key: cord-0740401-1hnghk72
authors: Espinosa-Paredes, G.; Rodriguez, E.; Alvarez-Ramirez, J.
title: A singular value decomposition entropy approach to assess the impact of Covid-19 on the informational efficiency of the WTI crude oil market
date: 2022-05-23
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2022.112238
sha: 3a0694e051bd13afa94919b60717761d01407e2e
doc_id: 740401
cord_uid: 1hnghk72

This work investigates the impact of the Covid-19 outbreak on crude oil market efficiency. The approach is based on the singular value decomposition (SVD) entropy. Iso-distributional surrogate data test was used to contrast the results against random patterns, and phase randomization based on Fourier transform was used to assess nonlinearities. The analysis considered the WTI market and focused on the Covid-19 pandemic period January 2020–November 2021 and contrasted with the long preceding period from January 2000 to date. It was found that the crude oil market was informationally efficient most of the time with small sporadic deviations from efficiency in the pre-Covid-19 years. The Covid-19 period exhibited the largest deviations from efficiency, mainly in the first months of the outbreak, accompanied by a marked reduction of nonlinear components. The analysis was conducted for different scales, and the results showed that the deviations from efficiency were more pronounced for quarterly scales. For the sake of comparison, the analysis was also carried out on the trading volume dynamics and the results showed that the market activity is not fully random. The dynamics of the trading volume exhibited significant deviations from the randomness behavior when the crude oil market was efficient, and a behavior that was consistent with nonlinear patterns. The opposite behavior was noted for stages when the crude oil market showed strong deviations from efficiency. Overall, the findings of this study suggest an increasing opportunity for crude oil price predictions and abnormal returns during the Covid-19 pandemic.

The Covid-19 pandemic has had a great impact on the social and economic activities in the recent 21 months. The outbreak started with the report of a cluster of cases of pneumonia with atypical symptoms by the Wuhan Municipal Health Commission on December 31, 2019 in Wuhan City, Hubei Province, China. Quickly, the infection spread to the whole world, and many countries took actions adjustments in human activity to reduce the impact on the population. By April 2020, about half of the world's population was under some form of lockdown. More than 4.0 billion people in more than 100 countries, including US and Europe, were asked or ordered to stay at home by their governments. The lockdown had an immediate negative impact on the economic activity, adding uncertainty and risk to inversions and consumption. The Covid-19 outbreak damaged the economic activity, and in 2020 most economies fell to recession. In late February and March 2020, the stock markets crashed, with major indices dropping 20-30%. The shock in the different sectors of the economy has been long-lasting, with its adverse effects still perceived in the current labor and world trade.

The impact of the Covid-19 pandemic on financial markets has been documented in the recent two years [1] . For instance, Zhang et al. [2] showed that the shock induced by the Covid-19 outbreak has had a persistent effect on the long-range structure of global markets. Mnif et al. [3] used multifractal detrended fluctuation analysis to show that the COVID-19 pandemic has impacted positively the cryptocurrency market efficiency. Lahmiri and Bekiros [4] used the largest Lyapunov exponent and approximate entropy computations to study the impact of the Covid-19 outbreak on the stability and sequential irregularity of equity and cryptocurrency markets. It was concluded that the Covid-19 pandemic imposed a severe shock on financial markets, such that investment in cryptocurrencies was highly risky. Choi [5] used detrended fluctuation analysis to analyze the efficiency of eleven sectors of the S&P-500 market. It was shown that the Covid-19 outbreak introduced a shift in the price return behavior, from anti-persistence to persistence. Recently, Assaf et al. [6] used a time-varying lifting method to estimate the Hurst exponent for cryptocurrencies. A

The uncertainty of the Covid-19 evolution and the induced risks in the financial and economic activities provoked the most severe downturn in the crude oil market in recent years. In April 2020, crude oil prices plunged to historic low levels as a consequence of a loss of 1/3 of global demand.

April 20 saw WTI prices decrease from 17.85 to -37.63 $/bbl, being the largest one-day drop for US crude in history. Promptly, the impact of the Covid-19 pandemic was a matter of intensive research.

