Analysis of Self-Similarity in Short and Long Movements of Crude Oil Prices by Combination of Stationary Wavelet Transform and Range-Scale Analysis: Effects of the COVID-19 Pandemic and Russia-Ukraine War

Abstract

This paper examines the self-similarity (long memory) in prices of crude oil markets, namely Brent and West Texas Instruments (WTI), by means of fractals. Specifically, price series are decomposed by stationary wavelet transform (SWT) to obtain their short and long oscillations. Then, the Hurst exponent is estimated from each resulting oscillation by rescaled analysis (R/S) to represent hidden fractals in the original price series. The analysis is performed during three periods: the calm period (before the COVID-19 pandemic), the COVID-19 pandemic, and the Russia-Ukraine war. In summary, prices of Brent and WTI exhibited significant increases in persistence in long movements during the COVID-19 pandemic and the Russia-Ukraine war. In addition, they showed a significant increase in anti-persistence in short movements during the pandemic and a significant decrease in anti-persistence during the Russia-Ukraine war. It is concluded that both COVID-19 and the Russia-Ukraine war significantly affected long memory in the short and long movements of Brent and WTI prices.

1. Introduction

Crude oil is a strategic energy resource and is vital to any industry in the world. In this regard, crude oil has a significant impact on economic sectors, commodity markets, and financial markets. Over the decades, crude oil prices have undergone significant fluctuations due, for instance, to economic crises, oil supply and demand, international commercial inventory and exchange rates, international military conflicts, and speculative trading activities. In this regard, crude oil markets have attracted a large amount of attention from economic agents, investors, and scholars across the world. Therefore, investigating the dynamics of the crude oil market seems to be crucial and necessary.

On this subject, for instance, recent works in the literature examined the impact of crude oil price on global stock market volatility [1], the spillover relationships between international crude oil markets and global energy stock markets [2], impact of macroeconomic factors on volatility spillovers between stock, bond, gold and crude oil [3], dynamic linkages between crude oil futures, renewable energy indices, carbon credit futures and US sector market indices [4], spillover effects between oil and grain markets in response to supply- and demand-side shocks [5], relationship between investors’ attention to climate change risk and crude oil futures price returns [6], impact of the launching of crude oil futures in China on corporate risk management [7], effect of global economic and financial conditions on different oil supply-demand shocks [8], the relationships among crude oil and metal markets with respect to fluctuations in the US dollar (USD) [9], and study of the ex-ante selective hedging strategies of crude oil futures contracts [10].

Other studies focused on crude oil market prediction. For example, the comparison of 16 deep and machine learning models in forecasting the daily price and its direction of WTI, Brent, gold, and silver markets [11,12], the effect of global financial uncertainty on the prediction of crude oil price returns [13], the role of environment, society, and governance in forecasting returns [14], the role of ocean temperature in predicting crude oil prices [15], the effect of climate change on price forecasting [16], the role of investor’s fear in price volatility prediction [17], and the effectiveness of machine learning on price forecasting [18,19].

