We model interconnectedness among markets via network models. A network model is a convenient class of multivariate analysis that uses graphs to represent statistical models [9] . They are formally represented by ( G , θ ) ∈ ( G × Θ ) , where G is a graph of relationships between variables, θ is the model parameter, G is the space of graphs and Θ is the parameter space. The graph, G, is defined by a set of vertices (nodes/variables) joined by a set of edges (links), describing the statistical relationships between a pair of variables. A typical multivariate multiple regression model is given by

Y = B X + U (1)

where X = ( X 1 , ⋯ , X n ) and Y = ( Y 1 , ⋯ , Y n ) are vector of exogenous and response variables respectively, B is a coefficient matrix and U is a vector of errors typically assumed to be multivariate normal. In this example, relationships between X and Y can be summarized by a weighted, A w ∈ ℝ n × n , or unweighted adjacency matrix, A ∈ { 0,1 } n × n , whose ij-th element

A i j = { 0, if G i j = 0, X j → Y i 1, otherwise , X j → Y i ,     A i j w = { 0, if B i j = 0 B i , j otherwise (2)

where X j → Y i means X j influences Y i , and A i j w specifies the weight of the relationship.

Let Y = ( Y 1 , ⋯ , Y n ) be return series of n-assets with risks σ Y = ( σ 1 , ⋯ , σ n ) and let A w be a weighted adjacency matrix. Assume S σ = σ Y σ ′ Y is the inner-product of risks, and Ω = ( I + A w ) ′ ( I + A w ) is their constrained precision matrix. Following [1] and [10] , we construct an index of equity market risk (EMR) as:

EMR = 1 n tr ( Ω S σ ) (3)

where EMR captures the average degree of unusual changes in asset returns and their interactions. The index signals widespread market turmoil and can be viewed as a measure of market-level fears (volatilities) that are amplified through interconnectedness.

The above index uses market interconnectedness ( A w ) and volatilities ( S σ ). The latter can be estimated directly from time series of stock market price/returns. The market interconnectedness measure ( A w ) is supposed to capture weighted channels of influence among markets. Thus, A w can be estimated by approximating the dynamics in observed multivariate financial time series as in (1) and (2). The commonly discussed methods for estimating A w are: Granger-causality [5] ; Lasso regularization [11] [12] ; forecast error variance decomposition [6] ; and Bayesian graph structural learning [13] [14] [15] . In this study, we derive the weighted A w via a Bayesian graph structural learning approach as in [10] [14] .

We model the marginal distribution of the EMR as ARIMA (1, 1, 1) and the conditional distribution of the VIX given EMR as ARIMAX (2, 1, 1).

Δ EMR t − Φ 1 Δ EMR t − 1 + Θ 1 ξ t − 1 e + ξ t e (4)

Δ VIX t = β Δ EMR t + Φ 1 Δ VIX t − 1 + Φ 2 Δ VIX t − 2 + Θ 1 ξ t − 1 v + ξ t v (5)

Δ VIX t = Φ 1 Δ VIX t − 1 + Θ 1 ξ t − 1 v + ξ t v (6)

where Δ EMR t is the first difference of equity market risk index at time t, Δ VIX t is the first difference of the VIX index at time t, ξ t e is the error term of the EMR model, and ξ t v is the error term of the VIX model. Φ 1 is a first-order autoregressive (AR) term, while Θ 1 is a first-order moving average (MA) term, and β captures the contemporaneous relationship between EMR and VIX. Thus, if β is significantly different from zero then EMR has significant information to improve the prediction of the VIX. The above model specification is such that (4) is the ARIMA (1, 1, 1) EMR model, (5) is the ARIMAX (2, 1, 1) VIX model with the EMR as an exogenous variable, and (6) is the ARIMA (1, 1, 1) VIX benchmark model.