Narayan [8] employed a threshold regression model and reported that the number of infections had a limited effect on the crude oil price in the first days of the Covid-19 outbreak. Gil-Alana and Monge [9] used the Whittle function in the frequency domain and concluded that the oil price series is mean reverting, implying that the shock will be transitory, albeit with long-lasting effects. Wang et al. [10] studied the cross-correlations between crude oil and agricultural futures markets (e.g., London Sugar and USA Cotton #2), concluding that the co-dependence of the agricultural futures with the crude oil market increased after the emergence of COVID-19. Zhang and Hamori [11] studied the return and volatility spillover between the COVID-19 pandemic in 2020, showing that the impact of Covid-19 on the volatility of the oil market exceeds that caused by the 2008 global financial crisis. Le et al. [12] considered autoregressive distributed lag bounds testing with a structural break to study the crude oil price behavior in the period January 17-September 14, 2020.

The results revealed that increases in Covid-19 pandemic cases, US economic policy uncertainty, and expected stock market volatility contributed to the fall in the WTI crude oil price. Recently, Gharib et al. [13] applied log-periodic power-law singularity bubble indicators to explore the dynamic bubbles of oil prices and predict their crash times. The analysis showed that the WTI market experienced a negative financial bubble during the COVID-19 outbreak.

Crude oil markets can be considered complex systems in the sense postulated by Kwapień and Drożdż [14] . Crude oil markets are open systems that interchange information with surrounding systems and adaptively modify their internal structure and behavior in the process of selforganization [15] . Flexibility and adaptation in the face of exogenous changing conditions are key characteristics of crude oil markets. A central issue in the operation of financial and commodity markets is the absence of exogenous bias and possible manipulations of the price outcomes. The efficient market hypothesis (EMH) is a central concept to deal with the informational fairness of

The study of the informational efficiency of crude oil markets is rich in results and controversy. Some salient works include Tabak and Cajueiro [17] which showed that the market is becoming more efficient with time, probably caused by market liberalization. Alvarez-Ramirez et al. [18] showed that the market efficiency was not uniform but depended on both scale and time over the period 1986-2009. In turn, this result was in line with the concept of the adaptive market hypothesis [15] that the market participants adapt their expectations and risks to the informational flow. Zhang et al. [19] utilized a time-varying GAR (1)-TGARCH (1,1) model and concluded that the market was extremely efficient up to 2014. Ghazani and Ebrahimi [20] used data covering the period 2003-2018 to analyze the crude oil market efficiency in the context of adaptive expectations. The results showed that the Brent and WTI markets possessed the highest efficiency levels. Kristoufek [21] used fractal rescaled range and detrended fluctuation analysis to characterize the crude oil market efficiency. The results, which were contrasted with the Geweke-Porter-Hudak estimator, showed that the market has been efficient most of the time with some deviations linked to the 2008 global financial crisis. Studies on the impact of the Covid-19 pandemic on the informational efficiency of the crude oil market are still scarce. Gil-Alana and Monge [9] used long-memory techniques to show that the crude oil market was consistent with the EMH in the pre-Covid-19 period and became inefficient in the pandemic outbreak. Mensi et al. [22] used asymmetric detrended fluctuation analysis to assess the effect of the oil price variability in the long memory and weak-form efficiency of stock markets. It was shown that asymmetric crude oil price dynamics is a key driver of the trend of stock dynamics. Wang et al. [23] used Hurst exponent computations to show that the informational efficiency of crude oil markets declined during the Covid-19 pandemic.

Okorie and Lin [24] pointed out that level of informational efficiency of markets is key to profiteering by strategic players. In this regard, exogenous shocks, such as the Covid-19 pandemic, play a significant role in the nature of informational efficiency. Out of the work by Gil-Alana and Monge [9] that focused on the first months of the pandemic, detailed studies of the impact of Covid-19 pandemic on the informational efficiency of the crude oil markets are lacking. The issue is important since Gil-Alana and Monge [9] hypothesized that the Covid-19 outbreak could have potential long-lasting effects. In this work, we try to fill the literature gap by examining the timevarying informational efficiency of the crude oil market, and in this way assess the impact of the also with phase randomized surrogate data to assess nonlinear behavior linked to the market dynamics.