Another hot topic is about examining efficiency in the crude oil market according to the efficient market hypothesis that states that prices should follow a random walk [20]. In an early study, the authors in [21] examined the random walk hypothesis by using variance ratio tests for the Brent and WTI using daily data over the period 1982–2008 and found that the Brent crude oil market is weak-form efficient while the WTI crude oil market seems to be inefficient in the 1994–2008 sub-period. They concluded that deregulation had not improved the efficiency of the WTI crude oil market in the sense of making returns less predictable. In recent years, several works have been conducted from an econophysics perspective. For instance, the authors in [22] used the detrended fluctuation analysis for the Brent crude oil returns and found that there is a presence of long-range correlation in returns. In addition, it was concluded that after the financial crisis of 2008, the Brent crude oil market becomes more persistent. In [23], the R/S analysis and detrended fluctuation analysis were used to study the temporal properties of price shock sequences in crude oil markets, and empirical results showed that crude oil price shock sequences exhibit time-clustering behavior and long-range correlations. The authors in [24] applied the multifractal detrended fluctuation analysis and found that the WTI market shows multifractality. In addition, the sources of multifractality are from long-range correlations and fat-tailed distributions. Furthermore, the long-range correlation has a bigger impact. In [25], the presence of chaos in crude oil markets Brent and WTI was examined by estimating the largest Lyapunov exponent before and after the 2008 international financial crisis. Empirical results showed no evidence of chaos in prices and returns in Brent and WTI before and after the financial crisis. Yet, strong evidence of chaotic dynamics was found in the volatility of Brent and WTI after the financial crisis. By applying the detrended fluctuation analysis and the detrending moving average analysis techniques, the authors in [26] examined the efficiency of the WTI crude oil futures prices during the period of 1983–2012. They concluded that the WTI oil futures market is weak-form efficient during the entire period; however, it is inefficient right after the crashes (1985, 2008) and the Gulf War.

Very recently, some interesting studies have focused on examining the impact of extreme events like the COVID-19 pandemic and the Russia-Ukraine war on crude oil markets. For instance, estimated correlation dimension, Lyapunov exponent, and approximate entropy were estimated for periods before and during the pandemic to examine efficiency in WTI, Brent, natural gas, heating oil, and gasoline markets [27]. It was found that during the pandemic, stability strongly decreased in WTI, irregularity decreased in all markets, and heating oil and gasoline markets appear to be unaffected by the COVID-19 pandemic compared to WTI, Brent, and gas markets. In [28], it was concluded that the COVID-19 pandemic significantly affected information transmission between fossil energy markets and altered the microstructure of fossil energy markets. The authors in [29] concluded that the WTI market was efficient most of the time in the years before the COVID-19 pandemic. However, it was not efficient in some major periods during the pandemic. In [30], the authors found that before the COVID-19 pandemic, the causal relationship predominantly flowed from gold and crude oil (Brent and WTI) toward the cryptocurrency markets. In contrast, during the COVID-19 period, the direction of causality shifted, with cryptocurrencies exerting influence on the gold and crude oil markets. The author in [31] examined multiscale fractals and entropy in fossil energy markets and found that all energy markets exhibit multifractal properties before and during the pandemic; multifractals intensified during the pandemic only in price returns of Brent, WTI, and gasoline markets, and multifractals decreased during the pandemic in price returns of the heating oil market. In addition, the analysis based on multiscale entropy showed strong evidence of a reduced irregularity in energy market price returns during the COVID-19 pandemic.

In addition, other scholars examined the impact of military conflicts like the Russia-Ukraine war on crude oil markets. For instance, it was found that the war affected speculation, amplified fluctuations in oil prices, intensified low inventories, and sustained an increase in oil prices [32]. However, it was concluded that the war had a small long-term effect on oil prices. In [33], the authors found evidence of the effect of the war on return spillovers across fossil energy markets and that Brent showed the highest dynamic gross directional return spillovers. The authors in [34] found that crude oil and equity markets were the major risk sources under COVID-19, while natural gas was the major risk source during the Russia-Ukraine conflict. The asymmetric effects of the war on energy, metal, and agricultural markets were examined by means of cross-quantilogram analysis in [35], and it was found that the war significantly affects metal, agricultural, and energy commodities. In [36], it was found that the war notably increased systemic risk in both the European gas and the Brent futures. In addition, the systemic risk increases quickly and reduces slowly in the gas market compared to Brent futures. Finally, during the war, the Brent futures market performed significantly and switched from the net transmitter to the net receiver from coal futures and natural gas [37].

According to the efficient markets hypothesis (EMH) [20], in a competitive market with rational, profit-maximizing agents, asset prices incorporate all available information. Therefore, asset prices are not predictable, and riskless profits cannot be generated by traders and investors [20]. In other words, market prices are efficient as they reflect all available information, so the price is a fair representation of asset value. In this regard, neither technical analysis (the study of historical prices to predict future prices) nor fundamental analysis (the study of fundamental data like financial and economic information) can be appropriate approaches to predicting prices of assets.

