Time-Varying Market Efficiency: A Focus on Crude Oil and Commodity Dynamics

Abstract

This study investigated market efficiency across 20 major commodity assets, including crude oil, utilizing fractal analysis. Additionally, a rolling window approach was employed to capture the time-varying nature of efficiency in these markets. A Granger causality test was applied to assess the influence of crude oil on other commodities. Key findings revealed significant inefficiencies in RBOB(Reformulated Blendstock for Oxygenated Blending) Gasoline, Palladium, and Brent Crude Oil, largely driven by geopolitical risks that exacerbated supply–demand imbalances. By contrast, Copper, Kansas Wheat, and Soybeans exhibited greater efficiency because of their stable market dynamics. The COVID-19 pandemic underscored the time-varying nature of efficiency, with short-term volatility causing price fluctuations. Geopolitical events such as the Russia–Ukraine War exposed some commodities to shocks, while others remained resilient. Brent Crude Oil was a key driver of market inefficiency. Our findings align with Fractal Fractional (FF) concepts. The MF-DFA method revealed self-similarity in market prices, while inefficient markets exhibited long-memory effects, challenging the Efficient Market Hypothesis. Additionally, rolling window analysis captured evolving market efficiency, influenced by external shocks, reinforcing the relevance of fractal fractional models in financial analysis. Furthermore, these findings can help traders, policymakers, and researchers, by highlighting Brent Crude Oil as a key market indicator and emphasizing the need for risk management and regulatory measures.

1. Introduction

Many countries have grown and developed through economic activities shaped by their unique geographical and topographical characteristics. Weather, climate, and the surrounding environment influence the availability and value of goods, leading to variations in the relative abundance and scarcity of commodities across regions. Consequently, trade activities based on raw materials have been integral to global interactions for centuries, sometimes sparking wars over access to these resources. Commodity markets trade raw materials, including agricultural products (cocoa, coffee, cotton, sugar, and oats), precious metals (copper, gold, palladium, and platinum), and energy resources (Brent crude oil, natural gas, and heating oil).

Price fluctuations in commodity markets play a critical role in national economic performance by influencing the activities of production and investment companies. These changes directly impact the profitability of trade-oriented firms, which subsequently affects the pricing of financial instruments such as stocks, bonds, and currencies in broader financial markets. Notably, prices of commodities such as crude oil, copper, and corn are closely linked to financial assets [1,2]. However, external shocks, such as the COVID-19 pandemic, and geopolitical events, such as the Russia–Ukraine War, can alter these dynamics. In particular, the Russia–Ukraine War disrupted energy and agricultural supplies, causing food price volatility and financial instability due to the reliance on Russian resources. It also affected the metal and energy markets, highlighting commodity market interconnections. Crude oil remains a key commodity, as seen during the COVID-19 pandemic, when global crude oil futures briefly turned negative, underscoring its critical role in global trade and economic stability.

A wide range of financial instruments are traded in financial markets, shaping market prices. Ref. [3] introduced the Efficient Market Hypothesis (EMH), which posits that the degree to which information is incorporated into price determination models defines market efficiency. According to the EMH, if a financial market is efficient, prices (and expected returns) adjust immediately to new information. Even if price changes occur because of new information, these effects are short-lived, as they are quickly reflected in market prices.

While numerous studies have sought to validate this model, significant challenges emerged, owing to the unrealistic assumptions underlying the EMH. One key assumption is that the price series follow a random walk, implying that market prices are temporally uncorrelated. This finding suggests that all available information about a financial asset has already been incorporated into its current price, making past information irrelevant for predicting future prices. However, empirical evidence contradicts this assumption, revealing long-term memory correlations in asset prices across various financial markets and challenging the notion of market efficiency.

This study investigates the efficiency of various assets in the commodity market, with a particular emphasis on crude oil. By adopting a time-varying perspective, we calculate and evaluate changes in asset efficiency over time. Furthermore, we examine the extent to which the efficiency of crude oil influences the efficiency of other commodities in the market. This study provides valuable insights into the evolving dynamics of commodity markets and their alignment with efficiency principles, thus contributing to a deeper understanding of the interrelationships among key market assets.

Numerous studies have examined the market efficiencies of different assets. These analyses explored various asset classes and applied a range of methodologies to assess the extent to which market prices reflect all available information. The various assets included the stock market [4,5,6,7], crude oil [8,9,10,11], commodities [12,13,14], and cryptocurrencies [15,16,17,18]. These findings contribute to a broader understanding of how markets operate efficiently, with implications for investors and policymakers. We briefly review previous studies on market efficiency in crude oil and commodities in Section 2.1 and Section 2.2, respectively. Furthermore, we analyze the effects of crude oil on other commodity assets in terms of market efficiency. This study is motivated by previous studies that revealed the significant influence of crude oil on the broader commodity market. Numerous studies have extensively examined the impact of crude oil on the commodity market, focusing on various aspects such as price dynamics, volatility patterns, correlation with other commodities, and spillover effects [19,20,21,22,23,24,25,26,27].

This study focuses on the commodity market for several reasons.

First, commodities in financial markets have strong hedging effectiveness, as they exhibit relatively low correlations with traditional financial assets such as stocks and bonds. This characteristic makes commodities highly beneficial for maximizing portfolio diversification and enhancing risk mitigation through portfolio allocation strategies. Commodities have proven to be effective hedges against inflation, a major concern in global financial markets, especially in the aftermath of the COVID-19 pandemic [28,29,30]. As commodity markets are driven by demands directly linked to real consumption, they offer valuable insights into long-term risk management and policy implications. Therefore, focusing on the market efficiency of commodity assets is of significant importance in both financial and economic contexts.

Second, understanding the efficiency of the major commodity markets is crucial, because of their significant role in global economic stability, inflation control, and portfolio diversification. However, recent global events such as the COVID-19 pandemic and geopolitical crises have introduced unprecedented levels of volatility and uncertainty, raising questions about the adaptive efficiency of these markets. While previous studies analyzed market fluctuations during crises, they often overlooked long-term structural shifts and behavioral responses. This study addresses this gap by using advanced econometric techniques to dynamically assess market efficiency, considering structural breaks and investor sentiment.

Finally, understanding market efficiency in the context of major commodities is critical for policymakers, institutional investors, and market participants, who rely on these assets for risk management and strategic planning. By identifying the inefficiencies and their underlying causes, this study seeks to provide actionable insights that can improve market stability and inform regulatory decisions.

Methodologically, we analyze the asset-level efficiency in commodity markets using a multifractal detrended fluctuation analysis (MF-DFA) approach. Long-range autocorrelation properties in financial series are often considered indicators of market inefficiency. We investigate the long-range autocorrelation and describe the fractal properties using MF-DFA, to determine the asset market’s inefficiency. MF-DFA has been widely used in research on the EMH and multifractality of financial assets, such as stock markets, foreign exchange markets, gold markets, and cryptocurrencies [31,32,33,34,35,36,37,38,39,40]. We provide a further review of studies analyzing financial markets using the MF-DFA method in Section 2.3.

We use MF-DFA to calculate market efficiency. Analyzing the efficiency of commodity assets using the MF-DFA methodology offers several advantages. First, MF-DFA plays a crucial role in examining one-dimensional financial time-series data, effectively capturing multifractal characteristics across various timescales. This allows a comprehensive analysis of market efficiency and long-term memory [41,42,43,44]. Second, compared to traditional single-fractal methods, such as range analysis (R/S), detrended fluctuation analysis (DFA), wavelet transform, and entropy-based approaches, MF-DFA provides a more reliable framework for handling nonstationary and nonlinear data. Minimizing biases and errors in correlation calculations enables a more accurate identification of long-term correlations and multifractal properties [45,46,47,48,49]. Consequently, these advantages establish MF-DFA as a suitable methodology for this study’s objective of analyzing the efficiency of commodity assets.

We also investigate the effect of crude oil on other commodity assets in terms of market efficiency. We employ the Granger causality test to analyze the relationship between the efficiency of crude oil and the efficiency of other commodity assets.

The fractal behavior of commodity assets is closely linked to their efficiency. In financial markets, fractal behavior refers to the self-similarity observed in asset price movements, which indicates that patterns can be defined and analyzed across various timescales. In other words, because asset prices exhibit fractal characteristics that recur over different periods, it is possible to predict trends and movements using technical charts or statistical analysis. This concept contrasts with the EMH, which asserts that new information is instantly reflected in market prices, making excess returns unattainable. The EMH assumes that asset prices follow a random walk process, meaning that they lack predictable patterns.