Entropy is an index associated with a state of disorder, randomness, or uncertainty of a system. In recent decades, the concept of entropy has been widely used for temporal complexity analysis of real-world signals. An important feature is that independent and identically distributed white noise is assumed to have maximal entropy and disorder. As a consequence, entropy is commonly used to assess the randomness of signals. The most used entropy measures include those introduced by Shannon [25] , Renyi [26] , Kolmogorov [27] , and Eckmann and Ruelle [28] . These entropy concepts were proposed from strong theoretical grounds and their practical implementation presents some troubles. In this regard, the approximate entropy (ApEn) by Pincus [29] , and the sample entropy (SampEn) by Richman and Moorman [30] were proposed as practical approaches to quantify the amount of regularity and the unpredictability of fluctuations over time-series data. The approaches rely on computing the occurrence and repeatability of patterns contained in a sequence. For realtime sequences, exact matching of patterns has a very low probability. Pincus [29] proposed to compute the approximate matching modulus of a given tolerance that should be adjusted to obtain reproducible results. ApEn and SampEn have been extensively used in many fields (e.g., physics, physiology and finance) to reveal the presence of complex patterns and their link to underlying dynamical mechanisms. In many instances, one is interested in computing the entropy over different time scales. The ApEn and SampEn computations are quite sensitive to the length of the sequence, such that establishing a scale for which the computations are stable offers some troubles. Costa and Goldberger [31] addressed this issue by considering the computation of entropy over coarse-grained time series obtained by averaging over a prescribed scale factor. By doing this, the coarse-grained time series is a smoothed (i.e., low-pass filtered) version of the original time series. The multi-scale approach solved some problems presented by the ApEn and the SampEn, although the computations for relatively small time series can be inaccurate. problem under scrutiny is to decide whether the above time series contains serial correlations. To this end, consider a sequence of size with leading time :

If the time series ( ) is affected by serial correlations, then the sequence (1) has some similarities with past sequences of the same size. To test the similarity of the sequence (1) with past sequences, consider the following square matrix of lagged subsequences:

The similarities of the sequence (1) with past events should be reflected in the structure of the matrix ( ; ). If similarities are not present at all, Eq. (2) corresponds to a random matrix.

Although a matrix ( ; ) containing correlations might be full rank ( ( ( ; )) = ), the presence of correlation indicates that the information tends to be aggregated around a subspace. In contrast, the absence of correlations would imply that a dimensionality reduction might lead to important information loss. That is, all row vectors in a non-correlated matrix contain the same amount of information, such that no one-row vector can be discarded without important information loss.

Based on the above, the analysis focuses on how the information on the dynamics of a process is preferentially aggregated about a subspace. The singular value decomposition (SVD) is a suitable tool to address the question of whether the dimensionality of the matrix ( ; ) can be reduced without a marked loss of information and hence to decide on the correlations contained in the sequence. The SVD is a factorization of real or complex matrices that generalizes the Eigenfactorization to any × matrix by extending the polar decomposition. The SVD entropy is a powerful tool that has been considered to analyze the complexity of financial signals [32] [33] [34] . This approach provides a method to quantify the order content in a time series [35] . In the case of the matrix given by Eq. (2), the SVD leads to factorization of the form The singular values of the matrix ( ; ) recover the correlation information of the time series for a horizon of discrete times. The entropy is commonly used to define indices of the degree of interdependence of the row/columns of a matrix [36] . Succinctly, the entropy is an index that reflects the average information contained in a process and is also a measure of the degree of randomness in the matrix. The higher the entropy, the higher the information required to reconstruct the underlying process dynamics. Entropy is estimated from the distribution of the singular values of the matrix ( ; ). In a first step, the singular values are normalized as follows:

Subsequently, the entropy of the matrix ( ; ) is computed by

For a perfectly non-correlated process (e.g., white noise), there are no preferential directions of information accumulation and * ( ; ) = 1/ , = 1, … , such that S ( ; ) = 1. For a matrix containing correlations and reflecting preferential information directions, one should get that S ( ; ) < 1.

