%3.74%3.74%5.40%6.54%16.56%Wheatμ2.704.773.437.0913.8627.7361.00ρ0.70∗∗∗0.72∗∗∗0.62∗∗∗0.51∗∗∗0.41∗∗∗0.31∗∗∗0.11∗∗0.22∗∗∗σ1216.97%8.23%3.04%4.08%3.48%4.14%11.59%27.15%σ2216.97%8.23%3.04%4.08%3.48%4.14%11.59%27.15%Goldμ2.634.554.629.5315.2538.1361.00ρ0.70∗∗∗0.69∗∗∗0.61∗∗∗0.53∗∗∗0.40∗∗∗0.28∗∗∗0.18∗∗∗0.21∗∗∗σ1219.07%5.17%4.57%4.88%4.33%6.13%6.35%18.73%σ2219.07%5.17%4.57%4.88%4.33%6.13%6.35%18.73%Leadμ2.755.005.9811.7323.4643.5776.25ρ0.81∗∗∗0.49∗∗∗0.30∗∗∗0.25∗∗∗0.15∗∗∗0.12∗∗0.09∗0.06σ1264.00%13.91%2.62%6.64%3.10%1.39%1.38%1.26%σ2264.00%13.91%2.62%6.64%3.10%1.39%1.38%1.26%Nickelμ2.804.777.0913.8627.7338.1376.25ρ0.77∗∗∗0.43∗∗∗0.33∗∗∗0.34∗∗∗0.22∗∗∗0.15∗∗∗0.12∗∗0.13∗∗σ1260.05%9.87%7.34%8.87%3.53%0.41%0.53%1.81%σ2260.05%9.87%7.34%8.87%3.53%0.41%0.53%1.81%Palladiumμ2.885.006.4911.7319.0633.8961.00ρ0.64∗∗∗0.47∗∗∗0.53∗∗∗0.42∗∗∗0.28∗∗∗0.08∗0.08∗0.02σ1249.32%8.92%15.82511.77%1.90%0.35%0.68%0.06%σ2249.32%8.92%15.82511.77%1.90%0.35%0.68%0.06%Zincμ2.754.305.009.5316.9433.8976.25ρ0.73∗∗∗0.69∗∗∗0.57∗∗∗∗∗∗0.51∗∗∗0.42∗∗∗0.32∗∗∗0.25∗∗∗0.19∗∗∗σ1216.89%5.69%8.19%2.29%4.10%5.39%7.04%15.09%σ2216.89%5.69%8.19%2.29%4.10%5.39%7.04%15.09%EPUμ2.825.006.4911.7321.7933.8976.25ρ0.86∗∗∗0.38∗∗∗0.28∗∗∗0.23∗∗∗0.11∗∗0.060.040.06σ1279.39%6.90%2.91%5.82%0.70%0.70%0.49%3.37%σ2279.39%6.90%2.91%5.82%0.70%0.70%0.49%3.37%Gvolatilityμ2.774.423.476.7813.2625.4276.25ρ0.71∗∗∗0.72∗∗∗0.61∗∗∗0.53∗∗∗0.41∗∗∗0.28∗∗∗0.17∗∗∗0.21∗∗∗σ1216.83%8.30%4.09%2.64%3.72%6.39%8.94%20.13%σ2216.83%8.30%4.09%2.64%3.72%6.39%8.94%20.13%OVXμ2.774.626.499.5315.2543.5776.25ρ0.69∗∗∗0.60∗∗∗0.46∗∗∗0.46∗∗∗0.39∗∗∗0.27∗∗∗0.15∗∗∗0.23∗∗∗σ1229.17%9.60%5.34%4.25%1.89%6.59%3.63%14.96%σ2229.17%9.60%5.34%4.25%1.89%6.59%3.63%14.96%VCRIXμ3.024.775.559.8421.7943.5750.83ρ0.63∗∗∗0.68∗∗∗0.56∗∗∗0.46∗∗∗0.36∗∗∗0.32∗∗∗0.20∗∗∗0.24∗∗∗σ1219.03%6.92%8.05%5.28%4.06%7.18%2.27%14.61%σ2219.03%6.92%8.05%5.28%4.06%7.18%2.27%14.61%VIXμ2.685.007.6312.2017.9443.5761ρ0.76∗∗∗0.46∗∗∗0.42∗∗∗0.33∗∗∗0.21∗∗∗0.25∗∗∗0.18∗∗∗0.03σ1256.30%11.39%5.34%3.68%3.42%5.43%0.81%0.74%σ2256.30%11.39%5.34%3.68%3.42%5.43%0.81%0.74%Note. μ, ρ, σ12, and σ22 denote mean period, Pearson product moment correlations, variance as % of observed, and variance as % of the sum of all IMFs and residue. ∗, ∗∗, and ∗∗∗ indicate significance at 10%, 5%, and 1% levels, respectively.Table 4
Multi-frequency entropy analysis of information flow between global commodities and uncertainties.
Flows towards uncertaintiesFlows towards global commoditiesCommoditiesHFQMFQLFQResidueHFQMFQLFQResidueEPUGlobal commoditiesAcommodity−0.063−0.047−0.106−0.116−0.126−0.082−0.027−0.114Aenergy0.091−0.050−0.057−0.0570.014−0.114∗0.003−0.111Ametals−0.047−0.047−0.012−0.0390.106−0.090−0.035−0.012Imetals0.0620.033−0.061−0.1210.163∗−0.0620.002−0.118Brent−0.0280.0360.014−0.116−0.075−0.063−0.081−0.115Gasoline0.0620.033−0.073−0.122−0.043−0.057−0.084−0.119Htoil−0.0090.027−0.065−0.124−0.122−0.063−0.080−0.124Ngas0.095−0.093−0.060−0.1220.098−0.086−0.060−0.109Petroleum0.035−0.004−0.033−0.113−0.027−0.136∗∗0.069−0.115Cocoa−0.079−0.046−0.065−0.118−0.021−0.0540.011−0.108Coffee−0.114−0.041−0.060−0.1110.069−0.116∗−0.026−0.110Corn−0.178∗∗−0.047−0.059−0.1200.057−0.056−0.068−0.117Cotton0.0120.035−0.069−0.1160.012−0.066−0.086−0.116Soybeans−0.030−0.038−0.068−0.1180.051−0.125∗∗−0.096−0.120Wheat−0.166∗∗−0.0490.015−0.1210.040−0.046−0.052−0.114Gold0.0010.029−0.182∗−0.1220.025−0.069−0.088−0.110Lead−0.079−0.055−0.054−0.023−0.058−0.100−0.051−0.015Nickel0.114−0.055−0.062−0.0740.017−0.027−0.066−0.070Palladium−0.0230.045−0.072−0.1130.035−0.098∗−0.071−0.119Zinc−0.0180.030−0.062−0.117−0.003−0.066−0.007−0.117GvolatilityGlobal commoditiesAcommodity−0.069−0.0600.011−0.106−0.072−0.0270.022−0.111Aenergy0.190∗∗−0.095−0.058−0.119−0.002−0.164∗∗−0.018−0.117Ametals−0.009−0.107∗−0.018−0.056−0.012−0.143∗−0.014−0.048Imetals0.231∗∗−0.104−0.063−0.077−0.0050.193∗∗∗−0.001−0.070Brent−0.062−0.148∗∗0.347∗∗∗−0.057−0.062∗0.007−0.064−0.120Gasoline0.048−0.110∗−0.248∗∗∗0.165∗∗−0.043−0.246∗∗∗−0.255∗∗∗−0.126∗Htoil−0.0300.005−0.046−0.065−0.104−0.151∗∗−0.025−0.077Ngas−0.027−0.063−0.057−0.121−0.001−0.050−0.064−0.117Petroleum0.045−0.017−0.063−0.1180.050−0.051−0.068−0.126Cocoa−0.092−0.080−0.051−0.1150.011−0.079−0.007−0.115Coffee−0.112−0.095−0.068−0.069−0.110−0.241∗∗∗−0.078−0.074Corn0.008−0.168∗∗−0.064−0.1190.001−0.077−0.011−0.116Cotton−0.029−0.178∗∗∗−0.057−0.073−0.057−0.098−0.019−0.082Soybeans0.033−0.173∗∗∗0.107−0.081−0.060−0.169∗∗∗−0.0320.040Wheat0.0710.113∗0.016−0.0850.051−0.090−0.044−0.064Gold0.080−0.126∗−0.102−0.123−0.117−0.190∗∗∗−0.0780.170∗∗Lead−0.151∗−0.060−0.058−0.063−0.199∗∗−0.018−0.062−0.048Nickel−0.062−0.011−0.063−0.110−0.025−0.062−0.052−0.114Palladium0.092−0.022−0.069−0.1140.100−0.136∗0.025−0.124Zinc0.181∗∗−0.121∗0.112−0.114−0.111−0.183∗∗∗−0.122∗−0.113OVXGlobal commoditiesAcommodity−0.152∗∗−0.021−0.063−0.125−0.107−0.144∗∗−0.066−0.118Aenergy−0.136∗0.098∗−0.059−0.115−0.185∗∗−0.180∗∗∗−0.086−0.110Ametals−0.035−0.068−0.061−0.0560.030−0.107−0.068−0.046Imetals0.017−0.1010.014−0.1220.034−0.084−0.073−0.119Brent0.0430.1200.017−0.0820.181∗∗−0.033−0.070−0.081Gasoline0.157∗0.113∗0.016−0.0740.169∗∗−0.010−0.074−0.074Htoil0.0200.1130.016−0.0910.116−0.036−0.073−0.070Ngas−0.037−0.069−0.076−0.118−0.036−0.043−0.032−0.107Petroleum−0.095−0.070−0.051−0.118−0.082−0.151∗∗∗−0.058−0.116Cocoa0.1340.181∗∗∗0.016−0.1210.153∗∗−0.114∗−0.074−0.115Coffee0.0520.1010.017−0.1300.051−0.177∗∗∗−0.0640.166∗∗Corn0.0450.182∗∗∗0.014−0.0670.082−0.013−0.063−0.103Cotton0.165∗∗0.194∗∗∗0.014−0.0900.038−0.019−0.072−0.072Soybeans0.175∗∗∗0.193∗∗∗0.0150.167∗∗0.049−0.066−0.073−0.121Wheat0.237∗∗∗−0.0700.058−0.1180.083−0.0010.069−0.113Gold0.0870.1120.0240.0450.147∗−0.025−0.097−0.082Lead−0.047−0.024−0.014−0.0550.057−0.119−0.109−0.047Nickel0.041−0.014−0.071−0.117∗0.060−0.055−0.133∗∗−0.114Palladium−0.060−0.026−0.017−0.120−0.016−0.126−0.055−0.115Zinc0.0760.110∗0.070−0.0650.257∗∗∗−0.0270.074−0.098VCRIXGlobal commoditiesAcommodity0.056−0.017−0.013−0.122∗0.031−0.063−0.020−0.115Aenergy0.0170.111∗0.195∗∗∗−0.118−0.027−0.105−0.119∗−0.117Ametals0.118−0.052−0.022−0.057−0.039−0.033−0.030−0.048Imetals0.045−0.153∗∗0.018−0.1250.228∗∗∗0.004−0.084−0.115Brent0.0420.208∗∗∗−0.090−0.1140.097−0.025−0.177∗∗∗−0.114Gasoline0.0590.115∗−0.031−0.1160.173∗∗−0.0140.105−0.120Htoil−0.0030.196∗∗∗0.190∗∗∗−0.1130.012−0.024−0.106−0.117Ngas0.215∗∗∗−0.055−0.058−0.1190.154∗−0.043−0.070−0.122Petroleum0.0450.003−0.055−0.1140.060−0.155∗∗−0.057−0.119Cocoa0.0900.324∗∗∗0.019−0.0980.062−0.086−0.102−0.073Coffee0.0810.103∗0.015−0.1140.059−0.096−0.086−0.124∗Corn−0.0540.126∗0.015−0.1160.114−0.130∗−0.091−0.112Cotton0.1270.181∗∗∗0.197∗∗∗−0.1220.188∗∗−0.010−0.119∗−0.113Soybeans0.0850.102∗−0.091−0.121−0.018−0.017−0.183∗∗∗−0.119Wheat0.048−0.0460.008−0.117−0.064−0.087−0.050−0.121Gold0.178∗∗0.104−0.003−0.1160.101−0.028−0.067−0.111Lead0.039−0.020−0.061−0.0540.087−0.123∗−0.063−0.048Nickel−0.091−0.028−0.059−0.113−0.004−0.108−0.055−0.118Palladium−0.024−0.0220.009−0.1170.051−0.039−0.067−0.117Zinc0.0710.111∗−0.094−0.1140.085−0.027−0.191∗∗∗−0.113VIXGlobal