However, if market movements exhibit fractal behavior, then asset prices may demonstrate non-normal distributions and long-memory properties, leading to identifiable patterns and localized predictability. This finding suggests that investors can leverage technical or quantitative analyses to optimize their portfolios and identify favorable trading opportunities, thereby potentially achieving excess returns. Evaluating whether asset prices adhere to fractal behavior can provide insights into market inefficiencies that challenge the principles of the EMH. Predictable asset price patterns would indicate a deviation from randomness and open the possibility of generating excess returns.

This study makes several significant contributions to the literature.

• This study examines the efficiency of 20 commodity assets, providing a more comprehensive analysis based on a richer dataset than previous research. By assessing the market efficiency of major commodities, including crude oil, we offer valuable insights into the overall efficiency of the commodity market.
• To ensure a comprehensive evaluation, we employ both static analysis, which assesses efficiency over the entire period, and a rolling window approach, which captures dynamic fluctuations over time. This dual methodology provides a holistic perspective on commodity market efficiency, from both static and dynamic viewpoints. By analyzing the entire period for all 20 commodity assets, we identify the most and least efficient markets, while also examining variations in efficiency across individual assets, offering a detailed assessment of these changes.
• Furthermore, we analyze the influence of crude oil on the broader commodity market from an efficiency perspective, rather than the conventional focus on price or volatility. This novel approach offers market participants new insights into the interrelationships within the commodity market.

Furthermore, the implications of this study extend to both policymakers and market participants, offering actionable recommendations to enhance market transparency, refine regulatory frameworks, and develop asset-specific risk-mitigation strategies. By bridging theoretical concepts with empirical findings, this study provides practical insights applicable to portfolio management, hedging strategies, and regulatory oversight.

The structure of the paper is as follows. The next section provides a brief review of prior studies on market efficiency in the commodity market. Section 3 outlines the commodity market data and elaborates on the MF-DFA methodology and research framework. Section 4 presents the empirical findings. Finally, Section 5 discusses the results and offers our concluding remarks.

2. Literature Review

This study first examines efficiency studies related to crude oil, and then explores efficiency studies concerning other commodity assets. Finally, we address the impact of crude oil on other commodity assets.

2.1. Crude Oil Market Efficiency

Ref. [50] analyzed the weak-form efficiency of the UK Brent USA WTI(West Texas Intermediate) markets using data from 1982 to 2008, applying non-parametric and wild-bootstrap variance ratio tests. The results showed that the Brent market exhibited weak-form efficiency, whereas the WTI market appeared to be inefficient after 1994. This finding suggests that deregulation does not enhance return predictability in the WTI market. The inefficiency of the WTI market can be attributed to storage capacity and logistical issues.

Ref. [8] analyzed the efficiency of crude oil markets using lagged detrended DFA to investigate autocorrelations across different time scales. Data from 1986 to 2009 showed that the market did not follow a random walk, with significant deviations from efficiency owing to lagged autocorrelations. A strong mean reversion tendency was observed in both the short- and long-term horizons, emphasizing the need for appropriate modeling. To improve forecasting accuracy, incorporating delay effects and multiscale patterns into models is essential.

Ref. [9] evaluated the weak-form efficiency of the WTI and Brent crude oil markets using modified Shannon entropy and symbolic time series analysis. Their results showed time-varying efficiencies in both markets, with the Brent market being the most efficient. The efficiency changes differed between the two, affecting oil price forecasting, volatility management, and portfolio risk management. During periods of high efficiency, prices serve as accurate signals, whereas periods of low efficiency offer speculative opportunities for excess returns.

Ref. [10] analyzed the weak-form efficiency and structural breaks of the WTI and Brent crude oil markets from 1990 to 2012 using the Hurst exponent and Shannon entropy. The results indicated that Brent was more efficient than WTI, with the Hurst exponent effectively detecting extreme events such as financial crises and geopolitical shocks. Structural break tests highlighted the significant impact of events like OPEC(Organization of the Petroleum Exporting Countries)’s production adjustments on volatility and returns, offering insights for risk management and forecasting.

Ref. [51] examined the weak-form efficiency of the WTI crude oil futures market from 1983 to 2012 using nonparametric methods (DMA(Detrending moving average analysis)/DFA(Detrended fluctuation analysis)) and bootstrapping. While generally efficient, inefficiencies arose during external shocks such as the Gulf War, Iraq War, and the 1985 and 2008 oil price crashes. A moving window analysis showed fluctuating efficiency, with temporary inefficiencies following major events, suggesting that market efficiency dynamically adjusted to external disruptions.

Ref. [52] investigated the efficiency of major crude oil spot markets in Europe, the USA, the United Arab Emirates, and China using a time-varying Generalized Autoregressive Regression (1)—Threshold Generalized Autoregressive Conditional Heteroskedasticity (1,1) model. Weekly data indicated that markets were mostly efficient, while daily data revealed intermittent efficiency. Daqing and WTI were more stable, whereas Brent exhibited inefficiency, and Dubai had the lowest efficiency. The 2008 financial crisis significantly impacted market efficiency, with all markets responding more strongly to negative news. Additionally, strong co-movement and volatility spillover were observed between Dubai and Daqing.

Ref. [11] explored the nonlinear relationship between efficiency and multifractality in the WTI crude oil market. Cross-correlation and nonlinear Granger causality analyses confirmed that increased multifractality reduced efficiency, with inefficiency and multifractality interacting in a nonlinear manner. A positive correlation and nonlinear Granger causality were found between efficiency and multifractality, providing insights for oil price forecasting and portfolio management.

Ref. [53] analyzed the impact of COVID-19 on WTI market efficiency using singular value decomposition entropy. The market was generally efficient pre-pandemic, but significant inefficiencies emerged during the early months, especially at medium time scales. Trading volume analysis indicated some predictability, with increased opportunities for abnormal returns reducing market efficiency and increasing complexity.

Ref. [54] examined the efficiency of the Chinese crude oil futures market (CCOFM) and its relationship with market depth, liquidity, and noise trading using data from 2018 to 2023. CCOFM showed weak-form efficiency from June 2020 to February 2022, but efficiency declined after the Russia–Ukraine War. Market depth and information asymmetry improved efficiency, while speculation had a negative effect. The Trader Suitability System enhanced efficiency, and policies supporting market stability and international participation were recommended.

2.2. The Market Efficiency of Other Commodities

Ref. [12] analyzed the efficiency of agricultural commodity futures markets, evaluating both short- and long-term market efficiency for the live cattle, lean hog, corn, and soybean meal markets. Using the Johansen cointegration test, the results confirmed that all four markets were efficient and unbiased in the long term; however, inefficiencies and price biases were observed in the short term. Notably, the live cattle and corn markets exhibited time-varying risk premiums and inefficiencies, as evidenced by the GQARCH-M-ECM(Generalized Quadratic Autoregressive Conditional Heteroskedasticity-in-Mean Error Correction Mode) analysis, which accounted for the nonlinearity and dynamic structure of the conditional variance. In contrast, the soybean meal market was efficient and unbiased, even in the short term.

In [13], five estimation methods (wavelet-based estimator, GPH(Geweke and Porter-Hudak) semi-parametric method, least squares and absolute deviation regression based on periodic functions, and quasi-maximum likelihood method) were used to quantify the degree of fractional integration of the Dow Jones-AIG commodity futures index. The analysis revealed that absolute returns exhibited high persistence. However, anti-persistence was observed in the broad precious metal index, whereas persistence was found in natural gas, lean hogs, and corn, thus rejecting the weak-form efficient market hypothesis. The results indicated that market participants for certain commodity indices failed to price assets efficiently.

Ref. [55] analyzed the market efficiency of 25 commodity futures including metals, energy, soft commodities, grains, and other agricultural products. Using the efficiency index, the results showed that cotton, wheat, and coffee were efficient commodities, whereas live and feeder cattle were inefficient. Subsequently, a group-based efficiency analysis revealed that energy commodities were the most efficient, whereas other agricultural products were the most inefficient. Furthermore, a non-standard positive relationship between the fractal dimension and the Hurst index was observed, which the authors attributed to the unique characteristic of commodity futures reverting to their fundamental prices.

Ref. [56] analyzed the weak-form efficiency of gold, silver, and platinum in relation to the adaptive market hypothesis. Using the automatic Portmanteau and variance ratio tests, the study found that return predictability in these markets fluctuated over time due to political conditions. Gold and silver exhibited a downward trend in predictability, suggesting increasing market efficiency, with gold being the most efficient among them.

Ref. [14] investigated excess comovement in commodity futures returns and examined how the characteristics of commodities with high excess comovement values affected futures returns. Excess comovement was defined as the dependency between commodity prices after removing common macroeconomic and market factors. The results showed that cross-market open interest changes significantly influenced the energy and livestock sectors, while cross-market inventory changes had little effect on the energy and grain sectors.

