Network density.

The network density measures the connectedness of a network. Intuitively, the more connections or more edges a BN has, the higher the network density will be. The network density at time t is defined as (3) where v_t and e_t are, respectively, the number of nodes and the number of edges of the network . It is the ratio of the number of edges in to the maximum possible number of edges, , implying that lies between 0 and 1. As edges in a BN represent directed links among variables, , which gives the proportion of edges in the BN, quantifies the complexity of the BN to represent hidden conditional dependence structures among variables. In the structural learning of Bayesian networks, we often impose a restriction on the maximum number of parents for each node for the sake of computational efficiency. Under this restriction, the maximum possible number of edges is much smaller than . Suppose that the maximum number of parents of each node is M (M ≤ v_t). It can be shown that with the restriction, the maximum possible number of edges in a BN is v_tM−M(M + 1)/2. More details of this maximum number of edges are provided in the S2 Appendix in S1 File. We define the modified network density with the maximum parent size M as (4)

The network density in Eq (4) extends that in Eq (3). For the special case of M = v_t−1, in which there is no restriction on the maximum number of parents, is reduced to . In the empirical study, we take M = 13, since the computation will be very slow when we choose a larger M. We will see how Eq (4) can be used to quantify systemic risk from the connectedness of the stocks in the empirical study.

Topological orders.

Bayesian networks are also known as causal networks because the network configuration may give us information on the “cause-and-effect” interpretation from a probabilistic perspective. For example, the factorization in Eq (2) for the BN in Fig 2 example gives us an idea of which variables are the “causes” and/or “effects” of the other variables. In the joint distribution in Eq (2), X₁ appears in P(X₁), P(X₃|X₁, X₂), and P(X₄|X₃, X₁). We can interpret that X₁ as a “cause” of X₃ but not vice versa because X₁ appears in the conditional distribution of X₃ as a given variable in the factorization. Similarly, X₁ and X₃ can be regarded as “causes” of X₄ according to the network structure in Fig 2 example. If we take the potential cause-and-effect structure into consideration, we can say a “cause” variable is lower than an “effect” variable. We can then talk about topological order as a topological feature of a BN.

A topological order of a BN defined in [29] refers to the ranking of the variables in the network. In a Bayesian network, if X_i precedes X_j in the topological order, then an edge from X_j to X_i is not allowed or X_j is not a parent of X_i. For example, in Fig 2 example, one possible topological order is (X₂, X₁, X₃, X₄) according to the rule stated above. Let T(X_i) be the position of X_i in a topological order. Then, the topological order (X₂, X₁, X₃, X₄) in Fig 2 example corresponds to T(X₁) = 2, T(X₂) = 1, T(X₃) = 3 and T(X₄) = 4. The topological order (X₁, X₂, X₃, X₄) is also allowed since neither X₁ nor X₂ is a parent of each other. For this topological order, we have T(X₁) = 1, T(X₂) = 2, T(X₃) = 3 and T(X₄) = 4. As there are multiple topological orders corresponding to a given BN, a fairer way to rank the variables is to take the median of all possible topological orders. It can be very computationally intensive to list out all possible topological orders when the network size (i.e., the number of nodes v_t) is large. Instead of identifying all topological orders of a Bayesian network at time t, , an alternative method is to sample a subset of topological orders from , the space of all possible topological orders corresponding to . Then, we can estimate using the sampled topological orders. We use the Kahn’s algorithm [47] to help generate samples from . The Kahn’s algorithm is a node sorting algorithm. We need to input an initial topological order of the nodes into this algorithm, then the algorithm will start to sort the nodes based on the initial topological order. For each initial topological order, the algorithm returns a topological order that is compatible with . To generate samples of topological orders from , we can input different randomly generated initial topological orders to the algorithm. There are v_t! ways to input the initial topological orders. In the empirical study, we will randomly pick K = 100 initial topological orders to input into the algorithm for obtaining samples of topological orders for each and to estimate M_t(X_i) using the samples.