commoditiesAcommodity−0.131−0.089−0.061−0.114−0.134∗−0.0700.010−0.122Aenergy0.0310.001−0.054−0.1120.020−0.085−0.013−0.121Ametals0.037−0.117∗∗−0.058−0.063−0.037−0.179∗∗∗0.015−0.045Imetals0.167∗∗0.016−0.064−0.1180.0150.036−0.081−0.118Brent−0.051−0.150∗∗0.196∗∗∗−0.119−0.131∗−0.098−0.106∗−0.113Gasoline0.113−0.047−0.089−0.115−0.046−0.123∗∗−0.184∗∗−0.124Htoil0.036−0.0960.029−0.124−0.114−0.021−0.091−0.119Ngas−0.054−0.041−0.056−0.1190.020−0.055−0.068−0.119Petroleum−0.005−0.038−0.023−0.1160.030−0.022−0.026−0.123Cocoa−0.0040.017−0.062−0.1000.0270.047−0.015−0.075Coffee−0.015−0.056−0.002−0.119−0.037−0.081−0.020−0.120Corn−0.133−0.048−0.059−0.1150.116−0.029−0.022−0.110Cotton0.027−0.107−0.014−0.1170.013−0.0010.079−0.121Soybeans0.040−0.0440.159∗∗−0.115−0.045−0.0890.071−0.117Wheat−0.022−0.070−0.069−0.1190.018−0.076−0.119∗−0.114Gold−0.003−0.061−0.085−0.121−0.137∗−0.0630.023−0.112Lead−0.054−0.058−0.055−0.053−0.150∗−0.078−0.057−0.051Nickel−0.102−0.076−0.068−0.1140.004−0.062−0.056−0.118Palladium0.012−0.031−0.013−0.116−0.134−0.167∗∗∗−0.044−0.117Zinc0.143∗−0.0630.102−0.1150.060−0.062−0.031−0.114Note. ∗, ∗∗, and ∗∗∗ indicate significance at 10%, 5%, and 1% levels, respectively.Based on the mean periods of each IMFs, we characterise the IMFs into high frequency (sum of IMFs 1–4), medium frequency (sum of IMFs 5 and 6) and low frequency (IMF 7) using cluster analysis, representing the short, medium, and long term. The high, medium, and low frequencies for most variables have mean frequency of less than 15 days, between 15 and 50 days, and more than 50 days, respectively (a detailed presentation of mean periods, correlations, and variances is shown in Table3). The residue is the non-oscillating drift of the data, which is not influenced by short-to-medium term fluctuations [25, 28, 92]. It represents the long-term trend of commodities and uncertainties dictated by factors such as earning base and valuation multiple [91]; the technical factors that encompass diverse macroeconomic factors [93]; market sentiment factors—animal spirits of investors and environmental context [94]; and market anomalies [95]. The correlation coefficients and variances of the IMFs and residue show that the high frequency which measures the short-term dynamics of the data (IMFs 1–4) is dominating (this is shown in Table 3).Table5 shows the descriptive statistics of the reconstructed series and the residue obtained through the CEEMDAN. The Pearson correlation coefficient, Kendall tau-b, and variance analysis are presented for all the 25 variables for each of the multi-frequencies. The descriptive measures for the individual IMFs (see Tables 3 and 4) corroborate with those of Table 5. Analysis from Table 5 indicates that the high frequency is dominant in all cases. This contradicts the outcome of Adam et al. [71] who found the residue as the dominating mode. The variance of each multi-frequency as a percentage of the original series and the sum of the entire IMFs and residue extracted with the CEEMDAN corroborate with each other. This is because the CEEMDAN technique solves the residual noise in the reconstructed signals within the EEMD by appending the noise to the residual of prior iteration instead of the original signal [77]. The Pearson correlation and Kendall tau-b coefficients among the commodities as well uncertainty indices depict similar behaviour at each multi-frequency, but the strength and significance of each multi-frequency vary among themselves. In this case, most markets demonstrate high levels of efficiency among themselves at a specific frequency. However, as the markets evolve—due to events and structural changes, and adapt—due to varying levels of markets at diverse investment horizons or intrinsic times, they become heterogeneous and adaptive. This is partly due to rational, albeit irrational investors’ relentless search for competing rewards and risks to satisfy their portfolio goals.Table 5
Descriptive statistics of the reconstructed series and the residue obtained through CEEMDAN (HFQ, MFQ, LFQ, and RESID denote high frequency, medium frequency, low frequency, and residue, respectively;∗, ∗∗, and ∗∗∗ indicate significance at 10%, 5%, and 1% levels, respectively).
Commodities and uncertaintiesPearson correlation coefficientKendall tau-bVariance as % of observedVariance as % of the sum of all IMFs and residueHFRQMFRQLFRQRESIDHFRQMFRQLFRQRESIDHFRQ (%)MFRQ (%)LFRQ (%)RESID (%)HFRQ (%)MFRQ (%)LFRQ (%)RESID (%)Acommodity0.94∗∗∗0.23∗∗∗0.20∗∗∗0.16∗∗∗0.75∗∗∗0.11∗∗∗0.13∗∗∗0.09∗∗∗88.686.091.752.9088.686.091.752.90Aenergy0.89∗∗∗0.39∗∗∗0.16∗∗∗0.23∗∗∗0.70∗∗∗0.10∗∗∗−0.020.06∗66.8112.425.0617.5766.8112.425.0617.57Ametals0.95∗∗∗0.26∗∗∗0.16∗∗∗0.080.80∗∗∗0.14∗∗∗0.11∗∗∗0.05∗92.004.701.710.6092.004.701.710.60Imetals0.88∗∗∗0.36∗∗∗0.12∗∗0.26∗∗∗0.74∗∗∗0.12∗∗∗−0.070.12∗∗∗70.0010.874.1521.447010.874.1521.44Brent0.87∗∗∗0.43∗∗∗0.17∗∗∗0.19∗∗∗0.65∗∗∗0.10∗∗∗0.10∗∗∗−0.0863.4513.849.1822.5263.4513.849.1822.52Gasoline0.88∗∗∗0.42∗∗∗0.19∗∗∗0.19∗∗∗0.67∗∗∗0.15∗∗∗0.04−0.0564.0013.348.9720.6964.0013.348.9720.69Htoil0.88∗∗0.43∗∗∗0.19∗∗∗0.20∗∗∗0.69∗∗∗0.07∗∗0.07∗∗−0.0863.9212.917.5119.1663.9212.917.5119.16Ngas0.97∗∗∗0.18∗∗∗0.050.070.82∗∗∗0.08∗∗0.030.0595.625.421.890.8295.625.421.890.82Petroleum0.95∗∗∗0.13∗∗∗0.17∗∗∗0.12∗∗0.74∗∗∗0.07∗∗0.16∗∗∗0.09∗∗92.397.271.963.6692.397.271.963.66Cocoa0.88∗∗∗0.42∗∗∗0.20∗∗∗0.220.72∗∗∗0.20∗∗∗−0.020.0464.1614.665.9516.0164.1614.665.9516.01Coffee0.88∗∗∗0.41∗∗∗0.17∗∗∗0.23∗∗∗0.59∗∗∗0.10∗∗∗−0.010.0464.0511.628.3022.1064.0511.628.3022.10Corn0.88∗∗∗0.42∗∗∗0.18∗∗∗0.24∗∗∗0.58∗∗∗0.14∗∗∗−0.040.07∗∗63.1113.136.2818.6363.1113.136.2818.63Cotton0.89∗∗∗0.40∗∗∗0.19∗∗∗0.23∗∗∗0.63∗∗∗0.10∗∗∗−0.060.07∗∗64.5212.166.8717.9164.5212.166.8717.91Soybeans0.89∗∗∗0.41∗∗∗0.21∗∗∗0.23∗∗∗0.55∗∗∗0.03−0.040.09∗∗63.7212.576.5416.5663.7212.576.5416.56Wheat0.89∗∗∗0.42∗∗∗0.11∗∗0.22∗∗∗0.62∗∗∗0.12∗∗∗−0.050.06∗63.2711.1011.5927.1563.2711.1011.5927.15Gold0.88∗∗∗0.40∗∗∗0.18∗∗∗0.21∗∗∗0.74∗∗∗0.050.06∗−0.0465.1314.866.3518.7365.1314.866.3518.73Lead0.96∗∗∗0.19∗∗∗0.09∗0.060.81∗∗∗0.11∗∗∗0.08∗∗0.0396.214.831.381.2696.214.831.381.26Nickel0.95∗∗∗0.23∗∗∗0.12∗∗0.13∗∗0.80∗∗∗0.11∗∗∗0.06∗0.07∗∗93.774.690.531.8193.774.690.531.81Palladium0.98∗∗∗0.27∗∗∗0.08∗0.020.82∗∗∗0.17∗∗∗0.08∗∗0.0393.342.580.680.0693.342.580.680.06Zinc0.87∗∗∗0.43∗∗∗0.25∗∗∗0.19∗∗∗0.73∗∗∗0.13∗∗∗0.10∗∗∗−0.0764.6113.577.0415.0964.6113.577.0415.09EPU0.98∗∗∗0.10∗∗0.040.060.88∗∗∗0.02−0.01−0.0198.952.080.493.3798.952.080.493.37Gvolatility0.89∗∗∗0.41∗∗∗0.17∗∗∗0.21∗∗∗0.74∗∗∗0.020.06∗∗−0.0363.2613.118.9420.1363.2613.118.9420.13OVX0.89∗∗∗0.35∗∗∗0.15∗∗∗0.23∗∗∗0.74∗∗∗0.11∗∗∗−0.010.08∗∗73.5212.573.6314.9673.5212.573.6314.96VCRIX0.87∗∗∗0.40∗∗∗0.20∗∗∗0.24∗∗∗0.41∗∗∗0.13∗∗∗0.030.0968.1815.822.2714.6168.1815.822.2714.61VIX0.95∗∗∗0.34∗∗∗0.18∗∗∗0.030.81∗∗∗0.14∗∗∗0.04−0.0587.158.150.810.7487.158.150.810.74


