key: cord-0876735-anfh0ouv
authors: Chu, Amanda M.Y.; Chan, Lupe S.H.; So, Mike K.P.
title: Stochastic actor‐oriented modelling of the impact of COVID‐19 on financial network evolution
date: 2021-08-18
journal: Stat (Int Stat Inst)
DOI: 10.1002/sta4.408
sha: a2b3d7a5a10b10c61230abcc67ccf2a4e499687a
doc_id: 876735
cord_uid: anfh0ouv

The coronavirus disease 2019 (COVID‐19) pandemic has led to tremendous loss of human life and has severe social and economic impacts worldwide. The spread of the disease has also caused dramatic uncertainty in financial markets, especially in the early stages of the pandemic. In this paper, we adopt the stochastic actor‐oriented model (SAOM) to model dynamic/longitudinal financial networks with the covariates constructed from the network statistics of COVID‐19 dynamic pandemic networks. Our findings provide evidence that the transmission risk of the COVID‐19, measured in the transformed pandemic risk scores, is a main explanatory factor of financial network connectedness from March to May 2020. The pandemic statistics and transformed pandemic risk scores can give early signs of the intense connectedness of the financial markets in mid‐March 2020. We can make use of the SAOM approach to predict possible financial contagion using pandemic network statistics and transformed pandemic risk scores of the COVID‐19 and other pandemics.

through network modelling. These network analyses of COVID-19 have prompted us to model financial networks by incorporating pandemic risk information in our modelling. present evidence of significant changes in financial market connectedness in the Hong Kong stock market during the early stage of COVID-19. study the financial connectedness and systemic risk during the COVID-19. In terms of social and financial anxiety, panic can spread faster than COVID-19, especially when people have little knowledge of the pandemic (Depoux et al., 2020) . It is thus particularly meaningful to examine how COVID-19 affects, and perhaps may explain, financial market contagions.

From the findings in Shehzad et al. (2020) , there is evidence that COVID-19 has a substantial impact on financial markets and the world economy. Big financial events and global public health disasters, like the COVID-19 pandemic, often trigger financial contagion and induce systemic risk in financial markets (Haldane & May, 2011; Elliott et al., 2014; Guo et al., 2021) . To understand and model the impact of the COVID-19 pandemic on financial markets, it is natural to incorporate information regarding the pandemic situation in financial modelling. In this paper, we introduce the use of the pandemic network statistics in So, Chu, Tiwari, et al. (2021) , who have showed that the pandemic network statistics provide early warning signals of the pandemic. Specifically, we adopt the stochastic actor-oriented model (SAOM) in Snijders (2002) to incorporate the pandemic statistics in modelling financial network evolution.

The SAOM has been applied to various fields. Boda et al. (2020) investigate how assigning fresh undergraduate cohorts into small groups two months prior to their first day at university could influence their friendship network. Cao et al. (2017) apply the SAOM to study the evolution of the macro structure of project-based collaborative networks. The SAOM regards longitudinal data as snapshots of a continuous-time Markov process, which can be represented by networks. The nodes in the networks are also known as actors. The connections among the actors are represented by edges or ties (which can be directed or undirected). We model financial networks with undirected edges using the SAOM. The actors in the SAOM are the composite market indices of major financial markets. Each composite index is associated with several pandemic network statistics and a pandemic risk score of the region to which the composite index belongs. Related network analysis of financial markets during COVID-19 can be found in the literature. Billio et al. (2021) apply a semiparametric matrix regression model to the spread of COVID-19 in financial networks. Lai & Hu (2021) study the systemic risk of global stock markets under COVID-19 based on complex financial networks. The main advantage of using the SAOM to model financial networks is that we can take their dynamic features into consideration while incorporating information from the pandemic networks explicitly to model network evolution.