However, in complex economic systems, fractal theory provides a powerful tool for assessing the complex nonlinear properties of market price data. For instance, compared with the theory of EMH, the fractal market hypothesis (FMH) [38] is more helpful in explaining the complexity of financial and commodity markets. Indeed, Mandelbrot [39] showed that significant complexity in the financial market could be described by fractal theory. In this regard, based on the work of Mandelbrot [39] and using the fractal objects whose disparity parts are self-similar, Peters [38] proposed the FMH to explain variations and non-periodicity that describe stock markets.

In this paper, we aim to study self-similarity in short and long variations in Brent and WTI prices under three different periods: the regular period (before the COVID-19 pandemic and Russia-Ukraine war), the COVID-19 period, and the Russia-Ukraine war period. We make the hypothesis that such extreme events may alter self-similarity in the short and long movements of prices differently. For instance, short variations in prices would represent the immediate response to these extreme shocks, whilst long variations would represent the delayed response. In addition, the response would differ with the scale (short or long variation) and the type of extreme event.

In this work, short- and long-term variations used to describe sudden fluctuations and trends in original prices are extracted by means of stationary wavelet transform (SWT) [40] thanks to its effectiveness in signal representation, separation, detection, and filtering [41]. Specifically, unlike the traditional discrete wavelet transform, the SWT preserves more information after applying the transform. In this regard, it was successful in various problems, including examining chaos in exchange rates [42], clustering of international financial and commodity markets [43], short-term load forecasting for energy management [44], prediction of fault in electric distribution systems [45], and automatic detection of diabetic retinopathy [46]. Finally, self-similarity in short and long movements in prices of crude oil markets is assessed by the R/S fractal estimator of the Hurst exponent due to its effectiveness in various business [43,47] and engineering [48,49,50] problems. The main contributions of our study are as follows:

(a)

Self-similarity is examined in prices of two major crude oil markets, namely Brent and WTI.

(b)

The effect of the COVID-19 pandemic and the Russia-Ukraine war on self-similarity is examined. A calm period is considered a reference period.

(c)

We propose to combine SWT and R/S analysis to better analyze the data by estimating the Hurst exponent in the SWT domain.

(d)

Self-similarity is estimated from short and long movements in prices for better understanding of the dynamics of price series under different extreme events.

(e)

The recent literature [1,2,3,4,5] on crude oil markets is enriched by the effect of extreme events on self-similarity in short and long movements in prices of crude oil markets.

(f)

The study should provide traders and investors with insightful directions regarding investment in crude oil markets during times of major crises.

The paper is structured as follows: Section 2 introduces the SWT and R/S techniques. Section 3 describes the data and provides the empirical results. Section 4 discusses and concludes the work.

2. Materials and Methods

The flow chart of the proposed approach for the analysis of crude oil prices is shown in Figure 1. First, the price time series are decomposed by SWT to obtain details and approximation coefficients used to represent short and long (trend) movements (oscillations) in the original data. Then, the R/S analysis is applied to each category of coefficients to estimate the Hurst exponent (HE) and to reveal how fractals affect short and long oscillations. Therefore, a detailed analysis of nonlinear dynamics in price data can be achieved by applying the R/S in the SWT domain. Finally, the obtained Hurst exponent from detail and approximation coefficients can be interpreted. Specifically, the computed value of HE should be compared to 0.5. For instance, if HE is equal to 0.5, then price series obey a random process and cannot be predicted. If HE is less than 0.5, then price series are anti-persistent. In this regard, a decrease (increase) in price is most likely followed by an increase (decrease) in price. However, if HE is larger than 0.5, then price series are persistent. Accordingly, a decrease (increase) in price is most likely followed by a decrease (increase) in price. The SWT and R/S analyses are described next.