Ref. [57] analyzed the weak-form efficiency of nine major agricultural commodity futures using high-frequency data before and after a market merger. The weak-form efficiency hypothesis was rejected for holding periods of 5, 10, and 15 min, indicating short-term inefficiencies. However, for 30-min and 60-min intervals, some commodity futures followed a random walk, suggesting mixed results. These findings indicate that market imperfections persisted in agricultural commodity futures, despite the merger.

Ref. [58] examined the impact of speculative activity on informational efficiency in 19 major commodity futures markets from 1992 to 2019. While financialization was associated with declining market quality, no systematic changes in efficiency were observed. Panel regression analysis revealed that speculative activity negatively affected informational efficiency, particularly in the energy and grain sectors. Long-short speculator activity played a significant role, whereas index investors had a relatively limited impact.

Ref. [59] analyzed the connectedness between cryptocurrencies and traditional financial assets in China using a time–frequency perspective. Unlike previous studies that focused on traditional assets such as gold and oil, this study highlighted the role of investor expectations in driving risk transmission. Cryptocurrencies exhibited strong short-term spillover effects, particularly during events like COVID-19, amplifying market inefficiencies and positioning them as key risk transmitters.

Ref. [60] examined how online word-of-mouth (eWOM) influences hotel booking prices to assess market efficiency. Key eWOM factors, including review depth, ratings, and positive word usage, were moderated by the number of reviews and helpfulness scores. Using regression analysis and ensemble methods (bagging and boosting), the study found that eWOM factors significantly predicted prices. Ensemble methods outperformed traditional approaches, suggesting their applicability in analyzing market efficiency across various commodities.

2.3. Financial Market Analysis Using the MF-DFA Method

In [45], econophysics and the Efficient Market Hypothesis (EMH) were combined to analyze the multifractal characteristics of Islamic and developed markets using MF-DFA on 22 indices. The results showed that developed markets are more efficient in the short term, whereas Islamic markets demonstrate strong efficiency during crises, with Malaysia, Indonesia, and Turkey performing well. The analysis highlights the significant role of market development in efficiency and suggests that advancing Islamic markets could positively affect economic growth and financial sector development. This study provided key insights into the link between market efficiency rankings and development, and offered valuable guidance to policymakers and investors.

In [61], the asymmetric MF-DFA (A-MF-DFA) method was applied to analyze EU carbon emission trading market data from 4 August 2005, to 31 December 2019. The results showed distinct multifractal characteristics between uptrends and downtrends, with informational efficiency improving across Phases I to III. In Phase III, inefficiency was higher during uptrends than during downtrends, and although the overall efficiency improved, it remained below the level required by the efficient market hypothesis. This study highlights the need for stricter regulations and system improvements to enhance efficiency and prevent manipulation of the EU ETS (European Union Emissions Trading Scheme), along with support for renewable and clean energy initiatives to better address climate change.

In [62], the A-MF-DFA method was used to analyze the multifractal characteristics and inefficiencies of China’s clean energy stock market, along with its cross-correlation with the crude oil market. The market was found to be inefficient, with significant asymmetries between the upward and downward trends. Market deficiency measure analysis revealed that long-term trends displayed greater asymmetry, while cross-correlations with the crude oil market indicated persistent long-term impacts. These findings underscore the importance of tailored hedging strategies and provide insights for improving investment efficiency and risk management.

Ref. [63] analyzed skewed multifractality in financial time series by detecting change-points and examining multifractal properties before and after them. Using MF-DFA, the study found increased multifractality in the USA, Japan, and the Eurozone during COVID-19, while China’s market stabilized. These results highlight regional differences in market structures, suggesting the need for further research on long-term multifractality variations.

Ref. [64] assessed market efficiency in four S&P Kensho Fintech indices using MF-DFA. Deviations from the Hurst exponent (0.5) indicated inefficiencies caused by extreme values and autocorrelation. External factors like COP26 regulations, transaction costs, and knowledge diffusion were also examined. The study provided insights into market efficiency and demonstrated the applicability of MF-DFA in financial analysis.

Ref. [65] applied MF-DFA to analyze the multifractal characteristics of Brent and WTI crude oil markets. Nonlinear correlation was identified as the primary source of multifractality. WTI exhibited greater persistence in a monofractal sense but had a narrower multifractal spectrum than Brent. The study highlighted the need for further research on how market participants’ expectations influence multifractality.

Ref. [66] investigated the multifractality and weak-form efficiency of GCC stock markets using MF-DFA. The results showed persistent multifractal characteristics, with stronger short-term inefficiencies. GCC markets were less efficient than their global, regional, and Islamic counterparts, with Kuwait being the least inefficient. These findings offer insights into market forecasting and policy reforms.

Ref. [67] examined market efficiency in seven Central and Eastern European (CEE) stock markets. The MF-DFA results revealed that none exhibited weak-form efficiency, with Bulgaria and the Czech Republic being the least efficient. These findings suggest that efficiency may improve with market development, highlighting the need for dynamic Hurst exponents and sliding window techniques in future research.

Ref. [68] studied the impact of fuel cell introduction on the Korean electricity market using MF-DFA. After fuel cells were introduced, market volatility decreased, long-term correlations weakened, and efficiency improved. The market deficiency measure dropped from 0.536 to 0.267, indicating greater stability. Future research should explore regional variations and long-term effects of fuel cell technology.

Ref. [69] proposed a decision-making system integrating fractal market analysis and volatility forecasting for Brent crude oil. Combining MF-DFA with GARCH (Generalized Autoregressive Conditional Heteroskedasticity), ARFIMA (Autoregressive Fractionally Integrated Moving Average), and neural networks (BPNN (Backpropagation Neural Network), CNN (Convolutional Neural Network), WNN (Wavelet Neural Network), LSTM (Long Short-Term Memory)), the system enhanced short-term predictive performance. Findings showed that small fluctuations exhibited stronger persistence than large ones, improving forecasting accuracy and aiding risk management and investment strategies.

Based on this review, numerous studies have examined the efficiency of commodity assets, including crude oil. The MF-DFA methodology has been widely applied to analyze financial markets and assess their efficiency. Building on these findings, this study explores the efficiency of multiple commodity assets, including crude oil, with a focus on their time-varying characteristics, using the MF-DFA method. We also investigate the influence of crude oil on the efficiency of other commodity assets, offering deeper insights into market efficiency dynamics.

3. Data Description and Method

3.1. Commodity Market Data

In this study, we selected 20 major commodities traded in the commodity market. These commodities include Cocoa, Coffee, Cotton, Sugar, Oats, Rough Rice, Soybeans, Kansas Wheat, Copper, Gold, Palladium, Platinum, Silver, Brent Crude Oil, Natural Gas, Heating Oil, RBOB Gasoline, Feeder Cattle, Lean Hogs, and Live Cattle. The data were sourced from Yahoo Finance (https://finance.yahoo.com, accessed on 1 July 2024).

The sample period was January 2018 to June 2024. Figure 1 graphically represents the returns on commodity assets during this period.

3.2. MF-DFA

Employing the MF-DFA technique, which characterizes multifractal attributes in financial time series, we can effectively evaluate and rank market efficiency. As demonstrated by [70], this approach involves five distinct steps.

Let $\{x_k,k=1,\cdots ,N\}$ be a time series, where N represents its length.

• Step 1: Compute the Profile

The first step involves calculating the profile $Y\left(i\right)$ for $i=1,2,\cdots ,N$:

$$Y\left(i\right)=\sum _{k=1}^i\left(x\left(k\right)-\overline{x}\right),$$

(1)

where

$$\overline{x}={\displaystyle \frac{1}{N}}\sum _{k=1}^{N}x\left(k\right).$$

(2)

• Step 2: Segment the Profile

Segment the profile ${\left\{Y\left(i\right)\right\}}_{i=1}^N$ into $N_s$ non-overlapping parts, each containing s data points. To cover the entire sample, repeat the segmentation from the end, resulting in $2N_s$ segments:

$$\begin{array}{cc}\hfill & {\left\{Y[(\nu -1)s+i]\right\}}_{i=1}^s,\nu =1,2,\dots ,N_s,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & {\left\{Y[N-(\nu -N_s)s+i]\right\}}_{i=1}^s,\nu =N_s+1,N_s+2,\dots ,2N_s.\hfill \end{array}$$

(4)

• Step 3: Calculate the Variance

For each of the $2N_s$ segments, determine the local trend using a least-squares polynomial of the order m and calculate the variance:

$$F^2(s,\nu )=\left\{\begin{array}{cc}{\displaystyle \frac{1}{s}}\sum _{i=1}^s{\left\{Y[(\nu -1)s+i]-{\widehat{Y}}_{\nu }^m\left(i\right)\right\}}^2,\hfill & \nu =1,2,\dots ,N_s,\hfill \\ {\displaystyle \frac{1}{s}}\sum _{i=1}^s{\left\{Y[N-(\nu -N_s)s+i]-{\widehat{Y}}_{\nu }^m\left(i\right)\right\}}^2,\hfill & \nu =N_s+1,\dots ,2N_s.\hfill \end{array}\right.$$

(5)

Here, ${\widehat{Y}}_{\nu }^m\left(i\right)$ is a fitting polynomial. Linear ($m=1$), quadratic ($m=2$), or cubic ($m=3$) polynomials are typically used [7,71,72]. However, linear polynomials ($m=1$) are often employed to avoid overfitting and to simplify the calculations [73,74].