### 3.2. Multi-Frequency Entropy

We present an analysis for 20 commodities and 5 uncertainty returns through the multi-frequency entropy approach (the transfer entropies for Figures2–6 are provided in Table 4) at the 95% confidence bounds. The multi-frequencies indicate the importance of multi-scales in financial time series. The decomposed returns series based on CEEMDAN are further reconstructed into high, medium, and low frequencies, representing short, medium, and long terms, respectively. In addition, we present the residue which denotes the non-oscillating drift of the data, which is not influenced by short-to-medium term fluctuations; however, it is influenced by the structural variations in the data generation process.Figure 2
Multi-frequency information transfer between commodities and EPU. (a) Renyi’s effective transfer entropy between global commodities and EPU-(HFQ). (b) Renyi’s effective transfer entropy between global commodities and EPU-(MFQ). (c) Renyi’s effective transfer entropy between global commodities and EPU-(LFQ). (d) Renyi’s effective transfer entropy between global commodities and EPU-(residue).
(a)(b)(c)(d)Figure 3
Multi-frequency information transfer between commodities and Gvolatility. (a) Renyi’s effective transfer entropy between global commodities and Gvolatility-(HFQ). (b) Renyi’s effective transfer entropy between global commodities and Gvolatility-(MFQ). (c) Renyi’s effective transfer entropy between global commodities and Gvolatility-(LFQ). (d) Renyi’s effective transfer entropy between global commodities and Gvolatility-(residue).
(a)(b)(c)(d)Figure 4
Multi-frequency information transfer between commodities and OVX. (a) Renyi’s effective transfer entropy between global commodities and OVX-(HFQ). (b) Renyi’s effective transfer entropy between global commodities and OVX-(MFQ). (c) Renyi’s effective transfer entropy between global commodities and OVX-(LFQ). (d) Renyi’s effective transfer entropy between global commodities and OVX-(residue).
(a)(b)(c)(d)Figure 5
Multi-frequency information transfer between commodities and VCRIX. (a) Renyi’s effective transfer entropy between global commodities and VCRIX-(HFQ). (b) Renyi’s effective transfer entropy between global commodities and VCRIX-(MFQ). (c) Renyi’s effective transfer entropy between global commodities and VCRIX-(LFQ). (d) Renyi’s effective transfer entropy between global commodities and VCRIX-(residue).
(a)(b)(c)(d)Figure 6
Multi-frequency information transfer between commodities and VIX. (a) Renyi’s effective transfer entropy between global commodities and VIX-(HFQ). (b) Renyi’s effective transfer entropy between global commodities and VIX-(MFQ). (c) Renyi’s effective transfer entropy between global commodities and VIX-(LFQ). (d) Renyi’s effective transfer entropy between global commodities and VIX-(residue).
(a)(b)(c)(d)The presence of a negative ETE implies that awareness of one variable suggests a higher risk coverage for the other variable, whereas a positive ETE designates that the knowledge of one variable reduces the risk of the other variable [96]. The knowledge in the tails is assigned a high weight for low values of q, resulting in a significant effective transfer entropy result in the current situation. For this reason, we set q from the Rényi effective transfer entropy to 0.3 to offer more weights to the tails, which bears direct implications. The ETE decreases and even becomes negative as the weight is reduced. Since transfer entropy is a non-parametric estimate and has a higher likelihood of determining statistical interdependence between time series, we present the discussion between global commodities and uncertainties following the information flow theory, financial instability, modern portfolio theory, theory of interdependencies, the CMH, etc. and put our DVMCES phenomenon into perspective.


#### 3.2.1. Information Flow between Commodities and Economic Policy Uncertainty

Figure2 provides a mixture of negative and positive information flow between global commodities and EPU from high, medium, and low frequencies. This establishes the unstable nature of commodities to external uncertainty shocks [57]. However, we notice more negative flow of information from EPU to the global commodities. This explains the reason why EPU has an adverse impact on most financial markets [52, 53, 70, 97].Specifically, in the high frequency, cotton, petroleum, gasoline, Imetals, Aenergy, nickel, and Ngas transmit positive shocks to EPU, and the reverse is true for flows from EPU to these commodities. We notice more negative information flows for most of the variables after the high frequency. This implies that awareness of EPU suggests a higher risk coverage for global commodities. Accordingly, global commodities are more vulnerable to external shocks, especially in the long term, where most financial markets are experiencing market rebound [90]. Investors would therefore have to rebalance their portfolio to minimise their risk due to changing market conditions.


#### 3.2.2. Information Flow between Commodities and Global Volatility Index

Figure3 provides a mixture of negative and positive information flow between global commodities and Gvolatility from high, medium, and low frequencies. However, we notice more negative flow of information from Gvolatility to the global commodities. The negative influence of Gvolatility on global commodities highlights that fluctuations in market volatilities influence investors’ portfolio choices either by altering the trade-off between risk and return or their predictions of future market performance. As a result, investors may capitalise on this by hedging against market volatility since rising volatility does not incentivize investment opportunities [38]. This is necessary because phases of high volatility tend to correspond to drawdowns in markets [39].Specifically, in the high frequency, almost half of the global commodities transmit positive shocks to Gvolatility, but flow information flows from Gvolatility to most of the commodities are negative. In the short term, zinc, Aenergy and Imetals transmit positive flows to Gvolatility index which is not pleasant for diversification potentials. Investors can hedge against volatilities within lead and Brent using the Gvolatility index at high frequency. Generally, we find more negative information flows between the variables from the high to low frequency and the residue. This implies that awareness of any of the global commodities and Gvolatility suggests a higher risk coverage for each other. In this regard, global commodities, except gasoline and gold, are vulnerable to Gvolatility, especially in the long term, where most financial markets are experiencing market rebound [90]. Investors would therefore have to rebalance their portfolio or hedge their portfolio risks using the Gvolatility index to minimise their overall investment risks.


#### 3.2.3. Information Flow between Commodities and Volatilities in Crude Oil

From Figure4, we find a bi-directional causality of both negative and positive information flow between global commodities and investor fear and expectations (OVX) mostly in the high frequency. However, we notice more negative flows of information from the OVX to the global commodities in the medium and low frequencies and residue. The negative influence of OVX on global commodities shows that fluctuations in market volatilities influence investors’ portfolio choices either by altering the trade-off between risk and return or their predictions of future market performance. As a result, investors may benefit by hedging against market volatility since rising volatility does not incentivize investment opportunities [38].The outcome from the study implies that awareness of any of the global commodities and OVX suggests a higher risk coverage for each other. As a result, global commodities are susceptible to shocks from the OVX, especially in the medium and long term, where most financial markets are experiencing market rebound [90]. Investors would have to rebalance their portfolio or hedge their portfolio risks using the OVX index to minimise their overall investment risks. This makes global commodities more preferred to receiving safe haven benefits from the OVX futures [98] during market stress.


#### 3.2.4. Information Flow between Commodities and Cryptocurrency Volatility Index

We find a mixture of negative and positive information flow with between global commodities and VCRIX from high, medium, and low frequencies from Figure5. However, we notice more positive flows from the Cryptocurrency Volatility Index to global commodities in the short term but more negative flows of information from the medium and low frequencies and the residue. The positive flows from the VCRIX are indicative of plunging diversification benefits in the short term which is at the early part of COVID-19 pandemic. This shows that the knowledge of VCRIX minimises risk transmission to global commodities in the short term.On the other hand, the negative influence of VCRIX on global commodities highlights that fluctuations in market volatilities influence investors’ portfolio choices either by altering the trade-off between risk and return or their predictions of future market performance. Investors who wish to minimise their portfolio risks can employ VCRIX as a hedged instrument as shown from the medium and low frequencies and residue estimates. This supports the assertion of Conlon et al. [98] that investors can hedge against volatility in financial market using cryptocurrencies. Accordingly, volatility in global commodities occasioned by VCRIX would require investors to reconstruct their portfolio to include VCRIX. Accordingly, the finding provided by Yarovaya et al. [19] that speculative bubbles in cryptocurrency marketplaces might spread contagion and compromise financial stability comes to bear. Specifically, for information flows from commodities, investors may less likely shelter their portfolios against VCRIX for commodities such as cotton, gold, Ngas (at HFQ); soybean, coffee, Aenergy, zinc, gasoline, corn, cotton, Htoil, Brent, and cocoa (at MFQ); and Htoil, Aenergy, and cotton (at LFQ).


