COVID-19: A Game-changer to Equity Markets?

This article applies the effective transfer entropy methodology to quantify the information flow between equities in major global equity markets in Australia, Brazil, Canada, China, Germany, Iran, Japan, Qatar, Saudi Arabia, South Africa, South Korea, United Kingdom, and the United States—a pool of 2200 companies is included. To account for COVID-19 impacts, the period of the study was extended over two years. The results show changes to the information flow pattern after COVID-19, with the largest changes being encountered in Australia, Brazil, Canada, Japan, and the United States for their largest market participants (by market capitalization). In comparison, the Asian markets' information flow patterns show less noticeable changes following COVID-19. On a sector level, most of the markets studied have seen substantial changes in the functionality of their sectors—in terms of being a transmitter or receiver of information—after COVID-19's appearance. The fraction of sectors with a complete change in their influencing role since COVID-19 has been over 70% in Australia, Canada, South Africa, and the United States. The financial services sector has retained its role as being the most influential sector in 6 out of 13 markets considered after COVID-19. For most of the markets, the basic materials, communications, energy, and utilities sectors have retained an intermediate position in the information flow diagram after COVID-19. The German market has been the only market, in which the main information transmitter and receiver sectors have remained unchanged since COVID-19. The results suggest drastic moves in major global equity markets, which have been concurrent with


Introduction
Many of us think of the world in two time periods -before and after the COVID-19 pandemic.This thinking might have been initiated because of the complex elements of the COVID-19 issue involving both known and unknown.Apart from its medical complexities, the COVID-19 crisis has left us with ambiguities about its other impact dimensions as well.Thus far, studies have found that this global disruption has influenced several affairs with respect to the psychological well-being of people [1], the environment [2], education [3], and geopolitics [4], to name a few.
In terms of finance and capital markets, researches have been performed to study the link between the COVID-19 pandemic and financial aspects in the scope of banking [5][6][7][8][9], cryptocurrency [10][11][12][13][14][15][16], exchange rates [17,18], financial system [19], insurance [20], interest rates [21] and equity markets .In the case of equity markets, the effects of the COVID-19 outbreak have been investigated on the return [27-28, 41-43, 46] and volatility [26,32,[42][43][44] of the market in developed [22,29] and developing [39,45] economies.The bulk of these studies have found marked differences in the market behavior before and after the pandemic announcement -an increase in volatility and a decrease in returns after the pandemic outbreak.Nevertheless, this market reaction is found to vary across countries [32,44].For example, Harjoto et al. [22] found the stock returns, volatility, and trading volume of equity markets in developed countries to be affected by the news about COVID-19 mortality rates, while in the case of developing economies, both mortality rates and the number of infections were found to be influential.Grouping the COVID-19 deaths and cases into an anxiety index, Yu et al. [27] found a dynamic correlation between stock returns and the anxiety index, which has been decreasing since the introduction of the mRNA-based vaccine.Fry-McKibbin et al. [29] suggest the global equity markets are exposed to pandemic risk factors and find evidence of contagion across multiple channels from the U.S. to G20 equity markets.Bissoondoyal-Bheenick et al. [42] conclude that the stock return and volatility connectedness in G20 markets will increase as the COVID-19 crisis escalates.Benlagha and El Omari [31] report that the linkage between the major global equity markets (China, Germany, Japan, U.K., and U.S.) has increased after the COVID-19 outbreak.In addition, the risk spillovers from equity markets in the American and European regions are found to have increased following the COVID-19 epidemic [33], while Asian markets have been more resilient to the situation [34]-showing a decrease in the risk contagion effects [33].In this respect, the negative effects of the COVID-19 pandemic on Asian equity markets are reported to be less severe compared to other regions of the world.For example, Salman and Ali [45] studied the equity markets in several Asian countries (Bahrain, Kuwait, Oman, Qatar, Saudi Arabia, and the United Arab Emirates) and found a negative short-term impact on these markets from the COVID-19 crisis.Some studies [30] have also pointed out the weak integration of some Asian markets (China and Saudi Arabia) into the global market reaction that abounds following the COVID-19 appearance.
To further elucidate the equity market response to the COVID-19 issue, some empirical studies have focused on the market performance through its constituents, on a sector level [37,38,40].For example, He et al. [38] reported the manufacturing, information technology, education, and health-care industries in China as being the most resilient sectors within the pandemic timeline, while the transportation, mining, electricity, and environmental industries are found to be the most vulnerable to the COVID-19 crisis in the Chinese market.Alam et al. [40] inspected the sectoral performance of the Australian equity market in the immediate days following the COVID-19 announcement and found good performance for the telecommunications, pharmaceuticals, and healthcare sectors in this period.Buszko et al. [37] analyzed the sector constituents of the Warsaw Stock Exchange during the COVID-19 pandemic and found the construction, information technology, game developers, biotechnology, telecoms, and new technology sectors to exhibit positive relative changes in their average prices in the medium-term period after the crisis began.While some other sectors-automobiles, banking, chemicals, clothes, energy, food, mining, oil & gas, pharmaceuticals, and real estatewere found unprofitable within the same timeline in that market.
The mentioned literature should signal the radical changes undergoing the global equity markets in the aftermath of COVID-19 outbreak.Given the new circumstances, acquiring knowledge of the exact interaction between market constituents should be of utmost importance.Nevertheless, the methods adopted in some of the previous investigations seem to be barely adequate to provide us with a detailed picture of the market activity.For instance, the Granger causality method is unable to measure the nonlinear relations between financial time series and cannot ascertain the direction of information flow between the two sets [47].In this respect, the methodology of transfer entropy [48] can address both deficiencies -with the added benefits of being invariant to linear transformation and not being restricted to linear dynamics.The output of this new proxy -information flow between market constituents -can be used to determine the information-dominant component(s) in a market.This latter data can be directly used by stakeholders to react appropriately to information transfer dynamics by adjusting their market status based on the interaction hierarchy among the components in their portfolio.In addition, the acquisition of a true map of information flow among equities would enable us to reliably determine the performance of market sectors in a given period by accounting for the net information flow for the companies belonging to the given sectors.Although the concept of (effective) entropy transfer has been expensively applied in financial markets [47,[49][50][51][52][53][54][55], its application in analyzing the market status after COVID-19 outbreak seems to be un-attempted.
The present article contributes to the existing literature in this field in different ways.The article reports on the first application of the transfer entropy methodology to analyze the sector performance of several equity markets in the aftermath of COVID-19.In this respect, the article has reached a great milestone of quantifying the information flow between market constituents-comprised of 2200 companies-active in major equity markets worldwide.The number and distribution of companies/markets considered should further weight our results, to be a detailed map of the current market status within the COVID-19 crisis, in a global sense.The results presented and the methods applied herein are the first to be reported for some Asian markets, such as the Tehran Stock Exchange, for which previous studies are minimal.