• Step 4: Compute the Fluctuation Function

Take the average variances across all segments to obtain the q-th order fluctuation function:

$$F_q\left(s\right)=\left\{\begin{array}{cc}{\left[{\displaystyle \frac{1}{2N_s}}\sum _{\nu =1}^{2N_s}{\left(F^2(s,\nu )\right)}^{q/2}\right]}^{1/q},\hfill & q\ne 0,\hfill \\ exp\left[{\displaystyle \frac{1}{4N_s}}\sum _{\nu =1}^{2N_s}ln\left(F^2(s,\nu )\right)\right],\hfill & q=0.\hfill \end{array}\right.$$

(6)

• Step 5: Analyze the Scaling Properties

Examine the scaling properties of $F_q\left(s\right)$ by analyzing the log–log plots for various q values. In the presence of long-range power-law correlations, $F_q\left(s\right)$ scales as

$$F_q\left(s\right)\propto s^{h\left(q\right)},$$

(7)

where $h\left(q\right)$ is the generalized Hurst exponent. Taking logarithms yields

$$log\left(F_q\left(s\right)\right)=h\left(q\right)·log\left(s\right)+c,$$

(8)

where c is a constant.

If $h\left(q\right)$ is constant for all q, the series is monofractal; otherwise, it is multifractal. Notably, $h\left(2\right)$ corresponds to the Hurst exponent. When $h\left(2\right)=0.5$, the series follows a random walk, indicating weak market efficiency [75]. If $h\left(2\right)>0.5$, the series exhibits long-range dependence, whereas $h\left(2\right)<0.5$ indicates nonpersistent behavior.

The multifractal scaling exponents $\tau \left(q\right)$ are related to $h\left(q\right)$, as follows:

$$\tau \left(q\right)=qh\left(q\right)-1.$$

(9)

Using the Legendre transform, $\tau \left(q\right)$ is converted into a multifractal spectrum:

$$\begin{array}{cc}\hfill \alpha & ={\displaystyle \frac{d}{dq}}\tau \left(q\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill f\left(\alpha \right)& =\alpha q-\tau \left(q\right).\hfill \end{array}$$

(11)

The spectrum width $\Delta \alpha ={\alpha }_{\max }-{\alpha }_{\min }$ quantifies the degree of multifractality [76,77,78]. Larger $\Delta \alpha$ values indicate stronger multifractality and lower market efficiency [5,79,80].

4. Results

This section presents the empirical results. First, we examined the market efficiency measures for each commodity asset over the entire sample period. Next, we calculated the time-varying market efficiency measures. Finally, we analyzed the impact of crude oil market efficiency on other commodity assets.

4.1. Full Period

We calculated the market efficiency for each commodity asset from January 2018 to June 2024. To this end, we applied the MF-DFA method to measure the efficiency of each asset market. The market efficiency measures were based on the width of the multifractal spectrum ($\Delta \alpha$). A higher value of $\Delta \alpha$ indicated more pronounced multifractality within the data. Consequently, greater multifractality implied reduced efficiency in the corresponding market [47].

First, we present log−log plots illustrating the relationship between $F_q\left(s\right)$ and s across all the return series of commodity assets in Figure 2. These plots cover a range of q values, specifically from $-10$ to 10, in increments of $0.5$, using a fixed polynomial order of $m=1$. The order of the curves from the bottom to the top corresponds to increasing q values. The patterns observed in these plots indicate the presence of distinct scaling laws and their associated exponents, suggesting multifractal scaling behavior.

Additionally, Figure 3 presents the generalized Hurst exponents of the commodity asset returns. The trend reveals a decline in the generalized Hurst exponent of the return series as parameter q ranges from $-10$ to 10. This distinct pattern indicates the presence of multifractal characteristics in the return series of all the commodity assets. Notably, at $q=2$, the generalized Hurst exponent $h\left(q\right)$ coincides with the conventional Hurst exponent. Furthermore, none of the Hurst exponents reach a value of 0.5, which rules out any suggestion of random walk behavior.

Figure 4 illustrates the multifractal spectra of the returns on all commodity assets. These subfigures highlight the variations in the multifractal spectra across different assets, enabling a comparative analysis of their unique characteristics.

We calculated the $\Delta \alpha$ values for all commodity assets using the results in

Table 1. Multifractal parameters $\Delta \alpha$ for the return series of the commodity assets.

Asset	Brent Crude Oil	Cocoa	Coffee	Copper	Cotton
$\Delta \alpha$	0.7686	0.5229	0.5445	0.3358	0.4944
Asset	Feeder Cattle	Gold	Heating Oil	Kansas Wheat	Lean Hogs
$\Delta \alpha$	0.5724	0.5091	0.5824	0.4001	0.6863
Asset	Live Cattle	Natural Gas	Oats	Palladium	Platinum
$\Delta \alpha$	0.4908	0.5403	0.541	0.7999	0.6069
Asset	RBOB Gasoline	Rough Rice	Silver	Soybeans	Sugar
$\Delta \alpha$	0.8474	0.6604	0.6142	0.401	0.4841

. Additionally, Figure 5 depicts these results in descending order based on the $\Delta \alpha$ values.

According to the results in Figure 5, among the commodity assets, RBOB Gasoline, Palladium, and Brent Crude Oil exhibited relatively low market efficiency, whereas Soybeans, Kansas Wheat, and Copper demonstrated comparatively high market efficiency.

Based on the $\Delta \alpha$ values, we categorized the assets into inefficient, moderately efficient, and efficient markets. The asset with the highest $\Delta \alpha$ value was RBOB Gasoline ($\Delta \alpha$ = 0.8474), which was the most inefficient asset in the energy market. The complexity of gasoline increased owing to various factors such as crude oil prices, refining processes, and seasonal demand (e.g., increased transportation during summer). In particular, geopolitical risks and supply chain issues caused information asymmetry, showing the greatest volatility in the Fluctuation Function and Multifractal Spectrum. Palladium ($\Delta \alpha$ = 0.7999) was the second-most inefficient asset after RBOB Gasoline. They are primarily used as key materials in automotive catalysts, and their supply is concentrated in Russia and South Africa. This supply chain uncertainty significantly impacted prices. Brent Crude Oil ($\Delta \alpha$ = 0.7686), a benchmark asset in the global crude oil market, experienced market inefficiency owing to complex factors such as OPEC policies, geopolitical conflicts, and supply adjustments. Lean Hogs ($\Delta \alpha$ = 0.6863), a futures contract for pork, had price volatility, mainly due to breeding cycles and disease risks (e.g., African swine fever). Rough Rice ($\Delta \alpha$ = 0.6604), an agricultural commodity, was sensitive to climate change and supply–demand instability, and such supply risks caused inefficiency. Silver ($\Delta \alpha$ = 0.6142) and platinum ($\Delta \alpha$ = 0.6069), which are precious metals, had high price volatility due to the combination of investment demand and industrial use. In particular, the complexity of platinum increased owing to its demand in the automotive catalyst industry and concentrated supply chains. These assets had large $\Delta \alpha$ values, indicating that the markets did not operate efficiently because of information asymmetry and supply chain uncertainty.

Assets showing moderate efficiency included Heating Oil ($\Delta \alpha$ = 0.5824), Feeder Cattle ($\Delta \alpha$ = 0.5724), Coffee ($\Delta \alpha$ = 0.5445), Natural Gas ($\Delta \alpha$ = 0.5403), Oats ($\Delta \alpha$ = 0.5410), and Cocoa ($\Delta \alpha$ = 0.5229). The price fluctuations of Heating Oil, used as a heating fuel, are influenced by seasonal demand, but with a lower $\Delta \alpha$ value than RBOB Gasoline, which is a relatively stable market. Feeder Cattle, which are futures for calves intended for feeding, have prices determined by feed costs and slaughter demand, indicating a relatively stable market structure for livestock products. Coffee is sensitive to climate change and labor issues in major producing countries (Brazil and Vietnam), but because global demand is stable, it exhibits characteristics where efficiency and inefficiency coexist. Natural Gas is sensitive to seasonal heating demand and global energy supply changes, but within the energy market, it has a relatively low $\Delta \alpha$ value, indicating higher efficiency. Oats, a small-grain market, are sensitive to climate change and fluctuations in production volume; however, their small market size mitigates complexity. Cocoa is influenced by climate change and labor issues in West African-producing countries, and with a medium-level $\Delta \alpha$ value, it is relatively stable. These assets have smaller $\Delta \alpha$ values than assets in inefficient markets and were evaluated as markets in which stability and uncertainty coexist.