2.1. Stationary Wavelet Transform

The stationary wavelet transform (SWT) [40] is like the discrete wavelet transform (DWT) [51], where wavelet coefficients are obtained through the application of convolution followed by a decimation, but downsampling the scales is omitted. For instance, the SWT of a signal X is obtained by the convolution product of X and filters (low-pass followed by high-pass) that procures the approximation C_j,k and detail coefficients W_j,k sequentially at level j. Specifically, the approximation and detail coefficients are computed by applying two scale functions φ_k and Ψ_k in the time domain as expressed in Equations (1)–(4):

$$C_{j,k}=〈X\left(t\right){\phi }_k\left(t\right)〉$$

(1)

$$W_{j,k}=〈X\left(t\right){\mathsf{\Psi }}_k\left(t\right)〉$$

(2)

$${\phi }_k\left(t\right)=2^{-j}\phi \left(2^{-j}\left(t-k\right)\right)$$

(3)

$${\mathsf{\Psi }}_k\left(t\right)=2^{-j}\mathsf{\Psi }\left(2^{-j}\left(t-k\right)\right)$$

(4)

In this study, the Symlet-4 wavelet is chosen for the analysis of the data, and the level of decomposition is set to two.

2.2. R/S Analysis

The R/S analysis [52,53] is a commonly used fractal estimation method [43,47,48,49,50]. For instance, at a given scale n, the mean value of the signal X is computed as:

$${\overline{\mathrm{u}}}_n\left(t\right)={\displaystyle \frac{1}{n}}{\sum }_{X=1}^{n}u\left(x\right)$$

(5)

The total accumulative deviation is expressed as follows:

$$u\left(X,n\right)={\sum }_{X=1}^n\left[u\left(x\right)-{\overline{u}}_n\right]$$

(6)

The extreme difference R(n) is calculated as:

$$R\left(n\right)=\underset{1\le X<n}{\max }u\left(X,n\right)-\underset{1\le X<n}{\min }u\left(X,n\right)$$

(7)

The standard deviation is given by:

$$S\left(n\right)=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{X=1}^n{\left[u\left(x\right)-{\overline{u}}_n\right]}^2}$$

(8)

Finally, based on the relationship $R\left(n\right)/S\left(n\right)\propto n^H$, the Hurst exponent H can be calculated by fitting a linear regression of log(n) on log(R(n)/S(n)).

If the signal X is characterized by long-range correlation features when it is not random. For instance, when 0 < H < 0.5, X is anti-persistent; so, a large (small) variation is more likely to be followed by a small (large) variation. When 0.5 < H < 1, X is persistent; so, a large (small) variation is more likely to be followed by a large (small) variation. Lastly, when H = 0.5, the dynamics of X follow a random walk, and X is difficult to predict. Recall that when H ≥ 1, the autocorrelation exists and ceases to be a power-law form.

3. Results

We collected daily price data on Brent and West Texas Intermediate (WTI) for the period from 2 January 2018 to 30 September 2024 from the Federal Reserve Economic Data (FRED) database of the Federal Reserve Bank of St. Louis, USA [54]. The full period is split into three subperiods: prior to the COVID-19 pandemic (2 January 2018 to 31 December 2019), the COVID-19 pandemic (2 January 2020 to 31 December 2021), and the Russia-Ukraine war (3 January 2022 to 30 September 2024). Figure 2 exhibits the approximation and detail coefficients of Brent and WTI through the full sample. The estimated values of the Hurst exponent used to represent self-similarity in approximation (long trend movements or general trend) and detail (short movements) coefficients are shown in Figure 3 and Figure 4.

As shown in Figure 3, the detail coefficients used to represent short movements in prices of Brent are anti-persistent during all periods. In addition, the approximation coefficients used to represent long movements (trend) in prices of Brent are persistent during all periods. In addition, for Brent prices, anti-persistence in short movements increased during the COVID-19 pandemic and decreased during the Russia-Ukraine war. Similarly, persistence in long movements increased during the COVID-19 pandemic and decreased during the Russia-Ukraine war. Furthermore, the Hurst exponent in short movements during the war is lower than that in the calm period. Thus, short movements in Brent prices are less anti-persistent during the war compared to the calm and pandemic periods.

