The Global Equity Market Reactions of the Oil & Gas Midstream and Marine Shipping Industries to COVID-19: An Entropy Analysis

This article quantifies the information flow between major equities in the Oil & Gas Midstream and Marine Shipping industries, on the basis of the effective transfer entropy methodology. In addition, the article provides the first analysis of the investors` fear and market expectations in these sectors, according to Rényi entropy approach. The period of study was extended over five years, to fully capture the pre/post-COVID situations. The entropy results reveal a major change in the underlying information flow pattern among equities in the Oil & Gas Midstream and Marine Shipping sectors, in the aftermath of COVID-19. According to the new (post-COVID) paradigm, the stocks in the Oil & Gas Midstream and Integrated Freight & Logistics industries have gained momentum in occupying six of the ten positions within the list of most-influencing equities in the market, in terms of information transmission. The disorder and randomness has decreased for over 89% of the studied equities, after virus outbreak. For the equities detected with high informationtransmission standing, the Rényi entropy results indicate that investors more likely showed higher level of future expectations and lower level of fear regarding frequent market events, within the post-COVID timeline.


Introduction
The world has witnessed a different scenery since the emergence of the Coronavirus (COVID- 19). One of such major changes has been the implementation of worldwide Non-Pharmaceutical Interventions (NPI)mainly in the form of mandatory quarantines, business closures and international travel restrictions -in order to control the spread of the virus. Although proven effective in reducing the rate of virus transmission [1,2], the implementation of such largescale containment measures has had negative economic consequences [3] which varies depending on their scale and severity of implementation. Among the repercussions of NPI, the diminishing international trade [4] -caused jointly by reduced production and market demand-should logically impact the transportation industry, in a sequel. As a matter of fact, the disruption in global supply chain, resulted from the COVID-19 emergence, drove the transportation industry to a near halt [5], particularly during the early months of the crisis.
A growing body of literature have focused on the impact of COVID-19 issue on the marine transportation sector, in terms of performance [6][7][8][9][10] and equity market reactions [5]. For example, Xu et al. (2021) [6] conducted a structural equation modelling analysis of confirmatory factor analysis and path assessment to study the impact of COVID-19 on the transportation and logistics sectors in China, and found statistically insignificant correlation between COVID-19 and the ocean freight in that country. Verschuur et al. (2021) [10] conducted an investigation on a global level and used the empirical vessel tracking information -as a high-frequency indicator of economic activity -to study the impact of NPI measures on maritime trade and found worldwide port-level trade losses, following the COVID-19 emergence, for which the ports in China, the Middle East and Western Europe were detected with the largest absolute losses. Furthermore, it was estimated that the reduction in maritime trade became as low as -9.6% in the first eight months of the crisis [10]. With regards to the equity market reactions, Kamal et al. (2021) [5] applied an event study methodology to assess the market reactions of selected shipping stocks -listed under New York Stock Exchange (NYSE)to several COVID-related news of optimistic and pessimistic nature. They found positive market reactions of marine transportation equities to the announcement of optimistic events -such as approval of the first COVID-19 vaccine or the proposal of economic stimulus plans-and adverse market reactions to pessimistic news. However, the number of such investigationslinking COVID-19 and transportation equities-seem to be quite limited, compared to the existing bulk literature on the COVID-19 impacts on global equity markets . As stock markets can be considered as a set of inter-connected and correlated equities, it is conceivable that the internal force of the markets be formed through the cumulative interactions of their listed firms.
As such, understanding the mutual information between equities should be important in analyzing the markets. However, such an information on connectivity (between equity participants) should be complemented by the information on the underlying directionality, in order to provide a complete image. Such a binary information set can be obtained by applying the concept of Transfer Entropy (TE), which is derived upon the formulation of conditional mutual information [37]. The transfer entropy methodology effectively quantifies the reduction in uncertaintyprovided by past values of variablesin predicting the dependent variable, as it is conditioned on these past values, and is considered as a model-free statistic capable of measuring the time-directed transfer of information between stochastic variables as well as providing the asymmetric information transfer measures in multivariate distributions [37]. A number of previous investigations have applied the TE methodology to analyze the financial markets [11,38,39].
For instance, Golmohammadi & Fazelabdolabadi (2021) [11] mapped the information transfer paradigm between 2200 equitiesglobally distributed within major financial marketsfor the periods before and after the COVID-19 outbreak. They report on drastic changes in major global equity markets in the aftermath of COVID-19 emergence, which was based on the changes in the underlying information flow pattern -derived from effective transfer entropywithin the markets studied [11] -Australia, Brazil, Canada, China, Germany, Iran, Japan, Qatar, Saudi Arabia, South Africa, South Korea, United Kingdom, and the United States. In addition, they report on substantial changes (nearly 70%) in the functionality of the market sectorsin terms of being a transmitter or receiver of informationencountered after COVID-19 emergence. Given the new circumstances that abound the global financial markets, it may be necessary to conduct an investigation to thoroughly understand the current standing of equities in the marine shipping and Oil & Gas midstream sectors. In this respect, the present work makes a two-fold contribution to the existing literatureproviding the first information transfer map between equities in the marine shipping and Oil & Gas midstream sectors (in a cross-market domain) and quantifying the market expectations and investor fear for selected equities in these sectors.