The paper is structured as follows. Section 2 sets out the detailed methodology, including details of the construction of the financial networks, the SAOM modelling framework and the pandemic related variables used as our predictors in the SAOM. In Section 3, we describe the data analysis, visualization of the fitted model results and our main findings. Section 4 sets out our conclusions.

2 | METHODOLOGY 2.1 | Dynamic financial networks A major objective of this paper is to model the impact of the COVID-19 pandemic on financial networks using the SAOM. In particular, we will incorporate pandemic network information in the prediction of financial network evolution. Specifically, we adopt the statistics of the pandemic networks as the characteristics of the financial markets in the SAOM framework, which is a useful way to predict the formation of connections within financial networks. A network is formulated by nodes and connections which link the nodes together. We denote the set of vertices at time t by V(t) and the set of edges at time t by E(t). We first illustrate how to construct a dynamic financial network, which is a time series of networks GðtÞ ¼ ðVðtÞ, EðtÞÞ constructed based on partial correlations.

Suppose that we have data from N + G market indices in T days. Let P it be the adjusted closing price of the ith index in day

where the first N indices represent the stocks or composite indices of the international financial markets. The remaining G indices are the benchmark market indices used to represent the common market factors explaining the co-movement of the first N indices . The variables of interest are the continuously compounded returns, R it ¼ logP it À logP iðtÀ1Þ , for t ¼ 2,…, T. We denote the vector of returns of the first N indices by R 

After standardizing Σ SjM, t , we obtain the desired partial correlation matrix, ρ SjM,t .

To model the impact of the COVID-19 pandemic on financial network connectedness, we first construct time-varying financial networks by estimating the partial correlation matrix derived from Σ SjM, t . The partial correlation matrix of R ðSÞ t , given R ðMÞ t , can capture the dependence between stock market index returns which cannot be explained by global market movement. The connectedness due to this type of dependence can help us to assess the systemic risk which can be attributed to unusually severe stock market co-movement. Following Xu et al. (2017) and , we use a moving-window approach to estimate Σ SjM, t , the partial covariance matrix at time t. Effectively, for a window of width w, we estimate the partial covariance matrix at time t for t = w, w + 1, … , T by calculating the sample covariance matrix,Σ SS,t ðwÞ ¼ 1 wÀ1 P t s¼tÀwþ1 ðR ðSÞ s Àμ t ðwÞÞðR ðSÞ s Àμ t ðwÞÞ T , whereμ t ðwÞ is the sample mean of R ðSÞ s , s = t À w + 1, … , t. Similarly, we calculate the sample estimatesΣ MM,t ðwÞ andΣ SM,t ðwÞ based on the window of w observations R ðSÞ s and R ðMÞ s , respectively, for s = t À w + 1, … , t. Then, the partial covariance and the partial correlation matrix ρ SjM,t can be estimated byΣ SjM,t ðwÞ ¼Σ SS,t ðwÞ ÀΣ SM,t ðwÞΣ MM ðwÞ À1Σ MS,t ðwÞ andρ SjM,t ðwÞ ¼ diagfΣ SjM,t ðwÞg À 1 2Σ SjM,t ðwÞdiagfΣ SjM,t ðwÞg À 1 2 : Having computed the correlation matrices, we can define the (i, j)th entry of A t , the adjacency matrix of G(t) (i.e., the financial network at time t) as

where ½ρ SjM,t ðwÞ ij is the (i, j)th element ofρ SjM,t ðwÞ and r is the threshold. We take r ¼ 0:5 in this paper, and A ii, t ≡ 0 for convention.

To construct these dynamic financial networks, we collect from Bloomberg the adjusted closing prices (P (MSCI Inc., 2021a) . We believe that these two MSCI indices give good proxies to explain possible co-movement among composite indices due to global market factors. Using a window width of w ¼ 40 for the composite index and global amrket index data, we calculate the partial correlation matricesΣ SjM,t for t = w, w + 1, … , T, or from 12 February to 27 May 2020 (T À w þ 1 ¼ 115 À 40 þ 1 ¼ 76 trading days). Then, we form the edge set E(t) on each day t for the pairs with correlations on day t that are greater than the threshold r.