#### 3.2.5. Information Flow between Commodities and Investor Fear and Expectations

Figure6 illustrates a combination of negative and positive information flow between global commodities and investor fear and expectations (VIX) from high, medium, and low frequencies. However, we notice more negative flow of information from the VIX to the global commodities. The negative influence of VIX on global commodities documents that fluctuations in market volatilities influence investors’ portfolio choices either by altering the trade-off between risk and return or their predictions of future market performance. As a result, investors may capitalise on this by hedging against market volatility since rising volatility does not incentivize investment opportunities [38].Generally, we find more negative information flows between the variables from the high to low frequency and the residue. This implies that awareness of any of the global commodities and VIX suggests a higher risk coverage for each other. In this regard, global commodities are vulnerable to VIX, especially in the long term, where most financial markets are experiencing market rebound [90]. Investors would therefore have to reconstruct their portfolio or hedge their portfolio risks using the VIX index to minimise their overall investment risks. This makes global commodities more preferred to receiving safe haven benefits from the VIX futures [99] during market stress.


## 3.1. Reconstruction of IMFs

Analysis of the study is performed using 7 IMFs and a residue decomposed through the CEEMDAN technique for the global commodities and uncertainties. The mean period of each IMF expressed as a ratio of total number of points to the number of peaks, Pearson product moment correlation between each IMF and the original series, the variance of each IMF as a percentage of the original series, and the sum of the entire IMFs and residue are presented in Tables3 and 4. Specifically, the mean frequency depicts the average frequency of each IMF, and the correlation illustrates the degree of connectedness of each IMF to the original series, whereas the variance of each IMF as a percentage of the original series elucidates the influence of each IMF to the total volatility of the original series [71, 72, 81]. We do these to provide a substantive information about the global commodity and uncertainty dynamics.Table 3
Measures of IMFs and residue obtained through CEEMDAN.
VariablesIMF1IMF2IMF3IMF4IMF5IMF6IMF7ResidueAcommodityμ2.775.006.9312.7125.4233.8976.25ρ0.73∗∗∗0.38∗∗∗0.40∗∗∗0.35∗∗∗0.27∗∗∗0.030.20∗∗∗0.16∗∗∗σ1260.37%7.52%11.18%5.78`%3.85%2.52%1.75%2.90%σ2260.37%7.52%11.18%5.78%3.85%2.52%1.75%2.90%Aenergyμ2.684.844.369.2416.0530.561.00ρ0.71∗∗∗0.68∗∗∗0.59∗∗∗0.49∗∗∗0.39∗∗∗0.28∗∗∗0.16∗∗∗0.23∗∗∗σ1222.56%6.22%3.83%4.63%3.02%6.45%5.87%17.57%σ2222.56%6.22%3.83%4.63%3.02%6.45%5.87%17.57%Ametalsμ2.635.007.6311.7320.3350.8350.83ρ0.72∗∗∗0.48∗∗∗0.49∗∗∗0.24∗∗∗0.23∗∗∗0.11∗∗0.16∗∗∗0.08∗σ1254.47%11.29%12.79%3.79%3.80%1.02%1.71%0.60%σ2254.47%11.29%12.79%3.79%3.80%1.02%1.71%0.60%Imetalsμ2.684.425.1710.1721.7927.7361.00ρ0.70∗∗∗0.67∗∗∗0.60∗∗∗0.50∗∗∗0.37∗∗∗0.26∗∗∗0.12∗∗0.26∗∗∗σ1222.55%3.63%7.32%4.89%2.66%5.23%4.15%21.44%σ2222.55%3.63%7.32%4.89%2.66%5.23%4.15%21.44%Brentμ2.774.493.727.8212.7127.7376.25ρ0.71∗∗∗0.71∗∗∗0.61∗∗∗0.53∗∗∗0.39∗∗∗0.30∗∗∗0.17∗∗∗0.19∗∗∗σ1218.34%5.08%4.67%3.54%5.52%5.16%9.18%22.52%σ2218.34%5.08%4.67%3.54%5.52%5.16%9.18%22.52%Gasolineμ2.684.693.397.2613.2630.576.25ρ0.70∗∗∗0.72∗∗∗0.61∗∗∗0.53∗∗∗0.41∗∗∗0.30∗∗∗0.19∗∗∗0.19∗∗∗σ1219.17%6.55%4.27%2.83%4.63%4.99%8.97%20.69%σ2219.17%6.55%4.27%2.83%4.63%4.99%8.97%20.69%Htoilμ2.804.553.597.2612.7130.576.25ρ0.70∗∗∗0.72∗∗∗0.62∗∗∗0.54∗∗∗0.41∗∗∗0.31∗∗∗0.19∗∗∗0.20∗∗∗σ1217.55%7.36%3.78%3.13%4.40%5.11%7.51%19.16%σ2217.55%7.36%3.78%3.13%4.40%5.11%7.51%19.16%Ngasμ2.805.657.0912.7123.4643.5776.25ρ0.77∗∗∗0.45∗∗∗0.43∗∗∗0.27∗∗∗0.20∗∗∗0.10∗∗0.050.07σ1266.18%9.77%8.70%3.83%1.62%3.40%1.89%0.82%σ2266.18%9.77%8.70%3.83%1.62%3.40%1.89%0.82%Petroleumμ2.754.776.2210.8920.3338.1376.25ρ0.72∗∗∗0.35∗∗∗0.38∗∗∗0.33∗∗∗0.23∗∗∗0.020.17∗∗∗0.12∗∗σ1266.64%9.63%10.30%9.39%1.84%5.06%1.96%3.66%σ2266.64%9.63%10.30%9.39%1.84%5.06%1.96%3.66%Cocoaμ2.824.494.629.5316.9427.7350.83ρ0.73∗∗∗0.69∗∗∗0.62∗∗∗0.53∗∗∗0.44∗∗∗0.31∗∗∗0.20∗∗∗0.22∗∗∗σ1213.52%6.98%3.77%5.76%3.07%7.06%5.95%16.01%σ2213.52%6.98%3.77%5.76%3.07%7.06%5.95%16.01%Coffeeμ2.824.553.727.4415.5227.7350.83ρ0.72∗∗∗0.71∗∗∗0.62∗∗∗0.52∗∗∗0.42∗∗∗0.30∗∗∗0.17∗∗∗0.23∗∗∗σ1215.47%8.26%3.29%4.08%2.23%6.40%8.30%22.10%σ2215.47%8.26%3.29%4.08%2.23%6.40%8.30%22.10%Cornμ2.804.693.968.9714.5238.1350.83ρ0.70∗∗∗0.70∗∗∗0.63∗∗∗0.51∗∗∗0.41∗∗∗0.29∗∗∗0.18∗∗∗0.24∗∗∗σ1217.18%7.60%3.43%3.96%4.11%5.49%6.28%18.63%σ2217.18%7.60%3.43%3.96%4.11%5.49%6.28%18.63%Cottonμ2.774.423.396.2212.2027.7361.00ρ0.71∗∗∗0.72∗∗∗0.63∗∗∗0.51∗∗∗0.41∗∗∗0.29∗∗∗0.19∗∗∗0.23∗∗∗σ1216.67%8.34%3.25%4.14%3.49%4.88%6.87%17.91%σ2216.67%8.34%3.25%4.14%3.49%4.88%6.87%17.91%Soybeansμ2.804.923.356.112.225.4261.00ρ0.70∗∗∗0.72∗∗∗0.63∗∗∗0.52∗∗∗0.40∗∗∗0.29∗∗∗0.21∗∗∗0.23∗∗∗σ1216.21%9.12%3.07%3.74%3.74%5.40%6.54%16.56%σ2216.21%9.12%3.07%3.74%3.74%5.40%6.54%16.56%Wheatμ2.704.773.437.0913.8627.7361.00ρ0.70∗∗∗0.72∗∗∗0.62∗∗∗0.51∗∗∗0.41∗∗∗0.31∗∗∗0.11∗∗0.22∗∗∗σ1216.97%8.23%3.04%4.08%3.48%4.14%11.59%27.15%σ2216.97%8.23%3.04%4.08%3.48%4.14%11.59%27.15%Goldμ2.634.554.629.5315.2538.1361.00ρ0.70∗∗∗0.69∗∗∗0.61∗∗∗0.53∗∗∗0.40∗∗∗0.28∗∗∗0.18∗∗∗0.21∗∗∗σ1219.07%5.17%4.57%4.88%4.33%6.13%6.35%18.73%σ2219.07%5.17%4.57%4.88%4.33%6.13%6.35%18.73%Leadμ2.755.005.9811.7323.4643.5776.25ρ0.81∗∗∗0.49∗∗∗0.30∗∗∗0.25∗∗∗0.15∗∗∗0.12∗∗0.09∗0.06σ1264.00%13.91%2.62%6.64%3.10%1.39%1.38%1.26%σ2264.00%13.91%2.62%6.64%3.10%1.39%1.38%1.26%Nickelμ2.804.777.0913.8627.7338.1376.25ρ0.77∗∗∗0.43∗∗∗0.33∗∗∗0.34∗∗∗0.22∗∗∗0.15∗∗∗0.12∗∗0.13∗∗σ1260.05%9.87%7.34%8.87%3.53%0.41%0.53%1.81%σ2260.05%9.87%7.34%8.87%3.53%0.41%0.53%1.81%Palladiumμ2.885.006.4911.7319.0633.8961.00ρ0.64∗∗∗0.47∗∗∗0.53∗∗∗0.42∗∗∗0.28∗∗∗0.08∗0.08∗0.02σ1249.32%8.92%15.82511.77%1.90%0.35%0.68%0.06%σ2249.32%8.92%15.82511.77%1.90%0.35%0.68%0.06%Zincμ2.754.305.009.5316.9433.8976.25ρ0.73∗∗∗0.69∗∗∗0.57∗∗∗∗∗∗0.51∗∗∗0.42∗∗∗0.32∗∗∗0.25∗∗∗0.19∗∗∗σ1216.89%5.69%8.19%2.29%4.10%5.39%7.04%15.09%σ2216.89%5.69%8.19%2.29%4.10%5.39%7.04%15.09%EPUμ2.825.006.4911.7321.7933.8976.25ρ0.86∗∗∗0.38∗∗∗0.28∗∗∗0.23∗∗∗0.11∗∗0.060.040.06σ1279.39%6.90%2.91%5.82%0.70%0.70%0.49%3.37%σ2279.39%6.90%2.91%5.82%0.70%0.70%0.49%3.37%Gvolatilityμ2.774.423.476.7813.2625.4276.25ρ0.71∗∗∗0.72∗∗∗0.61∗∗∗0.53∗∗∗0.41∗∗∗0.28∗∗∗0.17∗∗∗0.21∗∗∗σ1216.83%8.30%4.09%2.64%3.72%6.39%8.94%20.13%σ2216.83%8.30%4.09%2.64%3.72%6.39%8.94%20.13%OVXμ2.774.626.499.5315.2543.5776.25ρ0.69∗∗∗0.60∗∗∗0.46∗∗∗0.46∗∗∗0.39∗∗∗0.27∗∗∗0.15∗∗∗0.23∗∗∗σ1229.17%9.60%5.34%4.25%1.89%6.59%3.63%14.96%σ2229.17%9.60%5.34%4.25%1.89%6.59%3.63%14.96%VCRIXμ3.024.775.559.8421.7943.5750.83ρ0.63∗∗∗0.68∗∗∗0.56∗∗∗0.46∗∗∗0.36∗∗∗0.32∗∗∗0.20∗∗∗0.24∗∗∗σ1219.03%6.92%8.05%5.28%4.06%7.18%2.27%14.61%σ2219.03%6.92%8.05%5.28%4.06%7.18%2.27%14.61%VIXμ2.685.007.6312.2017.9443.5761ρ0.76∗∗∗0.46∗∗∗0.42∗∗∗0.33∗∗∗0.21∗∗∗0.25∗∗∗0.18∗∗∗0.03σ1256.30%11.39%5.34%3.68%3.42%5.43%0.81%0.74%σ2256.30%11.39%5.34%3.68%3.42%5.43%0.81%0.74%Note. μ, ρ, σ12, and