Methods
Used as the main processing stream in the present work, the method of transfer entropy, originally proposed by Shreiber (2000) [40], quantifies the asymmetric dynamics of two processes, using the conditional block entropy [41]. If the entropy is considered as a proxy to measure the uncertainty level inherent in optimally encoding the independent draws of a discrete random variable, the formulation of transfer entropy would be based on the premise of Shannon entropy [42]. Assuming as being a discrete random variable, with probability distribution function ( ), the Shannon entropy, , is defined as: If the random variable represents the event space of a time series, the sequence of its state outcomes until time , with back steps in time, becomes: If we denote the probability of observing the variable in state at time + 1 as ( +1 ∨ ( ) ) = ( +1 ∨ , . . . , − +1 ) then the average number of bits needed to encode the output state of the variable in time + 1 with known backstep valuesthe entropy of +1 -can be written as: where the summation runs over all the possible values of ( +1 , ( ) ), for a fixed time .
The value of the calculated entropy hence depends on the selection of the block length -referred to as conditional block entropywhich decreases along the increase in the length of the block, as long as − contains more information to predict +1 than − +1 [41].
For a bi-variate case, the value of transfer entropy can be obtained by accounting the deviation from the generalized Markov property. Considering a time series , the sequence of its observations until time , with back steps in time, can be taken as: An information flow from process to process exists, if the information in ( ) can be valuable in forecasting +1 , despite the information collected from ( ) . The transfer entropy, → ( , ), is then formulated by Shreiber (2000) [40] as Equation 5, to subtract the information already contained in ( ) : where ℎ , ( . )denotes the conditional entropy of , given the information of both ( ) and ( ) blocks.
The results of the transfer entropy may be subject to bias, due to small-sample effects. To correct for this bias, it is suggested [43] to compute the effective transfer entropy, → ( , ), between the two processes. The effective transfer entropy is calculated by subtracting the value of transfer entropy obtained from Equation 5 from the value obtained after conducting a shuffling operation on process , ℎ → ( , ). The shuffling procedure entails taking random draws from the distribution of and re-arrangement of the selected set to generate a new time series, in order to destroy statistical dependencies between the two processes as well as the time series dependencies of [42]: ℎ → ( , ) → 0 as the sample size increases and becomes non-zero in case small-sample effects exist.
The set of probability measures listed above are established over discretized values of the variables; therefore, the variables` data should be grouped into non-overlapping partitions, a priori. For this reason, the symbolic encoding scheme dominantly used would select the size of the bins, according to the 5% and 95% empirical quantiles of the data -[0.05] and [0.95] . As a result, the symbolically-encoded time series, , takes the following form: To account for frequent and rare events, signal complexities were assessed by incorporating the Rényi entropy (as Equation 9) for each time series considered.
where ( ≥ 0) represents the order of Rényi entropy, which favors rare events when < 1 and privileges frequents events as > 1 [44]. The estimation of the probabilities in Equation 9 was made through the Gaussian kernel functions.

Data Description
The information used as input in the present study, is comprised of the closing daily prices of stocks of 70 companies, which presumably represent the main equities in the Oil & Gas Midstream and Marine Shipping sectors worldwide. The names of the companies selected are listed in Table 1. Such a name selection also ensures a crossmarket inspection of the information transfer, as the equities are being traded in different financial markets. The input data was obtained from Yahoo Finance. The data was acquired for the time span between (2016-Aug-01 and 2021-Aug-01). This length was later divided into two periods, to account for prior/post-COVID timelines. The date used to set this division was taken to be 30-January-2020, on which the pandemic outbreak was officially declared by the World Health Organization [44].