To visualize and compare the topological order of the nodes on different days, since the numbers of nodes can change over time, the range of M_t(X_i) can be different. To facilitate the comparison, a relative topological order of the network is defined: (5)

Note that, if M_t(X_i) = 1 (i.e., at the head of the topological order), then RM_t(X_i) = 1; if M_t(X_i) = v_t (i.e., at the tail of the topological order), then RM_t(X_i) = 0. RM_t(X_i) always lies between 0 and 1, and a larger RM_t(X_i) indicates that X_i is relatively near the head of the topological order. As in M_t(X_i), we can estimate RM_t(X_i) by using sampled topological orders from .

In financial modeling, this relative topological order allows us to quantify the importance of each stock in a Bayesian network at different time points. If a stock is topologically ordered near the front, it can be regarded as the “cause” of the changes in other stocks in the Bayesian network, or as part of the sources of market fluctuation. On the other hand, if a stock is topologically ordered near the tail, the stock’s influence on other stocks’ fluctuation may be relatively smaller. In short, RM_t(X_i) helps measure the relative importance of the stocks in a BN over time. If RM_t(X_i) changes substantially in a short amount of time, it may indicate unexpected changes in the market fluctuation, or rapid increase of systemic risk in financial markets.

Order distance.

To trace the stability of financial markets over time, we propose measuring the changes in the topological orders. The rationale behind this is that the network topology is likely to be stable when the market is “quiet” or the volatility of the market is low. Conversely, the network topology can change substantially when there are severe adverse changes in the market. Therefore, being able to quantify changes in the topological order can help assess financial risk, especially for events leading to extreme market movement, which triggers systemic risk in financial markets. Let V_t be the set of nodes in the Bayesian network at time t. Consider the vector of median topological orders, , on two consecutive days: day t−1 and day t for the stock X_i. Since V_t and V_t−1 may have different nodes, we only consider the nodes in V_t−1∩V_t to define a topological order distance on day t. Note that the scale of M_t(X_i) and M_t−1(X_i) can also be different, since M_t(X_i) takes values from 1 to v_t. Following the idea of setting portfolio weights in asset allocation, we can define a vector of normalized topological orders on day t with respect to the set V_s as (6)

By construction, all elements in NT_t(V_s) lie between 0 and 1 and , where NT_t(X_i) is an element in NT_t(V_s). The elements in the vector in Eq (6) are analogous to the portfolio weights for investing in the stocks represented by the nodes in V_s, putting more weights on the stocks near the head of the topological orders.

The order distance between two consecutive networks on trading day t−1 and t is defined as (7) where ∥⋅∥₁ is the L¹-norm.

We use of L¹-norm to define the order distance because in NT_t(⋅), v_t − M_t(X_i) is normalized by the sum of its elements (which are all positive). This normalization can be regarded as a L¹-normalization. If we choose L^p-norms (p > 1), the corresponding order distance will tend to decrease when the dimension of NT_t(⋅) increases (which is undesirable for a regression predictor).

We use the order distance because we believe that there is a relationship between a high level of volatility and a rapid change in network connectedness, since a volatile market is often unstable. Specifically, we posit that the association between the volatility and order distance is more likely to be positive whenever the volatility is high. We provide empirical evidence of this phenomenon in S3 Appendix in S1 File. We fit moving-window regressions using the absolute daily returns of the Hang Seng Index, a proxy of the volatility, as the responses, and the estimated order distances in the previous trading day as the predictor. The modified network density and the lagged absolute returns are also included as control variables. We obtained the estimates of the coefficients of the order distances on each of the trading days over the study period. We partition the log-transformed absolute returns (log-transformation is conducted because we want to ensure the results can be easily visualized), and count the proportion of positive coefficients in each of the intervals. We found that the proportions are, on average, larger whenever the absolute returns are larger. The same analysis is conducted using the Dow Jones Industrial Average Index; the proportions show a similar phenomenon. The order distance is thus useful for predicting volatility in a period with high levels of volatility, and a high level of volatility is often accompanied by a large order distance. Based on this phenomenon, we will focus on extreme scenarios in this paper. We provide a more detailed discussion of these extreme scenarios in the LASSO Regression Prediction section later in this paper. More details of the above empirical evidence can be found in S3 Appendix in S1 File.