σ22 denote mean period, Pearson product moment correlations, variance as % of observed, and variance as % of the sum of all IMFs and residue. ∗, ∗∗, and ∗∗∗ indicate significance at 10%, 5%, and 1% levels, respectively.Table 4
Multi-frequency entropy analysis of information flow between global commodities and uncertainties.
Flows towards uncertaintiesFlows towards global commoditiesCommoditiesHFQMFQLFQResidueHFQMFQLFQResidueEPUGlobal commoditiesAcommodity−0.063−0.047−0.106−0.116−0.126−0.082−0.027−0.114Aenergy0.091−0.050−0.057−0.0570.014−0.114∗0.003−0.111Ametals−0.047−0.047−0.012−0.0390.106−0.090−0.035−0.012Imetals0.0620.033−0.061−0.1210.163∗−0.0620.002−0.118Brent−0.0280.0360.014−0.116−0.075−0.063−0.081−0.115Gasoline0.0620.033−0.073−0.122−0.043−0.057−0.084−0.119Htoil−0.0090.027−0.065−0.124−0.122−0.063−0.080−0.124Ngas0.095−0.093−0.060−0.1220.098−0.086−0.060−0.109Petroleum0.035−0.004−0.033−0.113−0.027−0.136∗∗0.069−0.115Cocoa−0.079−0.046−0.065−0.118−0.021−0.0540.011−0.108Coffee−0.114−0.041−0.060−0.1110.069−0.116∗−0.026−0.110Corn−0.178∗∗−0.047−0.059−0.1200.057−0.056−0.068−0.117Cotton0.0120.035−0.069−0.1160.012−0.066−0.086−0.116Soybeans−0.030−0.038−0.068−0.1180.051−0.125∗∗−0.096−0.120Wheat−0.166∗∗−0.0490.015−0.1210.040−0.046−0.052−0.114Gold0.0010.029−0.182∗−0.1220.025−0.069−0.088−0.110Lead−0.079−0.055−0.054−0.023−0.058−0.100−0.051−0.015Nickel0.114−0.055−0.062−0.0740.017−0.027−0.066−0.070Palladium−0.0230.045−0.072−0.1130.035−0.098∗−0.071−0.119Zinc−0.0180.030−0.062−0.117−0.003−0.066−0.007−0.117GvolatilityGlobal commoditiesAcommodity−0.069−0.0600.011−0.106−0.072−0.0270.022−0.111Aenergy0.190∗∗−0.095−0.058−0.119−0.002−0.164∗∗−0.018−0.117Ametals−0.009−0.107∗−0.018−0.056−0.012−0.143∗−0.014−0.048Imetals0.231∗∗−0.104−0.063−0.077−0.0050.193∗∗∗−0.001−0.070Brent−0.062−0.148∗∗0.347∗∗∗−0.057−0.062∗0.007−0.064−0.120Gasoline0.048−0.110∗−0.248∗∗∗0.165∗∗−0.043−0.246∗∗∗−0.255∗∗∗−0.126∗Htoil−0.0300.005−0.046−0.065−0.104−0.151∗∗−0.025−0.077Ngas−0.027−0.063−0.057−0.121−0.001−0.050−0.064−0.117Petroleum0.045−0.017−0.063−0.1180.050−0.051−0.068−0.126Cocoa−0.092−0.080−0.051−0.1150.011−0.079−0.007−0.115Coffee−0.112−0.095−0.068−0.069−0.110−0.241∗∗∗−0.078−0.074Corn0.008−0.168∗∗−0.064−0.1190.001−0.077−0.011−0.116Cotton−0.029−0.178∗∗∗−0.057−0.073−0.057−0.098−0.019−0.082Soybeans0.033−0.173∗∗∗0.107−0.081−0.060−0.169∗∗∗−0.0320.040Wheat0.0710.113∗0.016−0.0850.051−0.090−0.044−0.064Gold0.080−0.126∗−0.102−0.123−0.117−0.190∗∗∗−0.0780.170∗∗Lead−0.151∗−0.060−0.058−0.063−0.199∗∗−0.018−0.062−0.048Nickel−0.062−0.011−0.063−0.110−0.025−0.062−0.052−0.114Palladium0.092−0.022−0.069−0.1140.100−0.136∗0.025−0.124Zinc0.181∗∗−0.121∗0.112−0.114−0.111−0.183∗∗∗−0.122∗−0.113OVXGlobal commoditiesAcommodity−0.152∗∗−0.021−0.063−0.125−0.107−0.144∗∗−0.066−0.118Aenergy−0.136∗0.098∗−0.059−0.115−0.185∗∗−0.180∗∗∗−0.086−0.110Ametals−0.035−0.068−0.061−0.0560.030−0.107−0.068−0.046Imetals0.017−0.1010.014−0.1220.034−0.084−0.073−0.119Brent0.0430.1200.017−0.0820.181∗∗−0.033−0.070−0.081Gasoline0.157∗0.113∗0.016−0.0740.169∗∗−0.010−0.074−0.074Htoil0.0200.1130.016−0.0910.116−0.036−0.073−0.070Ngas−0.037−0.069−0.076−0.118−0.036−0.043−0.032−0.107Petroleum−0.095−0.070−0.051−0.118−0.082−0.151∗∗∗−0.058−0.116Cocoa0.1340.181∗∗∗0.016−0.1210.153∗∗−0.114∗−0.074−0.115Coffee0.0520.1010.017−0.1300.051−0.177∗∗∗−0.0640.166∗∗Corn0.0450.182∗∗∗0.014−0.0670.082−0.013−0.063−0.103Cotton0.165∗∗0.194∗∗∗0.014−0.0900.038−0.019−0.072−0.072Soybeans0.175∗∗∗0.193∗∗∗0.0150.167∗∗0.049−0.066−0.073−0.121Wheat0.237∗∗∗−0.0700.058−0.1180.083−0.0010.069−0.113Gold0.0870.1120.0240.0450.147∗−0.025−0.097−0.082Lead−0.047−0.024−0.014−0.0550.057−0.119−0.109−0.047Nickel0.041−0.014−0.071−0.117∗0.060−0.055−0.133∗∗−0.114Palladium−0.060−0.026−0.017−0.120−0.016−0.126−0.055−0.115Zinc0.0760.110∗0.070−0.0650.257∗∗∗−0.0270.074−0.098VCRIXGlobal commoditiesAcommodity0.056−0.017−0.013−0.122∗0.031−0.063−0.020−0.115Aenergy0.0170.111∗0.195∗∗∗−0.118−0.027−0.105−0.119∗−0.117Ametals0.118−0.052−0.022−0.057−0.039−0.033−0.030−0.048Imetals0.045−0.153∗∗0.018−0.1250.228∗∗∗0.004−0.084−0.115Brent0.0420.208∗∗∗−0.090−0.1140.097−0.025−0.177∗∗∗−0.114Gasoline0.0590.115∗−0.031−0.1160.173∗∗−0.0140.105−0.120Htoil−0.0030.196∗∗∗0.190∗∗∗−0.1130.012−0.024−0.106−0.117Ngas0.215∗∗∗−0.055−0.058−0.1190.154∗−0.043−0.070−0.122Petroleum0.0450.003−0.055−0.1140.060−0.155∗∗−0.057−0.119Cocoa0.0900.324∗∗∗0.019−0.0980.062−0.086−0.102−0.073Coffee0.0810.103∗0.015−0.1140.059−0.096−0.086−0.124∗Corn−0.0540.126∗0.015−0.1160.114−0.130∗−0.091−0.112Cotton0.1270.181∗∗∗0.197∗∗∗−0.1220.188∗∗−0.010−0.119∗−0.113Soybeans0.0850.102∗−0.091−0.121−0.018−0.017−0.183∗∗∗−0.119Wheat0.048−0.0460.008−0.117−0.064−0.087−0.050−0.121Gold0.178∗∗0.104−0.003−0.1160.101−0.028−0.067−0.111Lead0.039−0.020−0.061−0.0540.087−0.123∗−0.063−0.048Nickel−0.091−0.028−0.059−0.113−0.004−0.108−0.055−0.118Palladium−0.024−0.0220.009−0.1170.051−0.039−0.067−0.117Zinc0.0710.111∗−0.094−0.1140.085−0.027−0.191∗∗∗−0.113VIXGlobal