As shown in Figure 4, the detail coefficients corresponding to short movements in prices of WTI are anti-persistent in all periods. In addition, the approximation coefficients corresponding to long movements (trend) in prices of WTI are persistent in all periods. As well, for WTI prices, anti-persistence in short movements raised during the COVID-19 pandemic and reduced during the Russia-Ukraine war. Likewise, persistence in long movements intensified during the COVID-19 pandemic and diminished during the Russia-Ukraine war. Also, the Hurst exponent in short movements during the war is lower than that in the calm period. Hence, short movements in WTI prices are less anti-persistent during the war compared to the calm and pandemic periods.

In summary, prices of Brent and WTI exhibited significant increases in persistence in long movements during the COVID-19 pandemic and the Russia-Ukraine war. In addition, they showed a significant increase in anti-persistence in short movements during the pandemic and a significant decrease in anti-persistence during the Russia-Ukraine war.

For further analysis,

Table 1. Rate of change in the Hurst exponent as measured by the R/S technique across.

	Brent Movements		WTI Movements
	Long	Short	Long	Short
Periods
Calm	0.9339	0.2726	0.9024	0.3032
Pandemic	1.0223	0.3094	1.0062	0.342
War	0.9756	0.1931	0.9557	0.2875
Change rate
Calm–pandemic	9%	13%	12%	13%
Pandemic–war	−5%	−38%	−5%	−16%

provides the percentage rate of change in the Hurst exponent between two successive periods. As indicated, the rate of change in the Hurst exponent of long movements from the calm to the pandemic period is 9% for Brent and 12% for WTI. Then, the response of WTI to the pandemic is more pronounced than the response of Brent. In addition, the rate of change in the Hurst exponent of short movements from the calm to the pandemic period is 13% for both Brent and WTI. Therefore, the response of short movements in Brent and WTI to the pandemic is similar and higher compared to short movements. More importantly, the pandemic strongly affected self-similarity in short movements of Brent compared to its long movements: 9% versus 13%.

In addition, the rate of change in the Hurst exponent of long movements from the pandemic to the Russia-Ukraine war is −5% for both Brent and WTI. This finding suggests that the effect of the war on the self-similarity of long movements is similar across prices of Brent and WTI. In addition, the rate of change in the Hurst exponent in price short movements from the pandemic to the war period is −38% for Brent and −16% for WTI. Hence, the effect of the war is stronger on short movements than on long movements. Furthermore, the war period strongly affected self-similarity in short movements of Brent compared to short movements in WTI: −38% versus −16%. It is worth mentioning that the rate of change in the Hurst exponent is always negative during the war period.

4. Discussion and Conclusions

Crude oil plays a major role in the world economy and development. The occurrence of extreme events like the COVID-19 pandemic and the Russia-Ukraine war has significantly altered the connectedness between energy markets on one hand and energy markets and commodity markets on the other hand [27,28,29,30,31]. The main purpose of the current study is to examine the effects of the COVID-19 pandemic and Russia-Ukraine on self-similarity (long memory) in prices of Brent and WTI. We proposed employing SWT to extract short and long movements in price series, followed by the estimation of the Hurst exponent by R/S analysis. The estimated Hurst exponent is used to assess self-similarity in short and long movements in prices. The empirical results follow. First, prices of Brent and WTI exhibited significant increases in persistence in long movements during the COVID-19 pandemic and the Russia-Ukraine war. Second, prices of Brent and WTI showed a significant increase in anti-persistence in short movements during the pandemic and a significant decrease in anti-persistence during the Russia-Ukraine war.