Data.

To demonstrate our proposed Bayesian network method in assessing systemic risk using topological features of the network, we obtain the closing prices of the HSI’s constituent stocks [48] in the Hong Kong stock market and the closing prices of HSI from 2 January 2008 to 4 November 2021 (3,404 trading days). We use the HSI because the Hong Kong Stock Exchange is one of the most representative exchanges in the world [49]. We believe that the Hong Kong stock market is a useful example for studying systemic risk. Moreover, the constituent companies in the HSI represent about 65% of the capitalization of the Hong Kong Stock Exchange [50]. It is important to study the systemic risk of the constituent stocks in the HSI. A time series plot of the HSI is given in Fig 7. The constituent names, stock symbols and their sectors are listed in the S1 Appendix in S1 File. Let P_it and R_it = logP_it − logP_i(t−1) be the closing price and log-return, respectively, of the i-th stock on day t, where day 0 refers to 2 January 2008. The data on 2 January 2008 are used to calculate the log-return on 3 January 2008, so that there are altogether T = 3403 returns in our investigation. Since the composition of the HSI is reviewed regularly, both the numbers of constituent stocks and the list of constituent stocks change over time. In the period of the empirical study, the number of the constituent stocks increases gradually from 46 to 60. The construction of the time series of Bayesian networks is based on the HSI’s constituent stocks. Therefore, the number of nodes v_t ranges from 46 to 60 in the empirical study.

In this paper, we focus on dichotomous return variables in the Bayesian network construction though, in general, our methodology can also be applicable to continuous variables. From a practical perspective, it is also easier to model Bayesian networks using discrete data than continuous data. Specifically, we define the return indicator of the i-th stock on day t by (8) for t = 1, …, T, where t = 1 refers to 3 January 2008. X_it captures the direction of movement of stock i on day t. It is expected that learning the topological features of the stocks over time through the time series of Bayesian networks can provide useful information about the risk propagation in financial markets.

Structural learning.

We adopt a rolling-window approach and use the return indicators X_it to construct a time series of Bayesian networks. Let D_t be the set of return indicator variables of the HSI’s constituent stocks on day t. For each day t, we use the data sets D_(t−w+1):t = {D_t−w+1, …, D_t} to construct , the BN on day t, where w is referred to as the rolling-window size. Since the number of variables in D_t can be different on different day t, the network only includes variables that are common in all data sets in D_(t−w+1):t. We employ the Markov chain Monte Carlo (MCMC) structural learning algorithms, including the Structure MCMC algorithm [40] and the Order MCMC algorithm [41] to estimate by obtaining the network that maximizes the Bayesian Dirichlet equivalent uniform (BDeu) score [29] using the data set D_(t−w+1):t.

Structural learning refers to a task for learning the network structure of a set of variables from the data. There are mainly two classes of structural learning algorithm: score-based algorithms, which maximizes an objective function, and constraint-based algorithms, which tests for conditional independence structures in the data. MCMC is a score-based algorithm, allowing us to sample from the target distribution, which is difficult to generate directly. The most traditional MCMC structural learning algorithm is the Structure MCMC algorithm proposed by [40], conducts MCMC sampling by adding, deleting, or reversing an edge in the network. Specifically, on a day t, let and be two structures of a Bayesian network, where is obtained from by adding, deleting, or reversing an edge. The Structure MCMC is a Metropolis-Hastings algorithm with an acceptance ratio, in our case, where is the BDeu score for the structure using the data set D_(t−w+1):t, and is the transition kernel from the structure to . We take the structure with the highest BDeu score in the Structure MCMC as .

Another algorithm we used in the study is the Order MCMC algorithm proposed by [41], which conducts MCMC sampling on the topological order space. The procedures of the Order MCMC algorithm are similar to those in the Structure MCMC algorithm, except we need to modify the BDeu score. We combine both algorithms to enhance the efficiency in terms of searching networks with high BDeu scores. The main advantage of using MCMC algorithms in our study is that it enables us to avoid becoming tapped in local maximum [51], making it more efficient to learn networks compared to some other optimization methods, such as greedy hill-climbing. This feature is especially important when it is necessary to learn many networks across thousands of trading days, as in our rolling-window modeling.