commoditiesAcommodity−0.131−0.089−0.061−0.114−0.134∗−0.0700.010−0.122Aenergy0.0310.001−0.054−0.1120.020−0.085−0.013−0.121Ametals0.037−0.117∗∗−0.058−0.063−0.037−0.179∗∗∗0.015−0.045Imetals0.167∗∗0.016−0.064−0.1180.0150.036−0.081−0.118Brent−0.051−0.150∗∗0.196∗∗∗−0.119−0.131∗−0.098−0.106∗−0.113Gasoline0.113−0.047−0.089−0.115−0.046−0.123∗∗−0.184∗∗−0.124Htoil0.036−0.0960.029−0.124−0.114−0.021−0.091−0.119Ngas−0.054−0.041−0.056−0.1190.020−0.055−0.068−0.119Petroleum−0.005−0.038−0.023−0.1160.030−0.022−0.026−0.123Cocoa−0.0040.017−0.062−0.1000.0270.047−0.015−0.075Coffee−0.015−0.056−0.002−0.119−0.037−0.081−0.020−0.120Corn−0.133−0.048−0.059−0.1150.116−0.029−0.022−0.110Cotton0.027−0.107−0.014−0.1170.013−0.0010.079−0.121Soybeans0.040−0.0440.159∗∗−0.115−0.045−0.0890.071−0.117Wheat−0.022−0.070−0.069−0.1190.018−0.076−0.119∗−0.114Gold−0.003−0.061−0.085−0.121−0.137∗−0.0630.023−0.112Lead−0.054−0.058−0.055−0.053−0.150∗−0.078−0.057−0.051Nickel−0.102−0.076−0.068−0.1140.004−0.062−0.056−0.118Palladium0.012−0.031−0.013−0.116−0.134−0.167∗∗∗−0.044−0.117Zinc0.143∗−0.0630.102−0.1150.060−0.062−0.031−0.114Note. ∗, ∗∗, and ∗∗∗ indicate significance at 10%, 5%, and 1% levels, respectively.Based on the mean periods of each IMFs, we characterise the IMFs into high frequency (sum of IMFs 1–4), medium frequency (sum of IMFs 5 and 6) and low frequency (IMF 7) using cluster analysis, representing the short, medium, and long term. The high, medium, and low frequencies for most variables have mean frequency of less than 15 days, between 15 and 50 days, and more than 50 days, respectively (a detailed presentation of mean periods, correlations, and variances is shown in Table3). The residue is the non-oscillating drift of the data, which is not influenced by short-to-medium term fluctuations [25, 28, 92]. It represents the long-term trend of commodities and uncertainties dictated by factors such as earning base and valuation multiple [91]; the technical factors that encompass diverse macroeconomic factors [93]; market sentiment factors—animal spirits of investors and environmental context [94]; and market anomalies [95]. The correlation coefficients and variances of the IMFs and residue show that the high frequency which measures the short-term dynamics of the data (IMFs 1–4) is dominating (this is shown in Table 3).Table5 shows the descriptive statistics of the reconstructed series and the residue obtained through the CEEMDAN. The Pearson correlation coefficient, Kendall tau-b, and variance analysis are presented for all the 25 variables for each of the multi-frequencies. The descriptive measures for the individual IMFs (see Tables 3 and 4) corroborate with those of Table 5. Analysis from Table 5 indicates that the high frequency is dominant in all cases. This contradicts the outcome of Adam et al. [71] who found the residue as the dominating mode. The variance of each multi-frequency as a percentage of the original series and the sum of the entire IMFs and residue extracted with the CEEMDAN corroborate with each other. This is because the CEEMDAN technique solves the residual noise in the reconstructed signals within the EEMD by appending the noise to the residual of prior iteration instead of the original signal [77]. The Pearson correlation and Kendall tau-b coefficients among the commodities as well uncertainty indices depict similar behaviour at each multi-frequency, but the strength and significance of each multi-frequency vary among themselves. In this case, most markets demonstrate high levels of efficiency among themselves at a specific frequency. However, as the markets evolve—due to events and structural changes, and adapt—due to varying levels of markets at diverse investment horizons or intrinsic times, they become heterogeneous and adaptive. This is partly due to rational, albeit irrational investors’ relentless search for competing rewards and risks to satisfy their portfolio goals.Table 5
Descriptive statistics of the reconstructed series and the residue obtained through CEEMDAN (HFQ, MFQ, LFQ, and RESID denote high frequency, medium frequency, low frequency, and residue, respectively;∗, ∗∗, and ∗∗∗ indicate significance at 10%, 5%, and 1% levels, respectively).
Commodities and uncertaintiesPearson correlation coefficientKendall tau-bVariance as % of observedVariance as % of the sum of all IMFs and residueHFRQMFRQLFRQRESIDHFRQMFRQLFRQRESIDHFRQ (%)MFRQ (%)LFRQ (%)RESID (%)HFRQ (%)MFRQ (%)LFRQ (%)RESID (%)Acommodity0.94∗∗∗0.23∗∗∗0.20∗∗∗0.16∗∗∗0.75∗∗∗0.11∗∗∗0.13∗∗∗0.09∗∗∗88.686.091.752.9088.686.091.752.90Aenergy0.89∗∗∗0.39∗∗∗0.16∗∗∗0.23∗∗∗0.70∗∗∗0.10∗∗∗−0.020.06∗66.8112.425.0617.5766.8112.425.0617.57Ametals0.95∗∗∗0.26∗∗∗0.16∗∗∗0.080.80∗∗∗0.14∗∗∗0.11∗∗∗0.05∗92.004.701.710.6092.004.701.710.60Imetals0.88∗∗∗0.36∗∗∗0.12∗∗0.26∗∗∗0.74∗∗∗0.12∗∗∗−0.070.12∗∗∗70.0010.874.1521.447010.874.1521.44Brent0.87∗∗∗0.43∗∗∗0.17∗∗∗0.19∗∗∗0.65∗∗∗0.10∗∗∗0.10∗∗∗−0.0863.4513.849.1822.5263.4513.849.1822.52Gasoline0.88∗∗∗0.42∗∗∗0.19∗∗∗0.19∗∗∗0.67∗∗∗0.15∗∗∗0.04−0.0564.0013.348.9720.6964.0013.348.9720.69Htoil0.88∗∗0.43∗∗∗0.19∗∗∗0.20∗∗∗0.69∗∗∗0.07∗∗0.07∗∗−0.0863.9212.917.5119.1663.9212.917.5119.16Ngas0.97∗∗∗0.18∗∗∗0.050.070.82∗∗∗0.08∗∗0.030.0595.625.421.890.8295.625.421.890.82Petroleum0.95∗∗∗0.13∗∗∗0.17∗∗∗0.12∗∗0.74∗∗∗0.07∗∗0.16∗∗∗0.09∗∗92.397.271.963.6692.397.271.963.66Cocoa0.88∗∗∗0.42∗∗∗0.20∗∗∗0.220.72∗∗∗0.20∗∗∗−0.020.0464.1614.665.9516.0164.1614.665.9516.01Coffee0.88∗∗∗0.41∗∗∗0.17∗∗∗0.23∗∗∗0.59∗∗∗0.10∗∗∗−0.010.0464.0511.628.3022.1064.0511.628.3022.10Corn0.88∗∗∗0.42∗∗∗0.18∗∗∗0.24∗∗∗0.58∗∗∗0.14∗∗∗−0.040.07∗∗63.1113.136.2818.6363.1113.136.2818.63Cotton0.89∗∗∗0.40∗∗∗0.19∗∗∗0.23∗∗∗0.63∗∗∗0.10∗∗∗−0.060.07∗∗64.5212.166.8717.9164.5212.166.8717.91Soybeans0.89∗∗∗0.41∗∗∗0.21∗∗∗0.23∗∗∗0.55∗∗∗0.03−0.040.09∗∗63.7212.576.5416.5663.7212.576.5416.56Wheat0.89∗∗∗0.42∗∗∗0.11∗∗0.22∗∗∗0.62∗∗∗0.12∗∗∗−0.050.06∗63.2711.1011.5927.1563.2711.1011.5927.15Gold0.88∗∗∗0.40∗∗∗0.18∗∗∗0.21∗∗∗0.74∗∗∗0.050.06∗−0.0465.1314.866.3518.7365.1314.866.3518.73Lead0.96∗∗∗0.19∗∗∗0.09∗0.060.81∗∗∗0.11∗∗∗0.08∗∗0.0396.214.831.381.2696.214.831.381.26Nickel0.95∗∗∗0.23∗∗∗0.12∗∗0.13∗∗0.80∗∗∗0.11∗∗∗0.06∗0.07∗∗93.774.690.531.8193.774.690.531.81Palladium0.98∗∗∗0.27∗∗∗0.08∗0.020.82∗∗∗0.17∗∗∗0.08∗∗0.0393.342.580.680.0693.342.580.680.06Zinc0.87∗∗∗0.43∗∗∗0.25∗∗∗0.19∗∗∗0.73∗∗∗0.13∗∗∗0.10∗∗∗−0.0764.6113.577.0415.0964.6113.577.0415.09EPU0.98∗∗∗0.10∗∗0.040.060.88∗∗∗0.02−0.01−0.0198.952.080.493.3798.952.080.493.37Gvolatility0.89∗∗∗0.41∗∗∗0.17∗∗∗0.21∗∗∗0.74∗∗∗0.020.06∗∗−0.0363.2613.118.9420.1363.2613.118.9420.13OVX0.89∗∗∗0.35∗∗∗0.15∗∗∗0.23∗∗∗0.74∗∗∗0.11∗∗∗−0.010.08∗∗73.5212.573.6314.9673.5212.573.6314.96VCRIX0.87∗∗∗0.40∗∗∗0.20∗∗∗0.24∗∗∗0.41∗∗∗0.13∗∗∗0.030.0968.1815.822.2714.6168.1815.822.2714.61VIX0.95∗∗∗0.34∗∗∗0.18∗∗∗0.030.81∗∗∗0.14∗∗∗0.04−0.0587.158.150.810.7487.158.150.810.74