Results and Discussion
The effective transfer entropy was calculated, for each pair of the listed stocks (Table 1) along the both directions -→ and → . For each state in a given pair, the calculations were attempted over the periods, before and after the COVID-19 outbreak. The selection for the lag ordersand -was taken as unity, which is an appropriate choice when analyzing the financial markets [42]. The number of shuffling operations performed was set to one hundred, to ensure efficient removal of bias from the established results. Figures 1 to 4 depict the computed results for the values of the effective transfer entropy for the companies considered. To ease its visual inspection, the results are presented separately for entries 1-40 and 41-70 of the list (Table 1), as well as for the pre/post-COVID periods. With respect to the color interpretation of the results, a more positive number indicates more information transfer (from stock y to stock x) and zero is the case in which no information transfer has been detected, within the considered time span. The whole set of computed results for all the companies considered -including the transfer entropy, the effective transfer entropy and the corresponding statistical measures (standard deviations, p-values)can be obtained from the corresponding author, upon reasonable request.    Table 1

), after the COVID-19 outbreak
The effective transfer entropy results show the formation of a new information transfer paradigm, after COVID-19 emergence, among major equities in the Oil & Gas Midstream and Marine Shipping sectors. According to our results, the new price action of equities acts more sensitively to each other (with few exceptions) and the overall information transfer in the two sectors has increased after COVID-19 outbreak, even in the devised cross-market domain. Given the market capitalization of the selected equities, a general extension of this finding to the post-COVID status of these two sectors is plausible.
With respect to the information transmission, the market has seen an altered list of major players in the Oil & Gas Midstream and Marine Shipping sectors. As part of our analysis in the present paper, we have also studied the status of equities (in these sectors) with respect to their net information flow. An equity was then interpreted as being an information transmitter (receiver) if the net information outflow was positive (negative). In this context, a more positive net information outflow value rendered the equity as a holding a more influencing role in the market. Tables 2-3 list the main information transmitter equities in the Oil & Gas Midstream and Marine Shipping sectors, before and after COVID-19 respectively. As evident from the list, the Marine Shipping equities have lost grounds to other industries in the market, in the post-COVID timeline. This argument is based on the fact that six positions out of ten most influencing equities in these sectors were taken by the firms operating in the Oil & Gas Midstream and Integrated Freight & Logistics industries (Table 3)    In terms of market expectations and investor fear, the reactions have been mixed. Table 4 provides the net values of Rényi entropy for equities considered (Table 1), computed up to the order of 20. This net value was calculated as the Rényi entropy difference between the corresponding post/pre-COVID values. The results follow four distinct patterns, as described in Table 5.    Randomness and disorder has decreased in the post-COVID timeline. The level of information disorder in frequent events has increased during the pandemic, which indicates that investors showed higher level of fear and lower level of future expectations regarding most frequent events.

II
Randomness and disorder has decreased in the post-COVID timeline. The level of information disorder in frequent events has decreased during the pandemic, which indicates that investors showed lower level of fear and higher level of future expectations regarding most frequent events.

III
Randomness and disorder has increased in the post-COVID timeline. The level of information disorder in frequent events has increased during the pandemic, which indicates that investors showed higher level of fear and lower level of future expectations regarding most frequent events.

IV
Randomness and disorder has increased in the post-COVID timeline. The level of information disorder in frequent events has decreased during the pandemic, which indicates that investors showed lower level of fear and higher level of future expectations regarding most frequent events.
For the majority of the equities considered (over 89%) the randomness and disorder has decreased after the pandemic. The investors` expectations and level of fear for this group, however, was evenly distributed. In other words, for the most frequent events in the market, the investors showed both lower/higher level of future expectations. Table 6 reports the equities, according to their detected pattern. For the most influencing stocks (Table 3), the Rényi entropy pattern belonged to group II (Table 4), which indicates that investors had shown lower level of fear regarding frequent market events in these equities, in the post-COVID timeline.

Conclusion
The entropy analysis of equities in the Oil & Gas Midstream and Marine Shipping sectors reveals changes in its underlying information flow pattern, since the emergence of the COVID-19 virus. The post-COVID market action of equities in these two sectors behaves more sensitively to each other, as deducted from the effective transfer entropy results. According to the new (post-COVID) paradigm, the stocks in the Oil & Gas Midstream and Integrated Freight & Logistics industries have gained momentum in occupying six of the ten positions within the list of most-influencing equities in the market, in terms of information transmission. The disorder and randomness has generally decreased for the studied equities after the COVID-19 emergence. The investors' fear and future market expectations for the studied equities is found to be mixed. Nevertheless, the Rényi entropy results indicate that investors more likely showed lower level of fear regarding frequent market events in the equities possessing high information transmission status in the market.

Author Contributions
All authors have equally contributed towards Conceptualization, methodology, formal analysis, investigation, resources, writing-original draft preparation, writing-review and editing, visualization. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement
The data presented in this study are available in article.