Assets classified as efficient markets had small $\Delta \alpha$ values and prices quickly reflected new information. The most efficient asset was Copper ($\Delta \alpha$ = 0.3358), which is a key indicator of global economic growth as an industrial metal. Copper has a stable supply and demand structure, and its f($\alpha$) spectrum width is narrow, indicating market efficiency. Kansas Wheat ($\Delta \alpha$ = 0.4001), a major grain, reflects stable supply chains and predictable demand within the United States, and was evaluated as an efficient market. Soybeans ($\Delta \alpha$ = 0.4010), a global agricultural commodity, have a stable supply and demand, and their low $\Delta \alpha$ value is efficient. Sugar ($\Delta \alpha$ = 0.4841) and Cotton ($\Delta \alpha$ = 0.4944), which are key raw materials in bioethanol production and the textile industry, respectively, exhibit efficient market characteristics through stable supply and demand structures. Live Cattle ($\Delta \alpha$ = 0.4908) and slaughtered cattle reflect supply chain stability and predictable demand, and are thus efficient. Gold ($\Delta \alpha$ = 0.5091), a globally safe asset, is an efficient market, owing to its high trading volume and transparent information. These assets have small $\Delta \alpha$-values and demonstrate efficient market characteristics through stable market structures and transparent information.

Ref. [81] demonstrated that RBOB gasoline, palladium, and Brent crude oil were highly sensitive to geopolitical risks (GPR). First, RBOB gasoline was particularly affected in regions heavily reliant on refined petroleum products. This is because political instability or conflict can disrupt supply chains, leading to immediate imbalances in supply and demand, which, in turn, significantly increase price volatility.

For palladium, the authors highlighted that political tensions in major producing countries can greatly disrupt supply chains, resulting in heightened price volatility. This was attributed to palladium’s critical role in industrial applications, such as automotive catalytic converters, making the market highly sensitive to supply shocks caused by geopolitical risks.

Brent crude oil is closely linked to geopolitical events in oil-producing countries. As a globally essential energy resource, geopolitical risks that impact production and exports can trigger immediate market reactions. Brent crude oil serves as a global benchmark for oil prices, causing geopolitical events to have a more pronounced effect than other commodities.

In addition, Ref. [82] confirmed that these three commodities are closely connected to geopolitical risks. The study revealed that geopolitical risks affected the agricultural, metal, and energy markets, with higher sensitivity observed in the energy and metal markets. Th study emphasized that the volatility of RBOB gasoline, palladium, and Brent crude oil is amplified through both economic and financial channels under the influence of geopolitical risks. Notably, commodities with a higher global market share, such as these three, are more significantly affected by geopolitical risks in terms of price volatility.

The Paraguayan soybean market demonstrated stable supply and demand trends, owing to technological advancements and policy support. The introduction of SBR-resistant cultivars mitigated production losses caused by soybean rust (SBR), reduced fungicide costs, and maintained consistent yields. Additionally, over half of Paraguay’s soybean production is exported, while the remainder is processed domestically. A balance between exports and domestic use contributes to market stability. Government initiatives such as infrastructure improvements and export support policies have helped sustain export volumes, establishing Paraguay as a reliable supplier in the global market. These factors indicate that Paraguay’s soybean supply and demand remain sustainable and stable [83].

In [84], while the study did not explicitly state that Kansas Wheat supply and demand are ‘‘stable’’, key market characteristics and data indirectly suggest this. The Kansas Wheat market benefits from the activities of "hedgers" and "speculators", who provide liquidity and price stability. Inventory levels play a critical role in assessing supply stability. Additionally, the market’s connection to global financial fundamentals (e.g., the value of USD) shows that, despite external influences, the supply and demand system tends to maintain an overall balance.

According to [85], China’s copper market maintained supply and demand stability through diversified imports from multiple countries and through stable export markets. In 2015, 80% of raw copper materials were imported from resource-rich nations, and exports primarily targeted economically stable countries. The recycling of copper scrap complemented the domestic supply, while new technologies and mining exploration ensured steady domestic production. The ‘‘Belt and Road Initiative’’ initiative is expected to further increase global demand for Chinese copper products, contributing to long-term market stability. These factors indicate that China’s copper supply and demand are managed in a stable and sustainable manner.

Because the full-period analysis did not reflect changes in the market efficiency of commodity assets over time, we investigated the dynamics of market efficiency using a rolling window analysis in the following section.

4.2. Dynamic Market Efficiency

This section examines the time-varying efficiency of each commodity market. The analysis was conducted by calculating $\Delta \alpha$ using the rolling window approach. Figure 6 illustrates the time series of $\Delta \alpha$ for each commodity market. The $\Delta \alpha$ values were estimated using a rolling-window approach, with 400 observations. Initially, we aimed to select a window length that would encompass approximately one year of data preceding the COVID-19 pandemic. To satisfy this criterion, the window length had to be less than 500 observations. However, excessively short window lengths can lead to instability in the calculated $\Delta \alpha$ values. Considering these factors, we examined window lengths exceeding 300 observations and ultimately determined that 400 observations was the most appropriate choice. The following presents the results from a time-varying perspective.

From March 2019 to September 2024, the COVID-19 pandemic, the Russia–Ukraine War, and extreme weather events served as the major drivers of $\Delta \alpha$ volatility. The pandemic caused significant logistical disruptions, industrial slowdowns, and decreased consumption, which likely led to a sharp increase in $\Delta \alpha$ for most commodities. In 2022, the war exacerbated supply chain instabilities in energy, metals, and agricultural commodities, likely driving $\Delta \alpha$ volatility to extreme levels. In addition, extreme weather events in 2021 and 2023 significantly amplified $\Delta \alpha$ in agricultural commodities. These changes in $\Delta \alpha$ appear to highlight shifts in market efficiency and illustrate the interplay between macroeconomic shocks and sensitivity to individual commodities.

First, we conducted an individual analysis of efficiency changes in commodity assets based on $\Delta \alpha$. The $\Delta \alpha$ for Brent Crude Oil (Figure 6a) began in 2019 under relatively stable global oil market conditions. However, in early 2020, the spread of COVID-19 triggered a sharp drop in oil demand, and $\Delta \alpha$ reached its peak, likely due to the OPEC+ production-cut agreement. In April 2020, WTI futures prices turned negative for the first time in history (approximately −37 dollars per barrel), leading to extreme $\Delta \alpha$ volatility. In 2021, as the pandemic subsided and the global economy recovered, the oil market stabilized; however, the Russia–Ukraine War in 2022 prompted Europe to suspend imports of Russian oil, likely disrupting the oil supply chain and causing another spike in $\Delta \alpha$. From 2023 onward, $\Delta \alpha$ gradually declined amid global economic recovery and stabilized oil supply. These fluctuations appeared to reflect structural issues in the oil supply chain and were likely associated with geopolitical factors, as illustrated by the historic WTI futures price collapse in March 2020.

The $\Delta \alpha$ for Cocoa (Figure 6b) surged during the early stages of the pandemic in 2020, likely due to lockdowns and port or logistics disruptions in major producing countries such as Côte d’Ivoire and Ghana, resulting in a sharp decline in cocoa exports. In 2021, $\Delta \alpha$ stabilized somewhat as the economy recovered, but in 2022, the Russia–Ukraine War disrupted the supply of potassium and nitrogen fertilizers, significantly increasing agricultural production costs and amplifying $\Delta \alpha$ volatility. Cocoa, which is highly sensitive to fertilizer and logistics costs, was likely affected by these international events.

Coffee $\Delta \alpha$ (Figure 6c) increased during the initial stage of the 2020 pandemic, primarily due to logistical disruptions and reduced consumption in Brazil and Vietnam. Brazil, the world’s largest coffee producer, faced severe drought and frost in 2021, likely cutting Arabica coffee production by about 25 percent and pushing $\Delta \alpha$ volatility to its peak. Between 2020 and 2022, logistical disruptions and container shortages also hindered coffee exports. From 2022 to 2023, rising energy and logistics costs continued to drive $\Delta \alpha$ volatility, but improved supply conditions in 2024 helped reduce this. Coffee tends to be sensitive to climatic changes in major producing countries and logistics hurdles, factors that appeared to have a notable impact on its $\Delta \alpha$.