The entropy value S ( ; ) = 1 for uncorrelated sequences is a theoretical reference that holds asymptotically (i.e., for very long sequences). In practice, entropy analysis should deal with sequences of finite size. Also, one would like to explore the entropy for short sequences associated with relatively small scales (e.g., days for financial time series). In this way, the SVD entropy depends on the scale and should be smaller than one for sequences of finite size. and 44 observations. In turn, this advantage provides a more detailed view of the variations of the entropy over a given period.

The SVD entropy will be computed to analyze the temporal variations of the informational efficiency of the crude oil market. Most approaches for characterizing the informational efficiency are based on testing the weak form of the EMH. In this way, one should test the hypothesis that the price dynamics is consistent with a random walk behavior [15] . The reader is referred to the recent critical review by Degutis and Novickytė [37] and references therein for a description of the methods used to test informational efficiency. The existence of serial correlations via the Box-Pierce statistics is a widely used approach for testing the randomness of return time series. Run and variance ratio tests [38] remain among the most common approaches to checking market efficiency.

The computation of the scaling (e.g., Hurst) exponent via the rescaled-range (R/S) [15] and DFA methods are increasingly used for a wider characterization (e.g., adaptive market hypothesis) of return time series.

In terms of the SVD approach described above, one should decide whether the entropy of a tested return sequence ( ; ) corresponds to the entropy of a random sequence. If the probability distribution ( ) that generated the values of the sequence ( ; ) is available, a suitable approach is to generate many random sequences of size and to compute the statistics of the SVD entropy to obtain the corresponding confidence intervals (CI). However, the exact distribution is hardly available in practice for a given process. Bootstrapping estimates can be used by considering an approximate (i.e., empirical) distribution. The iso-distributional surrogate data approach proposed by Theiler et al. [39] was used to test randomness. In this way, the following procedure is proposed to estimate the CI for randomness: a) Compute ℎ shuffled sequences ℎ ( ; ) from the original sequence ( ; ). In principle, shuffling destroys serial correlations while retaining the statistical distribution of values. That is, the sequences ℎ ( ; ) and ( ; ) were generated from a common distribution ( ). b) Compute the SVD entropy for the shuffled sequences ℎ ( ; ), which reflects the entropy of a random sequence. c) Carry out the statistical analysis of the ℎ SVD entropy values to obtain the corresponding CI for randomness. The test used 5000 randomized sequences to compute the 80, 90 and 95% confidence intervals.

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The randomness test should provide insights on whether the crude oil market dynamics involve hidden predictable patterns. Diverse mechanisms may be responsible for deviations from random behavior. Wa̧torek et al. [40] showed that the crude oil market is driven by nonlinear mechanisms and the effects are reflected as multifractal dynamics. Shuffling destroys all the temporal correlations, both the linear and nonlinear ones. The point was addressed by carrying out a test for assessing the nonlinear nature of the crude oil market dynamics. Phase randomization is a strategy that preserves the amplitude distribution while destroying the nonlinear ordering contained in the phase structure [41] . The method is widely used in the field of nonlinear data analysis for testing for weak nonlinearities. ) to obtain the real-time sequence ℎ ℎ ( ; ). Apply SVD on ℎ ℎ ( ; ) to obtain the corresponding entropy. d) Apply the procedure for ℎ ℎ phase randomization realizations and obtain the corresponding confidence intervals CI. The test was carried out for 5000 phase randomized sequences to compute the 80, 90 and 95% confidence intervals.

The SVD entropy described above was implemented over an overlapping sliding window scheme window . In this way, the window sizes of 5, 22 and 66 business days corresponded to the weekly, monthly and quarterly scales, which are relevant for the market operation. In particular, the monthly scale is important since crude oil futures expire each month, on the third business day before the 25th calendar day of the month preceding the delivery month.