## 3.2. Multi-Frequency Entropy

We present an analysis for 20 commodities and 5 uncertainty returns through the multi-frequency entropy approach (the transfer entropies for Figures2–6 are provided in Table 4) at the 95% confidence bounds. The multi-frequencies indicate the importance of multi-scales in financial time series. The decomposed returns series based on CEEMDAN are further reconstructed into high, medium, and low frequencies, representing short, medium, and long terms, respectively. In addition, we present the residue which denotes the non-oscillating drift of the data, which is not influenced by short-to-medium term fluctuations; however, it is influenced by the structural variations in the data generation process.Figure 2
Multi-frequency information transfer between commodities and EPU. (a) Renyi’s effective transfer entropy between global commodities and EPU-(HFQ). (b) Renyi’s effective transfer entropy between global commodities and EPU-(MFQ). (c) Renyi’s effective transfer entropy between global commodities and EPU-(LFQ). (d) Renyi’s effective transfer entropy between global commodities and EPU-(residue).
(a)(b)(c)(d)Figure 3
Multi-frequency information transfer between commodities and Gvolatility. (a) Renyi’s effective transfer entropy between global commodities and Gvolatility-(HFQ). (b) Renyi’s effective transfer entropy between global commodities and Gvolatility-(MFQ). (c) Renyi’s effective transfer entropy between global commodities and Gvolatility-(LFQ). (d) Renyi’s effective transfer entropy between global commodities and Gvolatility-(residue).
(a)(b)(c)(d)Figure 4
Multi-frequency information transfer between commodities and OVX. (a) Renyi’s effective transfer entropy between global commodities and OVX-(HFQ). (b) Renyi’s effective transfer entropy between global commodities and OVX-(MFQ). (c) Renyi’s effective transfer entropy between global commodities and OVX-(LFQ). (d) Renyi’s effective transfer entropy between global commodities and OVX-(residue).
(a)(b)(c)(d)Figure 5
Multi-frequency information transfer between commodities and VCRIX. (a) Renyi’s effective transfer entropy between global commodities and VCRIX-(HFQ). (b) Renyi’s effective transfer entropy between global commodities and VCRIX-(MFQ). (c) Renyi’s effective transfer entropy between global commodities and VCRIX-(LFQ). (d) Renyi’s effective transfer entropy between global commodities and VCRIX-(residue).
(a)(b)(c)(d)Figure 6
Multi-frequency information transfer between commodities and VIX. (a) Renyi’s effective transfer entropy between global commodities and VIX-(HFQ). (b) Renyi’s effective transfer entropy between global commodities and VIX-(MFQ). (c) Renyi’s effective transfer entropy between global commodities and VIX-(LFQ). (d) Renyi’s effective transfer entropy between global commodities and VIX-(residue).
(a)(b)(c)(d)The presence of a negative ETE implies that awareness of one variable suggests a higher risk coverage for the other variable, whereas a positive ETE designates that the knowledge of one variable reduces the risk of the other variable [96]. The knowledge in the tails is assigned a high weight for low values of q, resulting in a significant effective transfer entropy result in the current situation. For this reason, we set q from the Rényi effective transfer entropy to 0.3 to offer more weights to the tails, which bears direct implications. The ETE decreases and even becomes negative as the weight is reduced. Since transfer entropy is a non-parametric estimate and has a higher likelihood of determining statistical interdependence between time series, we present the discussion between global commodities and uncertainties following the information flow theory, financial instability, modern portfolio theory, theory of interdependencies, the CMH, etc. and put our DVMCES phenomenon into perspective.


### 3.2.1. Information Flow between Commodities and Economic Policy Uncertainty

Figure2 provides a mixture of negative and positive information flow between global commodities and EPU from high, medium, and low frequencies. This establishes the unstable nature of commodities to external uncertainty shocks [57]. However, we notice more negative flow of information from EPU to the global commodities. This explains the reason why EPU has an adverse impact on most financial markets [52, 53, 70, 97].Specifically, in the high frequency, cotton, petroleum, gasoline, Imetals, Aenergy, nickel, and Ngas transmit positive shocks to EPU, and the reverse is true for flows from EPU to these commodities. We notice more negative information flows for most of the variables after the high frequency. This implies that awareness of EPU suggests a higher risk coverage for global commodities. Accordingly, global commodities are more vulnerable to external shocks, especially in the long term, where most financial markets are experiencing market rebound [90]. Investors would therefore have to rebalance their portfolio to minimise their risk due to changing market conditions.


### 3.2.2. Information Flow between Commodities and Global Volatility Index

Figure3 provides a mixture of negative and positive information flow between global commodities and Gvolatility from high, medium, and low frequencies. However, we notice more negative flow of information from Gvolatility to the global commodities. The negative influence of Gvolatility on global commodities highlights that fluctuations in market volatilities influence investors’ portfolio choices either by altering the trade-off between risk and return or their predictions of future market performance. As a result, investors may capitalise on this by hedging against market volatility since rising volatility does not incentivize investment opportunities [38]. This is necessary because phases of high volatility tend to correspond to drawdowns in markets [39].Specifically, in the high frequency, almost half of the global commodities transmit positive shocks to Gvolatility, but flow information flows from Gvolatility to most of the commodities are negative. In the short term, zinc, Aenergy and Imetals transmit positive flows to Gvolatility index which is not pleasant for diversification potentials. Investors can hedge against volatilities within lead and Brent using the Gvolatility index at high frequency. Generally, we find more negative information flows between the variables from the high to low frequency and the residue. This implies that awareness of any of the global commodities and Gvolatility suggests a higher risk coverage for each other. In this regard, global commodities, except gasoline and gold, are vulnerable to Gvolatility, especially in the long term, where most financial markets are experiencing market rebound [90]. Investors would therefore have to rebalance their portfolio or hedge their portfolio risks using the Gvolatility index to minimise their overall investment risks.


### 3.2.3. Information Flow between Commodities and Volatilities in Crude Oil

From Figure4, we find a bi-directional causality of both negative and positive information flow between global commodities and investor fear and expectations (OVX) mostly in the high frequency. However, we notice more negative flows of information from the OVX to the global commodities in the medium and low frequencies and residue. The negative influence of OVX on global commodities shows that fluctuations in market volatilities influence investors’ portfolio choices either by altering the trade-off between risk and return or their predictions of future market performance. As a result, investors may benefit by hedging against market volatility since rising volatility does not incentivize investment opportunities [38].The outcome from the study implies that awareness of any of the global commodities and OVX suggests a higher risk coverage for each other. As a result, global commodities are susceptible to shocks from the OVX, especially in the medium and long term, where most financial markets are experiencing market rebound [90]. Investors would have to rebalance their portfolio or hedge their portfolio risks using the OVX index to minimise their overall investment risks. This makes global commodities more preferred to receiving safe haven benefits from the OVX futures [98] during market stress.


### 3.2.4. Information Flow between Commodities and Cryptocurrency Volatility Index

We find a mixture of negative and positive information flow with between global commodities and VCRIX from high, medium, and low frequencies from Figure5. However, we notice more positive flows from the Cryptocurrency Volatility Index to global commodities in the short term but more negative flows of information from the medium and low frequencies and the residue. The positive flows from the VCRIX are indicative of plunging diversification benefits in the short term which is at the early part of COVID-19 pandemic. This shows that the knowledge of VCRIX minimises risk transmission to global commodities in the short term.On the other hand, the negative influence of VCRIX on global commodities highlights that fluctuations in market volatilities influence investors’ portfolio choices either by altering the trade-off between risk and return or their predictions of future market performance. Investors who wish to minimise their portfolio risks can employ VCRIX as a hedged instrument as shown from the medium and low frequencies and residue estimates. This supports the assertion of Conlon et al. [98] that investors can hedge against volatility in financial market using cryptocurrencies. Accordingly, volatility in global commodities occasioned by VCRIX would require investors to reconstruct their portfolio to include VCRIX. Accordingly, the finding provided by Yarovaya et al. [19] that speculative bubbles in cryptocurrency marketplaces might spread contagion and compromise financial stability comes to bear. Specifically, for information flows from commodities, investors may less likely shelter their portfolios against VCRIX for commodities such as cotton, gold, Ngas (at HFQ); soybean, coffee, Aenergy, zinc, gasoline, corn, cotton, Htoil, Brent, and cocoa (at MFQ); and Htoil, Aenergy, and cotton (at LFQ).


### 3.2.5. Information Flow between Commodities and Investor Fear and Expectations

Figure6 illustrates a combination of negative and positive information flow between global commodities and investor fear and expectations (VIX) from high, medium, and low frequencies. However, we notice more negative flow of information from the VIX to the global commodities. The negative influence of VIX on global commodities documents that fluctuations in market volatilities influence investors’ portfolio choices either by altering the trade-off between risk and return or their predictions of future market performance. As a result, investors may capitalise on this by hedging against market volatility since rising volatility does not incentivize investment opportunities [38].Generally, we find more negative information flows between the variables from the high to low frequency and the residue. This implies that awareness of any of the global commodities and VIX suggests a higher risk coverage for each other. In this regard, global commodities are vulnerable to VIX, especially in the long term, where most financial markets are experiencing market rebound [90]. Investors would therefore have to reconstruct their portfolio or hedge their portfolio risks using the VIX index to minimise their overall investment risks. This makes global commodities more preferred to receiving safe haven benefits from the VIX futures [99] during market stress.


## 3.2.1. Information Flow between Commodities and Economic Policy Uncertainty

Figure2 provides a mixture of negative and positive information flow between global commodities and EPU from high, medium, and low frequencies. This establishes the unstable nature of commodities to external uncertainty shocks [57]. However, we notice more negative flow of information from EPU to the global commodities. This explains the reason why EPU has an adverse impact on most financial markets [52, 53, 70, 97].Specifically, in the high frequency, cotton, petroleum, gasoline, Imetals, Aenergy, nickel, and Ngas transmit positive shocks to EPU, and the reverse is true for flows from EPU to these commodities. We notice more negative information flows for most of the variables after the high frequency. This implies that awareness of EPU suggests a higher risk coverage for global commodities. Accordingly, global commodities are more vulnerable to external shocks, especially in the long term, where most financial markets are experiencing market rebound [90]. Investors would therefore have to rebalance their portfolio to minimise their risk due to changing market conditions.


## 3.2.2. Information Flow between Commodities and Global Volatility Index

Figure3 provides a mixture of negative and positive information flow between global commodities and Gvolatility from high, medium, and low frequencies. However, we notice more negative flow of information from Gvolatility to the global commodities. The negative influence of Gvolatility on global commodities highlights that fluctuations in market volatilities influence investors’ portfolio choices either by altering the trade-off between risk and return or their predictions of future market performance. As a result, investors may capitalise on this by hedging against market volatility since rising volatility does not incentivize investment opportunities [38]. This is necessary because phases of high volatility tend to correspond to drawdowns in markets [39].Specifically, in the high frequency, almost half of the global commodities transmit positive shocks to Gvolatility, but flow information flows from Gvolatility to most of the commodities are negative. In the short term, zinc, Aenergy and Imetals transmit positive flows to Gvolatility index which is not pleasant for diversification potentials. Investors can hedge against volatilities within lead and Brent using the Gvolatility index at high frequency. Generally, we find more negative information flows between the variables from the high to low frequency and the residue. This implies that awareness of any of the global commodities and Gvolatility suggests a higher risk coverage for each other. In this regard, global commodities, except gasoline and gold, are vulnerable to Gvolatility, especially in the long term, where most financial markets are experiencing market rebound [90]. Investors would therefore have to rebalance their portfolio or hedge their portfolio risks using the Gvolatility index to minimise their overall investment risks.


## 3.2.3. Information Flow between Commodities and Volatilities in Crude Oil

From Figure4, we find a bi-directional causality of both negative and positive information flow between global commodities and investor fear and expectations (OVX) mostly in the high frequency. However, we notice more negative flows of information from the OVX to the global commodities in the medium and low frequencies and residue. The negative influence of OVX on global commodities shows that fluctuations in market volatilities influence investors’ portfolio choices either by altering the trade-off between risk and return or their predictions of future market performance. As a result, investors may benefit by hedging against market volatility since rising volatility does not incentivize investment opportunities [38].The outcome from the study implies that awareness of any of the global commodities and OVX suggests a higher risk coverage for each other. As a result, global commodities are susceptible to shocks from the OVX, especially in the medium and long term, where most financial markets are experiencing market rebound [90]. Investors would have to rebalance their portfolio or hedge their portfolio risks using the OVX index to minimise their overall investment risks. This makes global commodities more preferred to receiving safe haven benefits from the OVX futures [98] during market stress.