The $\Delta \alpha$ for Copper (Figure 6d) expanded in 2020, partly due to reduced industrial activity and global manufacturing slowdowns caused by COVID-19. However, in 2021, the green energy transition and higher demand for electric vehicles drove a surge in copper demand, maintaining $\Delta \alpha$ volatility. International copper prices reached record highs in 2021. In 2022, the Russia–Ukraine War appeared to have severely disrupted the copper supply chain, while labor disputes and political instability in major producing countries such as Chile and Peru further intensified $\Delta \alpha$ volatility. As a key material in electric vehicle batteries and power grid expansion, copper’s $\Delta \alpha$ appears to be driven by the interplay between potential geopolitical events and rising demand.

The $\Delta \alpha$ for Cotton (Figure 6e) jumped sharply in 2020 as major apparel brands halted production and closed stores, reducing apparel consumption and cotton demand. In 2021, $\Delta \alpha$ moderated with the recovery of the textile industry. However, in 2022 to 2023, rising energy and transportation costs triggered another spike in $\Delta \alpha$ volatility. The Russia–Ukraine War likely raised logistics costs for textile-producing countries, while higher energy and transportation expenses amplified the dual effects of recovering demand and surging costs.

The $\Delta \alpha$ for Feeder Cattle (Figure 6f) increased notably in 2020, likely due to disruptions in livestock transportation and feed supply during the pandemic. In 2021, $\Delta \alpha$ stabilized as consumption recovered, but it rose again in 2023 as droughts and higher feed prices emerged. Between 2022 and 2023, rising feed grain costs, especially for corn and soybeans, likely affected cattle feed expenses and significantly increased Feeder Cattle $\Delta \alpha$ volatility.

As a safe haven asset, Gold experienced a significant rise in $\Delta \alpha$ (Figure 6g) in 2020, likely due to heightened demand during the pandemic, with prices exceeding 2000 dollars per ounce. $\Delta \alpha$ continued to expand in 2022 to 2023 in response to rising inflation and interest rate hikes. The U.S. Federal Reserve’s rate hikes (initiated in 2022) appear to have influenced investment demand, while geopolitical uncertainty arising from the Russia–Ukraine War and inflation likely contributed to sustaining $\Delta \alpha$ volatility.

The $\Delta \alpha$ for Heating Oil (Figure 6h) spiked during the early pandemic, as energy demand plummeted. In 2022, the Russia–Ukraine War likely disrupted Russian natural gas supplies, prompting Europe to turn to heating oil as a substitute and significantly boosting demand, which likely drove $\Delta \alpha$ volatility. The restructuring of energy supply chains and increased heating oil demand appear to have kept $\Delta \alpha$ elevated.

The $\Delta \alpha$ for Kansas Wheat (Figure 6i) climbed in the early stages of the pandemic in 2020, largely due to logistical disruptions. In 2022, the Russia–Ukraine War restricted global wheat supply, sending $\Delta \alpha$ volatility to its peak. Ukraine and Russia account for approximately 30 percent of global wheat exports, and the war severely disrupted supply. By 2023, $\Delta \alpha$ decreased as the grain market stabilized. Wheat may be especially sensitive to warfare and climate change, factors that appear to significantly influence $\Delta \alpha$.

Lean Hogs (Figure 6j) rose sharply in 2020 due to slaughterhouse closures and supply chain disruptions, with around 25 percent of major United States meat processing plants shutting down. In 2021, $\Delta \alpha$ stabilized as the economy recovered; however, feed price increases and reduced demand in 2023 renewed $\Delta \alpha$ volatility. The livestock market is sensitive to feed grain prices and production costs, which can lead to significant $\Delta \alpha$ volatility.

The Live Cattle $\Delta \alpha$ (Figure 6k) also rose sharply during the pandemic, driven by slaughterhouse closures and livestock transportation restrictions. In 2020, about 30 percent of slaughter facilities in the United States and Canada were closed, causing supply shortages. In 2023, elevated feed prices and higher livestock production costs due to extreme weather led to another increase in $\Delta \alpha$ volatility.

The $\Delta \alpha$ for Natural Gas (Figure 6l) jumped during the pandemic, partly due to a steep decline in energy demand. In 2022, the Russia–Ukraine War severely disrupted European natural gas supplies—Russia provided around 40 percent of exports—likely driving $\Delta \alpha$ to its peak. In 2023, global liquefied natural gas exports expanded, stabilizing the market and reducing $\Delta \alpha$ volatility.

Oat $\Delta \alpha$ (Figure 6m) increased in 2020 because of logistics breakdowns and export restrictions prompted by the pandemic. Shipping costs during this period may have played a key role in shaping $\Delta \alpha$. In 2021, severe droughts in North America reportedly slashed oat production by 40 percent compared to the previous year, further intensifying $\Delta \alpha$ volatility.

The $\Delta \alpha$ for Palladium (Figure 6n) rose during the pandemic, driven by reduced automotive production. In 2022, the Russia–Ukraine War disrupted Russian palladium supply, which constituted 40 percent of global output, pushing $\Delta \alpha$ to its highest level. By 2023, escalating demand for electric vehicle catalytic converters sustained $\Delta \alpha$ volatility. Supply disruptions and rising demand appeared to affect $\Delta \alpha$ considerably.

Platinum $\Delta \alpha$ (Figure 6o) increased during the pandemic, as automotive production slowed. In 2022, mine production disruptions and the war drove $\Delta \alpha$ to its peak. Like palladium, platinum is used in automotive catalytic converters, and the pandemic and war likely had a significant impact on its supply and demand. By 2023, demand for hydrogen fuel cell vehicles continued to support $\Delta \alpha$.

The $\Delta \alpha$ for RBOB Gasoline (Figure 6p) surged in 2020 as global traffic volumes declined, reducing gasoline demand by more than 30 percent. In 2022, the war further disrupted energy supply chains, likely intensifying $\Delta \alpha$ volatility. The restructuring of energy supplies and ongoing logistical complications were key factors.

The Rough Rice $\Delta \alpha$ (Figure 6q) climbed during the pandemic as major exporters like India and Thailand restricted exports, likely creating global supply disruptions. Combined, these two countries account for about 50 percent of worldwide rice exports. In 2022, fertilizer price spikes, exacerbated by the war, drove additional $\Delta \alpha$ volatility.

Silver’s $\Delta \alpha$ (Figure 6r) soared during the pandemic, probably due to greater demand for safe haven assets. Rising silver prices reflected this and may have contributed to sustaining $\Delta \alpha$ volatility. In 2023, $\Delta \alpha$ was influenced by competing industrial and investment demands.

The Soybeans $\Delta \alpha$ (Figure 6s) rose sharply during the pandemic, largely because of logistics disruptions. As a key feed and oilseed crop, soybeans are especially vulnerable to supply chain issues. In 2021, droughts in the United States reduced production, further amplifying $\Delta \alpha$. In 2022, the war disrupted global grain supply chains, sustaining $\Delta \alpha$ volatility.

The Sugar $\Delta \alpha$ (Figure 6t) increased during the pandemic due to production and logistical challenges. Brazil, a major sugar producer, faced logistical hurdles during the pandemic, reducing output. In 2022, soaring energy costs and the Russia–Ukraine War likely fueled additional $\Delta \alpha$ volatility.

We summarize the notable features of the time-varying efficiency characteristics observed in these individual commodities as follows: First, the COVID-19 pandemic led to substantial price volatility and supply chain disruptions in global commodity markets. As indicated by the figures above, its impact was particularly strong in the energy and metal markets. During the early phase of the pandemic, Brent Crude Oil, Gold, and Natural Gas exhibited severe price declines, driven by a sharp decrease in economic activity. Subsequently, these commodities underwent rapid price increases fueled by expectations of economic recovery and supply chain adjustments. This sequence illustrates both the immediate shocks to commodity prices that a pandemic can cause and how post-recovery demand shifts can greatly influence commodity markets.

In particular, Brent Crude Oil experienced extreme price volatility, resulting from an oil price war and suppressed global demand during the early pandemic, underscoring the profound effects that supply–demand imbalances can have on markets. Meanwhile, Gold demand grew steadily, as it was sought as a safe haven asset during the pandemic. Consequently, the pandemic clearly revealed not only the short-term disruptions that commodity markets can face but also the potential for more enduring adjustments and shifts in market sentiment.

RBOB Gasoline, Palladium, and Brent Crude Oil are highly sensitive to geopolitical risk (GPR), and this sensitivity tends to become particularly prominent during certain periods and wartime. Notably, the Russia–Ukraine War and the Israel–Palestine conflict seem to have significantly influenced these commodity markets.

First, RBOB Gasoline seems to have been notably influenced by geopolitical events in regions heavily dependent on refined petroleum products. The Russia–Ukraine War in 2022 appears to have severely disrupted the supply chain of oil and refined products in Europe, likely causing a significant surge in RBOB Gasoline price volatility. Additionally, the Israel–Palestine conflict in 2023 seems to have heightened uncertainties in the RBOB Gasoline market by threatening supply chain stability in the Middle East.