We study the relationship between our EMR index and the VIX by using data from the Bloomberg database. The data consists of the daily market indices of 20 countries, including G10 economies, covering January 3, 2000, to June 30, 2020. We consider only one index per country, which typically contains the stock prices of the largest companies listed in the nation’s largest stock exchange. The considered countries can be grouped into three regions: the Americas (Brazil, Canada, Mexico, and the United States), Asia-Pacific (Australia, China, Hong Kong, India, Japan, and South Korea), and Europe (Belgium, France, Germany, Italy, the Netherlands, Portugal, Russia, Spain, Switzerland, and the United Kingdom). A description of the market indices chosen for the selected countries is presented in Table 1.

The selected market indices vary in terms of composition, in the sense that some have a smaller number of stocks compared to others. For instance, the U.S. is represented by the S & P 500, which contains the stocks of the top 500 large-cap corporations, whereas France is represented by CAC 40, which contains 40 stocks selected among the top 100 corporations.

Figure 1 reports the plot of daily closing prices on a logarithmic scale. Due to differences in the values, plotting the original prices would be difficult to visualize. We, therefore, scale the prices to a zero mean and unit variance and add the absolute minimum value of each series to avoid negative outcomes. This standardizes the scale of measurement for the different series. The plot shows that, amid many fluctuations, and some local specificities, stock markets are highly synchronized and have been affected by four major financial stresses characterized by huge drop in prices. These major financial stress periods are: 1) the tech-bubble crisis of 2000-2003 which was triggered by an endogenous shock fuelled via the adoption of the internet in the late 1990’s, especially in developed markets; 2) the sub-prime and global financial crisis of 2007-2009 which was triggered by the massive defaults of sub-prime borrowers in the US mortgage market; 3) the European sovereign debt crisis of 2010-2013 which emanated from the inability of a cluster of EU member states to repay or refinance their sovereign debt and bailout heavily leveraged financial institutions without recourse to third party assistance; and 4) the ongoing distress to the world economy and global financial markets caused by the novel coronavirus pandemic in 2020.

We compute daily returns as the log differences of successive daily closing prices. We obtain monthly estimates of the model parameters and construct the matrices Ω and S σ , which are the core components of our EMR index. To improve the efficiency of the estimates of Ω we aggregate monthly estimates in yearly rolling windows of about 240 trading days. We set the increments between successive rolling windows to one month, setting the first window of our study from February 1, 1999, to January 31, 2000, followed by March 1, 1999, to February 29, 2000; the last window is from July 1, 2019, to June 30, 2020. In total, we consider 246 rolling windows. To avoid over smoothing, S σ is instead estimated monthly, that is, using only the last month of each rolling window.

We report in Figure 2 the plot of the EMR and the VIX. The figure shows a striking resemblance between the EMR and VIX, both proving a similar signal about the “direction” of global market risk. Both indices indicate spikes during the tech-bubble crisis (2000-2003), the global financial crisis (GFC, 2007-2009), the Eurozone crisis (2010-2013), the Chinese stock market turbulence (2015-2016), and the recent COVID-19 pandemic (2020: 1H). The historical highest spike recorded for the COVID is only second to that of the GFC, which indicates evidence of strong contagion in the equities market during the GFC and the COVID pandemic, more than in the tech-bubble and Eurozone crisis. This suggests that the behavior of the stock markets during the pandemic is in several ways more similar to the GFC than it is to other periods of financial market stress.

We present in Table 2 the cross-correlation of the first difference of VIX with EMR. The table reports the highest cross-correlation occurring at lag 0, which suggests evidence of a strong positive contemporaneous relationship. Table 3¹ confirms the significant contemporaneous impact of the EMR on the VIX. Table 4 presents the out-of-sample forecast of the monthly VIX for the rest of the year 2020 compared with the realized values for the months of July-September 2020². The results indicate that not only does the ARIMAX (2, 1, 1) performs better than the benchmark—ARIMAX (1, 1, 1) in terms of the RMSE of the in-sample training set, but also the out-sample forecasts are much closer to the realized observation than the benchmark. In summary, we document evidence of a significant relationship between the EMR and VIX, with both indices providing similar signals about the direction of global market risk. In addition, the EMR improves the prediction of the VIX.