The financial networks on every Wednesday from 19 February to 27 May 2020 (a total of 15 snapshots) are selected as our observed moments in the longitudinal analysis using the SAOM, and the data on or after 12 February 2020 are used as the lagged predictors in the model.

Choosing Wednesdays allows us to focus on weekly changes in the financial networks rather than being influenced by after-the-weekend or weekend effects. Hence, we pick only the day in the middle of each week, that is, Wednesday, and use this to represent the network pattern in the corresponding week.

Using the method in Section 2.1, we construct financial networks using data from 12 February to 27 May 2020 on 41 market indices.

Details of the markets including the countries and regions in which they are located are shown in Table 1 . Figure 1a presents network diagrams on four selected days (for network illustration purposes), 4 March, 11 March, 15 April and 27 May 2020. Arcs indicate the connections between two financial market indices with the colours indicating the magnitudes of partial correlations. Note that the adjacency matrices do not take the magnitude of the partial correlations into account. The coloured arcs are for showing supplementary information on the strength of the partial correlations. It is not surprising to see that financial markets that are close to each other are highly connected (Asgharian et al., 2013) , especially during volatile periods (Solnik et al., 1996) . On 4 March 2020, at an early stage of COVID-19, we can see obvious connections in the financial network, and the network connectedness is quite high. Comparing the four selected days, the financial network connectedness is the highest on 11 March 2020, when WHO declared COVID-19 as a global pandemic. It then drops in April 2020 compared with March 2020 as shown in Figure 1a . On 15 April 2020, the financial network has sparse connections, which may be due to the partial relief of the COVID-19 pandemic. On 27 May 2020, the network connectedness increases, which may be due to the possible recurrence of the pandemic.

In this section, we define the SAOM to model the evolution of the dynamic financial networks defined in the previous section. In particular, our 'actors' at the nodes in the SAOM are the international financial market indices. We investigate the impact of the COVID-19 on the time series pattern of these financial networks by using the network statistics and the risk scores from the pandemic networks as covariate inputs to the SAOM. We assign pandemic network statistics and risk scores to each financial market according to physical locations.

The SAOM is a class of network models which mimics network evolution as individual actors/nodes creating, maintaining or terminating ties/ edges to other actors, which can be characterized by covariates or behaviours. We adopt the non-directed network setting to model the financial networks. Following Snijders (2017), we need to focus on the 'opportunity' and 'decision rule' about changing a tie/edge/connection, where the formation of edges can be described in two microsteps. For simplicity, we select the one-sided initiative; that is, one actor or financial market is selected and has a multinomial choice about changing one of the edges. The selected actor/node has the right to change an edge. We first state some of the underlying model assumptions (following Snijders, 2017) before proceeding to describe the SAOM construction. In the SAOM setting, we define a continuous time network characterized by the adjacency matrix at time t, A t ¼ ½A ij,t Nt i,j¼1 . The network is a continuous time stochastic 

The Markov property implies that A t is conditionally independent of A s and s < t m À 1 given A tmÀ1 .

From now on, we use the terms actor and node interchangeably to represent financial market indices. The SAOM takes a micro-step mechanism. At each time t, only one selected actor has the opportunity to make a change. The selected actor can decide whether to make a change in his/her edges. The network change process can be decomposed into two sub-processes: the opportunity to change and the actor's decision. The

Markov assumption implies that the waiting time between consecutive changes for an actor is exponentially distributed. Let λ i (x; ρ m ) be the rate of change of the network at time point t with network x, that is, the mean parameter of the exponential distribution, where t T m :¼ ½t m , t mþ1 Þ and ρ m is a parameter. We set T m to be the mth period. In general, λ i (x; ρ m ) takes the form (Snijders et al., 2007) 

where a ki (x) denotes the kth statistics of actor i determining the characteristics of the ith actor in network x and α k is the coefficient indicating dependence on the statistics a ki (A t ), i ¼ 1,2,…, N t . In the first microstep, actor i in network x is selected to have an opportunity to make changes in his/her edge with the probability (Snijders, 2017) Pðactor