For Palladium, around 40% of the global supply comes from Russia, and the Russia–Ukraine War seems to have caused severe disruptions in Russian production and exports, likely driving a rapid increase in Palladium prices and impacting industrial demand and supply chain balance. Although the Israel–Palestine conflict may not have directly affected the Palladium market, geopolitical instability in the Middle East appears to have indirectly influenced global financial and commodity markets, thereby intensifying Palladium’s price volatility.

Brent Crude Oil has historically shown a strong connection to geopolitical risks, and its heightened sensitivity was evident during both the Russia–Ukraine War and the Israel–Palestine conflict. The Russia–Ukraine War in 2022 appears to have caused supply shortages in the global energy market, due to sanctions on Russian oil exports and pipeline closures, which likely led to sharp increases in Brent Crude Oil prices. In 2023, the Israel–Palestine conflict seems to have increased uncertainties regarding oil production and exports in the Middle East, further expanding the volatility of Brent Crude Oil.

Additionally, these three commodities often appear among the most susceptible to geopolitical risks in the energy and metal markets, showing a tendency for volatility to be amplified through economic and financial channels. The Russia–Ukraine War and the Israel–Palestine conflict appear to have driven structural changes in the energy and metals markets, intensifying price volatility through the vulnerabilities of global supply chains and the interconnectedness of financial markets.

Despite large-scale events such as the COVID-19 pandemic and geopolitical challenges, copper, Kansas wheat, and soybeans appear to have preserved relatively stable supply and demand structures and may fulfill important roles in global markets.

Copper seems to have upheld a steady supply chain and demand system even during the pandemic, potentially assisting the continued global economic recovery and industrial expansion. China, a major consumer of copper, is thought to have bolstered its supply stability by diversifying import origins. As of 2015, China reportedly obtained approximately eighty percent of its raw copper materials from multiple resource-rich nations, thereby reducing reliance on specific countries and mitigating supply risks. In addition, China appears to have reinforced this stability by continually expanding domestic production through copper scrap recycling, new technologies, and the exploration of new mines. Even with widespread supply chain disruptions during the COVID-19 pandemic, China’s proactive policies and industrial oversight may have enabled the copper market to remain relatively stable.

Kansas wheat appears to have retained a reliable supply system, despite the pandemic and geopolitical uncertainties, likely owing to its agriculture-based infrastructure and government-backed support. The Kansas wheat market is thought to ensure liquidity and price stability through the involvement of hedgers and speculators, while inventory management combined with an efficient agricultural production framework may bolster supply stability. Even amid the heightened global food demand during the pandemic, Kansas wheat reportedly experienced minimal supply chain disruptions, through efficient logistics and government export assistance. Furthermore, despite fluctuations in global financial indicators such as changes in the United States dollar’s value, the Kansas wheat market seems to have maintained a relative equilibrium, indicating resilience.

Soybeans are considered a key agricultural product, and appear to have upheld stable supply and demand during the pandemic, partly due to advancements in production technology and the consistency of international trade. Paraguay reportedly adopted SBR (soybean rust)-resistant varieties to minimize production losses and maintain stable export and domestic processing ratios, aiming for market stability. More than half of Paraguay’s soybean output is exported, while the remainder is processed domestically, presumably helping to balance foreign sales and local consumption. Despite disruptions across global logistics and trade networks during the pandemic, Paraguay is said to have fostered global confidence by preserving a steady supply chain through improved government infrastructure and export support initiatives.

Consequently, copper, Kansas wheat, and soybeans can be seen as commodities that manage to sustain stable supply and demand frameworks amid large-scale economic and political uncertainties, indicating their potential as important resources for maintaining global market equilibrium and advancing sustainability.

In this study, we used the geopolitical risk (GPR) index to more clearly illustrate the impact of geopolitical risk on market efficiency. The GPR index was introduced by [86] and has been widely used in previous research as a measure of geopolitical risk [87,88,89,90,91]. We provide a scatter plot of the geopolitical risk index and $\Delta \alpha$ series to visually depict the relationship between the two in Figure 7.

An analysis of the graphs for each commodity and $\Delta \alpha$ reveals a common pattern: as the GPR index increases, the distribution of $\Delta \alpha$ narrows and converges to a specific range. When GPR exceeds 300–400, this effect becomes pronounced, indicating heightened risk aversion, reduced trading liquidity, and distorted information reflection, ultimately leading to lower market efficiency.

For RBOB Gasoline, $\Delta \alpha$ is broadly distributed (0.3–1.0) when GPR is below 100, but contracts to 0.3–0.5 as GPR surpasses 300, reflecting diminished trading activity and efficiency. Palladium follows a similar trend, with $\Delta \alpha$ initially spread across 0.3–0.7 and above 0.8, but tightening to 0.3–0.4 at high GPR levels, suggesting market rigidity under geopolitical uncertainty. Brent Crude Oil also exhibits a significant contraction in $\Delta \alpha$ (from 0.4–1.2 to 0.3–0.6) as GPR rises above 300, limiting excess return opportunities and distorting price movements.

Overall, geopolitical risk negatively impacts market efficiency, particularly in the energy and precious metals markets, by constraining price discovery and increasing investor risk aversion. We recognize the need for a more comprehensive analysis of the relationship between geopolitical risk and dynamic efficiency, and propose this as a subject for future research.

4.3. Granger Causality

In this section, we analyze the impact of crude oil on the efficiency of other commodity assets, building on the studies discussed in the Introduction. Specifically, we investigate the extent to which crude oil influences various commodity assets by considering the insights provided by previous research on its broader effects within the commodity market.

In the prior section, we demonstrated the nonlinear behavior of asset dynamics using the results of the MF-DFA analysis. Based on these findings, we conducted a Granger causality test to further explore the predictive causality and lead–lag relationships associated with asset inefficiency. This additional analysis assessed the degree to which crude oil affects the inefficiencies of other assets from a predictive causality perspective.

Table 2. Granger causality test results for Brent Crude Oil.

Cause	Effect	F-Statistic	p-Value
BRENT CRUDE OIL	COCOA	0.20579	0.81403
BRENT CRUDE OIL	COFFEE	0.06075	0.94106
BRENT CRUDE OIL	COPPER	5.34766	0.00487
BRENT CRUDE OIL	COTTON	0.42245	0.65554
BRENT CRUDE OIL	FEEDER CATTLE	2.1287	0.11945
BRENT CRUDE OIL	GOLD	3.32919	0.03616
BRENT CRUDE OIL	HEATING OIL	6.90573	0.00104
BRENT CRUDE OIL	KANSAS WHEAT	1.39097	0.24924
BRENT CRUDE OIL	LEAN HOGS	7.17581	0.00080
BRENT CRUDE OIL	LIVE CATTLE	3.31557	0.03665
BRENT CRUDE OIL	NATURAL GAS	2.13997	0.11811
BRENT CRUDE OIL	OATS	3.68344	0.02542
BRENT CRUDE OIL	PALLADIUM	1.0524	0.34942
BRENT CRUDE OIL	PLATINUM	5.37412	0.00475
BRENT CRUDE OIL	RBOB GASOLINE	17.1459	0.000000046
BRENT CRUDE OIL	ROUGH RICE	5.41588	0.00456
BRENT CRUDE OIL	SILVER	3.70143	0.02497
BRENT CRUDE OIL	SOYBEANS	0.84072	0.43166
BRENT CRUDE OIL	SUGAR	0.29164	0.74700

and

Table 3. Granger causality test results for the variables affecting Brent Crude Oil.

Cause	Effect	F-Statistic	p-Value
COCOA	BRENT CRUDE OIL	1.98176	0.13828
COFFEE	BRENT CRUDE OIL	6.57874	0.00144
COPPER	BRENT CRUDE OIL	3.11391	0.04479
COTTON	BRENT CRUDE OIL	1.46307	0.23194
FEEDER CATTLE	BRENT CRUDE OIL	0.82367	0.43907
GOLD	BRENT CRUDE OIL	4.98068	0.00701
HEATING OIL	BRENT CRUDE OIL	1.30044	0.27280
KANSAS WHEAT	BRENT CRUDE OIL	2.32782	0.09795
LEAN HOGS	BRENT CRUDE OIL	1.32917	0.26509
LIVE CATTLE	BRENT CRUDE OIL	1.52184	0.21874
NATURAL GAS	BRENT CRUDE OIL	0.72870	0.48275
OATS	BRENT CRUDE OIL	1.44697	0.23570
PALLADIUM	BRENT CRUDE OIL	7.72348	0.00046
PLATINUM	BRENT CRUDE OIL	8.16214	0.00030
RBOB GASOLINE	BRENT CRUDE OIL	6.50198	0.00155
ROUGH RICE	BRENT CRUDE OIL	0.04102	0.95981
SILVER	BRENT CRUDE OIL	13.6859	0.0000013
SOYBEANS	BRENT CRUDE OIL	0.54979	0.57722
SUGAR	BRENT CRUDE OIL	0.84196	0.43112

summarize the Granger causality results. A comparison of the two tables reveals that Brent crude oil significantly leads 10 commodity assets, while it is significantly led by 7 commodity assets. This suggests that Brent crude oil acts more as an influential asset in the commodity market rather than one that is primarily influenced by market efficiency.