The maximum number of parents for each node, M, is set to 13. We choose M = 13 because the computation will be very slow if we were to use M > 13. Since we need w days of data to learn the structures, we can only obtain a time series of networks . The workflow of the construction of the time series of networks is shown in the timeline in Fig 3a. The first estimated structure is on day w using the data set D_1:w. The next structure to be estimated is on day w + 1 using the data set D_2:(w+1), and so on.

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Fig 3. The timelines of the different steps in our study.

(a) Timeline for the time series of network construction over the period under study, (b) Timeline for the calculation of the topological order distance over the period under study, (c) Timeline for the T − (w + wGC) + 1 Granger-causality tests over the period under study, and (d) Timeline for the LASSO regression prediction for the hypothesis H_i over the period under study, where i ∈ {0, 1a, 1b, 2}.

https://doi.org/10.1371/journal.pone.0279888.g003

Granger-causality tests.

An objective of this paper is to apply the network statistics of Bayesian networks in financial risk management. We investigate if the modified network density in Eq (4) and the order distance in Eq (7) are helpful in predicting future losses and risks in the market. We use the HSI returns as a proxy of market losses. Let R_t be the return of HSI on day t and define the loss on day t by Loss_t = −min{R_t, 0}. Following [17], we use a recent sample of observations to study any lead-lag relationships between historical network statistics and the loss on day t. In the empirical data analysis, we include observations from day t − w_GC + 1 to day t to conduct the Granger-causality tests by fitting the following regression models, where w_GC = 40 is the sample size: (9) where a_i, b_j, c_k and d_ℓ are regression coefficients, ε_t’s are random errors in the regression models, and L is the number of lags included in the Granger-causality tests. The H₀ represents the null model, which contains lagged losses as the only independent variables. The H_1a and H_1b respectively test whether the lagged modified network density and the order distance are helpful for the prediction of the next-day loss, and H₂ includes both independent variables and their interaction terms. The Granger-causality tests in Eq (9) are conducted for L = 1, 2, 3, 4, 5. Note that, the sample size w_GC is always fixed. Fig 4 shows the use of data from day t−w_GC + 1 to t to perform one Granger-causality test. When we use a lag L, we use the first L observations as lagged values. Then, including the data on the current day, we use L + 1 days of values for each hypothesis in Eq (9). Thus, effectively, we have w_GC−L rows of data to input into each Granger-causality test.

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Fig 4. The data included when performing a Granger-causality test on day t, where w_GC = 40 and L = 1, 2, 3, 4, 5. d_t represents the data available on day t.

https://doi.org/10.1371/journal.pone.0279888.g004

Similarly, we also conduct another set of Granger-causality tests in regard to the usefulness of the network statistics in predicting the volatility. A high level of volatility is associated with a high level of instability in financial markets and, thus, being able to predict future volatility movement can help predict market risk. The increase in volatility to a very high level can be associated with potential systemic risks in financial markets. A common proxy of volatility is the absolute return [44], and the corresponding null hypothesis is (10) with similar alternative hypotheses as in Eq (9), replacing Loss_t−i with |R_t−i|, i = 0, 1, …, L. The workflow of the Granger-causality tests is summarized in Fig 3b and 3c. In Fig 3b, the first order distance that can be calculated is on day w + 1 since we need both NT_w(V_w) and NT_w+1(V_w) to calculate OD_w+1 according to Eq (7), and the first available normalized topological order is NT_w(V_w) from the first network . The next order distance OD_w+2 is calculated from NT_w+1(V_w+1) and NT_w+2(V_w+1), and so on. Then, the first day on which we can conduct a Granger-causality test is on day w + w_GC, using the risk indicators from day w + 1, the first day an order distance is available, to day w + w_GC, leading to, altogether, w_GC days of observations, as shown in Fig 3c. The next test is conducted on day w + w_GC + 1, and so on. There are in total T − (w + w_GC) + 1 days on which we conduct the Granger-causality tests.