In this paper, we take λ i ðx;ρ m Þ ¼ ρ m , meaning that the transition rates of all actors are the same in the interval [t m , t m + 1 ) for m ¼ 0,1, …, M À 1. Then, the probability of actor i being selected at time t [t m , t m + 1 ) is

that is, all actors have the same probability of being chosen. The change that takes place in the SAOM in a microstep is regarded as an actor's choice and so the model is 'actor-oriented'.

In the second microstep, the selected actor may create or drop one edge, or make no change. Let x ðAEijÞ f0,1g NtÂNt be the candidate network which is identical to x except for the edge A ij, t . The decision of actor i to make a change from x to x (± ij ) is based on utility theories suggesting that the action will be decided by maximizing the following utility function

where f i ðx ðAEijÞ , v; βÞ is an objective function capturing all related information from the current network and covariates v and ε i is a random component. A common assumption in utility theories is to set ε j $ Gumbelð0,1Þ, i.e., the pdf of ε j is e Àðεjþe Àε j Þ , to be identical and independent for j ¼ 1, 2,…,N t . Then, it can be shown that U i ðx ðAEijÞ , v; βÞ is maximized with the transition probability to change from x to x (± ij) given by (Snijders, 2017) 

where x (± ii) = x when there is no change being made. In the SAOM adopted in this paper, f i ðx ðAEijÞ ; βÞ takes a linear form

where s ki (x (± ij ) , v)'s are the K effects of the networks, which depend on the candidate network x (± ij ) and the covariates v only, 1 and β ℝ K is a vector of parameters. We specify below various choices for the effects based on the data. 1 In general, the effects could be dependent on the current network x.

To estimate the unknown parameters in the SAOM, we apply SIENA (Simulation Investigation of Empirical Network Analysis) using R for our financial network modelling. More details of SIENA can be found in Ripley et al. (2021) and Snijders (2019) . To study the temporal properties of the dynamic financial networks, we consider two different kinds of effects s ki (x (± ij ) , v). The following effects depend only on the network configurations, which are (1) density effect (degree in RSiena, the SIENA in R), defined as s 1i ðA t , vÞ ¼ A iþ,t , where A iþ,t ¼ P j A ij,t ; (2) transitive triads effect (transTriads in RSiena), defined as s 2i ðA t , vÞ ¼ P k < h A ik,t A kh,t A hi,t ; (3) out-isolation (outIsoin RSiena), defined as s 3i ðA t , vÞ ¼ IfA iþ,t ¼ 0g, and (4) out degree related activity effect (outAct in RSiena), defined as s 4i ðA t , vÞ ¼ A 2 iþ,t . To understand the implications of these effects, we can re-write (1) by dividing both the numerator and the denominator by expðf i ðx, v;βÞ: is the log-odds defined by

implying that the difference, Δs ki ¼ s ki ðA ðAEijÞ t Þ À s ki ðA t Þ, contributes linearly to the log-odds ratio. We now examine how the four Δs ki , k = 1, …, 4, affect the log-odds ratio.

(1) The term in LOðA t , A ðAEijÞ t Þ relating to the difference in the density is β 1 Δs 1i , where Δs 1i ¼ A ðAEijÞ iþ,t À A iþ,t . If β 1 > 0, then creating an additional link in the graph A ðAEijÞ t increases the log-odds, while deleting a link decreases it. The effect of Δs 1i is similar to the effect of a categorical variable in linear regressions.