Research Methodology
In this section, we outline the method of transfer entropy which was used as the main processing stream in the present work.Proposed by Schreiber [48], the transfer entropy methodology quantifies the asymmetric dynamics of two processes, using the conditional block entropy [55].In this context, the entropy is defined 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 is then based on the premise of Shannon entropy [47].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  + 1with 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  [55].
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 [55].
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 Schreiber [48] as Equation 5, to subtract the information already contained in   () : → (, ) = ℎ  () − ℎ , (.) 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 [52] 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  [47]  → (, ) =  → (, ) −   ℎ → (, ) 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: In order to analyze the susceptibility of results to this choice of bin size, a different discretization scheme has also been tested in the present work, which incorporates more empirical quantiles of the data into its encoding system -Equation 9:

Data Description
The information used as input in the present study, is comprised of the closing daily prices of stocks -belonging to a number of 2200 companies -being traded worldwide.Table 1  The input data was obtained from Yahoo Finance, with the exception of TSE data, which was downloaded directly from the TSE data repository.The data was acquired for the time span between [2019-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, which is date on which the pandemic outbreak was officially declared by the World Health Organization [56].

Results of Discussion
The effective transfer entropy was calculated, for each pair of the stocks in a given market/index, 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 orders -and was taken as unity, which is an appropriate choice when analyzing the financial markets [47].The number of shuffling operations performed was set to one hundred, to ensure efficient removal of bias from the established results.Figures 1 to 26 depict the computed results for the values of the effective transfer entropy for the largest companies (by market capitalization) in the markets considered.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 results are provided for the both periods, before and after the virus outbreak.In order to maintain the full integrity of this paper, the whole set of computed results for all the companies considered is also being distributed in the supplementary information to this articleincluding the effective transfer entropy and its corresponding statistical measures (standard error, p-value).
A quick visual inspection of the results should reveal formation of new paradigms in the studied markets, after COVID-19 appearance.For major market participants (Figures 1 to 26), the newly-formed information flow pattern is seen with different levels of integrity across markets/countries.For example, the information flow between equities has risen sharply in the S&P TSX after COVID-19 (Figure 24).This rise in information flow is also the case in BSE, FTSE, JSE, Nikkei, S&P500, and S&P ASX markets.Therefore, the price action of largest equities in these markets has become more sensitive to each other, after COVID-19.The largest shift in the information flow pattern has occurred in BSE market, where the effective transfer entropy was kept minimal before the virus crisis (Figure1).In other markets, the change in information flow structure after COVID-19 has not been noticeable.This should be the case for CSI, FSE, KSE, QSE, SASE, and TSE markets.Because the majority of the Asian markets considered herein -CSI, KSE, QSE, SASE, TSErely within this category, it can be further deducted that the Asian markets have been relatively insensitive to COVID-19 crisis, with the exception of Nikkei.As part of our analysis in the present paper, we have also studied the change in market functionality after COVID-19, on a sector level.For a selected sector in a given market, the (average) information inflow/outflow was determined by summing the effective transfer entropy inflow/outflow of the stocks in that sector and dividing the result by the number of stocks involved.The net information outflow, , was taken as the difference between the average information outflow and the average information inflow.A sector is then interpreted as being an information transmitter (receiver) if the net information outflow is positive (negative).Tables 2 to 15 lists the computed net information outflow for the sectors within the markets considered.For the TSE market, only the top 30 companies by market capitalization were studied.Therefore, the listed sectors in Tables 14 to 15 merely correspond to the sectors of the selected TSE stocks.In general, the results indicate a large number of shifts to have occurred in the status of sectorsin terms of being a receiver or transmitter of information -after COVID-19 in the studied markets.This change in sector functionalityshifting from an information transmitter to receiver, or vice versa-can provide valuable information on the general price action.