LASSO regression prediction.

In addition to the Granger-causality test, to investigate any significant leading effects of the modified network density and the order distance on the losses or absolute returns, we also evaluate the predictive performance using the models under four hypotheses in Eq (9) using the loss as the response, and that using the absolute return as response in Eq (10). A better prediction of volatility, as a trigger and a cause of financial crises, allows us to detect systemic risk events more accurately. On each day t, the most recent m = 40 days of observations (i.e., observations from day t − m + 1 to t) will be used to fit LASSO regression models for one-day predictions using the models under different hypotheses in Eq (9) and Eq (10). LASSO regressions may help reduce the variance and improve the prediction by shrinking some regression coefficients to zero [52], due to the penalty term. We will fix L = 5 for the models in H_1a and H_1b in the prediction but, unlike in Granger-causality tests, L = 1, 2, 3, 4 are not included because the LASSO regression will automatically shrink the coefficients to zero if some lags are irrelevant. For the models in H₂, we pick a smaller number of lags L = 3, since this model contains more parameters than the models in H_1a and H_1b. We need to set a smaller L, otherwise the degrees of freedom in the model will be too small, which may lead to overfitting. Similar to the mechanism within a Granger-causality test, we show the mechanism of fitting a LASSO regression model on a day t in Fig 5. The first L observations are used as lagged values. Including the observation on the current day, there are L + 1 days of values for each equation in Eq (9) and Eq (10). Thus, effectively, we have m−L rows of data to fit each LASSO regression model.

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Fig 5. The data included when fitting a LASSO regression model on day t, where m = 40, L = 5 for models under H_1a and H_1b, and L = 3 for models under H₂. d_t represents the data available on day t.

https://doi.org/10.1371/journal.pone.0279888.g005

Let y_t be any one of the Loss_t or R_t, as in the models in Eq (9) or Eq (10), respectively. We let y_t(1;H_i) be the one-day-ahead prediction of y_t+1 using the model under the hypothesis H_i, i ∈ {0, 1a, 1b, 2}, given the information up to time t. The root-mean-square error (RMSE) is used to evaluate the predictive performances (11)

The workflow of the regression prediction is summarized in Fig 3d. The first fitted model is on day w + m, using the data from day w + 1, the first day an order distance is available, to day w + m. We then conduct a one-day prediction for day w + m + 1. The next fitted model is on day w + m + 1, using the data from day w + 2 to w + m + 1. We then conduct a one-day prediction for day w + m + 2, and so on. The last prediction is on day T. Thus, there are a total of T − (w + m) predictions, as in the denominator of Eq (11).

On top of evaluating the overall performance, it is also meaningful to look into the predictive performance under tail scenarios (or extreme scenarios), which are more crucial to systemic risk. The following methodology (tail-event strategy) is specifically designed for developing a predictive strategy, where we believe that the network statistics based on the time series of Bayesian networks can be more effective in predicting market risk under tail scenarios. We propose using the predicted values of the loss or absolute return only when the observed loss or absolute return one day before the prediction is large, which may indicate the occurrence of tail scenarios in the next few days. In other words, we develop the following prediction scheme where we selectively use the predicted values generated by the network statistics to focus on tail scenarios, or to infer the systemic risk of the market. On each day t, we accept the prediction whenever the observed value, y_t, is larger than an m′-day rolling-window p-th percentile q_t(p) (i.e., the p-th percentile of {y_t, y_t−1, …, y_{max{1,t−m′+1}}}). Otherwise, we do not include the predicted value in the RMSE assessment. Let Q(p) = {t:y_t > q_t(p), t = w + m, …, T−1} be the set of time indexes when the predicted value y_t(1, H_i) is included in the RMSE assessment, as in (Eq (11)). We also use the RMSE to evaluate the predictive performance of the tail-event strategy. The RMSE only assesses the prediction on the days with large observed losses or absolute returns, the time indexes of which are in the set Q(p). The RMSE is then defined as (12)

Both RMSE(H_i) and RMSE_p(H_i) are used to assess the predictive performance of the modified network density and the order distance in regard to the loss and absolute return.