(2) The term in LOðA t , A ðAEijÞ t Þ relating to the difference in the transitive triads is β 2 Δs 2i , where Δs 2i is the number of additional closed triplets created in A ðAEijÞ t . If β 2 > 0, creating a link that forms additional triangles in the graph A ðAEijÞ t increases the log-odds and encourages the small-world property.

(3) The term in LOðA t , A ðAEijÞ t Þ relating to the difference in the out-isolation is β 3 Δs 3i , where Δs 3i ¼ IfA ðAEijÞ iþ,t ¼ 0g À IfA iþ,t ¼ 0g, the number of additional isolated nodes when the graph is changed from A t to A ðAEijÞ t . This effect is included because we want to test whether isolated nodes have different behavior than connected nodes on link formation.

(4) The term in LOðA t , A ðAEijÞ t Þ relating to the difference in the out degree related activity is β 4 Δs 4i , where Δs 4i ¼ ðA ðAEijÞ iþ,t Þ 2 À A 2 iþ,t : The purpose of including this effect is to see if the degree of node i itself, A i + , t , affects the log-odds rather than their difference. To understand this effect,

ij,t ¼ 0, then Δs 4i ¼ À2A iþ,t þ 1. This effect is a good proxy for including the degree of nodes in the log-odds.

A main objective of this paper is to investigate the impact of the COVID-19 on financial network evolution. Specifically, we test whether the pandemic information set out in So, Chu, Tiwari, et al. (2021) is useful in explaining changes in the financial network dynamic under the SAOM. In So, Chu, Tiwari, et al. (2021) , the dynamic pandemic networks based on changes in the number of COVID-19 confirmed cases are constructed, from which we can calculate the time series of network statistics (including network density, clustering coefficient and assortativity) and a pandemic risk score called the preparedness risk score (PRS) and use these as covariates in the SAOM. The PRS accounts for the risk of asymptomatic or presymptomatic transmission. We adopt it as a measure of the transmission risk, which potentially influences the financial networks.

The following details for the construction of the pandemic networks and the pandemic network statistics can be found in So, Chu, Tiwari, et al. (2021) . Let X i, t be the number of confirmed COVID-19 cases of country i in day t, i ¼ 1,2, …, 32 countries. We calculate the daily changes for each country as Y i,t ¼ ffiffiffiffiffiffi ffi X i,t p À ffiffiffiffiffiffiffiffiffiffiffi X i,tÀ1 p . The sample correlation between country i and country j's daily changes in day t,ρ ij,t , is calculated using (Y i, t À k , Y j, t À k ), for k ¼ 0,…, 13. We build the pandemic network at time t by defining A p ij,t , the (i, j)th element of the adjacency matrix (A p t ) of the pandemic network at time t:

where r p ¼ 0:5. Let E p t and V p t be the number of edges and the number of nodes (countries) in the pandemic network, respectively. The network density of the pandemic network at time t is defined as

, which measures how dense the pandemic network is at time t. The global clustering coefficient of the pandemic network is defined as

where k it is the number of neighbours (or degree) of node i, and c it ¼ ðnumber of triangles formed by node iÞ= kit 2 À Á . C p t measures how strong nodes, or countries in the pandemic network, at time t are clustered together. The assortativity of the pandemic network at time t is defined as

which measures the correlation of the degree of the nodes in the pandemic networks. The PRS of country i at time t is defined as

where ω t is the vector of the population size of each country subtracted by the total number of confirmed cases in each country up to time t. The For the pandemic risk score, we can use the country-wise time-varying PRS in Figure 1 to form the SAOM covariates for the financial markets by mapping financial market indices to their respective countries. For example, we use the PRS of the United States as a covariate for SP500. A problem in using the PRS data is that the distribution of the PRS is highly right skewed and the magnitude of the scores is extremely small, which leads to unstable estimation of the SAOM using RSiena. To address this, we define the transformed PRS as logð1 þ 10 9 Â PRSÞ, which has a more symmetrical distribution and is of similar magnitude to the network statistics. Note that the PRS plotted in the heat maps shown in Figure 1e are not standardized. For better convergence of the algorithm (Ripley et al., 2021) , we standardize all covariates before inputting them into the SAOM. Most of the covariates fall between À2 to 2 after standardization.