## 3.2.4. Information Flow between Commodities and Cryptocurrency Volatility Index

We find a mixture of negative and positive information flow with between global commodities and VCRIX from high, medium, and low frequencies from Figure5. However, we notice more positive flows from the Cryptocurrency Volatility Index to global commodities in the short term but more negative flows of information from the medium and low frequencies and the residue. The positive flows from the VCRIX are indicative of plunging diversification benefits in the short term which is at the early part of COVID-19 pandemic. This shows that the knowledge of VCRIX minimises risk transmission to global commodities in the short term.On the other hand, the negative influence of VCRIX on global commodities highlights that fluctuations in market volatilities influence investors’ portfolio choices either by altering the trade-off between risk and return or their predictions of future market performance. Investors who wish to minimise their portfolio risks can employ VCRIX as a hedged instrument as shown from the medium and low frequencies and residue estimates. This supports the assertion of Conlon et al. [98] that investors can hedge against volatility in financial market using cryptocurrencies. Accordingly, volatility in global commodities occasioned by VCRIX would require investors to reconstruct their portfolio to include VCRIX. Accordingly, the finding provided by Yarovaya et al. [19] that speculative bubbles in cryptocurrency marketplaces might spread contagion and compromise financial stability comes to bear. Specifically, for information flows from commodities, investors may less likely shelter their portfolios against VCRIX for commodities such as cotton, gold, Ngas (at HFQ); soybean, coffee, Aenergy, zinc, gasoline, corn, cotton, Htoil, Brent, and cocoa (at MFQ); and Htoil, Aenergy, and cotton (at LFQ).


## 3.2.5. Information Flow between Commodities and Investor Fear and Expectations

Figure6 illustrates a combination of negative and positive information flow between global commodities and investor fear and expectations (VIX) from high, medium, and low frequencies. However, we notice more negative flow of information from the VIX to the global commodities. The negative influence of VIX on global commodities documents that fluctuations in market volatilities influence investors’ portfolio choices either by altering the trade-off between risk and return or their predictions of future market performance. As a result, investors may capitalise on this by hedging against market volatility since rising volatility does not incentivize investment opportunities [38].Generally, we find more negative information flows between the variables from the high to low frequency and the residue. This implies that awareness of any of the global commodities and VIX suggests a higher risk coverage for each other. In this regard, global commodities are vulnerable to VIX, especially in the long term, where most financial markets are experiencing market rebound [90]. Investors would therefore have to reconstruct their portfolio or hedge their portfolio risks using the VIX index to minimise their overall investment risks. This makes global commodities more preferred to receiving safe haven benefits from the VIX futures [99] during market stress.


## 4. Theoretical and Practical Underpinnings

Findings from the study have indicated that the dynamics of information flows between commodities and uncertainties are heterogeneous and adaptive. We notice the diverse behaviours of commodity and uncertainty response to information especially in the high and medium frequencies. Notwithstanding, from the low frequency to the residue, there are similar dynamics (as obtained in Table5 for Pearson correlation coefficient and Kendall tau-b) of information flows for almost all the commodities mostly depicting negative flows. The negative information flows intensify tracing from short to long terms for all commodities. It can be inferred that in the long term, there are high tendencies of convergence within the commodity markets. Accordingly, there could be high integration among these commodities in light of uncertainties which may require further analysis through the interdependency techniques during the COVID-19 pandemic.Therefore, the markets evolve–due to events and structural changes, adapt–and market efficiency varies in degree at different times to echo the AMH [32]. This is as a result of the behavioural intensions of investors which is not the same across time, with fluctuating economic conditions to contradict the EMH. We advocate that the asymmetric and time-based varying behaviours of investors in line with the HMH [31] and the AMH [32] make the commodity markets and uncertainties inefficient. The more negative information flows at most frequencies reveal high volatility which may influence investors’ portfolio options by either changing the trade-off between risk and return or their prediction of future markets performance. Thus, the intensity of information flows between commodities and uncertainties exacerbated by rational, albeit irrational investors’ relentless search for competing rewards and risks to satisfy the portfolio goals establish the CMH as advanced by Owusu Junior et al. [28].As a result, the intensity of information flows between markets (similar and dissimilar) may lead to high uncertainties to which individual financial markets are susceptible. It is advisable that investors of commodities hedge against the persistent shocks using most of these uncertainty indices since high volatility tends to correspond to drawdowns in markets which may minimise investors’ confidence [39]. It is also obvious that information flows are causal flows from the mutual information shared by these variables during the COVID-19 pandemic by observing each other [40]. Our findings divulge that the competitiveness of markets intensifies the flow of information among them and the stable and booming markets drive blindness for increasing risks and eventually result in asymmetric volatility information flows between markets. The delayed effect of price response to information is eminent from this study due to market friction [43]. This is because information imperfections potentially hinder timely price discovery and are associated with delayed price adjustment to information [44, 45].We, therefore, provide that market volatilities act as effective hedges for global commodities and clearly act as balancing assets rather than substitutes [3–5] in the long-term dynamics of the COVID-19 pandemic. Investors who delayed in investing within financial markets, possibly commodities and market volatilities, are likely to minimise their portfolio risks to concur the assertion of Hou and Moskowitz [43]. As a result, the relentlessness of market frictions affecting an asset contingent on the delay with which its price responds to information during stressed conditions as well as shocks from past markets dynamics offers diversification potentials in the long term as indicated by prior studies [46, 47]. The delayed effect of prices response to shocks through uncertainties on individual markets as a result of intensity of information flows between markets which amplifies market risks, and the impact of other external shocks place the empirical analysis in perspective. As expected, diversification potentials increased monotonically from the short to long terms (delay in market dynamics) in times of stressed conditions. Hence, the proposed delayed volatility of market competitiveness and external shocks (DVMCES) hypothesis is amplified to enable investors minimise their portfolio risks in the long term due to the dynamics of the markets. This partly accentuates the long-term memory of financial asset returns that prices do not respond immediately to information flows to induce persistent shocks in the volatility process. Accordingly, the self-similar response of each variable to negative information flow in the long term depicts that the markets have now become saturated. At this point, relevant information about the COVID-19 pandemic has reflected in ascertaining current prices of financial assets. This therefore renders the markets to experience more dropdowns.


## 5. Conclusion

We investigated the multi-frequency information flow between global commodities and uncertainties through the CEEMDAN-based entropy approach during COVID-19. The multi-frequencies are carefully extracted through cluster analysis to reconstruct the IMFs into high, medium, and low frequencies for IMFs 1–7, respectively. Consequently, we quantify the direction and strength of information transfer between global commodities and uncertainties at multi-frequencies. In this way, we explored the multi-scale information that might be ignored to provide a substantive information about the global commodities and uncertainties dynamics. Due to the non-linearity, non-stationarity, and asymmetric relationships of most financial time series, we adopt a log-likelihood ratio transfer entropy which quantifies information from a probability density function. We setq from the Rényi transfer entropy to 0.3 to account for tail events within the sampled time series. This indicates that it is extreme event rather than observation in the centre that comes to light when information flow is utilised [25, 28, 74].We found from the multi-frequency series reconstruction that the IMFs of almost all the variables displayed similar behaviour of dominant frequency in the high frequency as provided by the correlation coefficients and variances of the IMFs and residue. This is when the COVID-19 pandemic impact on financial markets became intense to contribute to the heterogeneous behaviour of market participants. The correlation coefficients and variances declined with increasing frequencies (short-long) and eventually achieved normalcy as indicated by the fundamental feature. However, both the global commodities and uncertainties are highly influenced by short-lived market fluctuations during the COVID-19 pandemic. This accounted for the more positive significant bi-directional information flows at the high frequency between the variables, which minimises diversification potentials relative to other frequencies. This supports that markets evolve–due to events and structural changes, adapt–and market efficiency varies in degree at different times to reiterate the AMH [32].Findings from the study reveal that information flow between global commodities and uncertainties is mostly negative, especially in the long term. This is suggestive of diversification benefits during the COVID-19 pandemic, despite the fact that awareness of one variable suggests a higher risk coverage for the other variable. Investors who wish to diversify in times of uncertainties may consider it profitable. However, we notice both positive and negative flows of information between the variables in the short and medium term representing the high and medium frequencies. This is the period COVID-19 attained its peak which distorted global economic activities [8, 24, 26–28]. This is to say, negative information flow increases from the short to long term which indicates that investors can rebalance their portfolio effectively to maximise their returns. The increasing negative information flow clearly suggests that the financial market volatilities and global commodities act as safe haven for each other [76]. Consequently, the declining impact of the COVID-19 pandemic on financial markets has maximised portfolio diversification. This brings to light the delayed effect of price response to shocks through uncertainties on individual markets as a result of intensity of information flows between markets which amplifies market risks, while considering the impact of other external shocks. This is true for almost all the information flows between the commodities and uncertainties.Generally, the findings present pertinent inferences for portfolio diversification, policy decisions, investing risk, and risk management schemes in global commodities. For the sake of asset allocation and risk management, the support is incumbent on the negative information flow between global commodities and market volatilities at various investment horizons. The differences in information flow at multi-frequencies during COVID-19 provide evidence of contagion [25]. We, therefore, provide that market volatilities act as effective hedges for global commodities and clearly act as balancing assets rather than substitutes [3–5] in the long-term dynamics of the COVID-19 pandemic. Investors who delayed in investing within financial markets, possibly commodities and market volatilities, are likely to minimise their portfolio risks. Accordingly, the delayed volatility of market competitiveness and external shocks (DVMCES) hypothesis is amplified to enable investors minimise their portfolio risks due to the dynamics of the markets.The study was limited to the use of five uncertainty indices; however, future studies can employ other types of uncertainties to assess their information flow with global commodities. Also, the patterns of information flow could be assessed prior to the COVID-19 pandemic to reveal its impact on the commodity-uncertainty dynamics. The current study did not consider the outcome of each IMF but its aggregated impact based on multi-frequencies. A detailed analysis can be conducted for each IMF for comparison. Further studies can consider portfolio analysis at time-frequencies using wavelet techniques [100–103] for other financial markets.



---

*Source: 6499876-2022-05-11.xml*