The analysis of the crude oil market considered the West Texas Intermediate (WTI) prices and volume for the period from January 2000 to April 2022 (https://finance.yahoo.com/). The size of the time series is 5428 observations. Table 1 presents the results of the descriptive statistics for the price and volume log differences. Negative skewness and positive kurtosis were exhibited by the two differences. Also, the Shapiro-Wilk test rejects the normality of the log differences for a 1% level. Figure 2 .a presents the variations of the SVD entropy for the weekly scale. The gray band denotes the 95% confidence interval for randomness computed from 5000 randomized versions of the window data. SVD entropy variations within the gray band indicate that the crude oil market meets the weak-form of the EMH with 95% confidence. Otherwise, the market is not efficient with a probability 5% since the sequence of log returns is not consistent with a random pattern. It should be stressed that the points located above the gray band still reflect random sequences since the band was computed for 95% confidence. Those points above the band should correspond to the right tail (e.g., extreme events) of the entropy distribution obtained from randomized sequences. Kristoufek [21] reported similar deviations from efficiency by using the rescaled range analysis. The deviation from randomness in 2020:Q3 coincided with the start of the Covid-19 second wave US and Europe. A more detailed view of the SVD entropy variations in this period is presented in Figure 2 .b. The red arrows indicate two prominent peaks that occurred in 2020:Q3 and 2021:Q1. Interestingly, the entropy remained in the randomness band despite the decrease in its value. This means that the market dynamics have been unpredictable over weekly scales for the Covid-19 period. The lower bound of the confidence band can be seen as an index of the market complexity. In this way, the peaks visible in Figure 2 .b suggest that the crude oil market complexity was reduced by the effect of, e.g., price trends, while retaining a certain degree of unpredictability over weekly time scales. Figure 2 showed that the price return dynamics were consistent with the EMH and a linear behavior over the weekly scale. That is, the price dynamics were driven by linear mechanisms and the evolution was not predictable most of the time. A link between these deviations with socio-political and economic events is not clear at all. The red arrow in Figure 3 .a indicates important deviations from randomness, meaning that in the corresponding periods the crude oil price contained some degree of predictability. Two marked peaks can be observed in 2020:Q1 and 2020:Q2. The larger one occurred in February-March 2020 ( Figure 3 .b) and coincided with the first Covid-19 lockdown, which led to a major disruption of the economic activity. As a consequence of the uncertain course of the Covid-19 pandemic, the WTI price dropped to historical minimums of negative levels. It is noted that the Covid-19 had a major negative effect on the informational efficiency for monthly scales in the last 21 years. Smaller deviations from the informational efficiency were displayed in 2021, which may be linked to subsequent Covid-19 waves and the disruption of the economic supply chains. The variation of the SVD entropy relative to the phase randomization is shown in Figure 3 .c for the whole scrutinized period from 2000 to 2022. Except for a very small period at about March 2020 indicated by an arrow, the entropy of the price return was higher than the entropy of the phase randomized samples at a 95% CI. This means that the price return dynamics are driven by nonlinear mechanisms that incorporate the complexity to the market. Such nonlinear effects did not add predictability to the price dynamics but instead contributed to the fulfillment of the informational efficiency. [9] in the sense that the impact of the onset of Covid-19 has been perceived on a wide range of scales, from weeks to quarters. A sudden drop in prices impact may propagate to different scales, generating a shock effect that impacts the informational efficiency of the crude oil market. However, a drop change must be persistent to be visible its effects in the long run. Otherwise, the energy added by the sudden price drop is dissipated and the price variation is not propagated for relatively long times. The implementation of the SVD entropy computation on a sliding window aims to detect the propagation of sudden price drops. In this way, the sharp price drops in the Covid-19 outbreak were propagated over different scales, from weeks to quarters (figures 2 to 4). However, the sharp price drops exerted in the 2008 Great Recession did not affect the informational efficiency of the market (see Figure 4 .b).

The behavior of the SVD entropy for the phase randomized samples (Figure 4 .c) exhibited a pattern similar to that for the monthly scale (Figure 3 The results described above showed that the WTI crude oil market has been in general informationally efficient in the last years prior-Covid-19 years. Short-term periods of deviations from randomness were exhibited, which were linked to the occurrence of financial and socioeconomic events. Overall, the results are in line with previous studies on the informational efficiency of crude markets. For instance, Tabak and Cajueiro [17] showed that the efficiency of the market was increasing over time. Wang and Liu [45] showed that short-term, medium-term and long-term behaviors were generally turning into efficient behavior over time, which agrees with the results in the present study. Kristoufek [21] found that the crude oil markets have remained efficient except for some events until the outbreak of the Global Financial Crisis in 2008. Okoroafor and Leirvik [46] reported that the WTI market is persistently inefficient during financial crises, with the high volatility of the efficiency in such periods. Our results suggested that this conclusion holds for the Covid-19 outbreak.