The fifth and sixth effects included in the SAOM are defined by v it , the covariates defined by pandemic network statistics and their lagged values. The pandemic network statistics we consider are the network density, D P t ; the global clustering coefficient, C P t ; the assortativity, AS P t , and the PRS, S t . These two effects are known as (5) covariate-ego (egoX in RSiena), defined as s i5 ðA t Þ ¼ v it A iþ,t , and (6) covariate-ego  alter (egoXaltX in RSiena), defined as s i6 ðA t Þ ¼ P k A ik,t v it v kt , where v it is a covariate of market index i at time t. Similar to the first four effects, the corresponding log-odds are given as follows:

(5) The term in LOðA t , A ðAEijÞ t Þ relating to the difference of covariate-ego is β 5 Δs 5i , where Δs 5i ¼ v it ðA ðAEijÞ iþ,t À A iþ,t Þ ¼ v it ðA ðAEijÞ ij,t À A ij,t Þ: (This is because all the edges are the same in graph A t and A ðAEijÞ t except for the edge between actor i and j.) In this case, the log-odds is proportional to the pandemic covariate v i , and thus, the SAOM can test whether pandemic network statistics and the PRS for COVID-19 have any effect on financial network evolution.

(6) The term in LOðA t , A ðAEijÞ t Þ relating to the difference of covariate-ego  alter is β 6 Δs 6i , where

ij,t À A ij,t Þ: In this case, the log-odds will be affected by the product of both covariates v it and v jt .

The detailed estimation procedure, which is based on the method of moments, was described in Amati et al. (2015) and Ripley et al. (2021) . Briefly speaking, moment equations are established based on the sufficient statistics of the transition rates and the parameters of the effects discussed in Amati et al. (2015) , with Robbins-Monro stochastic approximation being used to solve the system of moment equations. The RSiena package in R provides necessary functions for the estimation.

To further explore the detailed changes in the financial networks during the COVID-19, Figure 2 shows a summary. The networks were stable from mid-February 2020 until 24 February 2020, when many connections were added. The growth trend reached its peak in mid-March 2020.

After that, a substantial number of old edges tended to be dropped in April 2020. The network evolution became active again in May 2020.

Following the rationale in For example, a higher PRS seems to be associated with higher financial network connectedness. Therefore, we adopt pandemic network covariates in the SOAM, that is, v it as defined in Section 2.5, to investigate whether or not the severity of the pandemic (using statistics from the pandemic networks, as well as the risk scores as proxies) can predict financial connectedness.

The full SAOM considered in this paper contains 14 rate of change parameters and four network effects (density, transitive triads, out-isolation and out degree related activity). To specify the two covariate effects, that is, covariate-ego and covariate-ego  alter as set out in Section 2.5, for each of the four pandemic network statistics (network density, global clustering coefficient, assortativity and transformed PRS), we include their five lagged values separately as v it to define multiple effects five and six. Thus, we have a total of 14 (for the rate parameters) + 4 (for effects one to four) + 4  5  2 (for effects five and six) = 58 parameters in the full model. We then conduct variable elimination to remove insignificant effects from the full model using F tests and eventually retain the 23 variables listed in Table 2 . All 14 rate parameters are kept and restricted to be positive in the estimation (Ripley et al., 2021) . The out-isolation and out degree effects defined in Section 2.4 are removed, implying that the isolation and the degree of nodes are not useful for predicting financial network connectedness and evolution. Seven pandemic covariates effects are retained. These represent the lagged effects of pandemic network density, global clustering coefficient, assortativity and the transformed PRS. From the seven significant pandemic covariate effects, we obtain strong statistical evidence that pandemic network properties and the related pandemic risk scores can help explain financial network evolution during the COVID-19. The impact may not be spontaneous and may take up to 5 days to become apparent. We will see how the propagation of pandemic risk can potentially lead to changes in financial market connectedness.