To sum up, these two extreme events have affected long memory in prices of Brent and WTI. In this regard, our findings confirm previous works [27,28,29,30,31] used to show that the COVID-19 pandemic and Russia-Ukraine war affected crude oil markets. For instance, it was found that the Russia-Ukraine war affected speculation and amplified fluctuations in oil prices [32], caused global energy market shocks [33], and the COVID-19 pandemic and the Russia-Ukraine military conflict altered stability in connectedness between various commodity markets, including Brent and WTI, and the effect of the pandemic is stronger [34]. In addition, the Russia-Ukraine war significantly affected asymmetric bi-directional predictability between energy commodities [35], and it significantly increased systemic risk in both the European gas and oil markets [36]. Furthermore, because of the war, oil markets dominated the transmission of returns spillovers among traditional/new energy, green finance, and corporates’ environmental, social, and governance (ESG) and changed from the net transmitter to the net receiver. In this regard, our findings are in accordance with previous works that showed that the COVID-19 pandemic and the Russia-Ukraine war significantly affected energy commodities, as can be seen in

Table 2. Summary of comparison with previous works on the same topic.

Study	Purpose	Period	Findings
[21]	Examine the random walk hypothesis in Brent and WTI returns.	June 1982 to July 2008	The Brent crude oil market is weak-form efficient, while the WTI crude oil market seems to be inefficient in the 1994–2008 sub-period.
[22]	Examine long-range correlation in Brent returns.	May 1987 to September 2012	Presence of long-range correlation in returns. After the financial crisis of 2008, the Brent crude oil market becomes more persistent.
[23]	Examine long-range correlation in Brent and WTI returns.	May 1987 to September 2012	Crude oil price shock sequences exhibit long-range correlations.
[24]	Examine multifractal in WTI	Apr 1986 to June 2018	The WTI market shows multifractality. Sources of multifractality are from long-range correlations and fat-tailed distributions.
[25]	Examine chaos in Brent and WTI	November 1998 to March 2016	No evidence of chaos in prices and returns in Brent and WTI before and after the financial crisis. Strong evidence of chaos in the volatility of Brent and WTI after the financial crisis.
[26]	Examine long memory in WTI crude oil futures	April 1983 to October 2012	The WTI oil futures market is weak-form efficient during the entire period. It is inefficient right after the crashes (1985, 2008) and the Gulf War.
[27]	Examine correlation dimension, Lyapunov exponent, and approximate entropy for the periods before and during the COVID-19 pandemic in Brent and WTI, among others.	November 2017 to November 2022	The COVID-19 pandemic affected WTI, Brent, and gas markets.
[28]	Investigate causality between international fossil energy markets and the effect of the COVID-19 pandemic on their clustering structures.	August 1993 to June 2023	The COVID-19 pandemic significantly affected information transmission between fossil energy markets and altered the microstructure of fossil energy markets.
[29]	Investigate the impact of the COVID-19 outbreak on crude oil market efficiency.	January 2000 to April 2022	The WTI market was efficient most of the time in the years before the COVID-19 pandemic. However, it was not efficient in some major periods during the pandemic.
[30]	Investigate the connectedness between the 10 most traded cryptocurrencies and gold as well as crude oil markets pre-COVID-19 and during COVID-19.	January 2019 to December 2020	Before the COVID-19 pandemic, the causal relationship predominantly flowed from gold and crude oil (Brent and WTI) toward the cryptocurrency markets. During the COVID-19 period, the direction of causality shifted, with cryptocurrencies exerting influence on the gold and crude oil markets.
[31]	Examine multiscale fractals and entropy in fossil energy markets.	November 2017 to November 2022	The COVID-19 pandemic affected multifractal behavior and multiscale entropy characteristics in WTI, Brent, gasoline, and heating oil markets.
Our study	To decompose price time series of Brent and WTI by SWT to obtain their corresponding long and short movements. To estimate long memory in the decomposed series for better characterization and understanding of their dynamics.	January 2018 to September 2024	Both COVID-19 and the Russia-Ukraine war significantly affected long memory in the short and long movements of Brent and WTI prices.