Mensi et al. [47] showed the negative impact of the Covid-19 pandemic on the crude oil market efficiency and, in line with our findings, that the market is sensitive to scales. Table 2 presents the percentage of the time the return dynamics is outside the randomness band for three different confidence intervals and the three scrutinized scales. The percentage decreased with the level of the confidence interval. More interestingly, the percentage depended on the scale [47] , with the quarterly scale showing the highest percentage of deviations from randomness. In turn, this trend suggests that the Covid-19 outbreak has had the strongest effect over mid-term scales than over short-term scales (i.e., weeks). Recently, Sui et al. [48] showed that the Covid-19 epidemic is a key factor influencing oil price, and that the effect may spread from the stock market due to speculative investor behavior. On the other hand, Le et al. [12] determined that the increases in Covid-19 pandemic cases, US economic policy uncertainty, and expected stock market volatility had an important contribution to the fall in the WTI crude oil prices in the first semester of 2020. The Russia-Saudi Arabia crude oil price war in March 2020 could exacerbate the effect of the Covid-19 pandemic, driving the prices to historical lows. Our results showed that the price fall and subsequent recovery were accompanied by a marked reduction of the price dynamics complexity as well as of the informational efficiency, which was reflected for mid-term scales (i.e., monthly and quarterly).

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The trading volume has been barely considered in the analysis of the predictability of the crude oil market. The trading volume can be seen as an indicator of the intensity of the market activity. The analysis of the predictability of the trading volume dynamics should provide valuable insights into the informational efficiency of the crude oil market.

The course of the trading volume SVD entropy for weekly scales in the last 22 years is shown in are not related to a particular market event. Interestingly, the 2008 Great Recession did not impact the trading volume predictability as the entropy remained in the randomness band. However, the Covid-19 had an important effect on the trading volume dynamics, which was reflected by the several peaks in Figure 5 .b, with the largest peak in April 2021. Similarly, Zhang and Hamori [11] reported that the impact of Covid-19 on the crude oil markets exceeded that of the 2008 financial It should be emphasized that, similar to the log price differences, the degree of unpredictability (i.e., distance to randomness) of the trading volume is sensitive to the scale. Table 2 shows that the percentage of the time the trading volume entropy is outside the confidence interval is visibly higher than that of the log price differences. The percentage values increased with the scale and can achieve values as high as 89.76% for 85% confidence interval and quarterly scale.

While crude oil price dynamics have received a lot of research attention, results on trading volume are scant. Abdullahi and Kouhy [49] did not find evidence of a strong link between trading volume and returns, suggesting that trading volume and returns are not driven by the same information flow. Our results pointed out in a similar direction by indicating that price tends to be unpredictable while trading volume dynamics contain some regularities. In contrast, Ji and Zhang [50] reported a significant Granger causal relationship from the return to trading volume (F-statistic = 3.61). The results presented in figures 6 to 8 showed that the Covid-19 outbreak has had a mixed impact on the predictability of the trading volume. While the entropy has decreased in the short term (i.e., weekly scale), it has been increased in the medium-term (i.e., quarterly scale). In turn, this would suggest that the Covid-19 outbreak has had a direct impact on short-term market activity through, for example, reduced trade volume, although its effects are spreading on longer scales. Some insights on the opposite pattern between price and trading volume during exogenous shocks (e.g., Covid-19 outbreak) can be drawn from the herding behavior effect [51] . Herding occurs in markets when investors follow the crowd instead of their analysis. Large, unfounded market rallies and sell-offs are often based on a lack of fundamental support. The fast sell-offs induce a strong market directionality, leading to sharp price drops. As a consequence, the market informational efficiency has deteriorated since the information flows are not uniformly distributed among the market participants. While price achieves a downturn directionality, the sell-off intensity, as reflected by the trading volume, increases its uncertainty as the market participants enter a phase of high uncertainty on when to sell their assets. Such uncertainty is reflected as an increased unpredictability of the trading volume. 

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