In the SAOM modelling, an essential objective is to investigate how pandemic networks possibly influence the formation of financial networks over time. To examine this, we focus on the two effects in Section 2.5 in the log-odds:

where β 5 and β 6 are unknown parameters corresponding to the covariate effects and v it is a pandemic network covariate of node i at time t in Section 2.5. To visualize the effects efficiently, consider specifically the effect of adding one edge between node i and j in A t to form A ðAEijÞ t , in this case, A ðAEijÞ ij,t À A ij,t ¼ 1. Since we standardize all v i before inputting them to the SAOM for model fitting, the log-odds in (3) due to the pandemic covariates v it and v jt can be written as a two-dimensional function of v it and v jt ,

where v and σ v are, respectively, the sample mean and standard deviation of the covariates v it . We call F(v it , v jt ) in Equation 4 the composite effect. Table 3 shows six heat maps of F(v it , v jt ) to visualize the composite effect of different pandemic network covariates v it and v jt on the evolution of the financial network. Note that we have seven significant effects in Table 2 but only six heat maps in Table 3 because we combine the two lag-five global clustering coefficient effects together to form a composite effect. In the heat maps in Table 3 , the x axis corresponds to the covariate (we call it v it ) of node i which is given an opportunity to make a change, and the y axis refers to the covariate (we call it v jt ) of node j that may be connected or disconnected by node i. From the covariates constructed from the pandemic network statistics, only network density shows a significant effect at lag one. When both v it and v jt are large/small, the chance of network link formulation in the financial networks tends to be higher. This observation is consistent with the result from Table 2 that the significant covariate-ego  alter effects described in Section 2.5 are all positive. Similarly, in the global clustering coefficient of the pandemic networks, simultaneously large or small v it and v jt may trigger financial network edge formation but the effect appears in lag-2. High lag-three pandemic network assortativity also encourages financial link formulation. Regarding the effect of the PRS, it takes three to 4 days to show a significant impact on financial network connectedness. Again, when the lag-3 PRS in node i and node j simultaneously increase or decrease, the financial network will tend to be denser. In short, when pandemic network connectedness or the PRS (reflecting the pandemic severity) in the two locations corresponding to nodes i and jmove in the same direction in the past few days, the financial networks tend to be more connected during the COVID-19. In terms of financial risk management, we can keep track of the pandemic severity at different locations to foresee how the financial networks will evolve.

To study the actual longitudinal impact of the pandemic covariates on financial market connectedness, we calculate F(v it , v jt ) for significant effects v it and all possible pairs of i and j (there are N(N À 1)/2 pairs, N ¼ 41 in our case), to obtain the distribution of F(v it , v jt ) on day t. Figure 3a -f presents these distributions using boxplots for all six covariates listed in Table 3 on each trading day. From Figure 3a -d, we

The number of network connections added or dropped compared to the previous trading day, from February to May 2020 observe that the boxplots of the four network statistics in mid-to late-February 2020 are wider than that after March 2020, meaning that they may have made a substantial contribution to the formation of financial network connections over that period in February. corresponding to the transformed PRS are predominately positive and large in late-February and early-March and quite highly volatile throughout the investigation period of February to May 2020. From the above findings, we can categorize the effect of the pandemic network statistics and the pandemic risk scores as short-and long-term effects, respectively, because the network statistics only affect the financial networks in the earlier stages but the transformed PRS affects them across all periods. In particular, the transmission risk of the COVID-19, measured using the transformed PRS, is a main explanatory factor of financial network connectedness for the period March to May 2020. More severe transmission risk may lead to further lockdown of businesses and cities which may harm the economy, thus affecting the financial markets. We can keep track of lagged pandemic network statistics and transformed PRS to evaluate financial risk through financial market connectedness.