, used to summarize some of the recent works on the topic. Indeed, as extreme events, both the COVID-19 pandemic and the Russia-Ukraine war affected the short and long movements in prices of Brent and WTI. These results were revealed thanks to the decomposition of their respective prices by STW into short and long variations and applying the R/S analysis into the decomposed series as proposed in Figure 1.

It is intriguing that anti-persistence in short movements of Brent and WTI prices during the Russia-Ukraine war is less than that during the calm period. Indeed, anti-persistence increased during the pandemic period and decreased during the war. More importantly, the value in the war period is less than the value recorded during the calm period. This situation could reflect a moment of panic in crude oil markets (Brent and WTI), where a large increase in prices is followed by a large decrease and vice versa.

In addition, as shown in Figure 3 and Figure 4, the persistence in both long (approximation) and short (detail) movements of prices in Brent and WTI has slightly decreased in the post-COVID-19 pandemic period as compared with the pandemic period. This could be explained by the fact that the world economy started recovering from the crisis caused by the pandemic.

In addition, based on empirical findings, both short and long variations in prices of Brent and WTI are predictable as their respective Hurst exponents are significantly not close to 0.5 in all periods. Specifically, to generate profits, long-term investors may look at trends of price and expect that a large (small) variation in trend is more likely to be followed by a large (small) variation. In contrast, to generate profits, short-term traders may look at sudden changes in prices and expect that a large (small) variation in price is more likely to be followed by a small (large) variation.

Indeed, empirical results would help investors, traders, and risk managers in the crude oil industry (i) in understanding the impact of extreme events like pandemics and geopolitical risks on crude oil prices (ii) and in providing central inputs for asset allocation and hedging strategies with respect to Brent and WTI, the major crude oil markets in the West.

In addition, revealing differences in predictability in short and long movements of crude oil prices during normal and stressful periods is timely and serves as a useful tool for policymakers and market regulators. For instance, the comparative analysis of long memory in short and long variations in crude oil prices is supposed to offer groundbreaking evidence to assess the market correctly. Specifically, as short movements are anti-persistent and long movements are persistent, policymakers and market regulators may set specific strategies to avoid the misallocation of resources that would have a negative impact on short- and long-term economic growth. Likewise, revealing differences in predictability in short and long movements of crude oil prices during normal and stressful periods would help to improve the efficiency of resource allocation channels across calm and extreme periods and reduce distortions in the economy in the short and long run. In summary, our findings may be of some use to investors and policymakers who are struggling to minimize the impacts of global and regional extreme events. In particular, for both Brent and WTI, policymakers such as government and regulators should expect that long-term movements in prices will follow the same trend during extreme events as an increase (decrease) in price is most likely to be followed by an increase (decrease). For instance, if crude oil prices increase during the extreme event period, they will keep increasing, and policymakers are expected to increase the budget used to buy crude oil accordingly. On the other hand, if crude oil prices decrease during the extreme event period, they will keep decreasing; hence, policymakers are expected to decrease the budget used to buy crude oil accordingly. In addition, for both Brent and WTI, traders and investors who are interested in maximizing short-term profits should expect that short-term movements in prices will change direction frequently during extreme events like the pandemic and geopolitical risks or wars. For instance, if crude oil prices increase (decrease) during the extreme event period, they will decrease (increase) the next day. Thus, on a daily basis, traders and investors are expected to sell (buy) crude oil when the price increases (decreases) to generate profits, as price increases (decreases) aremost likely to be followed by a price decrease (increase) the next day. Therefore, trading crude oil daily would increase profits of traders and investors during extreme periods.

In future works, we will implement multivariate rescaled range analysis [55] to better understand the Brent–WTI complex system being studied. Hence, one could scrutinize the co-movements among the component series (for instance, Brent and WTI) to highlight the latent effects on stability and forecasting.