Table 2 outlines the extent to which Brent crude oil inefficiency affected other assets. Specifically, it led gold, live cattle, oats, and silver at the 5% significance level. Furthermore, Brent crude oil strongly led Copper, Heating Oil, Lean Hogs, Platinum, RBOB Gasoline, and Rough Rice at the 1% significance level. However, considering the multifractal spectrum ($\Delta \alpha$) values of the commodity assets presented in Table 1, it is evident that, apart from Gold (0.5091), Brent crude oil only led Copper (0.3358) and Live Cattle (0.4908) among the assets with high efficiency. This indicates that Brent crude oil had a more substantial influence on assets characterized by market inefficiency.

Table 3 details the impact of the other commodity assets on Brent’s crude oil efficiency. Copper, an asset with high efficiency, led Brent crude oil significantly at the 5% level. Additionally, Coffee, Gold, Palladium, Platinum, RBOB Gasoline, and Silver significantly led Brent crude oil at the 1% level. However, when considering the multifractal spectrum ($\Delta \alpha$) values, except for Gold (0.5091) and Copper (0.3358), the influence of generally efficient assets on Brent crude oil was minimal.

In summary, Brent crude oil has a greater influence on inefficient commodity assets than on efficient ones, and can be considered a leading asset in relation to other commodities characterized by inefficiency.

To enhance the robustness of our findings, we conducted additional analyses using rolling window lengths of 350 and 450 days. Specifically, we generated scatterplots and performed Granger causality tests for each window length. The results indicated no significant differences from the main findings of this study, confirming the robustness of our conclusions.

5. Discussion and Concluding Remarks

We used price data from 20 major commodity assets, including crude oil, to measure the market efficiency of each asset. Additionally, we employed rolling window analysis to examine the time-varying characteristics of individual asset efficiencies. Finally, to assess the impact of crude oil’s efficiency on the efficiency of other commodity assets, we conducted a Granger causality test. Specifically, we employed the MF-DFA technique to analyze the fractal behavior of commodity assets, which can be linked to market efficiency. The fractal characteristics of these assets challenged the EMH by revealing self-similar price patterns across different timescales.

The key findings of our study are as follows: First, the market efficiency analysis revealed that RBOB Gasoline, Palladium, and Brent Crude Oil markets were inefficient over the observed period. These commodities were significantly influenced by geopolitical risks, and our findings suggest that such external shocks exacerbated supply–demand imbalances, leading to increased market inefficiency. This is also supported by existing literature [81,82,92].

By contrast, Copper, Kansas Wheat, and Soybeans exhibited the highest efficiency during the same period. Unlike the former group, these commodities maintained stable supply demand dynamics, which may have contributed to their superior efficiency levels [83,84,85].

Furthermore, the commodity market efficiencies varied over time. During the COVID-19 pandemic, the efficiency of all commodity markets exhibited significant fluctuations. This volatility can be attributed to the pandemic’s short-term negative impact on commodity prices and global shifts in demand during the subsequent economic recovery phase. Several studies have documented the impact of the pandemic on commodities [93,94,95].

Consistent with the overall efficiency analysis, RBOB Gasoline, Palladium, and Brent Crude Oil remained highly sensitive to geopolitical risks, as demonstrated during the Russia–Ukraine War and the Israel–Palestine conflict. Conversely, Copper, Kansas Wheat, and Soybeans demonstrated resilience and maintained stability, despite the challenges posed by the pandemic and geopolitical events.

Finally, the Granger causality test results indicated that Brent Crude Oil was a more influential asset in the commodity market. Rather than being affected by market efficiency, Brent Crude Oil played a significant role in shaping the efficiency dynamics. In summary, it exerted a stronger influence on inefficient commodity assets than on efficient ones, positioning it as a leading asset among commodities characterized by inefficiency. This finding represents the most significant contribution of this study to the relevant research fields.

Commodity assets that are closely tied to daily life often operate within relatively efficient market environments because of their consistent and ongoing demand. Examples include copper, Kansas wheat, and soybeans, which maintain a steady demand, even during crises or supply shortages. These commodities can rapidly adjust to supply–demand imbalances, because of the availability of market information. Additionally, their non-luxury status makes it relatively feasible to stockpile reserves in advance or resort to alternative products to meet the demand.

By contrast, commodities such as RBOB gasoline, palladium, and Brent crude oil face significant supply risks influenced by geopolitical and political factors. These assets lack effective substitutes, meaning that supply disruptions often lead to substantial demand reduction. Owing to the influence of non-economic factors such as geopolitical tensions, market dynamics can exhibit inefficiencies in price formation and supply stability. To mitigate the risks associated with these commodities, governments must adopt long-term strategies, including systematic stockpiling and proactive supply–demand management, to stabilize domestic economies.

Many nations rely entirely on imports of essential commodities such as crude oil. In these countries, ensuring stable and reliable supplies is critical for sustaining economic growth and development. This necessitates comprehensive, long-term monitoring of supply trends and preemptive efforts to address inefficiencies and mitigate the associated risks.

The results offer critical insights for financial market participants, policymakers, and researchers, especially regarding commodity trading and effective risk management strategies.

From a trading and risk management perspective, Brent Crude Oil plays a pivotal role in driving commodity market inefficiencies, offering traders and fund managers opportunities to optimize strategies by using it as a leading indicator of correlated assets. The inefficiencies observed in RBOB Gasoline, Palladium, and Brent Crude Oil highlight the need to integrate geopolitical risk factors into risk management frameworks. The adoption of dynamic hedging strategies and portfolio diversification can help mitigate the heightened sensitivity of these commodities to external shocks. Additionally, understanding market efficiency levels can enhance trading decisions and help correct inefficiencies. This study encourages further research on global commodity market dynamics.

For policymakers, the time-varying efficiency of commodity markets, especially during crises such as the COVID-19 pandemic, underscores the need to stabilize these markets. Regulatory measures such as enhancing transparency, fostering international cooperation, and creating mechanisms to protect supply–demand imbalances from geopolitical or economic shocks are essential. This study provides insights to help policymakers monitor global prices and risk signals, develop strategies to stabilize commodity imports, and establish long-term risk management frameworks, particularly for crude oil.

From an academic standpoint, these findings encourage further exploration of the intricate interplay between geopolitical risks, market efficiency, and commodity dynamics. Researchers can build upon this study by investigating cross-market interactions and focusing on how inefficiencies in one market, such as Brent Crude Oil, propagate to others. This line of inquiry may provide deeper insights into the mechanisms driving global commodity market dynamics and inform theoretical and practical applications.

Our research findings are closely related to the Fractal Fractional (FF) concepts.

First, the MF-DFA method is fundamentally based on fractal structures, which reveal that market prices exhibit self-similarity across different timescales. This suggests that financial time series are not purely random, but instead follow persistent patterns that repeat at different levels of observation. Second, our analysis of market efficiency in the commodity market indicates notable inefficiencies. Within the FF framework, inefficient markets often exhibit long-memory effects and persistent dependencies, both of which are key characteristics of fractal and fractional-order dynamics. These findings challenge the assumptions of the EMH, suggesting that market behavior is influenced by historical price patterns rather than being entirely unpredictable. Third, our use of rolling window analysis aligns with the FF approach, as it captures the non-stationary and evolving fractal nature of price movements. This method provides a dynamic perspective on market efficiency, demonstrating how it fluctuates over time in response to external factors such as geopolitical risks and economic shocks. These results further reinforce the applicability of fractal fractional models in understanding financial market behavior.

Based on the findings of this study, several recommendations for future research projects are proposed. First, an investigation into the impact of geopolitical risks on commodity markets is suggested. For example, a detailed analysis of how specific geopolitical events such as the Russia–Ukraine War and the Israel–Palestine conflict propagate through commodity markets would provide valuable insights. Similarly, this study identified supply–demand stability as a key factor influencing asset efficiency, highlighting the need for a more comprehensive methodological analysis to assess its impact. Second, we recommend an examination of efficiency changes over time during crises. This research focused on understanding the drivers of time-varying efficiency across different commodity categories during crises such as pandemics or economic recessions. Third, we analyzed the role of Brent crude oil in international commodity markets. Further exploration into why Brent crude significantly influences market inefficiencies in global commodity markets could yield meaningful contributions to the field.