We observe from Figure 3a -e that the implied F(v it , v jt ) from the pandemic networks is volatile in mid-February 2020, whereas the financial networks from Figure 2 are quite stable in mid-February 2020. The pandemic statistics and transformed PRS can give early signals of the intense connectedness of the financial markets in March 2020. This is useful for financial management since we can use lagged pandemic network statistics and transformed PRS to infer future financial connectedness and, thus, to monitor systemic risk in financial markets more effectively. Furthermore, we can use the SAOM approach to predict possible financial contagion using pandemic network statistics and transformed PRS of the COVID-19 and other pandemics. 

Using the SAOM with longitudinal financial and pandemic datasets, we investigate how financial networks evolve by applying pandemic network statistics and transformed PRS as predictors. The results provide evidence that financial markets where the pandemic statistics and prevalence of the COVID-19 co-move in the same direction tend to be more connected. Moreover, pandemic network statistics contribute to financial network connectedness in the short term in the early stages of the pandemic, while the long-term connectedness is driven by the pandemic risk. The results also show that we can detect the early signs of financial contagion by observing the lagged pandemic networks. Future research on modelling longitudinal pandemic and financial networks simultaneously is worthy of study. 

Online learning amid the COVID-19 pandemic: Students' perspectives

Estimation of stochastic actor-oriented models for the evolution of networks by generalized method of moments

A spatial analysis of international stock market linkages

COVID-19 spreading in financial networks: A semiparametric matrix regression model. Department of Economics, ca' Foscari University of Venice research paper series

Short-term and long-term effects of a social network intervention on friendships among university students

Dynamics of project-based collaborative networks for BIM implementation: Analysis based on stochastic actor-oriented models

Dynamic network analysis of COVID-19 with a latent pandemic space model

Factors for sustainable online learning in higher education during the COVID-19 pandemic

The pandemic of social media panic travels faster than the COVID-19 outbreak

Financial networks and contagion

Tail risk contagion between international financial markets during COVID-19 pandemic

Systemic risk in banking ecosystems

A study of systemic risk of global stock markets under COVID-19 based on complex financial networks

If the world fails to protect the economy, COVID-19 will damage health not just now but also in the future

Msci World Index Performance Report 2021

The structure and function of complex networks

Spillover of COVID-19: Impact on the global economy

Predicting stock returns in the presence of COVID-19 pandemic: The role of health news

COVID-19's disasters are perilous than global financial crisis: A rumor or fact? Finance Research Letters

The statistical evaluation of social network dynamics

Stochastic actor-oriented models for network dynamics

Siena algorithms

Modeling the coevolution of networks and behavior

Financial network connectedness and systemic risk during the COVID-19 pandemic

Efficient estimation of high-dimensional dynamic covariance by risk factor mapping: Applications for financial risk management

Impacts of the COVID-19 pandemic on financial market connectedness

On topological properties of COVID-19: Predicting and controling pandemic risk with network statistics

Visualizing COVID-19 pandemic risk through network connectedness

International market correlation and volatility

A statistical analysis of impact of COVID-19 on the global economy and stock index returns

Topological characteristics of the Hong Kong stock market: A test-based p-threshold approach to understanding network complexity

Stochastic actor-oriented modelling of the impact of COVID-19 on financial network evolution

T A B L E 3 Two-dimensional plots of the composite effects F(v i , v j ), where the subscript t is omitted for brevity

Lagged effect (I) Lagged effect (II)Network density (a) lag 1: 0:Transformed PRS (e) lag 3: 0:7757ðv i À vÞðv j À vÞ=σ 2 v (f) lag 4: 0:3920ðv i À vÞ=σ v Note: All heatmaps use the same colour scale.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Amanda M.Y. Chu https://orcid.org/0000-0002-9543-747XMike K.P. So https://orcid.org/0000-0003-0781-8166