Based on our results, the highest number of shifts in sector functionality has been encountered in JSE, S&P500, S&P ASX, and S&P TSX markets.In South Africa, for instance, the sector functionality of over 90% of JSE sectors has altered after COVID-19suggesting a complete re-shaping of its underlying information transmission network.In other words, the sectors which used to influence others, now tend to be influenced, and the reverse exists.The fraction of sectors changing functionality has also been high in the Australian and North American markets -73% (S&P ASX), 71% (S&P TSX), and 68% (S&P500).The European markets have been more stable, in this regards.For the markets in Germany and the U.K., less than half of the sectors have changed functionality after COVID-19 -43% (FSE), and 37% (FTSE).Since FTSE was earlier found with a large change in its information flow pattern (Figure 8), the fairly low number of changes in the status of its sectors should mean that the conceived post-COVID changes are mainly caused by an alteration in the intensity (and not the direction) of information flow between equities, in that market.For the market in Brazil, the situation resembles its North American counterparts -58% (BSE).The fraction of sectors changing functionality has been at/below the middle for the markets in Iran, Japan, Qatar, Saudi Arabia and South Korea -50% (TSE), 50% (Nikkei), 42% (QSE), 35% (SASE), 46% (KSE).For China, this fraction is slightly above the average -53% (CSI).Following the same analogy put forth for the FTSE market in the above, it can be concluded that the changes in CSI market after COVID-19, are mainly due to changes in the direction of information flow between equities, in that market.In other words, the intensity of effective transfer entropy has remained mostly unchanged after virus crisis, in the CSI market (Figure 4).
With respect to the performance of industries after COVID-19, the response of the markets are varied.The financial services, real states, utilities, and consumer cyclical sectors are generally found in the transmitter status in these markets.The financial services sector, for example, has maintained it influencing role after COVID-19 outbreak, in 9 out of 13 markets studiedwith the exception of FSE, KSE, QSE and TSE.The healthcare sector is found as being an information receiver in the majority of marketsexcept QSE, and S&P TSXafter COVID-19.For some sectorssuch as the Industrialsthe situation has been mixed; as it has functioned both as an information transmitter and an information receiver, in relatively equal number of markets studied, after COVID-19.The real estate sector seems to be the most influencing sector in some Asian markets; such as Qatar and Saudi Arabia, which has presumably received the least impacts from COVID-19.
In the period after COVID-19 outbreak, the pattern of markets has posed different sectors as their leading (mostinfluencing) ones.Based on our results, this updated list is not unanimous.Tables 16 to 19 list the names of the main information transmitter (receiver) sectors, for pre/after-COVID periods, in the markets studied.The names of the sectors is based on the names, which were attributed by the Yahoo Finance.The tables includes two entries for the TSE market, the second of which relates to the case when the effective transfer entropy is evaluated through the detailed encoding system -Equation 9.The basic materials, communications, energy, and utilities sectors seem to have an intermediate position in the information flow diagram in most of the markets studied after COVID-19.Based on our results, the German market -FSE-seems to be the only market where no change has occurred as for its main information transmitter/receiver sectors after the COVID-19 outbreak.As mentioned earlier, the results presented this far have been obtained using the prevalent encoding system-based on 5% and 95% quantiles of data-for its evaluation of probability (Equation 8).The sensitivity of results towards this choice was investigated for the case of the TSE market by evaluating the effective transfer entropy through a more detailed encoding system (Equation 9).As one may notice, the results tend to be sensitive to this choice, as the computed effective transfer entropy values and the resultant sector behavior might differ from the earlier choice.Since the bulk of literature adopts a similar encoding system (Equation 8) in their analysis, the results presented herein should form the basis for later comparison.Nevertheless, further investigations are suggested to analyze the issue on the premise of a more detailed encoding system.

Table 17 . The main information receiver sectors, before COVID-19 Market/Index Sector
* Evaluated through Equation9